From 145b92748d68a431526228907a66d91aade77eae Mon Sep 17 00:00:00 2001
From: "Pawel T. Jochym" <pawel.jochym@ifj.edu.pl>
Date: Fri, 2 Jan 2026 16:44:50 +0100
Subject: [PATCH 01/23] Book text into book branch

---
 book/BaTiO3.pdf         |  Bin 0 -> 37905 bytes
 book/Fe2SiO4-1.pdf      |  Bin 0 -> 186961 bytes
 book/Fe2SiO4-2.pdf      |  Bin 0 -> 228838 bytes
 book/FeSe.pdf           |  Bin 0 -> 312787 bytes
 book/Fxs.png            |  Bin 0 -> 31209 bytes
 book/Jacob.jpg          |  Bin 0 -> 127341 bytes
 book/brillouin.pdf      |  Bin 0 -> 41528 bytes
 book/crystal.pdf        |  Bin 0 -> 257306 bytes
 book/diagram.pdf        |  Bin 0 -> 109369 bytes
 book/gap-hyb.pdf        |  Bin 0 -> 52496 bytes
 book/lattice.pdf        |  Bin 0 -> 76786 bytes
 book/main.tex           | 4221 +++++++++++++++++++++++++++++++++++++++
 book/methods-old.tex    | 3269 ++++++++++++++++++++++++++++++
 book/pseudopots-new.pdf |  Bin 0 -> 642609 bytes
 book/rare-earths.pdf    |  Bin 0 -> 18368 bytes
 book/usp.pdf            |  Bin 0 -> 22135 bytes
 16 files changed, 7490 insertions(+)
 create mode 100644 book/BaTiO3.pdf
 create mode 100644 book/Fe2SiO4-1.pdf
 create mode 100644 book/Fe2SiO4-2.pdf
 create mode 100644 book/FeSe.pdf
 create mode 100644 book/Fxs.png
 create mode 100644 book/Jacob.jpg
 create mode 100644 book/brillouin.pdf
 create mode 100644 book/crystal.pdf
 create mode 100644 book/diagram.pdf
 create mode 100644 book/gap-hyb.pdf
 create mode 100644 book/lattice.pdf
 create mode 100644 book/methods-old.tex
 create mode 100644 book/pseudopots-new.pdf
 create mode 100644 book/rare-earths.pdf
 create mode 100644 book/usp.pdf

diff --git a/book/BaTiO3.pdf b/book/BaTiO3.pdf
new file mode 100644
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diff --git a/book/FeSe.pdf b/book/FeSe.pdf
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index 0000000000000000000000000000000000000000..ffb731dc0036574936426cc1d56c82a10a89105b
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diff --git a/book/main.tex b/book/main.tex
index e69de29..f8d0058 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -0,0 +1,4221 @@
+\documentclass[12pt,a4paper,openany,aps,pra]{book}
+\usepackage{graphicx}
+\usepackage{gensymb}
+\usepackage{longtable}
+\usepackage{epsfig}% Include figure files
+\usepackage{dcolumn}% Align table columns on decimal point
+\usepackage{bm}% bold math
+\usepackage{anysize}
+\usepackage[utf8x]{inputenc}
+\usepackage[T1]{fontenc}
+%\usepackage[polish]{babel}
+\usepackage{amsmath}
+\usepackage[urlcolor=blue,colorlinks=true,citecolor=blue,linkcolor=blue,pdfstartview={FitH},bookmarks=false]{hyperref}
+\def\mb{\bm}
+\usepackage[left=2cm,right=2cm,top=3cm,bottom=3cm]{geometry}
+\DeclareMathOperator{\Tr}{Tr}
+
+
+\begin{document}
+
+
+\begin{titlepage}
+	\centering
+	{\Large\scshape Przemys\l{}aw Piekarz\par}
+    \vspace{4cm}
+	{\scshape\bfseries\Huge Ab initio methods\par}
+	\vspace{1cm}
+	{\scshape\bfseries\Huge in solid state physics\par}	
+	\vfill
+	{\scshape\large Department of Computational Materials Science \\ Institute of Nuclear Physics \\ Polish Academy of Sciences\par}
+\end{titlepage}
+
+\newpage
+\thispagestyle{empty}
+%\thispagestyle{plain} % empty
+\mbox{}
+
+\tableofcontents
+
+
+\chapter*{Introduction}
+\addcontentsline{toc}{chapter}{Introduction}  
+
+First principles calculation methods, also called {\it ab initio} methods, are based on the laws of quantum mechanics.
+The application of the Schr\"{o}dinger equation~\cite{Schrodinger} to atoms and molecules enabled to explain the basis properties of orbitals and chemical bonds~\cite{HL,hartree28,mulliken,JC}. 
+Many-electron systems, such as atoms, molecules and solids, are subject to Fermi-Dirac statistics~\cite{fermi26,Dirac26}, which is a direct consequence of the Pauli exclusion principle~\cite{Pauli25}.
+Application of quantum statistics to a homogeneous electron gas, which is the simplest model of the metallic state,
+made it possible to explain the problems of the classical Drude-Lorentz theory of conductivity.
+The band theory of periodic systems, which arose thanks to the works of Bloch~\cite{Bloch}, Peierls~\cite{peierls} and Wilson~\cite{Wilson},
+explained the significant differences in the electrical conductivity of materials and created the basis for dividing all crystals into metals, insulators and semiconductors.
+
+Due to many-body interactions occurring in electronic systems, the Schr\"{o}dinger equation
+cannot be solved accurately, and therefore approximate methods are needed.
+One of the first approaches to quantitative calculations was the Hartree \cite{hartree28} approximation,
+where the wave function is given as a product of single-particle functions.
+In the single-particle model, the interaction of electrons is replaced by an effective potential,
+in which each electron moves. This potential is composed of the attractive interaction generated by atomic nuclei
+and from the repulsive part coming from the remaining electrons.
+Since the electronic part of the potential depends on the wave functions we want to obtain,
+the Schr\"{o}dinger equation must be solved in a self-consistent manner.
+Taking into account the Pauli exclusion principle for the multi-electron wave function, which in its simplest form is written
+is in the form of the Slater determinant, the Hartree-Fock equation can be derived~\cite{fock30,slater30,slater51}.
+In this equation, in addition to the usual Coulomb interaction, there is a non-local
+exchange interaction. This is an interaction that occurs only between electrons with the same spin direction,
+effectively reducing the Coulomb repulsion between them.
+The Hartree-Fock approach describes electron systems approximately because it does not properly take into account correlations,
+which occur in the many-electron quantum state \cite{wigner34,GB}.
+This is also related to improper screening of Coulomb interactions,
+which leads to significantly overestimated values of the energy gap in solids.
+Methods based on the Hartree-Fock approach, the so-called multi-determinant methods, correctly take into account electronic correlations, but require time-consuming computer calculations. They are mainly used in quantum chemistry to study molecular systems.
+
+The first calculations of the electronic structure of metals used methods in which independent electrons interact
+with periodic atomic potential.
+In the cell method, proposed by Wigner and Seitz~\cite{wigner33}, the crystal potential is the sum of
+spherically symmetric potentials generated by each atom.
+Using Bloch's theorem, all we need to do is find solutions to the Schr\"{o}dinger equation
+in a single primitive cell, with appropriate boundary conditions at the cell boundary.
+In 1937, Slater introduced the augmented plane wave (APW) method. In this approach, the entire crystal is divided
+to the areas around the atomic positions (defined by the radius of the atomic sphere) and the remaining interstitial part.
+In each of these areas, a different form of the wave function is used. Inside the atomic spheres, the potential changes rapidly and
+wave functions, which behave similarly to atomic orbitals, are calculated in the basis of spherical harmonics.
+However, in the interstitial area, both the potential and wave functions are slowly changing, and the natural basis are plane waves.
+Additionally, the condition of continuity at the border of areas must be met.
+In another approach, proposed by Korring~\cite{K47} and Kohn and Rostoker~\cite{KR54},
+Green's functions 
+and scattering theory are used to determine electronic states in the crystal.
+The main drawback of these methods was the approximate nature of the electronic potential.
+The most frequently used potential was the {\it muffin-tin} type, which has spherical symmetry in the area of the atomic core,
+and outside it takes a constant value.
+A step forward was the orthogonalized plane wave (OPW) approach proposed by Herring~\cite{herring40}, which became
+the basis of the pseudopotential method~\cite{Antoncik,KP}.
+In this approach, the exact atomic potential is replaced by an approximate pseudopotential, having a finite value in the region of the atomic core,
+which allows the calculation of an approximate wave function (pseudo wave function) in the basis of plane waves in the entire crystal region.
+ 
+The real breakthrough was the formulation of density functional theory (DFT) by Hohenberg, Kohn and Sham~\cite{kohn64,kohn65}.
+The basic theorem of this theory says that the total energy of a given system,
+taking into account exchange and correlation interactions, is a functional
+of electron density, which can be determined by minimizing the ground state energy.
+In practical applications, the electron density is determined from single-particle wave functions, calculated as
+self-consistent by solving the Kohn-Sham equation.
+The exact values of the exchange-correlation potential are not known and must be determined within known approximations.
+The basic approach is the local density approximation (LDA),
+in which the exchange and correlation energies are obtained from accurate calculations for a homogeneous electron gas \cite{CeperleyAlder80,PZ,VWN}.
+The next step was to account for non-local effects within the generalized gradient approximation
+(GGA) \cite{Langreth83,Pardew86,Becke88}.
+Since the formulation of DFT, many methods have been proposed to solve the Kohn-Sham equations, such as
+the method of associated plane waves \cite{Andersen75}, or approaches based on pseudopotentials \cite{HSC,BHS,Vanderbilt90}.
+Thanks to these improvements, density functional theory became the basis for calculating electronic structure~\cite{jones}, determining crystal parameters~\cite{payne}, studying lattice dynamics~\cite{parlinski, baroni} and many other material properties~\cite {martin}.
+For its formulation, Walter Kohn received the Nobel Prize in Chemistry in 1998.
+
+Istnieją jednak układy, w których przybliżenia stosowane w DFT załamują się i nie potrafią poprawnie
+opisać struktury elektronowej. Zgodnie z klasyczną teorią pasmową, wiele tlenków metali przejściowych, takich jak NiO lub MnO,
+powinno znajdować się w stanie metalicznym, ze względu na częściowe wypełnienie stanów $3d$.
+Materiały te są w rzeczywistości izolatorami, ponieważ silne oddziaływanie kulombowskie w stanach $3d$
+uniemożliwia swobodny ruch elektronów i prowadzi do ich lokalizacji \cite{peierls}.
+W odróznieniu od izolatorów pasmowych, materiały te nazywane są izolatorami Motta \cite{mott}.
+Takie lokalne oddziaływanie elektronów, opisywane w ramach modelu Hubbarda parametrem $U$ \cite{hubbard},
+jest znacznie silniejsze w stanach $d$ i $f$, niż w bardziej rozległych (zdelokalizoanych) stanach $s$ i $p$.
+Struktura elektronowa tlenków metali przejściowych, wyznaczana w ramach obliczeń LDA lub GGA,
+albo nie posiada przerwy (CoO) albo ta przerwa jest znacznie zaniżona (NiO) \cite{terakura}.
+Zbyt mała przerwa energetyczna jest także problemem w obliczeniach dla klasycznych półprzewodników,
+takich jak krzem lub german, co oznacza, że również w tych materiałach potencjał wymienno-korelacyjny nie jest
+dobrze opisany.
+
+Podstawowe przybliżenia stosowane do wyznaczenia energii wymienno-korelacyjnej są źródłem błedów, które prowadzą efektywnie do samooddziaływania elektronów.
+Ten efekt jest szczególnie silny dla stanów $d$ lub $f$ i prowadzi do niekorzystnego wzrostu energii przy lokalizacji elektronów.
+To jest główną przyczyną niepoprawnego stanu metalicznego lub zbyt małej przerwy energetycznej otrzymywanych
+w obliczeniach dla tlenków metali przejściowych.  
+Jedna z metod korekcji tego efektu (SIC) \cite{PZ} polega na bezpośrednim wyliczeniu
+energii samoodziaływania i odjęcia jej od funkcjonału energii całkowitej.
+Metoda ta umożliwia poprawienie struktury elektronowej, ale nie jest łatwa do zaimplementowania w obliczeniach samouzgodnionych.
+Stan elektronowy można również poprawić stosując funkcjonały hybrydowe, w których przybliżona energia wymiany jest częściowo
+zastąpiona dokładną wartością otrzymaną w ramach metody Hartree-Focka \cite{becke93}.
+Wynika to z faktu, że w przybliżeniu Hartree-Focka nie występuje efekt samoodziaływania.
+  
+Potencjał elektronowy dla zlokalizowanych stanów można również poprawić przez dodanie do funkcjonału DFT
+dodatkowego wyrazu proporcjonalnego do energii oddziaływania $U$. Można to zrobić w przybliżeniu średniego
+pola w ramach metody LDA+U \cite{anisimov}. Zastosowanie tej metody do tlenków metali przejściowych \cite{anisimov}
+lub nadprzewodników wysokotemperaturowych \cite{czyzyk} pozwoliło znacznie poprawić ich strukturę elektronową 
+i własności magnetyczne w porównaniu do standardowych obliczeń DFT.
+Również w przypadku innych złożonych materiałów, w których występują oddziaływania między spinowymi, orbitalnymi
+i sieciowymi stopniami swobody to podejście daje jakościowo lepsze wyniki \cite{orbital1,orbital2,orbital3}.
+W silnie skorelowanych metalach, jakimi są ziemie rzadkie i aktynowce,
+metoda LDA+U pozwala tylko częściowo uwzględnić efekty związane z lokalizacją elekronów.
+Dobrym przykładem jest kryształ plutonu, gdzie bardzo duża niezgodność wyliczanej teoretycznie
+objetości, a wartością eksperymentalną (około 35 \%) może być skorygowana przy użycie
+realistycznej wartości parametru $U$\cite{Pu-LDAU}.
+Niestety, efekty dynamiczne takie jak fluktuacja momentów magnetycznych w stanie paramagnetycznyn, czy efekt Kondo
+nie mogą być poprawnie opisane na poziomie teorii średniego pola.
+W bardziej zaawansowanej metodzie dynamicznego średniego pola (DMFT), efekty wielociałowe
+są uwzględniane przez rozwiązanie modelu domieszki Andersona w ramach dokładnej diagonalizacji
+lub kwantowego Monte Carlo \cite{Georges}. Rozwiązania zawierają informacje o energii własnej i czasie
+życia stanów elektronowych, które nie są dostępne w ramach obliczeń DFT.
+Sama metoda kwantowego Monte Carlo rozwijana jest obecnie bardzo intensywnie i stosowana jest do obliczeń
+struktury elektronowej molekół i ukladów krystalicznych.
+
+
+\chapter{Electron interactions}
+
+\section{Basic properties}
+
+Spośród wszystkich oddziaływań, które występują w przyrodzie, decydującymi o własnościach fizycznych i chemicznych układów atomowych
+są oddziaływania elektromagnetyczne. 
+Wiązania międzyatomowe, dzięki którym powstają cząsteczki i ciała stałe, 
+są efektem wzajemnych oddziaływań elektrostatycznych w układzie elektronów i jąder atomowych.
+Hamiltonian, który opisuje dowolny układ atomów ma postać
+%
+\begin{equation}
+H=-\frac{\hbar^2}{2m}\sum_{i}\nabla_i^2-\sum_{i,j} \frac{Z_{j}e^2}{|\bm{r}_i-\bm{R}_j|}+\frac{1}{2}\sum_{i\neq j}\frac{e^2}{|\bm{r}_i-\bm{r}_j|}
+-\sum_j \frac{\hbar^2}{2M_j}\nabla_j^2 -\frac{1}{2}\sum_{i\neq j} \frac{Z_iZ_je^2}{|\bm{R}_i-\bm{R}_j|},
+\label{hamiltonian}
+\end{equation}
+%
+gdzie $\bm{r}_i$ określają położenia elektronów, $\bm{R_j}$ położenia jąder atomowych, $Z_j$ ładunki jąder, $M_j$ masy jąder i $m$ jest masą elektronu.
+Pierwsze trzy wyrazy opisują odpowiednio energię kinetyczną elektronów, oddziaływanie elektronów z jądrami i oddziaływania
+między elektronami. Ostatnie dwa wyrazy to energia kinetyczna jąder atomowych i oddziaływanie między
+ładunkami jąder. Pełna funkcja falowa złożonego układu elektronów i jąder atomowych $\Psi(\bm{r}_i,\bm{R}_j)$ spełnia równanie Schr\"{o}dingera
+%
+\begin{equation}
+H\Psi(\bm{r}_i,\bm{R}_j) = E\Psi(\bm{r}_i,\bm{R}_j),
+\label{Sch}
+\end{equation}
+%
+gdzie $E$ jest całkowitą energią układu.
+
+Kwantowo-mechaniczny opis kryształu można uprościć wykorzystując duży stosunek masy jadrą atomowego do masy elektronu,
+który dla najlżejszego jądra wodoru wynosi $m_p/m_e=1836$.
+Powoduje to, że dynamika elektronów jest znacznie większa niż jader atomowych i stany elektronowe bardzo szybko (adiabatycznie) dopasowują się
+do aktualnego położenia atomów. Typowe energie związane z ruchem elektronów są rzędu 1 eV, natomiast średnie energie drgań atomowych
+w kryształach są na poziomie 10 meV. 
+Zatem w pierwszym przybliżeniu możemy rozwiązać równanie Schr\"{o}dingera dla podukładu elektronów przy zadanych położeniach atomowych i
+pominąć wpływ kwantowych cech dynamiki atomów na funkcje falowe elektronów.
+Podejście to nazywa się przybliżeniem adiabatycznym lub przybliżeniem Borna-Oppenheimera \cite{BO}.
+Wprowadzając funkcję falową całego układu w postaci iloczynu funkcji elektronowej $\Phi(\bm{r_i},\bm{R_j})$ i funkcji jąder atomowych $\chi(\bm{R}_j)$
+%
+\begin{equation}
+\Psi(\bm{r}_i,\bm{R}_j) = \Phi(\bm{r}_i,\bm{R}_j)\chi(\bm{R}_j)
+\label{wave}
+\end{equation}
+%
+można rozseparować (\ref{Sch}) na dwa równania 
+%
+\begin{equation}
+(-\frac{\hbar^2}{2m}\sum_{i=1}^N \nabla_i^2-\sum_{i,j} \frac{Z_{j}e^2}{|\bm{r}_i-\bm{R}_j|}+\frac{1}{2}\sum_{i\neq j}
+\frac{e^2}{|\bm{r}_i-\bm{r}_j|})\Phi(\bm{r}_i,\bm{R}_j)=E_n(\bm{R}_j)\Phi(\bm{r}_i,\bm{R}_j),
+\label{Sch-el}
+\end{equation}
+%
+\begin{equation}
+(-\sum_j \frac{\hbar^2}{2M_j} \nabla_j^2 -\sum_{i,j} \frac{Z_iZ_je^2}{|\bm{R}_i-\bm{R}_j|}+E_n(\bm{R}_j))\chi_{n\alpha}(\bm{R}_j)=\varepsilon_{n\alpha}\chi_{n\alpha}(\bm{R}_j).
+\end{equation}
+%
+Pierwsze równanie opisuje funkcje falowe i energie własne $E_n(\bm{R}_j)$ układu elektronowego przy ustalonych położeniach jąder atomowych, gdzie $n$ oznacza zdefiniowany zbiór liczb kwantowych stanu elektronowego. Z drugiego równania możemy otrzymać funkcje falowe i energie własne $\varepsilon_{n\alpha}$ związane z ruchem jąder atomowych,
+gdzie $\alpha$ jest kwantową liczbą, która charakteryzuje te wielkości. Energia potencjalna w równaniu opisującym ruch jąder atomowych zależy od wzajemnego oddziaływania między jądrami i od energii podukładu elektronowego $E_n(\bm{R}_j)$. Obie wielkości są funkcją aktualnego położenia wszystkich jąder atomowych.  
+
+Zapiszmy funkcję falową układu $N$ elektronów, przy zadanych położeniach jader atomowych, w formie $\Phi(\bm{r}_1,\bm{r}_2,...,\bm{r}_N)$. 
+Znajomość funcji falowej pozwala wyznaczyć wiele podstawowych wielkości fizycznych dla danego układu. W szczególności gęstość elektronowa w punkcie $\bm{r}$
+dana jest wzorem
+%
+\begin{equation}
+n(\bm{r})=N\int d\bm{r}_2...d\bm{r}_N \Phi^*(\bm{r},\bm{r}_2,...,\bm{r}_N) \Phi(\bm{r},\bm{r}_2,...,\bm{r}_N).
+\end{equation}
+%
+Podstawową własnością elektronowej funkcji falowej jest jej antysymetryczność. Oznacza, że zamiana miejscami dwóch elektronów powoduje zmianę jej znaku
+%
+\begin{equation}
+\Phi(\bm{r}_1,\bm{r}_2,...,\bm{r}_i,...,\bm{r}_j,...,\bm{r}_N)=-\Phi(\bm{r}_1,\bm{r}_2,...,\bm{r}_j,...,\bm{r}_i,...,\bm{r}_N).
+\label{anty}
+\end{equation}
+%
+Ta własność wynika z zakazu Pauliego, który mówi, że dwa fermiony (czyli cząstki o spinie połówkowym) nie mogą znajdować się w tym samym stanie kwantowym - w stanie opisanym takim samym zestawem liczb kwantowych.
+
+Równanie Schr\"{o}dingera (\ref{Sch-el}) zawiera oddziaływania wielociałowe i nie posiada dokładnych rozwiązań analitycznych, z wyjątkiem atomu wodoru lub innych układów jednoelektronowych. Dokładne rozwiązania numeryczne równania Schr\"{o}dingera można uzyskać jedynie dla pojedynczych atomów i małych molekuł.
+Jako przykład rozważmy prosty układ z dwoma elektronami, jakim jest cząsteczka wodoru H$_2$.
+Hamiltonian tego układu, po uwzlędnieniu przybliżenia Borna-Oppenheimera, przyjmuje postać
+%
+\begin{equation}
+H=-\frac{\hbar^2}{2m}\nabla_1^2-\frac{\hbar^2}{2m}\nabla_2^2-\frac{e^2}{r_{1A}}-\frac{e^2}{r_{1B}}-\frac{e^2}{r_{2A}}-\frac{e^2}{r_{2B}}+\frac{e^2}{r_{12}}+\frac{e^2}{r_{AB}},
+\end{equation}
+%  
+gdzie $r_{1A}$, $r_{1B}$, $r_{2A}$, $r_{1B}$ oznaczają odległości elektronów (1 i 2) od dwóch protonów ($A$ i $B$), $r_{12}$ jest odległością między elektronami
+i $r_{AB}$ odległością miedzy protonami. 
+W roku 1927, Heitler i London zaproponowali funkcję falową w formie \cite{HL}
+%
+\begin{equation}
+\Phi(\bm{r}_1,\bm{r}_2)=N_{\pm}[\psi_A(\bm{r}_1)\psi_B(\bm{r}_2)\pm \psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]\chi_{\sigma},
+\end{equation}
+%
+gdzie $N_{\pm}$ jest czynnikiem normalizacyjnym, $\chi_{\sigma}$ jest częścią spinową funkcji falowej, a cztery funkcje $\psi_{\alpha}(\bm{r})$
+sa orbitalami $1s$ dla stanu podstawowego atomu wodoru
+%
+\begin{equation}
+\psi_{\alpha}(\bm{r})=\frac{1}{\sqrt{\pi a_0^3}}e^{-\frac{|\bm{r}-\bm{r}_{\alpha}|}{a_0}},
+\end{equation}
+gdzie $a_0$ jest promieniem Bohra, a $\bm{r}_{\alpha}$ jest położeniem protonu $\alpha=A$ lub $B$.
+Biorąc pod uwagę warunek antysymetryczności (\ref{anty}), funkcja falowa przyjmuje jedną z dopuszczalnych postaci
+%
+\begin{equation}
+\Phi_S(\bm{r}_1,\bm{r}_2)=N_{+}[\psi_A(\bm{r}_1)\psi_B(\bm{r}_2)+\psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]\frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle),
+\end{equation}
+%
+dla całkowitego spinu $S=0$ (stan singletowy), oraz
+%
+\begin{equation}
+\Phi_T(\bm{r}_1,\bm{r}_2)=\begin{cases}
+N_{-}[\psi_A(\bm{r}_1)\psi_B(\bm{r}_2)-\psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]|\uparrow\uparrow\rangle, \\
+N_{-}[\psi_A(\bm{r}_1)\psi_B(\bm{r}_2)-\psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]\frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle), \\
+N_{-}[\psi_A(\bm{r}_1)\phi_B(\bm{r}_2)-\psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]|\downarrow\downarrow\rangle, 
+\end{cases}
+\end{equation}
+%
+dla spinu $S=1$ (stan trypletowy). Stanem podstawowym cząsteczki wodoru jest stan singletowy, którego energia
+$E_S$ jest niższa od energii stanu stanu trypletowego $E_T$ dla każdej odległości między protonami $r_{AB}$.
+Stan związany odpowiada minimum energetycznemu $E_S=\langle\Phi_S|H|\Phi_S\rangle$, które dostajemy dla $r_{AB}=0.87$ \AA. Ta odległość jest większa od wartości eksperymentalnej równej 0.74 \AA. Natomiast wyliczona energia dysocjacji cząsteczki na dwa atomy wodoru wynosi $E_d=3.14$ eV i jest mniejsza od energii zmierzonej 4.75 eV. Jest to przykład wiązania kowalencyjnego, w którym dwa elektrony o przeciwnych spinach zostają uwspólnione, co prowadzi
+do obniżenia energii całego układu w porównaniu do sumy energii dwóch osobnych atomów.  
+W odróżnieniu od stanu singletowego, stan trypletowy nie tworzy stanu związanego dwóch atomów wodoru.
+Dokładniejsze wartości $r_{AB}=0.74$ \AA\ i $E_d=3.63$ eV otrzymuje się w przybliżeniu Hartree-Focka, które będzie tematem następnego rozdziału.
+Najlepsze wyniki uzyskuje się metodą wariacyjną, w której funkcja falowa zapisana jest w ogólnej formie iloczynu symetrycznej funkcji przestrzennej i antysymetrycznej funkcji spinowej \cite{JC}
+%
+\begin{equation}
+\Phi_S(\bm{r}_1,\bm{r}_2)=\Psi(\bm{r}_1,\bm{r}_2)\chi_0.
+\end{equation} 
+%
+Część przestrzenna funkcji falowej zależy od $M$ parametrów wariacyjnych, $p_1$, $p_2$,..., $p_M$.
+Minimalizując energię układu dla $M=13$ otrzymuje się wartości $E_d=4.70$ eV i $r_{AB}=4.74$ \AA, w bardzo dobrej zgodności z danymi eksperymentalnymi \cite{JC}.   
+Dla większości badanych układów molekularnych i ciał stałych nie da się wyliczyć dokładnych rozwiązań równania Schr\"{o}dingera. 
+Zatem konieczne jest stosowanie metod przybliżonych, głównie numerycznych, które pozwalają wyznaczyć najlepsze możliwe funkcje falowe i energie stanów elektronowych
+w rozsądnym czasie obliczeniowym. 
+
+\section{Równanie Hartree-Focka}
+\label{sec:HF}
+
+Jedną z pierwszych metod zostosowaną do opisu układów molekularnych jest przybliżenie Hartree-Focka \cite{hartree28,fock30,slater30}. W tym podejściu funkcja falowa układu $N$ elektronów ma postać wyznacznika Slatera
+%
+\begin{equation}
+\Phi = \frac{1}{\sqrt{N!}} 
+\begin{vmatrix}
+\phi_1(\bm{r}_1,\sigma_1) &  \phi_2(\bm{r}_1,\sigma_1)  & \dots & \phi_N(\bm{r}_1,\sigma_1) \\ 
+\phi_1(\bm{r}_2,\sigma_2) &  \phi_2(\bm{r}_2,\sigma_2)  & \dots & \phi_N(\bm{r}_2,\sigma_2) \\
+\dots                &  \dots                 &       &\dots \\
+\phi_1(\bm{r}_N,\sigma_N) &  \phi_2(\bm{r_N},\sigma_N)  & \dots & \phi_N(\bm{r}_N,\sigma_N) \\
+\end{vmatrix},
+\label{slater}
+\end{equation}
+%
+gdzie funkcje jednoelektronowe są iloczynem części zależnej od położenia $\bm{r}_i$
+i od spinu $\sigma_i$
+%
+\begin{equation}
+\phi_i(\bm{r}_i,\sigma_i)=\psi_i^{\sigma_i}(\bm{r}_i)\xi_i(\sigma_i).
+\end{equation}
+%
+Funkcja falowa zapisana w postaci wyznacznika Slatera spełnia podstawowe
+własności układu nieodróżnialnych cząstek.
+Zamiana miejscami dwóch elektronów odpowiada zamianie dwóch kolumn w wyznaczniku, co powoduje
+zmianę znaku funkcji falowej zgodnie z warunkiem antysymetryczności.
+Jeżeli dwa elektrony znajdują się w tym samym stanie kwantowym to dwie kolumny
+są jednakowe i cały wyznacznik się zeruje, co zgodne jest z zakazem Pauliego. 
+
+Rozwiązanie równania Schr\"{o}dingera odpowiada funkcji falowej, dla której średnia
+z hamiltonianu przyjmuje wartość minimalną. Przy założeniu ortonormalności funkcji $\Phi$
+odpowiada to zasadzie wariacyjnej w postaci
+%
+\begin{equation}
+\delta(\langle\Phi|H|\Phi\rangle - E\langle\Phi|\Phi\rangle)=0,
+\label{var}
+\end{equation}
+%
+gdzie symbol $\delta$ oznacza pochodną wariacyjną.
+Wyliczając średnią z hamiltonianu dla podukładu elektronowego (\ref{Sch-el}) w stanie opisanym funkcją falową (\ref{slater}) otrzymujemy
+%
+\begin{equation}
+\langle\Phi|H|\Phi\rangle=\sum_{i,\sigma}\int d\mb{r} \psi_i^{\sigma*}(\mb{r})[-\frac{\hbar^2}{2m}\nabla_i^2+V_Z(\mb{r})]\psi_i^{\sigma}(\mb{r}) 
++E_H+E_x,
+\label{E_HF}
+\end{equation}
+%
+gdzie $V_Z$ jest potencjałem elektrostatycznym wytwarzanym przez jądra atomowe, a $E_H$ nazywa się energią Hartree 
+%
+\begin{equation}
+E_H=\frac{e^2}{2}\sum_{i,j,\sigma,\sigma'}
+\int d\mb{r} \int d\mb{r}' \frac{\psi_i^{\sigma*}(\mb{r})\psi_j^{\sigma'*}(\mb{r}')\psi_i^{\sigma}(\mb{r})\psi_j^{\sigma'}(\mb{r}')}{|\mb{r}-\mb{r}'|}.
+\end{equation}
+%
+Wprowadzając gęstość ładunkową w punkcie $\mb{r}$
+%
+\begin{equation}
+n(\mb{r})=e\sum_{i,\sigma} |\psi_i^{\sigma}(\mb{r})|^2,
+\label{dens}
+\end{equation}
+%
+możemy zapisać energię Hartree w prostszej formie
+%
+\begin{equation}
+E_H=\frac{1}{2} \int d\mb{r} \int d\mb{r}' \frac{n(\mb{r})n(\mb{r}')}{|\mb{r}-\mb{r}'|}, 
+\label{Hartree}
+\end{equation}
+%
+która odpowiada klasycznej energii oddziaływania kulombowskiego przy zadanym rozkładzie gęstości ładunku elektrycznego. 
+%
+$E_x$ nazywa się energią wymiany i wynosi
+%
+\begin{equation}
+E_x=-\frac{e^2}{2}\sum_{i,j,\sigma}
+\int d\mb{r} \int d\mb{r}' \frac{\psi_i^{\sigma*}(\mb{r})\psi_j^{\sigma*}(\mb{r}')\psi_i^{\sigma}(\mb{r}')\psi_j^{\sigma}(\mb{r})}{|\mb{r}-\mb{r}'|}.
+\label{exc}
+\end{equation}
+%
+Energia wymiany jest wielkością kwantową, która nie ma odpowiednika w fizyce klasycznej. 
+Oddziaływanie wymianne charakteryzują dwie ważne cechy, które są ze sobą ściśle powiązane. 
+Przede wszystkim, oddziaływanie to dotyczy tylko elektronów o takim samym kierunku spinu. 
+Zgodnie z zasadą Pauliego dwa elektrony o tym samym kierunku spinu nie moga mieć takich samych
+pozostałych liczb kwantowych, co wiązałoby się z obsadzeniem tego samego stanu kwantowego. 
+Efektywnie prowadzi to do powiększenia odległości między takimi elektronami i
+obniżenia energii oddziaływania kulombowskiego. Z tego wynika druga cecha
+oddziaływania wymiennego: wartość ujemna energii tego oddziaływania. Można interpretować oddziaływanie wymienne
+jako oddziaływanie kulombowskie elektronu z ładunkiem dodatnim tzw. {\it dziurą wymienną} ({\it ang. exchange hole}),
+która powoduje redukcję gestości ładunku ujemnego wokół każdego elektronu. 
+
+Po zastosowaniu metody wariacyjnej do (\ref{E_HF}) otrzymujemy równanie Hartree-Focka
+%
+\begin{equation}
+[-\frac{\hbar^2}{2m}\nabla_i^2+V_Z(\mb{r})+V_H(\mb{r})]\psi_i^{\sigma}(\mb{r})
+-\frac{e^2}{2}\sum_{j,\sigma}\int d\mb{r}' \frac{\psi_j^{\sigma*}(\mb{r}')\psi_i^{\sigma}(\mb{r}')}{|\mb{r}-\mb{r}'|}\psi_j^{\sigma}(\mb{r})=E_i\psi_i^{\sigma}(\mb{r}),
+\label{HF}
+\end{equation} 
+%
+gdzie $V_H$ oznacza potencjał Hartree
+%
+\begin{equation}
+V_H(\mb{r})=\int d\mb{r}' \frac{n(\mb{r}')}{|\mb{r}-\mb{r}'|}.
+\end{equation}
+%
+Równanie Hartree-Focka jest równaniem nieliniowymi ponieważ w ostatnim wyrazie Hamiltonianu występuje poszukiwana funkcja falowa.
+Slater zaproponował inną postać równań (\ref{HF}), która
+pozwala lepiej zrozumieć charakter oddziaływania wymiennego \cite{slater51}.
+Mnożąc i dzieląc wyraz wymienny przez $\psi_i^{\sigma}(r)$ otrzymujemy
+%
+\begin{equation}
+[-\frac{\hbar^2}{2m}\nabla_i^2+V_Z(\mb{r})+V_H(\mb{r})
+-\frac{e}{2}\int d\mb{r}' \frac{n(\mb{r},\mb{r}')}{|\mb{r}-\mb{r}'|}]\psi_i^{\sigma}(\mb{r})=E_i\psi_i^{\sigma}(\mb{r}),
+\label{HFS}
+\end{equation}
+%
+gdzie $n(\mb{r},\mb{r}')$ można interpretować jako gęstość ładunku wymiennego 
+%
+\begin{equation}
+n(\mb{r},\mb{r}')= e\sum_{j,\sigma}\frac{\psi_j^{\sigma*}(\mb{r}')\psi_i^{\sigma}(\mb{r}')\psi_j^{\sigma}(\mb{r})\psi_i^{\sigma*}(\mb{r})}{\psi_i^{\sigma*}(\mb{r})\psi_i^{\sigma}(\mb{r})},
+\label{EC}
+\end{equation}
+%
+która jest funkcją dwóch położeń $\mb{r}$ i $\mb{r}'$ i zależy od stanu kwantowego $i$. Całkowity ładunek wymienny
+równy jest ładunkowi pojedynczego elektronu, co łatwo pokazać wykonując całkowanie po $\mb{r}'$ i korzystaniu z ortogonalności 
+funkcji falowych
+%
+\begin{equation}
+q=e\sum_{j,\sigma}[\int d\mb{r}'\psi_j^{\sigma*}(\mb{r}')\psi_i^{\sigma}(\mb{r}')]\frac{\psi_j^{\sigma}(\mb{r})}{\psi_i^{\sigma}(\mb{r})}=
+e\sum_{j,\sigma}\delta_{ij}\frac{\psi_j^{\sigma}(\mb{r})}{\psi_i^{\sigma}(\mb{r})}=e.
+\end{equation}
+%
+Biorąc pod uwagę przypadek, gdy obydwa położenia są jednakowe $\bm{r}=\bm{r'}$ otrzymujemy gęstość ładunku wymiennego zgodną ze wzorem (\ref{dens}).
+Powoduje to, że wyraz opisujący oddziaływanie elektronu z jego własnym polem wymiennym ma taką sama postać jak potencjał Hartree, ale z przeciwny znakiem
+Dzięki tej własności, obydwa wyrazy się wzajemnie kasują i w przybliżeniu Hartree-Focka nie występuje problem oddziaływania elektronu z jego własnym polem ({\it samooddziaływanie, ang. self-interaction}). 
+W formie (\ref{HFS}) równanie Hartree-Focka ma postać jednoelektronowego równania Schr\"{o}dingera z potencjałem wymiennym wytwarzanym
+w miejscu znajdowania się elektronu przez ładunek wymienny.
+Zgodnie z twierdzeniem Koopmansa, wartości własne $E_i$ odpowiadają energiom koniecznym do usunięcia elektronu z orbitalu 
+$\psi_i^{\sigma}(\mb{r})$~\cite{Koopman}.
+
+
+\section{Metoda pola samouzgodnionego}
+
+Równania Hartree-Focka najczęściej rozwiązuje się numerycznie stosując metodę pola samouzgodnionego, nazywaną również przybliżeniem średniego pola.
+Przy zadanych położeniach atomów wybieramy początkowe funkcje falowe $\psi_{i0}^{\sigma}$, 
+które przykładowo mogą być orbitalami izolowanych atomów.
+Dla tych funkcji falowych wyliczamy gęstość elektronową $n(\mb{r})$ i potencjał $V(\mb{r})$ 
+w każdym punkcie przestrzeni badanego układu. 
+Dla tak wyliczonego potencjału rozwiązujemy równanie (\ref{HF}) wyznaczając zbiór jednoelektronowych funkcji falowych i energii 
+stanów elektronowych. Wyznaczone funkcje falowe stosujemy do ponownego wyliczenia gęstości elektronów
+i potencjału efektywnego. Procedurę powtarzamy do momentu, gdy otrzymane w kolejnych krokach energie i funkcje falowe
+są jednakowe lub różnią się o niewielką zadaną wielkość. Najczęściej parametrem służącym do badania zbieżności
+rachunku jest całkowita energia układu. Metoda pola samoouzgodnionego zastosowana była po raz pierwszy
+do rozwiązania równania Hartree, które możemy otrzymać z rownania Hartree-Focka po usunięciu
+oddziaływania wymiennego. Równanie Hartree odpowiada funkcji falowej w postaci
+iloczynu niezależnych funkcji jednocząstkowych, czyli opisuje zbiór niezależnych elektronów
+w efektywnym polu elektrostatycznym. Metoda pola samouzgodnionego stosowana jest obecnie w większości 
+metod obliczeniowych struktury elektronowej ciał stałych. 
+
+\section{Korelacje elektronowe}
+
+Podejście Hartree-Focka jest metodą przybliżoną, zatem wyliczona energia
+i funkcje falowe różnią się od wielkości dokładnych, które odpowiadają rozwiązaniom równania
+Schr\"{o}dingera dla funkcji wieloelektronowej.
+Uwzględnienie wszystkich efektów oddziaływań wielociałowych w układzie elektronowym umożliwia
+dodatkowe obniżenie energii całkowitej w porównaniu do energii Hartree-Focka.
+Różnica między dokładną wartością energii ($E_{\rm{exact}}$) i energią wyliczoną w przybliżeniu 
+Hartree-Focka ($E_{\rm{HF}}$), nosi nazwę energii korelacji
+%
+\begin{equation}
+E_c=E_{\rm{\text{exact}}}-E_{\rm{\text{HF}}}.
+\end{equation}
+%
+Żródłem korelacji elektronowych są oddziaływania kulombowskie, które dążą do przestrzennej separacji 
+elektronów. Elektron znajdujący się w położeniu $\bm{r}$ powoduje, że inne elektrony unikają tego położenia.
+Czyli prawdopodobieństwo znalezienia elektronu w danym punkcie zależy od położeń wszystkich pozostałych $N-1$
+elektronów. Materiały, w których lokalne oddziaływania kulombowskie mają decydujący wpływ na strukturę elektronową
+i własności transportowe nazywane są układami silnie skorelowanymi. 
+
+Metoda Hartree-Focka jest punktem wyjścia do bardziej zaawansowanych metod stosowanych głównie w chemii kwantowej.
+Można ogólnie zapisać funkcję falową układu $N$ elektronów w postaci rozwinięcia na skończoną ilość wyznaczników Slatera
+%
+\begin{equation}
+\Phi(\bm{r}_1,...,\bm{r}_N)=c_0\Phi_0(\bm{r}_1,...,\bm{r}_N)+\sum_{i=1}^{N_\text{det}} c_i\Phi_i(\bm{r}_1,...,\bm{r}_N),
+\end{equation}
+%
+gdzie $\Phi_0$ jest funkcją falową stanu podstawowego w przybliżeniu Hartree-Focka,
+a wyznaczniki $\Phi_i$ odpowiadają stanom wzbudzonym. Im większa liczba tych wyznaczników $N_{\text{det}}$, tym dokładniej
+opisane są korelacje elektronowe. Funkcje $\Phi_i$ konstruowane są przy pomocy orbitali jednoelektronowych, które nie są obsadzone
+w stanie podstawowym i mogą być zajęte przez elektrony wzbudzone. Wyznaczniki mogą opisywać pojedyncze przejścia elektronów do orbitali nieobsadzonych ($\Phi_i^{\text{S}}$),
+podwójne wzbudzenia elektronów ($\Phi_i^{\text{D}}$), potrójne wzbudzenia ($\Phi_i^{\text{T}}$) i tak dalej.
+Całkowita liczba wszystkich możliwych wzbudzonych konfiguracji rośnie bardzo szybko wraz z liczbą elektronów $N$ i orbitali $M$ (zajętych i pustych)
+według wzoru 
+%
+\begin{equation}
+N_{\text{det}}=\frac{(M+1)!}{N!(M+1-N)!}
+\end{equation}
+%
+W najprostszym podejściu nazywanym metodą oddziaływania konfiguracji ({\it ang. configuration interaction} - CI) \cite{GTO},
+wyznaczniki konstruuje się z zajętych i pustych orbitali jednoelektronowych hamiltonianu Hartree-Focka.
+Funkcja falowa przyjmuje postać szeregu wyznaczników o zwiększającej się ilości wzbudzonych elektronów
+%
+\begin{equation}
+\Phi_{\text{CI}}=c_0\Phi_0+\sum_{i=1}^{N_\text{S}} c^\text{S}_i \Phi_i^{\text{S}}+\sum_{i=1}^{N_\text{D}} c^\text{D}_i \Phi_i^{\text{D}}+\sum_{i=1}^{N_\text{T}} c^\text{T}_i \Phi_i^{\text{T}}+...,
+\label{phiCT}
+\end{equation}
+% 
+gdzie $N_\text{S}$, $N_\text{D}$, $N_\text{T}$ określają ilość wyznaczników dla danej liczby wzbudzonych orbitali.
+Stan podstawowy znajduje się przez wyznaczenie współczynników $c_i$, które minimalizuje energię
+%
+\begin{equation}
+E_\text{CI}=\int \Phi^*(\bm{r}_1,...,\bm{r}_N)H\Phi(\bm{r}_1,...,\bm{r}_N)d\bm{r}_1d\bm{r}_2...d\bm{r}_N,
+\end{equation}
+%
+gdzie $H$ jest hamiltonianem układu elektronowego (\ref{Sch-el}).
+Minimalizacja przeprowadzana jest przy warunku normalizacji funkcji falowej, co przekłada się na warunek dla wszystkich współczynników rozwinięcia
+$\sum_ic_i^2=1$.
+W granicy nieskończonej liczby wyznaczników tak otrzymana funkcja falowa odpowiada dokładnej funkcji wieloelektronowej.
+W praktyce stosuje się rozwinięcia do kilku największych wyrazów szeregu (\ref{phiCT}), które
+obejmują podwójne, potrójne lub poczwórne wzbudzenia.  
+Następnym krokiem jest umożliwienie optymalizacji również orbitali jednoelektronowych w połączeniu z
+optymalizacją współczynników rozwinięcia w ramach wielokonfiguracyjnej metody pola samouzgodnionego  ({\it ang. multi-configuration self-consistant field }- MCSCF). 
+Metody wielowyznacznikowe pozwalają wyznaczyć bardzo dokładnie funkcje falowe i energie stanów elektronowych, 
+jednak czas obliczeniowy skaluje się eksponencjalnie z rozmiarem układu,
+więc stosowane jest jedynie do małych układów molekularnych.
+
+\section{Gaz elektronowy}
+
+Jednorodny gaz elektronowy jest dobrym przybliżeniem rzeczywistej struktury
+elektronowej prostych metali i jest często wykorzystywany w wielu podstawowych
+metodach obliczeniowych (np. w przybliżeniu lokalnej gęstości).
+Najprostszy model gazu elektronowego składa się z nieoddziałujących cząstek
+zamkniętych w sześcianie o objętości $V=L^3$. Długość elektronowej fali de'Broglie'a
+rozchodzącej się w każdym kierunku musi spełniać warunek periodyczności ($n\lambda=L$),
+co prowadzi to kwantowania wektora falowego w kierunkach $x$, $y$ i $z$
+%
+\begin{equation}
+\mb{k}=(\frac{2\pi n_x}{L},\frac{2\pi n_x}{L},\frac{2\pi n_x}{L}),
+\end{equation}
+%
+gdzie $n_x$, $n_y$, $n_z$ są liczbami całkowitymi.
+Energie i funkcje falowe dyskretnych stanów elektronowych dane są wzorami
+%
+\begin{equation}
+E_k=\frac{\hbar^2 k^2}{2m},
+\end{equation}
+%
+\begin{equation}
+\psi_k(\mb{r})=\frac{1}{\sqrt{V}}e^{i\mb{kr}},
+\label{pw}
+\end{equation}
+%
+Zakładamy, że układ nie jest spolaryzowany magnetycznie i liczba elektronów ze spinami skierowanymi 
+do góry i na dół jest jednakowa, $N_{\uparrow}=N_{\downarrow}=\frac{1}{2}N$.
+Zgodnie z zakazem Pauliego w stanie kwantowym o danym wektorze falowym $\mb{k}$ mogą
+znajdować się maksymalnie dwa elektrony o przeciwnych kierunkach spinu.
+W temperaturze $T=0$ K elektrony obsadzają kolejne stany od najniższego do
+maksymalnej wartości $E_F$ nazywanej energią Fermiego. 
+Całkowita energia układu $N$ elektronów wynosi
+%
+\begin{equation}
+E=2\sum_{k<k_F} E_k = \frac{2V}{(2\pi)^3}\int d\bm{k} \frac{\hbar^2 k^2}{2m} = \frac{V}{(2\pi)^3}\frac{4\pi\hbar^2}{5m} k_F^5, 
+\label{E}
+\end{equation}
+%
+gdzie $(2\pi)^3/V$ jest objętością w przestrzeni odwrotnej przypadającą na jeden stan elektronowy, 
+a $k_F$ jest wektorem falowym Fermiego, który jest promieniem kuli w przestrzeni odwrotnej
+zawierającej zajęte stany elektronowe. W ogólnym przypadku powierzchnia obejmująca zajęte stany elektronowe,
+nazywana powierzchnią Fermiego, może mieć dowolny kształt i składać się z wielu odrębnych części.
+Liczba elektronów znajdująca się w objętości $V$ wyraża się wzorem
+%
+\begin{equation}
+N=\frac{2V}{(2\pi)^3}\int d\bm{k}=\frac{2V}{(2\pi)^3}\frac{4\pi}{3} k_F^3.  
+\label{N}
+\end{equation}
+%
+Dzieląc (\ref{E}) przez (\ref{N}) otrzymujemy średnią energię przypadającą
+na pojedynczy elektron, wyrażoną przez energię Fermiego
+%
+\begin{equation}
+\frac{E}{N}=\frac{3}{5} \frac{\hbar^2 k_F^2}{2m} = \frac{3}{5} E_F.
+\label{EN}
+\end{equation}  
+%
+W przypadku jednorodnego gazu gęstość elektronowa dana jest wyrażeniem
+%
+\begin{equation}
+n = \frac{N}{V} = \frac{k_F^3}{3\pi^2}.
+\label{nkf}
+\end{equation}
+%
+Często używaną wielkością jest objętość kuli w przestrzeni rzeczywistej przypadająca na jeden elektron 
+%
+\begin{equation}
+V_s=\frac{4\pi}{3}r_{s}^3=\frac{1}{n},
+\end{equation}
+%
+gdzie $r_s$ jest jej promieniem
+%
+\begin{equation}
+r_s=(\frac{3}{4\pi n})^{1/3}.
+\end{equation}
+%
+
+Wyliczymy teraz energię gazu elektronowego w przybliżeniu Hartree-Focka.
+Zakładamy, że ładunek dodatni rozłożony jest jednorodnie w całej przestrzeni $V$
+i ma gęstość taką samą jak gaz elektronów. Energia oddziaływania
+elektronów z takim polem kasuje się z energią Hartree i całkowita energia
+elektronu składa się tylko z części kinetycznej i oddziaływania wymiennego.
+Energię wymiany można łatwo wyliczyć przedstawiając potencjał kulombowski
+w postaci transformaty Fouriera
+%
+\begin{equation}
+\frac{1}{|\mb{r}-\mb{r}'|}=4\pi\int \frac{d\mb{q}}{(2\pi)^3}\frac{e^{i\mb{q}(\mb{r}-\mb{r}')}}{q^2},
+\end{equation}
+%
+Wstawiając funkcję falową w postaci (\ref{pw}) do wyrażenia na energię wymiany 
+i wykonując całkowanie otrzymujemy energie stanów jednoelektronowych
+%
+\begin{equation}
+E_k=\frac{\hbar^2 k^2}{2m} - \int_{\mb{k'}<\mb{k_F}} \frac{d\bm{k'}}{(2\pi)^3} \frac{4\pi}{|\mb{k}-\mb{k'}|^2}
+=\frac{\hbar^2 k^2}{2m} - \frac{k_F}{\pi}(1+\frac{k_F^2-k^2}{2kk_F}ln|\frac{k_F+k}{k_F-k}|).
+\end{equation}
+%
+Całkowitą energię układu elektronów dostajemy sumując to wyrażenie po wektorze falowym    
+%
+\begin{equation}
+E=\sum_{k<k_F}[\frac{\hbar^2 k^2}{2m} - \frac{k_F}{\pi}(1+\frac{k_F^2-k^2}{2kk_F}ln|\frac{k_F+k}{k_F-k}|)].
+\end{equation}
+%
+Wykorzystując (\ref{EN}) i zamieniając sumowanie w drugim wyrazie na całkę dostajemy
+%
+\begin{equation}
+E=N(\frac{3}{5}E_F - \frac{3k_F}{4\pi})=N(\frac{3}{5}E_F - \frac{3(3\pi^2n)^{\frac{1}{3}}}{4\pi}),
+\end{equation}
+gdzie skorzystaliśmy ze związku (\ref{nkf}). Energia wymiany przypadająca na jeden elektron wynosi 
+%
+\begin{equation}
+\varepsilon_x=\frac{E_x}{N}=-\frac{3}{4}\Big(\frac{3}{\pi}\Big)^{\frac{1}{3}}n^\frac{1}{3}.
+\label{exchange}
+\end{equation}
+%
+Ten wzór określający zależność energii wymieny od gęstości elektronowej
+wykorzystywany jest w teorii funkcjonału gęstości w ramach przybliżenia lokalnej gęstości.
+
+Dla gazu elektronowego można wyprowadzić wyrażenie opisujące rozkład ładunku dziury wymiennej (\ref{EC}),
+który zależy tylko od odległości $r$
+%
+\begin{equation}
+g^{\sigma}_x(r)=1 - [3\frac{sin(rk_F)-rk_Fcos(rk_F)}{(rk_F)^3}]^2.
+\end{equation}
+%
+Nie jest możliwe wyrażenie energii korelacji dla gazu swobodnych elektronów w postaci 
+funkcji analitycznej gęstości $n$. Pierwszy przybliżony wzór zaproponował Wigner \cite{wigner34}
+%
+\begin{equation}
+\varepsilon(r_s)=-\frac{0.44}{r_s+7.8}.
+\end{equation}
+%
+W granicy małych gęstości ($r_s\rightarrow \infty$), electrony tworzą tzw. kryształ Wignera, którego energia korelacji 
+pokrywa się z energią elektrostatyczną zlokalizowanych ładunków punktowych.
+W granicy dużej gęstości ($r_s\rightarrow 0$), Gellmann i Breuckner
+wyprowadzili wzór na gęstość energii korelacji niespolaryzowanego magnetycznie
+gazu elektronów \cite{GB}
+%
+\begin{equation}
+\varepsilon_c (r_s)=0.311\ln(r_s)+r_s(A\ln(r_s)+C)-0.048+... .
+\end{equation}
+%
+Energię korelacji gazu elektronowego najdokładniej wyznaczono numerycznie
+metodą kwantowego Monte Carlo (QMC) \cite{CeperleyAlder80}. 
+
+
+\chapter{Stany elektronowe w krysztale}
+
+\section{Sieć krystaliczna}
+
+Kryształy zbudowane są z atomów lub molekuł ułożonych w periodyczną sieć.
+Trójwymiarową sieć krystaliczną definiujemy jako zbiór punktów, których
+położenia określone są wektorem translacji
+%
+\begin{equation}
+\bm{R}_n=n_1\bm{a}_1+n_2\bm{a}_2+n_3\bm{a}_3,
+\label{trans}
+\end{equation}
+%
+gdzie wektory $\bm{a}_i$ nazywamy podstawowymi lub prymitywnymi wektorami translacji, a $n_i$ są liczbami całkowitymi.
+Zbiór punktów zdefiniowanych tym wzorem nazywamy siecią Bravais. Podobnie można zdefiniować sieć Bravais
+dla dowolnego wymiaru $d$ wybierając odpowiednią ilość wektorów bazowych: $\bm{a}_1,\bm{a}_2,...,\bm{a}_d$. 
+Z każdym punktem sieci Bravais można związać zbiór atomów zwanych bazą. Najmniejsza przestrzeń kryształu,
+która po przesunieciu o wszystkie translacje sieciowe wypełni całą przestrzeń nazywamy komórką prymitywną. 
+Komórka prymitywna zawiera dokładnie jeden węzeł sieci Bravais i zwykle zbudowana jest na podstawowych wektorach translacji sieci.
+Istnieje specjalny rodzaj komórki prymitywnej nazywany komórką Wignera-Seitza, która jest przestrzenią zawierającą punkty przestrzeni, 
+które znajdują się bliżej danego punktu sieci niż pozostałych. 
+Często zamiast komórki prymitywnej używa się komórki elementarnej, która zbudowana jest na innych wektorach bazowych
+i jej symetria jest taka sama jak symetria całego kryształu.
+Objętość komórki elementarnej jest całkowitą wielokrotnością objętości komórki prymitywnej.
+W strukturach prostych (bez centrowania powierzchniowego lub przestrzennego), komórka elementarna pokrywa się
+z komórką prymitywną. Różnice między poszczególnymi rodzajami komórek można przedstawić na przykładzie
+sieci dwuwymiarowej. Na rysunku \ref{fig:bravais} pokazana jest struktura centrowana prostokątna z zaznaczonymi dwoma wektorami podstawowaymi $\bm{a}_p$
+i $\bm{b}_p$, które definiują komórkę prymitywną, pokazaną na rysunku w dwóch wariantach. Komórka Wignera-Seitza pokazana
+jest po prawej stronie. Komórka elementarna wyznaczona jest przez dwa wektory bazowe $\bm{a}_e$ i $\bm{b}_e$.
+W dwóch wymiarach mamy w sumie pięć rodzajów sieci Bravais: skośna, prostokątna, prostokatna centrowana, heksagonalna i kwadratowa.
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.4]{lattice.pdf}
+\caption{Dwuwymiarowa sieć periodyczna.}\label{fig:bravais}
+\end{figure}
+
+W przypadku sieci trójwymiarowych, wszystkie możliwe struktury można podzielić na siedem układów krystalograficznych: 
+regularny, tetragonalny, rombowy, heksagonalny, trygonalny, jednoskośny i trójskośny, którym odpowiada czternaście możliwych sieci Bravais,
+które podzielone są na proste (P), centrowane powierzchniowo (F), centrowane na podstawach (C) i centrowane przestrzennie (I). 
+Wszystkie układy krystalograficzne z odpowiadającymi im sieciami Bravais przedstawione są w tabeli \ref{lattice}.
+Dla każdego układu pokazana jest zalezność między stałymi sieci ($a$, $b$, $c$) oraz kątami między kierunkami ($\alpha$, $\beta$, $\gamma$),
+które definiują krawędzie komórek elementarnych. 
+
+\begin{table}[h!]
+\caption{Układy krystalograficzne.}\label{lattice}
+\begin{center}
+\begin{tabular}{|c|c|c|c|}
+\hline
+Układ & Stałe sieci & Kąty & Sieci Bravais\\
+\hline
+regularny & $a=b=c$ & $\alpha=\beta=\gamma=90^{\degree}$ &  P, F, I \\ 
+tetragonaly & $a=b\ne c$ & $\alpha=\beta=\gamma=90^{\degree}$ &  P, I \\  
+rombowy & $a\ne b\ne c$ & $\alpha=\beta=\gamma=90^{\degree}$ &  P, F, C, I \\ 
+heksagonalny &  $a=b\ne c$ & $\alpha=\beta=90^{\degree}$, $\gamma=120^{\degree}$ &  P  \\
+trygonalny & $a=b\ne c$ & $\alpha=\beta=90^{\degree}$, $\gamma=120^{\degree}$ &  P \\
+jednoskośny & $a=b\ne c$ & $\alpha=\gamma=90^{\degree}$, $\beta\ne 90^{\degree}$ & P, C \\ 
+trójskośny & $a\ne b\ne c$ & $\alpha\ne \beta\ne \gamma\ne 90^{\degree}$ & P \\ \hline
+\end{tabular}
+\end{center}
+\end{table}
+
+Zbiór operacji, który pozostawia dany kryształ niezmienionym tworzy grupą przestrzenną.
+Składa się ona z translacji sieci krystalicznej opisanych wzorem (\ref{trans}) oraz symetrii punktowych: inwersji, obrotów i odbić.
+Dodatkowo występują operacje niesymorficzne, które są połączeniem operacji punktowych i ułamkowych translacji.
+Wszystkich grup przestrzennych jest 230, z których 73 to symorficzne i 157 niesymorficzne.
+
+\section{Przestrzeń odwrotna}
+
+Strukturę elektronową materiału analizuje się najczęściej wykorzystując przestrzeń wektorów falowych $\bm{k}$,
+którą nazywamy przestrzenią odwrotną. Każdej sieci Bravais w przestrzeni rzeczywistej odpowiada sieć punktów
+w przestrzeni odwrotnej zdefiniowana wektorami
+%
+\begin{equation}
+\bm{G}_m=m_1\bm{b}_1+m_2\bm{b}_2+m_3\bm{b}_3,
+\end{equation}
+%
+gdzie $m_j$ są liczbami całkowitymi, a $\bm{b}_j$ są wektorami odwrotnymi do wektorów bazowych $\bm{a}_i$,
+czyli spełniają warunek
+%
+\begin{equation}
+\bm{a}_i \cdot \bm{b}_j = 2\pi \delta_{ij}.
+\label{orto}
+\end{equation}
+%
+Wektory sieci odwrotnej można wyznaczyć z następujących zależności
+%
+\begin{eqnarray}
+\bm{b}_1=\frac{2\pi}{\Omega}\bm{a}_2\times \bm{a}_3, \\
+\bm{b}_2=\frac{2\pi}{\Omega}\bm{a}_3\times \bm{a}_1, \\
+\bm{b}_3=\frac{2\pi}{\Omega}\bm{a}_1\times \bm{a}_2, 
+\end{eqnarray}
+%
+gdzie $\Omega=\bm{a}_1\cdot(\bm{a}_2\times \bm{a}_3)$ jest objętością komórki prymitywnej.
+
+W przestrzeni odwrotnej również definiujemy komórkę prymitywną, która najczęściej ma kształt
+komórki Wignera-Seitza i nosi nazwę strefy Brillouina ({ang. Brillouin zone} - BZ).  
+Komórka, która obejmuje punkt $\bm{K}=0$, nazywany punktem $\Gamma$, nosi nazwę pierwszej strefy Brillouina.
+Na rysunku \ref{fig:lattice} pokazane są przykładowe strefy Brillouina dla trzech sieci regularnych. 
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=1.2]{brillouin.pdf}
+\caption{Strefy Brillouina dla sieci regularnych: prostej (P), centrowanej objetościowo (I) i centrowanej przestrzennie (F).}
+\label{fig:lattice}
+\end{figure}
+
+
+\section{Twierdzenie Blocha}
+
+W najprostszym opisie elektronów w periodycznym potencjale kryształu $V(\bm{r})$, stany elektronowe opisywane są przez niezależne jednocząstkowe funkcje falowe, bedące rozwiązaniami równania Schr\"{o}dingera w postaci
+%
+\begin{equation}
+[-\frac{\hbar^2\nabla^2}{2m}+V(\bm{r})]\psi^{\sigma}_{\bm{k}j}(\bm{r})=\varepsilon_{j\sigma}(\bm{k})\psi^{\sigma}_{\bm{k}j}(\bm{r}),
+\label{RS}
+\end{equation}
+%
+gdzie $\bm{k}$ jest wektorem falowym, $\sigma$ określa spin elektronu i indeks $j$ numeruje stany własne hamiltonianu odpowiadajace energiom $\varepsilon_{j\sigma}(\bm{k})$.
+Zgodnie z twierdzeniem Blocha, funkcja falowa elektronów w periodycznym potencjale ma postać
+%
+\begin{equation}
+\psi^{\sigma}_{\bm{k}j}(\bm{r})=e^{i\bm{k}\bm{r}}u^{\sigma}_{\bm{k}j}(\bm{r}),
+\end{equation}
+%
+gdzie $u^{\sigma}_{\bm{k}j}(\bm{r})$ jest funkcją periodyczną spełniającą warunek $u^{\sigma}_{\bm{k}j}(\bm{r})=u^{\sigma}_{\bm{k}j}(\bm{r}+\bm{R}_n)$ dla
+każdego wektora traslacji $\bm{R}_n$. Można pokazać, że funkcje Blocha są wektorami własnymi
+operatora translacji
+%
+\begin{equation}
+\hat{T}_n\psi^{\sigma}_{\bm{k}j}(\bm{r})=\psi^{\sigma}_{\bm{k}j}(\bm{r}+\bm{R}_n)=e^{i\bm{k}(\bm{r}+\bm{R}_n)}u^{\sigma}_{\bm{k}j}(\bm{r}+\bm{R}_n)=e^{i\bm{k}\bm{R}_n}\psi^{\sigma}_{\bm{k}j}(\bm{r}).
+\end{equation}
+%
+Hamiltonian jest niezmienniczy ze względu na działanie operatora translacji,
+co oznacza, że obydwa operatory ze sobą komutują. Funkcje Blocha są zatem również stanami własnymi hamiltonianu,
+co dowodzi prawdziwości twierdzenia Blocha. 
+
+Funkcja Blocha określona jest dla wektora falowego $\bm{k}$, który w układach periodycznych jest dobrze zdefiniowaną liczbą kwantową.
+Wektor falowy związany jest z kwazipędem elektronu w stanie $\psi_{\bm{k}j}$
+%
+\begin{equation}
+\bm{p}=\hbar\bm{k}.
+\end{equation}
+%
+Ponieważ wszystkie własności struktury elektronowej są translacyjnie niezmiennicze 
+ze względu na przesunięcie o dowolny wektor sieci odwrotnej, również kwazipęd nie zależy od takiego przesunięcia.
+
+\section{Fale płaskie}
+
+Metody obliczeniowe struktury pasmowej różnią sie między sobą stosowanymi reprezentacjami funkcji falowej.
+Ogólnie możemy zapisać funkcję falową w formie
+%
+\begin{equation}
+\psi^{\sigma}_{\bm{k}j}(\bm{r})=\sum_mc^{\sigma}_{jm}(\bm{k})\phi_{\bm{k}m}(\bm{r}),
+\end{equation}
+% 
+gdzie $\phi_{\bm{k}m}(\bm{r})$ są funkcjami bazowymi, a $c^{\sigma}_{jm}(\bm{k})$ to współczynniki rozwinięcia.
+W krysztale, ze względu na periodyczny potencjał, naturalną bazę stanowią fale płaskie. Jeżeli cześć periodyczną funkcji
+Blocha rozwiniemy w tej bazie, funkcja falowa przyjmuje postać
+%
+\begin{equation}
+\psi^{\sigma}_{\bm{k}j}(\bm{r})=e^{i\bm{k}\bm{r}}\sum_{m} c^{\sigma}_{jm}(\bm{k})e^{i\bm{G_m}\bm{r}}=\sum_{m} c^{\sigma}_{jm}(\bm{k})e^{i(\bm{k}+\bm{G_m})\bm{r}},
+\end{equation}
+% 
+gdzie $\bm{G_m}$ są wektorami sieci odwrotnej. Równanie Schr\"{o}dingera (\ref{RS}) przetransponowane do przestrzeni odwrotnej w bazie
+fal płaskich przyjmuje postać
+%
+\begin{equation}
+\sum_{n}H_{mn} c^{\sigma}_{jn}(\bm{k})=\varepsilon_{j\sigma}(\bm{k}) c^{\sigma}_{jm},
+\label{ham_rec}
+\end{equation}
+%
+gdzie elementy macierzowe Hamiltonianu dane są wyrażeniem
+%
+\begin{equation}
+H_{mn}=\frac{\hbar^2}{2m}|\bm{k}+\bm{G_m}|^2\delta_{mm'}+V(\bm{G_m}-\bm{G_n}).
+\end{equation}
+%
+Drugi wyraz jest transformatą Fouriera potencjału elektronowego
+%
+\begin{equation}
+V(\bm{G})=\frac{1}{\Omega}\int_{\Omega}\bm{dr}V(\bm{r})e^{-i\bm{Gr}},
+\end{equation}
+%
+gdzie całkowanie przebiega po komórce prymitywnej.
+
+\section{Periodyczne warunki brzegowe}
+
+Podobnie jak dla gazu elektronowego zamkniętego w pojemniku, również dla skończonego kryształu
+możemy wprowadzić periodyczne warunki brzegowe (warunki Borna-Karmana). Załóżmy, że rozmiary kryształu
+określone są przez liczbę komórek prymitywnych w każdym kierunku kryształu $(N_1,N_2,N_3)$.
+Zatem liczby całkowite, które definiują wektory translacyjne sieci (\ref{trans})
+zmieniają się w zakresie $n_i=0,1,2,...,N_i$, gdzie $i=1,2,3$.
+Zakładając, że funkcja falowa na dwóch przeciwległych brzegach układu (w każdym z trzech kierunków) przyjmuje
+takie same wartości i korzystając z twierdzenia Blocha otrzymujemy warunek
+%
+\begin{equation}
+\psi^{\sigma}_{\bm{k}j}(\bm{r})=\psi^{\sigma}_{\bm{k}j}(\bm{r}+N_i\bm{a}_i)=e^{i\bm{k}N_i\bm{a}_i}\psi^{\sigma}_{\bm{k}j}(\bm{r}),
+\label{pwb}
+\end{equation} 
+%
+który spełniony jest gdy 
+%
+\begin{equation}
+\bm{k}N_i\bm{a}_i=2\pi n_i.
+\end{equation}
+%
+Wykorzystując bazę wektorów prymitywnych sieci odwrotnej ($\bm{b}_1,\bm{b}_2,\bm{b}_3$) oraz biorąc pod uwagę zależność (\ref{orto}), dostajemy zbiór dozwolonych wektorów falowych
+%
+\begin{equation}
+\bm{k}=\frac{n_1}{N_1}\bm{b}_1+\frac{n_2}{N_2}\bm{b}_2+\frac{n_3}{N_3}\bm{b}_3.
+\label{vectk}
+\end{equation}
+%
+Wektory te określają funkcje falowe Blocha $\psi_{i,\bm{k}}$ i energie stanów $\varepsilon_i(\bm{k})$.
+Skończona ilość dostępnych stanów wynika zatem z ograniczonych rozmiarów kryształu i jest równa ilości komórek prymitywnych w układzie: $N=N_1N_2N_3$.  
+Ze wzoru (\ref{vectk}) wynika również, że dozwolone wektory falowe stanów elektronowych należą do pojedynczej komórki prymitywnej w przestrzeni odwrotnej.
+Najczęściej do analizy stanów elektronowych wybiera się wektory należące do pierwszej strefy Brillouina.
+
+\section{Sumowanie w przestrzeni odwrotnej}
+
+Wiele wielkości fizycznych wyznaczanych jest jako sumy po punktach w przestrzeni odwrotnej. 
+Przedstawiona powyżej analiza pokazuje, że wystarczy ograniczyć
+sie do wektorów falowych należących do pierwszej strefy Brillouina.
+Dodatkowo, można wykorzystać symetrię kryształu do zmniejszenia ilości punktów, które konieczne
+są do wyznaczenia danej wielkości fizycznej. 
+Oznaczmy operacje punktowe (symorficzne) przez $O_i$, a wektory translacji ułamkowej związane z daną symetrią $O_i$
+przez $\bm{t}_i$. W przestrzeni odwrotnej rozpatrujemy jedynie symorficzne operacje symetrii ponieważ translacje
+ułamkowe nie mają wpływu na przestrzeń odwrotną. Hamiltoniana opisujący kryształ jest niezmienniczy
+względem operacji punktowych: $\bm{r}\rightarrow O_i\bm{r}+\bm{t}_i$ oraz $\bm{k}\rightarrow O_i\bm{k}$.
+Oznacza to, że funkcje falowe otrzymane po tych transformacjach 
+%
+\begin{equation}
+\psi^{\sigma}_{O_i\bm{k}j}(O_i\bm{r}+\bm{t}_i)=\psi^{\sigma}_{\bm{k}j}(\bm{r}),
+\end{equation}  
+%
+są również funkcjami własnymi hamiltonianu z tą samą energią własną $\varepsilon_{j\sigma}(\bm{k})$.
+Oznacza to, że można ograniczyć sumowanie danej wielkości fizycznej $f(\bm{k})$ do wektorów $\bm{k}$ należących do mniejszego obszaru,
+który nazywany jest nieredukowalną częścią strefy Brillouina ({\it ang. irreducible Brillouin zone} - IBZ). 
+Obszar ten tworzą wektory $\bm{k}$, które są nierównoważne, czyli nie można ich wzajemnie połączyć operacją symetrii punktowej. 
+Wartości w pozostałych punktach dostajemy przez zastosowanie odpowiednich operacji symetrii $f(O_i\bm{k})=f(\bm{k})$.
+Do sumowania wykorzystujemy wagi $w_{\bm{k}}$, które zdefiniowane są jako ilości wektorów $\bm{k}$, powiązanych symetrią punktową 
+z wektorem należącym do IBZ (z uwzglednieniem tego wektora), podzielone przez całkowitą ilość wektorów $N_{\bm{k}}$.
+Przykładowo, wartość średnią wielkości $f$, możemy wyliczyć sumując po punktach należących do IBZ
+%
+\begin{equation}
+\bar{f}=\frac{1}{N_{\bm{k}}}\sum_{\bm{k}}^{BZ}f(\bm{k})=\sum_{\bm{k}}^{IBZ}w_{\bm{k}}f(\bm{k}).
+\end{equation}
+%
+
+Przy sumowaniu po przestrzeni odwrotnej można w zasadzie wybrać dowolne punkty należące do IBZ.
+Jednak wykorzystując własności funkcji periodycznych, można tak wybrać wektory $\bm{k}$,
+aby zminimalizować błędy przy obliczaniu sum lub całek.
+Dowolną funkcję periodyczną można rozwinąć przy pomocy transformaty Fouriera
+%
+\begin{equation}
+f(\bm{k})=\sum_n f(\bm{R}_n)e^{i\bm{k}\bm{R}_n},
+\end{equation} 
+%
+gdzie $\bm{R}_n$ są wektorami translacji sieci krystalicznej.
+Zbiór punktów, który jest optymalny do wyliczania sum w przestrzeni odwrotnej, 
+nazywany jest siecią Monkhorsta-Packa \cite{MP}. Wyznacza się go ze  wzoru
+%
+\begin{equation}
+\bm{k}(n_1,n_2,n_3)=\sum_i^3 \frac{2n_i-N_i-1}{2N_i}\bm{b}_i,
+\end{equation}
+%
+gdzie liczby $N_1$, $N_2$ i $N_3$ określają ilość punktów $\bm{k}$ w każdym kierunku: $n_i=1,2,...,N_i$.
+Tak zdefiniowane punkty tworzą jednorodną sieć w przestrzeni odwrotnej i charakteryzują się
+tym, że suma wartości periodycznej funkcji, która posiada komponenty Fouriera ograniczone w każdym kierunku do $\bm{R}_n=N_i\bm{a}_i$,
+równa jest dokładnej całce z tej funkcji.  
+
+\section{Energia Fermiego}
+
+Wielkości fizyczne, które dotyczą stanu podstawowego, wyliczane są zwykle dla obsadzonych stanów elektronowych.
+Czyli sumowanie w przestrzeni odwrotnej uwzglednia tylko zajęte stany.
+Energia najwyższego obsadzonego stanu określa energię Fermiego ($E_F$).
+W metalach pasma elektronowe przecinają $E_F$ i przy sumowaniu po dyskretnych punktach $\bm{k}$ pojawiają się nieciągłości
+w obsadzeniach stanów. Powoduje to wolniejszą zbieżność ze względu na ilość punktów $\bm{k}$. 
+Można łatwo rozwiązać ten problem przez wprowadznie ciagłej funkcji obsadzania stanów w pobliżu $E_F$.
+Naturalnym wyborem jest zastosowanie funkcji rozkładu Fermiego-Diraca dla skończonej temperatury
+%
+\begin{equation}
+F(\varepsilon,E_F,T)=[\exp(\frac{\varepsilon-E_F}{kT})+1]^{-1}.
+\end{equation}   
+%
+Dla $T=0$, dostajemy funkcję schodkową, która odpowiada rozkładowi w stanie podstawowym.
+Aby wyznaczyć $E_F$ dla dowolnej temperatury, korzystamy z warunku
+%
+\begin{equation}
+\sum_{\bm{k},j} w_{\bm{k}}F(\varepsilon,E_F,T)=n_{el},
+\end{equation}   
+%
+gdzie $n_{el}$ jest liczbą elektronów, a sumowanie wykonywane jest po wszystkich obsadzonych stanach
+uwzględniając polaryzację spinową.
+
+
+
+\section{Struktura pasmowa}
+
+Zbiór wszystkich stanów elektronowych $\varepsilon_{\bm{k}j}^{\sigma}$ otrzymanych w wyniku diagonalizacji hamiltonianiu 
+dla wektorów falowych $\bm{k}$ tworzy strukturę pasmową danego materiału.
+Dostępne energie stanów elektronowych, które odpowiadają funkcji własnej z indeksem $j$, grupują się w przedziały nazywane pasmami energetycznymi. 
+Rozwiązania równania (\ref{ham_rec}) w funkcji wektora falowego $\bm{k}$ są periodyczne ze względu na przesunięcie o 
+wektor sieci odwrotnej
+%
+\begin{equation}
+\varepsilon_{\bm{k}j}^{\sigma}=\varepsilon_{\bm{k}+\bm{G_m}j}^{\sigma}.
+\end{equation}   
+%
+Zatem przy prezentacji struktury pasmowej można ograniczyć się do pierwszej strefy Brillouina. Zwykle prezentuje się tylko stany elektronowe dla wektorów $\bm{k}$ należących do wybranych kierunków i punktów wysokiej symetrii w przestrzeni odwrotnej. Jeżeli energie pasm przyjmują różne wartości dla dwóch kierunków spinu,
+$\varepsilon_{\bm{k}j}^{\uparrow}\ne \varepsilon_{\bm{k}j}^{\downarrow}$, mamy doczynienia z rozszczepieniem spinowym (wymiennym), które związane jest z uporządkowaniem magnetycznym.
+
+Ważną wielkością, która charakteryzuje strukturą pasmową jest gęstość stanów elektronowych dla danego kierunku spinu $\rho_{\sigma}(E)$,
+która wyliczana jest jako suma po pierwszej strefie Brillouina
+%
+\begin{equation}
+\rho_{\sigma}(E)=\frac{1}{N_k}\sum_{j,\bm{k}}\delta[\varepsilon_{\bm{k}j}^{\sigma}-E],
+\end{equation}
+%
+gdzie $N_k$ jest ilością wektorów $\bm{k}$ w pierwszej strefie Brillouina.
+Całkowita gęstość stanów równa jest sumie obydwóch składowych spinowych, $\rho(E)=\rho_{\uparrow}(E)+\rho_{\downarrow}(E)$.
+Przy wyliczaniu gestości stanów często zamiast sieci Monkhorsta-Packa stosuje się metodę tetraedrów.
+Polega ona na podzieleniu całej strefy Brillouina na tetraedry i wyliczeniu energii stanów w punktach $\bm{k}$, które
+odpowiadają wierzchołkom tetraedrów. Energie pomiędzy tymi punktami wyznacza się stosując interpolację liniową.
+
+Struktura pasmowa decyduje o podstawowych własnościach danego materiału i jest jego wyróżnikiem, pozwalającym zakwalifikować go do określonej grupy materiałów. 
+Trzy podstawowe typy materiałów krystalicznych to metale, półprzewodniki i izolatory.
+Struktura pasmowa metali charakteryzuję się występowaniem powierzchni Fermiego, która odgranicza zajęte stany elektronowe od stanów pustych.
+Zatem w typowych metalach gęstość stanów na poziomie Fermiego jest niezerowa, $\rho(E_F)>0$.
+Tuż powyżej energii Fermiego ($E_F$) znajdują się stany, do których przechodzą elektrony wzbudzone termicznie lub w wyniku oddziaływania z innymi
+elektronami. W półprzewodnikach i izolatorach, powyżej ostatniego zajętego stanu elektronowego, znajduje się przerwa obejmująca
+zakres zabronionych energetycznie stanów.
+Pasmo, które znajduje się tuż poniżej przerwy energetycznej nazywa się pasmem walencyjnym, a to które jest powyżej
+pasmem przewodnictwa.
+Półprzewodniki zwykle charakteryzują się mniejszą przerwa energetyczną ($\sim$1-2 eV) niż izolatory.
+Występują też materiały nazywane półmetalami, w których przerwa energetyczna i $\rho(E_F)$ są równe zeru (np. grafen).
+Odrębną grupę materiałów stanowią izolatory Motta, w których przerwa energetyczna jest wynikiem silnych oddziaływań
+elektronowych.  
+
+
+
+\chapter{Teoria funkcjonału gęstości}
+
+\section{Twierdzenia Hohenberga-Kohna}
+
+Większość nowoczesnych metod obliczeniowych stosowanych do badania własności materiałów opiera się na teorii funkcjonału gestości ({\it ang. density functional theory} - DFT), która została sformułowana w pracach Hohenberga i Kohna \cite{kohn64}
+oraz Kohna i Shama \cite{kohn65}. Obecnie, DFT stosowana jest powszechnie do wyliczania struktury elektronowej,
+optymalizacji sieci krystalicznej, badania własności elastycznych i dynamiki sieci oraz wielu innych własności materiałowych.
+Jest praktycznie jedyną metodą umożliwiającą obliczenia kwantowo-mechaniczne dla układów zawierających setki,
+a nawet tysiące atomów.  
+Podobnie jak w opisie oddziaływania wymiennego zaproponowanym przez Slatera~\cite{slater51}
+i w teorii Thomasa-Fermiego~\cite{thomas,fermi1927,dirac} podstawową wielkością jest tutaj gęstość elektronowa zdefiniowana w każdym punkcie materiału $n(\bm{r})$.
+Główną ideą tego podejścia jest możliwość zastąpienie dokładnej funkcji falowej, układem stanów jednocząstkowych w efektywnym potencjale elektronowym, który
+daje taką samą gęstość elektronową jak opis wieloelektronowy.
+Takie podejście umożliwia wykorzystania dokładnej informacji o oddziaływaniach wymiennych i korelacyjnych
+otrzymanych dla jednorodnego gazu elektronowego do badania niejednorodnych układów atomowych.
+
+Praca Hohenberga i Kohna \cite{kohn64} zawiera dwa fundamentalne twierdzenia, które
+dotyczą relacji między gęstością elektronową $n(\bm{r})$, zewnętrznym potencjałem $V_{ext}(\bm{r})$ 
+oraz energią całkowitą $E[n]$, która jest funkcjonałem gęstości elektronowej. Zależność funkcjonalna oznacza, że dla dowolnego rozkładu 
+elektronów w całej przestrzeni $n(\bm{r})$ można jednoznacznie wyznaczyć energię całkowitą 
+układu $E[n]$. Zapiszmy hamiltonian układu oddziałujących elektronów
+w ogólnej formie
+
+\begin{equation}
+H=-\frac{\hbar^2}{2m}\sum_{i}\nabla_i^2+V_\text{ext}(\bm{r})+\frac{1}{2}\sum_{i\neq j}\frac{e^2}{|\bm{r}_i-\bm{r}_j|}.
+\label{hamil}
+\end{equation}
+%
+Twierdzenia Hohenberga-Kohna można sformułować następująco: 
+
+{\bf Twierdzenie I:} 
+Zewnętrzny potencjał układu oddziałujących elektronów $V_\text{ext}$ jest jednoznacznie określony (z dokładnością do stałej wartości) 
+przez gęstość elektronową w stanie podstawowym $n_0(\mb{r})$.
+
+{\bf Twierdzenie II:}
+Dla ustalonego potencjału zewnętrznego $V_\text{ext}(\bm{r})$, funkcjonał energii $E[n]$ osiąga
+minimalną wartość $E_0$ dla gęstości elektronowej w stanie podstawowym $n_0(\bm{r})$.
+
+Z twierdzeń tych wynikają następujace wnioski. Ponieważ hamiltonian jest jednoznacznie określony (z dokładnościa do stałej)
+przez $n_0(\bm{r})$, funkcje falowe wszystkich stanów elektronowych,
+jak również wszystkie własności układu są całkowicie zdeterminowane przez gęstość elektronową w stanie podstawowym.
+Znajomość funkcjonału energii $E[n]$ jest wystarczająca do wyznaczenia energii i gęstości elektronowej
+stanu podstawowego. 
+
+\section{Równanie Kohna-Shama}
+
+Twierdzenia Kohna-Shama pozwalają w sposób ścisły powiązać gestość elektronową w stanie podstawowym
+z równaniem Schr\"{o}dingera dla układu oddziałujacych elektronów, opisanego hamiltonianem (\ref{hamil}).
+W praktyce nie jest możliwe bezpośrednie rozwiązanie równania Schr\"{o}dingera i wyznaczenie wielociałowych funkcji falowych.
+Teoria funkcjonału gęstości pozwala zastąpić równanie wielociałowe nowym równaniem, nazywanym równaniem Kohna-Shama, które ma
+taką postać jak dla nieoddziałujących cząstek znajdujacych się w efektywnym polu $V_{eff}$.
+Znając jednoelektronowe rozwiązania równania Kohna-Shama $\psi_i^{\sigma}(\bm{r})$, zależne od położenia i spinu, 
+możemy wyznaczyć gęstość elektronową w każdym punkcie
+%
+\begin{equation}
+n(\bm{r})=\sum_{i,\sigma} f_{i\sigma}|\psi_i^{\sigma}(\bm{r})|^2,
+\label{density}
+\end{equation}
+%
+gdzie dla uproszczenia indeks $i$ określa zarówno numer stanu $j$, jak i wektor falowy $\bm{k}$, a $f_{i\sigma}$ są liczbami obsadzeń
+stanów, które w ogólnym przypadku mogą przyjmować ułamkowe wartości, np. zgodnie z rozkładem Fermiego-Diraca.
+Przyjmujemy również, że ładunek elementarny $e=1$, co oznacza, że gęstość ładunku jest tożsama z gęstością elektronową. 
+Twierdzenia Kohna-Shama zapewniają nam, że efektywny potencjał $V_{eff}$ jest jednoznacznie
+określony przez gęstość elektronową, jak również, że funkcjonał energii całkowitej osiąga
+minimum dla gęstości elektronowej w stanie podstawowym.
+Zatem stan podstawowy układu możemy wyznaczyć jeżeli znamy funkcjonał energii i potrafimy
+znaleźć jego minimum. Ogólnie możemy zapisać ten funkcjał w postaci 
+%
+\begin{equation}
+E[n]=T[n] + E_\text{ext}[n] + E_\text{H}[n] + E_\text{xc}[n],
+\label{EKS}
+\end{equation}
+%
+gdzie $T[n]$ jest energią kinetyczną nieoddziałujących elektronów
+%
+\begin{equation}
+T[n]=-\frac{\hbar^2}{2m}\sum_{i,\sigma} \int d\bm{r} \psi^{\sigma *}_i(\bm{r})\nabla_i^2 \psi^{\sigma}_i(\bm{r}),
+\end{equation}
+%
+$E_\text{ext}[n]$ jest energią oddziaływania z potencjałem zewnętrznym
+%
+\begin{equation}
+E_\text{ext}[n]=\int d\bm{r} V_\text{ext}(\bm{r}) n(\bm{r}),
+\end{equation}
+%
+$E_\text{H}[n]$ jest energią Hartree dana wzorem (\ref{Hartree}) i $E_\text{xc}[n]$ zawiera wszystkie pozostałe oddziaływania,
+czyli oddziaływanie wymienne i korelacje elektronowe, jak również różnicę między energią kinetyczną układu
+oddziałujących i nieoddziałujących elektronów. W skrócie ten wyraz nazywany jest funkcjonałem wymienno-korelacyjnym.
+
+Możemy teraz zastosować podejście wariacyjne uwzględniając warunek ortonormalności
+orbitali Kohna-Shama: $\langle\psi_i^{\sigma *}|\psi_j^{\sigma'}\rangle = \delta_{ij}\delta_{\sigma\sigma'}$
+i wprowadzając mnożniki Lagrange'a
+%
+\begin{equation}
+\frac{\delta}{\delta\psi_i^{\sigma *}}(E[n]-\sum_{i,\sigma}\varepsilon_{i\sigma}[\int d\bm{r} \psi_i^{\sigma *}(\bm {r})\psi_i^{\sigma}(\bm{r})-1]) = 0.
+\label{delta}
+\end{equation}
+%
+Wstawiając funkcjonał energii w postaci (\ref{EKS}) do (\ref{delta}) otrzymujemy 
+%
+\begin{equation}
+\frac{\delta T[n]}{\delta \psi_i^{\sigma *}}+\frac{\delta}{\delta n}(\int d\bm{r} V_{ext}(\bm{r}) n(\bm{r}) + E_H[n] + E_{xc}[n])\frac{\delta n}{\delta \psi_i^{\sigma *}}-\varepsilon_{i\sigma} \frac{\delta}{\delta\psi_i^{\sigma *}}\sum_{j,\sigma'}\int d\bm{r} \psi_j^{\sigma' *}\psi_j^{\sigma}=0,
+\end{equation}
+%
+gdzie zastosowalismy wzór na pochodna funkcji złozonej:
+% 
+\begin{equation}
+\frac{\delta}{\delta \psi_i^{\sigma *}}=\frac{\delta}{\delta n}\frac{\delta n}{\delta \psi_i^{\sigma *}}. 
+\end{equation}
+%
+Wykonując pochodne wariacyjne dostajemy równanie Kohna-Shama
+%
+\begin{equation}
+[-\frac{\hbar^2\nabla_i^2}{2m}+V_{KS}(\bm{r})]\psi_i^{\sigma}(\bm{r})=\varepsilon_{i\sigma}\psi_i^{\sigma}(\bm{r}),
+\end{equation}
+%
+które ma postać jednocząstkowego równania Schr\"{o}dingera z potencjałem Kohna-Shama złożonym z trzech wyrazów
+%
+\begin{equation}
+V_{KS}(\bm{r})=V_{ext}(\bm{r})+V_H(\bm{r})+V_{xc}(\bm{r})=V_{ext}(\bm{r})+\int d\bm{r'} \frac{n(\mb{r}')}{|\bm{r}-\bm{r'}|}+\frac{\delta E_{xc}[n]}{\delta n}.
+\label{VKS}
+\end{equation}
+%
+W potencjale tym nieznana jest dokładnie energia wymienno-korelacyjna, która musi być przybliżana
+odpowiednimi metodami, które będą omówione w kolejnych rozdziałach.
+Znając spektrum energetyczne rozwiązań równania Kohna-Shama, energię stanu podstawowego można wyrazić w postaci
+%
+\begin{equation}
+E[n,f_i]=\sum_{i\sigma}f_{i\sigma} \varepsilon_{i\sigma} - \int d\bm{r} n(\bm{r})V_{KS}(\bm{r}) + \int d\bm{r} V_{ext}(\bm{r}) n(\bm{r}) + E_H[n] + E_{xc}[n].
+\end{equation} 
+%
+Funkcje jednoelektronowe i energie stanów Kohna-Shama $\varepsilon_{i\sigma}$ nie mają jednoznacznej interpretacji fizycznej.
+Można je jednak powiązać ze zmianą energii całkowitej przy zmianie obsadzenia stanów
+%
+\begin{equation}
+\frac{\partial E}{\partial f_{i\sigma}}=\Big(\frac{\partial E}{\partial f_{i\sigma}}\Big)_n+\int d\bm{r} \frac{\delta E}{\delta n}\frac{\partial n}{\partial f_{i\sigma}}.
+\label{janak}
+\end{equation} 
+%
+Drugi wyraz po prawej stronie równania zeruje się dla stanu podstawowego ($\delta E/\delta n=0$),
+co prowadzi do wzoru
+%
+\begin{equation}
+\varepsilon_{i\sigma}=\frac{\partial E}{\partial f_{i\sigma}},
+\end{equation}
+%
+który nazywany jest twierdzeniem Janaka \cite{janak}.
+Zastosowanie tego wzoru do najwyższego obsadzonego stanu elektronowego daje energię jonizacji, czyli energię potrzebną do usunięcia pojedynczego elektronu 
+z danego układu atomowego. 
+
+
+
+\section{Funkcjonał wymienno-korelacyjny}
+
+\subsection{Przybliżenie lokalnej gęstości (LDA)}
+
+Funkcjonał wymienno-korelacyjny $E_{xc}$ i odpowiadający mu potencjał $V_{xc}$ nie są znane dokładnie i w ramach teorii
+funkcjonału gęstości muszą być opisywane w przybliżony sposób.
+Najprostszym podejściem jest przybliżenie lokalnej gęstości ({\it ang. local density approximation} - LDA).
+W przybliżeniu LDA przyjmujemy, że energia wymienno-korelacyjna w każdym punkcie przestrzeni, gdzie gęstość elektronowa wynosi $n(\mb{r})$,
+równa jest energii wymienno-korelacyjnej jednorodnego gazu elektronowego o tej samej gęstości, $n=n(\mb{r})$.
+Funkcjonał wymienno-korelacyjny może być wtedy zapisany w postaci
+%
+\begin{equation}
+E_{xc}[n]=\int d\mb{r} n(\mb{r}) \varepsilon_{xc}(n),
+\end{equation}
+%
+gdzie $\varepsilon_{xc}(n)$ jest energią wymienno-korelacyjną przypadającą na pojedynczy elektron w jednorodnym gazie elektronowym o gestości $n$.
+Można ją zapisać jako sumę części wymiennej i korelacyjnej, $\varepsilon_{xc}(n)=\varepsilon_x(n)+\varepsilon_c(n)$.   
+Przybliżenie LDA jest dokładne w granicy wolno zmieniającej się gęstości, co odpowiada warunkowi
+%
+\begin{equation}
+\frac{q}{k_F}\ll 1,
+\end{equation} 
+%
+gdzie $q$ jest miarą niejednorodności układu
+%
+\begin{equation}
+q=\frac{|\nabla k_F|}{2k_F},
+\end{equation}
+%
+a $k_F$ odpowiada wektorowi falowemu Fermiego dla gazu jednorodnego, który w punkcie o lokalnej gęstości $n(\bm{r})$ dany jest wzorem
+%
+\begin{equation}
+k_F=[3\pi^2n(\bm{r})]^{1/3}.
+\end{equation} 
+%
+
+Można łatwo uogólnić to przybliżenie do układów z polaryzacją spinową. Wtedy energia wymienno-korelacyjna jest funkcjonałem
+gęstości spinów skierowanych do góry $n_{\uparrow}$ i w dół $n_{\downarrow}$
+%
+\begin{equation}
+E_{xc}[n^{\uparrow},n^{\downarrow}]=\int d\bm{r} n(\mb{r}) \varepsilon_{xc}(n_{\uparrow},n_{\downarrow}).
+\label{xclda}
+\end{equation}
+%
+To uogólnienie nazywane jest czasem przybliżeniem lokalnej gestości spinowej ({\it ang. local spin density} - LSD).
+Potencjał wymienno-korelacyjny zależny od spinu $\sigma$ w tym przybliżeniu określony jest przez zależność
+%
+\begin{equation}
+V^{\sigma}_{xc}=\frac{\delta E_{xc}}{\delta n_{\sigma}}=\varepsilon_{xc}+n_{\sigma}\frac{\partial \varepsilon_{xc}}{\partial n_{\sigma}}.
+\end{equation}
+%
+Zgodnie ze wzorem (\ref{exchange}), część wymienna dana jest wzorem
+%
+\begin{equation}
+\varepsilon_x(n_{\sigma})=-\frac{3}{4}\Big(\frac{3}{\pi}\Big)^{\frac{1}{3}}n_{\sigma}^\frac{1}{3},
+\end{equation}
+%
+co prowadzi do potencjału wymiennego w formie 
+%
+\begin{equation}
+V^{\sigma}_x=-\Big(\frac{3}{\pi}\Big)^{\frac{1}{3}}n_{\sigma}^\frac{1}{3}. 
+\label{Vex}
+\end{equation}
+%
+Część korelacyjna może być wyznaczona bardzo dokładnie numerycznie metodą kwantowego Monte Carlo \cite{CeperleyAlder80}. 
+W praktyce stosuje się odpowiednie wyrażenia analityczne dofitowane do wyników numerycznych, które określają zależność energii korelacji 
+od gęstości elektronowej \cite{PZ,VWN}. 
+Przykładowo energia korelacji w parametryzacji Pardew-Zungera \cite{PZ} ma postać
+%
+\begin{eqnarray}
+\varepsilon_c(r_s)&=&-0.048+0.031 \ln(r_s) -0.0116 r_s+0.002r_s \ln(r_s), \quad r_s<1  \\
+             &=&-0.1423/(1+1.0529\sqrt{r_s}+0.3334 r_s), \quad r_s>1.
+\end{eqnarray}
+%
+
+\subsection{Uogólnione przybliżenie gradientowe (GGA)}
+\label{sec:GGA}
+
+Uwzględnienie lokalnych zmian gęstości poprzez rozwinięcie energii wymienno-korelacyjnej w szereg gradientów gęstości
+zaproponowano już w pracy Kohna i Shama z 1965 roku \cite{kohn65}. Podejście to jednak nie spełnia reguł sum i załamuje się dla większości układów atomowych, 
+w których zmiany gęstości elektronowej sa przeważnie bardzo duże.
+Zaproponowane zostało nowe podejście zwane uogólnionym przybliżeniem gradientowym ({\it ang. generalized gradient approximation} - GGA),
+gdzie energia wymienno-korelacyjna jest funkcjonałem gęstości elektronowej i jej gradientów \cite{Langreth83,Pardew86,Becke88}. 
+W ogólnej formie dla układu spolaryzowanego spinowo może być zapisana w formie
+%
+\begin{equation}
+E_{xc}[n_{\uparrow},n_{\downarrow}]=\int d\mb{r} f(n_{\uparrow},n_{\downarrow},\nabla n_{\uparrow},\nabla n_{\downarrow}) 
+\label{xcgga}
+\end{equation}
+%
+Część wymienną tego funkcjonału dla układu bez polaryzacji spinowej można zapisać w postaci
+%
+\begin{equation}
+E_x[n]=\int d\mb{r} n \varepsilon_x(n) F_x(s),
+\end{equation}
+%
+gdzie $s=|\nabla n|/2k_Fn$ jest przeskalowanym (bezwymiarowym) gradientem gęstości elektronowej. Rozszerzenie na układy z polaryzacja spinową uzyskuje się stosując następujący wzór,
+który spełniony jest dla dokładnej energii wymiennej
+%
+\begin{equation}
+E_x[n^{\uparrow},n^{\downarrow}]=\frac{1}{2}(E_x[2n^{\uparrow}]+E_x[2n^{\downarrow}]).
+\end{equation}
+%
+Wiele form funkcji $F_x(s)$ zostało zaproponawych, w tym te najczęściej stosowane, które opisane są w pracach: A. D. Becke (Becke88) \cite{Becke88}, 
+J. P. Perdew i Y. Wang (PW91) \cite{PW91} i  J. P. Perdew, K. Burke, and M. Ernzerhof (PBE) \cite{PBE}.
+Jako przykład omówię funkcjonał PBE, który ma prostą postać i dobrze zdefiniowane warunki graniczne  
+które musi spełniać funkcja $F_x(s)$:
+
+\begin{enumerate}
+
+\item{W granicy małych wartości gradientu ($s\rightarrow0$)
+spełniony jest warunek
+%
+\begin{equation}
+F_x(s)\rightarrow 1+\mu s^2,
+\end{equation}
+%
+gdzie $\mu =0.219$. Warunek ten zapewnia właściwe zachowanie się odpowiedzi liniowej jednorodnego gazu
+elektronów (liniowe przyczynki od energii wymiany i korelacji kasują się).}
+
+\item{Dla dużych wartości gradientu ($s\rightarrow\infty$) funkcja ograniczona jest od góry $F_x(s)\leq 1.804$.}
+
+Funkcja, która spełnia te warunki ma postać
+%
+\begin{equation}
+F_x(s)=1+\kappa-\frac{\kappa}{1+\frac{\mu s^2}{\kappa}},
+\end{equation} 
+%
+gdzie $\kappa=0.804$.
+\end{enumerate}
+
+Część korelacyjną można zapisać w ogólnej formie
+%
+\begin{equation}
+E_c[n^{\uparrow},n^{\downarrow}]=\int d\mb{r} n [\epsilon_c(r_s,\zeta)+H(r_s,\zeta,t)],
+\end{equation}
+%
+gdzie $\zeta=(n^{\uparrow}-n^{\downarrow})/n$ jest względną polaryzacją spinową, $t=|\nabla n|/2\phi k_s n$ jest bezwymiarowym gradientem,
+$\phi=\sqrt{(1+\zeta)^{2/3}+(1-\zeta)^{2/3})}/2$ jest spinową funkcją skalującą i $k_s=\sqrt{4k_F/\pi a_0}$ wektorem ekranowania Thomasa-Fermiego.
+Funkcja $H$ spełnia następujące warunki:
+
+\begin{enumerate}
+\item{Dla małych gradientów ($t\rightarrow 0$) funkcja dana jest wyrazem rozwinięcia drugiego rzędu
+%
+\begin{equation}
+H\rightarrow \frac{e^2}{a_0}\beta\phi^3 t^2,
+\end{equation}
+%
+gdzie $\beta=0.067$.}
+\item{Przy szybko zmieniających się gęstościach ($t\rightarrow \infty$) energia korelacji znika
+co spełnione jest przy warunku
+%
+\begin{equation}
+H\rightarrow -\epsilon_c.
+\end{equation}}
+%
+\item{W granicy dużych gęstości ($r_s\rightarrow 0$) energia korelacji dąży do stałej wartości.
+Funkcja $H$ musi kasować logarytmiczną osobliwość w $\epsilon_c$
+%
+\begin{equation}
+H\rightarrow \frac{e^2}{a_0}\gamma\phi^3 \ln t^2,
+\end{equation}
+%
+gdzie $\gamma=0.031$.}
+\end{enumerate}
+
+Warunki te spełnia funkcja w postaci
+%
+\begin{equation}
+H=\frac{e^2}{a_0}\gamma\phi^3\ln[1+\frac{\beta}{\gamma}t^2\frac{1+At^2}{1+At^2+A^2t^4}],
+\end{equation}
+%
+gdzie 
+%
+\begin{equation}
+A=\frac{\beta}{\gamma}[\exp\{-\epsilon_c/(\gamma\phi^3e^2/a_0)\}-1]^{-1}.
+\end{equation}
+
+
+W tabeli \ref{EXC} porównane są wartości energii wymiany i korelacji w przybliżeniu LDA i PBE
+z wartościami dokładnymi wyznaczonymi dla kilku wybranych atomów. Wartości bezwzględne energii wymiany są większe
+o około rząd wielkości od energii korelacji. W LDA, energia wymiany jest zaniżona średnio o $10\%$, a energia korelacji jest znacznie zawyżona,
+nawet powyżej $100\%$, w porównaniu do wartości dokładnych. Ze względu na przeciwne znaki odchyleń od wartości dokładnych, błędy energii wymiany 
+i korelacji w przybliżeniu LDA częściowo kasują się. 
+
+\begin{table}[h!]
+\caption{Porównanie energii wymiennej i korelacyjnej uzyskanych w kilku przybliżeniach z wartościami dokładnymi dla kilku wybranych atomów.}
+\label{EXC}
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|}
+\hline
+ & \multicolumn{2}{c|}{LDA} & \multicolumn{2}{c|}{PBE} & \multicolumn{2}{c|}{Dokładna wartość} \\ \hline
+Atom& $E_x$ & $E_c$ & $E_x$ & $E_c$  & $E_x$  & $E_c$ \\
+\hline
+  H & -0.2680 & -0.0222 &  -0.3059 & -0.0060 & -0.3125 & 0.0000\\
+  He & -0.8840 & -0.1125 &  -1.0136 & -0.0420 & -1.0258 & -0.0420\\
+  Be & -2.3124 & -0.2240 &  -2.6358 & -0.0856 & -2.6658 & -0.0950\\
+  N &  -5.9080 & -0.4268 &  -6.5521 & -0.1799 & -6.6044 & -0.1858\\
+  Ne & -11.0335 & -0.7428 &  -12.0667 & -0.3513 & -12.1050 & -0.3939\\ \hline
+\end{tabular}
+\end{center}
+\end{table}
+
+Te duże różnice w energii wymiany i korelacji wynikają z nieuwzględnienia wpływu lokalnych zmian 
+gęstości elektronowej, które są charakterystyczne dla atomów i molekuł \cite{Pardew92}.  
+W przybliżeniu GGA, wartości obydwóch energii poprawiają się. Energia wymiany jest w dalszym ciągu zaniżona, ale tylko
+na poziomie $\sim 1\%$. Energia korelacji w atomie wodoru jest znacznie zredukowana w porównaniu do LDA,
+a dla atomu helu jest bardzo dobrze odtworzona. W pozostałych przypadkach pokazanych w tabeli \ref{EXC} jest mniejsza
+od wartości dokładnej o kilka procent.
+
+Funkcjonał GGA w granicy zerowych gradientów, czyli dla jednorodnego gazu elektronowego, jest równoważny przybliżeniu LDA.
+Oznacza to, że dla niejednorodnych układów powinień dawać zawsze wyniki lepsze niż LDA. Tak się jednak nie dzieje, a wynika
+to z faktu, że dla niektórych układów częściowe redukowanie się błedów energii wymiany i korelacji jest większe w LDA niż w GGA.
+Przykładem są metale szlachetne (Ag, Au, Pt), dla których stałe sieci wyznaczone w LDA bardzo dobrze zgadzają się
+z eksperymentem, a w GGA są zawyżone. Większe wartości stałej sieci w przybliżeniu GGA wynikają z zaniżonej energii kohezji kryształów.
+LDA zwykle daje zawyżone wartości energii kohezji i zaniżone wartości stałych sieci. 
+Dla układów magnetycznych, czyli z otwartymi powłokami $d$ lub $f$, przybliżenie GGA daje zwykle lepsze wyniki niż LDA.
+Klasycznym przykładem jest żelazo, dla którego stanem podstawowym w przybliżeniu LDA jest struktura centrowana powierzchniowo ($fcc$) 
+z uporządkowaniem antyferromagnetycznym, a GGA daje poprawny wynik, czyli strukturę centrowaną objętościowo ($bcc$) z uporządkowaniem ferromagnetycznym.
+
+\section{Procedura iteracyjna}
+
+Równania Kohna-Shama (KS) rozwiązywane są metodą iteracyjną, której kolejne kroki przedstawione są na rysunku \ref{fig:diagram}.
+Ogólnie stosowane procedury pozwalają zminimalizować funkcjonał energii całkowitej przy zadanych położeniach atomów,
+jak również wyznaczyć położenia atomów i parametry sieci krystalicznej, które odpowiadają minimum energii całkowitej
+układu. Najpierw ustalamy początkowe położenia atomów, które można przyjąć na przykład na podstawie
+danych z eksperymentu dyfrakcyjnego. Przy ustalonych położeniach atomowych przyjmuje się początkowy
+rozkład gęstości elektronowej $n_0(\bm{r})$, dla którego wylicza się potencjał ze wzoru (\ref{VKS}).
+Dla tego potencjału rozwiązuje się równania KS i wyznacza jednoelektronowe funkcje falowe
+w odpowiedni wybranej bazie funkcyjnej. 
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.45]{diagram.pdf}
+\caption{Schemat interacyjnej procedury rozwiązywania równań Kohna-Sham.}
+\label{fig:diagram}
+\end{figure}
+
+Rozwiązywanie równań KS dla atomów w ciele stałym jest złożonym problemem i będzie dyskutowane w kolejnych rozdziałach.
+Znając funkcje falowe i energie własne równania KS wyznacza się gęstość elektronową w każdym punkcie przestrzeni 
+z zależności (\ref{density}) i całkowitą energię układu. 
+Nowy rozkład gęstości pozwala ponownie wyznaczyć potencjał KS i rozwiązać równania KS. Dostajemy nowy rozkład gęstości i zmienioną
+energię całkowitą. Jeżeli ta energia pokrywa się z energią w poprzednim
+kroku lub te energie różnią sie o małą wielkość (zdefiniowaną na początku obliczeń) to oznacza, że 
+otrzymaliśmy prawidłowo zbiegnięty rozkład gęstości. Zakończona jest w ten sposób pętla elektronowa, 
+która daje rozwiązanie równań KS i minimalną energię całkowitą przy zadanych położeniach atomowych.
+
+Procedurę minimalizacji można przyspieszyć wykorzystując odpowiednio gęstości elektronowe
+w każdym kroku iteracyjnym. W najprostszym podejściu początkową wartość gęstości elektronowej $n^p_{i+1}$ w kroku $i+1$ 
+można wyznaczyć jako liniową poprawkę do gęstości z poprzedniego kroku
+%
+\begin{equation}
+n^p_{i+1}=n^p_i+\alpha(n^k_i-n^p_i),
+\label{mixing}
+\end{equation}
+% 
+gdzie $\alpha$ jest współczynnikiem liniowym, $n^p_i$ i $n^k_i$ są początkową i końcową wartością gęstości w kroku $i$.
+Stała wartość współczynnika $\alpha$ nie daje optymalnej szybkości zbieżności.
+W najczęściej stosowanej metodzie Broydena, współczynnik liniowy wyznaczany jest z jakobianu układu $J_i$,
+który optymalizowany jest w każdym kroku iteracji, $\alpha=-J^{-1}_i$.
+
+Aby zoptymalizować układ ze względu na położenia atomowe należy odpowiednio przesuwać
+atomy w nowe położenia, tak aby całkowita energia obniżała się w kolejnych krokach. 
+Wykorzystuje się do tego bardzo efektywną metodę sprzężonego gradientu ({\it ang. conjugate gradient}),
+która jest również metodą iteracyjną (pętla jonowa).
+W każdym kroku tej procedury wykonuje się pełną optymalizację elektronową, aby wyznaczyć rozkład gęstości elektronowej 
+i energię układu dla aktualnego położenia atomów. Wyznaczona gęstość końcowa jest używana do wyznaczenia
+gęstości startowej w kolejnym kroku pętli jonowej według formuły (\ref{mixing}).
+Warunkiem zbieżności jest osiągnięcie stanu o minimalnej energii ze względu na położenia atomów
+i parametry sieci krystalicznej. Według najczęściej przyjmowanego kryterium zbieżność jest osiągnięta jeżeli różnice energii 
+całkowitej w dwóch kolejnych krokach jonowych jest mniejsza od zadanej na początku wartości.
+
+Stan podstawowy otrzymany w wyniku procedury minimalizacyjnej powinien spełniać warunki stanu równowagi badanego materiału.
+Warunki te są następujące: 
+
+\begin{enumerate}
+{\item
+Całkowita siła działająca na każdy atom zeruje się.
+Siły działające na atomy można wyznaczyć stosując twierdzenie Hellmanna-Feynmana \cite{Hellmann,Feynman}, 
+które mówi, że siła działająca na atom $i$ równa jest pochodnej energii całkowitej po położeniu
+tego atomu wziętej ze znakiem przeciwnym
+%
+\begin{equation}
+\bm{F_i}=-\frac{\partial E_{tot}}{\partial\bm{R_i}}.
+\end{equation}
+%
+Zgodnie z tym wzorem, w stanie podstawowym, który odpowiada minimum energii, wszystkie siły $\bm{F_i}$ są równe zeru.
+Wartości sił stanowią często dodatkowe kryterium zbieżności układu w procedurze optymalizacyjnej:
+zbieżność jest osiągnięta jeżeli największa siła działająca na atomy jest mniejsza od zadanej wartości.}
+
+{\item Makroskopowe naprężenia w układzie równe jest naprężeniu wywołanym ciśnieniem zewnętrznym.
+Średni tensor naprężeń określony jest przez pochodną energii całkowitej po tensorze deformacji 
+%
+\begin{equation}
+\sigma_{\alpha\beta}=-\frac{1}{V}\frac{\partial E}{\partial u_{\alpha\beta}}.
+\end{equation}
+gdzie $V$ jest objętością układu. Tensor deformacji jest symetrycznym tensorem pochodnych wektora przesunięcia
+$\bm{u}=\bm{r}-\bm{r'}$ po położeniu $\bm{r}$.
+Przy kompresji hydrostatycznej, ciśnienie wiąże się z tensorem naprężeń zależnością $P=-\frac{1}{2}\sum_{\alpha}\sigma_{\alpha\alpha}$.}
+\end{enumerate}
+
+
+
+\chapter{Metody wyznaczania struktury elektronowej}
+
+\section{Ogólna charakterystyka}
+
+Najważniejsze metody wyliczania struktury pasmowej oparte są na teorii DFT i polegają na
+wyznaczaniu jednocząstkowych funkcji falowych poprzez rozwiązanie równań Kohna-Shama
+lub podobnych równań zawierających efektywny potencjał elektronowy.  
+Podstawowym elementem odróżniającym poszczególne metody jest wybór funkcji bazowych, 
+które służą do rozwinięcia funkcji falowych w całym obszarze kryształu lub w poszczególnych jego częściach. 
+Najbardziej naturalną bazą w periodycznym krysztale są fale płaskie i obliczenia w tej bazie są bardzo efektywne oraz łatwe do implementacji. 
+Niestety, nie nadają sie do opisu elektronów w całym obszarze kryształu i wszystkich stanów elektronowych.
+Elektrony, które znajdują się najbliżej jąder atomowych i obsadzają najniższe stany energetyczne 
+wchodzą w skład rdzenia atomowego ({\it ang. atomic core}).
+W obszarze rdzeni atomowych, potencjał elektryczny jest silnie przyciągający i zbliżony jest do potencjału atomowego.
+Przez to stany elektronowe znajdujące się blisko jądra atomowego, zachowują sie podobnie do orbitali atomowych, czyli 
+silnie oscylują i zmieniają swój znak. Oznacza to również, że energia kinetyczna elektronów w tym obszarze jest duża.
+Funkcje falowe stanów rdzenia sa silnie zlokalizowane i szybko zanikają z odległością. 
+W tym przypadku znacznie lepszą zbieżność uzyskuje się stosując bazę zlokalizowanych funkcji, np. harmoniki sferyczne.
+Funkcja falowa elektronów walencyjnych zachowuje się odmiennie w obszarze rdzeni atomowych i w obszarze międzywęzłowym.
+Schematycznie potencjał periodyczny i funkcja falowa elektronów walencyjnych pokazana jest na rysunku \ref{fig:bloch}.
+W obszarze rdzeni, zaznaczonych kółkami o promieniu $r_c$, funkcja falowa, podobnie jak w przypadku stanów rdzenia,
+zmienia się bardzo szybko.
+W obszarze miedzywęzłowym ($r>r_c$), potencjał i gęstość elektronowa zmieniają się wolno w funkcji położenia. 
+Również funkcja falowa elektronów walencyjnych w tym obszarze jest wolnozmienna i 
+w tym przypadku można uzyskać szybką zbieżność w rozwinięciu na fale płaskie. 
+
+
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.5]{crystal.pdf}
+\caption{Potencjał periodyczny i funkcja falowa elektronów walencyjnych.}
+\label{fig:bloch}
+\end{figure}
+
+\newpage
+
+Najważniejsze metody wyznaczania struktury pasmowej można podzielić na trzy główne grupy:
+
+
+\begin{enumerate}
+\item{{\bf Pseudopotencjały.} Jest to grupa metod, która ogranicza ilość elektronów i rozwiązania równania Kohna-Shama tylko do stanów walencyjnych. 
+Ma to uzasadnienie wynikające z charakteru stanów rdzenia, które są silnie zlokalizowane blisko jądra atomowego i nie biorą udziału w wiązaniach atomowych.
+Przez to najważniejsze własności materiałów zdeterminowane są przez zachowanie elektronów walencyjnych. 
+W metodzie pseudopotencjału stosuje się rozwinięcie funkcji falowej w bazie fal płaskich. Aby zachować jednolity opis funkcji falowej w całym obszarze kryształu, stosuje się w tych metodach przybliżony potencjał działający na elektrony walencyjne w obszarze rdzenia atomowego, który nazywany jest pseudopotencjałem. 
+Dla elektronów walencyjnych wyznacza się pseudofunkcję falową, która jest rozwiązaniem jednocząstkowego równania Schr\"{o}dingera z odpowiednim pseudopotencjałem. W obszarze międzywęzłowym jest ona równa dokładnej funcji falowej, a w obszarze rdzenia jest wolnozmienną funkcją bez oscylacji i miejsc
+zerowych. Najdokładniejsze obliczenia w tej grupie metod stosują pseudopotencjały ultramiękkie i potencjały typu PAW.}
+\item{{\bf Zlokalizowane orbitale.} W tym podejściu funkcję falową elektronów zapisujemy w bazie orbitali zlokalizowanych na poszczególnych
+atomach. W najprostszym przybliżeniu ciasnego wiązania jedynymi istotnymi parametrami są elementy macierzowe Hamiltonianu, które
+opisują przekrywanie się lokalnych orbitali lub funkcji Wanniera. W dokładniejszych obliczeniach jako bazę stosuje się funkcje Gaussa
+lub orbitale Slatera.}
+\item{{\bf Stowarzyszone fale i atomowe sfery}. Ta grupa obejmuje metody obliczeniowe, które bazują na ogólnej zasadzie podziału
+kryształu na dwa obszary. Pierwszy obszar zawiera rdzenie atomowe, w których funkcje falowe zachowują cechy orbitali atomowych i drugi obszar między atomami,
+gdzie elektrony walencyjne opisane są wolnozmienną funkcją falową. W każdym z dwóch charakterystycznych obszarów stosuje się
+rozwinięcia funkcji falowej w różnych bazach funkcyjnych. Obszar rdzenia atomowego definiuje wartość promienia $r_c$. Funkcje falowe
+otrzymane dla odległości mniejszych i wiekszych od $r_c$ muszą spełniać odpowiednie warunki ciągłości na granicy rdzenia atomowego.}
+
+\end{enumerate}
+
+W kolejnych rozdziałach opisane zostaną metody obliczeniowe należące do tych trzech grup.
+  
+\section{Pseudopotencjały}
+
+Główną ideę pseudopotencjału ilustruje rysunek \ref{fig:pseudopot}. W obszarze rdzenia atomowego ($r<r_c$) dokładny potencjał $V$ przybliżany
+jest pseudopotencjałem $\tilde{V}$, który ma skończoną wartość dla $r=0$. Odpowiadająca mu pseudofunkcja falowa $\tilde{\psi}$
+ma gładkie zachowanie (bez oscylacji i miejsc zerowych) w tym obszarze. Dla $r>r_c$ pseudopotencjał pokrywa się z dokładnym potencjałem,
+a pseudofunkcja odpowiada dokładnej funkcji falowej $\psi$. Dzięki takiemu przybliżeniu, pseudofunkcja falowa może być rozwinięta
+na fale płaskie w całym obszarze kryształu.   
+
+  
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.1]{pseudopots-new.pdf}
+\caption{Psudopotencjał i pseudofunkcja falowa.}
+\label{fig:pseudopot}
+\end{figure}  
+  
+Metoda pseudopotencjału rozwinęła się z podejścia zortogonalizowanych fal płaskich ({\it ang. ortogonalized plane waves} - OPW)
+zaproponowanego przez Herringa w 1940 \cite{herring40}. 
+Można zdefiniować transformację, która prowadzi od dokładnego potencjału do pseudopotencjału, wprowadzając jednocześnie pseudofunkcję falową do opisu elektronów walencyjnych \cite{Antoncik,KP}.
+Wprowadzamy osobne oznaczenia dla stanów walencyjnych $|\psi_v\rangle$ i stanów rdzenia $|\psi_c\rangle$, które są stanami własnymi
+hamiltonianu $H$ z odpowiednimi energiami własnymi $\varepsilon_v$ i $\varepsilon_c$.
+Zakładamy, że dokładna funkcja falowa elektronów walencyjnych może być wyrażona przez gładką funkcję $\tilde{\psi}_v$ (pseudofunkcję),
+która jest ortogonalna do stanów rdzenia
+%
+\begin{equation}
+|\psi_v\rangle =|\tilde{\psi}_v\rangle +\sum_{\alpha,c} a_{\alpha} |\psi_{\alpha c}\rangle,
+\label{OPW}
+\end{equation}
+%
+gdzie sumowanie przebiega po atomach $\alpha$ i stanach rdzenia $c$. 
+Dla uproszczenia zapisu wskaźniki $v$ i $c$ określają zarówno numer stanu, jak i kierunek spinu elektronu.
+Współczynniki rozwinięcia dostajemy z warunku ortogonalności  
+%
+\begin{equation}
+a_{\alpha}=-\langle \psi_{\alpha c}|\tilde{\psi}_v\rangle.
+\label{a_alpha}
+\end{equation}
+% 
+W obszarze miedzywęzłowym dokładna funkcja falowa równa jest pseudofunkcji, która może być wyrażona w bazie fal płaskich.
+Wstawiając (\ref{OPW}) do równania $H|\psi_v\rangle=\varepsilon_v|\psi_v\rangle$ otrzymujemy
+%
+\begin{equation}
+H|\tilde{\psi}_v\rangle +\sum_{\alpha,c} a_{\alpha} \varepsilon_{\alpha c} |\psi_{\alpha c}\rangle =
+\varepsilon_v|\tilde{\psi}_v\rangle + \sum_{\alpha c} a_{\alpha} \varepsilon_v|\psi_{\alpha c}\rangle.
+\end{equation}
+%
+Przenosząc drugie równanie na prawą stronę i wykorzystując (\ref{a_alpha}) dostajemy
+%
+\begin{equation}
+(H +\sum_{\alpha,c}(\varepsilon_v - \varepsilon_{\alpha c})|\psi_{\alpha c}\rangle \langle \psi_{\alpha c}|)|\tilde{\psi}_v\rangle =
+\varepsilon_v|\tilde{\psi}_v\rangle.
+\end{equation}
+%
+Dostaliśmy równanie typu Schr\"{o}dingera z dodatkowym potencjałem 
+%
+\begin{equation}
+V^{R}=\sum_{\alpha,c}(\varepsilon_v - \varepsilon_{\alpha c})|\psi_{\alpha c}\rangle \langle \psi_{\alpha c}|.
+\end{equation}
+%
+Jeżeli dodamy to wyrażenie do oryginalnego potencjału $V$ dostaniemy wielkość nazywaną pseudopotencjałem $\tilde{V}=V+V^{R}$.
+W obszarze walencyjnym, pseudopotencjał $\tilde{V}$ pokrywa się z potencjałem $V$.  
+Silnie przyciagający potencjał atomowy w obszarze rdzenia jest znacznie osłabiony przez dodatnią wartość $V^R$.
+To umożliwia zbieżność pseudofunkcji falowej w bazie fal płaskich. 
+
+
+\subsection{Pseudopotencjały zachowujące normę}
+
+Dobre pseudopotencjały mają charakter uniwersalny. Wygenerowane w określonej configuracji atomowej
+powinny zachowywać się jednakowo dobrze w każdym innym ukladzie atomowym. 
+W roku 1979, Hamann, Schl\"{u}ter i Chiang \cite{HSC} zaproponowali pseudopotencjały zachowujące normę, 
+które spełniały  nastepujące warunki:
+
+\begin{enumerate}
+{\item Prawdziwa funkcja i pseudofuncja falowa są równe, $\psi_i(r)=\tilde{\psi}_i(r)$, dla $r\geq r_c$.}
+{\item Ich energie własne dla elektronów walencyjnych powinny być równe.}
+{\item Ładunek prawdziwy i pseudoładunek zawarty w obszarze o promieniu $r\geq r_c$ powinny się pokrywać (zachowanie normy)
+%
+\begin{equation}
+\int_0^r dr r^2 |\psi_i(r)|^2=\int_0^r dr r^2 |\tilde{\psi}_i(r)|^2.
+\end{equation}}
+%
+{\item Pochodna logarytmiczna prawdziwej funkcji i pseudofunkcji są równe dla $r\geq r_c$ 
+%
+\begin{equation}
+\frac{1}{\psi_i(r)}\frac{d\psi_i(r)}{dr}=\frac{1}{\tilde{\psi}_i(r)}\frac{d\tilde{\psi}_i(r)}{dr}.
+\end{equation}}
+
+\end{enumerate} 
+%
+Z tych własności wynika również spełnienie warunku równych pochodnych po energii funkcji dokładnej i pseudofunkcji dla $r\geq r_c$,
+co dodatkowo wzmacnia przenośny charakter tych pseudopotencjałów.
+Metody generowania pseudopotencjałów zachowujących normę zostały opisane szczegółowo w wielu pracach \cite{HSC,BHS}.
+
+Ponieważ pseudopotencjały w obszarze rdzenia zależą od orbitalnej liczby kwantowej $l$, ich charakter nie jest lokalny.
+Wartość charakterystycznego promienia $r_c$ również zależy od $l$.
+Ogólnie można podzielić cały pseudopotencjał na część lokalną i nielokalną
+%
+\begin{equation}
+V_l(r)=V_{lok}(r)+\delta V_l(r).
+\end{equation}
+%
+Nielokalny charakter dotyczy tylko obszaru rdzenia, więc $\delta V_l(r)=0$ dla $r>r_c$
+i wszystkie dalekozasięgowe efekty zależą tylko od $V_{lok}(r)$.
+Możliwość wyboru promienia $r_c$ daje pewną swobodę przy konstrukcji pseudopotencjałów.
+Dokładniejsze i bardziej uniwersalne pseudopotencjały charakteryzują sie mniejszymi wartościami
+$r_c$. Jednak większa dokładnosć wymaga uwzględnienia większej ilości fal płaskich
+w rozwinięciu pseudofunkcji falowej. Takie pseudopotencjały nazywane są twardymi. Miękkie pseudopotencjały odznaczają się
+większymi wartościami $r_c$, mniejszą ilość fal płaskich i bardziej gładkim charakterem pseudofunkcji falowej. 
+
+\subsection{Pseudopotencjały ultramiękkie}
+
+W roku 1990, Vanderbilt zaproponował nowy rodzaj pseupotencjałów, nazywane ultramiękkimi ({\it ang. ultrasoft pseudopotentials} - US-PP), które nie zachowują normy \cite{Vanderbilt90}. W tym podejściu, pseudofunkcje falowe spełniają ugólnione równanie własne w postaci
+%
+\begin{equation}
+H|\tilde{\psi}_i\rangle=\varepsilon_iS|\tilde{\psi}_i\rangle,
+\end{equation}
+%
+gdzie Hamiltonian ma ogólną postać 
+%
+\begin{equation}
+H=-\frac{\hbar^2}{2m}\nabla^2+V_{lok}+\delta V_{NL}, 
+\end{equation}
+a $S$ jest operatorem przekrywania
+%
+\begin{equation}
+S=1+\sum_{i,j} Q_{ij}|\beta_i\rangle\langle\beta_j|,
+\end{equation}
+%
+który określa uogólniony warunek normalizacji pseudofunkcji falowych $\langle\tilde{\psi}_i|S|\tilde{\psi}_j\rangle=\delta_{ij}$.
+Macierz $Q_{ij}$ opisuje różnicę w normalizacji funkcji i pseudofunkcji falowych
+%
+\begin{equation}
+Q_{ij}=\int_0^{r_c} dr Q_{ij}(\bm{r})=\int_0^{r_c} dr r^2[\psi_i^*(\bm{r})\psi_j(\bm{r})-\tilde{\psi}_i^*(\bm{r})\tilde{\psi}_j(\bm{r})].
+\end{equation}
+%
+Zbiór lokalnych funkcji falowych
+%
+\begin{equation}
+|\beta_i\rangle=\sum_j (B^{-1})_{ij}|\chi_j\rangle,
+\end{equation}
+%
+gdzie funkcje 
+%
+\begin{equation}
+|\chi_j\rangle=(\varepsilon_i+\frac{\hbar^2}{2m}\nabla^2-V_{loc})|\tilde{\psi}_j\rangle
+\end{equation}
+%
+zerują się w obszarze dla $r>r_c$. 
+Nielokalną część speudopotencjału otrzymuje się ze wzoru
+%
+\begin{equation}
+\delta V_{NL}=\sum_{ij}D_{ij}|\beta_i\rangle\langle\beta_j|,
+\end{equation}
+%
+gdzie $D_{ij}=B_{ij}+\varepsilon_j Q_{ij}$. Całkowitą gęstość elektonów walencyjnych
+w danym punkcie przestrzeni otrzymujemy ze wzoru 
+%
+\begin{equation}
+n_v(\bm{r})=\sum_i \tilde{\psi}_i^*(\bm{r})\tilde{\psi}_i(\bm{r})+\sum_{i,j}\sum_k\langle\beta_i|\tilde{\psi}_k\rangle\langle\tilde{\psi}_k|\beta_j\rangle Q_{ij}(\bm{r}),
+\end{equation}
+% 
+gdzie wyraz jest poprawką wynikającą z niezachowania normy przez pseudofunkcje falowe.
+
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=1]{usp.pdf}
+\caption{Porównanie dokładnej funkcji falowej $2p$ dla tlenu (linia ciągła) z pseudofunkcją zachowującą normę (linia kropkowana)
+i pseudofunkcją otrzymaną dla pseudopotancjału ultramiękkiego (linia przerywana). Rysunek z pracy \cite{Vanderbilt90}.}
+\label{fig:ups}
+\end{figure}
+
+Na rysunku \ref{fig:ups} pokazana jest dokładna funkcja falowa dla orbitalu $2p$ w tlenie oraz dwie pseudofunkcje otrzymane dla pseudopotencjału zachowujacego normę i pseudopotencjału ultramiękkiego. Promień obcięcia jest wyraźnie większy dla pseudopotencjału ultramiękkiego ($r_c\approx1.5$ a.u.) niż dla zachowującego normę ($r_c\approx0.5$ a.u.).
+
+\subsection{Metoda PAW}
+
+Omówione metody pseudopotencjałów pozwalaja efektywnie wyliczać wiele wielkości fizycznych bazując na pseudofukcjach falowych.
+W wielu przypadkach dokładniejsze obliczenia wymagają znajomości funkcji falowych elektronów walencyjnych
+również w obszarze rdzenia atomowego.
+W roku 1994, Bl\"{o}chl zaproponował nową metodę PAW ({\it ang. projector augmented-wave}),
+która łączy efektywność pseudopotencjałów i dokładność obliczeń porównywalną z metodami pełnego potencjału \cite{Blochl}. 
+Do opisu elektronów walencyjnych podejście to wykorzystuje również pseudofunkcje falowe $\tilde{\psi}_v$, które rozwijane są na fale płaskie i 
+w obszarze międzywęzłowym pokrywają sie z funkcjami dokładnymi $\psi_v$.
+Główną zaletą tej metody jest możliwość wyznaczenia dokładnej funkcji falowej poprzez transformację liniową pseudofukcji falowej
+%
+\begin{equation}
+|\psi_v\rangle=\mathcal{T}|\tilde{\psi}_v\rangle.
+\end{equation}
+% 
+Operator liniowy $\mathcal{T}$ działa w obszarach otoczonych sferą o promieniu $r_c$ wokół atomów w położeniach $\bm{R}_m$
+%
+\begin{equation}
+\mathcal{T}=1+\sum_m \mathcal{T}_m.
+\label{operator}
+\end{equation}
+% 
+Dokładna funkcja falowa i pseudofukcja rozwinięte są w obszarze atomowym na funkcje parcjalne $|\phi_m\rangle$ i $|\tilde{\phi}_m\rangle$
+%
+\begin{eqnarray}
+|\psi_v\rangle=\sum_m c_m |\phi_m\rangle, \label{eq1}\\
+|\tilde{\psi_v}\rangle=\sum_m c_m |\tilde{\phi}_m\rangle,
+\label{eq2}
+\end{eqnarray}
+%
+gdzie w obydwóch sumach występują te same współczynniki rozwinięcia $c_m$. Zatem te lokalne funkcje powiązane są tą samą
+transformacją
+%
+\begin{equation}
+|\phi_m\rangle=(1+\sum_m \mathcal{T}_m)|\tilde{\phi}_m\rangle.
+\end{equation}
+%
+Funkcje $|\phi_m\rangle$ są rozwiązaniami równania Schr\"{o}dingera dla dokładnego potencjału atomowego, które odpowiadają energiom $\varepsilon_m$
+i są ortogonalne do stanów rdzenia. Wskaźnik $m$ określa jednocześnie położenia atomów $\bm{R}$ i zbiór liczb kwantowych orbitali atomowych.
+Każdej parcjalnej funkcji dokładnej odpowiada jedna pseudofunkcja $|\tilde{\phi}_m\rangle$, z którą pokrywa się poza sferą o promieniu $r_c$.
+Oba rodzaje funkcji parcjalnych są funkcjami radialnymi, zdefiniowanymi na logarytmicznej siatce radialnej, przemnożonymi przez
+harmoniki sferyczne. 
+
+Odejmując stronami (\ref{eq1}) i (\ref{eq2}) dostajemy równanie
+%
+\begin{equation}
+|\psi_v\rangle=|\tilde{\psi}_v\rangle-\sum_m c_m |\tilde{\phi}_m\rangle + \sum_m c_m |\phi_m\rangle.
+\label{psiv}
+\end{equation}
+% 
+Aby operator $\mathcal{T}$ był liniowy współczynniki $c_m$ muszą być liniowymi funkcjonałami pseudofunkcji falowej $|\tilde{\psi}_v\rangle$
+%
+\begin{equation}
+c_m=\langle\tilde{p}_m|\tilde{\psi}_v\rangle,
+\label{coef}
+\end{equation}
+%
+gdzie $\langle\tilde{p}_m|$ są funkcjami rzutowymi dualnymi względem funkcji parcjalnych 
+$\langle\tilde{p}_m|\tilde{\phi}_n\rangle=\delta_{m,n}$.
+Dla każdej funkcji $|\tilde{\phi}_m\rangle$ mamy jedną funkcja rzutową $\langle\tilde{p}_m|$
+i dla nich spełniony jest warunek $\sum_m |\tilde{\phi}_m\rangle\langle\tilde{p}_m|=1$.
+Funkcje rzutowe są funkcjami radialnymi pomnożonymi przez harmoniki sferyczny, a następnie przetransformowane do bazy fal płaskich.
+Są przypisane do położeń atomów, ale nie zależą od potencjału atomowego.
+%
+Wstawiając (\ref{coef}) do (\ref{psiv}) otrzymujemy
+%
+\begin{equation}
+|\psi_v\rangle=|\tilde{\psi}_v\rangle+\sum_m (|\phi_m\rangle-|\tilde{\phi}_m\rangle)\langle\tilde{p}_m|)\tilde{\psi}_v\rangle
+=[1+\sum_m (|\phi_m\rangle-|\tilde{\phi}_m\rangle)\langle\tilde{p}_m|]|\tilde{\psi}_v\rangle,
+\label{psiv2}
+\end{equation}
+%
+gdzie wyrażenie w nawiasie kwadratowym jest wprowadzonym wcześniej operatorem liniowym
+%
+\begin{equation}
+\mathcal{T}=1+\sum_m \mathcal{T}_m=1+\sum_m (|\phi_m\rangle-|\tilde{\phi}_m\rangle)\langle\tilde{p}_m|.
+\end{equation}
+%
+
+Funkcje falowe rdzenia $|\psi_c\rangle$, podobnie do funkcji walencyjnych, są rozłożone na trzy składowe
+%
+\begin{equation}
+|\psi_c\rangle=|\tilde{\psi}_c\rangle+|\phi_c\rangle-|\tilde{\phi}_c\rangle,
+\end{equation}
+%
+gdzie kolejne wyrazy odpowiadają pseudofunkcji elektronów rdzenia, która pokrywa się z dokładną funkcją dla $r>r_c$,
+lokalnej (parcjalnej) funkcji rdzenia i lokalnej pseudofukcji rdzenia. Obydwie lokalne funcje wyrażone są jako funkcje radialne
+pomnożone przez harmoniki sferyczne.
+
+Średnia z operatora $A$ może być wyznaczona przy pomocy dokładnych funkcji lub pseudofunkcji
+%
+\begin{equation}
+\langle A \rangle = \sum_n f_n \langle \psi_n|A|\psi_n\rangle = \sum_n f_n \langle \tilde{\psi}_n|\tilde{A}|\tilde{\psi}_n\rangle,
+\end{equation}
+% 
+gdzie $f_n$ określa obsadzenie stanu $n$, a $\tilde{A}$ jest pseudooperatorem, który można otrzymać transformując operator $A$
+%
+\begin{equation}
+\tilde{A}=\mathcal{T}^{\dagger} A \mathcal{T}= A +\sum_{i,j} |\tilde{p}_i\rangle (\langle \phi_i |A|\phi_j\rangle - \langle \tilde{\phi}_i |A|\tilde{\phi}_j\rangle ) \langle\tilde{p}_j|.
+\end{equation}
+%
+Przykładowo, stosując te wyrażenia do operatora gęstości $n=|\bm{r}\rangle\langle\bm{r}|$,
+możemy wyrazić gęstość elektronową następującym wyrażeniem
+%
+\begin{equation}
+n(\bm{r})=\tilde{n}(\bm{r})+n^1(\bm{r})-\tilde{n}^1(\bm{r}),
+\end{equation}
+%
+gdzie
+%
+\begin{equation}
+\tilde{n}(\bm{r})=\sum_n f_n \langle \tilde{\psi}_n|\bm{r}\rangle\langle \bm{r}| \tilde{\psi}_n\rangle=\sum_n f_n |\tilde{\psi_n}(\bm{r})|^2,
+\end{equation}
+%
+\begin{equation}
+n^1(\bm{r})=\sum_{n,i,j} f_n \langle \tilde{\psi}_n|\tilde{p}_i\rangle\langle\phi_i|\bm{r}\rangle\langle \bm{r}|\phi_j\rangle\langle\tilde{p}_j |\tilde{\psi}_n\rangle,
+\end{equation}
+%
+\begin{equation}
+\tilde{n}^1(\bm{r})=\sum_{n,i,j} f_n \langle \tilde{\psi}_n|\tilde{p}_i\rangle\langle\tilde{\phi}_i|\bm{r}\rangle\langle \bm{r}|\tilde{\phi}_j\rangle\langle\tilde{p}_j |\tilde{\psi}_n\rangle.
+\end{equation}
+%
+W powyższych wzorach sumowanie obejmuje zarówno stany walencyjne, jak i stany rdzenia. 
+
+Metoda PAW należy obecnie do najdokładniejszych i najbardziej efektywnych metod. Została zaimplementowana
+w takich programach, jak Vienna Ab Initio Simulation Package (VASP) \cite{Vasp,PawVasp} i Quantum Espresso \cite{QE}.
+
+
+\section{Zlokalizowane orbitale}
+
+\subsection{Metoda ciasnego wiązania}
+
+W odróżnieniu od metod pseudopotencjału, które wykorzystują bazę fal płaskich, metody opisane w tym rozdziale
+stosują lokalne orbitale, które powiązane są z danymi atomami lub centrowane są w położeniach atomów.
+Opis przy pomocy lokalnych orbitali nadaje się dobrze do układów, gdzie występują zlokalizowane stany
+elektronowe, ale możliwy jest ruch elektronów poprzez przeskoki do sąsiednich atomów.
+Najprostszym opisem takich układów jest model ciasnego wiązania.
+Lokalne orbitale atomowe możemy zapisać w postaci $\phi_{\alpha}(\bm{r}-\bm{R}_{\alpha})$, 
+gdzie wektor $\bm{R}_{\alpha}$ oznacza położenie danego atomu. Wskaźnik $\alpha$ oznacza tutaj wszystkie liczby kwantowe, 
+które charakteryzują dany orbital ($\alpha=n,l,m$). Zakładamy dla uproszczenia, że z danym atomem związny jest tylko
+jeden orbital, ale model ciasnego wiązania można łatwo uogólnić na dowolną liczbę orbitali.
+Hamiltonian możemy zapisać jako
+%
+\begin{equation}
+H=-\frac{\hbar^2}{2m}\nabla^2+\sum_{\alpha}V_{\alpha}(\bm{r}-\bm{R}_{\alpha}),
+\end{equation}
+%
+gdzie $V_{\alpha}$ jest potencjałem atomowym wokół położenia $\bm{R}_{\alpha}$.
+Elementy macierzowe hamiltonianu możemy wyznaczyć ze wzoru
+%
+\begin{equation}
+H_{\alpha\beta}=\int d\bm{r} \phi_{\alpha}^*(\bm{r}-\bm{R}_{\alpha})H\phi_{\beta}(\bm{r}-\bm{R}_{\beta}).
+\label{matrix}
+\end{equation}
+% 
+Ponieważ orbitale atomowe należące do różnych atomów nie są wzajemnie ortogonalne
+wprowadza się również macierz przekrywania
+%
+\begin{equation}
+S_{\alpha\beta}=\int d\bm{r} \phi_{\alpha}^*(\bm{r}-\bm{R}_{\alpha})\phi_{\beta}(\bm{r}-\bm{R}_{\beta}).
+\end{equation}
+%
+Elementy diagonalne hamiltonianu $\varepsilon_{\alpha}=H_{\alpha\alpha}$ określają lokalną energię elektronu w danym orbitalu $\alpha$.
+Natomiast elementy pozadiagonalne $t_{\alpha\beta}=H_{\alpha\beta}$ nazywane są całkami przeskoku i określają prawdopodobieństwo
+przeskoku elektronu między dwoma atomami.
+
+Aby przetransformować elementy macierzowe hamiltonianu i macierzy $S$ do przestrzeni odwrotnej
+wprowadzamy bazę, która odpowiada wektorom falowym $\bm{k}$
+%
+\begin{equation}
+\phi_{\bm{k}\alpha}(\bm{r})=A_{\alpha\bm{k}}\sum_n e^{i\bm{k}\bm{T}_n}\phi_{\alpha}[\bm{r}-(\bm{R}_{\alpha}+\bm{T}_n)],
+\end{equation}
+%
+gdzie $A_{\bm{k}\alpha}$ są czynnikami normalizacji, a $\bm{T}_n$ są wektorami translacji sieci krystalicznej.
+W tej bazie możemy zapisać funkcję falową, która spełnia twierdzenie Blocha
+%
+\begin{equation}
+\psi_{\bm{k}i}=\sum_{\alpha} c_{i\alpha} \phi_{\bm{k}\alpha}(\bm{r}),
+\end{equation}
+%
+oraz elementy macierzowe hamiltonianu i macierzy $S$
+%
+\begin{equation}
+H_{\alpha\beta}(\bm{k})=\int d\bm{r} \phi_{\alpha\bm{k}}^*(\bm{r})H\phi_{\beta\bm{k}}(\bm{r})=\sum_n e^{i\bm{k}\bm{T}_n} H_{\alpha\beta},
+\end{equation}
+%
+\begin{equation}
+S_{\alpha\beta}(\bm{k})=\int d\bm{r} \phi_{\alpha\bm{k}}^*(\bm{r})\phi_{\beta\bm{k}}(\bm{r})=\sum_n e^{i\bm{k}\bm{T}_n} S_{\alpha\beta}.
+\end{equation}
+%
+Równanie na energie własne $\varepsilon_i(\bm{k})$ i współczynniki rozwinięcia funkcji falowej $c_{i,\alpha}(\bm{k})$ przyjmuje postać
+%
+\begin{equation}
+\sum_{\beta}[H_{\alpha\beta}(\bm{k})-\varepsilon_i(\bm{k})S_{\alpha\beta}(\bm{k})]c_{i\beta}(\bm{k})=0.
+\label{ham_loc} 
+\end{equation}
+%
+Dla orbitali wzajemnie ortogonalnych, mamy $S_{\alpha\beta}=\delta_{\alpha\beta}$ i równanie (\ref{ham_loc})
+przyjmuje postać taką samą jak równanie w bazie fal płaskich (\ref{ham_rec}).
+
+\subsection{Funkcje Wanniera}
+\label{sec:wannier}
+
+W metodzie ciasnego wiązania często wykorzystuje się bazę funkcji Waniera, które są zlokalizowanymi funkcjami, 
+centrowanymi na położeniach atomowych $\bm{R}_n$. 
+Funkcja Wanniera zdefiniowana jest jako transformata Fouriera funkcji Blocha powiązanej z danym pasmem $j$
+%
+\begin{equation}
+w_j(\bm{r}-\bm{R}_n)=\frac{1}{\sqrt{N}}\sum_{\bm{k}}e^{-i\bm{k}\bm{R}_n}\psi_{\bm{k}j}(\bm{r}). 
+\end{equation}
+%
+Spełniona jest również transformata odwrotna, która pozwala wyrazić funkcje Blocha przez funkcje
+Wanniera
+%
+\begin{equation}
+\psi_{\bm{k}j}(\bm{r})=\frac{1}{\sqrt{N}}\sum_n e^{i\bm{k}\bm{R}_n} w_j(\bm{r}-\bm{R}_n).
+\end{equation}
+%
+Funkcje Wanniera dla różnych położeń atomowych są ortogonalne
+%
+\begin{multline}
+\int d\bm{r} w_j^*(\bm{r}-\bm{R}_n)w_j(\bm{r}-\bm{R}_m)=\frac{1}{N}\sum_{\bm{k},\bm{k}'}\int d\bm{r}e^{i(\bm{k}\bm{R}_n-\bm{k}'\bm{R}_m)}\psi_{\bm{k}j}(\bm{r})\psi_{\bm{k}'j}(\bm{r}) \\
+=\frac{1}{N}\sum_{\bm{k}}e^{i\bm{k}(\bm{R}_n-\bm{R}_m)}=\delta_{mn},
+\end{multline}
+%
+gdzie wykorzystaliśmy ortogonalność funkcji Blocha. Spełniony jest również warunek zupełności
+%
+\begin{equation}
+\sum_n w_j^*(\bm{r}-\bm{R}_n)w_j(\bm{r}'-\bm{R}_n)=\delta^3(\bm{r}-\bm{r}').
+\end{equation}
+Zatem funkcje Wanniera tworzą zbiór zlokalizowanych funkcji bazowych.
+Energie pasma $j$ dla funkcji Blocha w tej bazie może być wyznaczona ze wzoru
+%
+\begin{equation}
+\varepsilon_{\bm{k}j}=\int d\bm{r} \psi_{\bm{k}j}^*(\bm{r})H\psi_{\bm{k}j}(\bm{r})
+=\frac{1}{N}\int d\bm{r}\sum_n w_j^*(\bm{r}-\bm{R}_n)e^{-i\bm{k}\bm{R}_n}H\sum_m w_j(\bm{r}-\bm{R}_m)e^{i\bm{k}\bm{R}_m}.
+\end{equation}
+%
+Sumowanie po wskaźnikach $n$ i $m$ możemy rozdzielić na dwa wyrazy dla $n=m$ i $n\neq m$
+%
+\begin{multline}
+\varepsilon_{\bm{k}j}=\frac{1}{N}\sum_n\int d\bm{r} w_j^*(\bm{r}-\bm{R}_n)Hw_j(\bm{r}-\bm{R}_n)
+\\ +\frac{1}{N}\sum_{n\neq m}e^{i\bm{k}(\bm{R}_m-\bm{R}_n)}\int d\bm{r} w_j^*(\bm{r}-\bm{R}_n)Hw_j(\bm{r}-\bm{R}_m).
+\end{multline}
+%
+Wykorzystując oznaczenia, które wprowadziliśmy dla elementów macierzowych hamiltonianu (\ref{matrix})
+dostajemy
+%
+\begin{equation}
+\varepsilon_{\bm{k}j}=\frac{1}{N}\sum_n \varepsilon_n+\frac{1}{N}\sum_{n\neq m}e^{i\bm{k}(\bm{R}_m-\bm{R}_n)}t_{nm}.
+\end{equation}
+%
+Wzór ten pozwala wyznaczyć strukturę pasmową w ramach metody ciasnego wiązania.
+Jeżeli ograniczymy się do modelu jednopasmowego i do przeskoków tylko między najbliższymi sąsiadami
+możemy dostać uproszczony model z jednym parametrem przeskoku $t$.
+Dla prostej struktury kubicznej dostajemy
+%
+\begin{equation}
+\varepsilon_{\bm{k}j}=\varepsilon_0+\frac{t}{N}\sum_{n}e^{i\bm{k}(\bm{R}_{n+1}-\bm{R}_n)}=\varepsilon_0+2t(cosk_xa+cosk_ya+cosk_za).
+\label{band}
+\end{equation}
+%
+gdzie $a$ jest stałą sieci, a $\varepsilon_0$ jest energią elektronu zlokalizowanego w stanie atomowym.
+Szerokość pasma w tym przybliżeniu dana jest wzorem $w=2z|t|$, gdzie $z$ jest liczbą najbliższych sąsiadów.  
+
+Metoda ciasnego wiązania, która wykorzystuje bazę orbitali atomowych lub funkcji Wanniera,
+często wykorzystywana jest w obliczeniach modelowych. W ramach tego podejścia można łatwo uwzględnić lokalne oddziaływania
+kulombowskie lub oddziaływania parujące w modelach nadprzewodników wysokotemperaturowych.
+Lokalne energie i całki przeskoku mogą być wyznaczone przez dofitowanie struktury elektronowej
+z modelu ciasnego wiązania do wyników eksperymentalnych lub do struktury pasmowej wyznaczonej z obliczeń DFT.
+W tym drugim przypadku stosuje się najczęściej bazę maksymalnie zlokalizowanych funkcji Wanniera.
+Wykorzystuje się swobodę wyboru fazy dla funkcji Blocha, którą możemy przemnożyć przez czynnik $e^{i\theta(\bm{k})}$,
+gdzie $\theta(\bm{k})$ jest dowolną funkcją rzeczywistą, bez wpływu na energie stanów elektronowych. 
+Taki czynnik ma jednak wpływ na kształt funkcji Wanniera, w szczególności na ich zasięg przestrzenny. 
+Można tak dobrać czynnik fazowy, aby funkcja Wanniera $w_i(\bm{r}-\bm{R}_n)$ zlokalizowana była
+wokół punktu $\bm{R}_n$ i szybko zanikała z odległością. 
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=1]{FeSe.pdf}
+\caption{Porównanie struktury pasmowej otrzymanej w ramach modelu ciasnego wiązania i obliczeń DFT dla {\bf a.} CeCoIn$_5$ i {\bf b.} FeSe.
+Rysunek z pracy \cite{FeSe}.}
+\label{fig:fese}
+\end{figure}
+
+Na rysunku \ref{fig:fese} pokazano przykłady struktur pasmowych dla dwóch związków, CeCoIn$_5$ i FeSe, wyliczonych w modelu ciasnego wiązania z bazą
+maksymalnie zlokalizowanych funkcji Wanniera. 
+Parametry modelu wyznaczono przez dofitowanie energii stanów elektronowych, dla każdego wektora falowego $\bm{k}$, 
+do struktur pasmowych otrzymanych w ramach obliczeń DFT \cite{FeSe}.
+Obliczenia wykonano programem Quantum Espresso \cite{QE}, stosując funkcjonał GGA \cite{PW91} w ramach metody PAW \cite{Blochl}.  
+Wyznaczone modele ciasnego wiązania zostały następnie wykorzystane do wyliczenia podatności tworzenia par Coopera
+i zbadania własności nadprzewodzących obydwóch materiałów \cite{FeSe}.
+ 
+
+\subsection{Funkcje Gaussa}
+
+Lokalne orbitale można również wykorzystać jako bazę do rozwiniecia funkcji falowej w ramach samozgodnej procedury rozwiązywania równań Kohna-Shama.
+Jeżeli funkcjami bazowymi są orbitale atomowe mówimy o metodzie liniowych kombinacji orbitali atomowych ({\it ang. linear combination
+of atomic orbitals} - LCAO). Zwykle jednak stosuje się atomo-podobne orbitale, których postać funkcyjna ułatwia implementację
+i przyspiesza obliczenia. Do najczęściej stosowanych należą orbitale typu Slatera \cite{STO1,STO2} i funkcje typu Gaussa (gaussiany) \cite{GTO}.
+Te drugie są szczególnie wygodne ponieważ całki z funkcji Gaussa można wyliczyć analitycznie.
+Naturalną reprezentacją dla gaussianów jest układ współrzędnych biegunowych, ale wygodniejszy do wyznaczania elementów macierzowych hamiltonianu jest układ kartezjański, w którym te funkcje mają postać
+%
+\begin{equation}
+G_{ijk}(\bm{r},\alpha,\bm{R}_n)=N(x-x_n)^i(y-y_n)^j(z-z_n)^ke^{-\alpha (\bm{r}-\bm{R}_n)^2},
+\end{equation}
+%
+gdzie $\alpha$ jest parametrem wariacyjnym, który umożliwia optymalizację bazy dla konkretnych atomów, $\bm{R}_n=(x_n,y_n,z_n)$ jest wektorem określającym centrowanie danej funkcji, a $N$ jest czynnikiem normalizacyjnym.
+Spełniona jest ponadto zależność $l=i+j+k$, gdzie $l$ jest orbitalną liczbą kwantową.
+W wyniku iloczynu dwóch fukcji typu $s$ ($l=0$) centrowanych w punktach $\bm{R}_n$ i $\bm{R}_m$ dostajemy
+%
+\begin{equation}
+G_s(\bm{r},\alpha,\bm{R}_n)G_s(\bm{r},\beta,\bm{R}_m)=e^{-\gamma(\bm{R}_n-\bm{R}_m)^2}G_s(\bm{r},\kappa,\bm{R}_p),
+\end{equation}
+%
+gdzie 
+%
+\begin{equation}
+\kappa=\alpha+\beta, \quad \gamma=\frac{\alpha\beta}{\alpha+\beta}, \quad \bm{R}_p=\frac{\alpha\bm{R}_n+\beta\bm{R}_m}{\alpha+\beta},
+\end{equation}
+%
+gdzie $\bm{R}_p$ jest wektorem centrowania wynikowej funkcji Gaussa.
+W ogólnym przypadku iloczyn dwóch gaussianów można zapisać w formie
+%
+\begin{multline}
+G_{i_1j_1k_1}(\bm{r},\alpha,\bm{R}_n)G_{i_2j_2k_2}(\bm{r},\beta,\bm{R}_m) = Ne^{-\gamma(\bm{R}_n-\bm{R}_m)^2}e^{-\kappa(\bm{r}-\bm{R}_p)^2} \\
+\times (x-x_n)^{i_1}(x-x_m)^{i_2}(y-y_n)^{j_1}(y-y_m)^{j_2}(z-z_n)^{k_1}(z-z_m)^{k_2}.
+\end{multline}
+%
+Te wzory pokazują, ze iloczyn funkcji Gaussa jest również funkcją Gaussa. Jest to główna zaleta tej bazy umożliwiająca
+analityczne wyliczenie złożonych całek zawierajacych kilka funkcji bazowych i w ten sposób znaczne przyspieszenie rachunków.
+W odróżnieniu od metod wykorzystujących fale płaskie, metoda funkcji Gaussa wymaga odpowiedniego wybrania funkcji bazowych
+dla danego układu atomowego. Metoda ta stosowana jest głównie przez chemików, a najbardziej popularnym programem jest GAUSSIAN.
+Głównym twórca tego programu jest John Pople, który w roku 1998 otrzymał wspólnie z Walterem Kohnem Nagrodę Nobla w dziedzinie chemii.
+
+\section{Stowarzyszone fale i atomowe sfery}
+
+Ostatnia grupa metod obliczeniowych łączy główne cechy dwóch poprzednich: pseudopotencjałów i lokalnych orbitali. 
+Do rozwinięcia funkcji falowych stosowane są dwie różne bazy w zależności od położenia w krysztale.
+Wokół rdzeni atomowych naturalnym wyborem sa lokalne funkcje, które można otrzymać z rozwiązania
+równania Schr\"{o}dingera w potencjale sferyczno-symetrycznym. W obszarze miedzywezłowym
+stosuje się rozwinięcie w bazie funkcji, które umożliwiają szybką zbieżność rachunków (np. fale płaskie).
+Centralnym problemem stają sie odpowiednie warunki brzegowe, które zapewniają ciągłość funkcji falowej i jej pochodnej 
+na granicy między tymi obszarami.
+Wszystkie metody omawiane w tym rozdziale stosowały na początku uproszczony potencjał typu {\it muffin-tin}, który w obszarze rdzenia
+ma charakter potencjału atomowego, a w obszarze międzywęzłowym jego wartość jest stała.
+Obecnie podejścia te zostały zaimplementowane w ramach procedury samozgodnej i zaliczane są do metod pełnego potencjału. 
+
+\subsection{Metoda KKR}
+
+W dwóch pracach, Korringa \cite{K47} oraz Kohn i Rostoker \cite{KR54} (KKR) zaproponowali metodę, która nazywana jest również metodą funcji Greena
+lub metodą wielokrotnego rozpraszania.
+Pierwsze obliczenia stałych sieci i modułów sztywności wykonano właśnie z wykorzystaniem tej metody \cite{Moruzzi77}. Była to przełomowa praca, która 
+pokazała możliwość zastosowania DFT do badania własności materiałowych.  
+
+Podstawową wielkością stosowaną tutaj jest funkcja Greena $G(E,\bm{r},\bm{r}')$, która opisuje propagację elektronu o energii $E$ między punktami $\bm{r}$ i $\bm{r}'$. Proces przemieszczania się elektronu można podzielić na swobodny ruch w obszarze międzywęzłowym i rozpraszanie na sferach atomowych
+w wyniku oddziaływania elektronu z potencjałem rdzenia atomowego.
+Dla elektronu opisanego jednocząstkowym równaniem Schr\"{o}dingera
+%
+\begin{equation}
+[-\frac{\hbar^2}{2m}\nabla^2+V(\bm{r})]\psi_i(\bm{r})=\varepsilon_i\psi_i(\bm{r}),
+\end{equation}
+%
+funkcja Greena spełnia równanie
+%
+\begin{equation}
+[-\frac{\hbar^2}{2m}\nabla^2+V(\bm{r})-E]G(E,\bm{r}-\bm{r}')=\delta(\bm{r}-\bm{r}'),
+\end{equation}
+%
+które posiada rozwiązanie w formie
+%
+\begin{equation}
+G(E,\bm{r},\bm{r}')=\sum_i\frac{\psi_i(\bm{r})\psi_i^*(\bm{r}')}{E-\varepsilon_i}.
+\end{equation}
+%
+Dla nieoddziałujących elektronów mamy $V(\bm{r})=0$ i funkcja Greena ma postać
+%
+\begin{equation}
+G_0(E,|\bm{r}-\bm{r}'|)= \frac{m}{2\pi\hbar^2}\frac{e^{ik_0|\bm{r}-\bm{r}'|}}{|\bm{r}-\bm{r}'|},
+\end{equation}
+%
+gdzie $k_0=\frac{\sqrt{2mE}}{\hbar}$.
+
+Funkcja Greena dla elektronu oddziałującego z siecią atomową może być zapisana w postaci rozwinięcia
+%
+\begin{equation}
+G=G_0+G_0tG_0+G_0tG_0tG_0+...=G_0+G_0tG,
+\end{equation}
+%
+które prowadzi do zależności
+%
+\begin{equation}
+G=(G_0^{-1}-t)^{-1},
+\end{equation}
+%
+gdzie $t$ jest macierzą rozpraszania elektronów na pojedynczych atomach sieci krystalicznej.
+Można wprowadzić również macierz całkowitego, wielokrotnego rozpraszania elektronów $T$, która zdefiniowana jest wzorem
+%
+\begin{equation}
+G=G_0+G_0TG_0=G_0+G_0(t+tG_0T)G_0,
+\end{equation}
+%
+z którego dostajemy
+%
+\begin{equation}
+T=(t^{-1}-G_0)^{-1}.
+\end{equation}
+%
+Stany stacjonarne, które odpowiadają rozwiązaniom równania Schr\'{o}dingera, są biegunami macierzy $T$
+i mogą być wyznaczone jako miejsca zerowe wyznacznika
+%
+\begin{equation}
+\det(t^{-1}-G_0)=0.
+\end{equation}
+%
+Równanie to dostarcza rozwiązania problemu wielokrotnego rozpraszania elektronów w przybliżeniu potencjału {\it muffin-tin}.
+Pełna informacja o rozpraszaniu elektronu na pojedynczej sferze atomowej zawarta jest w macierzy rozpraszania $t_l(E)$, 
+która zależy tylko od energii stanu elektronowego $E$ i orbitalnego krętu $l$.
+Korzystając z teorii rozpraszania, można wyrazić macierz rozpraszania jako funkcję przesunięcia fazy funkcji falowej $\eta_l(E)$
+%
+\begin{equation}
+t_l(E)=-\frac{1}{k_0}e^{i\eta_l(E)}\sin(\eta_l(E)).
+\label{fshift}
+\end{equation} 
+%
+Funkcja Greena $G_0$ zależy tylko od geometrii sieci krystalicznej i energii $E$.
+Wykorzystując rozwinięcie fal płaskich w bazie harmonik sferycznych, funcję Greena
+możemy zapisać w formie
+%
+\begin{equation}
+G_0(E,|\bm{r}-\bm{r}'|)=\sum_{L,L'} i^lj_l(k_0r)Y_L(\hat{\bm{r}})B_{LL'}(E,\bm{R}-\bm{R}')(-i)^{l'}j_{l'}(k_0r')Y^*_{L'}(\hat{\bm{r}}'),
+\end{equation} 
+%
+gdzie $j_l(k_0r)$ są funkcjami Bessela, $B_{LL'}$ są stałymi struktury KKR i $L=\{l,m\}$.
+Wektory położenia $\bm{r}$ i $\bm{r}'$ odnoszą się do dwóch sfer atomowych, których środki
+określają wektory $\bm{R}$ i $\bm{R}'$. Przy założeniu, że rozpraszanie w każdym węźle atomowym jest takie samo,
+co odpowiada sieci złożonej z jednego rodzaju atomów, możemy wprowadzić zależność stałych struktury od wektora falowego
+% 
+\begin{equation}
+B_{LL'}(E_{\bm{k}},\bm{k})=\sum_{m} B_{LL'}(E_{\bm{k}},\bm{T}_m)e^{-i\bm{k}\bm{T}_m},
+\end{equation}
+%
+gdzie $\bm{T}_m$ są wektorami translacji sieci krystalicznej, a energie $E_{\bm{k}}$
+są rozwiązaniami równania
+%
+\begin{equation}
+\det[t_l^{-1}(E_{\bm{k}})\delta_{LL'}-B_{LL'}(E_{\bm{k}},\bm{k})]=0.
+\end{equation}
+%
+Korzystając z wyrażenia (\ref{fshift}) dostajemy podstawowe równania na strukturę pasmową w metodzie KKR
+%
+\begin{equation}
+\sum_{L'}[B_{LL'}(E_{\bm{k}},\bm{k})+k_0\cot(\eta_l(E_{\bm{k}}))\delta_{LL'}]a_{L'}(\bm{k})=0.
+\label{KKR}
+\end{equation}
+%
+Rozwiązania tego równania pozwalają nam wyznaczyć funkcje falowe wewnątrz sfer atomowych
+%
+\begin{equation}
+\psi_{\bm{k}}(\bm{r})=\sum_{L'}a_{L'}(\bm{k})u_l(r,E_{\bm{k}})Y_L(\hat{\bm{r}}),
+\end{equation}
+%
+gdzie $u_l(r,E_{\bm{k}})$ są rozwiązaniami równania radialnego wewnątrz sfery, umówionego dokładnie w następnym rozdziale.
+Funkcje falowe w obszarze międzywęzłowym są rozwiązaniami następującego równania całkowego
+%
+\begin{equation}
+\psi_{\bm{k}}(\bm{r})=-\int d\bm{r}'G_0(E_{\bm{k}},|\bm{r}-\bm{r}'|)V(\bm{r}')\psi_{\bm{k}}(\bm{r}').
+\end{equation} 
+%
+Funkcje falowe w obydwóch obszarach muszą spełniać warunki brzegowe na każdej sferze atomowej. 
+
+W ogólnym przypadku macierz rozpraszania zależy od położenia atomu $t_l(E,\bm{R})$
+i równanie na funkcje Greena przyjmuje postać
+%
+\begin{equation}
+[G_{LL'}(E,\bm{R},\bm{R}')]^{-1}=\Big[[B_{LL'}(E,\bm{R}-\bm{R}')]^{-1}-t_l(E,\bm{R})\delta_{\bm{R}\bm{R}'}\delta_{LL'}]\Big],
+\end{equation}
+%
+a stany stacjonarne otrzymujemy z warunku
+%
+\begin{equation}
+\det\Big[t_l^{-1}(E,\bm{R})\delta_{\bm{R}\bm{R}'}\delta_{LL'}-B_{LL'}(E,\bm{R}-\bm{R}')\Big]=0.
+\end{equation}
+
+Funkcja Greena pozwala wyznaczyć różne wielkości fizycznych. Na przykład, część urojona funkcji Greena zwiazana jest z lokalną gęstością stanów 
+%
+\begin{equation}
+n_L(E,\bm{R})=-\frac{1}{\pi}\operatorname{Im}G_{LL}(E+i\delta,\bm{R}).
+\end{equation}
+%
+Podobnie część urojona transformaty Fouriera funkcji Green daje nam gęstość spektralną,
+czyli gęstość stanów elektronowych dla danej energii i wektora falowego
+%
+\begin{equation}
+n_L(E,\bm{k})=-\frac{1}{\pi}\operatorname{Im}G_{LL}(E+i\delta,\bm{k}).
+\end{equation}
+%
+
+Metoda funkcji Greena umożliwia wprowadzenie efektywnego (koherentnego) potencjału, który łączy własności dwóch różnych atomów.
+Jest to przybliżenie koherentnego potencjału ({\it ang. coherent potential approximation} - CPA), które stosowane jest do badania
+stopów metalicznych lub układów domieszkowanych \cite{CPA1,CPA2}.
+Metoda polega na uśrednieniu własności rozpraszania dwóch atomów umieszczonych w pobliżu efektywnego potencjału.
+Warunek równoważności średniej ważonej po obydwóch atomach i efektywnego potencjału prowadzi do zespolonego,
+zależnego od energii potencjału CPA. 
+
+
+\subsection{Metoda LAPW}
+
+Metoda zlinearyzowanych stowarzyszonych fal płaskich ({\it ang. linearized augmented plane wave} - LAPW)
+jest modyfikacją oryginalnego podejścia rozszerzonych fal płaskich (APW) zaproponowanego przez Slatera w roku 1937 \cite{slater37}.
+Podobnie jak w metodzie pseudopotencjału, cały kryształ podzielony jest na obszary wewnątrz sfer otaczających atomy ($r<r_c$)
+i obszar międzywęzłowy ($r>r_c$). Funkcje falowe w otoczeniu jądra atomowego charakteryzują sie dużą zmiennością i
+w przybliżeniu maja kształt sferyczny. Z tego względu naturalnym wyborem bazy sa funkcje radialne, będące rozwiązaniami równania Schr\"{o}dingera z
+potencjałem sferycznie symetrycznym. W obszarze więdzywęzłowym, gdzie potencjał zmienia sie bardzo wolno, rozwiązania
+funkcji falowej rozwijane są w bazie fal płaskich. Funkcję falową w metodzie APW można
+zapisać w postaci
+%
+\begin{equation}
+\psi_{\bm{k}j}(\bm{r})=\sum_{\bm{G}}c_{\bm{G}j}\phi_{\bm{k}+\bm{G}}(\bm{r}),
+\end{equation}
+%
+gdzie $\bm{G}$ oznaczają wektory sieci odwrotnej. Funkcje bazowe mogą być zapisane w formie 
+%
+\begin{equation} 
+\phi_{\bm{k}+\bm{G}}(\bm{r})=\begin{cases}
+        e^{i(\bm{k}+\bm{G})\bm{r}} & r>r_c \\
+       \sum_{lm}A_{lm}(\bm{k}+\bm{G})u_l(r,E_l)Y_{lm}(\theta,\varphi) &  r<r_c,
+             \end{cases}    
+\label{APWbase}                      
+\end{equation}
+%
+gdzie $A_{lm}$ są współczynnikami rozwinięcia, $Y_{lm}(\theta,\varphi)$ są harmonikami sferycznymi i $u_l(r,E_l)$ są rozwiązaniami równania
+%
+\begin{equation}
+[-\frac{d^2}{dx^2}+\frac{l(l+1)}{r^2}+V(r)-E_l]ru_l(r,E_l)=0,
+\label{radial}
+\end{equation}
+%
+gdzie $V(r)$ jest potencjałem wewnątrz sfery, a $E_l$ pełni rolę parametru i niekoniecznie jest to energia własna. 
+W przybliżeniu potencjału {\it muffin-tin} (MT), w obszarze atomowym jest on sferycznie
+symetryczny ($V(\bm{r})=V(r)$), a w obszarze międzywezłowym ma wartość stałą ($V(\bm{r})=V_0$) lub zero. Wtedy funkcje falowe $\psi(\bm{r})$ mogą być wyrażone przez rozwiazania równania Schr\"{o}dingera w poszczególnych obszarach kryształu, odpowiednio przez harmoniki sferyczne wokół atomów i fale płaskie między atomami. 
+Współczynniki $A_{lm}$ wyznacza się tak, aby spełniony był warunek ciągłości funkcji falowej na granicy sfery (warunek ciągłości pierwszych pochodnych nie jest spełniony). w ten sposób współczynniki $A_{lm}$ stają się zależne od vectora falowego.  
+Współczynniki $c_{\bm{G}}$ i liczby $E_l$ są parametrami wariacyjnymi metody APW. Rozwiązania, które numerowane są wektorami $\bm{G}$, składające sie z pojedynczych fal płaskich w obszarze miedzywęzłowym i dopasowane do nich rozwiązania radialne wewnątrz sfery nazywane są stowarzyszonymi falami płaskimi.
+
+Funkcje bazowe wewnątrz sfery zależą od energii $E_l$, co powoduje, że odpowiednie równanie własne na funkcje falowe jest nieliniowe. Jest to główny problem
+podejścia APW, uniemożliwiający łatwe rozszerzenie stosowalności tej metody na dowolne potencjały. W szczególności potencjały wyznaczane metodą samozgodną w ramach teorii funkcjonału gęstość.
+Rozwiązaniem tego problemu była zaproponowana przez Andersena w roku 1975 linearyzacja równań APW, która prowadzi do metody LAPW \cite{Andersen75}.
+W tym podejściu, modyfikuje się funkcje bazowe wewnątrz sfery do postaci
+%
+\begin{equation}
+\phi_{\bm{k}+\bm{G}}(\bm{r})=\sum_{lm}[A_{lm}(\bm{k}+\bm{G})u_l(r,E_l)Y_{lm}(\theta,\varphi)+B_{lm}(\bm{k}+\bm{G})\dot{u}_l(r,E_l)Y_{lm}(\theta,\varphi)],
+\end{equation}    
+%
+gdzie współczynniki rozwinięcia $A_{lm}$ i $B_{lm}$ wyznacza się z warunku ciągłości funkcji falowej i jej pochodnej na sferze, a pochodna $\dot{u}_l(r,E_l)=\partial u_l/\partial E$ spełnia równanie
+%
+\begin{equation}
+[-\frac{d^2}{dx^2}+\frac{l(l+1)}{r^2}+V(r)-E_l]r\dot{u}_l(r,E_l)=ru_l(r,E_l).
+\end{equation}
+Dodatkowo narzuca sie warunek normalizacji funkcji $u_l$
+%
+\begin{equation}
+\int_0^{r_c}dr[ru_l(r,E_l)]^2=1,
+\end{equation}
+%
+oraz ortogonalności funkcji $u_l$ i $\dot{u}_l$
+%
+\begin{equation}
+\int_0^{r_c}drr^2u_l(r,E_l)\dot{u}_l(r.E_l)=0.
+\end{equation}
+
+Znajomość gradientu $\dot{u}_l$ umożliwia wyznaczenie funkcji radialnej dla danej energii pasma $\varepsilon$, 
+uwzlędniając poprawkę liniową
+%
+\begin{equation}
+u_l(r,\varepsilon)=u_l(r,E_l)+(\varepsilon-E_l)\dot{u}_l(r,E_l)+O((\varepsilon-E_l)^2),
+\end{equation}
+%
+gdzie $O((\varepsilon-E_l)^2)$ oznacza błąd, który jest rzędu kwadratu różnicy tych dwóch energii.
+
+W ramach procedury samozgodnej, pełny potencjał wyliczany jest w obydwóch obszarach kryształu,
+stosując rozwinięcia w odpowiednich bazach
+%
+\begin{equation} 
+V(\bm{r})=\begin{cases}
+        \sum_{\rm{G}}V_{\bm{G}} e^{i\bm{G}\bm{r}} & r>r_c \\
+       \sum_{lm}V_{lm}Y_{lm}(\theta,\phi) &  r<r_c.
+             \end{cases}   
+\label{fullpot}                       
+\end{equation}
+%
+Podobnie wylicza się gęstość ładunku wewnątrz sfery korzystając ze współczynników rozwinięcia funkcji falowych
+w bazie harmonic sferycznych oraz w obszarze międzywęzłowym w bazie fal płaskich.
+
+W odróżnieniu od metody pseudopetcjału, gdzie uwzględniane są tylko elektrony walencyjne, w metodzie LAPW wyznaczane są
+zarówno stany walencyjne, jak i stany rdzenia.
+Ponieważ funkcje falowe stanów rdzenia sa zlokalizowane blisko jądra atomowego, wylicza się je stosując tylko 
+potencjał centrosymetryczny. Jeżeli zasięg funkcji falowej stanu rdzenia przekracza granicę strefy atomowej,
+stosuje się gładkie przedłużenie potencjału centrosymetrycznego poza tę granice. 
+Do wyznaczenia stanów rdzenia stosuje sie podejście w pełni relatywistyczne,
+czyli rozwiązuje się równanie Diraca, zamiast równania Schr\"{o}dingera (\ref{radial}). 
+Stany walencyjne wyznaczone są dla pełnego potencjału w całym obszarze kryształu.
+
+W niektórych materiałach występują również stany elektronowe, mające wspólne cechy stanów rdzenia i walencyjnych, które nazywane są
+pseudordzeniowymi. Przykładem są wysoko położone i wzlędnie rozległe stany $5d$ w metalach ziem rzadkich.
+Występowanie tych stanów wymaga modyfikacji metody LAPW. W standardowej wersji, baza konstruowana
+jest tak, aby jak najdokładniej opisać stany elektronowe bliskie energii $E_l$
+i zwykle wybiera się tę wartość blisko środka pasma walencyjnego.
+Jeżeli występują dodatkowe stany w innym zakresie energii, wybór wartości $E_l$ nie jest oczywisty. 
+Najlepszym podejściem do opisu stanów pseudordzeniowych jest rozszerzenie bazy przez wprowadzenie dodatkowych 
+lokalnych orbitali (LO), które mieszczą się wewnątrz atomowej sfery. Ta modyfikacja nazywa się metodą APW+LO \cite{singh91}. 
+Ponieważ istnieje swoboda wyboru bazy, pochodną funkcji radialnej $\dot{u}_l$ usuwa sie z głównej bazy i dołacza do LO.
+Zatem całkowita baza w tym podejściu składa się z oryginalnej bazy APW (\ref{APWbase})  
+i dodatkowych orbitali w formie
+%
+\begin{equation}
+\chi^{lo}_L(\bm{r})=[a^{lo}_{lm}u_l(r,E_l)+b^{lo}_{lm}\dot{u}_l(r,E_l)]Y(\theta,\varphi),
+\end{equation}
+%
+gdzie $r<r_c$, a współczynniki $a^{lo}_{lm}$ i $b^{lo}_{lm}$ są tak dobrane, aby LO
+znikały na sferze. Liczba orbitalna ograniczona jest w tym przypadku do wartości, które odpowiadają 
+fizycznym stanom kwantowym ($l\le 3$). W odróżnieniu od LAPW, w metodzie APW+LO nie ma ograniczenia na
+wartości pochodnych funkcji bazowych na sferze atomowej. 
+  
+
+\subsection{Metoda LMTO}
+
+W metodzie zlinearyzowanych orbitali {\it muffin-tin} ({\it ang. linearized muffin-tin orbitals} - LMTO)
+również wprowadza się podział kryształu na obszary wewnątrz sfer atomowych i obszar miedzywęzłowy.
+Funkcja falowa odpowiadająca energii $\varepsilon$ może być zapisana w ogólnej formie 
+%
+\begin{equation}
+\psi_{\bm{k}}(\varepsilon,\bm{r})=\sum_{lm} a_{lm}(\bm{k}) \phi_{lm}(\varepsilon,\kappa,\bm{r}),
+\end{equation}
+%
+gdzie funkcje bazowe w poszczególnych obszarach mają postać
+%
+\begin{equation}
+\phi_{lm}(\varepsilon,\kappa,\bm{r})=i^lY_{lm}(\theta,\phi)\begin{cases}
+        u_l(\varepsilon,r) +\kappa cot(\eta_l(\varepsilon))J_l(\kappa,r) & r<r_c \\
+       \kappa N_l(\kappa,r) &  r>r_c.
+             \end{cases}    
+\end{equation}
+%
+Funkcja $J_l(\kappa,r)$ jest tak skonstruowana, aby funkcje bazowe wewnątrz sfery nie zależały od energii, 
+czyli spełnione było równanie
+%
+\begin{equation}
+\frac{d}{d\varepsilon}\phi_{lm}(\varepsilon,\kappa,\bm{r})=i^lY_{lm}[\dot{u}_l(\varepsilon,r) +\kappa\frac{d}{d\varepsilon} cot(\eta_l(\varepsilon))J_l(\kappa,r)]=0
+\end{equation}
+%
+dla energii $\varepsilon=E_{\nu}$ odpowiadającej średniej energii danego orbitalu LMTO.
+Ten warunek prowadzi do wzoru
+%
+\begin{equation}
+J_l(\kappa,r)=-\frac{\dot{u}_l(E_{\nu},r)}{\kappa\frac{d}{d\varepsilon}cot(\eta_l(E_{\nu}))}.
+\end{equation}
+%
+W obszarze międzywęzłowym funkcje bazowe centrowane w położeniu danej sfery $\bm{R}$ otrzymuje się przez rozwinięcie na funkcji $J_l$
+powiązane z sąsiednimi sferami w położeniach $\bm{R}'$
+%
+\begin{equation}
+N_l(\kappa,\bm{r}-\bm{R})=4\pi\sum_{l',l''}C_{l',l'',l'''}n^*_{l''}(\kappa,\bm{R}-\bm{R'})J_{l'}(\kappa,\bm{r}-\bm{R}').
+\end{equation}
+%
+Taka konstrukcja powoduje, że funkcje LMTO są kombinacją liniową funkcji $u_l$ i $\dot{u}_l$ wewnątrz danej sfery
+i są gładko przedłużane do obszaru międzywęzłowego, łącząc się w sposób ciagły z funkcjami $\dot{u}_l$ z każdej
+sąsiedniej sfery.
+
+Metodę LMTO można uprościć poprzez pominięcie obszaru międzywęzłowego i ograniczenie się do optymalizacji funkcji falowych tylko wewnątrz sfer.
+W podejściu nazywanym przybliżeniem atomowych sfer ({\it ang. atomic sphere approximation} - ASA)
+stosuje się sfery, które częściowo na siebie nachodzą i ich sumaryczna objętość równa się całkowitej objetości kryształu. 
+Zastoswanie takiego podejścia ma uzasadnienie jedynie dla kryształów z gęstym upakowaniem atomów.
+
+\chapter{Funkcjonały i potencjały zależne od orbitali}
+
+\section{Problemy lokalnych funkcjonałów}
+\label{sec:mgga}
+
+Teoria funkcjonału gęstości odniosła wiele spektakularnych sukcesów w badaniach własności ciał stałych.
+Posiada jednak braki, które nie pozwalają opisać poprawnie struktury elektronowej wielu materiałów.
+Te problemy wynikają zasadniczo z przybliżeń stosowanych do wyliczenia energii wymienno-korelacyjnej, 
+której dokładna postać funkcjonalna nie jest znana.  
+Główną zaletą przybliżeń stosowanych w DFT, pozwalająca na obliczenia nawet dla dużych układów, jest lokalny lub kwazilokalny 
+charakter potencjału wymienno-korelacyjnego. Dokładny potencjał jest zwykle nielokalny i zależny jest od funkcji falowych (orbitali), a nie tylko gęstości elektronowej. Przykładem jest dokładna energia oddziaływania wymiennego, która wyliczana jest z orbitali jednocząstkowych w przybliżeniu Hartree-Focka. 
+Standardowe funkcjonały wymienno-korelacyjnych stosowane w DFT nie zależą od rodzaju obsadzonych orbitali. 
+Mówiąc ściślej, nie odróżniają stanów o różnej liczbie kwantowej $m$.
+Wszystkie orbitale traktowane są jednakowo i nie ma możliwości wyróżnienia cech, które decydują o nierównomiernym ich obsadzeniu.
+Taka możliwość często decyduje o charakterze stanu podstawowego i jest szczególnie istotna w układach silnie skorelowanych. 
+
+W przybliżeniach LDA i GGA występują dwa główne problemy, które są ze sobą ściśle powiązane.
+Pierwszym z nich jest brak nieciągłości pochodnej funkcjonalnej energii całkowitej po gęstości elektronowej, która jest równa potencjałowi chemicznemu. 
+Ta nieciągłość jest cechą charakterystyczną funkcjonału wymienno-korelacyjnym, ale występuje również w wyrazie opisującym energię kinetyczną elektronów. 
+Ponieważ nieciągłość potencjału chemicznego powiązana jest z przerwą energetyczną, to jej brak skutkuje zaniżonymi wartościami, a nawet
+zerowaniem się tej wielkości w izolatorach i półprzewodnikach.
+Problem przerwy energetycznej będzie dokładniej omawiany w rozdziale~(\ref{sec:insulators}).
+
+Drugim problemem jest występowanie samoodziaływania elektronów, czyli niezerowej energii oddziaływania elektronu 
+z jego własnym polem kulombowskim. Jest to efekt niefizyczny, który wynika z niedokładnego kasowania się występujących w hamiltonianie 
+przyczynków do oddziaływania elektronu z jego własnym polem elektrycznym.   
+W dokładnym opisie ujemny potencjał wymienno-korelacyjny wytwarzany przez każdy elektron powinien dokładnie kasować jego dodatni potencjał Hartree.
+W przypadku przybliżenia Hartree-Focka samoodziaływanie nie występuje, ponieważ oddziaływanie wymienne i Hartree wzajemnie kasują się.
+W przypadku DFT, potencjał Hartree wyliczany jest dokładnie, natomiast potencjał wymienno-korelacyjny jest przybliżony,
+stając się źródłem samooddziaływania. 
+Nawet w przypadku układu jednoelektronowego, energia Hartree nie kasuje się z energią wymienną wyznaczoną w przybliżeniu LDA lub GGA.
+Na przykładzie atomu wodoru, dyskutowanym w rozdziale \ref{sec:GGA}, dobrze widać efekt jaki powodują różnice między wartościami dokładnymi i przybliżonymi.
+Ponieważ wartość bezwględna energii wymiany jest mniejsza niż wartość dokładna, nie kasuje całkowicie dodatniej energii Hartree. 
+Wprawdzie częściowo różnica ta redukowana jest przez niezerową i ujemną energię korelacji, ale pozostaje skończona. 
+Ta dodatkowa energia oddziaływania elektronu z własnym polem elektrycznym może powodować duże błędy w obliczeniach.
+W zależności od rodzaju materiału, samoodziaływanie przyjmuje bardzo różne wartości. Jest ono zaniedbywalnie małe dla rozległych stanów elektronowych,
+zdelokalizowanych w dużym obszarze danego materiału. Dlatego wpływ samoodziaływania na stany elektronowe w prostych metalach, 
+gdzie dominują stany walencyjne typu $s$ i $p$, może być pominięty. 
+W materiałach, w których występują stany elektronowe mające tendencję do lokalizacji (np. stany {\it 3d} i {\it 4f}),
+jest ono szczególnie silne i często prowadzi do wyników jakościowo niezgodnych z eksperymentem.
+W stanach zlokalizowanych, oddziaływanie elektronu z jego własnym polem kulombowskim, które ma charakter odpychający,
+może prowadzić do znacznego wzrostu energii. Zredukowanie tej energie jest możliwe tylko poprzez zwiększenie delokalizacji
+ładunku i poszerzenie pasm elektronowych. Prowadzi to efektywnie do zmniejszenia przerwy energetycznej
+otrzymywanej w obliczeniach dla półprzewodników i izolatorów. W skrajnym przypadku błędy samoodziaływania powodują delokalizację ładunków w stanach $3d$ 
+i zamknięcie przerwy energetycznej, co skutkuje niewłaściwym stanem metalicznym w materiałach, które w rzeczywistości są izolatorami Motta. 
+
+Problemy te mogą być wyeliminowane lub częściowo zredukowane przy zastosowaniu funkcjonałów zależnych od orbitali elektronowych~\cite{kummel}.
+W kolejnych rozdziałach omawiane są metody, które uwzlędniają zależność orbitalną w funkcjonale wymienno-korelacyjnym
+lub wprowadzają poprawki do energii całkowitej zależne od obsadzenia orbitali.  
+
+\section{Funkcjonały OEP}
+\label{sec:mgga}
+
+Podejście nazywane metodą zoptymalizowanego potencjału efektywnego ({\it ang. optimized effective potential} - OEP)  
+zostało zaproponowane prez Sharpa i Hortona w roku 1953, jeszcze zanim powstała teoria funkcjonału gęstości~\cite{Sharp}. 
+Ta metoda zakłada funkcjonalną zależność energii wymienno-korelacyjnej od orbitali $\phi_{j\sigma}$, przy zachowanym lokalnym charakterze potencjału $V^{OEP}_{\sigma}$. Równanie OEP zostało rozwiązane numerycznie po raz pierwszy przez Talmana i Shadwicka~\cite{Talman}. 
+Energię całkowitą Kohna-Shama możemy zapisać jako funkcjonał orbitali
+%
+\begin{equation}
+E[\{\phi_{j\sigma}\}]=T[\{\phi_{j\sigma}\}]+\int d\bm{r} V_{ext}(\bm{r})n(\bm{r})+\frac{1}{2}\int \int d\bm{r} d\bm{r}' \frac{n(\bm{r})n(\bm{r}')}{|\bm{r}-\bm{r}'|}+E_{xc}[\{\phi_{j\sigma}\}].
+\end{equation}
+%
+Metoda OEP ma charakter wariacyjny i opiera się na dwóch równaniach. Pierwsze określa warunek minium (stacjonarności) energii całkowitej
+względem efektywnego potencjału
+%
+\begin{equation}
+\frac{\delta E[\{\phi_{j\sigma}\}]}{\delta V^{OEP}_{\sigma}}=0,
+\end{equation} 
+%
+który jest równoważny zasadzie wariacyjne Kohna-Shama $\delta E_\text{tot}/\delta n=0$~\cite{Sahni}.
+Drugie równanie ma postać równania Kohna-Shama z efektywnym potencjałem zależnym od orbitali
+%
+\begin{equation}
+\Big[-\frac{\hbar^2}{2m}\nabla^2+V^{OEP}_{\sigma}[\{\phi_{j\sigma}\}](\bm{r})\Big]\phi_{j\sigma}(\bm{r})=\varepsilon_{j\sigma}\phi_{j\sigma}(\bm{r}).
+\end{equation}
+%
+Część potencjału opisującą oddziaływanie wymienno-korelacyjne można zapisać w postaci podwójnej całki wykorzystując własności pochodnych funkcjonalnych
+%
+\begin{equation}
+V^{OEP}_{xc\sigma}(\bm{r})=\frac{\delta E_{xc}[\{\phi_{j\sigma}\}]}{\delta n_{\sigma}(\bm{r})}=\sum_{\nu\mu\i}\int\int d\bm{r}' d\bm{r}''\frac{\delta E_{xc}}{\delta \phi_{i\nu}(\bm{r}')}\frac{\delta \phi_{i\nu}(\bm{r}')}{\delta V^{OEP}_{\mu}(\bm{r}'')}\frac{\delta V^{OEP}_{\mu}(\bm{r}'')}{\delta n_{\sigma}(\bm{r})}.
+\label{voep}
+\end{equation}
+%
+Drugą z pochodnych funkcjonalnych pod całką można wyrazić w pierwszym rzędzie rozwinięcia pertubacyjnego 
+%
+\begin{equation}
+\frac{\delta \phi_{i\nu}(\bm{r}')}{\delta V^{OEP}_{\mu}(\bm{r}'')}=\delta_{\nu\mu}\sum_{k\neq i}\frac{\phi^*_{k\mu}(\bm{r}')\phi_{k\mu}(\bm{r}'')}{\varepsilon_{i\mu}-\varepsilon_{k\mu}}\phi_{i\mu}(\bm{r}'')=\delta_{\nu\mu}G^R_{i\mu}(\bm{r}',\bm{r}'')\phi_{i\mu}(\bm{r}''),
+\end{equation}
+%
+gdzie wprowadzono funkcję Greena dla nieodziałujących elektronów
+%
+\begin{equation}
+G^R_{i\mu}(\bm{r}',\bm{r}'')=\sum_{k\neq i}\frac{\phi^*_{k\mu}(\bm{r}')\phi_{k\mu}(\bm{r}'')}{\varepsilon_{i\mu}-\varepsilon_{k\mu}}.
+\label{fgreena}
+\end{equation} 
+%
+Trzecia z pochodnych jest odwrotnością liniowej funkcji odpowiedzi, którą również możemy wyrazić przez funkcję Greena
+%
+\begin{equation}
+\chi_{\sigma}(\bm{r},\bm{r}'')=\delta_{\sigma\mu}\frac{\delta n_{\sigma}(\bm{r})}{\delta V^{OEP}_{\mu}(\bm{r}'')}=\delta_{\sigma\mu}\sum_i G^R_{i\sigma}(\bm{r}'',\bm{r})\phi_{i\sigma}(\bm{r}'')\phi^*_{i\sigma}(\bm{r}).
+\label{vchi}
+\end{equation}
+%
+Mnożąc równanie (\ref{voep}) przez $\chi_{\sigma}(\bm{r}',\bm{r})$, całkując wzlędem $\bm{r}'$ i wykorzystując (\ref{vchi}) dostajemy 
+równanie całkowe metody OEP
+%
+\begin{equation}
+\sum_i\int d\bm{r}' \phi^*_{i\sigma}(\bm{r})[V^{OEP}_{xc\sigma}(\bm{r})-u^{OEP}_{xci\sigma}(\bm{r})]G^R_{i\sigma}(\bm{r}',\bm{r})\phi_{i\sigma}(\bm{r}) + c.c.=0,
+\label{oepint}
+\end{equation}
+%
+gdzie
+%
+\begin{equation}
+u^{OEP}_{xci\sigma}(\bm{r})=\frac{1}{\phi^*_{i\sigma}(\bm{r})}\frac{\delta E_{xc}[\{\phi_{j\sigma}\}]}{\delta \phi_{i\sigma}(\bm{r})}.
+\end{equation}
+%
+Jeżeli funkcjonał wymienno-korelacyjny zależy tylko od gęstości, wtedy $u^{OEP}_{xci\sigma}(\bm{r})=V^{OEP}_{xc\sigma}(\bm{r})$ i równanie (\ref{oepint})
+jest automatycznie spełnione.
+W ramach metody OEP, część wymienną można traktować dokładnie, tak jak w przybliżeniu Hartree-Focka, i wtedy potencjał zależny od orbitali
+ma postać
+%
+\begin{equation}
+u^{OEP}_{xi\sigma}=-\frac{1}{\phi^*_{i\sigma}(\bm{r})}\sum_j \phi^*_{j\sigma}\int d\bm{r}'\frac{\phi^*_{i\sigma}(\bm{r}')\phi_{j\sigma}(\bm{r}')}{|\bm{r}-\bm{r}'|}.
+\end{equation} 
+%
+Ze względu na całkowy charakter równań OEP, wyznaczenie samozgodnych rozwiązań, nawet przy zastosowaniu dodatkowych przybliżeniach, wymaga bardzo czasochłonnych obliczeń~\cite{kummel}. Z tego powodu potencjały OEP rzadko stosowane są w praktyce.
+Znacznie efektywniejsze są potencjały zależne od orbitali omówione w kolejnych rozdziałach. 
+
+\section{Korekcja samoodziaływania (SIC)}
+\label{sec:sic}
+
+W roku 1981, Pardew i Zunger zaproponowali metodę korekcji samoodziaływania ({\it ang. self-interaction correction} - SIC) \cite{PZ}.
+Metoda ta dotyczy stanów zlokalizowanych, więc rozpatrujemy zbiór orbitali $\phi_{\alpha\sigma}(\bm{r})$, gdzie $\alpha$ oznacza numer orbitalu,
+a $\sigma$ kierunek spinu. Oznaczamy gęstość elektronową pojedynczego orbitalu przez 
+%
+\begin{equation}
+n_{\alpha\sigma}(\bm{r})=f_{\alpha\sigma}|\phi_{\alpha\sigma}(\bm{r})|^2,
+\end{equation}
+%
+gdzie liczba $f_{\alpha\sigma}$ określa obsadzenie danego orbitalu. 
+Warunkiem braku samoodziaływania w każdym orbitalu jest kompletne wzajemne kasowanie się energii Hartree danej wzorem (\ref{Hartree}) 
+i energii wymienno-korelacyjnej
+%
+\begin{equation}
+E_H[n_{\alpha\sigma}]+E_{xc}[n_{\alpha\sigma}]=0,
+\label{nosic}
+\end{equation}
+%
+gdzie obydwie energie są funkcjonałami gęstości pojedynczego orbitalu.
+Dla pojedynczego elektronu spełniony jest również warunek kasowania
+się energii Hartree i dokładnej energii wymiany wylicznej w podejściu Hartree-Focka
+%
+\begin{equation}
+E_H[n_{\alpha\sigma}]+E_{x}[n_{\alpha\sigma}]=0.
+\end{equation}
+%
+Ponieważ energia korelacji $E_c=E_{xc}-E_{x}$, z tych dwóch warunków wynika, że energia korelacji dla pojedynczego orbitalu musi się zerować
+%
+\begin{equation}
+E_c[n_{\alpha\sigma}]=0.
+\end{equation}
+%
+Warunki te nie są spełnione dla energii wymienno-korelacyjnej danej wzorami w przybliżeniu LDA (\ref{xclda})
+lub GGA (\ref{xcgga}), co prowadzi do samoodziaływania elektronów. Oznaczmy ogólnie przez $E_{xc}^{DFT}[n_{\uparrow},n_{\downarrow}]$ funkcjonał wymienno-korelacyjny wyznaczonony w jednym z tych przybliżeń. Wtedy wzór na poprawioną energię wymienno-korelacyjną można zapisać w formie
+%
+\begin{equation}
+E_{xc}^{SIC}[n_{\alpha\sigma}]=E_{xc}^{DFT}[n_{\uparrow},n_{\downarrow}]-\sum_{\alpha\sigma}\delta_{\alpha\sigma},
+\label{xcsic}
+\end{equation}
+%
+gdzie
+%
+\begin{equation}
+\delta_{\alpha\sigma}=E_H[n_{\alpha\sigma}]+E_{xc}^{DFT}[n_{\alpha\sigma}],
+\end{equation}
+%
+a sumowanie jest tylko po zajętych orbitalach $\alpha$.
+Dla dokładnej wartości energii wymienno-korelacyjnej mielibyśmy $\delta_{\alpha\sigma}=0$,
+czyli spełniony byłby warunek (\ref{nosic}). 
+Energia wymienno-korelacyjna zdefiniowana wzorem (\ref{xcsic}) nie jest funkcjonałem gęstości elektronowej,
+tylko funkcjonałem orbitali atomowych. Powoduje to, że również funkcjonał energii całkowitej (\ref{EKS}) staje się zależny od orbitali
+% 
+\begin{equation}
+E^{SIC}[n_{\alpha\sigma}]=\sum_{\alpha\sigma}\langle\phi_{\alpha\sigma}|-\frac{\hbar^2}{2m}\nabla^2|\phi_{\alpha\sigma}\rangle+E_{ext}[n]+E_H[n]+E_{xc}^{SIC}[n_{\alpha\sigma}].
+\label{esic}
+\end{equation}
+% 
+Podobnie, jak w przypadku funkcjonału Kohna-Shama, możemy zastosować metodę wariacyjną do zmninimalizowania tej energię względem
+zbioru orbitali $\phi_{\alpha\sigma}$, przy założonym warunku ortonormalności 
+%
+\begin{equation}
+\langle\phi_{\alpha\sigma}|\phi_{\alpha'\sigma'}\rangle=\delta_{\alpha\alpha'}\delta_{\sigma\sigma'}.
+\end{equation}
+%
+Dostajemy wtedy równania jednocząstkowe analogiczne do równań Kohna-Shama
+%
+\begin{equation}
+[-\frac{\hbar^2}{2m}\nabla^2+V_{eff}(n_{\alpha\sigma},\bm{r})]\phi_{\alpha\sigma}(\bm{r})=\varepsilon_{\alpha\sigma}^{SIC}\phi_{\alpha\sigma}(\bm{r}),
+\label{eqsic}
+\end{equation} 
+%
+z efektywnym potencjałem danym wzorem
+%
+\begin{equation}
+V_{eff}(n_{\alpha\sigma},\bm{r})=V^{DFT}(\bm{r})-V_H(n_{\alpha\sigma},\bm{r})-V_{xc}(n_{\alpha\sigma},\bm{r}),
+\end{equation}
+%
+gdzie ostatnie dwa wyrazy to korekcja samoodziaływania do efektywnego potencjału elektronowego.
+Podobnie jak w przypadku stanów Kohna-Shama spełniona jest zależność
+%
+\begin{equation}
+\varepsilon_{\alpha\sigma}^{SIC}=\frac{\partial E^{SIC}}{\partial f_{\alpha\sigma}}.
+\end{equation}
+
+Równanie (\ref{eqsic}) pozwala wyznaczyć orbitale i energie własne zlokalizowanych stanów elektronowych. 
+Jednak w rzeczywistych materiałach, mamy często doczynienia z współistnieniem stanów zlokalizowanych i zdelokalizowanych, co powoduje, że rozwiązania tego równania nie muszą odpowiadać globalnemu minimum. Pełna optymalizacja funkcji falowych powinna 
+uwzlędniać zarówno stany zlokalizowane (typu Wanniera) i zdelokalizowane (typu Blocha).
+W praktyce polega to na wyznaczeniu różnych możliwych konfiguracji stanów zlokalizowanych i zdelokalizowanych,
+a następnie wybraniu tej o najniższej energii.    
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=1.2]{rare-earths.pdf}
+\caption{Różnice energii całkowitej dla walencyjności II i III wyliczone dla ziemiach rzadkich (o), porównane z eksperymentem (linia przerywana)
+i wartościami dla związków ziem rzadkich z siarką (+). Rysunek z pracy \cite{RE}.}
+\label{fig:rare}
+\end{figure}
+
+
+Dobrym przykładem są metale ziem rzadkich, w których występują zarówno stany zlokalizowane ($4f$),
+jak i zdelokalizowane ($5d$ i $6s$). Dodatkowo występują dwa rodzaje elektronów $4f$. Cześć z nich jest zlokalizowana,
+tak jak elektrony rdzenia, mając decydujący wpływ na moment magnetyczny i walencyjność danego atomu. 
+Pozostałe, w wyniku hybrydyzacji ze stanami $5d$ i $6s$, tworzą pasmo elektronowe i biorą udział w wiązaniu metalicznym.
+W ramach tradycyjnych przybliżeń energii wymienno-korelacyjnej nie można opisać prawidłowo tak złożonej
+struktury elektronowej ziem rzadkich. W pracy \cite{RE} zastosowano metodę SIC do wyliczenia obsadzeń
+stanów $4f$ i walencyjności we wszystkich atomach ziem rzadkich (rysunek \ref{fig:rare}).
+Walencyjność zdefiniowana jest jako ilość stanów zdelokalizowanych przypadająca na jeden atom i wyliczana jest ze wzoru
+%
+\begin{equation}
+N_{val}=Z-N_{core}-N_{SIC},
+\end{equation}
+%
+gdzie $Z$ jest liczbą atomową, $N_{core}$ ilością elektronów w rdzeniu i $N_{SIC}$ liczbą stanów zlokalizowanych $4f$,
+dla których odjęto energię samoodziaływania. Dla ziem rzadkich walencyjność przyjmuje dwie możliwe wartości:
+$N_{val}=2$ (atomy dwuwartościowe, II) lub $N_{val}=3$ (atomy trójwartościowe, III).
+Dla tych dwóch przypadków wyliczone różnice ich energii całkowitych $E_{II}-E_{III}$ pokazane są na rysunku \ref{fig:rare}.
+Otrzymane wartości bardzo dobrze zgadzają się z danymi eksperymentalnymi. 
+Jak widać, oprócz Eu i Yb, które są dwuwartościowe, wszystkie pozostałe ziemie rzadkie są trójwartościowe. 
+Podobnie dobrą zgodność z eksperymentem otrzymano dla promieni Wignera-Seitza i stałych sieci kryształów ziem rzadkich \cite{RE}.
+
+Ze względu na jawną zależność od orbitali, funkcjonały SIC wykazują nieciągłość pochodnej po gęstości elektronowej.
+Dzięki temu metoda ta znacznie poprawia wartości przerwy energetycznej i momentów magnetycznych
+w układach silnie skorelowanych~\cite{svane1990,temmerman2001}. 
+On the other hand, the SIC does not provide the systamic improvement in description of molecular
+systems~\cite{kummel}.
+
+
+\section{Funkcjonały hybrydowe}
+\label{sec:hybrid}
+
+Problem samoodziaływania nie występuje w przybliżeniu Hartree-Focka, gdzie dla każdego elektronu
+jego własne oddziaływanie kulombowskie kasowane jest przez oddziaływanie wymienne.
+Zatem częściowe uwzlędnienie dokładnej wartości energii wymiany w funkcjonale wymienno-korelacyjnym powinno
+zmniejszyć efekt samoodziaływania i w ten sposób poprawić całkowitą energię układu. 
+Takie funkcjonały, w których energia wymiany otrzymana w przybliżeniu LDA lub GGA jest częściowo zastąpiona
+dokładną wartością otrzymaną z metody Hartree-Focka, nazywane są funkcjonałami hybrydowymi.  
+Pierwszy funkcjonał hybrydowych zaproponował Becke w roku 1993 \cite{becke93}, uzasadniając jego postać
+na podstawie tzw. formuły adiabatycznego połączenia ({\it ang. adiabatic connection formula}).
+Rozważmy ogólny hamiltoniana w postaci
+%
+\begin{equation}
+H_{\lambda}=T+\lambda V,
+\end{equation}
+%
+gdzie $T$ jest operatorem energii kinetycznej, $V$ jest potencjałem oddziaływania między elektronami, a $\lambda$ jest parametrem, który
+określa siłę tego ddziaływania ($0\leq\lambda\leq 1$). 
+Funkcje falowe $\psi_{\lambda}$ i energie własne $E_{\lambda}$ tego hamiltonianu są zależne od parametru $\lambda$.
+Dla nich spełnione jest równanie
+%
+\begin{equation}
+\frac{dE_{\lambda}}{d\lambda}=\langle\psi_{\lambda}|V|\psi_{\lambda}\rangle,
+\end{equation} 
+%
+z którego dostajemy
+%
+\begin{equation}
+E_{\lambda=1}=E_{\lambda=0}+\int_0^1 d\lambda \langle\psi_{\lambda}|V|\psi_{\lambda}\rangle.
+\end{equation}
+%
+Energia dla $\lambda=0$ odpowiada energii kinetycznej, natomiast $\lambda=1$ daje energie z pełnym oddziaływaniem elektronowym $V$.
+Ta formuła określa właśnie adiabatyczne połączenie między układem nieoddziałujących i oddziałujących elektronów.
+Po zastosowaniu tej formuły do funkcjonału energii (z pominięciem potencjału zewnętrznego $V_{ext}$) dostajemy 
+%
+\begin{equation}
+E[n]=T[n]+\int_0^1 d\lambda \langle\psi_{\lambda}|V|\psi_{\lambda}\rangle.
+\end{equation}
+%
+Biorąc pod uwagę funkcjonał Kohna-Shama (\ref{EKS}), możemy napisać wzór na energię wymienno-korelacyjną w postaci
+%
+\begin{equation}
+E_{xc}[n]=\int_0^1 d\lambda \langle\psi_{\lambda}|V|\psi_{\lambda}\rangle-E_H[n]=\int_0^1d\lambda E_{xc,\lambda}[n].
+\label{xcn}
+\end{equation}
+%
+Powyższą całkę można przybliżyć, rozpatrując jedynie wartości końcowe. Dla $\lambda=0$, dostajemy energie wymiany
+w przybliżeniu Hartree-Focka, $E_{xc,0}=E_x^{HF}$. Becke zaproponował, aby dla $\lambda=1$ zastosować jedno z przybliżeń stosowanych w DFT (LDA lub GGA),
+$E_{xc,1}=E_{xc}^{DFT}$, a dla pośrednich wartości $\lambda$ zastosować interpolację liniową
+%  
+\begin{equation}
+E_{xc,\lambda}[n]=(1-\lambda)E_x^{HF}[n]+\lambda E_{xc}^{DFT}[n].
+\end{equation}
+%
+Po wstawieniu tej zależności do (\ref{xcn}) i wyliczeniu całki dostajemy funkcjonał hybrydowy 
+%
+\begin{equation}
+E_{xc}^{hyb}[n]=\frac{1}{2}E_x^{HF}+\frac{1}{2}E_{xc}^{DFT}[n].
+\label{becke}
+\end{equation} 
+%
+Funkcjonał ten można ulepszyć poprawiając przybliżone całkowanie $E_{xc,\lambda}[n]$.
+Dla dowolnego układu, można znaleźć parametr $b$, który najlepiej opisuje
+energię wymienną
+%
+\begin{equation}
+E_{xc}^{hyb}[n]=bE_x^{HF}+(1-b)E_{xc}^{DFT}[n].
+\end{equation}
+W praktyce stosuje się funkcjonały, w których liniowo miksowane są tylko dokładna i przybliżona część
+energii wymiennej, a część korelacyjna jest przybliżona w ramach LDA lub GGA.
+Parametr $b$ można wybrać empirycznie przez dofitowanie różnych wielkości, na przykład energii
+atomizacji, do eksperymentalnych wartości dla dużej grupy materiałów.
+Optymalne zgodności otrzymuje się zwykle dla wartości z przedziału $b=[0.15,30]$.
+
+Wybór parametru $b$ można dodatkowo uzasadnić stosująć podejście, które zaproponowali Pardew, Ernzerhof i Burke w roku 1996~\cite{PBE0}.  
+Zapisując zależność energii wymienno-korelacyjnej od $\lambda$ w formie rozwinięcia perturbacyjnego Mollera-Plesseta, 
+dostaje się zależność
+%
+\begin{equation}
+E_{xc,\lambda}[n]=E_{xc}^{DFT}[n]+(E_x^{HF}[n]-E_x^{DFT}[n])(1-\lambda)^{m-1},
+\end{equation}
+%
+która prowadzi do funkcjonału hybrydowego w postaci
+%
+\begin{equation}
+E_{xc}^{hyb}[n]=E_{xc}^{DFT}[n]+\frac{1}{m}(E_x^{HF}[n]-E_x^{DFT}[n]).
+\end{equation}
+%
+Dla $m=1$, energia wymiany jest równa dokładnej wartości $E_x^{HF}$. W tym przypadku dodanie przybliżonej wartość energii korelacji $E_c^{DFT}$ 
+poprawia wyniki w porównaniu do przybliżenia Hartree-Focka. Wyniki są jednak gorsze niż w przybliżeniach LDA lub GGA, w których błędy energii wymiany i korelacji mają przeciwne znaki, co powoduję ich częściowe kasowanie się. Przypadek $m=2$ odpowiada funkcjonałowi, który dany jest wzorem~(\ref{becke}). 
+Za optymalny uznano wybór $m=4$~\cite{PBE0}, wskazując na bardzo dobrą zgodnością wyników otrzymanych w ramach teorii perturbacyjnej Mollera-Plesseta czwartego rzędu z danymi eksperymentalnymi \cite{pople89}. Co więcej, w tym przypadku zarówno wartości $E_{xc,\lambda}$ i $E_{xc}^{DFT}$, jak również ich pierwsze oraz drugie pochodne wzajemnie pokrywają się dla $\lambda=1$. Parametr $b$ związany jest z wykładnikiem $m$ prostą zależnością $b=\frac{1}{m}$, co daje $b=0.25$, bliskie wartościom empirycznym.
+
+Funkcjonał hybrydowy jest specjalnym przypadkiem funkcjonału zależnego od orbitali, w którym 
+potencjał Kohna-Shama składa się z części lokalnej (kwazilokalnej) i części
+potencjału Focka opisującego dokładną energię wymiany. 
+Oddziaływanie wymienne ma charakter krótkozasięgowy i można uwzględnić wkład od dokładnej wartości jedynie
+w pewnym zakresie odległości. Uwzględniając ten efekt Heyd, Scuseria i Enzerhof (HSE) \cite{HSE} zaproponowali nowy funkcjonał
+w postaci
+%
+\begin{equation}
+E_{xc}^{hyb}(\mu) = (1-b)E_{x}^{GGA,SR}(\mu)+bE_x^{HF,SR}(\mu) + E_x^{GGA,LR} + E_c^{GGA},
+\end{equation}
+%
+gdzie $E_x^{GGA,SR}$ i $E_x^{GGA,LR}$ odpowiadają krótkozasięgowemu i długozasięgowemu oddziaływaniu wymiennemu w przybliżeniu GGA,
+$E_x^{HF,SR}$ jest częścią krótkozasięgowego dokladnego oddziaływania wymiennego, a parametr $\mu$
+określa zasięg oddziaływania krótkozasięgowego. .
+
+
+\section{Funkcjonały i potencjały meta-GGA}
+\label{mgga}
+
+
+Dwa podstawowe przybliżenia stosowane do opisu funkcjonału wymienno-korelacyjnego uwzlędniają tylko gęstość ładunku w danym punkcie
+przestrzeni (LDA) lub dodatkowo jego gradient (GGA). 
+Kolejnym krokiem pozwalającym poprawić ten funkcjonał jest uwzlędnienie jego zależności od gradientu orbitali, czyli 
+części kinetycznej całkowitej energii. 
+
+%
+\begin{figure}[h]
+\centering
+\includegraphics[scale=1]{Jacob.jpg}
+\caption{Drabina Jakuba opisująca kolejne poziomy dokładności opisu funkcjonału wymienno-korelacyjnego w DFT.}
+\label{fig:jacob}
+\end{figure}
+%
+
+Takie funkcjonały noszą nazwę meta-GGA i zapisywane są w ogólnej postaci
+%
+\begin{equation}
+E_{xc}[n_{\uparrow},n_{\downarrow}]=\int d\bm{r} n(\bm{r}) \varepsilon_{xc}(n_{\uparrow},n_{\downarrow},\nabla n_{\uparrow},\nabla n_{\downarrow},\tau_{\uparrow},\tau_{\downarrow}),
+\label{metaGGA}
+\end{equation}
+%
+gdzie $\tau_{\sigma}$ jest gęstością energii kinetycznej wyznaczoną dla obsadzonych stanów Kohna-Shama 
+%
+\begin{equation}
+\tau_{\sigma}(\bm{r})= \frac{1}{2}\sum_i |\nabla \psi_{i\sigma}(\bm{r})|^2,
+\end{equation}
+%
+w jednostkach $\hbar=m_\text{e}=1$. Uzasadnieniem włączenia $\tau_{\sigma}$ do funkcjonału jest występowanie tej wielkości w rozwinięciu Taylora
+uśrednionego sferycznie ładunku wymiennego wokół elektronu~\cite{Becke1998}. Drugą korzystną cecha jest możliwość zredukowania samoodziaływania
+w części korelacyjnej funkcjonału.
+
+
+\subsection{Potencjał Becke-Johnsona}
+
+Becke and Johnson (BJ) zastosowali rozszerzenie potencjału oddziaływania wymiennego Slatera w postaci~\cite{Becke2006}
+%
+\begin{equation}
+V^\text{BJ}_{\text{x}\sigma}(\bm{r})=V_{\text{x}\sigma}^\text{Slater}(\bm{r})+\frac{1}{\pi}\sqrt{\frac{5}{12}}\sqrt{\frac{2\tau_{\sigma}(\bm{r})}{n_{\sigma},(\bm{r})}}.
+\label{BJ}
+\end{equation} 
+%
+gdzie omawiany w rozdziale (\ref{sec:HF}) potencjał Slatera dany jest wzorem
+%
+\begin{equation}
+V_{\text{x}\sigma}^\text{Slater}(\bm{r})=-\frac{e}{2}\int d\mb{r}' \frac{n(\mb{r},\mb{r}')}{|\mb{r}-\mb{r}'|}, 
+\label{SlatPot}
+\end{equation}
+% 
+w którym występuje ładunek wymienny $n(\mb{r},\mb{r}')$ opisany wzorem (\ref{EC}). Potencjał BJ znacznie poprawia energie odziaływania wymiennego
+w porównaniu do przybliżenia LDA i daje wyniki porównywalne z potencjałem OEP~\cite{Becke2006}.  
+Tran i Blaha zaproponowali modyfikację potencjału wymiennego BJ w postaci~\cite{Tran2009}
+%
+\begin{equation}
+V^\text{mBJ}_{\text{x}\sigma}(\bm{r})=cV^\text{BR}_{\text{x}\sigma}(\bm{r})+(3c-2)\frac{1}{\pi}\sqrt{\frac{5}{12}}\sqrt{\frac{2\tau_{\sigma}(\bm{r})}{n_{\sigma}(\bm{r})}},
+\label{mBJ}
+\end{equation}
+%
+gdzie zamiast potencjału Slatera występuje potencjał Becke-Roussel'a
+%
+\begin{equation}
+V^\text{BR}_{\text{x}\sigma}(\bm{r})=-\frac{1}{b_{\sigma}(\bm{r})}(1-e^{-x_{\sigma}(\bm{r})}-\frac{1}{2}x_{\sigma}(\bm{r})e^{-x_{\sigma}(\bm{r})}).
+\end{equation}
+%  
+Wielkość $x_{\sigma}(\bm{r})$ zależy od gęstości elektronowej i jej gradientu oraz od $\tau_{\sigma}(\bm{r})$, natomiast
+%
+\begin{equation}
+b_{\sigma}(\bm{r})=[\frac{x^3_{\sigma}(\bm{r})e^{-x_{\sigma}(\bm{r})}}{8\pi n_{\sigma}(\bm{r})}]^{\frac{1}{3}}.       
+\end{equation}
+%
+Parametr $c$ wyraża się następującym wzorem
+%
+\begin{equation}
+c=\alpha+\beta[\frac{1}{V_\text{cell}}\int d\bm{r}'\frac{|\nabla n(\bm{r}')|}{n(\bm{r}')}]^{\frac{1}{2}},
+\end{equation}
+%
+gdzie $\alpha$ i $\beta$ wyznaczone są tak, aby zminimalizować odstępstwa wyliczanych przerw energetycznych
+od wartości eksperymentalnych dla dużej grupy materiałów.
+Potencjał mBJ (\ref{mBJ}) przechodzi w potencjał BJ (\ref{BJ}) dla $c=1$ oraz dobrze
+przybliża potencjał wymienny LDA dla stałej gęstości elektronowej dany wzorem (\ref{Vex}),
+niezależnie od wartości parametru $c$.
+W porównaniu do przybliżeń LDA lub GGA, obliczenia z potencjałem mBJ znacznie poprawiają wartości 
+przerwy energetycznej w gazach szlachetnych, półprzewodnikach i izolatorach~\cite{Tran2009}.
+Wartości te są porównywalne z wynikami otrzymanymi przy zastosowaniu funkcjonałów hybrydowych lub metody GW,
+które wymagają bardzo czasochłonnych obliczeń.
+Ponieważ w tym podejściu poprawiany jest potencjał wymienny, a nie funkcjonał wymienny, nie można go zastosować
+do optymalizacji położeń atomowych i stałych sieci krystalicznej.  
+  
+
+\subsection{Funkcja lokalizacji elektronów (ELF)}
+
+Funkcjonały meta-GGA używają bezwymiarowego parametru 
+%
+\begin{equation}
+\alpha=\frac{\tau-\tau^w}{\tau^{u}},
+\end{equation}
+%
+gdzie $\tau^w$ jest gęstością energii kinetycznej Weizsäckera
+%
+\begin{equation}
+\tau^w=\frac{|\nabla n|^2}{8n},
+\end{equation}
+%
+która określa dokładną wartość $\tau$ dla pojedynczego orbitalu, a $\tau^u$ jest jego wartością dla jednorodnego gazu
+%
+\begin{equation}
+\tau^u=\frac{3}{10}\Big{(}\frac{3}{\pi^2}\Big{)}^{\frac{2}{3}}n^{\frac{5}{3}}.
+\end{equation}
+%
+Parametr $\alpha$ wykorzystywany jest do charakteryzacji rozkładu gęstości elektronowej i pozwala odróżnić
+układy z wolnozmienną gestością elektronową, typową dla metali ($\alpha\approx 1$), 
+materiały z wiązaniami kowalencyjnymi między pojedynczymi orbitalami ($\alpha=0$) oraz słabymi wiązaniami niekowalencyjnymi między
+zamkniętymi powłokami atomowymi ($\alpha\rightarrow\infty$).
+Jest on bezpośrednio powiązany z funkcją lokalizacji elektronów ({\it ang. electron localization function} - ELF)~\cite{ELF1}
+%
+\begin{equation}
+\textrm{ELF} = \frac{1}{1+\alpha^2},
+\end{equation}     
+%
+która przyjmuje wartości z przedziału $[0,1]$ i jest używana do opisu wiązań chemicznych~\cite{ELF2,ELF3}.
+$\text{ELF}=1$ odpowiada idealnej lokalizacji elektronu, a $\text{ELF}=\frac{1}{2}$ jest wartością dla gazu elektronowego.  
+
+
+\subsection{Funkcjonał SCAN}
+
+W omówionych potencjałach BJ i mBJ zmiania się tylko część opisująca oddziaływanie wymienne, natomiast
+część korelacyjna pozostaje bez zmiany i może być wyznaczona w ramach przybliżeń LDA lub GGA.
+Funkcjonały meta-GGA, których ogólna postać wyraża się wzorem (\ref{metaGGA}), uwzględniają poprawki w części
+wymiennej i korelacyjnej. Zaproponowano kilka wersji funkcjonałów meta-GGA, których nazwy określane są skrótami
+PKZB~\cite{metaGGA}, TPSS~\cite{TPSS}, revTPSS~\cite{revTPSS} lub SCAN~\cite{SCAN}.
+Tutaj omówimy krótko funkcjonał SCAN ({\it ang. strongly contrained and appropriately normed}), który 
+spełnia 17 znanych warunków i norm dla funkcjonału wymienno-korelacyjnego.
+Jednym z nich jest warunek dla współczynnika określającego stosunek energii wymiany do odpowiedniej wartości w LDA, $F_x=E_x/E^{LDA}_x$, który
+nie może przekraczać wartości 1.174. Poprawnie opisuje również układy, dla których dokładne wartości są znane, np. jednorodny gaz elektronowy.  
+
+Część wymienną funkcjonału można zapisać w postaci
+%
+\begin{equation}
+E_x[n]=\int d\bm{r} n(\bm{r}) \varepsilon_x(n)F_x(s,\alpha),
+\end{equation}
+%
+gdzie $\varepsilon_x(n)$ jest energią wymiany dla jednorodnego gazu przypadającą na pojedynczy elektron,
+$\alpha$ jest parametrem zdefiniowanym w poprzednim rozdziale, a $s$ jest bezwymiarowym gradientem gęstości
+%
+\begin{equation}
+s=\frac{|\nabla n|}{2(3\pi^2)^{1/3}n^{4/3}}.
+\end{equation}
+%
+Rysunek~\ref{fig:Fxs} pokazuje zależność współczynnika $F_x$ od $s$ dla trzech charakterystycznych wartości $\alpha$.
+$F_x$ spełnia odpowiednie warunki zarówno dla małych $s\rightarrow0$, jak i dużych wartości $s\rightarrow\infty$.  
+Dla $\alpha=1$ i małych $s$, $F_x$ pokrywa się z wartościami dla funkcjonału PBE.
+Wartości $F_x$ dla innych wartości $\alpha$ uzyskuje się przez odpowiednią interpolację między $\alpha=0$ i $\alpha=1$ oraz ekstrapolację
+do $\alpha\rightarrow\infty$.
+%
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.5]{Fxs.png}
+\caption{Współczynnik $F_x$ w funkcji $s$ dla różnych wartości $\alpha$. Rysunek z pracy \cite{SCAN}.}
+\label{fig:Fxs}
+\end{figure}
+%
+
+Podobnie skonstruowany jest całkowity fukcjonał wymienno-korelacyjny $F_{xc}=F_{x}+F_{c}$, który w granicy
+dużych gęstości niespolaryzowanego gazu przyjmuje wartość energii wymiennej $F_x$.
+Dla małych gęstości funkcjonał spełnia warunek ograniczenia Lieba-Oxforda, $F_{xc}\leq 2.215$.
+ 
+
+\section{Metoda LDA+U}
+\label{sec:ldau}
+
+Metoda LDA+U zaproponowana została przez Anisimova, Zaanena i Andersena w roku 1991~\cite{anisimov}.
+Jednym z pierwszych jej zastosowań było wyliczenie struktury elektronowej dla nadprzewodnika wysokotemperaturowego La$_2$CuO$_4$~\cite{czyzyk}.
+Metoda ta polega na połączeniu podejścia DFT z modelem Hubbarda, który stosowany jest od lat sześćdziesiątych ubiegłego wieku do opisu układów silnie skorelowanych elektronów~\cite{hubbard}. Zacznę od omówienia tego modelu. 
+
+W najbardziej tradycyjnym podejściu, stany elektronowe w krysztale dzielimy na dwie grupy: stany zlokalizowane, które znajdują się w obrębie rdzenia atomowego i posiadają cechy orbitali atomowych oraz stany zdelokalizowane, które rozciągają się w całej przestrzeni kryształu.
+O własnościach transportowych danego materiału decydują głównie elektrony należące do drugiej grupy.
+Natomiast jeżeli chodzi o własności magnetyczne, to zarówno elektrony zlokalizowane, jak i wędrowne mogą oddziaływać wymiennie
+i decydować o rodzaju uporządkowania magnetycznego. W pierwszym przypadku mówimy o zlokalizowanych momentach magnetycznych, 
+których oddziaływanie można opisać w ramach modelu Heisenberga. W drugim mamy doczynienia z magnetyzmem pasmowym, który wynika
+z oddziaływania między mobilnymi elektronami walencyjnymi, które można opisać przy pomocy funkcji Blocha.
+
+Występują również stany elektronowe, które posiadają cechy obydwu tych grup. Stany te tworzą pasma, zachowując jednocześnie
+cechy orbitali atomowych. Ruch elektronów w takiej sytuacji polega na przeskokach między lokalnymi orbitali i odpowiadajace im pasma są znacznie węższe od
+typowych pasm metalicznych. Żródłem takiego zachowania są efekty korelacyjne, które uniemożliwiają swobodny przepływ elektronów w krysztale.
+Najlepszym przykładem są stany należące do niezapełnionej powłoki $3d$ w metalach przejściowych.
+Zgodnie z zakazem Pauliego, w każdym orbitalu mogą znajdować się tylko dwa elektrony z przeciwnymi kierunkami spinu. 
+Jeżeli występuje silne, lokalne oddziaływanie kulombowskie między ładunkami, to stany kwantowe, w których dwa elektrony   
+znajdują się w tym samym orbitalu są niekorzystne energetycznie. Zatem ruch elektronu jest bardzo utrudniony i dla odpowiednio dużego oddziaływania kulombowskiego
+elektrony zostają zlokalizowane. Materiały, w których występuje taki mechanizm lokalizacji to izolatory Motta.
+
+Hubbard zaproponował model do opisu stanów elektronowych, które posiadają cechy zlokalizowanych orbitali atomowych,
+wynikające z korelacji elektronowych~\cite{hubbard}.
+Jest to przykład  modelu ciasnego wiązania z orbitalami Wanniera, który opisany jest w rozdziale~\ref{sec:wannier}. 
+W najprostszym przypadku rozważamy pojedyncze orbitale typu $s$, zlokalizowane w węzłach atomowych. Silne korelacje elektronów znajdujących
+się w tym samym orbitalu można uwzględnić dodają do hamiltonianu lokalne oddziaływanie kulombowskie o ustalonej energii $U$. 
+Hamiltonian modelu Hubbarda zapisujemy stosując formalizm drugiego kwantowania
+%
+\begin{equation}
+H=\sum_{i,j,\sigma}t_{ij}c^{\dagger}_{i\sigma}c_{j\sigma}+U\sum_i n_{i\uparrow}n_{i\downarrow},
+\label{hubbard}
+\end{equation}
+%
+gdzie $c^{\dagger}_{i\sigma}$ i $c_{i\sigma}$ to operatory kreacji i anihilacji elektronu ze spinem $\sigma$ w zlokalizowanym orbitalu Wanniera $w(\bm{r}-\bm{R}_i)$,
+a całki przeskoku dane są wzorem
+%
+\begin{equation}
+t_{ij}=\int d\bm{r} w^*(\bm{r}-\bm{R}_i)H_0w(\bm{r}-\bm{R}_j).
+\end{equation}
+%
+$H_0$ jest jednocząstkowym hamiltonianem złożonym z energii kinetycznej i efektywnego potencjału atomowego. 
+Najczęście uwzlędnia się tylko całki przeskoku między najbliższymi sąsiadami. 
+Operator liczby cząstek ze spinem $\sigma$ na węźle $i$ dany jest wyrażeniem $n_{i\sigma}=c^{\dagger}_{i\sigma}c_{i\sigma}$.
+Energię oddziaływania kulombowskiego dwóch elektronów w tym samym orbitalu można wyliczyć ze wzoru
+%
+\begin{equation}
+U=\int d\bm{r}d\bm{r}' |w(\bm{r}-\bm{R}_i)|^2\frac{e^2}{|\bm{r}-\bm{r}'|}|w(\bm{r}-\bm{R}_i)|^2.
+\end{equation}
+%
+Jeżeli wykonamy transformatę Fouriera operatorów kreacji i anihilacji,
+pierwszy wyraz hamiltonianu (\ref{hubbard}) przyjmie postać $\sum_{\bm{k}\sigma}\varepsilon_{\bm{k}}n_{\bm{k}\sigma}$,
+gdzie $\varepsilon_{\bm{k}}$ opisuje strukturę pasmową w modelu ciasnego wiązania,
+a $n_{\bm{k}\sigma}=c^{\dagger}_{\bm{k}\sigma}c_{\bm{k}\sigma}$ jest operatorem liczby obsadzeń w stanie $\bm{k}$ i spinie $\sigma$.
+
+Zaproponowano kilka sformułowań metody LDA+U \cite{anisimov,orbital1,dudarev}, które różnią się opisem lokalnych 
+oddziaływań eletronowych. Ogólny schemat używany do wyliczenia całkowitej energii ma postać
+%
+\begin{equation}
+E_\text{tot}=E_\text{DFT}+E_U-E_\text{dc},
+\label{eldau}
+\end{equation}
+%
+gdzie $E_\text{DFT}$ jest energią układu otrzymaną w ramach metody DFT (w przybliżeniu LDA lub GGA), $E_U$ jest energią lokalnych oddziaływań elektronowych,
+a $E_\text{dc}$ odpowiada energii oddziaływań kulombowskich w przybliżeniu średniego pola.
+Ten ostatni wyraz jest konieczny, aby odjąć przybliżoną energię oddziaływań elektronowych, która uwzlędniona jest w $E_\text{DFT}$.
+W pracy \cite{orbital1} zaproponowano uogólnioną postać energii $E_U$, która uwzglądnia orbitalną zależność oddziaływań elektronowych
+%
+\begin{multline}
+E_U=\frac{1}{2}\sum_{i,\{m\},\sigma}[\langle m,m''|V_{ee}|m',m'''\rangle n_{i\sigma}^{mm'}n_{i-\sigma}^{m''m'''}
+\\+(\langle m,m''|V_{ee}|m',m'''\rangle - \langle m,m''|V_{ee}|m''',m'\rangle  )n_{i\sigma}^{mm'}n_{i\sigma}^{m''m'''}],
+\end{multline}  
+%
+gdzie $i$ numeruje węzły sieci,  $\{m\}=(m, m',m'', m''')$ są magnetycznymi liczbami kwantowymi,
+a elementy macierzowe oddziaływania kulombowskiego dane są wzorem
+%
+\begin{equation}
+\langle m,m''|V_{ee}|m',m'''\rangle=\int d\bm{r} \int \bm{r}' \phi_{lm}^*(\bm{r})\phi_{lm'}(\bm{r})\frac{e^2}{|\bm{r}-\bm{r}'|}\phi_{lm''}^*(\bm{r}')\phi_{lm'''}(\bm{r}').
+\label{integral}
+\end{equation}
+%
+Występujące pod całką atomowe funkcje falowe określone są dla orbitalnej liczby kwantowej $l$, która definiują zakres magnetycznych liczb kwantowych $-l\le \{m\} \le l$. Liczby obsadzeń orbitali atomowych tworzą tensor drugiego rzędu, którego elementy wyznaczane są przez rzutowanie funkcji falowych Kohna-Shama $\psi_{\bm{k}j}^{\sigma}$ na orbitale atomowe
+%
+\begin{equation}
+n_{i\sigma}^{mm'}=\sum_{\bm{k},j} f_{\bm{k}j}^{\sigma}\langle\psi_{\bm{k}j}^{\sigma}|\phi_{lm'}\rangle\langle\phi_{lm}|\psi_{\bm{k}j}\rangle, 
+\end{equation}
+%
+gdzie współczynniki $f_{\bm{k}j}^{\sigma}$ okreśkaja obsadzenia stanów $j$ z wektorem falowym $\bm{k}$ i spinie $\sigma$.
+Całkę (\ref{integral}) można przedstawić w postaci sumy iloczynów części kątowej i radialnej 
+%
+\begin{equation}
+\langle m,m''|V_{ee}|m',m'''\rangle=\sum_p a_p(m,m',m'',m''')F^p,
+\label{vee}
+\end{equation}
+%
+gdzie $p$ jest liczbą parzystą z przedziału $0\le p \le 2l$. Część kątowa $a_p$ wyliczana jest przy pomocy iloczynów współczynników Clebsha-Gordana 
+%
+\begin{equation}
+a_p(m,m',m'',m''')=\frac{4\pi}{2p+1}\sum_{q=-p}^p \langle lm|Y_{pq}|lm'\rangle\langle lm''|Y_{pq}^*|lm'''\rangle.
+\end{equation}
+%
+$F^p$ nazywane są całkami Slatera i wyznaczane są ze wzoru
+%
+\begin{equation}
+F^p=e^2\int d\bm{r}\int d\bm{r}' r^2 r'^2 R_{nl}^2(\bm{r})\frac{r_1^p}{r_2^{p+1}} R_{nl}^2(\bm{r}'),
+\end{equation}
+%
+gdzie $R_{nl}(\bm{r})$ są częścią radialną atomu funkcji falowej. 
+Dla stanów $d$ potrzebne są trzy całki Slatera $F^0$, $F^2$ i $F^4$, a dla stanów $f$ dodatkowo $F^6$, aby wyznaczyć elementy macierzowe 
+potencjału kulombowskiego (\ref{vee}).
+Efektywne parametry, które określają lokalne oddziaływanie kulombowskie ($U$) i wymienne ($J$), można wyrazić przy pomocy całek Slatera
+%
+\begin{eqnarray}
+U&=&\frac{1}{(2l+1)^2}\sum_{m.m'} \langle m,m'|V_{ee}|m,m'\rangle=F^0,\\
+J&=&\frac{1}{2l(2l+1)}\sum_{m\ne m'} \langle m,m'|V_{ee}|m',m\rangle=\frac{F^2+F^4}{14}.
+\label{HundJ}
+\end{eqnarray} 
+%
+Wykorzystując te parametry, możemy zapisać ostatni wyraz we wzorze (\ref{eldau}) w następującej formie
+%
+\begin{equation}
+E_{dc}=\frac{1}{2}\Big[\sum_i Un_i(n_i-1)-J[n_{i\uparrow}(n_{i\uparrow}-1)+n_{i\downarrow}(n_{i\downarrow}-1)]\Big],
+\end{equation}
+%
+gdzie $n_{i\sigma}=\Tr(n_{i\sigma}^{mm'})$ i $n_i=n_{i\uparrow}+n_{i\downarrow}$ jest całkowitym obsadzeniem orbitali w węźle $i$.
+Tak wyznaczone parametry $U$ i $J$ odnoszą sie do izolowanych atomów. Efektywne wartości parametru kulombowskiego $U$,
+stosowane dla danego rodzaju atomu, uwzględniaja efekty ekranowania elektronowego i zależą od rodzaju materiału.
+Można go wyznaczyć w ramach metody odpowiedzi liniowej poprzez wyliczenie zmiany obsadzenia stanów elektronowych na wybranym atomie
+pod wpływem przyłożonego lokalnego potencjału \cite{coco1,coco2}.
+Drugi z parametrów $J$, nazywany energią wymiany Hunda, znacznie słabiej zależy od rodzaju materiału i często stosowana jest jego wartość atomowa (\ref{HundJ}). 
+
+
+\chapter{Izolatory i półprzewodniki}
+\label{sec:insulators}
+
+\section{Przerwa energetyczna}
+\label{sec:bandgap}
+
+Fundamentalna przerwa energetyczna zdefiniowana jest jako różnica między energią jonizacji ($I$), czyli procesu usunięcia elektronu z danego materiału, 
+a powinowactwem elektronowym ($A$), czyli energią uzyskaną z dodania elektronu
+%
+\begin{eqnarray}
+E_g=I-A&=&[E(N-1)-E(N)]-[E(N)-E(N+1)] \nonumber \\ 
+       &=&E(N+1)+E(N-1)-2E(N),
+\label{energygap}       
+\end{eqnarray}
+%
+gdzie $E(N)$, $E(N-1)$ i $E(N+1)$ to energie całkowite wyznaczone kolejno dla układu neutralnego składającego się z $N$ elektronów oraz
+układów z odjętym i dodanym elektronem. Każda z tych energii może być wyznaczona jako energia stanu podstawowego układu z odpowiednią ilością
+elektronów. Wartość przerwy możemy również zapisać używając energii stanów Kohna-Shama 
+%
+\begin{equation}
+E_g=E(N+1)-E(N)-[E(N)-E(N-1)]=\varepsilon_{N+1}(N+1)-\varepsilon_N(N),
+\label{truegap}
+\end{equation}
+%
+gdzie $\varepsilon_N(N)$ jest energią najwyższego poziomu pasma walencyjnego w układzie z $N$ elektronami,
+a $\varepsilon_{N+1}(N+1)$ energią najniższego stanu pasma przewodnictwa w układzie z $N+1$ elektronami.
+Czyli wartość przerwy energetycznej jest teoretycznie osiągalna w ramach DFT, 
+pod warunkiem, że znana jest dokladna zależność $E(N)$. Stosując funkcjonały LDA i GGA możemy otrzymać ze wzoru (\ref{truegap}) jedynie przybliżone wartości przerwy fundamentalnej. 
+   
+Zwykle wykonujemy obliczenia dla układu z ustaloną liczbą elektronów $N$ i otrzymujemy przerwę w spektrum energetycznym Kohna-Shama, 
+która wynosi
+%
+\begin{equation}
+\Delta_{KS}=\varepsilon_{N+1}(N)-\varepsilon_N(N).
+\label{ksgap}
+\end{equation}
+%
+Porównując wzory (\ref{truegap}) i (\ref{ksgap}) dostajemy związek między tymi dwiema przerwami
+%
+\begin{equation}
+E_g=\Delta_{KS}+\varepsilon_{N+1}(N+1)-\varepsilon_{N+1}(N)=\Delta_{KS}+\Delta_{xc},
+\end{equation}
+%
+gdzie różnica między nimi $\Delta_{xc}$ wynika ze zmiany nachylenia liniowej funkcji $E(N)$ przy całkowitej liczbie elektronów \cite{pardew82}.
+Ta nieciągłość ma swoje źródło w potencjale wymienno-korelacyjnym, którego pochodna zmienia się skokowo przy zmianie ilości elektronów
+%
+\begin{equation}
+\Delta_{xc}=\frac{\delta E_{xc}[n]}{\delta n(\bm{r})}|_{N+\delta}-\frac{\delta E_{xc}[n]}{\delta n(\bm{r})}|_{N-\delta},
+\label{deltaxc}
+\end{equation}
+%
+gdzie pochodne funkcjonalne wyznaczone są dla gęstości elektronów $n(\bm{r})$, których całki po całej przestrzeni równe są $N+\delta$ i $N-\delta$, w granicy $\delta\rightarrow 0$. 
+
+Występowanie tej nieciągłości jest podstawową cechą DFT, ale w stosowanych przybliżeniach LDA i GGA, które charakteryzują się analityczną zależnością
+energii od gęstości elektronowej, te nieciągłości nie występują. Przybliżony charakter energii stanów jednocząstkowych,
+z których wyznaczana jest wartość $\Delta_{KS}$, w połączeniu z brakiem nieciągłości powoduje, że przerwy energetyczne wyznaczone w przybliżeniu LDA są średnio zaniżone o $40\%$. W GGA te wartości są zwykle poprawione, ale również znacznie odbiegają od wartości eksperymentalnych. 
+
+\section{Izolatory pasmowe i półprzewodniki}
+
+W większości półprzewodników i izolatorów przerwa energetyczna wynika z istnienia całkowicie zapełnionych pasm walencyjnych
+i pustych pasm przewodzących. Rozróżnienie między półprzewodnikami i izolatorami nie jest jednoznacznie określone.
+Zwykle półprzewodniki mają mniejszą przerwę energetyczną i ich przewodnictwo elektryczne wykazuje znaczną zależność
+od temperatury w wyniku występowania indukowanych termicznie nośników prądu w paśmie przewodnictwa.
+W tych materiałach korelacje elektronowe zwykle nie są silne, ale mimo tego wyznaczana w ramach standardowych przybliżeń DFT 
+przerwa energetyczna jest znacznie zaniżona. Wynika to z braku uwzglednienia efektów wielociałowych, które występują
+przy wzbudzaniu elektronu z pasma walencyjnego do pasma przewodnictwa. Przykładem jest oddziaływanie elektron-dziura
+wystepujace dla stanów wzbudzonych.
+W przypadku tych materiałów, metoda LDA+U nie jest skuteczna i lepiej nadają się metody SIC i funkcjonały hybrydowe,
+jak również metody wielociałowe typu GW. 
+W metodzie SIC, energia całkowita zależy od obsadzenia orbitali atomowych, co daje możliwość odtworzenie nieciągłości
+pochodnych funkcjonału wymienno-korelacyjnego (\ref{deltaxc}) i poprawienia wartości przerwy energetycznej.
+SIC daje dobre wyniki dla izolatorów z dużą przerwa takich, jak kryształy gazów szlachetnych \cite{PZ}, czy kryształy jonowe. Przykładowo dla NaCl przerwa energetyczna wynosi $E_g^{SIC}=9.2$~eV \cite{norman83}, co dobrze zgadza się z wartością eksperymentalną $E_g^{exp}=8.97$~eV.  
+
+Funkcjonały hybrydowe, które częściowo uwzlędniają dokladną wartość oddziaływania wymiennego i przez to redukują błąd samooddziaływania elektronów, 
+również poprawiają wartość przerwy energetycznej. Na rysunku~\ref{fig:gaphyb} porównano eksperymentalne wartości przerwy energetycznej
+z wyliczonymi w ramach czterech funkcjonałów hybrydowych dla wybranej grupy półprzewodników i tlenków metali przejściowych (FeO, CoO, NiO, MnO, and VO$_2$)~\cite{Gap-hyb}. Dwa z tych funkcjonałów zawierają 20 \% dokładnej wartości energii wymiany (B3PW91 i B3LYP), a dwa pozostałe 25 \% (PBE0 i HSE). Dla wiekszości materiałów 
+z przerwą mniejszą niź 5 eV, zgodność z eksperymentem jest bardzo dobra dla wszystkich funkcjonałów. Jednocześnie widzimy, że wybrane funkcjonały znacznie zaniżają wartości przerwy dla trzech półprzewodników o dużej przerwie (NaCl, LiCl i LiF).
+Dokładna analiza błędów pokazała, że najlepszą zgodność uzyskano dla funkcjonału HSE, który ma tendencję do niewielkiego zaniżania wartości przerwy (średnio o -0.24 eV). Dwa funkcjonały B3PW91 i B3LYP lekko zawyżają przerwę (średnie błedy wynoszą 0.14 eV i 0.13 eV). Największe odstępstwo obserwujemy dla funkcjonału PBE0, który zawyża wartość przerwy średnio o 0.43 eV.
+
+\begin{figure}[t!]
+\centering
+\includegraphics[scale=0.5]{gap-hyb.pdf}
+\caption{Porównanie eksperymentalnych wartości przerwy energetycznej z obliczonymi dla różnych funkcjonałów hybrydowych. Rysunek z pracy \cite{Gap-hyb}.}
+\label{fig:gaphyb}
+\end{figure}
+
+W tabeli~\ref{gap} zebrane są wartości przerwy energetycznej dla wybranych materiałów, wyliczone różnymi metodami
+i porównane z danymi eksperymentalnymi. LDA i GGA we wszystkich przypadkach znacznie zaniżają wartość przerwy.
+Dotyczy to zarówno półprzewodników, szczególnie germanu, gdzie przerwa jest równa zeru, jak i izolatorów.
+Duże rozbieżności w porównaniu do eksperymentu widzimy dla trzech ostatnich materiałów, zaliczanych do izolatorów Motta.
+Zastosowanie funkcjonału hybrydowego HSE poprawia wartości przerwy i dla większości półprzewodników zgodność z eksperymentem jest bardzo dobra.
+Dla izolatorów, ta zgodność jest lepsza lub gorsza w zależności od materiału.
+W tabeli~\ref{gap} pokazano również wyniki otrzymane w ramach metody GW i jej uproszczonej wersji G$_0$W$_0$, które również dają
+lepsze wartości przerwy. Metoda GW, która uwzlędnia oddziaływania wielociałowe w ramach rozwinięcia perturbacyjnego, będzie tematem jednego z
+kolejnych rozdziałów. 
+
+\begin{table}[h!]
+\caption{Wartości przerwy energetycznej wyliczone różnymi metodami i porównane z danymi eksperymentalnymi.}
+\label{gap}
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|}
+\hline
+ Materiał & LDA & PBE & HSE & G$_0$W$_0$ & GW & Eksperyment \\ \hline
+  C & 4.14 & 4.17  & 4.94 & 5.50 & 6.18 & 5.50 \\
+  Si & 0.60 & 0.71  & 1.11 & 1.12 & 1.41 & 1.17 \\
+  Ge & 0.00 &  & 0.83 & 0.66 & 0.95 & 0.74 \\
+  MgO & 4.70 & 4.74  & 6.46 & 7.25 & 9.16 & 7.90 \\
+  GaAs & 0.30 & 0.53  & 1.41 & 1.30 & 1.85 & 1.52 \\
+  SiC & 1.35 &    & 2.40 & 2.27 & 2.88 & 2.40 \\ 
+  GaP & 1.53 & 1.69 & 2.09 &   &   & 2.35  \\
+  CdS & 0.96 & 1.23 & 2.27 &   &   & 2.48 \\
+  InP & 0.50 & 0.72 & 1.52 &   &   &  1.42 \\
+  BN  & 4.42 & 4.53 & 5.39 &   &   &  6.20 \\
+  NaCl & 4.70 & 5.08 & 6.42 &  &   & 8.97  \\
+  LiF  & 8.84 & 9.04 & 11.4 &  &   & 13.6  \\  
+  MnO & 0.76 &    & 2.80 &  & 3.50 & 3.90 \\ 
+  FeO & 0 &    & 2.2 &  &  & 2.4 \\ 
+  NiO & 0.42 &    & 4.2 & 1.10 & 4.80 & 4.30 \\ \hline 
+\end{tabular}
+\end{center}
+\end{table}
+
+\section{Izolatory Motta}
+
+Szczególną klasę materiałów stanowią izolatory Motta, których własności elektronowe wynikają z silnych oddziaływaniań kulombowskich miedzy elektronami.
+W tym przypadku wielkość charakteryzująca nieciągłość wyliczana w ramach standardowych przybliżeń $\Delta_{KS}$ jest równe zeru 
+lub jest bardzo małą wielkością. Przerwa energetyczna wynika zasadniczo z nieciągłości funkcjonału energii ($\Delta_{xc}>0$), powstającej przy zmianie obsadzenia stanów orbitalnych. 
+Ta nieciągła zmiana energii jest proporcjonalna do energii oddziaływania kulombowskiego $U$. Dla stanów $d$ w metalach przejściowych parametr $U$ można zdefiniować
+jako energię związaną z transferem elektronu między dwoma atomami z początkowym obsadzeniem $d^n$,
+który prowadzi do zwiększenia ilości elektronów na jednym atomie $d^{n+1}$ i zmniejszenia na drugim $d^{n-1}$
+%
+\begin{equation}
+U=E(d^{n+1})+E(d^{n-1})-2E(d^n),
+\end{equation} 
+%
+gdzie $E(d^n)$ jest całkowitą energią układu, w którym pojedynczy atom posiada obsadzenie $d^n$.
+Ta relacja oznacza, że sama przerwa energetyczna dana wzorem (\ref{energygap}) powinna być liniową funkcją $U$.
+Potwierdzają to obliczenia przeprowadzone w ramach metody LDA+U.
+
+Jako przykład omówię wyniki obliczeń gestości stanów elektronowych dla izolatora Motta Fe$_2$SiO$_4$, otrzymane metodą GGA+U \cite{Mariana}.
+Na rysunku 6.2 pokazane sa wyniki dla stanu antyferromagnetycznego (AFM).
+Dla $U=0$ dostajemy nieprawidłowy stan metaliczny ($\Delta_{KS}=0$) z energią Fermiego przecinającą pasmo $t_{2g}$. 
+Pasmo to obsadzone jest przez elektrony z mniejszościowym kierunkiem spinów (kierunek przeciwny do momentu magnetycznego danego atomu). 
+Obliczenia dla $U=4.5$ eV dają prawidłowy stan elektronowy z przerwą energetyczną równą około 2 eV,
+która rozdziela pasmo $t_{2g}$ na dwie części: obsadzone dolne pasmo Hubbarda i nieobsadzone górne pasmo Hubbarda.
+
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.21]{Fe2SiO4-2.pdf}
+\caption{Gęstość stanów elektronowych w Fe$_2$SiO$_4$ dla $U=0$ (górny panel) i dla $U=4.5$~eV (dolny panel). Rysunek z pracy \cite{Mariana}.}
+\label{fig:mott}
+\end{figure}
+
+\begin{figure}[t!]
+\centering
+\includegraphics[scale=0.18]{Fe2SiO4-1.pdf}
+\caption{Zależność przerwy energetycznej od parametru oddziaływania kulombowskiego $U$ w Fe$_2$SiO$_4$ dla stanu antyferromagnetycznego (czerwone kółka)
+i ferromagnetycznego (niebieskie kwadraty). Rysunek z pracy \cite{Mariana}.}
+\label{fig:gap}
+\end{figure}
+
+Wartość przerwy energetycznej zależy od dokładnej wartości parametru $U$, co pokazano na rysunku 6.3.
+Dla małych wartości $U\leq 2$ eV, przerwa energetyczna równa jest zeru. Wynika to z kryterium, które mówi, że w izolatorach Motta, 
+niezerowa przerwa tworzy się dla wartości energii oddziaływania kulombowskiego większych od szerokości
+pasma elektronowego, $U>W$. Rzeczywiście, z rysunku 6.2 wynika, że szerokość pasma $t_{2g}$ jest równa z dobrym przybliżeniem $W=2$ eV.
+Gdy energia oddziaływania kulombowskiego $U$ przewyższa $W$, przerwa energetyczna rośnie praktycznie liniowo z jej wartością. 
+Jak widać z rysunku 6.3 wartości przerwy zależą od rodzaju uporządkowania magnetycznego i dla stanu AFM są większe niż dla stanu FM.
+
+\chapter{Polaryzacja elektryczna}
+
+\section{Teoria fazy Berry'ego}
+
+Polaryzacja elektryczna odgrywa ważną rolę w wielu zjawiskach fizycznych występujacych w dielektrykach. 
+Wyznaczenie wartości polaryzacji elektrycznej w materiałach periodycznych jakimi są kryształy stanowiło duży problem teoretyczny. 
+W obliczeniach z pierwszych zasad, dzięki periodycznym warunkom brzegowym, takie układy traktowane sa jako nieskończone
+i zwykle nie bierze sie pod uwagę wpływu powierzchni materiału. 
+W układzie skończonym można wyznaczyć polaryzację elektryczną obliczając całkowity moment dipolowy złożony z części jonowej
+i elektronowej
+%
+\begin{equation}
+\bm{P}=\frac{e}{\Omega}\sum_jZ_j\bm{R}_j+\frac{1}{\Omega}\int d\bm{r} n(\bm{r}) \bm{r},
+\label{polaryzacja}
+\end{equation}
+% 
+gdzie zarówno sumowanie po wszystkich jonach o ładunku $Z_je$, jak i całkowanie gestości elektronowej $n(\bm{r})$ 
+odbywa się po całej objetości $\Omega$. Z tego samego wzoru można wyliczyć zmianę polaryzacji $\Delta \bm{P}$
+wywołaną zmianą rozkładu gęstości elektronowej $\Delta n(\bm{r})$, np. na skutek przemieszczenia się atomów. 
+
+W układach periodycznych własności całego kryształu zależą od rozmieszczenia atomów w komórce prymitywnej.
+Wydaje się, że całkowity moment dipolowy kryształu i polaryzację elektryczną można wyznaczyć stosując wzór (\ref{polaryzacja}),
+do pojedynczej komórki. Jednak takie podejście możliwe jest tylko w skrajnych przepadkach kryształów jonowych
+lub molekularnych i odpowiada klasycznemu opisowi typu Clausiusa-Mossottiego, w którym występują izolowane dipole elektryczne.
+Gęstość elektronowa jest wielkością ciągłą i nie jest możliwe podzielenia całego obszaru kryształu w sposób jednoznaczny.  
+Co więcej, wiązania kowalencyjne, ktore występuja w ferroelektrykach, mają naturę kwantową i poprawny opis polaryzacji elektrycznej
+wymaga formalizmu kwantowego. 
+
+Punktem startowym, który umożliwia wyznaczenie polaryzacji w układach periodycznych jest 
+fundamentalna zależność między lokalną wartością polaryzacji elektronowej i przepływającym prądem
+%
+\begin{equation}
+\bm{P}_e(\bm{r},t)=\int_{t_0}^t dt' j_e(\bm{r},t').
+\label{delP}
+\end{equation}
+%
+Wzór ten umożliwia również pomiar zmiany polaryzacji $\Delta \bm{P}$ przez wyznaczenie przepływającego prądu.
+Przykładowo, mierząc zmiany polaryzacji w funkcji deformacji kryształu można wyznaczyć stałą piezoelektryczną.
+Przełomem w teorii były prace, w których pokazano związek polaryzacji elektrycznej z geometryczną fazą Berry'ego~\cite{berry1,berry2}. 
+W obliczeniach zamiast czasu wprowadza się parametr $\lambda\in [0,1]$, który  
+opisuje adiabatyczną transformację układu generującą zmianę polaryzacji
+%
+\begin{equation}
+\Delta\bm{P}_e = \int_0^1 d\lambda \frac{\partial\bm{P}_e}{\partial\lambda}. 
+\end{equation}
+%
+Przykładowo parametr $\lambda$ może być współrzędną, która określa położenie atomu w sieci krystalicznej.  
+Stosując wyrażenie z rachunku zaburzeń można powiązać pochodną polaryzacji ze strukturą elektronową 
+%
+\begin{equation}
+\frac{\partial\bm{P}_e}{\partial\lambda}=-i\frac{e\hbar}{\Omega m_e}\sum_{\bm{k}}\sum_{n=1}^{occ}\sum_{=1}^{emp}\frac{\langle\psi^{\lambda}_{\bm{k}n}|\bm{p}
+|\psi^{\lambda}_{\bm{k}m}\rangle\langle\psi^{\lambda}_{\bm{k}m}|\partial V^{\lambda}_{\text{KS}}/\partial\lambda|\psi^{\lambda}_{\bm{k}n}\rangle}{(\varepsilon^{\lambda}_{\bm{k}n}-\varepsilon^{\lambda}_{\bm{k}m})^2}+c.c.,
+\label{partialP}
+\end{equation}
+%
+gdzie sumowanie wykonywane jest po wszystkich stanach zajętych i pustych z pierwszej strefy Brillouina. 
+Elementy maciorzowe operatora pędu można wyrazić zależnością
+%
+\begin{equation}
+\langle\psi^{\lambda}_{\bm{k}n}|\bm{p}|\psi^{\lambda}_{\bm{k}m}\rangle=\frac{m_e}{\hbar}\langle u^\lambda_{\bm{k}n}|[\nabla_{\bm{k}},H^\lambda_{\bm{k}}]|u^\lambda_{\bm{k}m}\rangle,
+\end{equation}
+%
+w której $u^\lambda_{\bm{k}m}$ jest częścią periodyczną funkcji Blocha, a hamiltonian dany jest wyrażeniem
+%
+\begin{equation}
+H^\lambda_{\bm{k}}=\frac{1}{2m_e}(-i\hbar\nabla+\hbar\bm{k})^2+V^\lambda_\text{KS}.
+\end{equation}
+%
+Podobnie można wyrazić elementy macierzowe pochodnej potencjału Kohna-Shama
+%
+\begin{equation}
+\langle\psi^{\lambda}_{\bm{k}n}|\frac{\partial V^{\lambda}_{\text{KS}}}{\partial\lambda}|\psi^{\lambda}_{\bm{k}m}\rangle=\frac{m_e}{\hbar}\langle u^\lambda_{\bm{k}n}|[\frac{\partial}{\partial\lambda},H^\lambda_{\bm{k}}]|u^\lambda_{\bm{k}m}\rangle.
+\end{equation}
+%
+Po wstawieniu elementów macierzowych do (\ref{partialP}) otrzymujemy
+%
+\begin{equation}
+\Delta \bm{P}_e=\frac{-ie}{8\pi^3}\sum_{n=1}^{occ}\int_\text{BZ}d\bm{k}\int_0^1d\lambda[\langle\nabla u^\lambda_{\bm{k}n}|                  
+\frac{\partial u^\lambda_{\bm{k}n}}{\partial\lambda}\rangle-\langle\frac{\partial u^\lambda_{\bm{k}n}}{\partial\lambda}|                  
+\nabla_{\bm{k}} u^\lambda_{\bm{k}n}\rangle].
+\end{equation}
+Wykonując całkowanie przez części i wykorzystując periodyczność funkcji $u_{\bm{k}n}$ dostajemy
+%
+\begin{equation}
+\Delta \bm{P}_e=\bm{P}^{\lambda=1}_e-\bm{P}^{\lambda=0}_e,
+\label{pol1}
+\end{equation}
+%
+gdzie
+%
+\begin{equation}
+\bm{P}^\lambda_e=\frac{ie}{8\pi^2}\sum_{n=1}^{occ}\int_\text{BZ}d\bm{k}\langle u^\lambda_{\bm{k}n}|\nabla_{\bm{k}}|u^\lambda_{\bm {k}n}\rangle.
+\label{pol2}
+\end{equation}
+%
+Występująca w tym wzorze całka jest bezpośrednio związana z fazą Berry'ego stanów elektronowych~\cite{Zak}.
+Wielkość wektorowa $\bm{A}(\bm{k})=i\langle u^\lambda_{\bm{k}n}|\nabla_{\bm{k}}|u^\lambda_{\bm {k}n}\rangle$
+nazywa się potencjałem cechowania lub połączeniem Berry'ego, a całka tej wielkość
+po zamkniętym obszarze jest fazą Berry'ego, którą dla danego pasma $n$ i kierunku $j$ można zapisać w formie
+%
+\begin{equation}
+\phi^\lambda_{nj}=\frac{i}{\Omega_\text{BZ}}\int_\text{BZ} d\bm{k}\langle u^\lambda_{\bm{k}n}|\bm{G}_j\nabla_{\bm{k}}|u^\lambda_{\bm {k}n}\rangle,
+\end{equation}
+%
+gdzie $\bm{G}_j$ jest wektorem sieci odwrotnej. 
+Polaryzacja związana z pasmem $n$ wyraża się wzorem
+%
+\begin{equation}
+\bm{P}^\lambda_n=\frac{e}{2\pi\Omega}\sum_j\phi^\lambda_{nj}\bm{R}_j.
+\end{equation}
+%
+Otrzymany wynik, który pokazuje związek polaryzacji elektrycznej z fazą funkcji falowej nie jest zaskoczeniem.
+W mechanice kwantowej, faza funkcji falowej jest bezpośrednio związana z prądem elektrycznym
+i zgodnie ze wzorem (\ref{delP}) określa również polaryzację elektryczną. 
+Tak jak każda faza, również polaryzacja elektryczna nie jest wartością zdefiniowaną jednoznaczne, a jedynie określoną {\it modulo}
+stała wartość. Żeby określić tę wartość zakładamy, że stan początkowy ($\lambda=0$) i końcowy ($\lambda=1$) opisane
+są tym samym hamiltonianem. Wtedy funkcje falowe dla tych stanów mogą różnić się tylko wartością fazy
+%
+\begin{equation}
+u^{\lambda=1}_{\bm{k}n}(\bm{r})=e^{i\theta_{\bm{k}n}}u^{\lambda=0}_{\bm{\bm{k}n}}(\bm{r}).
+\end{equation}
+%
+Zmianę fazy można zapisać w ogólnej formie
+%
+\begin{equation}
+\theta_{\bm{k}n}=\beta_{\bm{k}n}+\bm{k}\bm{R}_n,
+\end{equation}
+%
+gdzie $\beta_{\bm{k}n}$ jest funkcją periodyczną w $\bm{k}$. Zgodnie ze wzorem (\ref{pol2}), zmiana polaryzacji w tym przypadku wynosi
+%
+\begin{equation}
+\Delta\bm{P}_e=-\frac{e}{8\pi^3}\sum_{n=1}^{occ}\int_\text{BZ}d\bm{k}\nabla_{\bm{k}}\theta_{\bm{k}n}=\frac{e}{\Omega}\sum_{n=1}^{occ}\bm{R}_n=\frac{e}{\Omega}\bm{R}.
+\end{equation}
+%
+Dla każdej sieci krystalicznej, można określić najmniejszą wartość $e\bm{R}_1/\Omega$, 
+która definiuje kwant polaryzacji, związany z symetrią translacyjną potencjału elektronowego, $V(\bm{r})=V(\bm{r}-\bm{R}_1)$.
+Ze względu na periodyczność kryształu wartość bezwględna polaryzacji nie ma większego sensu i istotna jest zmiana polaryzacji między dwoma stanami.
+Przesunięcia atomów, które zwykle są brane pod uwagę, są znacznie mniejsze niż odległości między atomami i wyliczana zmiana polaryzacji
+jest dużo mniejsza niż kwant polaryzacji, $\Delta\bm{P}_e\ll e\bm{R}_1/\Omega$. Pozwala to wyeliminować niejednoznaczność wartości polaryzacji
+w praktycznych zastosowaniach.  
+
+Polaryzację elektryczna można wyrazić również w reprezentacji funkcji Wanniera,
+które sa transformatami Fouriera funkcji Blocha
+%
+\begin{equation}
+\psi^{\lambda}_{\bm{k}j}=e^{i\bm{k}{r}}u^\lambda_{\bm{k}n}(\bm{r})=\frac{1}{\sqrt{N}}\sum_je^{i\bm{k}\bm{R}_j}w^\lambda_n(\bm{r}-\bm{R}_j),
+\end{equation}
+%
+gdzie $\bm{R}_j$ są wektorami sieci krystalicznej. Wyliczają z tej zależności $u^\lambda_{\bm{k}n}$ i wstawiając do wzoru (\ref{pol2}) otrzymujemy 
+%
+\begin{equation}
+\Delta\bm{P}_e=-\frac{e}{\Omega}\sum_{n=1}^{occ} [\int d\bm{r} \bm{r}|w^{\lambda=1}_n(\bm{r})|^2-\int d\bm{r} \bm{r}|w^{\lambda=0}_n(\bm{r})|^2].
+\end{equation}
+%
+Ten wzór pokazuje, że zmiana polaryzacji jest proporcjonalna do przesunięcia środka ciężkości rozkładu gestości ładunku,
+zlokalizowanego na orbitalach Wanniera, wywołanego adiabatyczną zmianą potencjału elektronowego.
+
+
+\section{Ładunki efektywne}
+
+Zgodnie ze wzorem (\ref{polaryzacja}), całkowita zmiana polaryzacji wzdłuż kierunku $\alpha$
+składa się z części jonowej i elektronowej
+%
+\begin{equation}
+(\Delta\bm{P})_\alpha=(\Delta\bm{P}_\text{ion})_\alpha+(\Delta\bm{P}_e)_\alpha.
+\label{pol3}
+\end{equation}
+% 
+Część elektronowa polaryzacji dana jest wyrażeniami (\ref{pol1}) i (\ref{pol2}), natomiast część jonowa wyliczana jest ze wzoru
+%
+\begin{equation}
+(\Delta\bm{P}_\text{ion})_\alpha=\frac{e}{\Omega}Z_\text{ion}u_\alpha,
+\label{pol4}
+\end{equation}
+%
+gdzie $Z_{ion}$ jest ładunkiem rdzenia atomowego, a $u_\alpha$ wartością przesunięcia atomów.
+Zmianę całkowitej polaryzacji można zapisać w formie
+%
+\begin{equation}
+(\Delta\bm{P})_\alpha = \frac{\partial P_\alpha}{\partial u_\beta}u_\beta = \frac{e}{\Omega}Z^*_{\alpha\beta} u_\beta,
+\label{pol5}
+\end{equation}
+%
+gdzie $Z^*_{\alpha\beta}$ jest tensorem ładunków efektywnych Borna.
+Liczba niezależnych i niezerowych składowych tensora zależy od symetrii kryształu i ilości nierównowaznych położeń atomowych. 
+Wszystkie składowe tensora $Z^*_{\alpha\beta}$ można wyznaczyć korzystając ze wzoru (\ref{pol5}), wyliczając
+zmianę polaryzacji dla wychyleń atomów w różnych kierunkach. 
+
+Ładunki efektywne przyjmują niezerowe wartości we wszystkich izolatorach i odgrywają ważną rolę przy opisie
+dynamiki sieci. Przesunięcia jonów związane z podłużnymi modami optycznymi (LO) w centrum strefy Brillouina ($\bm{k}=0$), które aktywne są w podczerwieni ({\it ang. infrared modes}), generują makroskopową polaryzację elektryczną i w ten sposób modyfikują siły międzyatomowe.
+Te dodatkowe siły powodują, że energia modów LO blisko punktu $\Gamma$ jest większa niż poprzecznych modów optycznych (TO).
+Powoduje to rozszczepienie LO-TO, które może być wyliczone jeżeli znamy ładunki efektywne jonów i stałą dielektryczną materiału~\cite{PCM}.
+
+ 
+\section{Ferroelektryki}
+
+Ferroelektryki należą do osobnej grupy izolatorów, w których spontanicznie pojawia się polaryzacja elektryczna w określonych
+warunkach termodynamicznych. Praktyczne wykorzystanie ferroelektryków wynika z możliwości przełączania kierunku polaryzacji
+po przyłożeniu pola elektrycznego. W metalach indukowane są prądy elektronowe, które ekranują pole elektryczne 
+i dlatego lokalizacja elektronów jest warunkiem koniecznym występowania makroskopowej polaryzacji elektrycznej.  
+W izolatorach elektrony nie mogą poruszać się swobodnie w krysztale, natomiast biorą udział w prądach polaryzacji, 
+powiązanych z lokalną wartością polaryzacji elektrycznej.  
+W ferroelektrykach ważną role odgrywają wiązania kowalencyjne i hybrydyzacja między różnymi stanami elektronowymi 
+umożliwiająca transfer ładunku między jonami.
+Najbardziej podstawowy mechanizm tworzenia się fazy ferroelektrycznej związany jest ze strukturalnym przejściem fazowym. 
+W fazie paraelektrycznej, występującej zwykle w wyższych temperaturach, sieć krystaliczna posiada symetrię centrosymetryczną,
+w której średnia polaryzacja $\bm{P}=0$. Aby powstała spontaniczna polaryzacja, musi zajść transformacja układu
+do struktury bez centrum symetrii, czyli takiej, która nie jest niezmiennicza wzlędem operacji ($\bm{r}\rightarrow -\bm{r}$).
+To jest warunek konieczny występowania fazy ferroelektrycznej, gdzie $\bm{P}\neq 0$.
+
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=1]{BaTiO3.pdf}
+\caption{Deformacja sieci krystalicznej ferroelektryka o strukturze perowskitu ABO$_3$.}
+\label{fig:batio3}
+\end{figure}
+
+Przykładem ferroelektryków są materiały o strukturze perowskitu ABO$_3$, pokazanej na rysunku (\ref{fig:batio3}).
+Do tej grupy należy tytanian baru BaTiO$_3$ i wiele innych tlenków zawierajacych metale przejściowe.
+W narożach komórki znajdują się kationy A, natomiast znajdujący się w środku kation B otoczony jest oktaedrem anionów O.
+W wysokich temperaturach kryształ posiada centrosymetryczną symetrię regularną. Przy obniżaniu temperatury w materiale pojawia się
+niestabilność sieci, związana z wystepowaniem miękkiego modu fononowego, którego energia dąży do zera przy zbliżaniu się do przejścia fazowego.
+Wychylenia atomów w miękkim modzie, pokazane schematycznie na rysunku (\ref{fig:batio3}), prowadzą do statycznej 
+deformacji sieci poniżej temperatury krytycznej. W strukturze tetragonalnej złamana jest symetria centrosymetryczna 
+i pojawia się niezerowa polaryzacja elektryczna, której żródłem jest względne przesunięcie anionów (O),
+względem kationów (A i B). Zmiana polaryzacji spowodowana jest nie tylko sztywnym przesunięciem ładunków zlokalizowanych na jonach,
+ale również, jak to zostało wyjaśnione w poprzednich rozdziałach, przepływem prądów elektronowych wzdłuż wiązań kowalencyjnych. 
+ 
+Wartość polaryzacji elektrycznej w ferroelektrykach można wyliczyć startując ze stanu początkowego ($\lambda=0$), który odpowiada położeniom jonów w strukturze 
+centrosymetrycznej ($\bm{P}=0$).
+Wtedy zmiana polaryzacji (\ref{pol3}), indukowana przysunięciem jonów do położeń w strukturze niecentrosymetrycznej,
+odpowiada wartości spontanicznej polaryzacji, która pojawia się w wyniku przejścia fazowego.
+Wartość polaryzacji wyliczona dla ferroelektryka KNbO$_3$ wynosi 0.35 C/m$^2$~\cite{resta1993}
+i bardzo dobrze zgadza się z wartością wyznaczoną eksperymentalnie 0.37 C/m$^2$. 
+ 
+W ferroelektrykach występują szczególnie duże wartości ładunków efektywnych, często znacznie przewyższające ładunki statyczne jonów. 
+Przykładowo w strukturze regularnej BaTiO$_3$ ładunki efektywne jonów znajdujacych się w czterech nierównoważnych położeniach przyjmują wartości: 
+$Z^*_\text{Ba}=2.75$, $Z^*_\text{Ti}=7.06$, $Z^*_\text{O1}=-5.83$ i $Z^*_\text{O2}=-2.11$~\cite{zhong1994}.
+Znacznie podwyższone wartości ładunków efektywnych jonów Ti i O1 wynikają z silnej hybrydyzacji między stanami $3d$ i $2p$ 
+w wiązaniu kowalencyjnym Ti-O1. Zmiana odległości między tymi jonami generuje transfer ładunku (prąd polaryzacji)
+i zmianę polaryzacji zgodnie ze wzorem (\ref{delP}). Całkowita zmiana polaryzacji składa się z części jonowej, zależnej od ładunków statycznych~(\ref{pol4}),
+i części elektronowej, która powoduje zwiększenie wartości ładunków efektywnych.
+Duże ładunki efektywne powodują, że rozszczepienie LO-TO przyjmuje bardzo duże wartości w ferroelektrykach~\cite{zhong1994}.
+
+ 
+\chapter{Oddziaływanie van der Waalsa}
+\label{sec:vdw}
+
+\section{Podstawowe własności}
+
+Oddziaływanie van der Waalsa (vdW) nazywane również dyspersyjnym jest efektem korelacji elektronowych, które nie są uwzględnione
+w standardowych funkcjonałach wymienno-korelacyjnych.
+Jest to oddziaływanie przyciągające odgrywające ważną role w kryształach molekularnych, takich jak
+kryształy gazów szlachetnych, w których atomy mają zamknięte powłoki elektronowe,
+oraz materiały składające się z neutralnych cząsteczek. 
+Ważne jest też w układach biologicznych, np. ma duży wpływ na prawidłowy kształt i działanie białek.
+Żródłem tego oddziaływania są fluktacje kwantowe dipola elektrycznego na jednym atomie (molekule), które indukują dipol elektryczny na innym atomie (molekule).
+Wzajemne oddziaływanie tych dipoli prowadzi do słabego przyciągania atomów lub molekuł. 
+W klasycznym opisie, pole elektryczne wytworze przez moment dipolowy $p_1$ w odległości $R$ można wyrazić wzorem 
+%
+\begin{equation}
+E=\frac{p_1}{R^3}.
+\end{equation}
+% 
+Jeżeli pole elektryczne oddziałuje na cząsteczkę o polaryzowalności $\alpha$ to indukuje moment dipolowy
+%
+\begin{equation}
+p_2=\frac{\alpha p_1}{R^3}.
+\end{equation}
+%
+Średnia energia oddziaływania tych dwóch dipoli wynosi
+%
+\begin{equation}
+U=-\frac{\alpha p_1^2}{R^6}.
+\end{equation}
+%
+Uwzglednienie korelacji elektronowych w ramach teorii perturbacyjneje prowadzi do takiej samej zależność oddziaływania 
+od odległości $\sim R^{-6}$, ale poprawne wyznaczenie polaryzowalności i siły oddziaływania wymaga wyjścia poza najprostsze 
+przybliżenie faz przypadkowych ({\it ang. random-phase approximation} - RPA).   
+ 
+\section{Poprawki do funkcjonału energii}
+ 
+\subsection{Metody D2 i D3}  
+ 
+W obliczeniach DFT można uwzlędnić przybliżoną wartość energii oddziaływania vdW jako poprawkę do całkowitej energii układu
+%
+\begin{equation}
+E_\text{tot}=E_\text{KS}+E_\text{vdW}
+\end{equation}
+%
+gdzie $E_\text{KS}$ jest wyznaczonym w procedurze samozgodnej funkcjonałem Kohna-Shama.
+W takim podejściu oddziaływanie vdW nie wpływa bezpośrednio na strukturę elektronową, 
+jedynie poprzez modyfikację odległości między atomami.
+ 
+W wersji, którą zaproponował Grimme~\cite{grimme2004,grimme2006}, zależność energii vdW od odległości między atomami $R_{ij}$
+ma postać
+%
+\begin{equation}
+E_\text{vdW}=-\frac{1}{2}\sum_{i,j}f_\text{d}(R_{ij})\frac{C_{6ij}}{R_{ij}^6},
+\end{equation}
+% 
+gdzie $f_\text{d}$ jest funkcją tłumiącą, która określa asymptotyczne zachowanie dla $R_{ij}\rightarrow 0$  
+i wyklucza podwójne uzwględnianie efektów korelacyjnych przy pośrednich odległościach.
+Jej rolą jest zapewnienie, że oddziaływanie vdW ograniczone jest do odległości większych niż 
+długości typowych wiązań, poprawnie opisywanych w ramach standardowych funkcjonałów.
+Współczynniki dyspersyjne $C_{6ij}$ zdefiniowane są dla każdej pary atomów
+jako średnie geometryczne parametrów atomowych
+%
+\begin{equation}
+C_{6ij}=\sqrt{C_{6i}C_{6j}}.
+\end{equation}
+%
+Dla każdego atomu określa się parametr $C_6=0.05NI_\text{p}\alpha$, gdzie $I_\text{p}$ jest potencjałem jonizującym, $\alpha$ jest
+statyczną polaryzowalnościa dipolową, a $N=2, 10, 18, 36, 54$ dla kolejnych rzędów układu okresowego pierwiastków.  
+Współczynnik skalujący wyznaczonony jest tak, aby zreprodukować odległości międzyatomowe i energie wiązania
+w odpowiedniej grupie pierwiastków. Funkcję tłumiącą można zapisać w formie \cite{grimme2006}
+%
+\begin{equation}
+f_\text{d}(R_{ij})=\frac{s_6}{1+e^{-d(R_{ij}/R_{0ij}-1)}},
+\label{damping}
+\end{equation}
+%
+gdzie $s_6$ jest współczynnikiem określonym dla danego funkcjonału DFT, np. $s_6=0.75$ dla PBE,
+$d$ jest współczynnikim tłumienia, którego typowa wartość wynosi 20, a $R_{0ij}=R_{0i}+R_{0j}$ jest promieniem odcięcia
+i wyraża się sumą promieni vdW dla danych atomów. 
+Parametry $C_{6i}$ i $R_{0i}$ są zdefiniowane dla wszystkch atomów i nie zależą od rodzaju materiału. 
+Dodatkowo w obliczeniach określa się maksymalny zasięg oddziaływania vdW. 
+Opisany schemat (DFT-D2) umożliwia poprawienie wyliczanych sił i odległości międzyatomowych w kryształach gazów szlachetnych oraz
+układach molekularnych~\cite{bucko2010}. Oddziaływanie vdW odgrywa również ważną rolę w materiałach warstwowych,
+dla których standardowe obliczenia DFT dają zwykle zaniżoną wartość siły wiążącej między warstwami. 
+Przykładowo, stała sieci $c$ w graficie, której wartość w przybliżeniu PBE ($c=8.84$~\AA)
+jest znacznie zawyżona w porównaniu do wartości eksperymentalnej ($6.71$~\AA), wyliczona z poprawką D2 wynosi $6.45$~\AA~\cite{bucko2010}.     
+
+W kolejnym podejściu (D3), Grimme uwzględnił dodatkowo drugi wyraz w wzorze na energię oddziaływania proporcjonalny do $R^{-8}$~\cite{grimme2010}
+%
+\begin{equation}
+E_\text{vdW}=-\frac{1}{2}\sum_{i,j}[f_{\text{d},6}(R_{ij})\frac{C_{6ij}}{R_{ij}^6}+f_{\text{d},8}(R_{ij})\frac{C_{8ij}}{R_{ij}^8}].
+\end{equation}
+%
+Współczynniki dyspersyjne $C_{6ij}$ wyznaczane są ze wzoru Casimira-Poldera 
+%
+\begin{equation}
+C_{6ij}=\frac{3}{\pi} \int d\omega \alpha_i(\omega)\alpha_j(\omega), 
+\label{CP}
+\end{equation}
+%
+gdzie $\alpha_i(\omega)$ jest średnią polaryzowalnością dipolową dla atomu $i$.
+Współczynniki $C_{8ij}$ wyliczane są z zależności $C_{8ij}=3C_{6ij}\sqrt{Q_iQ_j}$, gdzie $Q_i=s\sqrt{Z_i}\langle r^4\rangle_i/\langle r^2\rangle_i$.
+Funkcje tłumiące mają postać
+%
+\begin{equation}
+f_{\text{d},n}(R_{ij})=\frac{s_n}{1+6(R_{ij}/(s_{R,n}R_{0ij}))^{-\alpha_n}}.
+\end{equation}
+%
+W tym wzorze optymalizowane są parametry $s_6$, $s_8$ i $s_{R,6}$, a pozostałe mają ustalone wartości ($s_{R,8}=1$, $\alpha_6=14$, $\alpha_8=16$). 
+Alternatywnie można zastosować funkcję tłumiącą Becke-Johnsona (BJ) w postaci
+%
+\begin{equation}
+f_{\text{d},n}(R_{ij})=\frac{s_nR_{ij}^n}{R_{ij}^n+(a_1R_{0ij}+a_2)^n},
+\end{equation}
+%
+gdzie $s_n$, $a_1$ i $a_2$ sa parametrami dopasowania. Promień odcięcia można wyznaczyć znając współczynniki dyspersyjne
+$R_{0ij}=\sqrt{C_{8ij}/C_{6ij}}$. Analiza porównawcza przeprowadzona dla dużej grupy materiałów
+pokazała niewielki wpływ rodzaju funkcji tłumiącej na wyliczone odległości między atomami i trochę lepszą dokładność 
+przy zastosowaniu funkcji BJ do własności termodynamicznych~\cite{grimme2011}. 
+
+\subsection{Metoda TS}
+
+W metodzie zaproponowanej przez Tkatchenko i Schefflera (TS)~\cite{tkatchenko2009} bierze się pod uwagę tylko wyraz proporcjonalny do $R^{-6}$, ale
+uwzględnia się zależność współczynników dyspersyjnych i promienia odcięcią od gęstości ładunku.
+Wprowdzamy przybliżoną zależność polaryzowalności od częstości
+%
+\begin{equation}
+\alpha_i(\omega)=\frac{\alpha^0_i}{1-(\omega/\eta_i)^2},
+\label{alpha}
+\end{equation}
+%
+gdzie $\alpha^0_i$ jest statyczną polaryzowalnością i $\eta_i$ jest efektywną częstością dla atomu $i$.
+Po wstawieniu (\ref{alpha}) do (\ref{CP}) otrzymujemy formułę Londona
+%
+\begin{equation}
+C_{6ij}=\frac{3\eta_i\eta_j}{2(\eta_i+\eta_j)}\alpha_i^0\alpha_j^0.
+\end{equation}
+%
+Dla jednakowych atomów ($i=j$) otrzymujemy wzór
+%
+\begin{equation}
+\eta_i=\frac{4C_{6i}}{3(\alpha_i^0)^2},
+\end{equation}
+%
+który prowadzi do zależności
+%
+\begin{equation}
+C_{6ij}=\frac{2C_{6i}C_{6j}}{\frac{\alpha^0_j}{\alpha^0_i}C_{6i}+\frac{\alpha^0_i}{\alpha^0_j}C_{6j}}.
+\end{equation}
+%
+Współczynniki dyspersyjne dla poszczególnych atomów znajdujących się wewnątrz cząsteczek lub ciał stałych
+wyznacza się z wartości dla izolowanych atomów używając następującej formuły $C_{6i}=\nu_i^2C^{free}_{6i}$, gdzie $\nu_i$
+określa efektywną objętość atomu (podział Hirshfelda)
+%
+\begin{equation}
+\nu_i=\frac{V_i^{eff}}{V_i^{free}}=\frac{\int d\bm{r} r^3 w_i(\bm{r}) n(\bm{r})}{\int d\bm{r} r^3 n_i^{free}(\bm{r})},
+\end{equation}
+%
+z wagami Hirshfelda, które wyliczane są dla uśrednionych sferycznie gęstości elektronowych w izolowanych atomach
+%
+\begin{equation}
+w_i(\bm{r})=\frac{n_i^{free}(\bm{r})}{\sum_j n_i^{free}(\bm{r})}.
+\end{equation}
+%
+Podobnie znając wartości polaryzowalności i promieni vdW dla izolowanych atomów, można wyznaczyć te wartości
+dla atomów w cząsteczkach i kryształach: $\alpha_i=\nu_i\alpha^{free}$ i $R_{0i}=\nu_i^{\frac{1}{3}}R_{0i}^{free}$.
+Promienie vdW służą do wyznaczenia promienia odcięcia $R_{0ij}$ i funkcji tłumiacej $f_{\text{d},n}$ (\ref{damping}).
+
+W omówionych podejściach zakłada się jedynie oddziaływania dwuciałowe, nie biorąc pod uwagę modyfikacji oddziaływania vdW przez otaczające atomy lub molekuły. 
+W udoskonalonej wersji metody TS uwzglednia się dalekozasięgowe ekranowanie elektryczne w sposób samozgodny ({\it ang. self-consistent screening} - SCS)~\cite{tkatchenko2012}. Odpowiednie równanie samozgodne pozwalające wyznaczyć zmodyfikowaną polaryzowalność ma postać
+%
+\begin{equation}
+\alpha_i^\text{SCS}(\omega)=\alpha_i(\omega)-\alpha_i(\omega)\sum_{i\neq j}\tau_{ij}\alpha_i^\text{SCS}(\omega),
+\end{equation}
+%
+gdzie $\tau_{ij}$ jest tensorem oddziaływania dipol-dipol. Energię dyspersyjną wyznacza się z tych samych równań co w metodzie TS
+z odpowiednio zmodyfikowanymi współczynnikami dyspersyjnymi i promieniami odcięcia
+%
+\begin{equation}
+C_{6i}=\frac{3}{\pi}\int_0^\infty d\omega[\alpha^\text{SCS}_i(\omega)]^2,
+\end{equation}
+%
+\begin{equation}
+R_{0i}^\text{SCS}=\Big{(}\frac{\alpha_i^\text{SCS}}{\alpha_i}\Big{)}^{\frac{1}{3}}R_{0i}.
+\end{equation}
+
+%\subsection{Metoda MBD}
+
+\chapter{Efekty wielociałowe i stany wzbudzone}
+
+\section{Funkcje Greena i energia własna}
+
+Dokładny opis stanów elektronowych wymaga wyjścia poza przybliżenie niezależnych cząstek.   
+Stosując formalizm drugiego kwantowania, zapiszemy hamiltonian układu oddziałujących elektronów 
+w ogólnej formie
+%
+\begin{equation}
+H=\int d^3r \psi^\dagger(\bm{r},t) h_0(\bm{r}) \psi(\bm{r},t) +\frac{1}{2}\int d^3rd^3r' \psi^\dagger(\bm{r},t)\psi^\dagger(\bm{r}',t)v(\bm{r}-\bm{r}') \psi^\dagger(\bm{r}',t)\psi^\dagger(\bm{r},t),
+\end{equation}
+%  
+gdzie $h_0(x)$ opisuje energię kinetyczną oraz oddziaływanie elektronów z zewnętrznym polem, $v(\bm{r}-\bm{r}')$ jest
+potencjałem oddziaływania kulombowskiego między elektronami, a $\psi(\bm{r},t)$ jest operatorem pola w obrazie Heisenberga 
+%
+\begin{equation}
+\psi(\bm{r},t)=e^{\frac{i}{\hbar}Ht}\psi(\bm{r})e^{-\frac{i}{\hbar}Ht}.
+\label{field}
+\end{equation}
+%
+Operatory pola $\psi^\dagger(\bm{r})$ i $\psi(\bm{r})$ opisują kreacje i anihilację elektronu w punkcie $\bm{r}$.
+
+Stan podstawowy $|\Phi_N\rangle$ układu $N$ oddziałujących elektronów spełnia równanie 
+%
+\begin{equation}
+H|\Phi_N\rangle=E|\Phi_N\rangle.
+\end{equation}
+%
+Dalej zakładamy, że stan podstawowy jest unormowany $\langle\Phi_N|\Phi_N\rangle=1$.
+
+W kwantowej teorii układów wielu czastek można powiązać spektrum wzbudzeń z jednocząstkowymi funkcjami Greena, 
+zwanymi również propagatorami, które definiujemy
+%
+\begin{equation} 
+iG(x,x')=\langle \Phi_N|T\psi(x)\psi^\dagger(x')|\Phi_N\rangle=\begin{cases}
+        \langle \Phi_N|\psi(x)\psi^\dagger(x')|\Phi_N\rangle & t>t',\\
+        -\langle \Phi_N|T\psi^\dagger(x')\psi(x)|\Phi_N\rangle & t'>t,
+             \end{cases}    
+\label{greenf}                      
+\end{equation}
+% 
+gdzie wprowadzono oznaczenie $x=(\bm{r},t)$, a $T$ jest operatorem uporządkowania w czasie. 
+Funkcja Greena jest amplitudą prawdopodobieństwa propagacji elektronu z punktu $x'$ do punktu $x$ dla $t>t'$ 
+lub dziury z punktu $x$ do $x'$ dla $t'>t$. Te dwa przypadki opisują odpowiednio proces fotoemisji i odwrotnej fotoemisji.
+Funkcje Greena dla elektronów są macierzami ze względu na spinowe stopnie swobody. Dla uproszczenia zapisu pomijamy
+tutaj wskaźniki numerujące stany spinowe.
+Zajomość funkcji Green pozwala wyznaczyć energię stanu podstawowego, wartości oczekiwanego jednocząstkowych operatorów
+w stanie podstawowym oraz jednoelektronowe spektrum wzbudzeń.
+Funkcje Greena spełniaja równanie ruchu
+%
+\begin{equation}
+[i\frac{\partial}{\partial t}-H_0(x))]G(x,x')-\int dx''\Sigma(x,x'')G(x'',x')=\delta(x-x'),
+\end{equation}
+%
+gdzie $H_0 =h_0+V_H$ ($V_H$ jest potencjałem Hartree), a $\Sigma$ nosi nazwę energii własnej i opisuje oddziaływania wymienno-korelacyjne.
+
+Dla układu jednorodnego o objętości $V$, funkcję Greena możemy zapisać w formie transformaty Fouriera
+%
+\begin{equation}
+G(x,x')=\sum_{\bm{k}}\int\frac{d\omega}{2\pi V}e^{i\bm{k}(\bm{r}-\bm{r}')}e^{-i\omega(t-t')}G(\bm{k},\omega).
+\label{fourier}
+\end{equation}
+%
+$G(\bm{k},\omega)$ jest funkcją Greena, która opisuje propagację elektronu o zdefiniowanym wektorze falowym $\bm{k}$.
+Rozważmy na początku układ $N$ nieodziałujących elektronów. Operatory pola można zapisać
+%
+\begin{equation}
+\psi(\bm{r})=\frac{1}{V}\sum_{\bm{k}}e^{-i\bm{k}\bm{r}}c_{\bm{k}},
+\label{free}
+\end{equation}
+%
+gdzie $c_{\bm{k}}$ jest operatorem anihilacji electronu, dla którego $c_{\bm{k}}|\Phi_N\rangle=0$.
+Odpowiednio $c^\dagger_{k}$ jest operatorem kreacji elektronu o wektorze falowym $\bm{k}$.
+Można przedefiniować operatory kreacji i anihilacji wprowadzając osobne oznaczenia dla cząstek i dziur
+%
+\begin{equation}
+c_{\bm{k}}=\begin{cases}
+a_{\bm{k}} & k>k_F,\\
+b^\dagger_{-\bm{k}} & k<k_F,\end{cases}
+\end{equation}
+%
+gdzie $k_F$ jest wektorem falowym Fermiego. Hamiltonian w tej reprezentacji przyjmuje postać
+%
+\begin{equation}
+H_0=\sum_{\bm{k}}\hbar\omega_{\bm{k}} c^\dagger_{\bm{k}}c_{\bm{k}}=\sum_{\bm{k}>\bm{k}_F}\hbar\omega_{\bm{k}} a^\dagger_{\bm{k}}a_{\bm{k}}-\sum_{\bm{k}<\bm{k}_F}\hbar\omega_{\bm{k}} b^\dagger_{\bm{k}}b_{\bm{k}}+\sum_{\bm{k}<\bm{k}_F}\hbar\omega_{\bm{k}}.
+\end{equation}
+%
+Wykorzystując wzory (\ref{free}) i (\ref{field}) otrzymujemy zależność dla funkcji Greena swobodnych elektronów
+%
+\begin{equation}
+iG_0(x,x')=\frac{1}{(2\pi)^3}\int d^3k e^{i\bm{k}(\bm{r}-\bm{r}')}e^{-i\omega_{\bm{k}}(t-t')}[\theta(t-t')\theta(k-k_F)-\theta(t'-t)\theta(k_F-k)].
+\end{equation}
+%
+gdzie sumowanie po stanach $\bm{k}$ zostało zamienione na całkę.
+Funkcję schodkową $\theta(t-t')$  można zapisać przy pomocy wzoru
+%
+\begin{equation}
+\theta(t-t')=-\int_{-\infty}^{\infty}\frac{d\omega}{2\pi i}\frac{e^{-i\omega(t-t')}}{\omega+i\eta}=\begin{cases}1 & t>t',\\0 & t<t',\end{cases}
+\end{equation}
+% 
+gdzie $\eta$ jest infinitezymalnie małą liczbą. Zastosowanie tego wyrażenie po prostych przekształceniach
+prowadzi do wzoru
+%
+\begin{equation}
+G_0(x,x')=\frac{1}{(2\pi)^4}\int d^3k e^{i\bm{k}(\bm{r}-\bm{r}')}e^{-i\omega(t-t')}\Big{[}\frac{\theta(k-k_F)}{\omega-\omega_{\bm{k}}+i\eta}+\frac{\theta(k_F-k)}{\omega-\omega_{\bm{k}}-i\eta}\Big{]}.
+\end{equation}
+%
+Porównujące ten wzór z (\ref{fourier}) dostajemy wyrażenie na funkcję Greena swobodnych elektronów
+w przestrzeni pędów
+%
+\begin{equation}
+G_0(\bm{k},\omega)=\frac{\theta(k-k_F)}{\omega-\omega_{\bm{k}}+i\eta}+\frac{\theta(k_F-k)}{\omega-\omega_{\bm{k}}-i\eta},
+\label{G0}
+\end{equation}
+%
+Występujące w tym wzorze dwa wyrazy odpowiadają odpowiednio propagacji elektronu dla wektora falowego $k$
+większego do wektor falowego Fermiego $k_F$ i dziury dla $k<k_F$. 
+
+Funckję Greena dla oddziałujących elektronów można zapisać w postaci szeregu perturbacyjnego
+%
+\begin{equation}
+G(\bm{k},\omega)=G_0(\bm{k},\omega)+G_0(\bm{k},\omega)\Sigma_{\bm{k}}(\omega)G_0(\bm{k},\omega)+G_0(\bm{k},\omega)\Sigma_{\bm{k}}(\omega) G_0(\bm{k},\omega)\Sigma_{\bm{k}}(\omega) G_0(\bm{k},\omega)+...
+\end{equation}
+%
+gdzie $\Sigma_{\bm{k}}(\omega)$ jest transformatą Fouriera energii własnej. 
+Szereg ten można zapisać w formie równania Dysona
+%
+\begin{equation}
+G(\bm{k},\omega)=G_0(\bm{k},\omega)+G_0(\bm{k},\omega)\Sigma_{\bm{k}}(\omega)G(\bm{k},\omega),
+\end{equation}
+%
+lub po przekształceniu
+%
+\begin{equation}
+\Sigma_{\bm{k}}(\omega)=G_0^{-1}(\bm{k},\omega)-G^{-1}(\bm{k},\omega).
+\end{equation}
+%
+Łącząc ten wzór z wyrażeniem (\ref{G0}) dla swobodnych elektronów ($k>k_F$)
+dostajemu wzór na funkcję Greena
+%
+\begin{equation}
+G(\bm{k},\omega)=\frac{1}{\omega-\omega_{\bm{k}}-\Sigma_{\bm{k}}(\omega)}.
+\end{equation}
+%
+Z tego wzoru wynika, że jednocząstkowe stany o energii $\omega_{\bm{k}}$ są modyfikowane zarówno
+przez rzeczywistą część energii własnej, która przesuwa wartości energii, jak również przez
+jej część urojoną. 
+
+Znając funkcję Greena można wyznaczyć funkcję spektralną, która opisuje widmo energetyczne elektronów w funkcji wektora falowego,
+wyznaczane w pomiarach metodą kątowo-rozdzielczej spektroskopii fotoemisyjnej (ARPES).
+Funkcja spektralna wyznaczana jest części urojonej funkcji Greena
+%
+\begin{equation}
+A_{\bm{k}}(\omega)=\frac{1}{\pi}|\text{Im} G_{\bm{k}}(\omega)|=\frac{1}{\pi}\frac{|\text{Im} \Sigma_{\bm{k}}(\omega)|}{[\omega-\omega_{\bm{k}}-\text{Re} \Sigma_{\bm{k}}(\omega)]^2+[\text{Im} \Sigma_{\bm{k}}(\omega)]^2}.
+\end{equation}
+%
+Dla układu nieoddziałujących elektronów ($\Sigma=0$) funkcja spektralna składa się z delt Diraca w położeniach,
+które odpowiadają energiom $\omega_{\bm{k}}$.
+Część rzeczywista energii własnej powoduje przesunięcie energii elektronów
+%
+\begin{equation}
+\varepsilon_{\bm{k}}=\omega_{\bm{k}}+\Sigma_{\bm{k}}(\varepsilon_{\bm{k}}),
+\label{e_k}
+\end{equation}
+%
+natomiast część urojona powoduje poszerzenie pików, co związane jest z redukcją czasu życia stanów elektronowych.
+Jeżeli część urojona powoduje tylko niewielkie zaburzenie, istnieje jednoznaczna relacja między stanami
+elektronów swobodnych i elektronów oddziałujących. Takie stany elektronowe lub dziurowe, które występują blisko energii Fermiego nazywamy {\it kwasicząstkami}.
+Przy zbliżaniu się do poziomu Fermiego czas życia stanów elektronów rośnie ze względu na zmniejszajacą się ilość dostępnych stanów do których elektron
+może się rozproszyć. 
+
+Blisko energii Fermiego część rzeczywista energii wałsnej zależy liniowo od energii i można ją rozwinąć do liniowego wyrazu
+%
+\begin{equation}
+\text{Re} \Sigma_{\bm{k}}(\omega)=\text{Re} \Sigma_{\bm{k}}(\omega_{\bm{k}})+(\omega-\omega_{\bm{k}})\frac{\partial\text{Re}\Sigma_{\bm{k}}(\omega)}{\partial\omega}\Big{|}_{\omega=\omega_{\bm{k}}}.
+\end{equation}
+%
+Wprowadzając czynnik renormalizacyjny, który jest miarą charakteryzującą spektrum kwasicząstek
+%
+\begin{equation}
+Z_{\bm{k}}=\frac{1}{1-\frac{\partial\text{Re}\Sigma_{\bm{k}}(\omega)}{\partial\omega}\Big{|}_{\omega=\omega_{\bm{k}}}}
+\end{equation}
+%
+możemy zapisać wzór na energię (\ref{e_k}) w postaci
+%
+\begin{equation}
+\varepsilon_{\bm{k}}=\omega_{\bm{k}}+Z_{\bm{k}}\text{Re}\Sigma_{\bm{k}}(\omega_{\bm{k}}).
+\end{equation}
+%
+Również gęstość spektralną kwasicząstek można przybliżyć wzorem
+%
+\begin{equation}
+A_{\bm{k}}(\omega)=Z_{\bm{k}}\frac{Z_{\bm{k}}\text{Im}\Sigma_{\bm{k}}(\varepsilon_{\bm{k}})}{(\omega-\varepsilon_{\bm{k}})^2+[Z_{\bm{k}}\text{Im}\Sigma_{\bm{k}}(\varepsilon_{\bm{k}})]^2},
+\end{equation}
+% 
+która ma kształt linii Lorentza w położeniu $\varepsilon_{\bm{k}}$, szerokości $Z_{\bm{k}}\text{Im}\Sigma_{\bm{k}}(\varepsilon_{\bm{k}})$
+i amplitudzie $Z_{\bm{k}}$, nazywanej również wagą sprektralna kwasiczastek. $Z_{\bm{k}}$ określa m.in. natężenie (amplitudę) rozpraszania fotoemisyjnego,
+zmianę masy efektywnej kwazicząstek, jak również skok w obsadzeniu stanów na powierzchni Fermiego. 
+Dla układu, w którym część rzeczywista energii własnej nie zależy od energii $Z_{\bm{k}}=1$, ta nieciągłość pokrywa się z rozkładem Fermiego-Diraca dla $T=0$.
+Energie kwasiczastek, które są funkcjami wektorów falowych, określają strukturę pasmową materiału.
+W układach silnie skorelowanych część gęstości spektralnej przekazywana jest do stanów leżących dalej od poziomu Fermiego
+i pojawiają się dodatkowe struktury (satelity), których nie da się zinterpretować jako wzbudzenia kwasicząstek.
+
+\section{Teoria perturbacyjna}
+
+Podstawowe podejście stosowane w teorii pola opiera się na rozwinięciu perturbacyjnym funkcji Greena~\cite{AGD}.
+Najwygodniejszym jest tutaj obraz oddziaływania, w którym hamiltonian zapisany jest w formie
+%
+\begin{equation}
+H(t)=H_0+V_\text{I}(t),
+\end{equation}
+%
+gdzie pierwszy wyraz opisuje układ swobodnych elektronów, a drugi jest zależnym od czasu operatorem oddziaływania
+%
+\begin{equation}
+V_\text{I}(t)=\frac{1}{2}\int d\bm{r}\int d\bm{r}'\psi_\text{I}^\dagger(\bm{r},t)\psi_\text{I}^\dagger(\bm{r}',t)v(\bm{r},\bm{r}')
+\psi_\text{I}(\bm{r}',t)\psi_\text{I}(\bm{r},t).
+\end{equation}
+%
+Występujace w tym wyrażeniu operatory pola otrzymujemy przez transformację
+%
+\begin{equation}
+\psi_\text{I}(\bm{r},t)=e^{iH_0(t-t_0)}\psi(\bm{r})e^{-iH_0(t-t_0)}.
+\end{equation}
+%
+W obrazie oddziaływania, ewolucja stanów opisana jest zależnością
+%
+\begin{equation}
+|t\rangle=U_\text{I}(t,t_0)|t_0\rangle,
+\end{equation}
+gdzie operator ewolucji może być zapisany w postaci szeregu
+%
+\begin{equation}
+U_\text{I}=1+\sum_{n=1}^{\infty}\frac{(-1)^n}{n!}\int_{t_0}^tdt_1...\int_{t_0}^tdt_nT\Big{[}V_\text{I}(t_1)...V_\text{I}(t_n)\Big{]}.
+\end{equation}
+%
+Wykorzystując ten operator i wprowadzając oznaczenie $S=U_\text{I}(\infty,-\infty)$
+możemy zapisać funkcję Greena w postaci rozwinięcia
+%
+\begin{equation}
+G(x,x')=-i\frac{\langle\Phi_0|T[\psi(x)\psi^{\dagger}(x')S]|\Phi_0\rangle}{\langle\Phi_0|S|\Phi_0\rangle},
+\end{equation}
+%
+gdzie $|\Phi_0\rangle$ jest stanem podstawowym dla układu nieoddziałujących elektronów opisanego hamiltonianem $H_0$. 
+Występujące w tym szeregu wyrazy można wyrazić przy pomocy funkcji Greena i wyrazów opisujących oddziaływania kulombowskie.
+Prowadzi to do wzoru, który można zapisać w zwartej formie przy pomocy sumy wyznaczników
+%
+\begin{equation}
+G(x,x')=\sum_{n=0}^\infty i^n\int v(x_1,x_1')...v(x_n,x'_n)\begin{vmatrix}
+G_0(x,x') &  G_0(x,x_1)  & \dots & G_0(x,x'_n) \\ 
+G_0(x_1,x') & G_0(x_1,x^+_1) & \dots & G_0(x,x'_n) \\
+\dots                &  \dots                 &       &\dots \\
+G_0(x'_n,x') & G_0(x'_n,x')  & \dots & G_0(x'_n,x'^+_n) \\
+\end{vmatrix}.
+\end{equation}
+%  
+Każdy z wyrazów rozwinięcia można otrzymać stosując metodę diagramów Feynmana opisaną szczegółowo w wielu podręcznikach \cite{AGD,FW,martin2}.
+
+
+
+
+
+\section{Metoda GW}
+\label{sec:gw}
+
+Energię własną można powiązać z funkcją Greena
+%
+\begin{equation}
+\Sigma(x_1,x_2)=i\int dx_3dx_4 G(x_1,x_3)W(x_1,x_4)\Lambda(x_3,x_2,x_4).
+\end{equation}
+%
+$W$ jest ekranowanym potencjałem kulombowskim
+%
+\begin{equation}
+W(x_1,x_2)=\int dx_3\epsilon^{-1}(x_1,x_3)V(\bm{r}_3-\bm{r}_2),
+\end{equation}
+%
+\begin{equation}
+\epsilon^{-1}(x_1,x_2)=\frac{\delta V(x_1)}{\delta \phi(x_2)},
+\end{equation}
+%
+gdzie $V$ jest sumą potencjału Hartree $V_H$ i potencjału zewnętrznego $\phi$.
+$\Lambda$ jest funkcją wierzchołkową
+%
+\begin{equation}
+\Lambda(x_1,x_2,x_3) = -\frac{\delta G^{-1}(x_1)}{\delta V(x_3)}.
+\end{equation}
+%
+
+
+
+%\section{Teoria dynamicznego średniego pola (DMFT)}
+%\label{sec:dmft}
+
+%\section{Kwantowe Monte Carlo (QMC)}
+%\label{sec:qmc}
+
+%\bibliographystyle{acm}
+%\bibliography{biblio.bib}
+
+
+
+\thebibliography{500}
+
+
+\bibitem{Schrodinger} E. Schr\"{o}dinger, {\it An undulatory theory of the mechanics of atoms and molecules}, Phys. Rev. {\bf 28}, 1049 (1926).
+
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+                                          
+\end{document}
+		     
+
+
+
+
+
diff --git a/book/methods-old.tex b/book/methods-old.tex
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--- /dev/null
+++ b/book/methods-old.tex
@@ -0,0 +1,3269 @@
+\documentclass[12pt,a4paper,openany]{book}
+\usepackage{graphicx}
+\usepackage{gensymb}
+\usepackage{longtable}
+\usepackage{epsfig}% Include figure files
+\usepackage{dcolumn}% Align table columns on decimal point
+\usepackage{bm}% bold math
+\usepackage{anysize}
+\usepackage[utf8x]{inputenc}
+\usepackage[T1]{fontenc}
+%\usepackage[polish]{babel}
+\usepackage{amsmath}
+\def\mb{\bm}
+\usepackage[left=2cm,right=2cm,top=3cm,bottom=3cm]{geometry}
+\DeclareMathOperator{\Tr}{Tr}
+
+\usepackage{color}
+\usepackage[urlcolor=blue,colorlinks=true,citecolor=blue,linkcolor=blue,pdfstartview={FitH},bookmarks=false]{hyperref}
+\usepackage{ulem}
+\definecolor{myblue}{rgb}{0.0, 0.5, 0.3}
+\newcommand{\SP}[1]{\textcolor{myblue}{#1}}
+
+\begin{document}
+
+
+\begin{titlepage}
+	\centering
+	{\Large\scshape Przemys\l{}aw Piekarz\par}
+    \vspace{4cm}
+	{\scshape\bfseries\Huge Electronic structure of solids \par}
+	\vspace{1cm}
+	{\scshape\Large Introduction to ab initio methods\par}	
+	\vfill
+	{\scshape\large Department of Computational Materials Science \\ Institute of Nuclear Physics \\ Polish Academy of Sciences\par}
+\end{titlepage}
+
+\newpage
+\thispagestyle{empty}
+%\thispagestyle{plain} % empty
+\mbox{}
+
+\tableofcontents
+
+
+\chapter{Introduction}
+
+First-principles calcution methods, called also {\it ab initio} methods, are based on the laws of quantum mechanics.
+The application of the Schr\"{o}dinger equation~\cite{Schrodinger} to atoms and molecules enabled to explain the basis properties of chemical bonds~\cite{HL,hartree28,mulliken,JC}. 
+Many-electro, such as atoms, molecules and solids, are subject to Fermi-Dirac statistics~\cite{Fermi,Dirac26}, which is a direct consequence of the Pauli exclusion principle~\cite{Pauli25}.
+Application of quantum statistics to a homogeneous electron gas, which is the simplest model of the metallic state,
+made it possible to explain the problems of the classical Drude-Lorentz theory of conductivity.
+ 
+%Teoria pasmowa układów periodycznych, która powstała dzięki pracom Blocha~\cite{Bloch}, Peierlsa~\cite{Peierls} i Wilsona~\cite{Wilson}, wyjaśniła przyczyny znacznych różnic w przewodnictwie elektrycznym materiałów i stworzyła podstawy do podziału wszystkich kryształów na metale, izolatory i półprzewodniki. 
+
+\SP{The band theory of periodic systems, developed thanks to the works of Bloch~\cite{Bloch}, Peierls~\cite{Peierls}, and Wilson~\cite{Wilson}, explained the reasons for significant differences in the electrical conductivity of materials and laid the foundations for classifying all crystals into metals, insulators, and semiconductors.}
+
+
+Oddziaływania wielociałowe występujące w układach elektronowych uniemożliwiają dokładne rozwiązanie
+równania Schr\"{o}dingera i dlatego potrzebne są metody przybliżone.
+Jednym z pierwszych podejść umożliwiającym obliczenia ilościowe było przybliżenie Hartree \cite{hartree28}, 
+gdzie funkcja falowa przedstawiona jest w formie iloczynu funkcji jednocząstkowych.
+W modelu jednocząstkowym, wzajemne oddziaływanie elektronów zostaje zastąpione efektywnym potencjałem,
+w którym porusza się każdy z elektronów. Taki potencjał złożony jest z oddziaływania przyciągającego wytwarzanego przez jądra atomowe i 
+z części odpychającej pochodzącej od pozostałych elektronów.
+Ponieważ elektronowa część potencjału zależy od funkcji falowych, które chcemy wyznaczyć,
+równanie Schr\"{o}dingera musi być rozwiązywane w sposób samouzgodniony.
+Uwzględniając zakaz Pauliego do wieloelektronowej funkcji falowej, która w najprostszej postaci zapisywana
+jest w formie wyznacznika Slatera, można wyprowadzić równanie Hartree-Focka \cite{fock30,slater30,slater51}. 
+W równaniu tym, oprócz zwykłego oddziaływania kulombowskiego, pojawia się nielokalne 
+oddziaływanie wymienne. Jest to oddziaływanie występujące tylko między elektronami z tym samym kierunkiem spinu,
+powodujące efektywnie zmiejszenie odpychania kulombowskiego między elektronami.
+Podejście Hartree-Focka opisuje układy elektronowe w sposób przybliżony ponieważ nie uwzględnia poprawnie korelacji,
+które występują w wieloelektronowym stanie kwantowym \cite{wigner34,GB}.
+Wiąże się to również z niewłaściwym ekranowaniem oddziaływań kulombowskich,
+co prowadzi do znacznie zawyżonych wartości przerwy energetycznej w ciałach stałych.
+Metody rozwijane w oparciu o podejście Hartree-Focka, tzw. metody wielowyznacznikowe, uwzględniają poprawnie korelacje elektronowe, 
+ale wymagają czasochłonnych obliczeń komputerowych. Stosowane są głównie w chemii kwantowej do badania małych układów molekularnych.   
+
+W pierwszych obliczeniach struktury elektronowej metali stosowane były metody, w których niezależne elektrony oddziałują
+z periodycznym potencjałem atomowym.
+W metodzie komórkowej, zaproponowanej przez Wignera i Seitza \cite{wigner33}, potencjał kryształu jest sumą
+sferycznie symetrycznych potencjałów wytwarzanych przez każdy z atomów.
+Korzystając z twierdzenia Blocha, wystarczy znaleźć rozwiązania równania Schr\"{o}dingera
+w pojedynczej komórce prymitywnej, z odpowiednimi warunkami brzegowymi na granicy komórki.  
+W roku 1937, Slater wprowadził metodę stowarzyszonych fal płaskich. W tym podejściu cały kryształ podzielony jest
+na obszary wokół położeń atomowych (określone przez promień sfery atomowej) i pozostałą część miedzywęzłową.
+W każdym  z tych obszarów stosuje się inną postać funkcji falowej. Wewnątrz sfer atomowych potencjał zmienia się szybko i
+funkcje falowe, które zachowują się podobnie do orbitali atomowych, wyliczane są w bazie harmonic sferycznych. 
+Natomiast w obszarze międzywęzłowym, zarówno potencjał, jak i funkcje falowe są wolnozmienne, więc naturalną bazą sa fale płaskie.
+Dodatkowo musi być spelniony warunek ciągłości na granicy obszarów.
+W innym podejściu, zaproponowanym przez Korringa~\cite{K47} oraz Kohna i Rostokera~\cite{KR54},
+wykorzystuje się funkcje Greena i teorię rozpraszania do wyznaczenia stanów elektronowych w krysztale.
+Głównym mankamentem tych metod był przybliżony charakter potencjału elektronowego. 
+Najczęściej stosowano potencjał typu {\it muffin-tin}, który w obszarze rdzenia atomowego ma symetrię sferyczna,
+a poza nim przyjmuje wartość stałą.
+Krokiem naprzód było podejście zortogonalizowanych fal płaskich zaproponowane przez Herringa~\cite{herring40}, które stało się
+podstawą metody pseudopotencjałów~\cite{Antoncik,KP}.
+W tym podejściu dokładny potencjał atomowy zastępuje się przybliżonym pseudopotencjałem, mającym skończoną wartość w obszarze rdzenia atomowego,
+co umożliwia wyliczenie przybliżonej funkcji falowej (pseudofunkcji falowej) w bazie fal płaskich w całym obszarze kryształu.
+ 
+Prawdziwym przełomem było sformułowanie przez Hohenberga, Kohna i Shama teorii funkcjonału gęstości (DFT) \cite{kohn64,kohn65}.
+Podstawowe twierdzenie tej teorii mówi, że całkowita energia danego układu,
+z uwzględnieniem oddziaływania wymiennego i korelacyjnego, jest funkcjonałem
+gęstości elektronowej, która może być wyznaczona przez minimalizację energii stanu podstawowego.
+W praktycznych zastosowaniach, gęstość elektronowa wyznaczana jest z jednocząstkowych funkcji falowych, wyliczanych w sposób
+samouzgodniony poprzez rozwiazanie równań Kohna-Shama. 
+Dokładne wartości potencjału wymienno-korelacyjnego nie są znane i muszą być wyznaczane w ramach znanych przybliżeń.
+Podstawowym podejściem jest przybliżenie lokalnej gęstości (LDA), 
+w którym energia wymiany i korelacji otrzymane są z dokładnych obliczeń dla jednorodnego gazu elektronowego \cite{CeperleyAlder80,PZ,VWN}.
+Kolejnym krokiem było uwzględnienie efektów nielokalnych w ramach uogólnionego przybliżenia gradientowego
+(GGA) \cite{Langreth83,Pardew86,Becke88}.
+Od czasu sformułowania DFT, zaproponowano wiele metod rozwiązywania równań Kohna-Shama, takich jak
+metoda stowarzyszonych fal płaskich \cite{Andersen75}, czy podejścia oparte na pseudopotencjałach \cite{HSC,BHS,Vanderbilt90}.
+W ten sposób, teoria funkcjonału gęstości stała się podstawą do wyliczania struktury elektronowej dla wielu rodzajów materiałów \cite{jones,payne,martin}, 
+a za jej sformułowanie Walter Kohn otrzymał Nagrodę Nobla w dziedzinie chemii w roku 1998.
+
+Istnieją jednak układy, w których przybliżenia stosowane w DFT załamują się i nie potrafią poprawnie
+opisać struktury elektronowej. Zgodnie z klasyczną teorią pasmową, wiele tlenków metali przejściowych, takich jak NiO lub MnO,
+powinno znajdować się w stanie metalicznym, ze względu na częściowe wypełnienie stanów $3d$.
+Materiały te są w rzeczywistości izolatorami, ponieważ silne oddziaływanie kulombowskie w stanach $3d$
+uniemożliwia swobodny ruch elektronów i prowadzi do ich lokalizacji \cite{peierls}.
+W odróznieniu od izolatorów pasmowych, materiały te nazywane są izolatorami Motta \cite{mott}.
+Takie lokalne oddziaływanie elektronów, opisywane w ramach modelu Hubbarda parametrem $U$ \cite{hubbard},
+jest znacznie silniejsze w stanach $d$ i $f$, niż w bardziej rozległych (zdelokalizoanych) stanach $s$ i $p$.
+Struktura elektronowa tlenków metali przejściowych, wyznaczana w ramach obliczeń LDA lub GGA,
+albo nie posiada przerwy (CoO) albo ta przerwa jest znacznie zaniżona (NiO) \cite{terakura}.
+Zbyt mała przerwa energetyczna jest także problemem w obliczeniach dla klasycznych półprzewodników,
+takich jak krzem lub german, co oznacza, że również w tych materiałach potencjał wymienno-korelacyjny nie jest
+dobrze opisany.
+
+Podstawowe przybliżenia stosowane do wyznaczenia energii wymienno-korelacyjnej 
+są źródłem błedów, które prowadzą efektywnie do samooddziaływania elektronów.
+Nawet w przypadku układu jednoelektronowego, np. atom wodoru, dodatnia energia oddziaływania elektrostatycznego (typu Hartree),
+która wyliczana jest dokładnie, nie kasuje się z ujemną energią wymienną wyznaczoną w przybliżeniu LDA lub GGA.
+Ten efekt jest szczególnie silny dla stanów $d$ lub $f$ i prowadzi do niekorzystnego wzrostu energii przy lokalizacji elektronów.
+To jest główną przyczyną niepoprawnego stanu metalicznego lub zbyt małej przerwy energetycznej otrzymywanych
+w obliczeniach dla tlenków metali przejściowych.  
+Zaproponowano metodę korekcji tego efektu (SIC) \cite{PZ}, która polega na bezpośrednim wyliczeniu
+energii samoodziaływania i odjęcia jej od funkcjonału energii całkowitej.
+Metoda ta umożliwia poprawienie struktury elektronowej, ale nie jest łatwa do zaimplementowania w obliczeniach samouzgodnionych.
+Stan elektronowy można również poprawić stosując funkcjonały hybrydowe, w których przybliżona energia wymiany jest częściowo
+zastąpiona dokładną wartością otrzymaną w ramach metody Hartree-Focka \cite{becke93}.
+Wynika to z faktu, że w przybliżeniu Hartree-Focka nie występuje efekt samoodziaływania.
+  
+Potencjał elektronowy dla zlokalizowanych stanów można również poprawić przez dodanie do funkcjonału DFT
+dodatkowego wyrazu proporcjonalnego do energi oddziaływania $U$. Można to zrobić w przybliżeniu średniego
+pola w ramach metody LDA+U \cite{anisimov}. Zastosowanie tej metody do tlenków metali przejściowych \cite{anisimov}
+lub nadprzewodników wysokotemperaturowych \cite{czyzyk} pozwoliło znacznie poprawić ich strukturę elektronową 
+i własności magnetyczne w porównaniu do standardowych obliczeń DFT.
+Również w przypadku innych złożonych materiałów, w których występują oddziaływania między spinowymi, orbitalnymi
+i sieciowymi stopniami swobody to podejście daje jakościowo lepsze wyniki \cite{orbital1,orbital2,orbital3}.
+
+W silnie skorelowanych metalach, jakimi są ziemie rzadkie i aktynowce,
+metoda LDA+U pozwala tylko częściowo uwzględnić efekty związane z lokalizacją elekronów.
+Dobrym przykładem jest kryształ plutonu, gdzie bardzo duża niezgodność wyliczanej teoretycznie
+objetości, a wartością eksperymentalną (około 35 \%) może być skorygowana przy użycie
+realistycznej wartości parametru $U$\cite{Pu-LDAU}.
+Niestety, efekty dynamiczne takie jak fluktuacja momentów magnetycznych w stanie paramagnetycznyn, czy efekt Kondo
+nie mogą być poprawnie opisane na poziomie teorii średniego pola.
+W bardziej zaawansowanej metodzie dynamicznego średniego pola (DMFT), efekty wielociałowe
+są uwzględniane przez rozwiązanie modelu domieszki Andersona w ramach dokładnej diagonalizacji
+lub kwantowego Monte Carlo \cite{Georges}. Rozwiązania zawierają informacje o energii własnej i czasie
+życia stanów elektronowych, które nie są dostępne w ramach obliczeń DFT.
+Sama metoda kwantowego Monte Carlo rozwijana jest obecnie bardzo intensywnie i stosowana jest do obliczeń
+struktury elektronowej molekół i ukladów krystalicznych.
+
+
+
+\chapter{Oddziaływania elektronowe}
+
+\section{Podstawowe własności}
+
+Wiązania międzyatomowe, dzięki którym powstają cząsteczki i ciała stałe, 
+są efektem wzajemnych oddziaływań w układzie elektronów i jąder atomowych.
+Hamiltonian, który opisuje dowolny układ atomów ma postać
+%
+\begin{equation}
+H=-\frac{\hbar^2}{2m}\sum_{i}\nabla_i^2-\sum_{i,j} \frac{Z_{j}e^2}{|\bm{r}_i-\bm{R}_j|}+\frac{1}{2}\sum_{i\neq j}\frac{e^2}{|\bm{r}_i-\bm{r}_j|}
+-\sum_j \frac{\hbar^2}{2M_j}\nabla_j^2 -\frac{1}{2}\sum_{i\neq j} \frac{Z_iZ_je^2}{|\bm{R}_i-\bm{R}_j|},
+\label{hamiltonian}
+\end{equation}
+%
+gdzie $\bm{r}_i$ określają położenia elektronów, $\bm{R_j}$ położenia jąder atomowych, $Z_j$ ładunki jąder, $M_j$ masy jąder i $m$ jest masą elektronu.
+Pierwsze trzy wyrazy opisują odpowiednio energię kinetyczną elektronów, oddziaływanie elektronów z jądrami i oddziaływania
+między elektronami. Ostatnie dwa wyrazy to energia kinetyczna jąder atomowych i oddziaływanie między
+ładunkami jąder. Pełna funkcja falowa złożonego układu elektronów i jąder atomowych $\Psi(\bm{r}_i,\bm{R}_j)$ spełnia równanie Schr\"{o}dingera
+%
+\begin{equation}
+H\Psi(\bm{r}_i,\bm{R}_j) = E\Psi(\bm{r}_i,\bm{R}_j),
+\label{Sch}
+\end{equation}
+%
+gdzie $E$ jest całkowitą energią układu.
+
+Kwantowo-mechaniczny opis kryształu można uprościć wykorzystując duży stosunek masy jadrą atomowego do masy elektronu,
+który dla najlżejszego jądra wodoru wynosi $m_p/m_e=1836$.
+Powoduje to, że energia kinetyczna jąder jest znacznie mniejsza niż energia kinetyczna elektronów i może być pominięta
+przy wyznaczaniu funkcji falowej podukładu elektronowego $\Phi(\bm{r_i},\bm{R_j})$.
+Podejście to nazywa się przybliżeniem adiabatycznym lub przybliżeniem Borna-Oppenheimera \cite{BO}.
+Wprowadzając funkcję falową całego układu w postaci iloczynu funkcji elektronowej i funkcji jąder atomowych $\chi(\bm{R}_j)$
+%
+\begin{equation}
+\Psi(\bm{r}_i,\bm{R}_j) = \Phi(\bm{r}_i,\bm{R}_j)\chi(\bm{R}_j)
+\label{wave}
+\end{equation}
+%
+można rozseparować (\ref{Sch}) na dwa równania 
+%
+\begin{equation}
+(-\frac{\hbar^2}{2m}\sum_{i=1}^N \nabla_i^2-\sum_{i,j} \frac{Z_{j}e^2}{|\bm{r}_i-\bm{R}_j|}+\frac{1}{2}\sum_{i\neq j}
+\frac{e^2}{|\bm{r}_i-\bm{r}_j|})\Phi(\bm{r}_i,\bm{R}_j)=E_n(\bm{R}_j)\Phi(\bm{r}_i,\bm{R}_j),
+\label{Sch-el}
+\end{equation}
+%
+\begin{equation}
+(-\sum_j \frac{\hbar^2}{2M_j} \nabla_j^2 -\sum_{i,j} \frac{Z_iZ_je^2}{|\bm{R}_i-\bm{R}_j|}+E_n(\bm{R}_j))\chi_{n\alpha}(\bm{R}_j)=\varepsilon_{n\alpha}\chi_{n\alpha}(\bm{R}_j).
+\end{equation}
+%
+Pierwsze równanie opisuje funkcje falowe i energie własne $E_n$ układu elektronowego przy ustalonych położeniach jąder atomowych, gdzie $n$ oznacza zdefiniowany zbiór liczb kwantowych stanu elektronowego. Z drugiego równania możemy otrzymać funkcje falowe i energie własne $\varepsilon_{n\alpha}$ związane z ruchem jąder atomowych,
+gdzie $\alpha$ jest kwantową liczbą, która charakteryzuje te wielkości. Energia potencjalna w równaniu opisującym ruch jąder atomowych zależy od wzajemnego oddziaływania między jądrami i od energii podukładu elektronowego $E_n(\bm{R}_j)$. Obie wielkości są funkcją aktualnego położenia wszystkich jąder atomowych.  
+
+Zapiszmy funkcję falową układu $N$ elektronów, przy zadanych położeniach jader atomowych, w formie $\Phi(\bm{r}_1,\bm{r}_2,...,\bm{r}_N)$. 
+Znajomość funcji falowej pozwala wyznaczyć wiele podstawowych wielkości fizycznych dla danego układu. W szczególności gęstość elektronowa w punkcie $\bm{r}$
+dana jest wzorem
+%
+\begin{equation}
+n(\bm{r})=N\int d\bm{r}_2...d\bm{r}_N \Phi^*(\bm{r},\bm{r}_2,...,\bm{r}_N) \Phi(\bm{r},\bm{r}_2,...,\bm{r}_N).
+\end{equation}
+%
+Podstawową własnością elektronowej funkcji falowej jest jej antysymetryczność. Oznacza, że zamiana miejscami dwóch elektronów powoduje zmianę jej znaku
+%
+\begin{equation}
+\Phi(\bm{r}_1,\bm{r}_2,...,\bm{r}_i,...,\bm{r}_j,...,\bm{r}_N)=-\Phi(\bm{r}_1,\bm{r}_2,...,\bm{r}_j,...,\bm{r}_i,...,\bm{r}_N).
+\label{anty}
+\end{equation}
+%
+Ta własność wynika z zakazu Pauliego, który mówi, że dwa fermiony (czyli cząstki o spinie połówkowym) nie mogą znajdować się w tym samym stanie kwantowym - w stanie opisanym takim samym zestawem liczb kwantowych.
+
+Równanie Schr\"{o}dingera (\ref{Sch-el}) zawiera oddziaływania wielociałowe i nie posiada dokładnych rozwiązań analitycznych dla dwóch lub większej ilości elektronów/
+Konieczne jest zastosowanie metod przybliżonych, które pozwalają wyznaczyć funkcje falowe i energie stanów elektronowych. 
+Jako przykład rozważmy prosty układ z dwoma elektronami, jakim jest cząsteczka wodoru H$_2$.
+Hamiltonian tego układu, po uwzlędnieniu przybliżenia Borna-Oppenheimera, przyjmuje postać
+%
+\begin{equation}
+H=-\frac{\hbar^2}{2m}\nabla_1^2-\frac{\hbar^2}{2m}\nabla_2^2-\frac{e^2}{r_{1A}}-\frac{e^2}{r_{1B}}-\frac{e^2}{r_{2A}}-\frac{e^2}{r_{2B}}+\frac{e^2}{r_{12}}+\frac{e^2}{r_{AB}},
+\end{equation}
+%  
+gdzie $r_{1A}$, $r_{1B}$, $r_{2A}$, $r_{1B}$ oznaczają odległości elektronów (1 i 2) od dwóch protonów ($A$ i $B$), $r_{12}$ jest odległością między elektronami
+i $r_{AB}$ odległością miedzy protonami. 
+W roku 1927, Heitler i London zaproponowali funkcję falową w formie \cite{HL}
+%
+\begin{equation}
+\Phi(\bm{r}_1,\bm{r}_2)=N_{\pm}[\psi_A(\bm{r}_1)\psi_B(\bm{r}_2)\pm \psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]\chi_{\sigma},
+\end{equation}
+%
+gdzie $N_{\pm}$ jest czynnikiem normalizacyjnym, $\chi_{\sigma}$ jest częścią spinową funkcji falowej, a cztery funkcje $\psi_{\alpha}(\bm{r})$
+sa orbitalami $1s$ dla stanu podstawowego atomu wodoru
+%
+\begin{equation}
+\psi_{\alpha}(\bm{r})=\frac{1}{\sqrt{\pi a_0^3}}e^{-\frac{|\bm{r}-\bm{r}_{\alpha}|}{a_0}},
+\end{equation}
+gdzie $a_0$ jest promieniem Bohra, a $\bm{r}_{\alpha}$ jest położeniem protonu $\alpha=A$ lub $B$.
+Biorąc pod uwagę warunek antysymetryczności (\ref{anty}), funkcja falowa przyjmuje jedną z dopuszczalnych postaci
+%
+\begin{equation}
+\Phi_S(\bm{r}_1,\bm{r}_2)=N_{+}[\psi_A(\bm{r}_1)\psi_B(\bm{r}_2)+\psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]\frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle),
+\end{equation}
+%
+dla całkowitego spinu $S=0$ (stan singletowy), oraz
+%
+\begin{equation}
+\Phi_T(\bm{r}_1,\bm{r}_2)=\begin{cases}
+N_{-}[\psi_A(\bm{r}_1)\psi_B(\bm{r}_2)-\psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]|\uparrow\uparrow\rangle, \\
+N_{-}[\psi_A(\bm{r}_1)\psi_B(\bm{r}_2)-\psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]\frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle), \\
+N_{-}[\psi_A(\bm{r}_1)\phi_B(\bm{r}_2)-\psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]|\downarrow\downarrow\rangle, 
+\end{cases}
+\end{equation}
+%
+dla spinu $S=1$ (stan trypletowy). Stanem podstawowym cząsteczki wodoru jest stan singletowy, którego energia
+$E_S$ jest niższa od energii stanu stanu trypletowego $E_T$ dla każdej odległości między protonami $r_{AB}$.
+Stan związany odpowiada minimum energetycznemu $E_S=\langle\Phi_S|H|\Phi_S\rangle$, które dostajemy dla $r_{AB}=0.87$ \AA. Ta odległość jest większa od wartości eksperymentalnej równej 0.74 \AA. Natomiast wyliczona energia dysocjacji cząsteczki na dwa atomy wodoru wynosi $E_d=3.14$ eV i jest mniejsza od energii zmierzonej 4.75 eV. Jest to przykład wiązania kowalencyjnego, w którym dwa elektrony o przeciwnych spinach zostają uwspólnione, co prowadzi
+do obniżenia energii całego układu w porównaniu do sumy energii dwóch osobnych atomów.  
+W odróżnieniu od stanu singletowego, stan trypletowy nie tworzy stanu związanego dwóch atomów wodoru.
+
+Dokładniejsze wartości $r_{AB}=0.74$ \AA\ i $E_d=3.63$ eV otrzymuje się w przybliżeniu Hartree-Focka, które będzie tematem następnego rozdziału.
+Najlepsze wyniki uzyskuje się metodą wariacyjną, w której funkcja falowa zapisana jest w ogólnej formie iloczynu symetrycznej funkcji przestrzennej i antysymetrycznej funkcji spinowej \cite{JC}
+%
+\begin{equation}
+\Phi_S(\bm{r}_1,\bm{r}_2)=\Psi(\bm{r}_1,\bm{r}_2)\chi_0.
+\end{equation} 
+%
+Część przestrzenna funkcji falowej zależy od $M$ parametrów wariacyjnych, $p_1$, $p_2$,..., $p_M$.
+Minimalizując energię układu dla $M=13$ otrzymuje się wartości $E_d=4.70$ eV i $r_{AB}=4.74$ \AA, w bardzo dobrej zgodności z danymi eksperymentalnymi \cite{JC}.   
+
+
+\section{Równanie Hartree-Focka}
+
+Jedną z pierwszych metod zostosowaną do układów molekularnych jest przybliżenie Hartree-Focka \cite{hartree28,fock30,slater30}, w którym funkcja falowa dla $N$ elektronów ma postać wyznacznika Slatera
+%
+\begin{equation}
+\Phi = \frac{1}{\sqrt{N!}} 
+\begin{vmatrix}
+\phi_1(\bm{r}_1,\sigma_1) &  \phi_2(\bm{r}_1,\sigma_1)  & \dots & \phi_N(\bm{r}_1,\sigma_1) \\ 
+\phi_1(\bm{r}_2,\sigma_2) &  \phi_2(\bm{r}_2,\sigma_2)  & \dots & \phi_N(\bm{r}_2,\sigma_2) \\
+\dots                &  \dots                 &       &\dots \\
+\phi_1(\bm{r}_N,\sigma_N) &  \phi_2(\bm{r_N},\sigma_N)  & \dots & \phi_N(\bm{r}_N,\sigma_N) \\
+\end{vmatrix},
+\label{slater}
+\end{equation}
+%
+gdzie funkcje jednoelektronowe są iloczynem części zależnej od położenia $\bm{r}_i$
+i od spinu $\sigma_i$
+%
+\begin{equation}
+\phi_i(\bm{r}_i,\sigma_i)=\psi_i^{\sigma_i}(\bm{r}_i)\xi_i(\sigma_i).
+\end{equation}
+%
+Funkcja falowa zapisana w postaci wyznacznika Slater spełnia podstawowe
+własności układu nieodróżnialnych cząstek.
+Zamiana miejscami dwóch elektronów odpowiada zamianie dwóch kolumn w wyznaczniku, co powoduje
+zmianę znaku funkcji falowej zgodnie z warunkiem antysymetryczności.
+Jeżeli dwa elektrony znajdują się w tym samym stanie kwantowym to dwie kolumny
+są jednakowe i cały wyznacznik się zeruje, co zgodne jest z zakazem Pauliego. 
+
+Rozwiązanie równania Schr\"{o}dingera odpowiada funkcji falowej, dla której średnia
+z hamiltonianu przyjmuje wartość minimalną. Przy założeniu ortonormalności funkcji $\Phi$
+odpowiada to zasadzie wariacyjnej w postaci
+%
+\begin{equation}
+\delta(\langle\Phi|H|\Phi\rangle - E\langle\Phi|\Phi\rangle)=0,
+\label{var}
+\end{equation}
+%
+gdzie symbol $\delta$ oznacza pochodną wariacyjną.
+Wyliczając średnią z hamiltonianu (\ref{hamiltonian}) dla funkcji (\ref{slater}) otrzymujemy
+%
+\begin{equation}
+\langle\Phi|H|\Phi\rangle=\sum_{i,\sigma}\int d\mb{r} \psi_i^{\sigma*}(\mb{r})[-\frac{\hbar^2}{2m}\nabla_i^2+V_Z(\mb{r})]\psi_i^{\sigma}(\mb{r}) 
++E_H+E_x,
+\label{E_HF}
+\end{equation}
+%
+gdzie $V_Z$ jest potencjałem wytwarzanym przez jądra atomowe, a $E_H$ nazywa się energią Hartree 
+%
+\begin{equation}
+E_H=\frac{e^2}{2}\sum_{i,j,\sigma,\sigma'}
+\int d\mb{r} \int d\mb{r}' \frac{\psi_i^{\sigma*}(\mb{r})\psi_j^{\sigma'*}(\mb{r}')\psi_i^{\sigma}(\mb{r})\psi_j^{\sigma'}(\mb{r}')}{|\mb{r}-\mb{r}'|}.
+\end{equation}
+%
+Wprowadzając gęstość ładunkową w punkcie $\mb{r}$
+%
+\begin{equation}
+n(\mb{r})=e\sum_{i,\sigma} |\psi_i^{\sigma}(\mb{r})|^2,
+\label{dens}
+\end{equation}
+%
+możemy zapisać energię Hartree w prostszej formie
+%
+\begin{equation}
+E_H=\frac{1}{2} \int d\mb{r} \int d\mb{r}' \frac{n(\mb{r})n(\mb{r}')}{|\mb{r}-\mb{r}'|}, 
+\label{Hartree}
+\end{equation}
+%
+która odpowiada klasycznej energii oddziaływania kulombowskiego przy zadanym rozkładzie gęstości ładunku elektrycznego. 
+%
+$E_x$ nazywa się energią wymiany i wynosi
+%
+\begin{equation}
+E_x=-\frac{e^2}{2}\sum_{i,j,\sigma,\sigma'}
+\int d\mb{r} \int d\mb{r}' \frac{\psi_i^{\sigma*}(\mb{r})\psi_j^{\sigma*}(\mb{r}')\psi_i^{\sigma}(\mb{r}')\psi_j^{\sigma}(\mb{r})}{|\mb{r}-\mb{r}'|}.
+\label{exc}
+\end{equation}
+%
+Energia wymiany jest wielkością kwantową, która nie ma odpowiednika w fizyce klasycznej. 
+Oddziaływanie wymianne charakteryzują dwie ważne cechy, które są ze sobą ściśle powiązane. 
+Przede wszystkim, oddziaływanie to dotyczy tylko elektronów o takim samym kierunku spinu. 
+Zgodnie z zasadą Pauliego dwa elektrony o tym samym kierunku spinu nie moga mieć takich samych
+pozostałych liczb kwantowych, co wiązałoby się z obsadzeniem tego samego stanu kwantowego. 
+Efektywnie prowadzi to do powiększenia odległości między takimi elektronami i
+obniżenia energii oddziaływania kulombowskiego. Z tego wynika druga cecha
+oddziaływania wymiennego: wartość ujemna energii tego oddziaływania. Można interpretować oddziaływanie wymienne
+jako oddziaływanie kulombowskie elektronu z ładunkiem dodatnim tzw. {\it dziurą wymienną} ({\it ang. exchange hole}),
+która powoduje redukcję gestości ładunku ujemnego wokół każdego elektronu. 
+
+Po zastosowaniu metody wariacyjnej do (\ref{E_HF}) otrzymujemy równanie Hartree-Focka
+%
+\begin{equation}
+[-\frac{\hbar^2}{2m}\nabla_i^2+V_Z(\mb{r})+V_H(\mb{r})]\psi_i^{\sigma}(\mb{r})
+-\frac{e^2}{2}\sum_{j,\sigma}\int d\mb{r}' \frac{\psi_j^{\sigma*}(\mb{r}')\psi_i^{\sigma}(\mb{r}')}{|\mb{r}-\mb{r}'|}\psi_j^{\sigma}(\mb{r})=E\psi_i^{\sigma}(\mb{r}),
+\label{HF}
+\end{equation} 
+%
+gdzie $V_H$ oznacza potencjał Hartree
+%
+\begin{equation}
+V_H(\mb{r})=\int d\mb{r}' \frac{n(\mb{r}')}{|\mb{r}-\mb{r}'|}.
+\end{equation}
+%
+Równanie Hartree-Focka jest równaniem nieliniowymi ponieważ w ostatnim wyrazie Hamiltonianu występuje poszukiwana funkcja falowa.
+Slater zaproponował inną postać równań (\ref{HF}), która
+pozwala lepiej zrozumieć charakter oddziaływania wymiennego \cite{slater51}.
+Mnożąc i dzieląc wyraz wymienny przez $\psi_i^{\sigma}(r)$ otrzymujemy
+%
+\begin{equation}
+[-\frac{\hbar^2}{2m}\nabla_i^2+V_Z(\mb{r})+V_H(\mb{r})
+-\frac{e}{2}\int d\mb{r}' \frac{n(\mb{r},\mb{r}')}{|\mb{r}-\mb{r}'|}]\psi_i^{\sigma}(\mb{r})=E\psi_i^{\sigma}(\mb{r}),
+\label{HFS}
+\end{equation}
+%
+gdzie $n(\mb{r},\mb{r}')$ można interpretować jako gęstość ładunku wymiennego 
+%
+\begin{equation}
+n(\mb{r},\mb{r}')= e\sum_{j,\sigma}\frac{\psi_j^{\sigma*}(\mb{r}')\psi_i^{\sigma}(\mb{r}')\psi_j^{\sigma}(\mb{r})\psi_i^{\sigma*}(\mb{r})}{\psi_i^{\sigma*}(\mb{r})\psi_i^{\sigma}(\mb{r})},
+\label{EC}
+\end{equation}
+%
+która jest funkcją dwóch położeń $\mb{r}$ i $\mb{r}'$ i zależy od stanu kwantowego $i$. Całkowity ładunek dziury wymiennej
+równy jest ładunkowi pojedynczego elektronu, co łatwo pokazać wykonując całkowanie po $\mb{r}'$ i korzystaniu z ortogonalności 
+funkcji falowych
+%
+\begin{equation}
+q=e\sum_{j,\sigma}[\int d\mb{r}'\psi_j^{\sigma*}(\mb{r}')\psi_i^{\sigma}(\mb{r}')]\frac{\psi_j^{\sigma}(\mb{r})}{\psi_i^{\sigma}(\mb{r})}=
+e\sum_{j,\sigma}\delta_{ij}\frac{\psi_j^{\sigma}(\mb{r})}{\psi_i^{\sigma}(\mb{r})}=e.
+\end{equation}
+%
+Biorąc obydwa położenia równe $\bm{r}=\bm{r'}$ otrzymujemy gęstość ładunku wymiennego zgodną ze wzorem (\ref{dens}).
+Powoduje to, że człon opisujący oddziaływanie elektronu z jego własnym polem wymiennym ma taką sama postać jak oddziaływanie
+z potencjałem Hartree, ale przeciwny znak i wzajemnie się kasują. Dzięki tej własności w przybliżeniu Hartree-Focka nie występuje problem oddziaływania elektronu z jego własnym polem ({\it samooddziaływanie, ang. self-interaction}). 
+W formie (\ref{HFS}) równanie Hartree-Focka ma postać jednoelektronowego równania Schr\"{o}dingera z potencjałem wymiennym wytwarzanym
+w miejscu znajdowania się elektronu przez ładunek wymienny. 
+
+Podejście Hartree-Focka jest metodą przybliżoną, zatem wyliczona energia
+i funkcje falowe różnią się od wielkości dokładnych, które odpowiadają rozwiązaniom równania
+Schr\"{o}dingera dla funkcji wieloelektronowej.
+Uwzglednienie korelacji w wieloelektronowej funkcji falowej umożliwia
+dodatkowe obniżenie energii w porównaniu do energii Hartree-Focka.
+Różnica między dokładną wartością energii ($E_{\rm{exact}}$) i energią wyliczoną w przybliżeniu 
+Hartree-Focka ($E_{\rm{HF}}$), nosi nazwę energii korelacji
+%
+\begin{equation}
+E_c=E_{\rm{exact}}-E_{\rm{HF}}.
+\end{equation}
+%
+Energia korelacji wynika z faktu, że również elektrony o przeciwnych
+kierunkach spinów dążą do przestrzennej separacji i
+zmniejszenia energii oddziaływania kulombowskiego.
+Korelacje elektronowe można uwzględnić przez zastosowanie funkcji wielowyznacznikowych,
+w ramach metody oddziaływania konfiguracji ({\it ang. configuration interaction} - CI) \cite{GTO}.
+To podejście pozwala wyznaczyć bardzo dokładnie funkcję falową i energię stanu podstawowego, 
+jednak czas obliczeniowy skaluje się eksponencjalnie z rozmiarem układu,
+więc stosowane jest jedynie do małych układów molekularnych.
+
+\section{Metoda pola samouzgodnionego}
+
+Równania Hartree-Focka najczęściej rozwiązuje się numerycznie stosując metodę pola samouzgodnionego, nazywaną również przybliżeniem średniego pola.
+Przy zadanych położeniach atomów wybieramy początkowe funkcje falowe $\psi_{i0}^{\sigma}$, 
+które przykładowo mogą być orbitalami izolowanych atomów.
+Dla tych funkcji falowych wyliczamy gęstość elektronową $n(\mb{r})$ i potencjał $V(\mb{r})$ 
+w każdym punkcie przestrzeni badanego układu. 
+Dla tak wyliczonego potencjału rozwiązujemy równanie (\ref{HF}) wyznaczając zbiór jednoelektronowych funkcji falowych i energii 
+stanów elektronowych. Wyznaczone funkcje falowe stosujemy do ponownego wyliczenia gęstości elektronów
+i potencjału efektywnego. Procedurę powtarzamy do momentu, gdy otrzymane w kolejnych krokach energie i funkcje falowe
+są jednakowe lub różnią się o niewielką zadaną wielkość. Najczęściej parametrem służącym do badania zbieżności
+rachunku jest całkowita energia układu. Metoda pola samoouzgodnionego zastosowana była po raz pierwszy
+do rozwiązania równania Hartree, które możemy otrzymać z rownania Hartree-Focka po usunięciu
+oddziaływania wymiennego. Równanie Hartree odpowiada funkcji falowej w postaci
+iloczynu niezależnych funkcji jednocząstkowych, czyli opisuje zbiór niezależnych elektronów
+w efektywnym polu elektrostatycznym. Metoda pola samouzgodnionego stosowana jest obecnie w większości 
+metod obliczeniowych struktury elektronowej ciał stałych. 
+
+\section{Gaz elektronowy}
+
+Jednorodny gaz elektronowy jest dobrym przybliżeniem rzeczywistej struktury
+elektronowej prostych metali i jest często wykorzystywany w wielu podstawowych
+metodach obliczeniowych (np. w przybliżeniu lokalnej gęstości).
+Najprostszy model gazu elektronowego składa się z nieoddziałujących cząstek
+zamkniętych w sześcianie o objętości $V=L^3$. Długość elektronowej fali de'Broglie'a
+rozchodzącej się w każdym kierunku musi spełniać warunek periodyczności ($n\lambda=L$),
+co prowadzi to kwantowania wektora falowego w kierunkach $x$, $y$ i $z$
+%
+\begin{equation}
+\mb{k}=(\frac{2\pi n_x}{L},\frac{2\pi n_x}{L},\frac{2\pi n_x}{L}),
+\end{equation}
+%
+gdzie $n_x$, $n_y$, $n_z$ są liczbami całkowitymi.
+Energie i funkcje falowe dyskretnych stanów elektronowych dane są wzorami
+%
+\begin{equation}
+E_k=\frac{\hbar^2 k^2}{2m},
+\end{equation}
+%
+\begin{equation}
+\psi_k(\mb{r})=\frac{1}{\sqrt{V}}e^{i\mb{kr}},
+\label{pw}
+\end{equation}
+%
+Zakładamy, że układ nie jest spolaryzowany magnetycznie i liczba elektronów ze spinami skierowanymi 
+do góry i na dół jest jednakowa, $N_{\uparrow}=N_{\downarrow}=\frac{1}{2}N$.
+Zgodnie z zakazem Pauliego w stanie kwantowym o danym wektorze falowym $\mb{k}$ mogą
+znajdować się maksymalnie dwa elektrony o przeciwnych kierunkach spinu.
+W temperaturze $T=0$ K elektrony obsadzają kolejne stany od najniższego do
+maksymalnej wartości $E_F$ nazywanej energią Fermiego. 
+Całkowita energia układu $N$ elektronów wynosi
+%
+\begin{equation}
+E=2\sum_{k<k_F} E_k = \frac{2V}{(2\pi)^3}\int d\bm{k} \frac{\hbar^2 k^2}{2m} = \frac{V}{(2\pi)^3}\frac{4\pi\hbar^2}{5m} k_F^5, 
+\label{E}
+\end{equation}
+%
+gdzie $(2\pi)^3/V$ jest objętością w przestrzeni odwrotnej przypadającą na jeden stan elektronowy, 
+a $k_F$ jest wektorem falowym Fermiego, który jest promieniem kuli w przestrzeni odwrotnej
+zawierającej zajęte stany elektronowe. W ogólnym przypadku powierzchnia obejmująca zajęte stany elektronowe,
+nazywana powierzchnią Fermiego, może mieć dowolny kształt i składać się z wielu odrębnych części.
+Liczba elektronów znajdująca się w objętości $V$ wyraża się wzorem
+%
+\begin{equation}
+N=\frac{2V}{(2\pi)^3}\int d\bm{k}=\frac{2V}{(2\pi)^3}\frac{4\pi}{3} k_F^3.  
+\label{N}
+\end{equation}
+%
+Dzieląc (\ref{E}) przez (\ref{N}) otrzymujemy średnią energię przypadającą
+na pojedynczy elektron, wyrażoną przez energię Fermiego
+%
+\begin{equation}
+\frac{E}{N}=\frac{3}{5} \frac{\hbar^2 k_F^2}{2m} = \frac{3}{5} E_F.
+\label{EN}
+\end{equation}  
+%
+W przypadku jednorodnego gazu gęstość elektronowa dana jest wyrażeniem
+%
+\begin{equation}
+n = \frac{N}{V} = \frac{k_F^3}{3\pi^2}.
+\label{nkf}
+\end{equation}
+%
+Często używaną wielkością jest objętość kuli w przestrzeni rzeczywistej przypadająca na jeden elektron 
+%
+\begin{equation}
+V_s=\frac{4\pi}{3}r_{s}^3=\frac{1}{n},
+\end{equation}
+%
+gdzie $r_s$ jest jej promieniem
+%
+\begin{equation}
+r_s=(\frac{3}{4\pi n})^{1/3}.
+\end{equation}
+%
+
+Wyliczymy teraz energię gazu elektronowego w przybliżeniu Hartree-Focka.
+Zakładamy, że ładunek dodatni rozłożony jest jednorodnie w całej przestrzeni $V$
+i ma gęstość taką samą jak gaz elektronów. Energia oddziaływania
+elektronów z takim polem kasuje się z energią Hartree i całkowita energia
+elektronu składa się tylko z części kinetycznej i oddziaływania wymiennego.
+Energię wymiany można łatwo wyliczyć przedstawiając potencjał kulombowski
+w postaci transformaty Fouriera
+%
+\begin{equation}
+\frac{1}{|\mb{r}-\mb{r}'|}=4\pi\int \frac{d\mb{q}}{(2\pi)^3}\frac{e^{i\mb{q}(\mb{r}-\mb{r}')}}{q^2},
+\end{equation}
+%
+Wstawiając funkcję falową w postaci (\ref{pw}) do wyrażenia na energię wymiany 
+i wykonując całkowanie otrzymujemy energie stanów jednoelektronowych
+%
+\begin{equation}
+E_k=\frac{\hbar^2 k^2}{2m} - \int_{\mb{k'}<\mb{k_F}} \frac{d\bm{k'}}{(2\pi)^3} \frac{4\pi}{|\mb{k}-\mb{k'}|^2}
+=\frac{\hbar^2 k^2}{2m} - \frac{k_F}{\pi}(1+\frac{k_F^2-k^2}{2kk_F}ln|\frac{k_F+k}{k_F-k}|).
+\end{equation}
+%
+Całkowitą energię układu elektronów dostajemy sumując to wyrażenie po wektorze falowym    
+%
+\begin{equation}
+E=\sum_{k<k_F}[\frac{\hbar^2 k^2}{2m} - \frac{k_F}{\pi}(1+\frac{k_F^2-k^2}{2kk_F}ln|\frac{k_F+k}{k_F-k}|)].
+\end{equation}
+%
+Wykorzystując (\ref{EN}) i zamieniając sumowanie w drugim wyrazie na całkę dostajemy
+%
+\begin{equation}
+E=N(\frac{3}{5}E_F - \frac{3k_F}{4\pi})=N(\frac{3}{5}E_F - \frac{3(3\pi^2n)^{\frac{1}{3}}}{4\pi}),
+\end{equation}
+gdzie skorzystaliśmy ze związku (\ref{nkf}). Energia wymiany przypadająca na jeden elektron wynosi 
+%
+\begin{equation}
+\varepsilon_x=\frac{E_x}{N}=-\frac{3}{4}\Big(\frac{3}{\pi}\Big)^{\frac{1}{3}}n^\frac{1}{3}.
+\label{exchange}
+\end{equation}
+%
+Ten wzór określający zależność energii wymieny od gęstości elektronowej
+wykorzystywany jest w teorii funkcjonału gęstości w ramach przybliżenia lokalnej gęstości.
+
+Dla gazu elektronowego można wyprowadzić wyrażenie opisujące rozkład ładunku dziury wymiennej (\ref{EC}),
+który zależy tylko od odległości $r$
+%
+\begin{equation}
+g^{\sigma}_x(r)=1 - [3\frac{sin(rk_F)-rk_Fcos(rk_F)}{(rk_F)^3}]^2.
+\end{equation}
+%
+Nie jest możliwe wyrażenie energii korelacji dla gazu swobodnych elektronów w postaci 
+funkcji analitycznej gęstości $n$. Pierwszy przybliżony wzór zaproponowałem Wigner \cite{wigner34}
+%
+\begin{equation}
+\varepsilon(r_s)=-\frac{0.44}{r_s+7.8}.
+\end{equation}
+%
+W granicy małych gęstości ($r_s\rightarrow \infty$), electrony tworzą tzw. kryształ Wignera, którego energia korelacji 
+pokrywa się z energią elektrostatyczną zlokalizowanych ładunków punktowych.
+W granicy dużej gęstości ($r_s\rightarrow 0$), Gellmann i Breuckner
+wyprowadzili wzór na gęstość energii korelacji niespolaryzowanego magnetycznie
+gazu elektronów \cite{GB}
+%
+\begin{equation}
+\varepsilon_c (r_s)=0.311\ln(r_s)+r_s(A\ln(r_s)+C)-0.048+... .
+\end{equation}
+%
+Energię korelacji gazu elektronowego najdokładniej wyznaczono numerycznie
+metodą kwantowego Monte Carlo (QMC) \cite{CeperleyAlder80}. 
+
+
+
+\chapter{Stany elektronowe w krysztale}
+
+\section{Sieć krystaliczna}
+
+Kryształy zbudowane są z atomów lub molekuł ułożonych w periodyczną sieć.
+Trójwymiarową sieć krystaliczną definiujemy jako zbiór punktów, których
+położenia określone są wektorem translacji
+%
+\begin{equation}
+\bm{R}_n=n_1\bm{a}_1+n_2\bm{a}_2+n_3\bm{a}_3,
+\label{trans}
+\end{equation}
+%
+gdzie wektory $\bm{a}_i$ nazywamy podstawowymi lub prymitywnymi wektorami translacji, a $n_i$ są liczbami całkowitymi.
+Zbiór punktów zdefiniowanych tym wzorem nazywamy siecią Bravais. Podobnie można zdefiniować sieć Bravais
+dla dowolnego wymiaru $d$ wybierając odpowiednią ilość wektorów bazowych: $\bm{a}_1,\bm{a}_2,...,\bm{a}_d$. 
+Z każdym punktem sieci Bravais można związać zbiór atomów zwanych bazą. Najmniejsza przestrzeń kryształu,
+która po przesunieciu o wszystkie translacje sieciowe wypełni całą przestrzeń nazywamy komórką prymitywną. 
+Komórka prymitywna zawiera dokładnie jeden węzeł sieci Bravais i zwykle zbudowana jest na podstawowych wektorach translacji sieci.
+Istnieje specjalny rodzaj komórki prymitywnej nazywany komórką Wignera-Seitza, która jest przestrzenią zawierającą punkty przestrzeni, 
+które znajdują się bliżej danego punktu sieci niż pozostałych. 
+Często zamiast komórki prymitywnej używa się komórki elementarnej, która zbudowana jest na innych wektorach bazowych
+i jej symetria jest taka sama jak symetria całego kryształu.
+Objętość komórki elementarnej jest całkowitą wielokrotnością objętości komórki prymitywnej.
+W strukturach prostych (bez centrowania powierzchniowego lub przestrzennego), komórka elementarna pokrywa się
+z komórką prymitywną. Różnice między poszczególnymi rodzajami komórek można przedstawić na przykładzie
+sieci dwuwymiarowej. Na rysunku \ref{fig:bravais} pokazana jest struktura centrowana prostokątna z zaznaczonymi dwoma wektorami podstawowaymi $\bm{a}_p$
+i $\bm{b}_p$, które definiują komórkę prymitywną, pokazaną na rysunku w dwóch wariantach. Komórka Wignera-Seitza pokazana
+jest po prawej stronie. Komórka elementarna wyznaczona jest przez dwa wektory bazowe $\bm{a}_e$ i $\bm{b}_e$.
+W dwóch wymiarach mamy w sumie pięć rodzajów sieci Bravais: skośna, prostokątna, prostokatna centrowana, heksagonalna i kwadratowa.
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.4]{lattice.pdf}
+\caption{Dwuwymiarowa sieć periodyczna.}\label{fig:bravais}
+\end{figure}
+
+W przypadku sieci trójwymiarowych, wszystkie możliwe struktury można podzielić na siedem układów krystalograficznych: 
+regularny, tetragonalny, rombowy, heksagonalny, trygonalny, jednoskośny i trójskośny, którym odpowiada czternaście możliwych sieci Bravais,
+które podzielone są na proste (P), centrowane powierzchniowo (F), centrowane na podstawach (C) i centrowane przestrzennie (I). 
+Wszystkie układy krystalograficzne z odpowiadającymi im sieciami Bravais przedstawione są w tabeli \ref{lattice}.
+Dla każdego układu pokazana jest zalezność między stałymi sieci ($a$, $b$, $c$) oraz kątami między kierunkami ($\alpha$, $\beta$, $\gamma$),
+które definiują krawędzie komórek elementarnych. 
+
+\begin{table}[h!]
+\caption{Układy krystalograficzne.}\label{lattice}
+\begin{center}
+\begin{tabular}{|c|c|c|c|}
+\hline
+Układ & Stałe sieci & Kąty & Sieci Bravais\\
+\hline
+regularny & $a=b=c$ & $\alpha=\beta=\gamma=90^{\degree}$ &  P, F, I \\ 
+tetragonaly & $a=b\ne c$ & $\alpha=\beta=\gamma=90^{\degree}$ &  P, I \\  
+rombowy & $a\ne b\ne c$ & $\alpha=\beta=\gamma=90^{\degree}$ &  P, F, C, I \\ 
+heksagonalny &  $a=b\ne c$ & $\alpha=\beta=90^{\degree}$, $\gamma=120^{\degree}$ &  P  \\
+trygonalny & $a=b\ne c$ & $\alpha=\beta=90^{\degree}$, $\gamma=120^{\degree}$ &  P \\
+jednoskośny & $a=b\ne c$ & $\alpha=\gamma=90^{\degree}$, $\beta\ne 90^{\degree}$ & P, C \\ 
+trójskośny & $a\ne b\ne c$ & $\alpha\ne \beta\ne \gamma\ne 90^{\degree}$ & P \\ \hline
+\end{tabular}
+\end{center}
+\end{table}
+
+Zbiór operacji, który pozostawia dany kryształ niezmienionym tworzy grupą przestrzenną.
+Składa się ona z translacji sieci krystalicznej opisanych wzorem (\ref{trans}) oraz symetrii punktowych: inwersji, obrotów i odbić.
+Dodatkowo występują operacje niesymorficzne, które są połączeniem operacji punktowych i ułamkowych translacji.
+Wszystkich grup przestrzennych jest 230, z których 73 to symorficzne i 157 niesymorficzne.
+
+\section{Przestrzeń odwrotna}
+
+Strukturę elektronową materiału analizuje się najczęściej wykorzystując przestrzeń wektorów falowych $\bm{k}$,
+którą nazywamy przestrzenią odwrotną. Każdej sieci Bravais w przestrzeni rzeczywistej odpowiada sieć punktów
+w przestrzeni odwrotnej zdefiniowana wektorami
+%
+\begin{equation}
+\bm{G}_m=m_1\bm{b}_1+m_2\bm{b}_2+m_3\bm{b}_3,
+\end{equation}
+%
+gdzie $m_j$ są liczbami całkowitymi, a $\bm{b}_j$ są wektorami odwrotnymi do wektorów bazowych $\bm{a}_i$,
+czyli spełniają warunek
+%
+\begin{equation}
+\bm{a}_i \cdot \bm{b}_j = 2\pi \delta_{ij}.
+\label{orto}
+\end{equation}
+%
+Wektory sieci odwrotnej można wyznaczyć z następujących zależności
+%
+\begin{eqnarray}
+\bm{b}_1=\frac{2\pi}{\Omega}\bm{a}_2\times \bm{a}_3, \\
+\bm{b}_2=\frac{2\pi}{\Omega}\bm{a}_3\times \bm{a}_1, \\
+\bm{b}_3=\frac{2\pi}{\Omega}\bm{a}_1\times \bm{a}_2, 
+\end{eqnarray}
+%
+gdzie $\Omega=\bm{a}_1\cdot(\bm{a}_2\times \bm{a}_3)$ jest objętością komórki prymitywnej.
+
+W przestrzeni odwrotnej również definiujemy komórkę prymitywną, która najczęściej ma kształt
+komórki Wignera-Seitza i nosi nazwę strefy Brillouina ({ang. Brillouin zone} - BZ).  
+Komórka, która obejmuje punkt $\bm{K}=0$, nazywany punktem $\Gamma$, nosi nazwę pierwszej strefy Brillouina.
+Na rysunku \ref{fig:lattice} pokazane są przykładowe strefy Brillouina dla trzech sieci regularnych. 
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=1.2]{brillouin.pdf}
+\caption{Strefy Brillouina dla sieci regularnych: prostej (P), centrowanej objetościowo (I) i centrowanej przestrzennie (F).}
+\label{fig:lattice}
+\end{figure}
+
+
+\section{Twierdzenie Blocha}
+
+W najprostszym opisie elektronów w periodycznym potencjale kryształu $V(\bm{r})$, stany elektronowe opisywane są przez niezależne jednocząstkowe funkcje falowe, bedące rozwiązaniami równania Schr\"{o}dingera w postaci
+%
+\begin{equation}
+[-\frac{\hbar^2\nabla^2}{2m}+V(\bm{r})]\psi^{\sigma}_{\bm{k}j}(\bm{r})=\varepsilon_{j\sigma}(\bm{k})\psi^{\sigma}_{\bm{k}j}(\bm{r}),
+\label{RS}
+\end{equation}
+%
+gdzie $\bm{k}$ jest wektorem falowym, $\sigma$ określa spin elektronu i indeks $j$ numeruje stany własne hamiltonianu odpowiadajace energiom $\varepsilon_{j\sigma}(\bm{k})$.
+Zgodnie z twierdzeniem Blocha, funkcja falowa elektronów w periodycznym potencjale ma postać
+%
+\begin{equation}
+\psi^{\sigma}_{\bm{k}j}(\bm{r})=e^{i\bm{k}\bm{r}}u^{\sigma}_{\bm{k}j}(\bm{r}),
+\end{equation}
+%
+gdzie $u^{\sigma}_{\bm{k}j}(\bm{r})$ jest funkcją periodyczną spełniającą warunek $u^{\sigma}_{\bm{k}j}(\bm{r})=u^{\sigma}_{\bm{k}j}(\bm{r}+\bm{R}_n)$ dla
+każdego wektora traslacji $\bm{R}_n$. Można pokazać, że funkcje Blocha są wektorami własnymi
+operatora translacji
+%
+\begin{equation}
+\hat{T}_n\psi^{\sigma}_{\bm{k}j}(\bm{r})=\psi^{\sigma}_{\bm{k}j}(\bm{r}+\bm{R}_n)=e^{i\bm{k}(\bm{r}+\bm{R}_n)}u^{\sigma}_{\bm{k}j}(\bm{r}+\bm{R}_n)=e^{i\bm{k}\bm{R}_n}\psi^{\sigma}_{\bm{k}j}(\bm{r}).
+\end{equation}
+%
+Hamiltonian jest niezmienniczy ze względu na działanie operatora translacji,
+co oznacza, że obydwa operatory ze sobą komutują. Funkcje Blocha są zatem również stanami własnymi hamiltonianu,
+co dowodzi prawdziwości twierdzenia Blocha. 
+
+Funkcja Blocha określona jest dla wektora falowego $\bm{k}$, który w układach periodycznych jest dobrze zdefiniowaną liczbą kwantową.
+Wektor falowy związany jest z kwazipędem elektronu w stanie $\psi_{\bm{k}j}$
+%
+\begin{equation}
+\bm{p}=\hbar\bm{k}.
+\end{equation}
+%
+Ponieważ wszystkie własności struktury elektronowej są translacyjnie niezmiennicze 
+ze względu na przesunięcie o dowolny wektor sieci odwrotnej, również kwazipęd nie zależy od takiego przesunięcia.
+
+\section{Fale płaskie}
+
+Metody obliczeniowe struktury pasmowej różnią sie między sobą stosowanymi reprezentacjami funkcji falowej.
+Ogólnie możemy zapisać funkcję falową w formie
+%
+\begin{equation}
+\psi^{\sigma}_{\bm{k}j}(\bm{r})=\sum_mc^{\sigma}_{jm}(\bm{k})\phi_{\bm{k}m}(\bm{r}),
+\end{equation}
+% 
+gdzie $\phi_{\bm{k}m}(\bm{r})$ są funkcjami bazowymi, a $c^{\sigma}_{jm}(\bm{k})$ to współczynniki rozwinięcia.
+W krysztale, ze względu na periodyczny potencjał, naturalną bazę stanowią fale płaskie. Jeżeli cześć periodyczną funkcji
+Blocha rozwiniemy w tej bazie, funkcja falowa przyjmuje postać
+%
+\begin{equation}
+\psi^{\sigma}_{\bm{k}j}(\bm{r})=e^{i\bm{k}\bm{r}}\sum_{m} c^{\sigma}_{jm}(\bm{k})e^{i\bm{G_m}\bm{r}}=\sum_{m} c^{\sigma}_{jm}(\bm{k})e^{i(\bm{k}+\bm{G_m})\bm{r}},
+\end{equation}
+% 
+gdzie $\bm{G_m}$ są wektorami sieci odwrotnej. Równanie Schr\"{o}dingera (\ref{RS}) przetransponowane do przestrzeni odwrotnej w bazie
+fal płaskich przyjmuje postać
+%
+\begin{equation}
+\sum_{n}H_{mn} c^{\sigma}_{jn}(\bm{k})=\varepsilon_{j\sigma}(\bm{k}) c^{\sigma}_{jm},
+\label{ham_rec}
+\end{equation}
+%
+gdzie elementy macierzowe Hamiltonianu dane są wyrażeniem
+%
+\begin{equation}
+H_{mn}=\frac{\hbar^2}{2m}|\bm{k}+\bm{G_m}|^2\delta_{mm'}+V(\bm{G_m}-\bm{G_n}).
+\end{equation}
+%
+Drugi wyraz jest transformatą Fouriera potencjału elektronowego
+%
+\begin{equation}
+V(\bm{G})=\frac{1}{\Omega}\int_{\Omega}\bm{dr}V(\bm{r})e^{-i\bm{Gr}},
+\end{equation}
+%
+gdzie całkowanie przebiega po komórce prymitywnej.
+
+\section{Periodyczne warunki brzegowe}
+
+Podobnie jak dla gazu elektronowego zamkniętego w pojemniku, również dla skończonego kryształu
+możemy wprowadzić periodyczne warunki brzegowe (warunki Borna-Karmana). Załóżmy, że rozmiary kryształu
+określone są przez liczbę komórek prymitywnych w każdym kierunku kryształu $(N_1,N_2,N_3)$.
+Zatem liczby całkowite, które definiują wektory translacyjne sieci (\ref{trans})
+zmieniają się w zakresie $n_i=0,1,2,...,N_i$, gdzie $i=1,2,3$.
+Zakładając, że funkcja falowa na dwóch przeciwległych brzegach układu (w każdym z trzech kierunków) przyjmuje
+takie same wartości i korzystając z twierdzenia Blocha otrzymujemy warunek
+%
+\begin{equation}
+\psi^{\sigma}_{\bm{k}j}(\bm{r})=\psi^{\sigma}_{\bm{k}j}(\bm{r}+N_i\bm{a}_i)=e^{i\bm{k}N_i\bm{a}_i}\psi^{\sigma}_{\bm{k}j}(\bm{r}),
+\label{pwb}
+\end{equation} 
+%
+który spełniony jest gdy 
+%
+\begin{equation}
+\bm{k}N_i\bm{a}_i=2\pi n_i.
+\end{equation}
+%
+Wykorzystując bazę wektorów prymitywnych sieci odwrotnej ($\bm{b}_1,\bm{b}_2,\bm{b}_3$) oraz biorąc pod uwagę zależność (\ref{orto}), dostajemy zbiór dozwolonych wektorów falowych
+%
+\begin{equation}
+\bm{k}=\frac{n_1}{N_1}\bm{b}_1+\frac{n_2}{N_2}\bm{b}_2+\frac{n_3}{N_3}\bm{b}_3.
+\label{vectk}
+\end{equation}
+%
+Wektory te określają funkcje falowe Blocha $\psi_{i,\bm{k}}$ i energie stanów $\varepsilon_i(\bm{k})$.
+Skończona ilość dostępnych stanów wynika zatem z ograniczonych rozmiarów kryształu i jest równa ilości komórek prymitywnych w układzie: $N=N_1N_2N_3$.  
+Ze wzoru (\ref{vectk}) wynika również, że dozwolone wektory falowe stanów elektronowych należą do pojedynczej komórki prymitywnej w przestrzeni odwrotnej.
+Najczęściej do analizy stanów elektronowych wybiera się wektory należące do pierwszej strefy Brillouina.
+
+\section{Sumowanie w przestrzeni odwrotnej}
+
+Wiele wielkości fizycznych wyznaczanych jest jako sumy po punktach w przestrzeni odwrotnej. 
+Przedstawiona powyżej analiza pokazuje, że wystarczy ograniczyć
+sie do wektorów falowych należących do pierwszej strefy Brillouina.
+Dodatkowo, można wykorzystać symetrię kryształu do zmniejszenia ilości punktów, które konieczne
+są do wyznaczenia danej wielkości fizycznej. 
+Oznaczmy operacje punktowe (symorficzne) przez $O_i$, a wektory translacji ułamkowej związane z daną symetrią $O_i$
+przez $\bm{t}_i$. W przestrzeni odwrotnej rozpatrujemy jedynie symorficzne operacje symetrii ponieważ translacje
+ułamkowe nie mają wpływu na przestrzeń odwrotną. Hamiltoniana opisujący kryształ jest niezmienniczy
+względem operacji punktowych: $\bm{r}\rightarrow O_i\bm{r}+\bm{t}_i$ oraz $\bm{k}\rightarrow O_i\bm{k}$.
+Oznacza to, że funkcje falowe otrzymane po tych transformacjach 
+%
+\begin{equation}
+\psi^{\sigma}_{O_i\bm{k}j}(O_i\bm{r}+\bm{t}_i)=\psi^{\sigma}_{\bm{k}j}(\bm{r}),
+\end{equation}  
+%
+są również funkcjami własnymi hamiltonianu z tą samą energią własną $\varepsilon_{j\sigma}(\bm{k})$.
+Oznacza to, że można ograniczyć sumowanie danej wielkości fizycznej $f(\bm{k})$ do wektorów $\bm{k}$ należących do mniejszego obszaru,
+który nazywany jest nieredukowalną częścią strefy Brillouina ({\it ang. irreducible Brillouin zone} - IBZ). 
+Obszar ten tworzą wektory $\bm{k}$, które są nierównoważne, czyli nie można ich wzajemnie połączyć operacją symetrii punktowej. 
+Wartości w pozostałych punktach dostajemy przez zastosowanie odpowiednich operacji symetrii $f(O_i\bm{k})=f(\bm{k})$.
+Do sumowania wykorzystujemy wagi $w_{\bm{k}}$, które zdefiniowane są jako ilości wektorów $\bm{k}$, powiązanych symetrią punktową 
+z wektorem należącym do IBZ (z uwzglednieniem tego wektora), podzielone przez całkowitą ilość wektorów $N_{\bm{k}}$.
+Przykładowo, wartość średnią wielkości $f$, możemy wyliczyć sumując po punktach należących do IBZ
+%
+\begin{equation}
+\bar{f}=\frac{1}{N_{\bm{k}}}\sum_{\bm{k}}^{BZ}f(\bm{k})=\sum_{\bm{k}}^{IBZ}w_{\bm{k}}f(\bm{k}).
+\end{equation}
+%
+
+Przy sumowaniu po przestrzeni odwrotnej można w zasadzie wybrać dowolne punkty należące do IBZ.
+Jednak wykorzystując własności funkcji periodycznych, można tak wybrać wektory $\bm{k}$,
+aby zminimalizować błędy przy obliczaniu sum lub całek.
+Dowolną funkcję periodyczną można rozwinąć przy pomocy transformaty Fouriera
+%
+\begin{equation}
+f(\bm{k})=\sum_n f(\bm{R}_n)e^{i\bm{k}\bm{R}_n},
+\end{equation} 
+%
+gdzie $\bm{R}_n$ są wektorami translacji sieci krystalicznej.
+Zbiór punktów, który jest optymalny do wyliczania sum w przestrzeni odwrotnej, 
+nazywany jest siecią Monkhorsta-Packa \cite{MP}. Wyznacza się go ze  wzoru
+%
+\begin{equation}
+\bm{k}(n_1,n_2,n_3)=\sum_i^3 \frac{2n_i-N_i-1}{2N_i}\bm{b}_i,
+\end{equation}
+%
+gdzie liczby $N_1$, $N_2$ i $N_3$ określają ilość punktów $\bm{k}$ w każdym kierunku: $n_i=1,2,...,N_i$.
+Tak zdefiniowane punkty tworzą jednorodną sieć w przestrzeni odwrotnej i charakteryzują się
+tym, że suma wartości periodycznej funkcji, która posiada komponenty Fouriera ograniczone w każdym kierunku do $\bm{R}_n=N_i\bm{a}_i$,
+równa jest dokładnej całce z tej funkcji.  
+
+\section{Energia Fermiego}
+
+Wielkości fizyczne, które dotyczą stanu podstawowego, wyliczane są zwykle dla obsadzonych stanów elektronowych.
+Czyli sumowanie w przestrzeni odwrotnej uwzglednia tylko zajęte stany.
+Energia najwyższego obsadzonego stanu określa energię Fermiego ($E_F$).
+W metalach pasma elektronowe przecinają $E_F$ i przy sumowaniu po dyskretnych punktach $\bm{k}$ pojawiają się nieciągłości
+w obsadzeniach stanów. Powoduje to wolniejszą zbieżność ze względu na ilość punktów $\bm{k}$. 
+Można łatwo rozwiązać ten problem przez wprowadznie ciagłej funkcji obsadzania stanów w pobliżu $E_F$.
+Naturalnym wyborem jest zastosowanie funkcji rozkładu Fermiego-Diraca dla skończonej temperatury
+%
+\begin{equation}
+F(\varepsilon,E_F,T)=[\exp(\frac{\varepsilon-E_F}{kT})+1]^{-1}.
+\end{equation}   
+%
+Dla $T=0$, dostajemy funkcję schodkową, która odpowiada rozkładowi w stanie podstawowym.
+Aby wyznaczyć $E_F$ dla dowolnej temperatury, korzystamy z warunku
+%
+\begin{equation}
+\sum_{\bm{k},j} w_{\bm{k}}F(\varepsilon,E_F,T)=n_{el},
+\end{equation}   
+%
+gdzie $n_{el}$ jest liczbą elektronów, a sumowanie wykonywane jest po wszystkich obsadzonych stanach
+uwzględniając polaryzację spinową.
+
+
+
+\section{Struktura pasmowa}
+
+Zbiór wszystkich stanów elektronowych $\varepsilon_{\bm{k}j}^{\sigma}$ otrzymanych w wyniku diagonalizacji hamiltonianiu 
+dla wektorów falowych $\bm{k}$ tworzy strukturę pasmową danego materiału.
+Dostępne energie stanów elektronowych, które odpowiadają funkcji własnej z indeksem $j$, grupują się w przedziały nazywane pasmami energetycznymi. 
+Rozwiązania równania (\ref{ham_rec}) w funkcji wektora falowego $\bm{k}$ są periodyczne ze względu na przesunięcie o 
+wektor sieci odwrotnej
+%
+\begin{equation}
+\varepsilon_{\bm{k}j}^{\sigma}=\varepsilon_{\bm{k}+\bm{G_m}j}^{\sigma}.
+\end{equation}   
+%
+Zatem przy prezentacji struktury pasmowej można ograniczyć się do pierwszej strefy Brillouina. Zwykle prezentuje się tylko stany elektronowe dla wektorów $\bm{k}$ należących do wybranych kierunków i punktów wysokiej symetrii w przestrzeni odwrotnej. Jeżeli energie pasm przyjmują różne wartości dla dwóch kierunków spinu,
+$\varepsilon_{\bm{k}j}^{\uparrow}\ne \varepsilon_{\bm{k}j}^{\downarrow}$, mamy doczynienia z rozszczepieniem spinowym (wymiennym), które związane jest z uporządkowaniem magnetycznym.
+
+Ważną wielkością, która charakteryzuje strukturą pasmową jest gęstość stanów elektronowych dla danego kierunku spinu $\rho_{\sigma}(E)$,
+która wyliczana jest jako suma po pierwszej strefie Brillouina
+%
+\begin{equation}
+\rho_{\sigma}(E)=\frac{1}{N_k}\sum_{j,\bm{k}}\delta[\varepsilon_{\bm{k}j}^{\sigma}-E],
+\end{equation}
+%
+gdzie $N_k$ jest ilością wektorów $\bm{k}$ w pierwszej strefie Brillouina.
+Całkowita gęstość stanów równa jest sumie obydwóch składowych spinowych, $\rho(E)=\rho_{\uparrow}(E)+\rho_{\downarrow}(E)$.
+Przy wyliczaniu gestości stanów często zamiast sieci Monkhorsta-Packa stosuje się metodę tetraedrów.
+Polega ona na podzieleniu całej strefy Brillouina na tetraedry i wyliczeniu energii stanów w punktach $\bm{k}$, które
+odpowiadają wierzchołkom tetraedrów. Energie pomiędzy tymi punktami wyznacza się stosując interpolację liniową.
+
+Struktura pasmowa decyduje o podstawowych własnościach danego materiału i jest jego wyróżnikiem, pozwalającym zakwalifikować go do określonej grupy materiałów. 
+Trzy podstawowe typy materiałów krystalicznych to metale, półprzewodniki i izolatory.
+Struktura pasmowa metali charakteryzuję się występowaniem powierzchni Fermiego, która odgranicza zajęte stany elektronowe od stanów pustych.
+Zatem w typowych metalach gęstość stanów na poziomie Fermiego jest niezerowa, $\rho(E_F)>0$.
+Tuż powyżej energii Fermiego ($E_F$) znajdują się stany, do których przechodzą elektrony wzbudzone termicznie lub w wyniku oddziaływania z innymi
+elektronami. W półprzewodnikach i izolatorach, powyżej ostatniego zajętego stanu elektronowego, znajduje się przerwa obejmująca
+zakres zabronionych energetycznie stanów.
+Pasmo, które znajduje się tuż poniżej przerwy energetycznej nazywa się pasmem walencyjnym, a to które jest powyżej
+pasmem przewodnictwa.
+Półprzewodniki zwykle charakteryzują się mniejszą przerwa energetyczną ($\sim$1-2 eV) niż izolatory.
+Występują też materiały nazywane półmetalami, w których przerwa energetyczna i $\rho(E_F)$ są równe zeru (np. grafen).
+Odrębną grupę materiałów stanowią izolatory Motta, w których przerwa energetyczna jest wynikiem silnych oddziaływań
+elektronowych.  
+
+
+
+\chapter{Teoria funkcjonału gęstości}
+
+Większość nowoczesnych metod obliczeniowych stosowanych do badania własności materiałów opiera się na teorii funkcjonału gestości ({\it ang. density functional theory} - DFT), która została sformułowana w pracach Hohenberga i Kohna \cite{kohn64}
+oraz Kohna i Shama \cite{kohn65}. Obecnie, DFT stosowana jest powszechnie do wyliczania struktury elektronowej,
+optymalizacji sieci krystalicznej, badania własności elastycznych i dynamiki sieci oraz wielu innych własności materiałowych.
+Podobnie jak w opisie oddziaływania wymiennego zaproponowanym przez Slatera~\cite{slater51}
+i w teorii Thomasa-Fermiego~\cite{thomas,fermi,dirac} podstawową wielkością jest tutaj gęstość elektronowa zdefiniowana w każdym punkcie materiału $n(\bm{r})$.
+Główną ideą tego podejścia jest mozliwość zastąpienie dokładnej funkcji falowej, układem stanów jednocząstkowych w efektywnym potencjale elektronowym, który
+daje taką samą gęstość elektronową jak opis wieloelektronowy.
+Takie podejście daje możliwość wykorzystania dokładnej informacji o oddziaływaniach wymiennych i korelacyjnych
+otrzymanych dla jednorodnego gazu elektronowego do badania niejednorodnych układów atomowych.
+
+\section{Twierdzenia Hohenberga-Kohna}
+
+Praca Hohenberga i Kohna \cite{kohn64} zawiera dwa fundamentalne twierdzenia, które
+dotyczą relacji między gęstością elektronową $n(\bm{r})$, zewnętrznym potencjałem $V_{ext}(\bm{r})$ 
+oraz energią całkowitą układu elektronów $E[n]$, która jest funkcjonałem 
+gęstości elektronowej. Zależność funkcjonalna oznacza, że dla dowolnego rozkładu 
+elektronów w całej przestrzeni układu $n(\bm{r})$ można jednoznacznie wyznaczyć energię całkowitą 
+układu $E[n]$. Zapiszmy hamiltonian układu oddziałujących elektronów
+w ogólnej formie
+
+\begin{equation}
+H=-\frac{\hbar^2}{2m}\sum_{i}\nabla_i^2+V_{ext}(\bm{r})+\frac{1}{2}\sum_{i\neq j}\frac{e^2}{|\bm{r}_i-\bm{r}_j|}.
+\label{hamil}
+\end{equation}
+%
+Twierdzenia Hohenberga-Kohna można sformułować następująco: 
+
+{\bf Twierdzenie I:} 
+Zewnętrzny potencjał układu oddziałujących elektronów $V_{ext}$ jest jednoznacznie określony (z dokładnością do stałej wartości) 
+przez gęstość elektronową w stanie podstawowym $n_0(\mb{r})$.
+
+{\bf Twierdzenie II:}
+Dla ustalonego potencjału zewnętrznego $V_{ext}(\bm{r})$, funkcjonał energii $E[n]$ osiąga
+minimalną wartość $E_0$ dla gęstości elektronowej w stanie podstawowym $n_0(\bm{r})$.
+
+Z twierdzeń tych wynikają następujace wnioski. Ponieważ hamiltonian jest jednoznacznie określony (z dokładnościa do stałej)
+przez $n_0(\bm{r})$, funkcje falowe wszystkich stanów elektronowych,
+jak również wszystkie własności układu są całkowicie zdeterminowane przez gęstość elektronową w stanie podstawowym.
+Znajomość funkcjonału energii $E[n]$ jest wystarczająca do wyznaczenia dokładnej energii i gęstości elektronowej
+stanu podstawowego. 
+
+\section{Równanie Kohna-Shama}
+
+Twierdzenia Kohna-Shama pozwalają w sposób ścisły powiązać gestość elektronową w stanie podstawowym
+z równaniem Schr\"{o}dingera dla układu oddziałujacych elektronów, opisanego hamiltonianem (\ref{hamil}).
+W praktyce nie jest możliwe bezpośrednie rozwiązanie równania Schr\"{o}dingera i wyznaczenie wielociałowych funkcji falowych.
+Teoria funkcjonału gęstości pozwala zastąpić równanie wielociałowe nowym równaniem, nazywanym równaniem Kohna-Shama, które ma
+taką postać jak dla nieoddziałujących cząstek znajdujacych się w efektywnym polu $V_{eff}$.
+Znając jednoelektronowe rozwiązania równania Kohna-Shama $\psi_i^{\sigma}(\bm{r})$, zależne od położenia i spinu, 
+możemy wyznaczyć gęstość elektronową w każdym punkcie
+%
+\begin{equation}
+n(\bm{r})=\sum_{i,\sigma} f_{i\sigma}|\psi_i^{\sigma}(\bm{r})|^2,
+\label{density}
+\end{equation}
+%
+gdzie dla uproszczenia indeks $i$ określa zarówno numer stanu $j$, jak i wektor falowy $\bm{k}$, a $f_{i\sigma}$ są liczbami obsadzeń
+stanów, które w ogólnym przypadku mogą przyjmować ułamkowe wartości, np. zgodnie z rozkładem Fermiego-Diraca.
+Przyjmujemy również, że ładunek elementarny $e=1$, co oznacza, że gęstość ładunku jest tożsama z gęstością elektronową. 
+Twierdzenia Kohna-Shama zapewniają nam, że efektywny potencjał $V_{eff}$ jest jednoznacznie
+określony przez gęstość elektronową, jak również, że funkcjonał energii całkowitej osiąga
+minimum dla gęstości elektronowej w stanie podstawowym.
+Zatem stan podstawowy układu możemy wyznaczyć jeżeli znamy funkcjonał energii i potrafimy
+znaleźć jego minimum. W postaci ogólnej funkcjonał energii układu elektronów możemy
+zapisać 
+%
+\begin{equation}
+E[n]=T[n] + E_{ext}[n] + E_H[n] + E_{xc}[n],
+\label{EKS}
+\end{equation}
+%
+gdzie $T[n]$ jest energią kinetyczną elektronów
+%
+\begin{equation}
+T[n]=-\frac{\hbar^2}{2m}\sum_{i,\sigma} \int d\bm{r} \psi^{\sigma *}_i(\bm{r})\nabla_i^2 \psi^{\sigma}_i(\bm{r}),
+\end{equation}
+%
+$E_{ext}[n]$ jest energią oddziaływania z potencjałem zewnętrznym
+%
+\begin{equation}
+E_{ext}[n]=\int d\bm{r} V_{ext}(\bm{r}) n(\bm{r}),
+\end{equation}
+%
+$E_H[n]$ jest energią Hartree (\ref{Hartree}) i $E_{xc}[n]$ jest energią wymienno-korelacyjną.
+
+Możemy teraz zastosować podejście wariacyjne uwzględniając warunek ortonormalności
+orbitali Kohna-Shama: $\langle\psi_i^{\sigma *}|\psi_j^{\sigma'}\rangle = \delta_{ij}\delta_{\sigma\sigma'}$
+i wprowadzając mnożniki Lagrange'a
+%
+\begin{equation}
+\frac{\delta}{\delta\psi_i^{\sigma *}}(E[n]-\sum_{i,\sigma}\varepsilon_{i\sigma}[\int d\bm{r} \psi_i^{\sigma *}(\bm {r})\psi_i^{\sigma}(\bm{r})-1]) = 0.
+\label{delta}
+\end{equation}
+%
+Wstawiając funkcjonał energii w postaci (\ref{EKS}) do (\ref{delta}) otrzymujemy 
+%
+\begin{equation}
+\frac{\delta T[n]}{\delta \psi_i^{\sigma *}}+\frac{\delta}{\delta n}(\int d\bm{r} V_{ext}(\bm{r}) n(\bm{r}) + E_H[n] + E_{xc}[n])\frac{\delta n}{\delta \psi_i^{\sigma *}}-\varepsilon_{i\sigma} \frac{\delta}{\delta\psi_i^{\sigma *}}\sum_{j,\sigma'}\int d\bm{r} \psi_j^{\sigma' *}\psi_j^{\sigma}=0,
+\end{equation}
+%
+gdzie zastosowalismy wzór na pochodna funkcji złozonej:
+% 
+\begin{equation}
+\frac{\delta}{\delta \psi_i^{\sigma *}}=\frac{\delta}{\delta n}\frac{\delta n}{\delta \psi_i^{\sigma *}}. 
+\end{equation}
+%
+Wykonując pochodne wariacyjne dostajemy równanie Kohna-Shama
+%
+\begin{equation}
+[-\frac{\hbar^2\nabla_i^2}{2m}+V_{KS}(\bm{r})]\psi_i^{\sigma}(\bm{r})=\varepsilon_{i\sigma}\psi_i^{\sigma}(\bm{r}),
+\end{equation}
+%
+które ma postać jednocząstkowego równania Schr\"{o}dingera z potencjałem Kohna-Shama złożonym z trzech wyrazów
+%
+\begin{equation}
+V_{KS}(\bm{r})=V_{ext}(\bm{r})+V_H(\bm{r})+V_{xc}(\bm{r})=V_{ext}(\bm{r})+\int d\bm{r'} \frac{n(\mb{r}')}{|\bm{r}-\bm{r'}|}+\frac{\delta E_{xc}[n]}{\delta n}.
+\label{VKS}
+\end{equation}
+%
+W potencjale tym nieznana jest dokładnie energia wymienno-korelacyjna, która musi być przybliżana
+odpowiednimi metodami, które będą omówione w kolejnych rozdziałach.
+Znając spektrum energetyczne rozwiązań równania Kohna-Shama, energię stanu podstawowego można wyrazić w postaci
+%
+\begin{equation}
+E[n,f_i]=\sum_{i\sigma}f_{i\sigma} \varepsilon_{i\sigma} - \int d\bm{r} n(\bm{r})V_{KS}(\bm{r}) + \int d\bm{r} V_{ext}(\bm{r}) n(\bm{r}) + E_H[n] + E_{xc}[n].
+\end{equation} 
+%
+Funkcje jednoelektronowe i energie stanów Kohna-Shama $\varepsilon_{i\sigma}$ nie mają jednoznacznej interpretacji fizycznej.
+Można je jednak powiązać ze zmianą energii całkowitej przy zmianie obsadzenia stanów
+%
+\begin{equation}
+\frac{\partial E}{\partial f_{i\sigma}}=\Big(\frac{\partial E}{\partial f_{i\sigma}}\Big)_n+\int d\bm{r} \frac{\delta E}{\delta n}\frac{\partial n}{\partial f_{i\sigma}}.
+\label{janak}
+\end{equation} 
+%
+Drugi wyraz po prawej stronie równania zeruje się dla stanu podstawowego ($\delta E/\delta n=0$),
+co prowadzi do wzoru
+%
+\begin{equation}
+\varepsilon_{i\sigma}=\frac{\partial E}{\partial f_{i\sigma}},
+\end{equation}
+%
+który nazywany jest twierdzeniem Janaka \cite{janak}.
+Zastosowanie tego wzoru do najwyższego obsadzonego stanu elektronowego daje energię jonizacji, czyli energię potrzebną do usunięcia pojedynczego elektronu 
+z danego układu atomowego. 
+
+
+
+\section{Funkcjonał wymienno-korelacyjny}
+
+\subsection{Przybliżenie lokalnej gęstości (LDA)}
+
+Funkcjonał wymienno-korelacyjny $E_{xc}$ i odpowiadający mu potencjał $V_{xc}$ nie są znane dokładnie i w ramach teorii
+funkcjonału gęstości muszą być opisywane w przybliżony sposób.
+Najprostszym podejściem jest przybliżenie lokalnej gęstości ({\it ang. local density approximation} - LDA).
+W przybliżeniu LDA przyjmujemy, że energia wymienno-korelacyjna w każdym punkcie przestrzeni, gdzie gęstość elektronowa wynosi $n(\mb{r})$,
+równa jest energii wymienno-korelacyjnej jednorodnego gazu elektronowego o tej samej gęstości, $n=n(\mb{r})$.
+Funkcjonał wymienno-korelacyjny może być wtedy zapisany w postaci
+%
+\begin{equation}
+E_{xc}[n]=\int d\mb{r} n(\mb{r}) \varepsilon_{xc}(n),
+\end{equation}
+%
+gdzie$\varepsilon_{xc}(n)$ jest energią wymienno-korelacyjną przypadającą na pojedynczy elektron w jednorodnym gazie elektronowym o gestości $n$.
+Można ją zapisać jako sumę części wymiennej i korelacyjnej, $\varepsilon_{xc}(n)=\varepsilon_x(n)+\varepsilon_c(n)$.   
+Przybliżenie LDA jest dokładne w granicy wolno zmieniającej się gęstości, co odpowiada warunkowi
+%
+\begin{equation}
+\frac{q}{k_F}\ll 1,
+\end{equation} 
+%
+gdzie $q$ jest miarą niejednorodności układu
+%
+\begin{equation}
+q=\frac{|\nabla k_F|}{2k_F},
+\end{equation}
+%
+a $k_F$ odpowiada wektorowi falowemu Fermiego dla gazu jednorodnego, który w punkcie o lokalnej gęstości $n(\bm{r})$ dany jest wzorem
+%
+\begin{equation}
+k_F=[3\pi^2n(\bm{r})]^{1/3}.
+\end{equation} 
+%
+
+Można łatwo uogólnić to przybliżenie do układów z polaryzacją spinową. Wtedy energia wymienno-korelacyjna jest funkcjonałem
+gęstości spinów skierowanych do góry $n_{\uparrow}$ i w dół $n_{\downarrow}$
+%
+\begin{equation}
+E_{xc}[n^{\uparrow},n^{\downarrow}]=\int d\bm{r} n(\mb{r}) \varepsilon_{xc}(n_{\uparrow},n_{\downarrow}).
+\label{xclda}
+\end{equation}
+%
+To uogólnienie nazywane jest czasem przybliżeniem lokalnej gestości spinowej ({\it ang. local spin density} - LSD).
+Potencjał wymienno-korelacyjny zależny od spinu $\sigma$ w tym przybliżeniu określony jest przez zależność
+%
+\begin{equation}
+V^{\sigma}_{xc}=\frac{\delta E_{xc}}{\delta n_{\sigma}}=\varepsilon_{xc}+n_{\sigma}\frac{\partial \varepsilon_{xc}}{\partial n_{\sigma}}.
+\end{equation}
+%
+Zgodnie ze wzorem (\ref{exchange}), część wymienna dana jest wzorem
+%
+\begin{equation}
+\varepsilon_x(n_{\sigma})=-\frac{3}{4}\Big(\frac{3}{\pi}\Big)^{\frac{1}{3}}n_{\sigma}^\frac{1}{3},
+\end{equation}
+%
+co prowadzi do potencjału wymiennego w formie 
+%
+\begin{equation}
+V^{\sigma}_x=-\Big(\frac{3}{\pi}\Big)^{\frac{1}{3}}n_{\sigma}^\frac{1}{3}. 
+\end{equation}
+%
+Część korelacyjna może być wyznaczona bardzo dokładnie numerycznie metodą kwantowego Monte Carlo \cite{CeperleyAlder80}. 
+W praktyce stosuje się odpowiednie wyrażenia analityczne dofitowane do wyników numerycznych, które określają zależność energii korelacji 
+od gęstości elektronowej \cite{PZ,VWN}. 
+Przykładowo energia korelacji w parametryzacji Pardew-Zungera \cite{PZ} ma postać
+%
+\begin{eqnarray}
+\varepsilon_c(r_s)&=&-0.048+0.031 \ln(r_s) -0.0116 r_s+0.002r_s \ln(r_s), \quad r_s<1  \\
+             &=&-0.1423/(1+1.0529\sqrt{r_s}+0.3334 r_s), \quad r_s>1.
+\end{eqnarray}
+%
+
+\subsection{Uogólnione przybliżenie gradientowe (GGA)}
+\label{sec:GGA}
+
+Uwzględnienie lokalnych zmian gęstości poprzez rozwinięcie energii wymienno-korelacyjnej w szereg gradientów gęstości
+zaproponowano już w pracy Kohna i Shama z 1965 roku \cite{kohn65}. Podejście to jednak nie spełnia reguł sum i załamuje się dla większości układów atomowych, 
+w których zmiany gęstości elektronowej sa przeważnie bardzo duże.
+Zaproponowane zostało nowe podejście zwane uogólnionym przybliżeniem gradientowym ({\it ang. generalized gradient approximation} - GGA),
+gdzie energia wymienno-korelacyjna jest funkcjonałem gęstości elektronowej i jej gradientów \cite{Langreth83,Pardew86,Becke88}. 
+W ogólnej formie dla układu spolaryzowanego spinowo może być zapisana w formie
+%
+\begin{equation}
+E_{xc}[n_{\uparrow},n_{\downarrow}]=\int d\mb{r} f(n_{\uparrow},n_{\downarrow},\nabla n_{\uparrow},\nabla n_{\downarrow}) 
+\label{xcgga}
+\end{equation}
+%
+Część wymienną tego funkcjonału dla układu bez polaryzacji spinowej można zapisać w postaci
+%
+\begin{equation}
+E_x[n]=\int d\mb{r} n \varepsilon_x(n) F_x(s),
+\end{equation}
+%
+gdzie $s=|\nabla n|/2k_Fn$ jest przeskalowanym (bezwymiarowym) gradientem gęstości elektronowej. Rozszerzenie na układy z polaryzacja spinową uzyskuje się stosując następujący wzór,
+który spełniony jest dla dokładnej energii wymiennej
+%
+\begin{equation}
+E_x[n^{\uparrow},n^{\downarrow}]=\frac{1}{2}(E_x[2n^{\uparrow}]+E_x[2n^{\downarrow}]).
+\end{equation}
+%
+Wiele form funkcji $F_x(s)$ zostało zaproponawych, w tym te najczęściej stosowane, które opisane są w pracach: A. D. Becke (Becke88) \cite{Becke88}, 
+J. P. Perdew i Y. Wang (PW91) \cite{PW91} i  J. P. Perdew, K. Burke, and M. Ernzerhof (PBE) \cite{PBE}.
+Jako przykład omówię funkcjonał PBE, który ma prostą postać i dobrze zdefiniowane warunki graniczne  
+które musi spełniać funkcja $F_x(s)$:
+
+\begin{enumerate}
+
+\item{W granicy małych wartości gradientu ($s\rightarrow0$)
+spełniony jest warunek
+%
+\begin{equation}
+F_x(s)\rightarrow 1+\mu s^2,
+\end{equation}
+%
+gdzie $\mu =0.219$. Warunek ten zapewnia właściwe zachowanie się odpowiedzi liniowej jednorodnego gazu
+elektronów (liniowe przyczynki od energii wymiany i korelacji kasują się).}
+
+\item{Dla dużych wartości gradientu ($s\rightarrow\infty$) funkcja ograniczona jest od góry $F_x(s)\leq 1.804$.}
+
+Funkcja, która spełnia te warunki ma postać
+%
+\begin{equation}
+F_x(s)=1+\kappa-\frac{\kappa}{1+\frac{\mu s^2}{\kappa}},
+\end{equation} 
+%
+gdzie $\kappa=0.804$.
+\end{enumerate}
+
+Część korelacyjną można zapisać w ogólnej formie
+%
+\begin{equation}
+E_c[n^{\uparrow},n^{\downarrow}]=\int d\mb{r} n [\epsilon_c(r_s,\zeta)+H(r_s,\zeta,t)],
+\end{equation}
+%
+gdzie $\zeta=(n^{\uparrow}-n^{\downarrow})/n$ jest względną polaryzacją spinową, $t=|\nabla n|/2\phi k_s n$ jest bezwymiarowym gradientem,
+$\phi=\sqrt{(1+\zeta)^{2/3}+(1-\zeta)^{2/3})}/2$ jest spinową funkcją skalującą i $k_s=\sqrt{4k_F/\pi a_0}$ wektorem ekranowania Thomasa-Fermiego.
+Funkcja $H$ spełnia następujące warunki:
+
+\begin{enumerate}
+\item{Dla małych gradientów ($t\rightarrow 0$) funkcja dana jest wyrazem rozwinięcia drugiego rzędu
+%
+\begin{equation}
+H\rightarrow \frac{e^2}{a_0}\beta\phi^3 t^2,
+\end{equation}
+%
+gdzie $\beta=0.067$.}
+\item{Przy szybko zmieniających się gęstościach ($t\rightarrow \infty$) energia korelacji znika
+co spełnione jest przy warunku
+%
+\begin{equation}
+H\rightarrow -\epsilon_c.
+\end{equation}}
+%
+\item{W granicy dużych gęstości ($r_s\rightarrow 0$) energia korelacji dąży do stałej wartości.
+Funkcja $H$ musi kasować logarytmiczną osobliwość w $\epsilon_c$
+%
+\begin{equation}
+H\rightarrow \frac{e^2}{a_0}\gamma\phi^3 \ln t^2,
+\end{equation}
+%
+gdzie $\gamma=0.031$.}
+\end{enumerate}
+
+Warunki te spełnia funkcja w postaci
+%
+\begin{equation}
+H=\frac{e^2}{a_0}\gamma\phi^3\ln[1+\frac{\beta}{\gamma}t^2\frac{1+At^2}{1+At^2+A^2t^4}],
+\end{equation}
+%
+gdzie 
+%
+\begin{equation}
+A=\frac{\beta}{\gamma}[\exp\{-\epsilon_c/(\gamma\phi^3e^2/a_0)\}-1]^{-1}.
+\end{equation}
+
+
+W tabeli \ref{EXC} porównane są wartości energii wymiany i korelacji w przybliżeniu LDA i PBE
+z wartościami dokładnymi wyznaczonymi dla kilku wybranych atomów. Wartości bezwzględne energii wymiany są większe
+o około rząd wielkości od energii korelacji. W LDA, energia wymiany jest zaniżona średnio o $10\%$, a energia korelacji jest znacznie zawyżona,
+nawet powyżej $100\%$, w porównaniu do wartości dokładnych. Ze względu na przeciwne znaki odchyleń od wartości dokładnych, błędy energii wymiany 
+i korelacji w przybliżeniu LDA częściowo kasują się. 
+
+\begin{table}[h!]
+\caption{Porównanie energii wymiennej i korelacyjnej uzyskanych w kilku przybliżeniach z wartościami dokładnymi dla kilku wybranych atomów.}
+\label{EXC}
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|}
+\hline
+ & \multicolumn{2}{c|}{LDA} & \multicolumn{2}{c|}{PBE} & \multicolumn{2}{c|}{Dokładna wartość} \\ \hline
+Atom& $E_x$ & $E_c$ & $E_x$ & $E_c$  & $E_x$  & $E_c$ \\
+\hline
+  H & -0.2680 & -0.0222 &  -0.3059 & -0.0060 & -0.3125 & 0.0000\\
+  He & -0.8840 & -0.1125 &  -1.0136 & -0.0420 & -1.0258 & -0.0420\\
+  Be & -2.3124 & -0.2240 &  -2.6358 & -0.0856 & -2.6658 & -0.0950\\
+  N &  -5.9080 & -0.4268 &  -6.5521 & -0.1799 & -6.6044 & -0.1858\\
+  Ne & -11.0335 & -0.7428 &  -12.0667 & -0.3513 & -12.1050 & -0.3939\\ \hline
+\end{tabular}
+\end{center}
+\end{table}
+
+Te duże różnice w energii wymiany i korelacji wynikają z nieuwzględnienia wpływu lokalnych zmian 
+gęstości elektronowej, które są charakterystyczne dla atomów i molekuł \cite{Pardew92}.  
+W przybliżeniu GGA, wartości obydwóch energii poprawiają się. Energia wymiany jest w dalszym ciągu zaniżona, ale tylko
+na poziomie $\sim 1\%$. Energia korelacji w atomie wodoru jest znacznie zredukowana w porównaniu do LDA,
+a dla atomu helu jest bardzo dobrze odtworzona. W pozostałych przypadkach pokazanych w tabeli \ref{EXC} jest mniejsza
+od wartości dokładnej o kilka procent.
+
+Funkcjonał GGA w granicy zerowych gradientów, czyli dla jednorodnego gazu elektronowego, jest równoważny przybliżeniu LDA.
+Oznacza to, że dla niejednorodnych układów powinień dawać zawsze wyniki lepsze niż LDA. Tak się jednak nie dzieje, a wynika
+to z faktu, że dla niektórych układów częściowe redukowanie się błedów energii wymiany i korelacji jest większe w LDA niż w GGA.
+Przykładem są metale szlachetne (Ag, Au, Pt), dla których stałe sieci wyznaczone w LDA bardzo dobrze zgadzają się
+z eksperymentem, a w GGA są zawyżone. Większe wartości stałej sieci w przybliżeniu GGA wynikają z zaniżonej energii kohezji kryształów.
+LDA zwykle daje zawyżone wartości energii kohezji i zaniżone wartości stałych sieci. 
+Dla układów magnetycznych, czyli z otwartymi powłokami $d$ lub $f$, przybliżenie GGA daje zwykle lepsze wyniki niż LDA.
+Klasycznym przykładem jest żelazo, dla którego stanem podstawowym w przybliżeniu LDA jest struktura centrowana powierzchniowo ($fcc$) 
+z uporządkowaniem antyferromagnetycznym, a GGA daje poprawny wynik, czyli strukturę centrowaną objętościowo ($bcc$) z uporządkowaniem ferromagnetycznym.
+
+\section{Procedura iteracyjna}
+
+Równania Kohna-Shama (KS) rozwiązywane są metodą iteracyjną, której kolejne kroki przedstawione są na rysunku \ref{fig:diagram}.
+Ogólnie stosowane procedury pozwalają zminimalizować funkcjonał energii całkowitej przy zadanych położeniach atomów,
+jak również wyznaczyć położenia atomów i parametry sieci krystalicznej, które odpowiadają minimum energii całkowitej
+układu. Najpierw ustalamy początkowe położenia atomów, które można przyjąć na przykład na podstawie
+danych z eksperymentu dyfrakcyjnego. Przy ustalonych położeniach atomowych przyjmuje się początkowy
+rozkład gęstości elektronowej $n_0(\bm{r})$, dla którego wylicza się potencjał ze wzoru (\ref{VKS}).
+Dla tego potencjału rozwiązuje się równania KS i wyznacza jednoelektronowe funkcje falowe
+w odpowiedni wybranej bazie funkcyjnej. 
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.45]{diagram.pdf}
+\caption{Schemat interacyjnej procedury rozwiązywania równań Kohna-Sham.}
+\label{fig:diagram}
+\end{figure}
+
+Rozwiązywanie równań KS dla atomów w ciele stałym jest złożonym problemem i będzie dyskutowane w kolejnych rozdziałach.
+Znając funkcje falowe i energie własne równania KS wyznacza się gęstość elektronową w każdym punkcie przestrzeni 
+z zależności (\ref{density}) i całkowitą energię układu. 
+Nowy rozkład gęstości pozwala ponownie wyznaczyć potencjał KS i rozwiązać równania KS. Dostajemy nowy rozkład gęstości i zmienioną
+energię całkowitą. Jeżeli ta energia pokrywa się z energią w poprzednim
+kroku lub te energie różnią sie o małą wielkość (zdefiniowaną na początku obliczeń) to oznacza, że 
+otrzymaliśmy prawidłowo zbiegnięty rozkład gęstości. Zakończona jest w ten sposób pętla elektronowa, 
+która daje rozwiązanie równań KS i minimalną energię całkowitą przy zadanych położeniach atomowych.
+
+Procedurę minimalizacji można przyspieszyć wykorzystując odpowiednio gęstości elektronowe
+w każdym kroku iteracyjnym. W najprostszym podejściu początkową wartość gęstości elektronowej $n^p_{i+1}$ w kroku $i+1$ 
+można wyznaczyć jako liniową poprawkę do gęstości z poprzedniego kroku
+%
+\begin{equation}
+n^p_{i+1}=n^p_i+\alpha(n^k_i-n^p_i),
+\label{mixing}
+\end{equation}
+% 
+gdzie $\alpha$ jest współczynnikiem liniowym, $n^p_i$ i $n^k_i$ są początkową i końcową wartością gęstości w kroku $i$.
+Stała wartość współczynnika $\alpha$ nie daje optymalnej szybkości zbieżności.
+W najczęściej stosowanej metodzie Broydena, współczynnik liniowy wyznaczany jest z jakobianu układu $J_i$,
+który optymalizowany jest w każdym kroku iteracji, $\alpha=-J^{-1}_i$.
+
+Aby zoptymalizować układ ze względu na położenia atomowe należy odpowiednio przesuwać
+atomy w nowe położenia, tak aby całkowita energia obniżała się w kolejnych krokach. 
+Wykorzystuje się do tego bardzo efektywną metodę sprzężonego gradientu ({\it ang. conjugate gradient}),
+która jest również metodą iteracyjną (pętla jonowa).
+W każdym kroku tej procedury wykonuje się pełną optymalizację elektronową, aby wyznaczyć rozkład gęstości elektronowej 
+i energię układu dla aktualnego położenia atomów. Wyznaczona gęstość końcowa jest używana do wyznaczenia
+gęstości startowej w kolejnym kroku pętli jonowej według formuły (\ref{mixing}).
+Warunkiem zbieżności jest osiągnięcie stanu o minimalnej energii ze względu na położenia atomów
+i parametry sieci krystalicznej. Według najczęściej przyjmowanego kryterium zbieżność jest osiągnięta jeżeli różnice energii 
+całkowitej w dwóch kolejnych krokach jonowych jest mniejsza od zadanej na początku wartości.
+
+Stan podstawowy otrzymany w wyniku procedury minimalizacyjnej powinien spełniać warunki stanu równowagi badanego materiału.
+Warunki te są następujące: 
+
+\begin{enumerate}
+{\item
+Całkowita siła działająca na każdy atom zeruje się.
+Siły działające na atomy można wyznaczyć stosując twierdzenie Hellmanna-Feynmana \cite{Hellmann,Feynman}, 
+które mówi, że siła działająca na atom $i$ równa jest pochodnej energii całkowitej po położeniu
+tego atomu wziętej ze znakiem przeciwnym
+%
+\begin{equation}
+\bm{F_i}=-\frac{\partial E_{tot}}{\partial\bm{R_i}}.
+\end{equation}
+%
+Zgodnie z tym wzorem, w stanie podstawowym, który odpowiada minimum energii, wszystkie siły $\bm{F_i}$ są równe zeru.
+Wartości sił stanowią często dodatkowe kryterium zbieżności układu w procedurze optymalizacyjnej:
+zbieżność jest osiągnięta jeżeli największa siła działająca na atomy jest mniejsza od zadanej wartości.}
+
+{\item Makroskopowe naprężenia w układzie równe jest naprężeniu wywołanym ciśnieniem zewnętrznym.
+Średni tensor naprężeń określony jest przez pochodną energii całkowitej po tensorze deformacji 
+%
+\begin{equation}
+\sigma_{\alpha\beta}=-\frac{1}{V}\frac{\partial E}{\partial u_{\alpha\beta}}.
+\end{equation}
+gdzie $V$ jest objętością układu. Tensor deformacji jest symetrycznym tensorem pochodnych wektora przesunięcia
+$\bm{u}=\bm{r}-\bm{r'}$ po położeniu $\bm{r}$.
+Przy kompresji hydrostatycznej, ciśnienie wiąże się z tensorem naprężeń zależnością $P=-\frac{1}{2}\sum_{\alpha}\sigma_{\alpha\alpha}$.}
+\end{enumerate}
+
+
+
+\chapter{Metody wyznaczania struktury elektronowej}
+
+Najważniejsze metody wyliczania struktury pasmowej oparte są na teorii DFT i polegają na
+wyznaczaniu jednocząstkowych funkcji falowych poprzez rozwiązanie równań Kohna-Shama
+lub podobnych równań zawierających efektywny potencjał elektronowy.  
+Podstawowym elementem odróżniającym poszczególne metody jest wybór funkcji bazowych, 
+które służą do rozwinięcia funkcji falowych w całym obszarze kryształu lub w poszczególnych jego częściach. 
+Najbardziej naturalną bazą w periodycznym krysztale są fale płaskie i obliczenia w tej bazie są bardzo efektywne oraz łatwe do implementacji. 
+Niestety, nie nadają sie do opisu elektronów w całym obszarze kryształu i wszystkich stanów elektronowych.
+Elektrony, które znajdują się najbliżej jąder atomowych i obsadzają najniższe stany energetyczne 
+wchodzą w skład rdzenia atomowego ({\it ang. atomic core}).
+W obszarze rdzeni atomowych, potencjał elektryczny jest silnie przyciągający i zbliżony jest do potencjału atomowego.
+Przez to stany elektronowe znajdujące się blisko jądra atomowego, zachowują sie podobnie do orbitali atomowych, czyli 
+silnie oscylują i zmieniają swój znak. Oznacza to również, że energia kinetyczna elektronów w tym obszarze jest duża.
+Funkcje falowe stanów rdzenia sa silnie zlokalizowane i szybko zanikają z odległością. 
+W tym przypadku znacznie lepszą zbieżność uzyskuje się stosując bazę zlokalizowanych funkcji, np. harmoniki sferyczne.
+Funkcja falowa elektronów walencyjnych zachowuje się odmiennie w obszarze rdzeni atomowych i w obszarze międzywęzłowym.
+Schematycznie potencjał periodyczny i funkcja falowa elektronów walencyjnych pokazana jest na rysunku \ref{fig:bloch}.
+W obszarze rdzeni, zaznaczonych kółkami o promieniu $r_c$, funkcja falowa, podobnie jak w przypadku stanów rdzenia,
+zmienia się bardzo szybko.
+W obszarze miedzywęzłowym ($r>r_c$), potencjał i gęstość elektronowa zmieniają się wolno w funkcji położenia. 
+Również funkcja falowa elektronów walencyjnych w tym obszarze jest wolnozmienna i 
+w tym przypadku można uzyskać szybką zbieżność w rozwinięciu na fale płaskie. 
+
+
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.5]{crystal.pdf}
+\caption{Potencjał periodyczny i funkcja falowa elektronów walencyjnych.}
+\label{fig:bloch}
+\end{figure}
+
+\newpage
+
+Najważniejsze metody wyznaczania struktury pasmowej można podzielić na trzy główne grupy:
+
+
+\begin{enumerate}
+\item{{\bf Pseudopotencjały.} Jest to grupa metod, która ogranicza ilość elektronów i rozwiązania równania Kohna-Shama tylko do stanów walencyjnych. 
+Ma to uzasadnienie wynikające z charakteru stanów rdzenia, które są silnie zlokalizowane blisko jądra atomowego i nie biorą udziału w wiązaniach atomowych.
+Przez to najważniejsze własności materiałów zdeterminowane są przez zachowanie elektronów walencyjnych. 
+W metodzie pseudopotencjału stosuje się rozwinięcie funkcji falowej w bazie fal płaskich. Aby zachować jednolity opis funkcji falowej w całym obszarze kryształu, stosuje się w tych metodach przybliżony potencjał działający na elektrony walencyjne w obszarze rdzenia atomowego, który nazywany jest pseudopotencjałem. 
+Dla elektronów walencyjnych wyznacza się pseudofunkcję falową, która jest rozwiązaniem jednocząstkowego równania Schr\"{o}dingera z odpowiednim pseudopotencjałem. W obszarze międzywęzłowym jest ona równa dokładnej funcji falowej, a w obszarze rdzenia jest wolnozmienną funkcją bez oscylacji i miejsc
+zerowych. Najdokładniejsze obliczenia w tej grupie metod stosują pseudopotencjały ultramiękkie i potencjały typu PAW.}
+\item{{\bf Zlokalizowane orbitale.} W tym podejściu funkcję falową elektronów zapisujemy w bazie orbitali zlokalizowanych na poszczególnych
+atomach. W najprostszym przybliżeniu ciasnego wiązania jedynymi istotnymi parametrami są elementy macierzowe Hamiltonianu, które
+opisują przekrywanie się lokalnych orbitali lub funkcji Wanniera. W dokładniejszych obliczeniach jako bazę stosuje się funkcje Gaussa
+lub orbitale Slatera.}
+\item{{\bf Stowarzyszone fale i atomowe sfery}. Ta grupa obejmuje metody obliczeniowe, które bazują na ogólnej zasadzie podziału
+kryształu na dwa obszary. Pierwszy obszar zawiera rdzenie atomowe, w których funkcje falowe zachowują cechy orbitali atomowych i drugi obszar między atomami,
+gdzie elektrony walencyjne opisane są wolnozmienną funkcją falową. W każdym z dwóch charakterystycznych obszarów stosuje się
+rozwinięcia funkcji falowej w różnych bazach funkcyjnych. Obszar rdzenia atomowego definiuje wartość promienia $r_c$. Funkcje falowe
+otrzymane dla odległości mniejszych i wiekszych od $r_c$ muszą spełniać odpowiednie warunki ciągłości na granicy rdzenia atomowego.}
+
+\end{enumerate}
+
+W kolejnych rozdziałach opisane zostaną metody obliczeniowe należące do tych trzech grup.
+  
+\section{Pseudopotencjały}
+
+Główną ideę pseudopotencjału ilustruje rysunek \ref{fig:pseudopot}. W obszarze rdzenia atomowego ($r<r_c$) dokładny potencjał $V$ przybliżany
+jest pseudopotencjałem $\tilde{V}$, który ma skończoną wartość dla $r=0$. Odpowiadająca mu pseudofunkcja falowa $\tilde{\psi}$
+ma gładkie zachowanie (bez oscylacji i miejsc zerowych) w tym obszarze. Dla $r>r_c$ pseudopotencjał pokrywa się z dokładnym potencjałem,
+a pseudofunkcja odpowiada dokładnej funkcji falowej $\psi$. Dzięki takiemu przybliżeniu, pseudofunkcja falowa może być rozwinięta
+na fale płaskie w całym obszarze kryształu.   
+
+  
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.1]{pseudopots-new.pdf}
+\caption{Psudopotencjał i pseudofunkcja falowa.}
+\label{fig:pseudopot}
+\end{figure}  
+  
+Metoda pseudopotencjału rozwinęła się z podejścia zortogonalizowanych fal płaskich ({\it ang. ortogonalized plane waves} - OPW)
+zaproponowanego przez Herringa w 1940 \cite{herring40}. 
+Można zdefiniować transformację, która prowadzi od dokładnego potencjału do pseudopotencjału, wprowadzając jednocześnie pseudofunkcję falową do opisu elektronów walencyjnych \cite{Antoncik,KP}.
+Wprowadzamy osobne oznaczenia dla stanów walencyjnych $|\psi_v\rangle$ i stanów rdzenia $|\psi_c\rangle$, które są stanami własnymi
+hamiltonianu $H$ z odpowiednimi energiami własnymi $\varepsilon_v$ i $\varepsilon_c$.
+Zakładamy, że dokładna funkcja falowa elektronów walencyjnych może być wyrażona przez gładką funkcję $\tilde{\psi}_v$ (pseudofunkcję),
+która jest ortogonalna do stanów rdzenia
+%
+\begin{equation}
+|\psi_v\rangle =|\tilde{\psi}_v\rangle +\sum_{\alpha,c} a_{\alpha} |\psi_{\alpha c}\rangle,
+\label{OPW}
+\end{equation}
+%
+gdzie sumowanie przebiega po atomach $\alpha$ i stanach rdzenia $c$. 
+Dla uproszczenia zapisu wskaźniki $v$ i $c$ określają zarówno numer stanu, jak i kierunek spinu elektronu.
+Współczynniki rozwinięcia dostajemy z warunku ortogonalności  
+%
+\begin{equation}
+a_{\alpha}=-\langle \psi_{\alpha c}|\tilde{\psi}_v\rangle.
+\label{a_alpha}
+\end{equation}
+% 
+W obszarze miedzywęzłowym dokładna funkcja falowa równa jest pseudofunkcji, która może być wyrażona w bazie fal płaskich.
+Wstawiając (\ref{OPW}) do równania $H|\psi_v\rangle=\varepsilon_v|\psi_v\rangle$ otrzymujemy
+%
+\begin{equation}
+H|\tilde{\psi}_v\rangle +\sum_{\alpha,c} a_{\alpha} \varepsilon_{\alpha c} |\psi_{\alpha c}\rangle =
+\varepsilon_v|\tilde{\psi}_v\rangle + \sum_{\alpha c} a_{\alpha} \varepsilon_v|\psi_{\alpha c}\rangle.
+\end{equation}
+%
+Przenosząc drugie równanie na prawą stronę i wykorzystując (\ref{a_alpha}) dostajemy
+%
+\begin{equation}
+(H +\sum_{\alpha,c}(\varepsilon_v - \varepsilon_{\alpha c})|\psi_{\alpha c}\rangle \langle \psi_{\alpha c}|)|\tilde{\psi}_v\rangle =
+\varepsilon_v|\tilde{\psi}_v\rangle.
+\end{equation}
+%
+Dostaliśmy równanie typu Schr\"{o}dingera z dodatkowym potencjałem 
+%
+\begin{equation}
+V^{R}=\sum_{\alpha,c}(\varepsilon_v - \varepsilon_{\alpha c})|\psi_{\alpha c}\rangle \langle \psi_{\alpha c}|.
+\end{equation}
+%
+Jeżeli dodamy to wyrażenie do oryginalnego potencjału $V$ dostaniemy wielkość nazywaną pseudopotencjałem $\tilde{V}=V+V^{R}$.
+W obszarze walencyjnym, pseudopotencjał $\tilde{V}$ pokrywa się z potencjałem $V$.  
+Silnie przyciagający potencjał atomowy w obszarze rdzenia jest znacznie osłabiony przez dodatnią wartość $V^R$.
+To umożliwia zbieżność pseudofunkcji falowej w bazie fal płaskich. 
+
+
+\subsection{Pseudopotencjały zachowujące normę}
+
+Dobre pseudopotencjały mają charakter uniwersalny. Wygenerowane w określonej configuracji atomowej
+powinny zachowywać się jednakowo dobrze w każdym innym ukladzie atomowym. 
+W roku 1979, Hamann, Schl\"{u}ter i Chiang \cite{HSC} zaproponowali pseudopotencjały zachowujące normę, 
+które spełniały  nastepujące warunki:
+
+\begin{enumerate}
+{\item Prawdziwa funkcja i pseudofuncja falowa są równe, $\psi_i(r)=\tilde{\psi}_i(r)$, dla $r\geq r_c$.}
+{\item Ich energie własne dla elektronów walencyjnych powinny być równe.}
+{\item Ładunek prawdziwy i pseudoładunek zawarty w obszarze o promieniu $r\geq r_c$ powinny się pokrywać (zachowanie normy)
+%
+\begin{equation}
+\int_0^r dr r^2 |\psi_i(r)|^2=\int_0^r dr r^2 |\tilde{\psi}_i(r)|^2.
+\end{equation}}
+%
+{\item Pochodna logarytmiczna prawdziwej funkcji i pseudofunkcji są równe dla $r\geq r_c$ 
+%
+\begin{equation}
+\frac{1}{\psi_i(r)}\frac{d\psi_i(r)}{dr}=\frac{1}{\tilde{\psi}_i(r)}\frac{d\tilde{\psi}_i(r)}{dr}.
+\end{equation}}
+
+\end{enumerate} 
+%
+Z tych własności wynika również spełnienie warunku równych pochodnych po energii funkcji dokładnej i pseudofunkcji dla $r\geq r_c$,
+co dodatkowo wzmacnia przenośny charakter tych pseudopotencjałów.
+Metody generowania pseudopotencjałów zachowujących normę zostały opisane szczegółowo w wielu pracach \cite{HSC,BHS}.
+
+Ponieważ pseudopotencjały w obszarze rdzenia zależą od orbitalnej liczby kwantowej $l$, ich charakter nie jest lokalny.
+Wartość charakterystycznego promienia $r_c$ również zależy od $l$.
+Ogólnie można podzielić cały pseudopotencjał na część lokalną i nielokalną
+%
+\begin{equation}
+V_l(r)=V_{lok}(r)+\delta V_l(r).
+\end{equation}
+%
+Nielokalny charakter dotyczy tylko obszaru rdzenia, więc $\delta V_l(r)=0$ dla $r>r_c$
+i wszystkie dalekozasięgowe efekty zależą tylko od $V_{lok}(r)$.
+Możliwość wyboru promienia $r_c$ daje pewną swobodę przy konstrukcji pseudopotencjałów.
+Dokładniejsze i bardziej uniwersalne pseudopotencjały charakteryzują sie mniejszymi wartościami
+$r_c$. Jednak większa dokładnosć wymaga uwzględnienia większej ilości fal płaskich
+w rozwinięciu pseudofunkcji falowej. Takie pseudopotencjały nazywane są twardymi. Miękkie pseudopotencjały odznaczają się
+większymi wartościami $r_c$, mniejszą ilość fal płaskich i bardziej gładkim charakterem pseudofunkcji falowej. 
+
+\subsection{Pseudopotencjały ultramiękkie}
+
+W roku 1990, Vanderbilt zaproponował nowy rodzaj pseupotencjałów, nazywane ultramiękkimi ({\it ang. ultrasoft pseudopotentials} - US-PP), które nie zachowują normy \cite{Vanderbilt90}. W tym podejściu, pseudofunkcje falowe spełniają ugólnione równanie własne w postaci
+%
+\begin{equation}
+H|\tilde{\psi}_i\rangle=\varepsilon_iS|\tilde{\psi}_i\rangle,
+\end{equation}
+%
+gdzie Hamiltonian ma ogólną postać 
+%
+\begin{equation}
+H=-\frac{\hbar^2}{2m}\nabla^2+V_{lok}+\delta V_{NL}, 
+\end{equation}
+a $S$ jest operatorem przekrywania
+%
+\begin{equation}
+S=1+\sum_{i,j} Q_{ij}|\beta_i\rangle\langle\beta_j|,
+\end{equation}
+%
+który określa uogólniony warunek normalizacji pseudofunkcji falowych $\langle\tilde{\psi}_i|S|\tilde{\psi}_j\rangle=\delta_{ij}$.
+Macierz $Q_{ij}$ opisuje różnicę w normalizacji funkcji i pseudofunkcji falowych
+%
+\begin{equation}
+Q_{ij}=\int_0^{r_c} dr Q_{ij}(\bm{r})=\int_0^{r_c} dr r^2[\psi_i^*(\bm{r})\psi_j(\bm{r})-\tilde{\psi}_i^*(\bm{r})\tilde{\psi}_j(\bm{r})].
+\end{equation}
+%
+Zbiór lokalnych funkcji falowych
+%
+\begin{equation}
+|\beta_i\rangle=\sum_j (B^{-1})_{ij}|\chi_j\rangle,
+\end{equation}
+%
+gdzie funkcje 
+%
+\begin{equation}
+|\chi_j\rangle=(\varepsilon_i+\frac{\hbar^2}{2m}\nabla^2-V_{loc})|\tilde{\psi}_j\rangle
+\end{equation}
+%
+zerują się w obszarze dla $r>r_c$. 
+Nielokalną część speudopotencjału otrzymuje się ze wzoru
+%
+\begin{equation}
+\delta V_{NL}=\sum_{ij}D_{ij}|\beta_i\rangle\langle\beta_j|,
+\end{equation}
+%
+gdzie $D_{ij}=B_{ij}+\varepsilon_j Q_{ij}$. Całkowitą gęstość elektonów walencyjnych
+w danym punkcie przestrzeni otrzymujemy ze wzoru 
+%
+\begin{equation}
+n_v(\bm{r})=\sum_i \tilde{\psi}_i^*(\bm{r})\tilde{\psi}_i(\bm{r})+\sum_{i,j}\sum_k\langle\beta_i|\tilde{\psi}_k\rangle\langle\tilde{\psi}_k|\beta_j\rangle Q_{ij}(\bm{r}),
+\end{equation}
+% 
+gdzie wyraz jest poprawką wynikającą z niezachowania normy przez pseudofunkcje falowe.
+
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=1]{usp.pdf}
+\caption{Porównanie dokładnej funkcji falowej $2p$ dla tlenu (linia ciągła) z pseudofunkcją zachowującą normę (linia kropkowana)
+i pseudofunkcją otrzymaną dla pseudopotancjału ultramiękkiego (linia przerywana). Rysunek z pracy \cite{Vanderbilt90}.}
+\label{fig:ups}
+\end{figure}
+
+Na rysunku \ref{fig:ups} pokazana jest dokładna funkcja falowa dla orbitalu $2p$ w tlenie oraz dwie pseudofunkcje otrzymane dla pseudopotencjału zachowujacego normę i pseudopotencjału ultramiękkiego. Promień obcięcia jest wyraźnie większy dla pseudopotencjału ultramiękkiego ($r_c\approx1.5$ a.u.) niż dla zachowującego normę ($r_c\approx0.5$ a.u.).
+
+\subsection{Metoda PAW}
+
+Omówione metody pseudopotencjałów pozwalaja efektywnie wyliczać wiele wielkości fizycznych bazując na pseudofukcjach falowych.
+W wielu przypadkach dokładniejsze obliczenia wymagają znajomości funkcji falowych elektronów walencyjnych
+również w obszarze rdzenia atomowego.
+W roku 1994, Bl\"{o}chl zaproponował nową metodę PAW ({\it ang. projector augmented-wave}),
+która łączy efektywność pseudopotencjałów i dokładność obliczeń porównywalną z metodami pełnego potencjału \cite{Blochl}. 
+Do opisu elektronów walencyjnych podejście to wykorzystuje również pseudofunkcje falowe $\tilde{\psi}_v$, które rozwijane są na fale płaskie i 
+w obszarze międzywęzłowym pokrywają sie z funkcjami dokładnymi $\psi_v$.
+Główną zaletą tej metody jest możliwość wyznaczenia dokładnej funkcji falowej poprzez transformację liniową pseudofukcji falowej
+%
+\begin{equation}
+|\psi_v\rangle=\mathcal{T}|\tilde{\psi}_v\rangle.
+\end{equation}
+% 
+Operator liniowy $\mathcal{T}$ działa w obszarach otoczonych sferą o promieniu $r_c$ wokół atomów w położeniach $\bm{R}_m$
+%
+\begin{equation}
+\mathcal{T}=1+\sum_m \mathcal{T}_m.
+\label{operator}
+\end{equation}
+% 
+Dokładna funkcja falowa i pseudofukcja rozwinięte są w obszarze atomowym na funkcje parcjalne $|\phi_m\rangle$ i $|\tilde{\phi}_m\rangle$
+%
+\begin{eqnarray}
+|\psi_v\rangle=\sum_m c_m |\phi_m\rangle, \label{eq1}\\
+|\tilde{\psi_v}\rangle=\sum_m c_m |\tilde{\phi}_m\rangle,
+\label{eq2}
+\end{eqnarray}
+%
+gdzie w obydwóch sumach występują te same współczynniki rozwinięcia $c_m$. Zatem te lokalne funkcje powiązane są tą samą
+transformacją
+%
+\begin{equation}
+|\phi_m\rangle=(1+\sum_m \mathcal{T}_m)|\tilde{\phi}_m\rangle.
+\end{equation}
+%
+Funkcje $|\phi_m\rangle$ są rozwiązaniami równania Schr\"{o}dingera dla dokładnego potencjału atomowego, które odpowiadają energiom $\varepsilon_m$
+i są ortogonalne do stanów rdzenia. Wskaźnik $m$ określa jednocześnie położenia atomów $\bm{R}$ i zbiór liczb kwantowych orbitali atomowych.
+Każdej parcjalnej funkcji dokładnej odpowiada jedna pseudofunkcja $|\tilde{\phi}_m\rangle$, z którą pokrywa się poza sferą o promieniu $r_c$.
+Oba rodzaje funkcji parcjalnych są funkcjami radialnymi, zdefiniowanymi na logarytmicznej siatce radialnej, przemnożonymi przez
+harmoniki sferyczne. 
+
+Odejmując stronami (\ref{eq1}) i (\ref{eq2}) dostajemy równanie
+%
+\begin{equation}
+|\psi_v\rangle=|\tilde{\psi}_v\rangle-\sum_m c_m |\tilde{\phi}_m\rangle + \sum_m c_m |\phi_m\rangle.
+\label{psiv}
+\end{equation}
+% 
+Aby operator $\mathcal{T}$ był liniowy współczynniki $c_m$ muszą być liniowymi funkcjonałami pseudofunkcji falowej $|\tilde{\psi}_v\rangle$
+%
+\begin{equation}
+c_m=\langle\tilde{p}_m|\tilde{\psi}_v\rangle,
+\label{coef}
+\end{equation}
+%
+gdzie $\langle\tilde{p}_m|$ są funkcjami rzutowymi dualnymi względem funkcji parcjalnych 
+$\langle\tilde{p}_m|\tilde{\phi}_n\rangle=\delta_{m,n}$.
+Dla każdej funkcji $|\tilde{\phi}_m\rangle$ mamy jedną funkcja rzutową $\langle\tilde{p}_m|$
+i dla nich spełniony jest warunek $\sum_m |\tilde{\phi}_m\rangle\langle\tilde{p}_m|=1$.
+Funkcje rzutowe są funkcjami radialnymi pomnożonymi przez harmoniki sferyczny, a następnie przetransformowane do bazy fal płaskich.
+Są przypisane do położeń atomów, ale nie zależą od potencjału atomowego.
+%
+Wstawiając (\ref{coef}) do (\ref{psiv}) otrzymujemy
+%
+\begin{equation}
+|\psi_v\rangle=|\tilde{\psi}_v\rangle+\sum_m (|\phi_m\rangle-|\tilde{\phi}_m\rangle)\langle\tilde{p}_m|)\tilde{\psi}_v\rangle
+=[1+\sum_m (|\phi_m\rangle-|\tilde{\phi}_m\rangle)\langle\tilde{p}_m|]|\tilde{\psi}_v\rangle,
+\label{psiv2}
+\end{equation}
+%
+gdzie wyrażenie w nawiasie kwadratowym jest wprowadzonym wcześniej operatorem liniowym
+%
+\begin{equation}
+\mathcal{T}=1+\sum_m \mathcal{T}_m=1+\sum_m (|\phi_m\rangle-|\tilde{\phi}_m\rangle)\langle\tilde{p}_m|.
+\end{equation}
+%
+
+Funkcje falowe rdzenia $|\psi_c\rangle$, podobnie do funkcji walencyjnych, są rozłożone na trzy składowe
+%
+\begin{equation}
+|\psi_c\rangle=|\tilde{\psi}_c\rangle+|\phi_c\rangle-|\tilde{\phi}_c\rangle,
+\end{equation}
+%
+gdzie kolejne wyrazy odpowiadają pseudofunkcji elektronów rdzenia, która pokrywa się z dokładną funkcją dla $r>r_c$,
+lokalnej (parcjalnej) funkcji rdzenia i lokalnej pseudofukcji rdzenia. Obydwie lokalne funcje wyrażone są jako funkcje radialne
+pomnożone przez harmoniki sferyczne.
+
+Średnia z operatora $A$ może być wyznaczona przy pomocy dokładnych funkcji lub pseudofunkcji
+%
+\begin{equation}
+\langle A \rangle = \sum_n f_n \langle \psi_n|A|\psi_n\rangle = \sum_n f_n \langle \tilde{\psi}_n|\tilde{A}|\tilde{\psi}_n\rangle,
+\end{equation}
+% 
+gdzie $f_n$ określa obsadzenie stanu $n$, a $\tilde{A}$ jest pseudooperatorem, który można otrzymać transformując operator $A$
+%
+\begin{equation}
+\tilde{A}=\mathcal{T}^{\dagger} A \mathcal{T}= A +\sum_{i,j} |\tilde{p}_i\rangle (\langle \phi_i |A|\phi_j\rangle - \langle \tilde{\phi}_i |A|\tilde{\phi}_j\rangle ) \langle\tilde{p}_j|.
+\end{equation}
+%
+Przykładowo, stosując te wyrażenia do operatora gęstości $n=|\bm{r}\rangle\langle\bm{r}|$,
+możemy wyrazić gęstość elektronową następującym wyrażeniem
+%
+\begin{equation}
+n(\bm{r})=\tilde{n}(\bm{r})+n^1(\bm{r})-\tilde{n}^1(\bm{r}),
+\end{equation}
+%
+gdzie
+%
+\begin{equation}
+\tilde{n}(\bm{r})=\sum_n f_n \langle \tilde{\psi}_n|\bm{r}\rangle\langle \bm{r}| \tilde{\psi}_n\rangle=\sum_n f_n |\tilde{\psi_n}(\bm{r})|^2,
+\end{equation}
+%
+\begin{equation}
+n^1(\bm{r})=\sum_{n,i,j} f_n \langle \tilde{\psi}_n|\tilde{p}_i\rangle\langle\phi_i|\bm{r}\rangle\langle \bm{r}|\phi_j\rangle\langle\tilde{p}_j |\tilde{\psi}_n\rangle,
+\end{equation}
+%
+\begin{equation}
+\tilde{n}^1(\bm{r})=\sum_{n,i,j} f_n \langle \tilde{\psi}_n|\tilde{p}_i\rangle\langle\tilde{\phi}_i|\bm{r}\rangle\langle \bm{r}|\tilde{\phi}_j\rangle\langle\tilde{p}_j |\tilde{\psi}_n\rangle.
+\end{equation}
+%
+W powyższych wzorach sumowanie obejmuje zarówno stany walencyjne, jak i stany rdzenia. 
+
+Metoda PAW należy obecnie do najdokładniejszych i najbardziej efektywnych metod. Została zaimplementowana
+w takich programach, jak Vienna Ab Initio Simulation Package (VASP) \cite{Vasp,PawVasp} i Quantum Espresso \cite{QE}.
+
+
+\section{Zlokalizowane orbitale}
+
+\subsection{Metoda ciasnego wiązania}
+
+W odróżnieniu od metod pseudopotencjału, które wykorzystują bazę fal płaskich, metody opisane w tym rozdziale
+stosują lokalne orbitale, które powiązane są z danymi atomami lub centrowane są w położeniach atomów.
+Opis przy pomocy lokalnych orbitali nadaje się dobrze do układów, gdzie występują zlokalizowane stany
+elektronowe, ale możliwy jest ruch elektronów poprzez przeskoki do sąsiednich atomów.
+Najprostszym opisem takich układów jest model ciasnego wiązania.
+Lokalne orbitale atomowe możemy zapisać w postaci $\phi_{\alpha}(\bm{r}-\bm{R}_{\alpha})$, 
+gdzie wektor $\bm{R}_{\alpha}$ oznacza położenie danego atomu. Wskaźnik $\alpha$ oznacza tutaj wszystkie liczby kwantowe, 
+które charakteryzują dany orbital ($\alpha=n,l,m$). Zakładamy dla uproszczenia, że z danym atomem związny jest tylko
+jeden orbital, ale model ciasnego wiązania można łatwo uogólnić na dowolną liczbę orbitali.
+Hamiltonian możemy zapisać jako
+%
+\begin{equation}
+H=-\frac{\hbar^2}{2m}\nabla^2+\sum_{\alpha}V_{\alpha}(\bm{r}-\bm{R}_{\alpha}),
+\end{equation}
+%
+gdzie $V_{\alpha}$ jest potencjałem atomowym wokół położenia $\bm{R}_{\alpha}$.
+Elementy macierzowe hamiltonianu możemy wyznaczyć ze wzoru
+%
+\begin{equation}
+H_{\alpha\beta}=\int d\bm{r} \phi_{\alpha}^*(\bm{r}-\bm{R}_{\alpha})H\phi_{\beta}(\bm{r}-\bm{R}_{\beta}).
+\label{matrix}
+\end{equation}
+% 
+Ponieważ orbitale atomowe należące do różnych atomów nie są wzajemnie ortogonalne
+wprowadza się również macierz przekrywania
+%
+\begin{equation}
+S_{\alpha\beta}=\int d\bm{r} \phi_{\alpha}^*(\bm{r}-\bm{R}_{\alpha})\phi_{\beta}(\bm{r}-\bm{R}_{\beta}).
+\end{equation}
+%
+Elementy diagonalne hamiltonianu $\varepsilon_{\alpha}=H_{\alpha\alpha}$ określają lokalną energię elektronu w danym orbitalu $\alpha$.
+Natomiast elementy pozadiagonalne $t_{\alpha\beta}=H_{\alpha\beta}$ nazywane są całkami przeskoku i określają prawdopodobieństwo
+przeskoku elektronu między dwoma atomami.
+
+Aby przetransformować elementy macierzowe hamiltonianu i macierzy $S$ do przestrzeni odwrotnej
+wprowadzamy bazę, która odpowiada wektorom falowym $\bm{k}$
+%
+\begin{equation}
+\phi_{\bm{k}\alpha}(\bm{r})=A_{\alpha\bm{k}}\sum_n e^{i\bm{k}\bm{T}_n}\phi_{\alpha}[\bm{r}-(\bm{R}_{\alpha}+\bm{T}_n)],
+\end{equation}
+%
+gdzie $A_{\bm{k}\alpha}$ są czynnikami normalizacji, a $\bm{T}_n$ są wektorami translacji sieci krystalicznej.
+W tej bazie możemy zapisać funkcję falową, która spełnia twierdzenie Blocha
+%
+\begin{equation}
+\psi_{\bm{k}i}=\sum_{\alpha} c_{i\alpha} \phi_{\bm{k}\alpha}(\bm{r}),
+\end{equation}
+%
+oraz elementy macierzowe hamiltonianu i macierzy $S$
+%
+\begin{equation}
+H_{\alpha\beta}(\bm{k})=\int d\bm{r} \phi_{\alpha\bm{k}}^*(\bm{r})H\phi_{\beta\bm{k}}(\bm{r})=\sum_n e^{i\bm{k}\bm{T}_n} H_{\alpha\beta},
+\end{equation}
+%
+\begin{equation}
+S_{\alpha\beta}(\bm{k})=\int d\bm{r} \phi_{\alpha\bm{k}}^*(\bm{r})\phi_{\beta\bm{k}}(\bm{r})=\sum_n e^{i\bm{k}\bm{T}_n} S_{\alpha\beta}.
+\end{equation}
+%
+Równanie na energie własne $\varepsilon_i(\bm{k})$ i współczynniki rozwinięcia funkcji falowej $c_{i,\alpha}(\bm{k})$ przyjmuje postać
+%
+\begin{equation}
+\sum_{\beta}[H_{\alpha\beta}(\bm{k})-\varepsilon_i(\bm{k})S_{\alpha\beta}(\bm{k})]c_{i\beta}(\bm{k})=0.
+\label{ham_loc} 
+\end{equation}
+%
+Dla orbitali wzajemnie ortogonalnych, mamy $S_{\alpha\beta}=\delta_{\alpha\beta}$ i równanie (\ref{ham_loc})
+przyjmuje postać taką samą jak równanie w bazie fal płaskich (\ref{ham_rec}).
+
+\subsection{Funkcje Wanniera}
+\label{sec:wannier}
+
+W metodzie ciasnego wiązania często wykorzystuje się bazę funkcji Waniera, które są zlokalizowanymi funkcjami, 
+centrowanymi na położeniach atomowych $\bm{R}_n$. 
+Funkcja Wanniera zdefiniowana jest jako transformata Fouriera funkcji Blocha powiązanej z danym pasmem $j$
+%
+\begin{equation}
+w_j(\bm{r}-\bm{R}_n)=\frac{1}{\sqrt{N}}\sum_{\bm{k}}e^{-i\bm{k}\bm{R}_n}\psi_{\bm{k}j}(\bm{r}). 
+\end{equation}
+%
+Spełniona jest również transformata odwrotna, która pozwala wyrazić funkcje Blocha przez funkcje
+Wanniera
+%
+\begin{equation}
+\psi_{\bm{k}j}(\bm{r})=\frac{1}{\sqrt{N}}\sum_n e^{i\bm{k}\bm{R}_n} w_j(\bm{r}-\bm{R}_n).
+\end{equation}
+%
+Funkcje Wanniera dla różnych położeń atomowych są ortogonalne
+%
+\begin{multline}
+\int d\bm{r} w_j^*(\bm{r}-\bm{R}_n)w_j(\bm{r}-\bm{R}_m)=\frac{1}{N}\sum_{\bm{k},\bm{k}'}\int d\bm{r}e^{i(\bm{k}\bm{R}_n-\bm{k}'\bm{R}_m)}\psi_{\bm{k}j}(\bm{r})\psi_{\bm{k}'j}(\bm{r}) \\
+=\frac{1}{N}\sum_{\bm{k}}e^{i\bm{k}(\bm{R}_n-\bm{R}_m)}=\delta_{mn},
+\end{multline}
+%
+gdzie wykorzystaliśmy ortogonalność funkcji Blocha. Spełniony jest również warunek zupełności
+%
+\begin{equation}
+\sum_n w_j^*(\bm{r}-\bm{R}_n)w_j(\bm{r}'-\bm{R}_n)=\delta^3(\bm{r}-\bm{r}').
+\end{equation}
+Zatem funkcje Wanniera tworzą zbiór zlokalizowanych funkcji bazowych.
+Energie pasma $j$ dla funkcji Blocha w tej bazie może być wyznaczona ze wzoru
+%
+\begin{equation}
+\varepsilon_{\bm{k}j}=\int d\bm{r} \psi_{\bm{k}j}^*(\bm{r})H\psi_{\bm{k}j}(\bm{r})
+=\frac{1}{N}\int d\bm{r}\sum_n w_j^*(\bm{r}-\bm{R}_n)e^{-i\bm{k}\bm{R}_n}H\sum_m w_j(\bm{r}-\bm{R}_m)e^{i\bm{k}\bm{R}_m}.
+\end{equation}
+%
+Sumowanie po wskaźnikach $n$ i $m$ możemy rozdzielić na dwa wyrazy dla $n=m$ i $n\neq m$
+%
+\begin{multline}
+\varepsilon_{\bm{k}j}=\frac{1}{N}\sum_n\int d\bm{r} w_j^*(\bm{r}-\bm{R}_n)Hw_j(\bm{r}-\bm{R}_n)
+\\ +\frac{1}{N}\sum_{n\neq m}e^{i\bm{k}(\bm{R}_m-\bm{R}_n)}\int d\bm{r} w_j^*(\bm{r}-\bm{R}_n)Hw_j(\bm{r}-\bm{R}_m).
+\end{multline}
+%
+Wykorzystując oznaczenia, które wprowadziliśmy dla elementów macierzowych hamiltonianu (\ref{matrix})
+dostajemy
+%
+\begin{equation}
+\varepsilon_{\bm{k}j}=\frac{1}{N}\sum_n \varepsilon_n+\frac{1}{N}\sum_{n\neq m}e^{i\bm{k}(\bm{R}_m-\bm{R}_n)}t_{nm}.
+\end{equation}
+%
+Wzór ten pozwala wyznaczyć strukturę pasmową w ramach metody ciasnego wiązania.
+Jeżeli ograniczymy się do modelu jednopasmowego i do przeskoków tylko między najbliższymi sąsiadami
+możemy dostać uproszczony model z jednym parametrem przeskoku $t$.
+Dla prostej struktury kubicznej dostajemy
+%
+\begin{equation}
+\varepsilon_{\bm{k}j}=\varepsilon_0+\frac{t}{N}\sum_{n}e^{i\bm{k}(\bm{R}_{n+1}-\bm{R}_n)}=\varepsilon_0+2t(cosk_xa+cosk_ya+cosk_za).
+\label{band}
+\end{equation}
+%
+gdzie $a$ jest stałą sieci, a $\varepsilon_0$ jest energią elektronu zlokalizowanego w stanie atomowym.
+Szerokość pasma w tym przybliżeniu dana jest wzorem $w=2z|t|$, gdzie $z$ jest liczbą najbliższych sąsiadów.  
+
+Metoda ciasnego wiązania, która wykorzystuje bazę orbitali atomowych lub funkcji Wanniera,
+często wykorzystywana jest w obliczeniach modelowych. W ramach tego podejścia można łatwo uwzględnić lokalne oddziaływania
+kulombowskie lub oddziaływania parujące w modelach nadprzewodników wysokotemperaturowych.
+Lokalne energie i całki przeskoku mogą być wyznaczone przez dofitowanie struktury elektronowej
+z modelu ciasnego wiązania do wyników eksperymentalnych lub do struktury pasmowej wyznaczonej z obliczeń DFT.
+W tym drugim przypadku stosuje się najczęściej bazę maksymalnie zlokalizowanych funkcji Wanniera.
+Wykorzystuje się swobodę wyboru fazy dla funkcji Blocha, którą możemy przemnożyć przez czynnik $e^{i\theta(\bm{k})}$,
+gdzie $\theta(\bm{k})$ jest dowolną funkcją rzeczywistą, bez wpływu na energie stanów elektronowych. 
+Taki czynnik ma jednak wpływ na kształt funkcji Wanniera, w szczególności na ich zasięg przestrzenny. 
+Można tak dobrać czynnik fazowy, aby funkcja Wanniera $w_i(\bm{r}-\bm{R}_n)$ zlokalizowana była
+wokół punktu $\bm{R}_n$ i szybko zanikała z odległością. 
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=1]{FeSe.pdf}
+\caption{Porównanie struktury pasmowej otrzymanej w ramach modelu ciasnego wiązania i obliczeń DFT dla {\bf a.} CeCoIn$_5$ i {\bf b.} FeSe.
+Rysunek z pracy \cite{FeSe}.}
+\label{fig:fese}
+\end{figure}
+
+Na rysunku \ref{fig:fese} pokazano przykłady struktur pasmowych dla dwóch związków, CeCoIn$_5$ i FeSe, wyliczonych w modelu ciasnego wiązania z bazą
+maksymalnie zlokalizowanych funkcji Wanniera. 
+Parametry modelu wyznaczono przez dofitowanie energii stanów elektronowych, dla każdego wektora falowego $\bm{k}$, 
+do struktur pasmowych otrzymanych w ramach obliczeń DFT \cite{FeSe}.
+Obliczenia wykonano programem Quantum Espresso \cite{QE}, stosując funkcjonał GGA \cite{PW91} w ramach metody PAW \cite{Blochl}.  
+Wyznaczone modele ciasnego wiązania zostały następnie wykorzystane do wyliczenia podatności tworzenia par Coopera
+i zbadania własności nadprzewodzących obydwóch materiałów \cite{FeSe}.
+ 
+
+\subsection{Funkcje Gaussa}
+
+Lokalne orbitale można również wykorzystać jako bazę do rozwiniecia funkcji falowej w ramach samozgodnej procedury rozwiązywania równań Kohna-Shama.
+Jeżeli funkcjami bazowymi są orbitale atomowe mówimy o metodzie liniowych kombinacji orbitali atomowych ({\it ang. linear combination
+of atomic orbitals} - LCAO). Zwykle jednak stosuje się atomo-podobne orbitale, których postać funkcyjna ułatwia implementację
+i przyspiesza obliczenia. Do najczęściej stosowanych należą orbitale typu Slatera \cite{STO1,STO2} i funkcje typu Gaussa (gaussiany) \cite{GTO}.
+Te drugie są szczególnie wygodne ponieważ całki z funkcji Gaussa można wyliczyć analitycznie.
+Naturalną reprezentacją dla gaussianów jest układ współrzędnych biegunowych, ale wygodniejszy do wyznaczania elementów macierzowych hamiltonianu jest układ kartezjański, w którym te funkcje mają postać
+%
+\begin{equation}
+G_{ijk}(\bm{r},\alpha,\bm{R}_n)=N(x-x_n)^i(y-y_n)^j(z-z_n)^ke^{-\alpha (\bm{r}-\bm{R}_n)^2},
+\end{equation}
+%
+gdzie $\alpha$ jest parametrem wariacyjnym, który umożliwia optymalizację bazy dla konkretnych atomów, $\bm{R}_n=(x_n,y_n,z_n)$ jest wektorem określającym centrowanie danej funkcji, a $N$ jest czynnikiem normalizacyjnym.
+Spełniona jest ponadto zależność $l=i+j+k$, gdzie $l$ jest orbitalną liczbą kwantową.
+W wyniku iloczynu dwóch fukcji typu $s$ ($l=0$) centrowanych w punktach $\bm{R}_n$ i $\bm{R}_m$ dostajemy
+%
+\begin{equation}
+G_s(\bm{r},\alpha,\bm{R}_n)G_s(\bm{r},\beta,\bm{R}_m)=e^{-\gamma(\bm{R}_n-\bm{R}_m)^2}G_s(\bm{r},\kappa,\bm{R}_p),
+\end{equation}
+%
+gdzie 
+%
+\begin{equation}
+\kappa=\alpha+\beta, \quad \gamma=\frac{\alpha\beta}{\alpha+\beta}, \quad \bm{R}_p=\frac{\alpha\bm{R}_n+\beta\bm{R}_m}{\alpha+\beta},
+\end{equation}
+%
+gdzie $\bm{R}_p$ jest wektorem centrowania wynikowej funkcji Gaussa.
+W ogólnym przypadku iloczyn dwóch gaussianów można zapisać w formie
+%
+\begin{multline}
+G_{i_1j_1k_1}(\bm{r},\alpha,\bm{R}_n)G_{i_2j_2k_2}(\bm{r},\beta,\bm{R}_m) = Ne^{-\gamma(\bm{R}_n-\bm{R}_m)^2}e^{-\kappa(\bm{r}-\bm{R}_p)^2} \\
+\times (x-x_n)^{i_1}(x-x_m)^{i_2}(y-y_n)^{j_1}(y-y_m)^{j_2}(z-z_n)^{k_1}(z-z_m)^{k_2}.
+\end{multline}
+%
+Te wzory pokazują, ze iloczyn funkcji Gaussa jest również funkcją Gaussa. Jest to główna zaleta tej bazy umożliwiająca
+analityczne wyliczenie złożonych całek zawierajacych kilka funkcji bazowych i w ten sposób znaczne przyspieszenie rachunków.
+W odróżnieniu od metod wykorzystujących fale płaskie, metoda funkcji Gaussa wymaga odpowiedniego wybrania funkcji bazowych
+dla danego układu atomowego. Metoda ta stosowana jest głównie przez chemików, a najbardziej popularnym programem jest GAUSSIAN.
+Głównym twórca tego programu jest John Pople, który w roku 1998 otrzymał wspólnie z Walterem Kohnem Nagrodę Nobla w dziedzinie chemii.
+
+\section{Stowarzyszone fale i atomowe sfery}
+
+Ostatnia grupa metod obliczeniowych łączy główne cechy dwóch poprzednich: pseudopotencjałów i lokalnych orbitali. 
+Do rozwinięcia funkcji falowych stosowane są dwie różne bazy w zależności od położenia w krysztale.
+Wokół rdzeni atomowych naturalnym wyborem sa lokalne funkcje, które można otrzymać z rozwiązania
+równania Schr\"{o}dingera w potencjale sferyczno-symetrycznym. W obszarze miedzywezłowym
+stosuje się rozwinięcie w bazie funkcji, które umożliwiają szybką zbieżność rachunków (np. fale płaskie).
+Centralnym problemem stają sie odpowiednie warunki brzegowe, które zapewniają ciągłość funkcji falowej i jej pochodnej 
+na granicy między tymi obszarami.
+Wszystkie metody omawiane w tym rozdziale stosowały na początku uproszczony potencjał typu {\it muffin-tin}, który w obszarze rdzenia
+ma charakter potencjału atomowego, a w obszarze międzywęzłowym jego wartość jest stała.
+Obecnie podejścia te zostały zaimplementowane w ramach procedury samozgodnej i zaliczane są do metod pełnego potencjału. 
+
+\subsection{Metoda KKR}
+
+W dwóch pracach, Korringa \cite{K47} oraz Kohn i Rostoker \cite{KR54} (KKR) zaproponowali metodę, która nazywana jest również metodą funcji Greena
+lub metodą wielokrotnego rozpraszania.
+Pierwsze obliczenia stałych sieci i modułów sztywności wykonano właśnie z wykorzystaniem tej metody \cite{Moruzzi77}. Była to przełomowa praca, która 
+pokazała możliwość zastosowania DFT do badania własności materiałowych.  
+
+Podstawową wielkością stosowaną tutaj jest funkcja Greena $G(E,\bm{r},\bm{r}')$, która opisuje propagację elektronu o energii $E$ między punktami $\bm{r}$ i $\bm{r}'$. Proces przemieszczania się elektronu można podzielić na swobodny ruch w obszarze międzywęzłowym i rozpraszanie na sferach atomowych
+w wyniku oddziaływania elektronu z potencjałem rdzenia atomowego.
+Dla elektronu opisanego jednocząstkowym równaniem Schr\"{o}dingera
+%
+\begin{equation}
+[-\frac{\hbar^2}{2m}\nabla^2+V(\bm{r})]\psi_i(\bm{r})=\varepsilon_i\psi_i(\bm{r}),
+\end{equation}
+%
+funkcja Greena spełnia równanie
+%
+\begin{equation}
+[-\frac{\hbar^2}{2m}\nabla^2+V(\bm{r})-E]G(E,\bm{r}-\bm{r}')=\delta(\bm{r}-\bm{r}'),
+\end{equation}
+%
+które posiada rozwiązanie w formie
+%
+\begin{equation}
+G(E,\bm{r},\bm{r}')=\sum_i\frac{\psi_i(\bm{r})\psi_i^*(\bm{r}')}{E-\varepsilon_i}.
+\end{equation}
+%
+Dla nieoddziałujących elektronów mamy $V(\bm{r})=0$ i funkcja Greena ma postać
+%
+\begin{equation}
+G_0(E,|\bm{r}-\bm{r}'|)= \frac{m}{2\pi\hbar^2}\frac{e^{ik_0|\bm{r}-\bm{r}'|}}{|\bm{r}-\bm{r}'|},
+\end{equation}
+%
+gdzie $k_0=\frac{\sqrt{2mE}}{\hbar}$.
+
+Funkcja Greena dla elektronu oddziałującego z siecią atomową może być zapisana w postaci rozwinięcia
+%
+\begin{equation}
+G=G_0+G_0tG_0+G_0tG_0tG_0+...=G_0+G_0tG,
+\end{equation}
+%
+które prowadzi do zależności
+%
+\begin{equation}
+G=(G_0^{-1}-t)^{-1},
+\end{equation}
+%
+gdzie $t$ jest macierzą rozpraszania elektronów na pojedynczych atomach sieci krystalicznej.
+Można wprowadzić również macierz całkowitego, wielokrotnego rozpraszania elektronów $T$, która zdefiniowana jest wzorem
+%
+\begin{equation}
+G=G_0+G_0TG_0=G_0+G_0(t+tG_0T)G_0,
+\end{equation}
+%
+z którego dostajemy
+%
+\begin{equation}
+T=(t^{-1}-G_0)^{-1}.
+\end{equation}
+%
+Stany stacjonarne, które odpowiadają rozwiązaniom równania Schr\'{o}dingera, są biegunami macierzy $T$
+i mogą być wyznaczone jako miejsca zerowe wyznacznika
+%
+\begin{equation}
+\det(t^{-1}-G_0)=0.
+\end{equation}
+%
+Równanie to dostarcza rozwiązania problemu wielokrotnego rozpraszania elektronów w przybliżeniu potencjału {\it muffin-tin}.
+Pełna informacja o rozpraszaniu elektronu na pojedynczej sferze atomowej zawarta jest w macierzy rozpraszania $t_l(E)$, 
+która zależy tylko od energii stanu elektronowego $E$ i orbitalnego krętu $l$.
+Korzystając z teorii rozpraszania, można wyrazić macierz rozpraszania jako funkcję przesunięcia fazy funkcji falowej $\eta_l(E)$
+%
+\begin{equation}
+t_l(E)=-\frac{1}{k_0}e^{i\eta_l(E)}\sin(\eta_l(E)).
+\label{fshift}
+\end{equation} 
+%
+Funkcja Greena $G_0$ zależy tylko od geometrii sieci krystalicznej i energii $E$.
+Wykorzystując rozwinięcie fal płaskich w bazie harmonik sferycznych, funcję Greena
+możemy zapisać w formie
+%
+\begin{equation}
+G_0(E,|\bm{r}-\bm{r}'|)=\sum_{L,L'} i^lj_l(k_0r)Y_L(\hat{\bm{r}})B_{LL'}(E,\bm{R}-\bm{R}')(-i)^{l'}j_{l'}(k_0r')Y^*_{L'}(\hat{\bm{r}}'),
+\end{equation} 
+%
+gdzie $j_l(k_0r)$ są funkcjami Bessela, $B_{LL'}$ są stałymi struktury KKR i $L=\{l,m\}$.
+Wektory położenia $\bm{r}$ i $\bm{r}'$ odnoszą się do dwóch sfer atomowych, których środki
+określają wektory $\bm{R}$ i $\bm{R}'$. Przy założeniu, że rozpraszanie w każdym węźle atomowym jest takie samo,
+co odpowiada sieci złożonej z jednego rodzaju atomów, możemy wprowadzić zależność stałych struktury od wektora falowego
+% 
+\begin{equation}
+B_{LL'}(E_{\bm{k}},\bm{k})=\sum_{m} B_{LL'}(E_{\bm{k}},\bm{T}_m)e^{-i\bm{k}\bm{T}_m},
+\end{equation}
+%
+gdzie $\bm{T}_m$ są wektorami translacji sieci krystalicznej, a energie $E_{\bm{k}}$
+są rozwiązaniami równania
+%
+\begin{equation}
+\det[t_l^{-1}(E_{\bm{k}})\delta_{LL'}-B_{LL'}(E_{\bm{k}},\bm{k})]=0.
+\end{equation}
+%
+Korzystając z wyrażenia (\ref{fshift}) dostajemy podstawowe równania na strukturę pasmową w metodzie KKR
+%
+\begin{equation}
+\sum_{L'}[B_{LL'}(E_{\bm{k}},\bm{k})+k_0\cot(\eta_l(E_{\bm{k}}))\delta_{LL'}]a_{L'}(\bm{k})=0.
+\label{KKR}
+\end{equation}
+%
+Rozwiązania tego równania pozwalają nam wyznaczyć funkcje falowe wewnątrz sfer atomowych
+%
+\begin{equation}
+\psi_{\bm{k}}(\bm{r})=\sum_{L'}a_{L'}(\bm{k})u_l(r,E_{\bm{k}})Y_L(\hat{\bm{r}}),
+\end{equation}
+%
+gdzie $u_l(r,E_{\bm{k}})$ są rozwiązaniami równania radialnego wewnątrz sfery, umówionego dokładnie w następnym rozdziale.
+Funkcje falowe w obszarze międzywęzłowym są rozwiązaniami następującego równania całkowego
+%
+\begin{equation}
+\psi_{\bm{k}}(\bm{r})=-\int d\bm{r}'G_0(E_{\bm{k}},|\bm{r}-\bm{r}'|)V(\bm{r}')\psi_{\bm{k}}(\bm{r}').
+\end{equation} 
+%
+Funkcje falowe w obydwóch obszarach muszą spełniać warunki brzegowe na każdej sferze atomowej. 
+
+W ogólnym przypadku macierz rozpraszania zależy od położenia atomu $t_l(E,\bm{R})$
+i równanie na funkcje Greena przyjmuje postać
+%
+\begin{equation}
+[G_{LL'}(E,\bm{R},\bm{R}')]^{-1}=\Big[[B_{LL'}(E,\bm{R}-\bm{R}')]^{-1}-t_l(E,\bm{R})\delta_{\bm{R}\bm{R}'}\delta_{LL'}]\Big],
+\end{equation}
+%
+a stany stacjonarne otrzymujemy z warunku
+%
+\begin{equation}
+\det\Big[t_l^{-1}(E,\bm{R})\delta_{\bm{R}\bm{R}'}\delta_{LL'}-B_{LL'}(E,\bm{R}-\bm{R}')\Big]=0.
+\end{equation}
+
+Funkcja Green pozwala wyznaczyć różne wielkości fizycznych. Na przykład, część urojona funkcji Greena zwiazana jest z lokalną gęstością stanów 
+%
+\begin{equation}
+n_L(E,\bm{R})=-\frac{1}{\pi}\operatorname{Im}G_{LL}(E+i\delta,\bm{R}).
+\end{equation}
+%
+Podobnie część urojona transformaty Fouriera funkcji Green daje nam gęstość spektralną,
+czyli gęstość stanów elektronowych dla danej energii i wektora falowego
+%
+\begin{equation}
+n_L(E,\bm{k})=-\frac{1}{\pi}\operatorname{Im}G_{LL}(E+i\delta,\bm{k}).
+\end{equation}
+%
+
+Metoda funkcji Greena umożliwia wprowadzenie efektywnego (koherentnego) potencjału, który łączy własności dwóch różnych atomów.
+Jest to przybliżenie koherentnego potencjału ({\it ang. coherent potential approximation} - CPA), które stosowane jest do badania
+stopów metalicznych lub układów domieszkowanych \cite{CPA1,CPA2}.
+Metoda polega na uśrednieniu własności rozpraszania dwóch atomów umieszczonych w pobliżu efektywnego potencjału.
+Warunek równoważności średniej ważonej po obydwóch atomach i efektywnego potencjału prowadzi do zespolonego,
+zależnego od energii potencjału CPA. 
+
+
+\subsection{Metoda LAPW}
+
+Metoda zlinearyzowanych stowarzyszonych fal płaskich ({\it ang. linearized augmented plane wave} - LAPW)
+jest modyfikacją oryginalnego podejścia rozszerzonych fal płaskich (APW) zaproponowanego przez Slatera w roku 1937 \cite{slater37}.
+Podobnie jak w metodzie pseudopotencjału, cały kryształ podzielony jest na obszary wewnątrz sfer otaczających atomy ($r<r_c$)
+i obszar międzywęzłowy ($r>r_c$). Funkcje falowe w otoczeniu jądra atomowego charakteryzują sie dużą zmiennością i
+w przybliżeniu maja kształt sferyczny. Z tego względu naturalnym wyborem bazy sa funkcje radialne, będące rozwiązaniami równania Schr\"{o}dingera z
+potencjałem sferycznie symetrycznym. W obszarze więdzywęzłowym, gdzie potencjał zmienia sie bardzo wolno, rozwiązania
+funkcji falowej rozwijane są w bazie fal płaskich. Funkcję falową w metodzie APW można
+zapisać w postaci
+%
+\begin{equation}
+\psi_{\bm{k}j}(\bm{r})=\sum_{\bm{G}}c_{\bm{G}j}\phi_{\bm{k}+\bm{G}}(\bm{r}),
+\end{equation}
+%
+gdzie $\bm{G}$ oznaczają wektory sieci odwrotnej. Funkcje bazowe mogą być zapisane w formie 
+%
+\begin{equation} 
+\phi_{\bm{k}+\bm{G}}(\bm{r})=\begin{cases}
+        e^{i(\bm{k}+\bm{G})\bm{r}} & r>r_c \\
+       \sum_{lm}A_{lm}(\bm{k}+\bm{G})u_l(r,E_l)Y_{lm}(\theta,\varphi) &  r<r_c,
+             \end{cases}    
+\label{APWbase}                      
+\end{equation}
+%
+gdzie $A_{lm}$ są współczynnikami rozwinięcia, $Y_{lm}(\theta,\varphi)$ są harmonikami sferycznymi i $u_l(r,E_l)$ są rozwiązaniami równania
+%
+\begin{equation}
+[-\frac{d^2}{dx^2}+\frac{l(l+1)}{r^2}+V(r)-E_l]ru_l(r,E_l)=0,
+\label{radial}
+\end{equation}
+%
+gdzie $V(r)$ jest potencjałem wewnątrz sfery, a $E_l$ pełni rolę parametru i niekoniecznie jest to energia własna. 
+W przybliżeniu potencjału {\it muffin-tin} (MT), w obszarze atomowym jest on sferycznie
+symetryczny ($V(\bm{r})=V(r)$), a w obszarze międzywezłowym ma wartość stałą ($V(\bm{r})=V_0$) lub zero. Wtedy funkcje falowe $\psi(\bm{r})$ mogą być wyrażone przez rozwiazania równania Schr\"{o}dingera w poszczególnych obszarach kryształu, odpowiednio przez harmoniki sferyczne wokół atomów i fale płaskie między atomami. 
+Współczynniki $A_{lm}$ wyznacza się tak, aby spełniony był warunek ciągłości funkcji falowej na granicy sfery (warunek ciągłości pierwszych pochodnych nie jest spełniony). w ten sposób współczynniki $A_{lm}$ stają się zależne od vectora falowego.  
+Współczynniki $c_{\bm{G}}$ i liczby $E_l$ są parametrami wariacyjnymi metody APW. Rozwiązania, które numerowane są wektorami $\bm{G}$, składające sie z pojedynczych fal płaskich w obszarze miedzywęzłowym i dopasowane do nich rozwiązania radialne wewnątrz sfery nazywane są stowarzyszonymi falami płaskimi.
+
+Funkcje bazowe wewnątrz sfery zależą od energii $E_l$, co powoduje, że odpowiednie równanie własne na funkcje falowe jest nieliniowe. Jest to główny problem
+podejścia APW, uniemożliwiający łatwe rozszerzenie stosowalności tej metody na dowolne potencjały. W szczególności potencjały wyznaczane metodą samozgodną w ramach teorii funkcjonału gęstość.
+Rozwiązaniem tego problemu była zaproponowana przez Andersena w roku 1975 linearyzacja równań APW, która prowadzi do metody LAPW \cite{Andersen75}.
+W tym podejściu, modyfikuje się funkcje bazowe wewnątrz sfery do postaci
+%
+\begin{equation}
+\phi_{\bm{k}+\bm{G}}(\bm{r})=\sum_{lm}[A_{lm}(\bm{k}+\bm{G})u_l(r,E_l)Y_{lm}(\theta,\varphi)+B_{lm}(\bm{k}+\bm{G})\dot{u}_l(r,E_l)Y_{lm}(\theta,\varphi)],
+\end{equation}    
+%
+gdzie współczynniki rozwinięcia $A_{lm}$ i $B_{lm}$ wyznacza się z warunku ciągłości funkcji falowej i jej pochodnej na sferze, a pochodna $\dot{u}_l(r,E_l)=\partial u_l/\partial E$ spełnia równanie
+%
+\begin{equation}
+[-\frac{d^2}{dx^2}+\frac{l(l+1)}{r^2}+V(r)-E_l]r\dot{u}_l(r,E_l)=ru_l(r,E_l).
+\end{equation}
+Dodatkowo narzuca sie warunek normalizacji funkcji $u_l$
+%
+\begin{equation}
+\int_0^{r_c}dr[ru_l(r,E_l)]^2=1,
+\end{equation}
+%
+oraz ortogonalności funkcji $u_l$ i $\dot{u}_l$
+%
+\begin{equation}
+\int_0^{r_c}drr^2u_l(r,E_l)\dot{u}_l(r.E_l)=0.
+\end{equation}
+
+Znajomość gradientu $\dot{u}_l$ umożliwia wyznaczenie funkcji radialnej dla danej energii pasma $\varepsilon$, 
+uwzlędniając poprawkę liniową
+%
+\begin{equation}
+u_l(r,\varepsilon)=u_l(r,E_l)+(\varepsilon-E_l)\dot{u}_l(r,E_l)+O((\varepsilon-E_l)^2),
+\end{equation}
+%
+gdzie $O((\varepsilon-E_l)^2)$ oznacza błąd, który jest rzędu kwadratu różnicy tych dwóch energii.
+
+W ramach procedury samozgodnej, pełny potencjał wyliczany jest w obydwóch obszarach kryształu,
+stosując rozwinięcia w odpowiednich bazach
+%
+\begin{equation} 
+V(\bm{r})=\begin{cases}
+        \sum_{\rm{G}}V_{\bm{G}} e^{i\bm{G}\bm{r}} & r>r_c \\
+       \sum_{lm}V_{lm}Y_{lm}(\theta,\phi) &  r<r_c.
+             \end{cases}   
+\label{fullpot}                       
+\end{equation}
+%
+Podobnie wylicza się gęstość ładunku wewnątrz sfery korzystając ze współczynników rozwinięcia funkcji falowych
+w bazie harmonic sferycznych oraz w obszarze międzywęzłowym w bazie fal płaskich.
+
+W odróżnieniu od metody pseudopetcjału, gdzie uwzględniane są tylko elektrony walencyjne, w metodzie LAPW wyznaczane są
+zarówno stany walencyjne, jak i stany rdzenia.
+Ponieważ funkcje falowe stanów rdzenia sa zlokalizowane blisko jądra atomowego, wylicza się je stosując tylko 
+potencjał centrosymetryczny. Jeżeli zasięg funkcji falowej stanu rdzenia przekracza granicę strefy atomowej,
+stosuje się gładkie przedłużenie potencjału centrosymetrycznego poza tę granice. 
+Do wyznaczenia stanów rdzenia stosuje sie podejście w pełni relatywistyczne,
+czyli rozwiązuje się równanie Diraca, zamiast równania Schr\"{o}dingera (\ref{radial}). 
+Stany walencyjne wyznaczone są dla pełnego potencjału w całym obszarze kryształu.
+
+W niektórych materiałach występują również stany elektronowe, mające wspólne cechy stanów rdzenia i walencyjnych, które nazywane są
+pseudordzeniowymi. Przykładem są wysoko położone i wzlędnie rozległe stany $5d$ w metalach ziem rzadkich.
+Występowanie tych stanów wymaga modyfikacji metody LAPW. W standardowej wersji, baza konstruowana
+jest tak, aby jak najdokładniej opisać stany elektronowe bliskie energii $E_l$
+i zwykle wybiera się tę wartość blisko środka pasma walencyjnego.
+Jeżeli występują dodatkowe stany w innym zakresie energii, wybór wartości $E_l$ nie jest oczywisty. 
+Najlepszym podejściem do opisu stanów pseudordzeniowych jest rozszerzenie bazy przez wprowadzenie dodatkowych 
+lokalnych orbitali (LO), które mieszczą się wewnątrz atomowej sfery. Ta modyfikacja nazywa się metodą APW+LO \cite{singh91}. 
+Ponieważ istnieje swoboda wyboru bazy, pochodną funkcji radialnej $\dot{u}_l$ usuwa sie z głównej bazy i dołacza do LO.
+Zatem całkowita baza w tym podejściu składa się z oryginalnej bazy APW (\ref{APWbase})  
+i dodatkowych orbitali w formie
+%
+\begin{equation}
+\chi^{lo}_L(\bm{r})=[a^{lo}_{lm}u_l(r,E_l)+b^{lo}_{lm}\dot{u}_l(r,E_l)]Y(\theta,\varphi),
+\end{equation}
+%
+gdzie $r<r_c$, a współczynniki $a^{lo}_{lm}$ i $b^{lo}_{lm}$ są tak dobrane, aby LO
+znikały na sferze. Liczba orbitalna ograniczona jest w tym przypadku do wartości, które odpowiadają 
+fizycznym stanom kwantowym ($l\le 3$). W odróżnieniu od LAPW, w metodzie APW+LO nie ma ograniczenia na
+wartości pochodnych funkcji bazowych na sferze atomowej. 
+  
+
+\subsection{Metoda LMTO}
+
+W metodzie zlinearyzowanych orbitali {\it muffin-tin} ({\it ang. linearized muffin-tin orbitals} - LMTO)
+również wprowadza się podział kryształu na obszary wewnątrz sfer atomowych i obszar miedzywęzłowy.
+Funkcja falowa odpowiadająca energii $\varepsilon$ może być zapisana w ogólnej formie 
+%
+\begin{equation}
+\psi_{\bm{k}}(\varepsilon,\bm{r})=\sum_{lm} a_{lm}(\bm{k}) \phi_{lm}(\varepsilon,\kappa,\bm{r}),
+\end{equation}
+%
+gdzie funkcje bazowe w poszczególnych obszarach mają postać
+%
+\begin{equation}
+\phi_{lm}(\varepsilon,\kappa,\bm{r})=i^lY_{lm}(\theta,\phi)\begin{cases}
+        u_l(\varepsilon,r) +\kappa cot(\eta_l(\varepsilon))J_l(\kappa,r) & r<r_c \\
+       \kappa N_l(\kappa,r) &  r>r_c.
+             \end{cases}    
+\end{equation}
+%
+Funkcja $J_l(\kappa,r)$ jest tak skonstruowana, aby funkcje bazowe wewnątrz sfery nie zależały od energii, 
+czyli spełnione było równanie
+%
+\begin{equation}
+\frac{d}{d\varepsilon}\phi_{lm}(\varepsilon,\kappa,\bm{r})=i^lY_{lm}[\dot{u}_l(\varepsilon,r) +\kappa\frac{d}{d\varepsilon} cot(\eta_l(\varepsilon))J_l(\kappa,r)]=0
+\end{equation}
+%
+dla energii $\varepsilon=E_{\nu}$ odpowiadającej średniej energii danego orbitalu LMTO.
+Ten warunek prowadzi do wzoru
+%
+\begin{equation}
+J_l(\kappa,r)=-\frac{\dot{u}_l(E_{\nu},r)}{\kappa\frac{d}{d\varepsilon}cot(\eta_l(E_{\nu}))}.
+\end{equation}
+%
+W obszarze międzywęzłowym funkcje bazowe centrowane w położeniu danej sfery $\bm{R}$ otrzymuje się przez rozwinięcie na funkcji $J_l$
+powiązane z sąsiednimi sferami w położeniach $\bm{R}'$
+%
+\begin{equation}
+N_l(\kappa,\bm{r}-\bm{R})=4\pi\sum_{l',l''}C_{l',l'',l'''}n^*_{l''}(\kappa,\bm{R}-\bm{R'})J_{l'}(\kappa,\bm{r}-\bm{R}').
+\end{equation}
+%
+Taka konstrukcja powoduje, że funkcje LMTO są kombinacją liniową funkcji $u_l$ i $\dot{u}_l$ wewnątrz danej sfery
+i są gładko przedłużane do obszaru międzywęzłowego, łącząc się w sposób ciagły z funkcjami $\dot{u}_l$ z każdej
+sąsiedniej sfery.
+
+Metodę LMTO można uprościć poprzez pominięcie obszaru międzywęzłowego i ograniczenie się do optymalizacji funkcji falowych tylko wewnątrz sfer.
+W podejściu nazywanym przybliżeniem atomowych sfer ({\it ang. atomic sphere approximation} - ASA)
+stosuje się sfery, które częściowo na siebie nachodzą i ich sumaryczna objętość równa się całkowitej objetości kryształu. 
+Zastoswanie takiego podejścia ma uzasadnienie jedynie dla kryształów z gęstym upakowaniem atomów.
+
+\chapter{Poprawki do funkcjonału energii}
+
+Teoria funkcjonału gęstości odniosła wiele spektaklarnych sukcesów w badaniach własności materiałowych.
+Posiada jednak braki, które nie pozwalają opisać poprawnie struktury elektronowej i tych własności, które są konsekwencją wzajemnych oddziaływań elektronowych. 
+Niektóre z tych niedoskonałości wynikają bezpośrednio z przybliżeń stosowanych do wyliczenia energii wymienno-korelacyjnej, 
+której dokładna postać funkcjonalna nie jest znana i prawdopodobnie w ogóle nie istnieje.  
+Do tej grupy należą błedy wynikające z samoodziaływania, czyli niedokładnego zerowania się energii oddziaływania elektronu z jego własnym polem kulombowskim.
+Metoda, która pozwala dokonać korekcji samoodziaływania omówiona będzie w rozdziale~\ref{sec:sic}. 
+Można częściowo zmniejszyć efekt samoodziaływania przez zastosowanie funkcjonału hybrydowego, czyli połączenia energii wymiany wyznaczanej z podejścia Hartree-Focka i 
+przybliżenia stosowango w DFT, co będzie tematem rozdziału~\ref{sec:hybrid}.  
+Drugim ważnym defektem jest brak zależności standardowych funkcjonałów wymienno-korelacyjnych od rodzaju orbitalu. Mówiąc ściślej, 
+stosowane funkcjonały wymienno-korelacyjne nie odróżniają stanów o różnej liczbie kwantowej $m$.
+Wszystkie orbitale traktowane są jednakowo i nie ma możliwości wyróżnienia cech, które decydują o nierównomiernym ich obsadzeniu.
+Taka możliwość często decyduje o charakterze stanu podstawowego i jest szczególnie istotna w układach silnie skorelowanych. 
+Wprowadzenie zależności orbitalnej umożliwia metoda LDA+U, która omówiona będzie w rozdziale~\ref{sec:ldau}.
+Jedną z wielkości, która nie jest poprawnie opisywana w ramach standardowych przyblizeń DFT jest przerwa energetyczna wystepujaca w półprzewodnikach i izolatorach. 
+Problem zaniżonej wartości przerwy energetycznej i metody jej poprawiania omówione będą w rozdziale \ref{sec:gapproblem}.
+
+
+\section{Korekcja samoodziaływania (SIC)}
+\label{sec:sic}
+
+Niedokładności w wartościach energii wymiany i korelacji, wynikające ze stosowania przybliżenia LDA lub GGA, są jednym z głównych źródeł błędów
+w obliczeniach opartych na teorii funkcjonału gęstości. Na przykładzie atomu wodoru, dyskutowanym w rozdziale \ref{sec:GGA}, 
+dobrze widać efekt jaki powodują różnice między wartościami dokładnymi i przybliżonymi.
+Ponieważ wartość bezwględna energii wymiany jest mniejsza niż wartość dokładna, nie kasuje całkowicie dodatniej energii Hartree. 
+Wprawdzie częściowo różnica ta redukowana jest przez niezerową i ujemną energię korelacji, ale pozostaje skończona. 
+Ta dodatkowa energia oddziaływania elektronu z samym sobą jest wielkością niefizyczną i może powodować duże błędy w obliczeniach.
+Samoodziaływanie występuje również w układach wieloelektronowych i przyjmuje bardzo różne wartości
+w zależności od rodzaju materiału. Jest ono zaniedbywalnie małe dla rozległych stanów elektronowych,
+zdelokalizowanych w dużym obszarze danego materiału. Dlatego wpływ samoodziaływania na stany elektronowe w prostych metalach, 
+gdzie dominują stany walencyjne typu $s$ i $p$, może być pominięty. 
+W materiałach, w których występują stany elektronowe mające tendencję do lokalizacji (np. stany {\it 3d} i {\it 4f}),
+jest ono szczególnie silne i często prowadzi do wyników jakościowo niezgodnych z eksperymentem.
+W stanach zlokalizowanych, oddziaływanie elektronu z jego własnym polem kulombowskim, które ma charakter odpychający,
+może prowadzić do znacznego wzrostu energii. Zredukowanie tej energie jest możliwe tylko poprzez zwiększenie delokalizacji
+ładunku i poszerzenie pasm elektronowych. Prowadzi to efektywnie do zmniejszenia przerwy energetycznej
+otrzymywanej w obliczeniach dla półprzewodników i izolatorów. W skrajnym przypadku błędy samoodziaływania powodują delokalizację ładunków w stanach $3d$ 
+i zamknięcie przerwy energetycznej, co skutkuje niewłaściwym stanem metalicznym w materiałach, które w rzeczywistości są izolatorami Motta. 
+
+W roku 1981, Pardew and Zunger zaproponowali metodę korekcji samoodziaływania ({\it ang. self-interaction correction} - SIC) \cite{PZ}.
+Metoda ta dotyczy stanów zlokalizowanych, więc rozpatrujemy zbiór orbitali $\phi_{\alpha\sigma}(\bm{r})$, gdzie $\alpha$ oznacza numer orbitalu,
+a $\sigma$ kierunek spinu. Oznaczamy gęstość elektronową pojedynczego orbitalu przez 
+%
+\begin{equation}
+n_{\alpha\sigma}(\bm{r})=f_{\alpha\sigma}|\phi_{\alpha\sigma}(\bm{r})|^2,
+\end{equation}
+%
+gdzie liczba $f_{\alpha\sigma}$ określa obsadzenie danego orbitalu. 
+Warunkiem braku samoodziaływania w każdym orbitalu jest kompletne wzajemne kasowanie się energii Hartree danej wzorem (\ref{Hartree}) 
+i energii wymienno-korelacyjnej
+%
+\begin{equation}
+E_H[n_{\alpha\sigma}]+E_{xc}[n_{\alpha\sigma}]=0,
+\label{nosic}
+\end{equation}
+%
+gdzie obydwie energie są funkcjonałami gęstości pojedynczego orbitalu.
+Dla pojedynczego elektronu spełniony jest również warunek kasowania
+się energii Hartree i dokładnej energii wymiany wylicznej w podejściu Hartree-Focka
+%
+\begin{equation}
+E_H[n_{\alpha\sigma}]+E_{x}[n_{\alpha\sigma}]=0.
+\end{equation}
+%
+Ponieważ energia korelacji $E_c=E_{xc}-E_{x}$, z tych dwóch warunków wynika, że energia korelacji dla pojedynczego orbitalu musi się zerować
+%
+\begin{equation}
+E_c[n_{\alpha\sigma}]=0.
+\end{equation}
+%
+Warunki te nie są spełnione dla energii wymienno-korelacyjnej danej wzorami w przybliżeniu LDA (\ref{xclda})
+lub GGA (\ref{xcgga}), co prowadzi do samoodziaływania elektronów. Oznaczmy ogólnie przez $E_{xc}^{DFT}[n_{\uparrow},n_{\downarrow}]$ funkcjonał wymienno-korelacyjny wyznaczonony w jednym z tych przybliżeń. Wtedy wzór na poprawioną energię wymienno-korelacyjną można zapisać w formie
+%
+\begin{equation}
+E_{xc}^{SIC}[n_{\alpha\sigma}]=E_{xc}^{DFT}[n_{\uparrow},n_{\downarrow}]-\sum_{\alpha\sigma}\delta_{\alpha\sigma},
+\label{xcsic}
+\end{equation}
+%
+gdzie
+%
+\begin{equation}
+\delta_{\alpha\sigma}=E_H[n_{\alpha\sigma}]+E_{xc}^{DFT}[n_{\alpha\sigma}],
+\end{equation}
+%
+a sumowanie jest tylko po zajętych orbitalach $\alpha$.
+Dla dokładnej wartości energii wymienno-korelacyjnej mielibyśmy $\delta_{\alpha\sigma}=0$,
+czyli spełniony byłby warunek (\ref{nosic}). 
+Energia wymienno-korelacyjna zdefiniowana wzorem (\ref{xcsic}) nie jest funkcjonałem gęstości elektronowej,
+tylko funkcjonałem orbitali atomowych. Powoduje to, że również funkcjonał energii całkowitej (\ref{EKS}) staje się zależny od orbitali
+% 
+\begin{equation}
+E^{SIC}[n_{\alpha\sigma}]=\sum_{\alpha\sigma}\langle\phi_{\alpha\sigma}|-\frac{\hbar^2}{2m}\nabla^2|\phi_{\alpha\sigma}\rangle+E_{ext}[n]+E_H[n]+E_{xc}^{SIC}[n_{\alpha\sigma}].
+\label{esic}
+\end{equation}
+%
+Podobnie, jak w przypadku funkcjonału Kohna-Shama, możemy zastosować metodę wariacyjną do zmninimalizowania tej energię względem
+zbioru orbitali $\phi_{\alpha\sigma}$, przy założonym warunku ortonormalności 
+%
+\begin{equation}
+\langle\phi_{\alpha\sigma}|\phi_{\alpha'\sigma'}\rangle=\delta_{\alpha\alpha'}\delta_{\sigma\sigma'}.
+\end{equation}
+%
+Dostajemy wtedy równania jednocząstkowe analogiczne do równań Kohna-Shama
+%
+\begin{equation}
+[-\frac{\hbar^2}{2m}\nabla^2+V_{eff}(n_{\alpha\sigma},\bm{r})]\phi_{\alpha\sigma}(\bm{r})=\varepsilon_{\alpha\sigma}^{SIC}\phi_{\alpha\sigma}(\bm{r}),
+\label{eqsic}
+\end{equation} 
+%
+z efektywnym potencjałem danym wzorem
+%
+\begin{equation}
+V_{eff}(n_{\alpha\sigma},\bm{r})=V^{DFT}(\bm{r})-V_H(n_{\alpha\sigma},\bm{r})-V_{xc}(n_{\alpha\sigma},\bm{r}),
+\end{equation}
+%
+gdzie ostatnie dwa wyrazy to korekcja samoodziaływania do efektywnego potencjału elektronowego.
+Podobnie jak w przypadku stanów Kohna-Shama spełniona jest zależność
+%
+\begin{equation}
+\varepsilon_{\alpha\sigma}^{SIC}=\frac{\partial E^{SIC}}{\partial f_{\alpha\sigma}}.
+\end{equation}
+
+Równanie (\ref{eqsic}) pozwala wyznaczyć orbitale i energie własne zlokalizowanych stanów elektronowych. 
+Jednak w rzeczywistych materiałach, mamy często doczynienia z współistnieniem stanów zlokalizowanych i zdelokalizowanych, co powoduje, że rozwiązania tego równania nie muszą odpowiadać globalnemu minimum. Pełna optymalizacja funkcji falowych powinna 
+uwzlędniać zarówno stany zlokalizowane (typu Wanniera) i zdelokalizowane (typu Blocha).
+W praktyce polega to na wyznaczeniu różnych możliwych konfiguracji stanów zlokalizowanych i zdelokalizowanych,
+a następnie wybraniu tej o najniższej energii.    
+
+\begin{figure}[h]
+\centering
+\includegraphics[scale=1]{rare-earths.pdf}
+\caption{Różnice energii całkowitej dla walencyjności II i III wyliczone dla ziemiach rzadkich (o), porównane z eksperymentem (linia przerywana)
+i wartościami dla związków ziem rzadkich z siarką (+). Rysunek z pracy \cite{RE}.}
+\label{fig:rare}
+\end{figure}
+
+
+Dobrym przykładem są metale ziem rzadkich, w których występują zarówno stany zlokalizowane ($4f$),
+jak i zdelokalizowane ($5d$ i $6s$). Dodatkowo występują dwa rodzaje elektronów $4f$. Cześć z nich jest zlokalizowana,
+tak jak elektrony rdzenia, mając decydujący wpływ na moment magnetyczny i walencyjność danego atomu. 
+Pozostałe, w wyniku hybrydyzacji ze stanami $5d$ i $6s$, tworzą pasmo elektronowe i biorą udział w wiązaniu metalicznym.
+W ramach tradycyjnych przybliżeń energii wymienno-korelacyjnej nie można opisać prawidłowo tak złożonej
+struktury elektronowej ziem rzadkich. W pracy \cite{RE} zastosowano metodę SIC do wyliczenia obsadzeń
+stanów $4f$ i walencyjności we wszystkich atomach ziem rzadkich (rysunek \ref{fig:rare}).
+Walencyjność zdefiniowana jest jako ilość stanów zdelokalizowanych przypadająca na jeden atom i wyliczana jest ze wzoru
+%
+\begin{equation}
+N_{val}=Z-N_{core}-N_{SIC},
+\end{equation}
+%
+gdzie $Z$ jest liczbą atomową, $N_{core}$ ilością elektronów w rdzeniu i $N_{SIC}$ liczbą stanów zlokalizowanych $4f$,
+dla których odjęto energię samoodziaływania. Dla ziem rzadkich walencyjność przyjmuje dwie możliwe wartości:
+$N_{val}=2$ (atomy dwuwartościowe, II) lub $N_{val}=3$ (atomy trójwartościowe, III).
+Dla tych dwóch przypadków wyliczone różnice ich energii całkowitych $E_{II}-E_{III}$ pokazane są na rysunku \ref{fig:rare}.
+Otrzymane wartości bardzo dobrze zgadzają się z danymi eksperymentalnymi. 
+Jak widać, oprócz Eu i Yb, które są dwuwartościowe, wszystkie pozostałe ziemie rzadkie są trójwartościowe. 
+Podobnie dobrą zgodność z eksperymentem otrzymano dla promieni Wignera-Seitza, które decydują o stałej sieci kryształów ziem rzadkich \cite{RE}.
+
+
+\section{Funkcjonały hybrydowe}
+\label{sec:hybrid}
+
+Problem samoodziaływania nie występuje w przybliżeniu Hartree-Focka, gdzie dla każdego elektronu
+jego własne oddziaływanie kulombowskie kasowane jest przez oddziaływanie wymienne.
+Zatem częściowe uwzlędnienie dokładnej wartości energii wymiany w funkcjonale wymienno-korelacyjnym powinno
+zmniejszyć efekt samoodziaływania i w ten sposób poprawić całkowitą energię układu. 
+Takie funkcjonały, w których energia wymiany otrzymana w przybliżeniu LDA lub GGA jest częściowo zastąpiona
+dokładną wartością otrzymaną z metody Hartree-Focka, nazywane są funkcjonałami hybrydowymi.  
+Pierwszy funkcjonał hybrydowych zaproponował Becke w roku 1993 \cite{becke93}, uzasadniając jego postać
+na podstawie tzw. formuły adiabatycznego połączenia ({\it ang. adiabatic connection formula}).
+Rozważmy ogólny hamiltoniana w postaci
+%
+\begin{equation}
+H_{\lambda}=T+\lambda V,
+\end{equation}
+%
+gdzie $T$ jest operatorem energii kinetycznej, $V$ jest potencjałem oddziaływania między elektronami i $\lambda$ jest parametrem, który
+określa siłę tego ddziaływania ($0\leq\lambda\leq 1$). 
+Funkcje falowe $\psi_{\lambda}$ i energie własne $E_{\lambda}$ tego hamiltonianu są zależne od parametru $\lambda$.
+Dla nich spełnione jest równanie
+%
+\begin{equation}
+\frac{dE_{\lambda}}{d\lambda}=\langle\psi_{\lambda}|V|\psi_{\lambda}\rangle,
+\end{equation} 
+%
+z którego dostajemy
+%
+\begin{equation}
+E_{\lambda=1}=E_{\lambda=0}+\int_0^1 d\lambda \langle\psi_{\lambda}|V|\psi_{\lambda}\rangle.
+\end{equation}
+%
+Energia dla $\lambda=0$ odpowiada energii kinetycznej, natomiast $\lambda=1$ daje energie z pełnym oddziaływaniem elektronowym $V$.
+Ta formuła określa właśnie adiabatyczne połączenie między układem nieoddziałujących i oddziałujących elektronów.
+Po zastosowaniu tej formuły do funkcjonału energii (z pominięciem potencjału zewnętrznego $V_{ext}$) dostajemy 
+%
+\begin{equation}
+E[n]=T[n]+\int_0^1 d\lambda \langle\psi_{\lambda}|V|\psi_{\lambda}\rangle.
+\end{equation}
+%
+Biorąc pod uwagę funkcjonał Kohna-Shama (\ref{EKS}), możemy napisać wzór na energię wymienno-korelacyjną w postaci
+%
+\begin{equation}
+E_{xc}[n]=\int_0^1 d\lambda \langle\psi_{\lambda}|V|\psi_{\lambda}\rangle-E_H[n]=\int_0^1d\lambda E_{xc,\lambda}[n].
+\label{xcn}
+\end{equation}
+%
+Powyższą całkę można przybliżyć, rozpatrując jedynie wartości końcowe. Dla $\lambda=0$, dostajemy energie wymiany
+w przybliżeniu Hartree-Focka, $E_{xc,0}=E_x^{HF}$. Becke zaproponował, aby dla $\lambda=1$ zastosować jedno z przybliżeń stosowanych w DFT (LDA lub GGA),
+$E_{xc,1}=E_{xc}^{DFT}$, a dla pośrednich wartości $\lambda$ zastosować interpolację liniową
+%  
+\begin{equation}
+E_{xc,\lambda}[n]=(1-\lambda)E_x^{HF}[n]+\lambda E_{xc}^{DFT}[n].
+\end{equation}
+%
+Po wstawieniu tej zależności do (\ref{xcn}) i wyliczeniu całki dostajemy funkcjonał hybrydowy 
+%
+\begin{equation}
+E_{xc}^{hyb}[n]=\frac{1}{2}E_x^{HF}+\frac{1}{2}E_{xc}^{DFT}[n].
+\label{becke}
+\end{equation} 
+%
+
+Pardew, Ernzerhof i Burke w roku 1996 powiązali zależność energii wymienno-korelacyjnej od $\lambda$ \cite{PBE0} 
+z rozwinięciem perturbacyjnym Mollera-Plesseta i zaproponowali bardziej ogólną formę 
+%
+\begin{equation}
+E_{xc,\lambda}[n]=E_{xc}^{DFT}[n]+(E_x^{HF}[n]-E_x^{DFT}[n])(1-\lambda)^{m-1},
+\end{equation}
+%
+która prowadzi do funkcjonału w postaci
+%
+\begin{equation}
+E_{xc}^{hyb}[n]=E_{xc}^{DFT}[n]+\frac{1}{m}(E_x^{HF}[n]-E_x^{DFT}[n]).
+\end{equation}
+%
+Dla $m=1$, energia wymiany jest równa dokładnej wartości $E_x^{HF}$. W tym przypadku dodanie przybliżonej wartość energii korelacji $E_c^{DFT}$ 
+poprawia wyniki w porównaniu do przybliżenia Hartree-Focka. Wyniki są jednak gorsze niż w przybliżeniach LDA lub GGA, w których błędy energii wymiany i korelacji mają przeciwne znaki, co powoduję ich częściowe kasuwanie się. Przypadek $m=2$ odpowiada funkcjonałowi z pracy~\cite{becke93}, który dany jest wzorem~(\ref{becke}). 
+Za optymalny uznano wybór $m=4$~\cite{PBE0}, wskazując na bardzo dobrą zgodnością wyników otrzymanych w ramach teorii perturbacyjnej Mollera-Plesseta czwartego rzędu z danymi eksperymentalnymi \cite{pople89}. Co więcej, w tym przypadku zarówno wartości $E_{xc,\lambda}$ i $E_{xc}^{DFT}$, jak również ich pierwsze oraz drugie pochodne wzajemnie pokrywają się dla $\lambda=1$. Warunek ciągłości pochodnych nie jest spełniony dla $m=2$.
+
+Oddziaływanie wymienne ma charakter krótkozasięgowy i można uwzględnić wkład od dokładnej wartości jedynie
+w pewnym zakresie odległości. Uwzględniając ten efekt Heyd, Scuseria i Enzerhof (HSE) \cite{HSE} zaproponowali nowy funkcjonał
+w postaci
+%
+\begin{equation}
+E_{xc}^{hyb}(\mu) = (1-a)E_{x}^{GGA,SR}(\mu)+aE_x^{HF,SR}(\mu) + E_x^{GGA,LR} + E_c^{GGA},
+\end{equation}
+%
+gdzie $E_x^{GGA,SR}$ i $E_x^{GGA,LR}$ odpowiadają krótkozasięgowemu i długozasięgowemu oddziaływaniu wymiennemu w przybliżeniu GGA,
+$E_x^{HF,SR}$ jest częścią krótkozasięgowego dokladnego oddziaływania wymiennego, a parametr $\mu$
+określa zasięg oddziaływania krótkozasięgowego. Parametr $a$ związany jest z wykładnikiem $m$ prostą zależnością $a=\frac{1}{m}$.
+
+\section{Funkcjonały OEP}
+\label{sec:mgga}
+
+Główną zaletą funkcjonału Kohna-Shama, która umożliwia obliczenia nawet dla dużych układów, jest lokalny charakter potencjału
+wymienno-korelacyjnego. Dokładny potencjał jest nielokalny i ogólnie jest zależny od funkcji falowych.
+Taką formę ma energia oddziaływania wymiennego zależna od orbitali jednocząstkowych w ramach przybliżenia Hartree-Focka. 
+Omawiany wcześniej potencjał efektywny w metodzie SIC również jest zalezny od orbitali atomowych.
+Podejście nazywane metodą zoptymalizowanego potencjału efektywnego ({\it ang. optimized effective potential} - OEP)  
+zakłada funkcjonalną zależność energii wymienno-korelacyjnej od orbitali $\phi_{j\sigma}$, przy zachowanym lokalnym charakterze potencjału $V^{OEP}_{\sigma}$
+\cite{Sharp,Talman}. 
+Energię całkowitą Kohna-Shama możemy zapisać jako funkcjonał orbitali
+%
+\begin{equation}
+E[\{\phi_{j\sigma}\}]=T[\{\phi_{j\sigma}\}]+\int d\bm{r} V_{ext}(\bm{r})n(\bm{r})+\frac{1}{2}\int \int d\bm{r} d\bm{r}' \frac{n(\bm{r})n(\bm{r}')}{|\bm{r}-\bm{r}'|}+E_{xc}[\{\phi_{j\sigma}\}].
+\end{equation}
+Metoda OEP ma charakter wariacyjny i opiera się na dwóch równaniach. Pierwsze określa warunek minium (stacjonarności) energii całkowitej
+względem efektywnego potencjału
+%
+\begin{equation}
+\frac{\delta E[\{\phi_{j\sigma}\}]}{\delta V^{OEP}_{\sigma}}=0.
+\end{equation} 
+%
+Drugie z nich ma postać równania Kohna-Shama z efektywnym potencjałem zależnym od orbitali
+%
+\begin{equation}
+\Big[-\frac{\hbar^2}{2m}\nabla^2+V^{OEP}_{\sigma}[\{\phi_{j\sigma}\}](\bm{r})\Big]\phi_{j\sigma}(\bm{r})=\varepsilon_{j\sigma}\phi_{j\sigma}(\bm{r}).
+\end{equation}
+%
+Część potencjału opisującą oddziaływanie wymienno-korelacyjne moża wyrazić w formie:
+%
+\begin{equation}
+V^{OEP}_{xc\sigma}(\bm{r})=\frac{\delta E_{xc}[\{\phi_{j\sigma}\}]}{\delta n_{\sigma}(\bm{r})}=\sum_{\nu\mu\i}\int\int d\bm{r}' d\bm{r}''\frac{\delta E_{xc}}{\delta \phi_{i\nu}(\bm{r}')}\frac{\delta \phi_{i\nu}(\bm{r}')}{\delta V^{OEP}_{\mu}(\bm{r}'')}\frac{\delta V^{OEP}_{\mu}(\bm{r}'')}{\delta n_{\sigma}(\bm{r})}.
+\label{voep}
+\end{equation}
+%
+Drugą pochodną funkcjonalna pod całką można wyrazić w pierwszym rzędzie rozwinięcia pertubacyjnego przy pomocy funkcji Greena dla nieodziałujęcego układu elektronów
+%
+\begin{equation}
+\frac{\delta \phi_{i\nu}(\bm{r}')}{\delta V^{OEP}_{\mu}(\bm{r}'')}=\delta_{\nu\mu}G^R_{i\mu}(\bm{r}',\bm{r}'')\phi_{i\mu}(\bm{r}'')=\delta_{\nu\mu}\sum_{k\neq i}\Big[\frac{\phi^*_{k\mu}(\bm{r}')\phi_{k\mu}(\bm{r}'')}{\varepsilon_{i\mu}-\varepsilon_{k\mu}}\Big]\phi_{i\mu}(\bm{r}'').
+\end{equation}
+%
+Trzecia pochodna jest odwrotnością liniowej funkcji odpowiedzi, którą również możemy wyrazić przez funkcję Greena
+%
+\begin{equation}
+\chi_{\sigma}(\bm{r},\bm{r}'')=\delta_{\sigma\mu}\frac{\delta n_{\sigma}(\bm{r})}{\delta V^{OEP}_{\mu}(\bm{r}'')}=\delta_{\sigma\mu}\sum_i G^R_{i\sigma}(\bm{r}'',\bm{r})\phi_{i\sigma}(\bm{r}'')\phi^*_{i\sigma}(\bm{r}).
+\label{vchi}
+\end{equation}
+%
+Mnożąc równanie (\ref{voep}) przez $\chi_{\sigma}(\bm{r}',\bm{r})$, całkując wzlędem $\bm{r}'$ i wykorzystując (\ref{vchi}) dostajemy 
+równanie całkowe metody OEP
+%
+\begin{equation}
+\sum_i\int d\bm{r}' \phi^*_{i\sigma}(\bm{r})[V^{OEP}_{xc\sigma}(\bm{r})-u^{OEP}_{xci\sigma}(\bm{r})]G^R_{i\sigma}(\bm{r}',\bm{r})\phi_{i\sigma}(\bm{r}) + c.c.=0,
+\label{oepint}
+\end{equation}
+%
+gdzie
+%
+\begin{equation}
+u^{OEP}_{xci\sigma}(\bm{r})=\frac{1}{\phi^*_{i\sigma}(\bm{r})}\frac{\delta E_{xc}[\{\phi_{j\sigma}\}]}{\delta \phi_{i\sigma}(\bm{r})}.
+\end{equation}
+%
+Jeżeli funkcjonał wymienno-korelacyjny zależy tylko od gęstości, wtedy $u^{OEP}_{xci\sigma}(\bm{r})=V^{OEP}_{xc\sigma}(\bm{r})$ i równanie (\ref{oepint})
+jest automatycznie spełnione.
+W ramach metody OEP, część wymienną można traktować dokładnie, tak jak w przybliżeniu Hartree-Focka, i wtedy potencjał zależny od orbitali
+ma postać
+%
+\begin{equation}
+u^{OEP}_{xi\sigma}=-\frac{1}{\phi^*_{i\sigma}(\bm{r})}\sum_j \phi^*_{j\sigma}\int d\bm{r}'\frac{\phi^*_{i\sigma}(\bm{r}')\phi_{j\sigma}(\bm{r}')}{|\bm{r}-\bm{r}'|}.
+\end{equation} 
+%
+
+\section{Funkcjonały meta-GGA}
+\label{mgga}
+
+Dwa podstawowe przybliżenia stosowane do opisu funkcjonału wymienno-korelacyjnego uwzlędniają tylko gęstość ładunku w danym punkcie
+przestrzeni (LDA) lub dodatkowo jego gradient (GGA). 
+Kolejnym krokiem pozwalającym poprawić ten funkcjonał jest uwzlędnienie jego zależności od gradientu orbitali, czyli 
+części kinetycznej całkowitej energii. Takie funkcjonały noszą nazwę meta-GGA i zapisywane są w ogólnej postaci
+%
+\begin{equation}
+E_{xc}[n_{\uparrow},n_{\downarrow}]=\int d\bm{r} n(\bm{r}) \varepsilon_{xc}(n_{\uparrow},n_{\downarrow},\nabla n_{\uparrow},\nabla n_{\downarrow},\tau_{\uparrow},\tau_{\downarrow}),
+\end{equation}
+%
+gdzie $\tau_{\sigma}$ jest gęstością energii kinetycznej wyznaczoną dla obsadzonych stanów Kohna-Shama
+%
+\begin{equation}
+\tau_{\sigma}= \frac{1}{2}\sum_i |\nabla \psi_{i\sigma}|^2.
+\end{equation}
+%
+Uzasadnieniem włączenia $\tau_{\sigma}$ do funkcjonału jest występowanie tej wielkości w rozwinięciu Taylora
+uśrednionej sferycznie dziury wymiennej wokół elektronu~\cite{Becke1998}. Drugą korzystną cecha jest możliwość zredukowania samoodziaływania
+w części korelacyjnej funkcjonału.
+
+Od roku 2003 zaproponowano kilka wersji funkcjonałów meta-GGA~\cite{TPSS,Becke2006,revTPSS,Tran2009,SCAN}. 
+Najdokładniejszym z nich jest funkcjonał SCAN ({\it ang. strongly contrained and appropriately normed}), który 
+spełnia 17 znanych warunków i norm dla funkcjonału wymienno-korelacyjnego~\cite{SCAN}.
+Jednym z nich jest warunek dla współczynnika określającego stosunek energii wymiany do odpowiedniej wartości w LDA, $F_x=E_x/E^{LDA}_x$, który
+nie może przekraczać wartości 1.174.
+Poprawnie opisuje również układy, dla których dokładne wartości są znane, np. jednorodny gaz elektronowy.  
+Funkcjonał SCAN zależy od bezwymiarowego parametru 
+%
+\begin{equation}
+\alpha=\frac{\tau-\tau^w}{\tau^{u}},
+\end{equation}
+%
+gdzie $\tau^w=|\nabla n|^2/8n$ określa granicę $\tau$ dla pojedynczego orbitalu, a $\tau^u=(3/10)(3\pi^2)^{2/3}n^{5/3}$ jest jego wartością
+dla jednorodnego gazu.
+Parametr $\alpha$ wykorzystywany jest do charakteryzacji rozkładu gęstości elektronowej i pozwala odróżnić
+układy z wolnozmienną gestością elektronową, typową dla metali ($\alpha\approx 1$), 
+materiały z wiązaniami kowalencyjnymi między pojedynczymi orbitalami ($\alpha=0$) oraz słabymi wiązaniami niekowalencyjnymi między
+zamkniętymi powłokami atomowymi ($\alpha\rightarrow\infty$).
+Jest on bezpośrednio powiązany z funkcją lokalizacji elektronowej ({\it ang. electron localization function} - ELF)
+%
+\begin{equation}
+\textrm{ELF} = \frac{1}{1+\alpha^2},
+\end{equation}     
+%
+która przyjmuje wartości z przedziału $(0,1)$ i jest używana do opisu wiązań chemicznych~\cite{ELF1,ELF2,ELF3}.
+
+Część wymienną funkcjonału można zapisać w postaci
+%
+\begin{equation}
+E_x[n]=\int d\bm{r} n(\bm{r}) \varepsilon_x(n)F_x(s,\alpha),
+\end{equation}
+%
+gdzie $\varepsilon_x(n)$ jest energią wymiany dla jednorodnego gazu przypadającą na pojedynczy elektron,
+a $s$ jest bezwymiarowym gradientem gęstości
+%
+\begin{equation}
+s=\frac{|\nabla n|}{2(3\pi^2)^{1/3}n^{4/3}}.
+\end{equation}
+%
+Rysunek~\ref{fig:Fxs} pokazuje zależność współczynnika $F_x$ od $s$ dla trzech charakterystycznych wartości $\alpha$.
+$F_x$ spełnia odpowiednie warunki zarówno dla małych $s\rightarrow0$, jak i dużych wartości $s\rightarrow\infty$.  
+Dla $\alpha=1$ i małych $s$, $F_x$ pokrywa się z wartościami dla funkcjonału PBE.
+Wartości $F_x$ dla innych wartości $\alpha$ uzyskuje się przez odpowiednią interpolację między $\alpha=0$ i $\alpha=1$ oraz ekstrapolację
+do $\alpha\rightarrow\infty$.
+%
+\begin{figure}[h]
+\centering
+\includegraphics[scale=0.5]{Fxs.png}
+\caption{Współczynnik $F_x$ w funkcji $s$ dla różnych wartości $\alpha$. Rysunek z pracy \cite{SCAN}.}
+\label{fig:Fxs}
+\end{figure}
+%
+
+Podobnie skonstruowany jest całkowity fukcjonał wymienno-korelacyjny $F_{xc}=F_{x}+F_{c}$, który w granicy
+dużych gęstości niespolaryzowanego gazu przyjmuje wartość energii wymiennej $F_x$.
+Dla małych gęstości funkcjonał spełnia warunek ograniczenia Lieba-Oxforda, $F_{xc}\leq 2.215$.
+ 
+
+
+
+
+\section{Metoda LDA+U}
+\label{sec:ldau}
+
+Metoda LDA+U została zaproponowana przez Anisimova, Zaanena i Andersena w roku 1991~\cite{anisimov}.
+Jednym z pierwszych jej zastosowań było wyliczenie struktury elektronowej dla nadprzewodnika wysokotemperaturowego La$_2$CuO$_4$~\cite{czyzyk}.
+Metoda ta polega na połaczeniu podejścia DFT z modelem Hubbarda, który stosowany jest od lat sześćdziesiątych ubiegłego wieku do opisu układów silnie skorelowanych elektronów~\cite{hubbard}. Zacznę od omówienia tego modelu. 
+
+W najbardziej tradycyjnym podejściu, stany elektronowe w krysztale dzielimy na dwie grupy: stany zlokalizowane, które znajdują się w obrębie rdzenia atomowego i posiadają cechy orbitali atomowych oraz stany zdelokalizowane, które rozciągają się w całej przestrzeni kryształu.
+O własnościach transportowych danego materiału decydują głównie elektrony należące do drugiej grupy.
+Natomiast jeżeli chodzi o własności magnetyczne, to zarówno elektrony zlokalizowane, jak i wędrowne mogą oddziaływać wymiennie
+i decydować o rodzaju uporządkowania magnetycznego. W pierwszym przypadku mówimy o zlokalizowanych momentach magnetycznych, 
+których oddziaływanie można opisać w ramach modelu Heisenberga. W drugim mamy doczynienia z magnetyzmem pasmowym, który wynika
+z oddziaływania między mobilnymi elektronami walencyjnymi, które można opisać przy pomocy funkcji Blocha.
+
+Występują również stany elektronowe, które posiadają cechy obydwu tych grup. Stany te tworzą pasma, zachowując jednocześnie
+cechy orbitali atomowych. Ruch elektronów w takiej sytuacji polega na przeskokach między lokalnymi orbitali i odpowiadajace im pasma są znacznie węższe od
+typowych pasm metalicznych. Żródłem takiego zachowania są efekty korelacyjne, które uniemożliwiają swobodny przepływ elektronów w krysztale.
+Najlepszym przykładem są stany należące do niezapełnionej powłoki $3d$ w metalach przejściowych.
+Zgodnie z zakazem Pauliego, w każdym orbitalu mogą znajdować się tylko dwa elektrony z przeciwnymi kierunkami spinu. 
+Jeżeli występuje silne, lokalne oddziaływanie kulombowskie między ładunkami, to stany kwantowe, w których dwa elektrony   
+znajdują się w tym samym orbitalu są niekorzystne energetycznie. Zatem ruch elektronu jest bardzo utrudniony i dla odpowiednio dużego oddziaływania kulombowskiego
+elektrony zostają zlokalizowane. Materiały, w których występuje taki mechanizm lokalizacji to izolatory Motta.
+
+Hubbard zaproponował model do opisu stanów elektronowych, które posiadają cechy zlokalizowanych orbitali atomowych,
+wynikające z korelacji elektronowych~\cite{hubbard}.
+Jest to przykład  modelu ciasnego wiązania z orbitalami Wanniera, który opisany jest w rozdziale~\ref{sec:wannier}. 
+W najprostszym przypadku rozważamy pojedyncze orbitale typu $s$, zlokalizowane w węzłach atomowych. Silne korelacje elektronów znajdujących
+się w tym samym orbitalu można uwzględnić dodają do hamiltonianu lokalne oddziaływanie kulombowskie o ustalonej energii $U$. 
+Hamiltonian modelu Hubbarda zapisujemy stosując formalizm drugiego kwantowania
+%
+\begin{equation}
+H=\sum_{i,j,\sigma}t_{ij}c^{\dagger}_{i\sigma}c_{j\sigma}+U\sum_i n_{i\uparrow}n_{i\downarrow},
+\label{hubbard}
+\end{equation}
+%
+gdzie $c^{\dagger}_{i\sigma}$ i $c_{i\sigma}$ to operatory kreacji i anihilacji elektronu ze spinem $\sigma$ w zlokalizowanym orbitalu Wanniera $w(\bm{r}-\bm{R}_i)$,
+a całki przeskoku dane są wzorem
+%
+\begin{equation}
+t_{ij}=\int d\bm{r} w^*(\bm{r}-\bm{R}_i)H_0w(\bm{r}-\bm{R}_j).
+\end{equation}
+%
+$H_0$ jest jednocząstkowym hamiltonianem złożonym z energii kinetycznej i efektywnego potencjału atomowego. 
+Najczęście uwzlędnia się tylko całki przeskoku między najbliższymi sąsiadami. 
+Operator liczby cząstek ze spinem $\sigma$ na węźle $i$ dany jest wyrażeniem $n_{i\sigma}=c^{\dagger}_{i\sigma}c_{i\sigma}$.
+Energię oddziaływania kulombowskiego dwóch elektronów w tym samym orbitalu można wyliczyć ze wzoru
+%
+\begin{equation}
+U=\int d\bm{r}d\bm{r}' |w(\bm{r}-\bm{R}_i)|^2\frac{e^2}{|\bm{r}-\bm{r}'|}|w(\bm{r}-\bm{R}_i)|^2.
+\end{equation}
+%
+Jeżeli wykonamy transformatę Fouriera operatorów kreacji i anihilacji,
+pierwszy wyraz hamiltonianu (\ref{hubbard}) przyjmie postać $\sum_{\bm{k}\sigma}\varepsilon_{\bm{k}}n_{\bm{k}\sigma}$,
+gdzie $\varepsilon_{\bm{k}}$ opisuje strukturę pasmową w modelu ciasnego wiązania,
+a $n_{\bm{k}\sigma}=c^{\dagger}_{\bm{k}\sigma}c_{\bm{k}\sigma}$ jest operatorem liczby obsadzeń w stanie $\bm{k}$ i spinie $\sigma$.
+
+Zaproponowano kilka sformułowań metody LDA+U \cite{anisimov,orbital1,dudarev}, które różnią się opisem lokalnych 
+oddziaływań eletronowych. Ogólny schemat używany do wyliczenia całkowitej energii ma postać
+%
+\begin{equation}
+E_{tot}=E_{DFT}+E_U-E_{dc},
+\label{eldau}
+\end{equation}
+%
+gdzie $E_{DFT}$ jest energią układu otrzymaną w ramach metody DFT (w przybliżeniu LDA lub GGA), $E_U$ jest energią lokalnych oddziaływań elektronowych,
+a $E_{dc}$ odpowiada energii oddziaływań kulombowskich w przybliżeniu średniego pola.
+Ten ostatni wyraz jest konieczny, aby odjąć przybliżoną energię oddziaływań elektronowych, która uwzlędniona jest w $E_{DFT}$.
+W pracy \cite{orbital1} zaproponowano uogólnioną postać energii $E_U$, która uwzglądnia orbitalną zależność oddziaływań elektronowych
+%
+\begin{multline}
+E_U=\frac{1}{2}\sum_{i,\{m\},\sigma}[\langle m,m''|V_{ee}|m',m'''\rangle n_{i\sigma}^{mm'}n_{i-\sigma}^{m''m'''}
+\\+(\langle m,m''|V_{ee}|m',m'''\rangle - \langle m,m''|V_{ee}|m''',m'\rangle  )n_{i\sigma}^{mm'}n_{i\sigma}^{m''m'''}],
+\end{multline}  
+%
+gdzie $i$ numeruje węzły sieci,  $\{m\}=(m, m',m'', m''')$ są magnetycznymi liczbami kwantowymi,
+a elementy macierzowe oddziaływania kulombowskiego dane są wzorem
+%
+\begin{equation}
+\langle m,m''|V_{ee}|m',m'''\rangle=\int d\bm{r} \int \bm{r}' \phi_{lm}^*(\bm{r})\phi_{lm'}(\bm{r})\frac{e^2}{|\bm{r}-\bm{r}'|}\phi_{lm''}^*(\bm{r}')\phi_{lm'''}(\bm{r}').
+\label{integral}
+\end{equation}
+%
+Występujące pod całką atomowe funkcje falowe określone są dla orbitalnej liczby kwantowej $l$, która definiują zakres magnetycznych liczb kwantowych $-l\le \{m\} \le l$. Liczby obsadzeń orbitali atomowych tworzą tensor drugiego rzędu, którego elementy wyznaczane są przez rzutowanie funkcji falowych Kohna-Shama $\psi_{\bm{k}j}^{\sigma}$ na orbitale atomowe
+%
+\begin{equation}
+n_{i\sigma}^{mm'}=\sum_{\bm{k},j} f_{\bm{k}j}^{\sigma}\langle\psi_{\bm{k}j}^{\sigma}|\phi_{lm'}\rangle\langle\phi_{lm}|\psi_{\bm{k}j}\rangle, 
+\end{equation}
+%
+gdzie współczynniki $f_{\bm{k}j}^{\sigma}$ okreśkaja obsadzenia stanów $j$ z wektorem falowym $\bm{k}$ i spinie $\sigma$.
+Całkę (\ref{integral}) można przedstawić w postaci sumy iloczynów części kątowej i radialnej 
+%
+\begin{equation}
+\langle m,m''|V_{ee}|m',m'''\rangle=\sum_p a_p(m,m',m'',m''')F^p,
+\label{vee}
+\end{equation}
+%
+gdzie $p$ jest liczbą parzystą z przedziału $0\le p \le 2l$. Część kątowa $a_p$ wyliczana jest przy pomocy iloczynów współczynników Clebsha-Gordana 
+%
+\begin{equation}
+a_p(m,m',m'',m''')=\frac{4\pi}{2p+1}\sum_{q=-p}^p \langle lm|Y_{pq}|lm'\rangle\langle lm''|Y_{pq}^*|lm'''\rangle.
+\end{equation}
+%
+$F^p$ nazywane są całkami Slatera i wyznaczane są ze wzoru
+%
+\begin{equation}
+F^p=e^2\int d\bm{r}\int d\bm{r}' r^2 r'^2 R_{nl}^2(\bm{r})\frac{r_1^p}{r_2^{p+1}} R_{nl}^2(\bm{r}'),
+\end{equation}
+%
+gdzie $R_{nl}(\bm{r})$ są częścią radialną atomu funkcji falowej. 
+Dla stanów $d$ potrzebne są trzy całki Slatera $F^0$, $F^2$ i $F^4$, a dla stanów $f$ dodatkowo $F^6$, aby wyznaczyć elementy macierzowe 
+potencjału kulombowskiego (\ref{vee}).
+Efektywne parametry, które określają lokalne oddziaływanie kulombowskie ($U$) i wymienne ($J$), można wyrazić przy pomocy całek Slatera
+%
+\begin{eqnarray}
+U&=&\frac{1}{(2l+1)^2}\sum_{m.m'} \langle m,m'|V_{ee}|m,m'\rangle=F^0,\\
+J&=&\frac{1}{2l(2l+1)}\sum_{m\ne m'} \langle m,m'|V_{ee}|m',m\rangle=\frac{F^2+F^4}{14}.
+\label{HundJ}
+\end{eqnarray} 
+%
+Wykorzystując te parametry, możemy zapisać ostatni wyraz we wzorze (\ref{eldau}) w następującej formie
+%
+\begin{equation}
+E_{dc}=\frac{1}{2}\Big[\sum_i Un_i(n_i-1)-J[n_{i\uparrow}(n_{i\uparrow}-1)+n_{i\downarrow}(n_{i\downarrow}-1)]\Big],
+\end{equation}
+%
+gdzie $n_{i\sigma}=\Tr(n_{i\sigma}^{mm'})$ i $n_i=n_{i\uparrow}+n_{i\downarrow}$ jest całkowitym obsadzeniem orbitali w węźle $i$.
+Tak wyznaczone parametry $U$ i $J$ odnoszą sie do izolowanych atomów. Efektywne wartości parametru kulombowskiego $U$,
+stosowane dla danego rodzaju atomu, uwzględniaja efekty ekranowania elektronowego i zależą od rodzaju materiału.
+Można go wyznaczyć w ramach metody odpowiedzi liniowej poprzez wyliczenie zmiany obsadzenia stanów elektronowych na wybranym atomie
+pod wpływem przyłożonego lokalnego potencjału \cite{coco1,coco2}.
+Drugi z parametrów $J$, nazywany energią wymiany Hunda, znacznie słabiej zależy od rodzaju materiału i często stosowana jest jego wartość atomowa (\ref{HundJ}). 
+
+\section{Oddziaływanie van der Waalsa}
+\label{sec:vdw}
+
+\chapter{Izolatory i półprzewodniki}
+\label{sec:gapproblem}
+
+\section{Przerwa energetyczna}
+
+Fundamentalna przerwa energetyczna zdefiniowana jest jako różnica między energią jonizacji ($I$), czyli procesu usunięcia elektronu z danego materiału, 
+a powinowactwem elektronowym ($A$), czyli energią uzyskaną z dodania elektronu
+%
+\begin{eqnarray}
+E_g=I-A&=&[E(N-1)-E(N)]-[E(N)-E(N+1)] \nonumber \\ 
+       &=&E(N+1)+E(N-1)-2E(N),
+\label{energygap}       
+\end{eqnarray}
+%
+gdzie $E(N)$, $E(N-1)$ i $E(N+1)$ to energie całkowite wyznaczone kolejno dla układu neutralnego składającego się z $N$ elektronów oraz
+układów z odjętym i dodanym elektronem. Każda z tych energii może być wyznaczona jako energia stanu podstawowego układu z odpowiednią ilością
+elektronów. Wartość przerwy możemy również zapisać używając energii stanów Kohna-Shama 
+%
+\begin{equation}
+E_g=E(N+1)-E(N)-[E(N)-E(N-1)]=\varepsilon_{N+1}(N+1)-\varepsilon_N(N),
+\label{truegap}
+\end{equation}
+%
+gdzie $\varepsilon_N(N)$ jest energią najwyższego poziomu pasma walencyjnego w układzie z $N$ elektronami,
+a $\varepsilon_{N+1}(N+1)$ energią najniższego stanu pasma przewodnictwa w układzie z $N+1$ elektronami.
+Czyli wartość przerwy energetycznej jest teoretycznie osiągalna w ramach DFT, 
+pod warunkiem, że znana jest dokladna zależność $E(N)$. Stosując funkcjonały LDA i GGA możemy otrzymać ze wzoru (\ref{truegap}) jedynie przybliżone wartości przerwy fundamentalnej. 
+   
+Zwykle wykonujemy obliczenia dla układu z ustaloną liczbą elektronów $N$ i otrzymujemy przerwę w spektrum energetycznym Kohna-Shama, 
+która wynosi
+%
+\begin{equation}
+\Delta_{KS}=\varepsilon_{N+1}(N)-\varepsilon_N(N).
+\label{ksgap}
+\end{equation}
+%
+Porównując wzory (\ref{truegap}) i (\ref{ksgap}) dostajemy związek między tymi dwiema przerwami
+%
+\begin{equation}
+E_g=\Delta_{KS}+\varepsilon_{N+1}(N+1)-\varepsilon_{N+1}(N)=\Delta_{KS}+\Delta_{xc},
+\end{equation}
+%
+gdzie różnica między nimi $\Delta_{xc}$ wynika ze zmiany nachylenia liniowej funkcji $E(N)$ przy całkowitej liczbie elektronów \cite{pardew82}.
+Ta nieciągłość ma swoje źródło w potencjale wymienno-korelacyjnym, którego pochodna zmienia się skokowo przy zmianie ilości elektronów
+%
+\begin{equation}
+\Delta_{xc}=\frac{\delta E_{xc}[n]}{\delta n(\bm{r})}|_{N+\delta}-\frac{\delta E_{xc}[n]}{\delta n(\bm{r})}|_{N-\delta},
+\label{deltaxc}
+\end{equation}
+%
+gdzie pochodne funkcjonalne wyznaczone są dla gęstości elektronów $n(\bm{r})$, których całki po całej przestrzeni równe są $N+\delta$ i $N-\delta$, w granicy $\delta\rightarrow 0$. 
+
+Występowanie tej nieciągłości jest podstawową cechą DFT, ale w stosowanych przybliżeniach LDA i GGA, które charakteryzują się analityczną zależnością
+energii od gęstości elektronowej, te nieciągłości nie występują. Przybliżony charakter energii stanów jednocząstkowych,
+z których wyznaczana jest wartość $\Delta_{KS}$, w połączeniu z brakiem nieciągłości powoduje, że przerwy energetyczne wyznaczone w przybliżeniu LDA są średnio zaniżone o $40\%$. W GGA te wartości są zwykle poprawione, ale również znacznie odbiegają od wartości eksperymentalnych. 
+
+\section{Izolatory Motta}
+
+Szczególną klasę materiałów stanowią izolatory Motta, których własności elektronowe wynikają z silnych oddziaływaniań kulombowskich miedzy elektronami.
+W tym przypadku wielkość charakteryzująca nieciągłość wyliczana w ramach standardowych przybliżeń $\Delta_{KS}$ jest równe zeru 
+lub jest bardzo małą wielkością. Przerwa energetyczna wynika zasadniczo z nieciągłości funkcjonału energii ($\Delta_{xc}>0$), powstającej przy zmianie obsadzenia stanów orbitalnych. 
+Ta nieciągła zmiana energii jest proporcjonalna do energii oddziaływania kulombowskiego $U$. Dla stanów $d$ w metalach przejściowych parametr $U$ można zdefiniować
+jako energię związaną z transferem elektronu między dwoma atomami z początkowym obsadzeniem $d^n$,
+który prowadzi do zwiększenia ilości elektronów na jednym atomie $d^{n+1}$ i zmniejszenia na drugim $d^{n-1}$
+%
+\begin{equation}
+U=E(d^{n+1})+E(d^{n-1})-2E(d^n),
+\end{equation} 
+%
+gdzie $E(d^n)$ jest całkowitą energią układu, w którym pojedynczy atom posiada obsadzenie $d^n$.
+Ta relacja oznacza, że sama przerwa energetyczna dana wzorem (\ref{energygap}) powinna być liniową funkcją $U$.
+Potwierdzają to obliczenia przeprowadzone w ramach metody LDA+U.
+
+Jako przykład omówię wyniki obliczeń gestości stanów elektronowych dla izolatora Motta Fe$_2$SiO$_4$, otrzymane metodą GGA+U \cite{Mariana}.
+Na rysunku 6.2 pokazane sa wyniki dla stanu antyferromagnetycznego (AFM).
+Dla $U=0$ dostajemy nieprawidłowy stan metaliczny ($\Delta_{KS}=0$) z energią Fermiego przecinającą pasmo $t_{2g}$. 
+Pasmo to obsadzone jest przez elektrony z mniejszościowym kierunkiem spinów (kierunek przeciwny do momentu magnetycznego danego atomu). 
+Obliczenia dla $U=4.5$ eV dają prawidłowy stan elektronowy z przerwą energetyczną równą około 2 eV,
+która rozdziela pasmo $t_{2g}$ na dwie części: obsadzone dolne pasmo Hubbarda i nieobsadzone górne pasmo Hubbarda.
+
+\begin{figure}[t!]
+\centering
+\includegraphics[scale=0.21]{Fe2SiO4-2.pdf}
+\caption{Gęstość stanów elektronowych w Fe$_2$SiO$_4$ dla $U=0$ (górny panel) i dla $U=4.5$~eV (dolny panel). Rysunek z pracy \cite{Mariana}.}
+\label{fig:mott}
+\end{figure}
+
+\begin{figure}[b!]
+\centering
+\includegraphics[scale=0.18]{Fe2SiO4-1.pdf}
+\caption{Zależność przerwy energetycznej od parametru oddziaływania kulombowskiego $U$ w Fe$_2$SiO$_4$ dla stanu antyferromagnetycznego (czerwone kółka)
+i ferromagnetycznego (niebieskie kwadraty). Rysunek z pracy \cite{Mariana}.}
+\label{fig:gap}
+\end{figure}
+
+Wartość przerwy energetycznej zależy od dokładnej wartości parametru $U$, co pokazano na rysunku 6.3.
+Dla małych wartości $U\leq 2$ eV, przerwa energetyczna równa jest zeru. Wynika to z kryterium, które mówi, że w izolatorach Motta, 
+niezerowa przerwa tworzy się dla wartości energii oddziaływania kulombowskiego większych od szerokości
+pasma elektronowego, $U>W$. Rzeczywiście, z rysunku 6.2 wynika, że szerokość pasma $t_{2g}$ jest równa z dobrym przybliżeniem $W=2$ eV.
+Gdy energia oddziaływania kulombowskiego $U$ przewyższa $W$, przerwa energetyczna rośnie praktycznie liniowo z jej wartością. 
+Jak widać z rysunku 6.3 wartości przerwy zależą od rodzaju uporządkowania magnetycznego i dla stanu AFM są większe niż dla stanu FM.
+
+\section{Izolatory pasmowe i półprzewodniki}
+
+Metoda LDA+U skutecznie poprawia wartość przerwy energetycznie jedynie w izolatorach Motta.
+W przypadku izolatorów pasmowych i półprzewodników lepszymi podejściami są dwie omówione wcześniej 
+metody: SIC i funkcjonały hybrydowe. 
+W metodzie SIC, energia całkowita zależy od obsadzenia orbitali atomowych, co daje możliwość odtworzenie nieciągłości
+pochodnych funkcjonału wymienno-korelacyjnego (\ref{deltaxc}) i poprawienia wartości przerwy energetycznej.
+SIC daje dobre wyniki dla izolatorów z dużą przerwa takich, jak kryształy gazów szlachetnych \cite{PZ}, czy kryształy jonowe. Przykładowo dla NaCl przerwa energetyczna wynosi $E_g^{SIC}=9.2$~eV \cite{norman83}, co dobrze zgadza się z wartością eksperymentalną $E_g^{exp}=8.97$~eV.  
+
+Funkcjonały hybrydowe, które częściowo uwzlędniają dokladną wartość oddziaływania wymiennego i przez to redukują błąd samooddziaływania elektronów, 
+również poprawiają wartość przerwy energetycznej. Na rysunku~\ref{fig:gaphyb} porównano eksperymentalne wartości przerwy energetycznej
+z wyliczonymi w ramach czterech funkcjonałów hybrydowych dla wybranej grupy półprzewodników i tlenków metali przejściowych (FeO, CoO, NiO, MnO, and VO$_2$)~\cite{Gap-hyb}. Dwa z tych funkcjonałów zawierają 20 \% dokładnej wartości energii wymiany (B3PW91 i B3LYP), a dwa pozostałe 25 \% (PBE0 i HSE). Dla wiekszości materiałów 
+z przerwą mniejszą niź 5 eV, zgodność z eksperymentem jest bardzo dobra dla wszystkich funkcjonałów. Jednocześnie widzimy, że wybrane funkcjonały znacznie zaniżają wartości przerwy dla trzech półprzewodników o dużej przerwie (NaCl, LiCl i LiF).
+Dokładna analiza błędów pokazała, że najlepszą zgodność uzyskano dla funkcjonału HSE, który ma tendencję do niewielkiego zaniżania wartości przerwy (średnio o -0.24 eV). Dwa funkcjonały B3PW91 i B3LYP lekko zawyżają przerwę (średnie błedy wynoszą 0.14 eV i 0.13 eV). Największe odstępstwo obserwujemy dla funkcjonału PBE0, który zawyża wartość przerwy średnio o 0.43 eV.
+
+\begin{figure}[t!]
+\centering
+\includegraphics[scale=0.5]{gap-hyb.pdf}
+\caption{Porównanie eksperymentalnych wartości przerwy energetycznej z obliczonymi dla różnych funkcjonałów hybrydowych. Rysunek z pracy \cite{Gap-hyb}.}
+\label{fig:gaphyb}
+\end{figure}
+
+W tabeli~\ref{gap} zebrane są wartości przerwy energetycznej dla wybranych materiałów, wyliczone różnymi metodami
+i porównane z danymi eksperymentalnymi. LDA i GGA we wszystkich przypadkach znacznie zaniżają wartość przerwy.
+Dotyczy to zarówno półprzewodników, szczególnie germanu, gdzie przerwa jest równa zeru, jak i izolatorów.
+Duże rozbieżności w porównaniu do eksperymentu widzimy dla trzech ostatnich materiałów, zaliczanych do izolatorów Motta.
+Zastosowanie funkcjonału hybrydowego HSE poprawia wartości przerwy i dla większości półprzewodników zgodność z eksperymentem jest bardzo dobra.
+Dla izolatorów, ta zgodność jest lepsza lub gorsza w zależności od materiału.
+W tabeli~\ref{gap} pokazano również wyniki otrzymane w ramach metody GW i jej uproszczonej wersji G$_0$W$_0$, które również dają
+lepsze wartości przerwy. Metoda GW, która uwzlędnia oddziaływania wielociałowe w ramach rozwinięcia perturbacyjnego, będzie tematem jednego z
+kolejnych rozdziałów. 
+
+\begin{table}[h!]
+\caption{Wartości przerwy energetycznej wyliczone różnymi metodami i porównane z danymi eksperymentalnymi.}
+\label{gap}
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|}
+\hline
+ Materiał & LDA & PBE & HSE & G$_0$W$_0$ & GW & Eksperyment \\ \hline
+  C & 4.14 & 4.17  & 4.94 & 5.50 & 6.18 & 5.50 \\
+  Si & 0.60 & 0.71  & 1.11 & 1.12 & 1.41 & 1.17 \\
+  Ge & 0.00 &  & 0.83 & 0.66 & 0.95 & 0.74 \\
+  MgO & 4.70 & 4.74  & 6.46 & 7.25 & 9.16 & 7.90 \\
+  GaAs & 0.30 & 0.53  & 1.41 & 1.30 & 1.85 & 1.52 \\
+  SiC & 1.35 &    & 2.40 & 2.27 & 2.88 & 2.40 \\ 
+  GaP & 1.53 & 1.69 & 2.09 &   &   & 2.35  \\
+  CdS & 0.96 & 1.23 & 2.27 &   &   & 2.48 \\
+  InP & 0.50 & 0.72 & 1.52 &   &   &  1.42 \\
+  BN  & 4.42 & 4.53 & 5.39 &   &   &  6.20 \\
+  NaCl & 4.70 & 5.08 & 6.42 &  &   & 8.97  \\
+  LiF  & 8.84 & 9.04 & 11.4 &  &   & 13.6  \\  
+  MnO & 0.76 &    & 2.80 &  & 3.50 & 3.90 \\ 
+  FeO & 0 &    & 2.2 &  &  & 2.4 \\ 
+  NiO & 0.42 &    & 4.2 & 1.10 & 4.80 & 4.30 \\ \hline 
+\end{tabular}
+\end{center}
+\end{table}
+
+\section{Ferroelektryki}
+
+
+
+%\chapter{Efekty wielociałowe i stany wzbudzone}
+
+%\section{Oddziaływanie van der Waalsa}
+%\label{sec:vdw}
+
+%\section{Metoda GW}
+%\label{sec:gw}
+
+%\section{Teoria dynamicznego średniego pola (DMFT)}
+%\label{sec:dmft}
+
+%\section{Kwantowe Monte Carlo (QMC)}
+%\label{sec:qmc}
+
+
+
+
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+                          
+
+		     
+\end{thebibliography} 
+
+
+\end{document}
+
+
+
+
diff --git a/book/pseudopots-new.pdf b/book/pseudopots-new.pdf
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HcmV?d00001


From 39372ebdd198430e4306fd7eeed2abb24434bb5f Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Fri, 2 Jan 2026 16:26:56 +0000
Subject: [PATCH 02/23] Initial plan


From f312a17e03ea9bc4a2ced1899a18e1d55f68c6b0 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Fri, 2 Jan 2026 16:32:49 +0000
Subject: [PATCH 03/23] Translate Introduction and Basic Properties section to
 English

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 195 ++++++++++++++++++++++++++------------------------
 1 file changed, 101 insertions(+), 94 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index f8d0058..e4879d8 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -108,62 +108,66 @@ \chapter*{Introduction}
 Thanks to these improvements, density functional theory became the basis for calculating electronic structure~\cite{jones}, determining crystal parameters~\cite{payne}, studying lattice dynamics~\cite{parlinski, baroni} and many other material properties~\cite {martin}.
 For its formulation, Walter Kohn received the Nobel Prize in Chemistry in 1998.
 
-Istnieją jednak układy, w których przybliżenia stosowane w DFT załamują się i nie potrafią poprawnie
-opisać struktury elektronowej. Zgodnie z klasyczną teorią pasmową, wiele tlenków metali przejściowych, takich jak NiO lub MnO,
-powinno znajdować się w stanie metalicznym, ze względu na częściowe wypełnienie stanów $3d$.
-Materiały te są w rzeczywistości izolatorami, ponieważ silne oddziaływanie kulombowskie w stanach $3d$
-uniemożliwia swobodny ruch elektronów i prowadzi do ich lokalizacji \cite{peierls}.
-W odróznieniu od izolatorów pasmowych, materiały te nazywane są izolatorami Motta \cite{mott}.
-Takie lokalne oddziaływanie elektronów, opisywane w ramach modelu Hubbarda parametrem $U$ \cite{hubbard},
-jest znacznie silniejsze w stanach $d$ i $f$, niż w bardziej rozległych (zdelokalizoanych) stanach $s$ i $p$.
-Struktura elektronowa tlenków metali przejściowych, wyznaczana w ramach obliczeń LDA lub GGA,
-albo nie posiada przerwy (CoO) albo ta przerwa jest znacznie zaniżona (NiO) \cite{terakura}.
-Zbyt mała przerwa energetyczna jest także problemem w obliczeniach dla klasycznych półprzewodników,
-takich jak krzem lub german, co oznacza, że również w tych materiałach potencjał wymienno-korelacyjny nie jest
-dobrze opisany.
-
-Podstawowe przybliżenia stosowane do wyznaczenia energii wymienno-korelacyjnej są źródłem błedów, które prowadzą efektywnie do samooddziaływania elektronów.
-Ten efekt jest szczególnie silny dla stanów $d$ lub $f$ i prowadzi do niekorzystnego wzrostu energii przy lokalizacji elektronów.
-To jest główną przyczyną niepoprawnego stanu metalicznego lub zbyt małej przerwy energetycznej otrzymywanych
-w obliczeniach dla tlenków metali przejściowych.  
-Jedna z metod korekcji tego efektu (SIC) \cite{PZ} polega na bezpośrednim wyliczeniu
-energii samoodziaływania i odjęcia jej od funkcjonału energii całkowitej.
-Metoda ta umożliwia poprawienie struktury elektronowej, ale nie jest łatwa do zaimplementowania w obliczeniach samouzgodnionych.
-Stan elektronowy można również poprawić stosując funkcjonały hybrydowe, w których przybliżona energia wymiany jest częściowo
-zastąpiona dokładną wartością otrzymaną w ramach metody Hartree-Focka \cite{becke93}.
-Wynika to z faktu, że w przybliżeniu Hartree-Focka nie występuje efekt samoodziaływania.
+% **[REVIEW NEEDED]**: Technical terminology for DFT approximations and strongly correlated systems
+However, there are systems in which the approximations used in DFT break down and fail to correctly
+describe the electronic structure. According to classical band theory, many transition metal oxides, such as NiO or MnO,
+should be in a metallic state, due to the partial filling of $3d$ states.
+These materials are actually insulators, because the strong Coulomb interaction in the $3d$ states
+prevents the free movement of electrons and leads to their localization \cite{peierls}.
+In contrast to band insulators, these materials are called Mott insulators \cite{mott}.
+Such local electron interaction, described within the Hubbard model by the parameter $U$ \cite{hubbard},
+is much stronger in $d$ and $f$ states than in the more extended (delocalized) $s$ and $p$ states.
+The electronic structure of transition metal oxides, determined within LDA or GGA calculations,
+either has no gap (CoO) or the gap is significantly underestimated (NiO) \cite{terakura}.
+Too small an energy gap is also a problem in calculations for classical semiconductors,
+such as silicon or germanium, which means that the exchange-correlation potential is not
+well described in these materials either.
+
+% **[REVIEW NEEDED]**: Self-interaction correction terminology and hybrid functional description
+The basic approximations used to determine the exchange-correlation energy are a source of errors that effectively lead to electron self-interaction.
+This effect is particularly strong for $d$ or $f$ states and leads to an unfavorable increase in energy upon electron localization.
+This is the main reason for the incorrect metallic state or too small energy gap obtained
+in calculations for transition metal oxides.  
+One method for correcting this effect (SIC) \cite{PZ} consists of directly calculating
+the self-interaction energy and subtracting it from the total energy functional.
+This method enables improvement of the electronic structure, but is not easy to implement in self-consistent calculations.
+The electronic state can also be improved by using hybrid functionals, in which the approximate exchange energy is partially
+replaced by the exact value obtained within the Hartree-Fock method \cite{becke93}.
+This stems from the fact that in the Hartree-Fock approximation there is no self-interaction effect.
   
-Potencjał elektronowy dla zlokalizowanych stanów można również poprawić przez dodanie do funkcjonału DFT
-dodatkowego wyrazu proporcjonalnego do energii oddziaływania $U$. Można to zrobić w przybliżeniu średniego
-pola w ramach metody LDA+U \cite{anisimov}. Zastosowanie tej metody do tlenków metali przejściowych \cite{anisimov}
-lub nadprzewodników wysokotemperaturowych \cite{czyzyk} pozwoliło znacznie poprawić ich strukturę elektronową 
-i własności magnetyczne w porównaniu do standardowych obliczeń DFT.
-Również w przypadku innych złożonych materiałów, w których występują oddziaływania między spinowymi, orbitalnymi
-i sieciowymi stopniami swobody to podejście daje jakościowo lepsze wyniki \cite{orbital1,orbital2,orbital3}.
-W silnie skorelowanych metalach, jakimi są ziemie rzadkie i aktynowce,
-metoda LDA+U pozwala tylko częściowo uwzględnić efekty związane z lokalizacją elekronów.
-Dobrym przykładem jest kryształ plutonu, gdzie bardzo duża niezgodność wyliczanej teoretycznie
-objetości, a wartością eksperymentalną (około 35 \%) może być skorygowana przy użycie
-realistycznej wartości parametru $U$\cite{Pu-LDAU}.
-Niestety, efekty dynamiczne takie jak fluktuacja momentów magnetycznych w stanie paramagnetycznyn, czy efekt Kondo
-nie mogą być poprawnie opisane na poziomie teorii średniego pola.
-W bardziej zaawansowanej metodzie dynamicznego średniego pola (DMFT), efekty wielociałowe
-są uwzględniane przez rozwiązanie modelu domieszki Andersona w ramach dokładnej diagonalizacji
-lub kwantowego Monte Carlo \cite{Georges}. Rozwiązania zawierają informacje o energii własnej i czasie
-życia stanów elektronowych, które nie są dostępne w ramach obliczeń DFT.
-Sama metoda kwantowego Monte Carlo rozwijana jest obecnie bardzo intensywnie i stosowana jest do obliczeń
-struktury elektronowej molekół i ukladów krystalicznych.
+% **[REVIEW NEEDED]**: LDA+U method description and advanced many-body methods  
+The electronic potential for localized states can also be improved by adding to the DFT functional
+an additional term proportional to the interaction energy $U$. This can be done in the mean-field
+approximation within the LDA+U method \cite{anisimov}. Application of this method to transition metal oxides \cite{anisimov}
+or high-temperature superconductors \cite{czyzyk} has significantly improved their electronic structure 
+and magnetic properties compared to standard DFT calculations.
+Also in the case of other complex materials, in which interactions between spin, orbital,
+and lattice degrees of freedom occur, this approach gives qualitatively better results \cite{orbital1,orbital2,orbital3}.
+In strongly correlated metals, such as rare earths and actinides,
+the LDA+U method allows only partial inclusion of effects related to electron localization.
+A good example is plutonium crystal, where the very large discrepancy between the theoretically calculated
+volume and the experimental value (about 35\%) can be corrected using
+a realistic value of the parameter $U$\cite{Pu-LDAU}.
+Unfortunately, dynamic effects such as fluctuation of magnetic moments in the paramagnetic state, or the Kondo effect
+cannot be correctly described at the level of mean-field theory.
+In the more advanced dynamical mean-field theory (DMFT) method, many-body effects
+are included by solving the Anderson impurity model using exact diagonalization
+or quantum Monte Carlo \cite{Georges}. The solutions contain information about the self-energy and lifetime
+of electronic states, which are not available within DFT calculations.
+The quantum Monte Carlo method itself is currently being developed very intensively and is used for calculations
+of the electronic structure of molecules and crystalline systems.
 
 
 \chapter{Electron interactions}
 
 \section{Basic properties}
 
-Spośród wszystkich oddziaływań, które występują w przyrodzie, decydującymi o własnościach fizycznych i chemicznych układów atomowych
-są oddziaływania elektromagnetyczne. 
-Wiązania międzyatomowe, dzięki którym powstają cząsteczki i ciała stałe, 
-są efektem wzajemnych oddziaływań elektrostatycznych w układzie elektronów i jąder atomowych.
-Hamiltonian, który opisuje dowolny układ atomów ma postać
+% **[REVIEW NEEDED]**: Fundamental quantum mechanics terminology - electromagnetic interactions and Hamiltonian description
+Among all interactions that occur in nature, the decisive factors for the physical and chemical properties of atomic systems
+are electromagnetic interactions. 
+Interatomic bonds, which give rise to molecules and solids, 
+are the result of mutual electrostatic interactions in the system of electrons and atomic nuclei.
+The Hamiltonian, which describes an arbitrary system of atoms, has the form
 %
 \begin{equation}
 H=-\frac{\hbar^2}{2m}\sum_{i}\nabla_i^2-\sum_{i,j} \frac{Z_{j}e^2}{|\bm{r}_i-\bm{R}_j|}+\frac{1}{2}\sum_{i\neq j}\frac{e^2}{|\bm{r}_i-\bm{r}_j|}
@@ -171,34 +175,35 @@ \section{Basic properties}
 \label{hamiltonian}
 \end{equation}
 %
-gdzie $\bm{r}_i$ określają położenia elektronów, $\bm{R_j}$ położenia jąder atomowych, $Z_j$ ładunki jąder, $M_j$ masy jąder i $m$ jest masą elektronu.
-Pierwsze trzy wyrazy opisują odpowiednio energię kinetyczną elektronów, oddziaływanie elektronów z jądrami i oddziaływania
-między elektronami. Ostatnie dwa wyrazy to energia kinetyczna jąder atomowych i oddziaływanie między
-ładunkami jąder. Pełna funkcja falowa złożonego układu elektronów i jąder atomowych $\Psi(\bm{r}_i,\bm{R}_j)$ spełnia równanie Schr\"{o}dingera
+where $\bm{r}_i$ denote the positions of electrons, $\bm{R_j}$ the positions of atomic nuclei, $Z_j$ the nuclear charges, $M_j$ the nuclear masses, and $m$ is the electron mass.
+The first three terms describe respectively the kinetic energy of electrons, the interaction of electrons with nuclei, and the interactions
+between electrons. The last two terms are the kinetic energy of atomic nuclei and the interaction between
+nuclear charges. The complete wave function of the complex system of electrons and atomic nuclei $\Psi(\bm{r}_i,\bm{R}_j)$ satisfies the Schr\"{o}dinger equation
 %
 \begin{equation}
 H\Psi(\bm{r}_i,\bm{R}_j) = E\Psi(\bm{r}_i,\bm{R}_j),
 \label{Sch}
 \end{equation}
 %
-gdzie $E$ jest całkowitą energią układu.
+where $E$ is the total energy of the system.
 
-Kwantowo-mechaniczny opis kryształu można uprościć wykorzystując duży stosunek masy jadrą atomowego do masy elektronu,
-który dla najlżejszego jądra wodoru wynosi $m_p/m_e=1836$.
-Powoduje to, że dynamika elektronów jest znacznie większa niż jader atomowych i stany elektronowe bardzo szybko (adiabatycznie) dopasowują się
-do aktualnego położenia atomów. Typowe energie związane z ruchem elektronów są rzędu 1 eV, natomiast średnie energie drgań atomowych
-w kryształach są na poziomie 10 meV. 
-Zatem w pierwszym przybliżeniu możemy rozwiązać równanie Schr\"{o}dingera dla podukładu elektronów przy zadanych położeniach atomowych i
-pominąć wpływ kwantowych cech dynamiki atomów na funkcje falowe elektronów.
-Podejście to nazywa się przybliżeniem adiabatycznym lub przybliżeniem Borna-Oppenheimera \cite{BO}.
-Wprowadzając funkcję falową całego układu w postaci iloczynu funkcji elektronowej $\Phi(\bm{r_i},\bm{R_j})$ i funkcji jąder atomowych $\chi(\bm{R}_j)$
+% **[REVIEW NEEDED]**: Born-Oppenheimer approximation description and adiabatic treatment
+The quantum-mechanical description of a crystal can be simplified by exploiting the large ratio of the atomic nuclear mass to the electron mass,
+which for the lightest nucleus, hydrogen, equals $m_p/m_e=1836$.
+This causes the dynamics of electrons to be much faster than that of atomic nuclei, and electronic states adjust very quickly (adiabatically)
+to the current positions of atoms. Typical energies associated with electron motion are on the order of 1 eV, whereas average atomic vibration energies
+in crystals are at the level of 10 meV. 
+Thus, in the first approximation, we can solve the Schr\"{o}dinger equation for the electronic subsystem at fixed atomic positions and
+neglect the influence of quantum features of atomic dynamics on electronic wave functions.
+This approach is called the adiabatic approximation or the Born-Oppenheimer approximation \cite{BO}.
+Introducing the wave function of the entire system as a product of the electronic function $\Phi(\bm{r_i},\bm{R_j})$ and the nuclear function $\chi(\bm{R}_j)$
 %
 \begin{equation}
 \Psi(\bm{r}_i,\bm{R}_j) = \Phi(\bm{r}_i,\bm{R}_j)\chi(\bm{R}_j)
 \label{wave}
 \end{equation}
 %
-można rozseparować (\ref{Sch}) na dwa równania 
+one can separate (\ref{Sch}) into two equations 
 %
 \begin{equation}
 (-\frac{\hbar^2}{2m}\sum_{i=1}^N \nabla_i^2-\sum_{i,j} \frac{Z_{j}e^2}{|\bm{r}_i-\bm{R}_j|}+\frac{1}{2}\sum_{i\neq j}
@@ -210,56 +215,58 @@ \section{Basic properties}
 (-\sum_j \frac{\hbar^2}{2M_j} \nabla_j^2 -\sum_{i,j} \frac{Z_iZ_je^2}{|\bm{R}_i-\bm{R}_j|}+E_n(\bm{R}_j))\chi_{n\alpha}(\bm{R}_j)=\varepsilon_{n\alpha}\chi_{n\alpha}(\bm{R}_j).
 \end{equation}
 %
-Pierwsze równanie opisuje funkcje falowe i energie własne $E_n(\bm{R}_j)$ układu elektronowego przy ustalonych położeniach jąder atomowych, gdzie $n$ oznacza zdefiniowany zbiór liczb kwantowych stanu elektronowego. Z drugiego równania możemy otrzymać funkcje falowe i energie własne $\varepsilon_{n\alpha}$ związane z ruchem jąder atomowych,
-gdzie $\alpha$ jest kwantową liczbą, która charakteryzuje te wielkości. Energia potencjalna w równaniu opisującym ruch jąder atomowych zależy od wzajemnego oddziaływania między jądrami i od energii podukładu elektronowego $E_n(\bm{R}_j)$. Obie wielkości są funkcją aktualnego położenia wszystkich jąder atomowych.  
+The first equation describes the wave functions and eigenvalues $E_n(\bm{R}_j)$ of the electronic system at fixed positions of atomic nuclei, where $n$ denotes a defined set of quantum numbers of the electronic state. From the second equation we can obtain the wave functions and eigenvalues $\varepsilon_{n\alpha}$ associated with the motion of atomic nuclei,
+where $\alpha$ is a quantum number that characterizes these quantities. The potential energy in the equation describing the motion of atomic nuclei depends on the mutual interaction between nuclei and on the energy of the electronic subsystem $E_n(\bm{R}_j)$. Both quantities are functions of the current positions of all atomic nuclei.  
 
-Zapiszmy funkcję falową układu $N$ elektronów, przy zadanych położeniach jader atomowych, w formie $\Phi(\bm{r}_1,\bm{r}_2,...,\bm{r}_N)$. 
-Znajomość funcji falowej pozwala wyznaczyć wiele podstawowych wielkości fizycznych dla danego układu. W szczególności gęstość elektronowa w punkcie $\bm{r}$
-dana jest wzorem
+Let us write the wave function of a system of $N$ electrons, at fixed positions of atomic nuclei, in the form $\Phi(\bm{r}_1,\bm{r}_2,...,\bm{r}_N)$. 
+Knowledge of the wave function allows us to determine many basic physical quantities for a given system. In particular, the electron density at point $\bm{r}$
+is given by the formula
 %
 \begin{equation}
 n(\bm{r})=N\int d\bm{r}_2...d\bm{r}_N \Phi^*(\bm{r},\bm{r}_2,...,\bm{r}_N) \Phi(\bm{r},\bm{r}_2,...,\bm{r}_N).
 \end{equation}
 %
-Podstawową własnością elektronowej funkcji falowej jest jej antysymetryczność. Oznacza, że zamiana miejscami dwóch elektronów powoduje zmianę jej znaku
+% **[REVIEW NEEDED]**: Pauli exclusion principle and fermion antisymmetry
+The fundamental property of the electronic wave function is its antisymmetry. This means that exchanging the positions of two electrons causes a sign change
 %
 \begin{equation}
 \Phi(\bm{r}_1,\bm{r}_2,...,\bm{r}_i,...,\bm{r}_j,...,\bm{r}_N)=-\Phi(\bm{r}_1,\bm{r}_2,...,\bm{r}_j,...,\bm{r}_i,...,\bm{r}_N).
 \label{anty}
 \end{equation}
 %
-Ta własność wynika z zakazu Pauliego, który mówi, że dwa fermiony (czyli cząstki o spinie połówkowym) nie mogą znajdować się w tym samym stanie kwantowym - w stanie opisanym takim samym zestawem liczb kwantowych.
+This property follows from the Pauli exclusion principle, which states that two fermions (i.e., particles with half-integer spin) cannot occupy the same quantum state - a state described by the same set of quantum numbers.
 
-Równanie Schr\"{o}dingera (\ref{Sch-el}) zawiera oddziaływania wielociałowe i nie posiada dokładnych rozwiązań analitycznych, z wyjątkiem atomu wodoru lub innych układów jednoelektronowych. Dokładne rozwiązania numeryczne równania Schr\"{o}dingera można uzyskać jedynie dla pojedynczych atomów i małych molekuł.
-Jako przykład rozważmy prosty układ z dwoma elektronami, jakim jest cząsteczka wodoru H$_2$.
-Hamiltonian tego układu, po uwzlędnieniu przybliżenia Borna-Oppenheimera, przyjmuje postać
+% **[REVIEW NEEDED]**: Hydrogen molecule example - Heitler-London method terminology
+The Schr\"{o}dinger equation (\ref{Sch-el}) contains many-body interactions and has no exact analytical solutions, except for the hydrogen atom or other one-electron systems. Exact numerical solutions of the Schr\"{o}dinger equation can be obtained only for individual atoms and small molecules.
+As an example, let us consider a simple system with two electrons, which is the hydrogen molecule H$_2$.
+The Hamiltonian of this system, after including the Born-Oppenheimer approximation, takes the form
 %
 \begin{equation}
 H=-\frac{\hbar^2}{2m}\nabla_1^2-\frac{\hbar^2}{2m}\nabla_2^2-\frac{e^2}{r_{1A}}-\frac{e^2}{r_{1B}}-\frac{e^2}{r_{2A}}-\frac{e^2}{r_{2B}}+\frac{e^2}{r_{12}}+\frac{e^2}{r_{AB}},
 \end{equation}
 %  
-gdzie $r_{1A}$, $r_{1B}$, $r_{2A}$, $r_{1B}$ oznaczają odległości elektronów (1 i 2) od dwóch protonów ($A$ i $B$), $r_{12}$ jest odległością między elektronami
-i $r_{AB}$ odległością miedzy protonami. 
-W roku 1927, Heitler i London zaproponowali funkcję falową w formie \cite{HL}
+where $r_{1A}$, $r_{1B}$, $r_{2A}$, $r_{2B}$ denote the distances of electrons (1 and 2) from the two protons ($A$ and $B$), $r_{12}$ is the distance between electrons,
+and $r_{AB}$ is the distance between protons. 
+In 1927, Heitler and London proposed a wave function in the form \cite{HL}
 %
 \begin{equation}
 \Phi(\bm{r}_1,\bm{r}_2)=N_{\pm}[\psi_A(\bm{r}_1)\psi_B(\bm{r}_2)\pm \psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]\chi_{\sigma},
 \end{equation}
 %
-gdzie $N_{\pm}$ jest czynnikiem normalizacyjnym, $\chi_{\sigma}$ jest częścią spinową funkcji falowej, a cztery funkcje $\psi_{\alpha}(\bm{r})$
-sa orbitalami $1s$ dla stanu podstawowego atomu wodoru
+where $N_{\pm}$ is the normalization factor, $\chi_{\sigma}$ is the spin part of the wave function, and the four functions $\psi_{\alpha}(\bm{r})$
+are $1s$ orbitals for the ground state of the hydrogen atom
 %
 \begin{equation}
 \psi_{\alpha}(\bm{r})=\frac{1}{\sqrt{\pi a_0^3}}e^{-\frac{|\bm{r}-\bm{r}_{\alpha}|}{a_0}},
 \end{equation}
-gdzie $a_0$ jest promieniem Bohra, a $\bm{r}_{\alpha}$ jest położeniem protonu $\alpha=A$ lub $B$.
-Biorąc pod uwagę warunek antysymetryczności (\ref{anty}), funkcja falowa przyjmuje jedną z dopuszczalnych postaci
+where $a_0$ is the Bohr radius, and $\bm{r}_{\alpha}$ is the position of proton $\alpha=A$ or $B$.
+Taking into account the antisymmetry condition (\ref{anty}), the wave function takes one of the allowed forms
 %
 \begin{equation}
 \Phi_S(\bm{r}_1,\bm{r}_2)=N_{+}[\psi_A(\bm{r}_1)\psi_B(\bm{r}_2)+\psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]\frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle),
 \end{equation}
 %
-dla całkowitego spinu $S=0$ (stan singletowy), oraz
+for total spin $S=0$ (singlet state), and
 %
 \begin{equation}
 \Phi_T(\bm{r}_1,\bm{r}_2)=\begin{cases}
@@ -269,23 +276,23 @@ \section{Basic properties}
 \end{cases}
 \end{equation}
 %
-dla spinu $S=1$ (stan trypletowy). Stanem podstawowym cząsteczki wodoru jest stan singletowy, którego energia
-$E_S$ jest niższa od energii stanu stanu trypletowego $E_T$ dla każdej odległości między protonami $r_{AB}$.
-Stan związany odpowiada minimum energetycznemu $E_S=\langle\Phi_S|H|\Phi_S\rangle$, które dostajemy dla $r_{AB}=0.87$ \AA. Ta odległość jest większa od wartości eksperymentalnej równej 0.74 \AA. Natomiast wyliczona energia dysocjacji cząsteczki na dwa atomy wodoru wynosi $E_d=3.14$ eV i jest mniejsza od energii zmierzonej 4.75 eV. Jest to przykład wiązania kowalencyjnego, w którym dwa elektrony o przeciwnych spinach zostają uwspólnione, co prowadzi
-do obniżenia energii całego układu w porównaniu do sumy energii dwóch osobnych atomów.  
-W odróżnieniu od stanu singletowego, stan trypletowy nie tworzy stanu związanego dwóch atomów wodoru.
-Dokładniejsze wartości $r_{AB}=0.74$ \AA\ i $E_d=3.63$ eV otrzymuje się w przybliżeniu Hartree-Focka, które będzie tematem następnego rozdziału.
-Najlepsze wyniki uzyskuje się metodą wariacyjną, w której funkcja falowa zapisana jest w ogólnej formie iloczynu symetrycznej funkcji przestrzennej i antysymetrycznej funkcji spinowej \cite{JC}
+for spin $S=1$ (triplet state). The ground state of the hydrogen molecule is the singlet state, whose energy
+$E_S$ is lower than the energy of the triplet state $E_T$ for any distance between protons $r_{AB}$.
+The bound state corresponds to the energy minimum $E_S=\langle\Phi_S|H|\Phi_S\rangle$, which we obtain for $r_{AB}=0.87$ \AA. This distance is larger than the experimental value of 0.74 \AA. Meanwhile, the calculated dissociation energy of the molecule into two hydrogen atoms is $E_d=3.14$ eV and is smaller than the measured energy of 4.75 eV. This is an example of a covalent bond, in which two electrons with opposite spins become shared, which leads
+to a lowering of the energy of the entire system compared to the sum of energies of two separate atoms.  
+In contrast to the singlet state, the triplet state does not form a bound state of two hydrogen atoms.
+More accurate values $r_{AB}=0.74$ \AA\ and $E_d=3.63$ eV are obtained in the Hartree-Fock approximation, which will be the subject of the next chapter.
+The best results are obtained by the variational method, in which the wave function is written in the general form of a product of a symmetric spatial function and an antisymmetric spin function \cite{JC}
 %
 \begin{equation}
 \Phi_S(\bm{r}_1,\bm{r}_2)=\Psi(\bm{r}_1,\bm{r}_2)\chi_0.
 \end{equation} 
 %
-Część przestrzenna funkcji falowej zależy od $M$ parametrów wariacyjnych, $p_1$, $p_2$,..., $p_M$.
-Minimalizując energię układu dla $M=13$ otrzymuje się wartości $E_d=4.70$ eV i $r_{AB}=4.74$ \AA, w bardzo dobrej zgodności z danymi eksperymentalnymi \cite{JC}.   
-Dla większości badanych układów molekularnych i ciał stałych nie da się wyliczyć dokładnych rozwiązań równania Schr\"{o}dingera. 
-Zatem konieczne jest stosowanie metod przybliżonych, głównie numerycznych, które pozwalają wyznaczyć najlepsze możliwe funkcje falowe i energie stanów elektronowych
-w rozsądnym czasie obliczeniowym. 
+The spatial part of the wave function depends on $M$ variational parameters, $p_1$, $p_2$,..., $p_M$.
+Minimizing the system energy for $M=13$ yields values $E_d=4.70$ eV and $r_{AB}=4.74$ \AA, in very good agreement with experimental data \cite{JC}.   
+For most studied molecular systems and solids, it is not possible to calculate exact solutions of the Schr\"{o}dinger equation. 
+Therefore, it is necessary to use approximate methods, mainly numerical, which allow us to determine the best possible wave functions and energies of electronic states
+in a reasonable computational time. 
 
 \section{Równanie Hartree-Focka}
 \label{sec:HF}

From d7a908bebab971c28dfc126969fa7bf456edb127 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Fri, 2 Jan 2026 16:35:34 +0000
Subject: [PATCH 04/23] Translate Hartree-Fock and Self-consistent field
 sections

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 116 ++++++++++++++++++++++++++------------------------
 1 file changed, 60 insertions(+), 56 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index e4879d8..b2ebe3c 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -294,10 +294,11 @@ \section{Basic properties}
 Therefore, it is necessary to use approximate methods, mainly numerical, which allow us to determine the best possible wave functions and energies of electronic states
 in a reasonable computational time. 
 
-\section{Równanie Hartree-Focka}
+\section{Hartree-Fock equation}
 \label{sec:HF}
 
-Jedną z pierwszych metod zostosowaną do opisu układów molekularnych jest przybliżenie Hartree-Focka \cite{hartree28,fock30,slater30}. W tym podejściu funkcja falowa układu $N$ elektronów ma postać wyznacznika Slatera
+% **[REVIEW NEEDED]**: Slater determinant and Hartree-Fock approximation terminology
+One of the first methods used to describe molecular systems is the Hartree-Fock approximation \cite{hartree28,fock30,slater30}. In this approach, the wave function of a system of $N$ electrons has the form of a Slater determinant
 %
 \begin{equation}
 \Phi = \frac{1}{\sqrt{N!}} 
@@ -310,31 +311,31 @@ \section{Równanie Hartree-Focka}
 \label{slater}
 \end{equation}
 %
-gdzie funkcje jednoelektronowe są iloczynem części zależnej od położenia $\bm{r}_i$
-i od spinu $\sigma_i$
+where the single-electron functions are products of the part depending on position $\bm{r}_i$
+and on spin $\sigma_i$
 %
 \begin{equation}
 \phi_i(\bm{r}_i,\sigma_i)=\psi_i^{\sigma_i}(\bm{r}_i)\xi_i(\sigma_i).
 \end{equation}
 %
-Funkcja falowa zapisana w postaci wyznacznika Slatera spełnia podstawowe
-własności układu nieodróżnialnych cząstek.
-Zamiana miejscami dwóch elektronów odpowiada zamianie dwóch kolumn w wyznaczniku, co powoduje
-zmianę znaku funkcji falowej zgodnie z warunkiem antysymetryczności.
-Jeżeli dwa elektrony znajdują się w tym samym stanie kwantowym to dwie kolumny
-są jednakowe i cały wyznacznik się zeruje, co zgodne jest z zakazem Pauliego. 
+The wave function written in the form of a Slater determinant satisfies the basic
+properties of a system of indistinguishable particles.
+Exchanging two electrons corresponds to exchanging two columns in the determinant, which causes
+a sign change of the wave function in accordance with the antisymmetry condition.
+If two electrons are in the same quantum state, then two columns
+are identical and the entire determinant vanishes, which is consistent with the Pauli exclusion principle. 
 
-Rozwiązanie równania Schr\"{o}dingera odpowiada funkcji falowej, dla której średnia
-z hamiltonianu przyjmuje wartość minimalną. Przy założeniu ortonormalności funkcji $\Phi$
-odpowiada to zasadzie wariacyjnej w postaci
+The solution of the Schr\"{o}dinger equation corresponds to a wave function for which the average
+of the Hamiltonian takes a minimum value. Assuming orthonormality of the function $\Phi$,
+this corresponds to the variational principle in the form
 %
 \begin{equation}
 \delta(\langle\Phi|H|\Phi\rangle - E\langle\Phi|\Phi\rangle)=0,
 \label{var}
 \end{equation}
 %
-gdzie symbol $\delta$ oznacza pochodną wariacyjną.
-Wyliczając średnią z hamiltonianu dla podukładu elektronowego (\ref{Sch-el}) w stanie opisanym funkcją falową (\ref{slater}) otrzymujemy
+where the symbol $\delta$ denotes the variational derivative.
+Calculating the average of the Hamiltonian for the electronic subsystem (\ref{Sch-el}) in the state described by the wave function (\ref{slater}), we obtain
 %
 \begin{equation}
 \langle\Phi|H|\Phi\rangle=\sum_{i,\sigma}\int d\mb{r} \psi_i^{\sigma*}(\mb{r})[-\frac{\hbar^2}{2m}\nabla_i^2+V_Z(\mb{r})]\psi_i^{\sigma}(\mb{r}) 
@@ -342,30 +343,31 @@ \section{Równanie Hartree-Focka}
 \label{E_HF}
 \end{equation}
 %
-gdzie $V_Z$ jest potencjałem elektrostatycznym wytwarzanym przez jądra atomowe, a $E_H$ nazywa się energią Hartree 
+% **[REVIEW NEEDED]**: Hartree energy and exchange energy terminology
+where $V_Z$ is the electrostatic potential generated by atomic nuclei, and $E_H$ is called the Hartree energy 
 %
 \begin{equation}
 E_H=\frac{e^2}{2}\sum_{i,j,\sigma,\sigma'}
 \int d\mb{r} \int d\mb{r}' \frac{\psi_i^{\sigma*}(\mb{r})\psi_j^{\sigma'*}(\mb{r}')\psi_i^{\sigma}(\mb{r})\psi_j^{\sigma'}(\mb{r}')}{|\mb{r}-\mb{r}'|}.
 \end{equation}
 %
-Wprowadzając gęstość ładunkową w punkcie $\mb{r}$
+Introducing the charge density at point $\mb{r}$
 %
 \begin{equation}
 n(\mb{r})=e\sum_{i,\sigma} |\psi_i^{\sigma}(\mb{r})|^2,
 \label{dens}
 \end{equation}
 %
-możemy zapisać energię Hartree w prostszej formie
+we can write the Hartree energy in a simpler form
 %
 \begin{equation}
 E_H=\frac{1}{2} \int d\mb{r} \int d\mb{r}' \frac{n(\mb{r})n(\mb{r}')}{|\mb{r}-\mb{r}'|}, 
 \label{Hartree}
 \end{equation}
 %
-która odpowiada klasycznej energii oddziaływania kulombowskiego przy zadanym rozkładzie gęstości ładunku elektrycznego. 
+which corresponds to the classical Coulomb interaction energy for a given distribution of electric charge density. 
 %
-$E_x$ nazywa się energią wymiany i wynosi
+$E_x$ is called the exchange energy and equals
 %
 \begin{equation}
 E_x=-\frac{e^2}{2}\sum_{i,j,\sigma}
@@ -373,18 +375,19 @@ \section{Równanie Hartree-Focka}
 \label{exc}
 \end{equation}
 %
-Energia wymiany jest wielkością kwantową, która nie ma odpowiednika w fizyce klasycznej. 
-Oddziaływanie wymianne charakteryzują dwie ważne cechy, które są ze sobą ściśle powiązane. 
-Przede wszystkim, oddziaływanie to dotyczy tylko elektronów o takim samym kierunku spinu. 
-Zgodnie z zasadą Pauliego dwa elektrony o tym samym kierunku spinu nie moga mieć takich samych
-pozostałych liczb kwantowych, co wiązałoby się z obsadzeniem tego samego stanu kwantowego. 
-Efektywnie prowadzi to do powiększenia odległości między takimi elektronami i
-obniżenia energii oddziaływania kulombowskiego. Z tego wynika druga cecha
-oddziaływania wymiennego: wartość ujemna energii tego oddziaływania. Można interpretować oddziaływanie wymienne
-jako oddziaływanie kulombowskie elektronu z ładunkiem dodatnim tzw. {\it dziurą wymienną} ({\it ang. exchange hole}),
-która powoduje redukcję gestości ładunku ujemnego wokół każdego elektronu. 
+% **[REVIEW NEEDED]**: Exchange interaction and exchange hole concept
+The exchange energy is a quantum quantity that has no counterpart in classical physics. 
+The exchange interaction is characterized by two important features that are closely related to each other. 
+First of all, this interaction concerns only electrons with the same spin direction. 
+According to the Pauli principle, two electrons with the same spin direction cannot have the same
+remaining quantum numbers, which would be associated with occupying the same quantum state. 
+Effectively, this leads to an increase in the distance between such electrons and
+a reduction of the Coulomb interaction energy. From this follows the second feature
+of the exchange interaction: the negative value of this interaction energy. One can interpret the exchange interaction
+as the Coulomb interaction of an electron with a positive charge called the {\it exchange hole},
+which causes a reduction of the negative charge density around each electron. 
 
-Po zastosowaniu metody wariacyjnej do (\ref{E_HF}) otrzymujemy równanie Hartree-Focka
+After applying the variational method to (\ref{E_HF}), we obtain the Hartree-Fock equation
 %
 \begin{equation}
 [-\frac{\hbar^2}{2m}\nabla_i^2+V_Z(\mb{r})+V_H(\mb{r})]\psi_i^{\sigma}(\mb{r})
@@ -392,16 +395,16 @@ \section{Równanie Hartree-Focka}
 \label{HF}
 \end{equation} 
 %
-gdzie $V_H$ oznacza potencjał Hartree
+where $V_H$ denotes the Hartree potential
 %
 \begin{equation}
 V_H(\mb{r})=\int d\mb{r}' \frac{n(\mb{r}')}{|\mb{r}-\mb{r}'|}.
 \end{equation}
 %
-Równanie Hartree-Focka jest równaniem nieliniowymi ponieważ w ostatnim wyrazie Hamiltonianu występuje poszukiwana funkcja falowa.
-Slater zaproponował inną postać równań (\ref{HF}), która
-pozwala lepiej zrozumieć charakter oddziaływania wymiennego \cite{slater51}.
-Mnożąc i dzieląc wyraz wymienny przez $\psi_i^{\sigma}(r)$ otrzymujemy
+The Hartree-Fock equation is a nonlinear equation because the last term of the Hamiltonian contains the wave function we are looking for.
+Slater proposed a different form of equations (\ref{HF}), which
+allows better understanding of the nature of the exchange interaction \cite{slater51}.
+Multiplying and dividing the exchange term by $\psi_i^{\sigma}(r)$, we obtain
 %
 \begin{equation}
 [-\frac{\hbar^2}{2m}\nabla_i^2+V_Z(\mb{r})+V_H(\mb{r})
@@ -409,42 +412,43 @@ \section{Równanie Hartree-Focka}
 \label{HFS}
 \end{equation}
 %
-gdzie $n(\mb{r},\mb{r}')$ można interpretować jako gęstość ładunku wymiennego 
+where $n(\mb{r},\mb{r}')$ can be interpreted as the exchange charge density 
 %
 \begin{equation}
 n(\mb{r},\mb{r}')= e\sum_{j,\sigma}\frac{\psi_j^{\sigma*}(\mb{r}')\psi_i^{\sigma}(\mb{r}')\psi_j^{\sigma}(\mb{r})\psi_i^{\sigma*}(\mb{r})}{\psi_i^{\sigma*}(\mb{r})\psi_i^{\sigma}(\mb{r})},
 \label{EC}
 \end{equation}
 %
-która jest funkcją dwóch położeń $\mb{r}$ i $\mb{r}'$ i zależy od stanu kwantowego $i$. Całkowity ładunek wymienny
-równy jest ładunkowi pojedynczego elektronu, co łatwo pokazać wykonując całkowanie po $\mb{r}'$ i korzystaniu z ortogonalności 
-funkcji falowych
+which is a function of two positions $\mb{r}$ and $\mb{r}'$ and depends on the quantum state $i$. The total exchange charge
+equals the charge of a single electron, which can be easily shown by integrating over $\mb{r}'$ and using the orthogonality 
+of wave functions
 %
 \begin{equation}
 q=e\sum_{j,\sigma}[\int d\mb{r}'\psi_j^{\sigma*}(\mb{r}')\psi_i^{\sigma}(\mb{r}')]\frac{\psi_j^{\sigma}(\mb{r})}{\psi_i^{\sigma}(\mb{r})}=
 e\sum_{j,\sigma}\delta_{ij}\frac{\psi_j^{\sigma}(\mb{r})}{\psi_i^{\sigma}(\mb{r})}=e.
 \end{equation}
 %
-Biorąc pod uwagę przypadek, gdy obydwa położenia są jednakowe $\bm{r}=\bm{r'}$ otrzymujemy gęstość ładunku wymiennego zgodną ze wzorem (\ref{dens}).
-Powoduje to, że wyraz opisujący oddziaływanie elektronu z jego własnym polem wymiennym ma taką sama postać jak potencjał Hartree, ale z przeciwny znakiem
-Dzięki tej własności, obydwa wyrazy się wzajemnie kasują i w przybliżeniu Hartree-Focka nie występuje problem oddziaływania elektronu z jego własnym polem ({\it samooddziaływanie, ang. self-interaction}). 
-W formie (\ref{HFS}) równanie Hartree-Focka ma postać jednoelektronowego równania Schr\"{o}dingera z potencjałem wymiennym wytwarzanym
-w miejscu znajdowania się elektronu przez ładunek wymienny.
-Zgodnie z twierdzeniem Koopmansa, wartości własne $E_i$ odpowiadają energiom koniecznym do usunięcia elektronu z orbitalu 
+% **[REVIEW NEEDED]**: Self-interaction cancellation and Koopman's theorem
+Taking into account the case when both positions are the same $\bm{r}=\bm{r'}$, we obtain the exchange charge density consistent with formula (\ref{dens}).
+This causes the term describing the interaction of an electron with its own exchange field to have the same form as the Hartree potential, but with the opposite sign.
+Due to this property, both terms cancel each other out, and in the Hartree-Fock approximation there is no problem of an electron interacting with its own field ({\it self-interaction}). 
+In the form (\ref{HFS}), the Hartree-Fock equation has the form of a one-electron Schr\"{o}dinger equation with an exchange potential generated
+at the location of the electron by the exchange charge.
+According to Koopman's theorem, the eigenvalues $E_i$ correspond to the energies required to remove an electron from the orbital 
 $\psi_i^{\sigma}(\mb{r})$~\cite{Koopman}.
 
 
-\section{Metoda pola samouzgodnionego}
+\section{Self-consistent field method}
 
-Równania Hartree-Focka najczęściej rozwiązuje się numerycznie stosując metodę pola samouzgodnionego, nazywaną również przybliżeniem średniego pola.
-Przy zadanych położeniach atomów wybieramy początkowe funkcje falowe $\psi_{i0}^{\sigma}$, 
-które przykładowo mogą być orbitalami izolowanych atomów.
-Dla tych funkcji falowych wyliczamy gęstość elektronową $n(\mb{r})$ i potencjał $V(\mb{r})$ 
-w każdym punkcie przestrzeni badanego układu. 
-Dla tak wyliczonego potencjału rozwiązujemy równanie (\ref{HF}) wyznaczając zbiór jednoelektronowych funkcji falowych i energii 
-stanów elektronowych. Wyznaczone funkcje falowe stosujemy do ponownego wyliczenia gęstości elektronów
-i potencjału efektywnego. Procedurę powtarzamy do momentu, gdy otrzymane w kolejnych krokach energie i funkcje falowe
-są jednakowe lub różnią się o niewielką zadaną wielkość. Najczęściej parametrem służącym do badania zbieżności
+The Hartree-Fock equations are most often solved numerically using the self-consistent field method, also called the mean-field approximation.
+For fixed atomic positions, we choose initial wave functions $\psi_{i0}^{\sigma}$, 
+which can be, for example, orbitals of isolated atoms.
+For these wave functions, we calculate the electron density $n(\mb{r})$ and potential $V(\mb{r})$ 
+at each point in space of the studied system. 
+For the potential calculated in this way, we solve equation (\ref{HF}) determining the set of one-electron wave functions and energies 
+of electronic states. We use the determined wave functions to recalculate the electron density
+and effective potential. We repeat the procedure until the energies and wave functions obtained in successive steps
+are the same or differ by a small specified amount. Most often, the parameter used to test convergence
 rachunku jest całkowita energia układu. Metoda pola samoouzgodnionego zastosowana była po raz pierwszy
 do rozwiązania równania Hartree, które możemy otrzymać z rownania Hartree-Focka po usunięciu
 oddziaływania wymiennego. Równanie Hartree odpowiada funkcji falowej w postaci

From 3e454f029a51c31deb3479567094c676a05db630 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Fri, 2 Jan 2026 16:37:18 +0000
Subject: [PATCH 05/23] Translate Electronic correlations and start Electron
 gas section

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 113 ++++++++++++++++++++++++++------------------------
 1 file changed, 58 insertions(+), 55 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index b2ebe3c..2cb3cdf 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -449,96 +449,99 @@ \section{Self-consistent field method}
 of electronic states. We use the determined wave functions to recalculate the electron density
 and effective potential. We repeat the procedure until the energies and wave functions obtained in successive steps
 are the same or differ by a small specified amount. Most often, the parameter used to test convergence
-rachunku jest całkowita energia układu. Metoda pola samoouzgodnionego zastosowana była po raz pierwszy
-do rozwiązania równania Hartree, które możemy otrzymać z rownania Hartree-Focka po usunięciu
-oddziaływania wymiennego. Równanie Hartree odpowiada funkcji falowej w postaci
-iloczynu niezależnych funkcji jednocząstkowych, czyli opisuje zbiór niezależnych elektronów
-w efektywnym polu elektrostatycznym. Metoda pola samouzgodnionego stosowana jest obecnie w większości 
-metod obliczeniowych struktury elektronowej ciał stałych. 
+of the calculation is the total energy of the system. The self-consistent field method was first applied
+to solve the Hartree equation, which we can obtain from the Hartree-Fock equation by removing
+the exchange interaction. The Hartree equation corresponds to a wave function in the form of
+a product of independent single-particle functions, i.e., it describes a set of independent electrons
+in an effective electrostatic field. The self-consistent field method is currently used in most 
+computational methods for the electronic structure of solids. 
 
-\section{Korelacje elektronowe}
+\section{Electronic correlations}
 
-Podejście Hartree-Focka jest metodą przybliżoną, zatem wyliczona energia
-i funkcje falowe różnią się od wielkości dokładnych, które odpowiadają rozwiązaniom równania
-Schr\"{o}dingera dla funkcji wieloelektronowej.
-Uwzględnienie wszystkich efektów oddziaływań wielociałowych w układzie elektronowym umożliwia
-dodatkowe obniżenie energii całkowitej w porównaniu do energii Hartree-Focka.
-Różnica między dokładną wartością energii ($E_{\rm{exact}}$) i energią wyliczoną w przybliżeniu 
-Hartree-Focka ($E_{\rm{HF}}$), nosi nazwę energii korelacji
+% **[REVIEW NEEDED]**: Correlation energy definition and strongly correlated systems
+The Hartree-Fock approach is an approximate method, so the calculated energy
+and wave functions differ from exact quantities, which correspond to solutions of the Schr\"{o}dinger equation
+for the many-electron wave function.
+Taking into account all effects of many-body interactions in the electronic system enables
+an additional lowering of the total energy compared to the Hartree-Fock energy.
+The difference between the exact energy value ($E_{\rm{exact}}$) and the energy calculated in the 
+Hartree-Fock approximation ($E_{\rm{HF}}$) is called the correlation energy
 %
 \begin{equation}
 E_c=E_{\rm{\text{exact}}}-E_{\rm{\text{HF}}}.
 \end{equation}
 %
-Żródłem korelacji elektronowych są oddziaływania kulombowskie, które dążą do przestrzennej separacji 
-elektronów. Elektron znajdujący się w położeniu $\bm{r}$ powoduje, że inne elektrony unikają tego położenia.
-Czyli prawdopodobieństwo znalezienia elektronu w danym punkcie zależy od położeń wszystkich pozostałych $N-1$
-elektronów. Materiały, w których lokalne oddziaływania kulombowskie mają decydujący wpływ na strukturę elektronową
-i własności transportowe nazywane są układami silnie skorelowanymi. 
+The source of electronic correlations are Coulomb interactions, which tend to spatially separate 
+electrons. An electron located at position $\bm{r}$ causes other electrons to avoid this position.
+That is, the probability of finding an electron at a given point depends on the positions of all the remaining $N-1$
+electrons. Materials in which local Coulomb interactions have a decisive influence on the electronic structure
+and transport properties are called strongly correlated systems. 
 
-Metoda Hartree-Focka jest punktem wyjścia do bardziej zaawansowanych metod stosowanych głównie w chemii kwantowej.
-Można ogólnie zapisać funkcję falową układu $N$ elektronów w postaci rozwinięcia na skończoną ilość wyznaczników Slatera
+% **[REVIEW NEEDED]**: Configuration interaction and multi-determinant methods
+The Hartree-Fock method is the starting point for more advanced methods used mainly in quantum chemistry.
+One can generally write the wave function of a system of $N$ electrons as an expansion in a finite number of Slater determinants
 %
 \begin{equation}
 \Phi(\bm{r}_1,...,\bm{r}_N)=c_0\Phi_0(\bm{r}_1,...,\bm{r}_N)+\sum_{i=1}^{N_\text{det}} c_i\Phi_i(\bm{r}_1,...,\bm{r}_N),
 \end{equation}
 %
-gdzie $\Phi_0$ jest funkcją falową stanu podstawowego w przybliżeniu Hartree-Focka,
-a wyznaczniki $\Phi_i$ odpowiadają stanom wzbudzonym. Im większa liczba tych wyznaczników $N_{\text{det}}$, tym dokładniej
-opisane są korelacje elektronowe. Funkcje $\Phi_i$ konstruowane są przy pomocy orbitali jednoelektronowych, które nie są obsadzone
-w stanie podstawowym i mogą być zajęte przez elektrony wzbudzone. Wyznaczniki mogą opisywać pojedyncze przejścia elektronów do orbitali nieobsadzonych ($\Phi_i^{\text{S}}$),
-podwójne wzbudzenia elektronów ($\Phi_i^{\text{D}}$), potrójne wzbudzenia ($\Phi_i^{\text{T}}$) i tak dalej.
-Całkowita liczba wszystkich możliwych wzbudzonych konfiguracji rośnie bardzo szybko wraz z liczbą elektronów $N$ i orbitali $M$ (zajętych i pustych)
-według wzoru 
+where $\Phi_0$ is the ground state wave function in the Hartree-Fock approximation,
+and the determinants $\Phi_i$ correspond to excited states. The larger the number of these determinants $N_{\text{det}}$, the more accurately
+the electronic correlations are described. The functions $\Phi_i$ are constructed using single-electron orbitals that are not occupied
+in the ground state and can be occupied by excited electrons. The determinants can describe single transitions of electrons to unoccupied orbitals ($\Phi_i^{\text{S}}$),
+double electron excitations ($\Phi_i^{\text{D}}$), triple excitations ($\Phi_i^{\text{T}}$), and so on.
+The total number of all possible excited configurations grows very rapidly with the number of electrons $N$ and orbitals $M$ (occupied and empty)
+according to the formula 
 %
 \begin{equation}
 N_{\text{det}}=\frac{(M+1)!}{N!(M+1-N)!}
 \end{equation}
 %
-W najprostszym podejściu nazywanym metodą oddziaływania konfiguracji ({\it ang. configuration interaction} - CI) \cite{GTO},
-wyznaczniki konstruuje się z zajętych i pustych orbitali jednoelektronowych hamiltonianu Hartree-Focka.
-Funkcja falowa przyjmuje postać szeregu wyznaczników o zwiększającej się ilości wzbudzonych elektronów
+In the simplest approach called the configuration interaction (CI) method \cite{GTO},
+the determinants are constructed from occupied and empty single-electron orbitals of the Hartree-Fock Hamiltonian.
+The wave function takes the form of a series of determinants with an increasing number of excited electrons
 %
 \begin{equation}
 \Phi_{\text{CI}}=c_0\Phi_0+\sum_{i=1}^{N_\text{S}} c^\text{S}_i \Phi_i^{\text{S}}+\sum_{i=1}^{N_\text{D}} c^\text{D}_i \Phi_i^{\text{D}}+\sum_{i=1}^{N_\text{T}} c^\text{T}_i \Phi_i^{\text{T}}+...,
 \label{phiCT}
 \end{equation}
 % 
-gdzie $N_\text{S}$, $N_\text{D}$, $N_\text{T}$ określają ilość wyznaczników dla danej liczby wzbudzonych orbitali.
-Stan podstawowy znajduje się przez wyznaczenie współczynników $c_i$, które minimalizuje energię
+where $N_\text{S}$, $N_\text{D}$, $N_\text{T}$ determine the number of determinants for a given number of excited orbitals.
+The ground state is found by determining the coefficients $c_i$ that minimize the energy
 %
 \begin{equation}
 E_\text{CI}=\int \Phi^*(\bm{r}_1,...,\bm{r}_N)H\Phi(\bm{r}_1,...,\bm{r}_N)d\bm{r}_1d\bm{r}_2...d\bm{r}_N,
 \end{equation}
 %
-gdzie $H$ jest hamiltonianem układu elektronowego (\ref{Sch-el}).
-Minimalizacja przeprowadzana jest przy warunku normalizacji funkcji falowej, co przekłada się na warunek dla wszystkich współczynników rozwinięcia
+where $H$ is the Hamiltonian of the electronic system (\ref{Sch-el}).
+The minimization is carried out under the condition of wave function normalization, which translates into a condition for all expansion coefficients
 $\sum_ic_i^2=1$.
-W granicy nieskończonej liczby wyznaczników tak otrzymana funkcja falowa odpowiada dokładnej funkcji wieloelektronowej.
-W praktyce stosuje się rozwinięcia do kilku największych wyrazów szeregu (\ref{phiCT}), które
-obejmują podwójne, potrójne lub poczwórne wzbudzenia.  
-Następnym krokiem jest umożliwienie optymalizacji również orbitali jednoelektronowych w połączeniu z
-optymalizacją współczynników rozwinięcia w ramach wielokonfiguracyjnej metody pola samouzgodnionego  ({\it ang. multi-configuration self-consistant field }- MCSCF). 
-Metody wielowyznacznikowe pozwalają wyznaczyć bardzo dokładnie funkcje falowe i energie stanów elektronowych, 
-jednak czas obliczeniowy skaluje się eksponencjalnie z rozmiarem układu,
-więc stosowane jest jedynie do małych układów molekularnych.
-
-\section{Gaz elektronowy}
-
-Jednorodny gaz elektronowy jest dobrym przybliżeniem rzeczywistej struktury
-elektronowej prostych metali i jest często wykorzystywany w wielu podstawowych
-metodach obliczeniowych (np. w przybliżeniu lokalnej gęstości).
-Najprostszy model gazu elektronowego składa się z nieoddziałujących cząstek
-zamkniętych w sześcianie o objętości $V=L^3$. Długość elektronowej fali de'Broglie'a
-rozchodzącej się w każdym kierunku musi spełniać warunek periodyczności ($n\lambda=L$),
-co prowadzi to kwantowania wektora falowego w kierunkach $x$, $y$ i $z$
+In the limit of an infinite number of determinants, the wave function obtained in this way corresponds to the exact many-electron wave function.
+In practice, expansions are used up to a few largest terms of the series (\ref{phiCT}), which
+include double, triple, or quadruple excitations.  
+The next step is to enable optimization of single-electron orbitals as well, in combination with
+optimization of expansion coefficients within the multi-configuration self-consistent field (MCSCF) method. 
+Multi-determinant methods allow very accurate determination of wave functions and energies of electronic states, 
+however, the computational time scales exponentially with the system size,
+so it is used only for small molecular systems.
+
+\section{Electron gas}
+
+% **[REVIEW NEEDED]**: Free electron gas model and Fermi energy
+A homogeneous electron gas is a good approximation of the actual electronic structure
+of simple metals and is often used in many fundamental
+computational methods (e.g., in the local density approximation).
+The simplest model of an electron gas consists of non-interacting particles
+confined in a cube of volume $V=L^3$. The length of the electronic de Broglie wave
+propagating in each direction must satisfy the periodicity condition ($n\lambda=L$),
+which leads to quantization of the wave vector in the $x$, $y$, and $z$ directions
 %
 \begin{equation}
 \mb{k}=(\frac{2\pi n_x}{L},\frac{2\pi n_x}{L},\frac{2\pi n_x}{L}),
 \end{equation}
 %
-gdzie $n_x$, $n_y$, $n_z$ są liczbami całkowitymi.
-Energie i funkcje falowe dyskretnych stanów elektronowych dane są wzorami
+where $n_x$, $n_y$, $n_z$ are integers.
+The energies and wave functions of discrete electronic states are given by the formulas
 %
 \begin{equation}
 E_k=\frac{\hbar^2 k^2}{2m},

From cde5bfc8a820a931d525af9ec5ed57f725c782b4 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Fri, 2 Jan 2026 16:38:34 +0000
Subject: [PATCH 06/23] Continue translating Electron gas section - Fermi
 energy and density

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 34 +++++++++++++++++-----------------
 1 file changed, 17 insertions(+), 17 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index 2cb3cdf..27a35b6 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -552,52 +552,52 @@ \section{Electron gas}
 \label{pw}
 \end{equation}
 %
-Zakładamy, że układ nie jest spolaryzowany magnetycznie i liczba elektronów ze spinami skierowanymi 
-do góry i na dół jest jednakowa, $N_{\uparrow}=N_{\downarrow}=\frac{1}{2}N$.
-Zgodnie z zakazem Pauliego w stanie kwantowym o danym wektorze falowym $\mb{k}$ mogą
-znajdować się maksymalnie dwa elektrony o przeciwnych kierunkach spinu.
-W temperaturze $T=0$ K elektrony obsadzają kolejne stany od najniższego do
-maksymalnej wartości $E_F$ nazywanej energią Fermiego. 
-Całkowita energia układu $N$ elektronów wynosi
+We assume that the system is not magnetically polarized and the number of electrons with spins pointing 
+up and down is equal, $N_{\uparrow}=N_{\downarrow}=\frac{1}{2}N$.
+According to the Pauli exclusion principle, in a quantum state with a given wave vector $\mb{k}$, there can be
+a maximum of two electrons with opposite spin directions.
+At temperature $T=0$ K, electrons occupy consecutive states from the lowest to
+the maximum value $E_F$ called the Fermi energy. 
+The total energy of a system of $N$ electrons is
 %
 \begin{equation}
 E=2\sum_{k<k_F} E_k = \frac{2V}{(2\pi)^3}\int d\bm{k} \frac{\hbar^2 k^2}{2m} = \frac{V}{(2\pi)^3}\frac{4\pi\hbar^2}{5m} k_F^5, 
 \label{E}
 \end{equation}
 %
-gdzie $(2\pi)^3/V$ jest objętością w przestrzeni odwrotnej przypadającą na jeden stan elektronowy, 
-a $k_F$ jest wektorem falowym Fermiego, który jest promieniem kuli w przestrzeni odwrotnej
-zawierającej zajęte stany elektronowe. W ogólnym przypadku powierzchnia obejmująca zajęte stany elektronowe,
-nazywana powierzchnią Fermiego, może mieć dowolny kształt i składać się z wielu odrębnych części.
-Liczba elektronów znajdująca się w objętości $V$ wyraża się wzorem
+where $(2\pi)^3/V$ is the volume in reciprocal space per one electronic state, 
+and $k_F$ is the Fermi wave vector, which is the radius of a sphere in reciprocal space
+containing the occupied electronic states. In the general case, the surface encompassing occupied electronic states,
+called the Fermi surface, can have any shape and consist of many separate parts.
+The number of electrons in volume $V$ is expressed by the formula
 %
 \begin{equation}
 N=\frac{2V}{(2\pi)^3}\int d\bm{k}=\frac{2V}{(2\pi)^3}\frac{4\pi}{3} k_F^3.  
 \label{N}
 \end{equation}
 %
-Dzieląc (\ref{E}) przez (\ref{N}) otrzymujemy średnią energię przypadającą
-na pojedynczy elektron, wyrażoną przez energię Fermiego
+Dividing (\ref{E}) by (\ref{N}), we obtain the average energy per
+single electron, expressed through the Fermi energy
 %
 \begin{equation}
 \frac{E}{N}=\frac{3}{5} \frac{\hbar^2 k_F^2}{2m} = \frac{3}{5} E_F.
 \label{EN}
 \end{equation}  
 %
-W przypadku jednorodnego gazu gęstość elektronowa dana jest wyrażeniem
+In the case of a homogeneous gas, the electron density is given by the expression
 %
 \begin{equation}
 n = \frac{N}{V} = \frac{k_F^3}{3\pi^2}.
 \label{nkf}
 \end{equation}
 %
-Często używaną wielkością jest objętość kuli w przestrzeni rzeczywistej przypadająca na jeden elektron 
+A commonly used quantity is the volume of a sphere in real space per one electron 
 %
 \begin{equation}
 V_s=\frac{4\pi}{3}r_{s}^3=\frac{1}{n},
 \end{equation}
 %
-gdzie $r_s$ jest jej promieniem
+where $r_s$ is its radius
 %
 \begin{equation}
 r_s=(\frac{3}{4\pi n})^{1/3}.

From 9707d562b199e18dee2ae8818e6066241288cf0d Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Fri, 2 Jan 2026 16:39:42 +0000
Subject: [PATCH 07/23] Fix wave vector components in electron gas section

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/book/main.tex b/book/main.tex
index 27a35b6..d1be837 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -537,7 +537,7 @@ \section{Electron gas}
 which leads to quantization of the wave vector in the $x$, $y$, and $z$ directions
 %
 \begin{equation}
-\mb{k}=(\frac{2\pi n_x}{L},\frac{2\pi n_x}{L},\frac{2\pi n_x}{L}),
+\mb{k}=(\frac{2\pi n_x}{L},\frac{2\pi n_y}{L},\frac{2\pi n_z}{L}),
 \end{equation}
 %
 where $n_x$, $n_y$, $n_z$ are integers.

From cefed28877c6b4c6e8da24354300b9744d532149 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Fri, 2 Jan 2026 17:54:27 +0000
Subject: [PATCH 08/23] Complete translation of Section 1.5 Electron gas

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 54 ++++++++++++++++++++++++++-------------------------
 1 file changed, 28 insertions(+), 26 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index d1be837..83b12cb 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -603,74 +603,76 @@ \section{Electron gas}
 r_s=(\frac{3}{4\pi n})^{1/3}.
 \end{equation}
 %
-
-Wyliczymy teraz energię gazu elektronowego w przybliżeniu Hartree-Focka.
-Zakładamy, że ładunek dodatni rozłożony jest jednorodnie w całej przestrzeni $V$
-i ma gęstość taką samą jak gaz elektronów. Energia oddziaływania
-elektronów z takim polem kasuje się z energią Hartree i całkowita energia
-elektronu składa się tylko z części kinetycznej i oddziaływania wymiennego.
-Energię wymiany można łatwo wyliczyć przedstawiając potencjał kulombowski
-w postaci transformaty Fouriera
+% **[REVIEW NEEDED]**: Hartree-Fock treatment of electron gas and exchange energy derivation
+We will now calculate the energy of the electron gas in the Hartree-Fock approximation.
+We assume that the positive charge is distributed uniformly throughout the entire space $V$
+and has the same density as the electron gas. The interaction energy
+of electrons with such a field cancels with the Hartree energy, and the total energy
+of an electron consists only of the kinetic part and the exchange interaction.
+The exchange energy can be easily calculated by representing the Coulomb potential
+in the form of a Fourier transform
 %
 \begin{equation}
 \frac{1}{|\mb{r}-\mb{r}'|}=4\pi\int \frac{d\mb{q}}{(2\pi)^3}\frac{e^{i\mb{q}(\mb{r}-\mb{r}')}}{q^2},
 \end{equation}
 %
-Wstawiając funkcję falową w postaci (\ref{pw}) do wyrażenia na energię wymiany 
-i wykonując całkowanie otrzymujemy energie stanów jednoelektronowych
+Inserting the wave function in the form (\ref{pw}) into the expression for the exchange energy 
+and performing integration, we obtain the energies of single-electron states
 %
 \begin{equation}
 E_k=\frac{\hbar^2 k^2}{2m} - \int_{\mb{k'}<\mb{k_F}} \frac{d\bm{k'}}{(2\pi)^3} \frac{4\pi}{|\mb{k}-\mb{k'}|^2}
 =\frac{\hbar^2 k^2}{2m} - \frac{k_F}{\pi}(1+\frac{k_F^2-k^2}{2kk_F}ln|\frac{k_F+k}{k_F-k}|).
 \end{equation}
 %
-Całkowitą energię układu elektronów dostajemy sumując to wyrażenie po wektorze falowym    
+The total energy of the electron system is obtained by summing this expression over the wave vector    
 %
 \begin{equation}
 E=\sum_{k<k_F}[\frac{\hbar^2 k^2}{2m} - \frac{k_F}{\pi}(1+\frac{k_F^2-k^2}{2kk_F}ln|\frac{k_F+k}{k_F-k}|)].
 \end{equation}
 %
-Wykorzystując (\ref{EN}) i zamieniając sumowanie w drugim wyrazie na całkę dostajemy
+Using (\ref{EN}) and converting the summation in the second term to an integral, we obtain
 %
 \begin{equation}
 E=N(\frac{3}{5}E_F - \frac{3k_F}{4\pi})=N(\frac{3}{5}E_F - \frac{3(3\pi^2n)^{\frac{1}{3}}}{4\pi}),
 \end{equation}
-gdzie skorzystaliśmy ze związku (\ref{nkf}). Energia wymiany przypadająca na jeden elektron wynosi 
+where we used the relation (\ref{nkf}). The exchange energy per electron is 
 %
 \begin{equation}
 \varepsilon_x=\frac{E_x}{N}=-\frac{3}{4}\Big(\frac{3}{\pi}\Big)^{\frac{1}{3}}n^\frac{1}{3}.
 \label{exchange}
 \end{equation}
 %
-Ten wzór określający zależność energii wymieny od gęstości elektronowej
-wykorzystywany jest w teorii funkcjonału gęstości w ramach przybliżenia lokalnej gęstości.
+% **[REVIEW NEEDED]**: Exchange energy formula and local density approximation
+This formula determining the dependence of exchange energy on electron density
+is used in density functional theory within the local density approximation.
 
-Dla gazu elektronowego można wyprowadzić wyrażenie opisujące rozkład ładunku dziury wymiennej (\ref{EC}),
-który zależy tylko od odległości $r$
+For an electron gas, one can derive an expression describing the distribution of exchange hole charge (\ref{EC}),
+which depends only on distance $r$
 %
 \begin{equation}
 g^{\sigma}_x(r)=1 - [3\frac{sin(rk_F)-rk_Fcos(rk_F)}{(rk_F)^3}]^2.
 \end{equation}
 %
-Nie jest możliwe wyrażenie energii korelacji dla gazu swobodnych elektronów w postaci 
-funkcji analitycznej gęstości $n$. Pierwszy przybliżony wzór zaproponował Wigner \cite{wigner34}
+% **[REVIEW NEEDED]**: Correlation energy approximations - Wigner, Gell-Mann-Brueckner, and QMC
+It is not possible to express the correlation energy for a free electron gas in the form of 
+an analytical function of density $n$. The first approximate formula was proposed by Wigner \cite{wigner34}
 %
 \begin{equation}
 \varepsilon(r_s)=-\frac{0.44}{r_s+7.8}.
 \end{equation}
 %
-W granicy małych gęstości ($r_s\rightarrow \infty$), electrony tworzą tzw. kryształ Wignera, którego energia korelacji 
-pokrywa się z energią elektrostatyczną zlokalizowanych ładunków punktowych.
-W granicy dużej gęstości ($r_s\rightarrow 0$), Gellmann i Breuckner
-wyprowadzili wzór na gęstość energii korelacji niespolaryzowanego magnetycznie
-gazu elektronów \cite{GB}
+In the limit of low densities ($r_s\rightarrow \infty$), electrons form the so-called Wigner crystal, whose correlation energy 
+coincides with the electrostatic energy of localized point charges.
+In the limit of high density ($r_s\rightarrow 0$), Gell-Mann and Brueckner
+derived a formula for the correlation energy density of a magnetically unpolarized
+electron gas \cite{GB}
 %
 \begin{equation}
 \varepsilon_c (r_s)=0.311\ln(r_s)+r_s(A\ln(r_s)+C)-0.048+... .
 \end{equation}
 %
-Energię korelacji gazu elektronowego najdokładniej wyznaczono numerycznie
-metodą kwantowego Monte Carlo (QMC) \cite{CeperleyAlder80}. 
+The correlation energy of the electron gas was most accurately determined numerically
+by the quantum Monte Carlo (QMC) method \cite{CeperleyAlder80}. 
 
 
 \chapter{Stany elektronowe w krysztale}

From c89ec30775e3472247030830db357a87c3b56b88 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Fri, 2 Jan 2026 18:27:01 +0000
Subject: [PATCH 09/23] Replace "spin direction" with "parallel/opposite spins"
 terminology

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index 83b12cb..c57774a 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -61,7 +61,7 @@ \chapter*{Introduction}
 Taking into account the Pauli exclusion principle for the multi-electron wave function, which in its simplest form is written
 is in the form of the Slater determinant, the Hartree-Fock equation can be derived~\cite{fock30,slater30,slater51}.
 In this equation, in addition to the usual Coulomb interaction, there is a non-local
-exchange interaction. This is an interaction that occurs only between electrons with the same spin direction,
+exchange interaction. This is an interaction that occurs only between electrons with parallel spins,
 effectively reducing the Coulomb repulsion between them.
 The Hartree-Fock approach describes electron systems approximately because it does not properly take into account correlations,
 which occur in the many-electron quantum state \cite{wigner34,GB}.
@@ -378,8 +378,8 @@ \section{Hartree-Fock equation}
 % **[REVIEW NEEDED]**: Exchange interaction and exchange hole concept
 The exchange energy is a quantum quantity that has no counterpart in classical physics. 
 The exchange interaction is characterized by two important features that are closely related to each other. 
-First of all, this interaction concerns only electrons with the same spin direction. 
-According to the Pauli principle, two electrons with the same spin direction cannot have the same
+First of all, this interaction concerns only electrons with parallel spins. 
+According to the Pauli principle, two electrons with parallel spins cannot have the same
 remaining quantum numbers, which would be associated with occupying the same quantum state. 
 Effectively, this leads to an increase in the distance between such electrons and
 a reduction of the Coulomb interaction energy. From this follows the second feature
@@ -555,7 +555,7 @@ \section{Electron gas}
 We assume that the system is not magnetically polarized and the number of electrons with spins pointing 
 up and down is equal, $N_{\uparrow}=N_{\downarrow}=\frac{1}{2}N$.
 According to the Pauli exclusion principle, in a quantum state with a given wave vector $\mb{k}$, there can be
-a maximum of two electrons with opposite spin directions.
+a maximum of two electrons with opposite spins.
 At temperature $T=0$ K, electrons occupy consecutive states from the lowest to
 the maximum value $E_F$ called the Fermi energy. 
 The total energy of a system of $N$ electrons is

From 78ddb0ba88e4f05cf9d18106202575a895e7015a Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Fri, 2 Jan 2026 18:35:37 +0000
Subject: [PATCH 10/23] Fix terminology: "Pauli principle" to "Pauli exclusion
 principle"

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/book/main.tex b/book/main.tex
index c57774a..7ccd2a7 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -379,7 +379,7 @@ \section{Hartree-Fock equation}
 The exchange energy is a quantum quantity that has no counterpart in classical physics. 
 The exchange interaction is characterized by two important features that are closely related to each other. 
 First of all, this interaction concerns only electrons with parallel spins. 
-According to the Pauli principle, two electrons with parallel spins cannot have the same
+According to the Pauli exclusion principle, two electrons with parallel spins cannot have the same
 remaining quantum numbers, which would be associated with occupying the same quantum state. 
 Effectively, this leads to an increase in the distance between such electrons and
 a reduction of the Coulomb interaction energy. From this follows the second feature

From fb9f9b738086b13461d2d88dcd656ea9e5cd17d4 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Fri, 2 Jan 2026 18:50:08 +0000
Subject: [PATCH 11/23] Fix spelling: "Koopman's" to "Koopmans'" theorem

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index 7ccd2a7..506ad37 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -428,13 +428,13 @@ \section{Hartree-Fock equation}
 e\sum_{j,\sigma}\delta_{ij}\frac{\psi_j^{\sigma}(\mb{r})}{\psi_i^{\sigma}(\mb{r})}=e.
 \end{equation}
 %
-% **[REVIEW NEEDED]**: Self-interaction cancellation and Koopman's theorem
+% **[REVIEW NEEDED]**: Self-interaction cancellation and Koopmans' theorem
 Taking into account the case when both positions are the same $\bm{r}=\bm{r'}$, we obtain the exchange charge density consistent with formula (\ref{dens}).
 This causes the term describing the interaction of an electron with its own exchange field to have the same form as the Hartree potential, but with the opposite sign.
 Due to this property, both terms cancel each other out, and in the Hartree-Fock approximation there is no problem of an electron interacting with its own field ({\it self-interaction}). 
 In the form (\ref{HFS}), the Hartree-Fock equation has the form of a one-electron Schr\"{o}dinger equation with an exchange potential generated
 at the location of the electron by the exchange charge.
-According to Koopman's theorem, the eigenvalues $E_i$ correspond to the energies required to remove an electron from the orbital 
+According to Koopmans' theorem, the eigenvalues $E_i$ correspond to the energies required to remove an electron from the orbital 
 $\psi_i^{\sigma}(\mb{r})$~\cite{Koopman}.
 
 

From 3c8964634270632b636b047956b3c4187342988e Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Fri, 2 Jan 2026 18:56:11 +0000
Subject: [PATCH 12/23] Change "is equal" to "is the same" for stylistic
 preference

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/book/main.tex b/book/main.tex
index 506ad37..21b902e 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -553,7 +553,7 @@ \section{Electron gas}
 \end{equation}
 %
 We assume that the system is not magnetically polarized and the number of electrons with spins pointing 
-up and down is equal, $N_{\uparrow}=N_{\downarrow}=\frac{1}{2}N$.
+up and down is the same, $N_{\uparrow}=N_{\downarrow}=\frac{1}{2}N$.
 According to the Pauli exclusion principle, in a quantum state with a given wave vector $\mb{k}$, there can be
 a maximum of two electrons with opposite spins.
 At temperature $T=0$ K, electrons occupy consecutive states from the lowest to

From 18887df4f3877ec543c8f8c869204fdd1baf67a5 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Fri, 2 Jan 2026 19:00:08 +0000
Subject: [PATCH 13/23] Change "entire space" to "entire volume" per author
 preference

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/book/main.tex b/book/main.tex
index 21b902e..7db662a 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -605,7 +605,7 @@ \section{Electron gas}
 %
 % **[REVIEW NEEDED]**: Hartree-Fock treatment of electron gas and exchange energy derivation
 We will now calculate the energy of the electron gas in the Hartree-Fock approximation.
-We assume that the positive charge is distributed uniformly throughout the entire space $V$
+We assume that the positive charge is distributed uniformly throughout the entire volume $V$
 and has the same density as the electron gas. The interaction energy
 of electrons with such a field cancels with the Hartree energy, and the total energy
 of an electron consists only of the kinetic part and the exchange interaction.

From 85fd1ec81fa8d92848afa8a88ba2f2a31161619e Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Pawe=C5=82=20T=2E=20Jochym?=
 <jochym@users.noreply.github.com>
Date: Fri, 2 Jan 2026 20:32:58 +0100
Subject: [PATCH 14/23] Add GitHub Copilot instructions for repository (#24)
 (#27)

* Initial plan

* Add GitHub Copilot instructions for the repository



* Clarify academic tasks can be modified if directly requested



---------

Co-authored-by: Copilot <198982749+Copilot@users.noreply.github.com>
Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 .github/copilot-instructions.md | 95 +++++++++++++++++++++++++++++++++
 1 file changed, 95 insertions(+)
 create mode 100644 .github/copilot-instructions.md

diff --git a/.github/copilot-instructions.md b/.github/copilot-instructions.md
new file mode 100644
index 0000000..eb7508a
--- /dev/null
+++ b/.github/copilot-instructions.md
@@ -0,0 +1,95 @@
+# Copilot Instructions for Ab Initio Methods Repository
+
+## Repository Overview
+
+This repository contains an interactive academic book for computational solid state physics lectures aimed at PhD candidates. The materials combine theoretical lecture content with hands-on computational examples.
+
+## Purpose
+
+This is a work-in-progress (WIP) educational resource. Copilot is intended to help with the technical side of preparation for release of this material, not the academic content itself.
+
+## Repository Structure
+
+- **Lecture Materials**: PDF files in the `lecture/` directory covering topics from electron interactions to lattice thermodynamics
+- **Interactive Notebooks**: Numbered Jupyter notebooks (`.ipynb` files) providing hands-on examples:
+  - `00_Index.ipynb` - Main index and introduction
+  - `01_Hydrogen_Molecule.ipynb` through `11_Nanoparticle_MD.ipynb` - Progressive exercises
+- **Support Libraries**: 
+  - `abilib.py` - Core support functions for the exercises
+  - `ase_patch.py` - ASE (Atomic Simulation Environment) patches
+  - `make-am-files.py` - Utility script for file generation
+- **Data Directories**: `data/`, `psp/` contain supporting data files
+- **Configuration**: `anphon/` and related files for phonon calculations
+
+## Key Technologies
+
+- **Python 3**: Primary programming language
+- **Jupyter Notebooks**: Interactive computational environment
+- **ASE (Atomic Simulation Environment)**: Framework for atomic simulations
+- **NumPy/SciPy/Matplotlib**: Scientific computing and visualization libraries
+- **ABINIT**: Ab initio computational materials science calculator
+
+## Coding Guidelines
+
+### General Principles
+1. **Educational Focus**: Code should be clear and pedagogical, suitable for PhD students learning computational physics
+2. **Reproducibility**: Maintain consistency in computational examples and results
+3. **Documentation**: Comment code to explain physics concepts, not just implementation details
+4. **License Compliance**: All code is under CC BY-SA 4.0 license
+
+### Python Style
+- Follow PEP 8 conventions for Python code
+- Use clear, descriptive variable names that reflect physical quantities
+- Include docstrings for functions explaining both the code and the physics
+- Use type hints where they improve clarity
+
+### Notebook Guidelines
+- Ensure notebooks are self-contained and can run independently
+- Include markdown cells explaining the physics and methodology
+- Keep computational cells focused and well-commented
+- Ensure outputs are cleared before committing (to reduce repository size)
+
+### Dependencies
+- Minimize external dependencies beyond the core scientific stack
+- Document any new dependencies clearly
+- Ensure compatibility with JupyterLab/Binder environments
+
+## What Copilot Should Help With
+
+✅ **Technical tasks Copilot should assist with:**
+- Code refactoring and optimization
+- Bug fixes in Python code and notebooks
+- Improving code documentation and comments
+- Ensuring consistent coding style
+- File organization and repository structure
+- Build and deployment configurations
+- Testing and validation scripts
+- Integration with JupyterLab/Binder environments
+
+❌ **Academic tasks Copilot should NOT modify unless directly requested:**
+- Lecture content in PDF files
+- Physics equations and theoretical derivations
+- Scientific conclusions or interpretations
+- Bibliography and references
+- Academic authorship information
+
+## Testing and Validation
+
+- Test notebooks by running them in a clean Jupyter environment
+- Verify that computational examples produce expected physical results
+- Ensure backward compatibility with existing notebooks
+- Check that notebooks run successfully in Binder environment
+
+## Special Considerations
+
+- **Performance**: Some calculations are computationally intensive; maintain reasonable execution times
+- **Reproducibility**: Random seeds and computational parameters should be documented
+- **Educational Value**: Changes should enhance learning, not just optimize code
+- **Environment**: Must work in both local JupyterLab and cloud Binder environments
+
+## Contact and Context
+
+- **Authors**: Przemysław Piekarz, Paweł T. Jochym
+- **Institution**: Institute of Nuclear Physics PAN, Cracow, Poland
+- **Level**: PhD candidate coursework
+- **License**: CC BY-SA 4.0

From 71c0294aa8d496f91247b81cf575c1b8f65f9d64 Mon Sep 17 00:00:00 2001
From: Copilot <198982749+Copilot@users.noreply.github.com>
Date: Sat, 3 Jan 2026 00:16:20 +0100
Subject: [PATCH 15/23] Document book branch LaTeX manuscript structure and
 Quarto migration goal in copilot-instructions.md (#29)

* Initial plan

* Update copilot-instructions.md with comprehensive book branch documentation

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Clarify LaTeX guideline: allow chapter translation by user request

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Add Quarto migration goal to copilot instructions

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 .github/copilot-instructions.md | 55 +++++++++++++++++++++++++++++++++
 1 file changed, 55 insertions(+)

diff --git a/.github/copilot-instructions.md b/.github/copilot-instructions.md
index eb7508a..0e700d2 100644
--- a/.github/copilot-instructions.md
+++ b/.github/copilot-instructions.md
@@ -21,6 +21,26 @@ This is a work-in-progress (WIP) educational resource. Copilot is intended to he
 - **Data Directories**: `data/`, `psp/` contain supporting data files
 - **Configuration**: `anphon/` and related files for phonon calculations
 
+### Book Branch
+
+The book branch contains a comprehensive LaTeX manuscript that provides the theoretical foundation for the course:
+
+- **LaTeX Source Files**: 
+  - `book/main.tex` - Main manuscript file (246KB, ~4200 lines) containing comprehensive theoretical content
+  - `book/methods-old.tex` - Legacy manuscript file
+  
+- **Supporting Figures and Diagrams**:
+  - Crystal structure diagrams: `crystal.pdf`, `lattice.pdf`, `brillouin.pdf`
+  - Material-specific examples: `BaTiO3.pdf` (barium titanate), `Fe2SiO4-1.pdf` and `Fe2SiO4-2.pdf` (iron orthosilicate), `FeSe.pdf` (iron selenide)
+  - Methodology illustrations: `diagram.pdf`, `gap-hyb.pdf` (hybrid functionals), `pseudopots-new.pdf` (pseudopotentials), `rare-earths.pdf`, `usp.pdf` (ultrasoft pseudopotentials)
+  - Other assets: `Jacob.jpg`, `Fxs.png`
+
+- **Book Content Structure**:
+  - **Introduction**: Historical development of DFT and ab initio methods, from Schrödinger equation to modern density functional theory
+  - **Chapter 1 - Electron Interactions**: Fundamental quantum mechanics, Hartree-Fock approximation, electronic correlations, homogeneous electron gas, hydrogen molecule example (Heitler-London method)
+  - **Chapter 2 - Stany elektronowe w krysztale** (Electronic States in Crystals, in Polish): Crystal lattices, Bravais lattices, reciprocal space, Bloch's theorem, band structure theory
+  - Additional chapters covering DFT theory, electronic structure methods, orbital-dependent functionals, insulators/semiconductors, electric polarization, van der Waals interactions, and many-body effects
+
 ## Key Technologies
 
 - **Python 3**: Primary programming language
@@ -28,6 +48,8 @@ This is a work-in-progress (WIP) educational resource. Copilot is intended to he
 - **ASE (Atomic Simulation Environment)**: Framework for atomic simulations
 - **NumPy/SciPy/Matplotlib**: Scientific computing and visualization libraries
 - **ABINIT**: Ab initio computational materials science calculator
+- **LaTeX**: Document preparation system for the textbook manuscript
+- **Quarto** (migration target): Modern publishing system for unified scientific content (notebooks and manuscripts)
 
 ## Coding Guidelines
 
@@ -54,6 +76,15 @@ This is a work-in-progress (WIP) educational resource. Copilot is intended to he
 - Document any new dependencies clearly
 - Ensure compatibility with JupyterLab/Binder environments
 
+### LaTeX Guidelines
+- Follow standard LaTeX best practices for academic manuscripts
+- Maintain consistency in mathematical notation throughout the document
+- Use descriptive labels for equations, figures, and sections (e.g., `\label{eq:schrodinger}`, `\label{fig:crystal}`)
+- Ensure all references and citations are properly formatted using BibTeX
+- Keep figure files organized in the `book/` directory
+- Use comments to mark sections requiring review (e.g., `% **[REVIEW NEEDED]**: Description of issue`)
+- Preserve multi-language content structure (English and Polish sections), except when translating consecutive chapters by user request
+
 ## What Copilot Should Help With
 
 ✅ **Technical tasks Copilot should assist with:**
@@ -65,10 +96,16 @@ This is a work-in-progress (WIP) educational resource. Copilot is intended to he
 - Build and deployment configurations
 - Testing and validation scripts
 - Integration with JupyterLab/Binder environments
+- LaTeX formatting and structure improvements
+- Cross-referencing between notebooks and book content
+- Figure and diagram organization in the book directory
+- Build scripts for LaTeX compilation (if needed)
+- Migration to Quarto-based publishing system (unify content where sensible, but not forcefully)
 
 ❌ **Academic tasks Copilot should NOT modify unless directly requested:**
 - Lecture content in PDF files
 - Physics equations and theoretical derivations
+- Mathematical proofs and derivations in `main.tex`
 - Scientific conclusions or interpretations
 - Bibliography and references
 - Academic authorship information
@@ -79,6 +116,8 @@ This is a work-in-progress (WIP) educational resource. Copilot is intended to he
 - Verify that computational examples produce expected physical results
 - Ensure backward compatibility with existing notebooks
 - Check that notebooks run successfully in Binder environment
+- Verify LaTeX compilation succeeds without errors (for book branch)
+- Ensure all figures referenced in LaTeX are present in book directory
 
 ## Special Considerations
 
@@ -86,6 +125,22 @@ This is a work-in-progress (WIP) educational resource. Copilot is intended to he
 - **Reproducibility**: Random seeds and computational parameters should be documented
 - **Educational Value**: Changes should enhance learning, not just optimize code
 - **Environment**: Must work in both local JupyterLab and cloud Binder environments
+- **Multi-language Content**: The book manuscript contains sections in English and Polish (Polish sections start at Chapter 2/Rozdział 2)
+- **Branch Structure**: Main branch focuses on interactive materials; book branch adds comprehensive textbook manuscript
+- **Figure Management**: Keep book figures organized and ensure LaTeX references are correct
+- **Quarto Migration Goal**: The repository aims to migrate to a Quarto-based solution for unified publishing. Prioritize approaches that facilitate this transition and unify content structure where it makes sense, but avoid forced unification that compromises existing functionality
+
+## Branch-Specific Notes
+
+### Main Branch
+- Focus: Interactive Jupyter notebooks and lecture PDFs
+- Target: Hands-on computational exercises
+
+### Book Branch
+- Focus: Comprehensive LaTeX textbook manuscript with supporting materials
+- Target: Theoretical foundation and detailed methodology
+- Key File: `book/main.tex` (main manuscript source)
+- Build Process: LaTeX compilation to PDF (build system TBD)
 
 ## Contact and Context
 

From 472bf27aaadf7feaa10cb0f9ee3b7dce1761b221 Mon Sep 17 00:00:00 2001
From: Copilot <198982749+Copilot@users.noreply.github.com>
Date: Sat, 3 Jan 2026 06:31:49 +0100
Subject: [PATCH 16/23] Translate Chapter 2 (Electronic States in Crystals)
 from Polish to English (#30)

* Initial plan

* Translate Chapter 2 (Electronic states in crystals) from Polish to English

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Address review comments: remove review markers and simplify phrasing

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 330 +++++++++++++++++++++++++-------------------------
 1 file changed, 165 insertions(+), 165 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index 7db662a..73ee131 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -675,92 +675,92 @@ \section{Electron gas}
 by the quantum Monte Carlo (QMC) method \cite{CeperleyAlder80}. 
 
 
-\chapter{Stany elektronowe w krysztale}
+\chapter{Electronic states in crystals}
 
-\section{Sieć krystaliczna}
+\section{Crystal lattice}
 
-Kryształy zbudowane są z atomów lub molekuł ułożonych w periodyczną sieć.
-Trójwymiarową sieć krystaliczną definiujemy jako zbiór punktów, których
-położenia określone są wektorem translacji
+Crystals are built from atoms or molecules arranged in a periodic lattice.
+A three-dimensional crystal lattice is defined as a set of points whose
+positions are given by the translation vector
 %
 \begin{equation}
 \bm{R}_n=n_1\bm{a}_1+n_2\bm{a}_2+n_3\bm{a}_3,
 \label{trans}
 \end{equation}
 %
-gdzie wektory $\bm{a}_i$ nazywamy podstawowymi lub prymitywnymi wektorami translacji, a $n_i$ są liczbami całkowitymi.
-Zbiór punktów zdefiniowanych tym wzorem nazywamy siecią Bravais. Podobnie można zdefiniować sieć Bravais
-dla dowolnego wymiaru $d$ wybierając odpowiednią ilość wektorów bazowych: $\bm{a}_1,\bm{a}_2,...,\bm{a}_d$. 
-Z każdym punktem sieci Bravais można związać zbiór atomów zwanych bazą. Najmniejsza przestrzeń kryształu,
-która po przesunieciu o wszystkie translacje sieciowe wypełni całą przestrzeń nazywamy komórką prymitywną. 
-Komórka prymitywna zawiera dokładnie jeden węzeł sieci Bravais i zwykle zbudowana jest na podstawowych wektorach translacji sieci.
-Istnieje specjalny rodzaj komórki prymitywnej nazywany komórką Wignera-Seitza, która jest przestrzenią zawierającą punkty przestrzeni, 
-które znajdują się bliżej danego punktu sieci niż pozostałych. 
-Często zamiast komórki prymitywnej używa się komórki elementarnej, która zbudowana jest na innych wektorach bazowych
-i jej symetria jest taka sama jak symetria całego kryształu.
-Objętość komórki elementarnej jest całkowitą wielokrotnością objętości komórki prymitywnej.
-W strukturach prostych (bez centrowania powierzchniowego lub przestrzennego), komórka elementarna pokrywa się
-z komórką prymitywną. Różnice między poszczególnymi rodzajami komórek można przedstawić na przykładzie
-sieci dwuwymiarowej. Na rysunku \ref{fig:bravais} pokazana jest struktura centrowana prostokątna z zaznaczonymi dwoma wektorami podstawowaymi $\bm{a}_p$
-i $\bm{b}_p$, które definiują komórkę prymitywną, pokazaną na rysunku w dwóch wariantach. Komórka Wignera-Seitza pokazana
-jest po prawej stronie. Komórka elementarna wyznaczona jest przez dwa wektory bazowe $\bm{a}_e$ i $\bm{b}_e$.
-W dwóch wymiarach mamy w sumie pięć rodzajów sieci Bravais: skośna, prostokątna, prostokatna centrowana, heksagonalna i kwadratowa.
+where the vectors $\bm{a}_i$ are called the fundamental or primitive translation vectors, and $n_i$ are integers.
+The set of points defined by this formula is called a Bravais lattice. Similarly, a Bravais lattice
+can be defined for any dimension $d$ by choosing an appropriate number of basis vectors: $\bm{a}_1,\bm{a}_2,...,\bm{a}_d$. 
+With each Bravais lattice point, one can associate a set of atoms called the basis. The smallest region of the crystal
+that, when translated by all lattice translations, fills the entire space is called the primitive cell. 
+The primitive cell contains exactly one Bravais lattice point and is usually built on the primitive translation vectors of the lattice.
+There exists a special type of primitive cell called the Wigner-Seitz cell, which is the region containing all points in space 
+that are closer to a given lattice point than to any other. 
+Often, instead of the primitive cell, the conventional cell is used, which is built on different basis vectors
+and whose symmetry is the same as the symmetry of the entire crystal.
+The volume of the conventional cell is an integer multiple of the volume of the primitive cell.
+In simple structures (without face-centering or body-centering), the conventional cell coincides
+with the primitive cell. The differences between various types of cells can be illustrated using the example
+of a two-dimensional lattice. Figure \ref{fig:bravais} shows a centered rectangular structure with two primitive vectors $\bm{a}_p$
+and $\bm{b}_p$ marked, which define the primitive cell, shown in the figure in two variants. The Wigner-Seitz cell is shown
+on the right side. The conventional cell is determined by two basis vectors $\bm{a}_e$ and $\bm{b}_e$.
+In two dimensions, there are five types of Bravais lattices in total: oblique, rectangular, centered rectangular, hexagonal, and square.
 
 \begin{figure}[h]
 \centering
 \includegraphics[scale=0.4]{lattice.pdf}
-\caption{Dwuwymiarowa sieć periodyczna.}\label{fig:bravais}
+\caption{Two-dimensional periodic lattice.}\label{fig:bravais}
 \end{figure}
 
-W przypadku sieci trójwymiarowych, wszystkie możliwe struktury można podzielić na siedem układów krystalograficznych: 
-regularny, tetragonalny, rombowy, heksagonalny, trygonalny, jednoskośny i trójskośny, którym odpowiada czternaście możliwych sieci Bravais,
-które podzielone są na proste (P), centrowane powierzchniowo (F), centrowane na podstawach (C) i centrowane przestrzennie (I). 
-Wszystkie układy krystalograficzne z odpowiadającymi im sieciami Bravais przedstawione są w tabeli \ref{lattice}.
-Dla każdego układu pokazana jest zalezność między stałymi sieci ($a$, $b$, $c$) oraz kątami między kierunkami ($\alpha$, $\beta$, $\gamma$),
-które definiują krawędzie komórek elementarnych. 
+In the case of three-dimensional lattices, all possible structures can be divided into seven crystal systems: 
+cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, and triclinic, corresponding to fourteen possible Bravais lattices,
+which are divided into simple (P), face-centered (F), base-centered (C), and body-centered (I). 
+All crystal systems with their corresponding Bravais lattices are presented in Table \ref{lattice}.
+For each system, the relationship between the lattice constants ($a$, $b$, $c$) and the angles between directions ($\alpha$, $\beta$, $\gamma$)
+that define the edges of the conventional cells is shown. 
 
 \begin{table}[h!]
-\caption{Układy krystalograficzne.}\label{lattice}
+\caption{Crystal systems.}\label{lattice}
 \begin{center}
 \begin{tabular}{|c|c|c|c|}
 \hline
-Układ & Stałe sieci & Kąty & Sieci Bravais\\
+System & Lattice constants & Angles & Bravais lattices\\
 \hline
-regularny & $a=b=c$ & $\alpha=\beta=\gamma=90^{\degree}$ &  P, F, I \\ 
-tetragonaly & $a=b\ne c$ & $\alpha=\beta=\gamma=90^{\degree}$ &  P, I \\  
-rombowy & $a\ne b\ne c$ & $\alpha=\beta=\gamma=90^{\degree}$ &  P, F, C, I \\ 
-heksagonalny &  $a=b\ne c$ & $\alpha=\beta=90^{\degree}$, $\gamma=120^{\degree}$ &  P  \\
-trygonalny & $a=b\ne c$ & $\alpha=\beta=90^{\degree}$, $\gamma=120^{\degree}$ &  P \\
-jednoskośny & $a=b\ne c$ & $\alpha=\gamma=90^{\degree}$, $\beta\ne 90^{\degree}$ & P, C \\ 
-trójskośny & $a\ne b\ne c$ & $\alpha\ne \beta\ne \gamma\ne 90^{\degree}$ & P \\ \hline
+cubic & $a=b=c$ & $\alpha=\beta=\gamma=90^{\degree}$ &  P, F, I \\ 
+tetragonal & $a=b\ne c$ & $\alpha=\beta=\gamma=90^{\degree}$ &  P, I \\  
+orthorhombic & $a\ne b\ne c$ & $\alpha=\beta=\gamma=90^{\degree}$ &  P, F, C, I \\ 
+hexagonal &  $a=b\ne c$ & $\alpha=\beta=90^{\degree}$, $\gamma=120^{\degree}$ &  P  \\
+trigonal & $a=b\ne c$ & $\alpha=\beta=90^{\degree}$, $\gamma=120^{\degree}$ &  P \\
+monoclinic & $a=b\ne c$ & $\alpha=\gamma=90^{\degree}$, $\beta\ne 90^{\degree}$ & P, C \\ 
+triclinic & $a\ne b\ne c$ & $\alpha\ne \beta\ne \gamma\ne 90^{\degree}$ & P \\ \hline
 \end{tabular}
 \end{center}
 \end{table}
 
-Zbiór operacji, który pozostawia dany kryształ niezmienionym tworzy grupą przestrzenną.
-Składa się ona z translacji sieci krystalicznej opisanych wzorem (\ref{trans}) oraz symetrii punktowych: inwersji, obrotów i odbić.
-Dodatkowo występują operacje niesymorficzne, które są połączeniem operacji punktowych i ułamkowych translacji.
-Wszystkich grup przestrzennych jest 230, z których 73 to symorficzne i 157 niesymorficzne.
+The set of operations that leaves a given crystal unchanged forms a space group.
+It consists of crystal lattice translations described by formula (\ref{trans}) and point symmetries: inversion, rotations, and reflections.
+Additionally, there are nonsymmorphic operations, which are combinations of point operations and fractional translations.
+There are 230 space groups in total, of which 73 are symmorphic and 157 are nonsymmorphic.
 
-\section{Przestrzeń odwrotna}
+\section{Reciprocal space}
 
-Strukturę elektronową materiału analizuje się najczęściej wykorzystując przestrzeń wektorów falowych $\bm{k}$,
-którą nazywamy przestrzenią odwrotną. Każdej sieci Bravais w przestrzeni rzeczywistej odpowiada sieć punktów
-w przestrzeni odwrotnej zdefiniowana wektorami
+The electronic structure of a material is most often analyzed using the space of wave vectors $\bm{k}$,
+which is called reciprocal space. Each Bravais lattice in real space corresponds to a lattice of points
+in reciprocal space defined by vectors
 %
 \begin{equation}
 \bm{G}_m=m_1\bm{b}_1+m_2\bm{b}_2+m_3\bm{b}_3,
 \end{equation}
 %
-gdzie $m_j$ są liczbami całkowitymi, a $\bm{b}_j$ są wektorami odwrotnymi do wektorów bazowych $\bm{a}_i$,
-czyli spełniają warunek
+where $m_j$ are integers, and $\bm{b}_j$ are vectors reciprocal to the basis vectors $\bm{a}_i$,
+that is, they satisfy the condition
 %
 \begin{equation}
 \bm{a}_i \cdot \bm{b}_j = 2\pi \delta_{ij}.
 \label{orto}
 \end{equation}
 %
-Wektory sieci odwrotnej można wyznaczyć z następujących zależności
+The reciprocal lattice vectors can be determined from the following relations
 %
 \begin{eqnarray}
 \bm{b}_1=\frac{2\pi}{\Omega}\bm{a}_2\times \bm{a}_3, \\
@@ -768,251 +768,251 @@ \section{Przestrzeń odwrotna}
 \bm{b}_3=\frac{2\pi}{\Omega}\bm{a}_1\times \bm{a}_2, 
 \end{eqnarray}
 %
-gdzie $\Omega=\bm{a}_1\cdot(\bm{a}_2\times \bm{a}_3)$ jest objętością komórki prymitywnej.
+where $\Omega=\bm{a}_1\cdot(\bm{a}_2\times \bm{a}_3)$ is the volume of the primitive cell.
 
-W przestrzeni odwrotnej również definiujemy komórkę prymitywną, która najczęściej ma kształt
-komórki Wignera-Seitza i nosi nazwę strefy Brillouina ({ang. Brillouin zone} - BZ).  
-Komórka, która obejmuje punkt $\bm{K}=0$, nazywany punktem $\Gamma$, nosi nazwę pierwszej strefy Brillouina.
-Na rysunku \ref{fig:lattice} pokazane są przykładowe strefy Brillouina dla trzech sieci regularnych. 
+In reciprocal space, we also define a primitive cell, which most often has the shape
+of a Wigner-Seitz cell and is called the Brillouin zone (BZ).  
+The cell that includes the point $\bm{K}=0$, called the $\Gamma$ point, is called the first Brillouin zone.
+Figure \ref{fig:lattice} shows example Brillouin zones for three cubic lattices. 
 
 \begin{figure}[h]
 \centering
 \includegraphics[scale=1.2]{brillouin.pdf}
-\caption{Strefy Brillouina dla sieci regularnych: prostej (P), centrowanej objetościowo (I) i centrowanej przestrzennie (F).}
+\caption{Brillouin zones for cubic lattices: simple (P), body-centered (I), and face-centered (F).}
 \label{fig:lattice}
 \end{figure}
 
 
-\section{Twierdzenie Blocha}
+\section{Bloch's theorem}
 
-W najprostszym opisie elektronów w periodycznym potencjale kryształu $V(\bm{r})$, stany elektronowe opisywane są przez niezależne jednocząstkowe funkcje falowe, bedące rozwiązaniami równania Schr\"{o}dingera w postaci
+In the simplest description of electrons in the periodic potential of a crystal $V(\bm{r})$, electronic states are described by independent single-particle wave functions, which are solutions of the Schr\"{o}dinger equation in the form
 %
 \begin{equation}
 [-\frac{\hbar^2\nabla^2}{2m}+V(\bm{r})]\psi^{\sigma}_{\bm{k}j}(\bm{r})=\varepsilon_{j\sigma}(\bm{k})\psi^{\sigma}_{\bm{k}j}(\bm{r}),
 \label{RS}
 \end{equation}
 %
-gdzie $\bm{k}$ jest wektorem falowym, $\sigma$ określa spin elektronu i indeks $j$ numeruje stany własne hamiltonianu odpowiadajace energiom $\varepsilon_{j\sigma}(\bm{k})$.
-Zgodnie z twierdzeniem Blocha, funkcja falowa elektronów w periodycznym potencjale ma postać
+where $\bm{k}$ is the wave vector, $\sigma$ specifies the electron spin, and the index $j$ numbers the eigenstates of the Hamiltonian corresponding to energies $\varepsilon_{j\sigma}(\bm{k})$.
+According to Bloch's theorem, the wave function of electrons in a periodic potential has the form
 %
 \begin{equation}
 \psi^{\sigma}_{\bm{k}j}(\bm{r})=e^{i\bm{k}\bm{r}}u^{\sigma}_{\bm{k}j}(\bm{r}),
 \end{equation}
 %
-gdzie $u^{\sigma}_{\bm{k}j}(\bm{r})$ jest funkcją periodyczną spełniającą warunek $u^{\sigma}_{\bm{k}j}(\bm{r})=u^{\sigma}_{\bm{k}j}(\bm{r}+\bm{R}_n)$ dla
-każdego wektora traslacji $\bm{R}_n$. Można pokazać, że funkcje Blocha są wektorami własnymi
-operatora translacji
+where $u^{\sigma}_{\bm{k}j}(\bm{r})$ is a periodic function satisfying the condition $u^{\sigma}_{\bm{k}j}(\bm{r})=u^{\sigma}_{\bm{k}j}(\bm{r}+\bm{R}_n)$ for
+each translation vector $\bm{R}_n$. It can be shown that Bloch functions are eigenvectors
+of the translation operator
 %
 \begin{equation}
 \hat{T}_n\psi^{\sigma}_{\bm{k}j}(\bm{r})=\psi^{\sigma}_{\bm{k}j}(\bm{r}+\bm{R}_n)=e^{i\bm{k}(\bm{r}+\bm{R}_n)}u^{\sigma}_{\bm{k}j}(\bm{r}+\bm{R}_n)=e^{i\bm{k}\bm{R}_n}\psi^{\sigma}_{\bm{k}j}(\bm{r}).
 \end{equation}
 %
-Hamiltonian jest niezmienniczy ze względu na działanie operatora translacji,
-co oznacza, że obydwa operatory ze sobą komutują. Funkcje Blocha są zatem również stanami własnymi hamiltonianu,
-co dowodzi prawdziwości twierdzenia Blocha. 
+The Hamiltonian is invariant under the action of the translation operator,
+which means that both operators commute. Bloch functions are therefore also eigenstates of the Hamiltonian,
+which proves the validity of Bloch's theorem. 
 
-Funkcja Blocha określona jest dla wektora falowego $\bm{k}$, który w układach periodycznych jest dobrze zdefiniowaną liczbą kwantową.
-Wektor falowy związany jest z kwazipędem elektronu w stanie $\psi_{\bm{k}j}$
+A Bloch function is defined for the wave vector $\bm{k}$, which in periodic systems is a well-defined quantum number.
+The wave vector is related to the quasimomentum of an electron in state $\psi_{\bm{k}j}$
 %
 \begin{equation}
 \bm{p}=\hbar\bm{k}.
 \end{equation}
 %
-Ponieważ wszystkie własności struktury elektronowej są translacyjnie niezmiennicze 
-ze względu na przesunięcie o dowolny wektor sieci odwrotnej, również kwazipęd nie zależy od takiego przesunięcia.
+Since all properties of the electronic structure are translationally invariant 
+with respect to translation by any reciprocal lattice vector, the quasimomentum also does not depend on such translation.
 
-\section{Fale płaskie}
+\section{Plane waves}
 
-Metody obliczeniowe struktury pasmowej różnią sie między sobą stosowanymi reprezentacjami funkcji falowej.
-Ogólnie możemy zapisać funkcję falową w formie
+Computational methods for band structure differ in the representations of the wave function they employ.
+In general, we can write the wave function in the form
 %
 \begin{equation}
 \psi^{\sigma}_{\bm{k}j}(\bm{r})=\sum_mc^{\sigma}_{jm}(\bm{k})\phi_{\bm{k}m}(\bm{r}),
 \end{equation}
 % 
-gdzie $\phi_{\bm{k}m}(\bm{r})$ są funkcjami bazowymi, a $c^{\sigma}_{jm}(\bm{k})$ to współczynniki rozwinięcia.
-W krysztale, ze względu na periodyczny potencjał, naturalną bazę stanowią fale płaskie. Jeżeli cześć periodyczną funkcji
-Blocha rozwiniemy w tej bazie, funkcja falowa przyjmuje postać
+where $\phi_{\bm{k}m}(\bm{r})$ are basis functions, and $c^{\sigma}_{jm}(\bm{k})$ are expansion coefficients.
+In a crystal, due to the periodic potential, plane waves form a natural basis. If we expand the periodic part of the
+Bloch function in this basis, the wave function takes the form
 %
 \begin{equation}
 \psi^{\sigma}_{\bm{k}j}(\bm{r})=e^{i\bm{k}\bm{r}}\sum_{m} c^{\sigma}_{jm}(\bm{k})e^{i\bm{G_m}\bm{r}}=\sum_{m} c^{\sigma}_{jm}(\bm{k})e^{i(\bm{k}+\bm{G_m})\bm{r}},
 \end{equation}
 % 
-gdzie $\bm{G_m}$ są wektorami sieci odwrotnej. Równanie Schr\"{o}dingera (\ref{RS}) przetransponowane do przestrzeni odwrotnej w bazie
-fal płaskich przyjmuje postać
+where $\bm{G_m}$ are reciprocal lattice vectors. The Schr\"{o}dinger equation (\ref{RS}) transformed to reciprocal space in the plane wave basis
+takes the form
 %
 \begin{equation}
 \sum_{n}H_{mn} c^{\sigma}_{jn}(\bm{k})=\varepsilon_{j\sigma}(\bm{k}) c^{\sigma}_{jm},
 \label{ham_rec}
 \end{equation}
 %
-gdzie elementy macierzowe Hamiltonianu dane są wyrażeniem
+where the matrix elements of the Hamiltonian are given by the expression
 %
 \begin{equation}
 H_{mn}=\frac{\hbar^2}{2m}|\bm{k}+\bm{G_m}|^2\delta_{mm'}+V(\bm{G_m}-\bm{G_n}).
 \end{equation}
 %
-Drugi wyraz jest transformatą Fouriera potencjału elektronowego
+The second term is the Fourier transform of the electronic potential
 %
 \begin{equation}
 V(\bm{G})=\frac{1}{\Omega}\int_{\Omega}\bm{dr}V(\bm{r})e^{-i\bm{Gr}},
 \end{equation}
 %
-gdzie całkowanie przebiega po komórce prymitywnej.
+where the integration is performed over the primitive cell.
 
-\section{Periodyczne warunki brzegowe}
+\section{Periodic boundary conditions}
 
-Podobnie jak dla gazu elektronowego zamkniętego w pojemniku, również dla skończonego kryształu
-możemy wprowadzić periodyczne warunki brzegowe (warunki Borna-Karmana). Załóżmy, że rozmiary kryształu
-określone są przez liczbę komórek prymitywnych w każdym kierunku kryształu $(N_1,N_2,N_3)$.
-Zatem liczby całkowite, które definiują wektory translacyjne sieci (\ref{trans})
-zmieniają się w zakresie $n_i=0,1,2,...,N_i$, gdzie $i=1,2,3$.
-Zakładając, że funkcja falowa na dwóch przeciwległych brzegach układu (w każdym z trzech kierunków) przyjmuje
-takie same wartości i korzystając z twierdzenia Blocha otrzymujemy warunek
+Similarly to the electron gas enclosed in a container, we can also introduce periodic boundary conditions (Born-Karman conditions) for a finite crystal. 
+Let us assume that the size of the crystal
+is determined by the number of primitive cells in each direction of the crystal $(N_1,N_2,N_3)$.
+Thus, the integers that define the lattice translation vectors (\ref{trans})
+vary in the range $n_i=0,1,2,...,N_i$, where $i=1,2,3$.
+Assuming that the wave function on two opposite edges of the system (in each of the three directions) takes
+the same values and using Bloch's theorem, we obtain the condition
 %
 \begin{equation}
 \psi^{\sigma}_{\bm{k}j}(\bm{r})=\psi^{\sigma}_{\bm{k}j}(\bm{r}+N_i\bm{a}_i)=e^{i\bm{k}N_i\bm{a}_i}\psi^{\sigma}_{\bm{k}j}(\bm{r}),
 \label{pwb}
 \end{equation} 
 %
-który spełniony jest gdy 
+which is satisfied when 
 %
 \begin{equation}
 \bm{k}N_i\bm{a}_i=2\pi n_i.
 \end{equation}
 %
-Wykorzystując bazę wektorów prymitywnych sieci odwrotnej ($\bm{b}_1,\bm{b}_2,\bm{b}_3$) oraz biorąc pod uwagę zależność (\ref{orto}), dostajemy zbiór dozwolonych wektorów falowych
+Using the basis of primitive vectors of the reciprocal lattice ($\bm{b}_1,\bm{b}_2,\bm{b}_3$) and taking into account relation (\ref{orto}), we obtain the set of allowed wave vectors
 %
 \begin{equation}
 \bm{k}=\frac{n_1}{N_1}\bm{b}_1+\frac{n_2}{N_2}\bm{b}_2+\frac{n_3}{N_3}\bm{b}_3.
 \label{vectk}
 \end{equation}
 %
-Wektory te określają funkcje falowe Blocha $\psi_{i,\bm{k}}$ i energie stanów $\varepsilon_i(\bm{k})$.
-Skończona ilość dostępnych stanów wynika zatem z ograniczonych rozmiarów kryształu i jest równa ilości komórek prymitywnych w układzie: $N=N_1N_2N_3$.  
-Ze wzoru (\ref{vectk}) wynika również, że dozwolone wektory falowe stanów elektronowych należą do pojedynczej komórki prymitywnej w przestrzeni odwrotnej.
-Najczęściej do analizy stanów elektronowych wybiera się wektory należące do pierwszej strefy Brillouina.
+These vectors determine the Bloch wave functions $\psi_{i,\bm{k}}$ and state energies $\varepsilon_i(\bm{k})$.
+The finite number of available states therefore results from the limited size of the crystal and equals the number of primitive cells in the system: $N=N_1N_2N_3$.  
+From formula (\ref{vectk}) it also follows that the allowed wave vectors of electronic states belong to a single primitive cell in reciprocal space.
+Most often, vectors belonging to the first Brillouin zone are chosen for the analysis of electronic states.
 
-\section{Sumowanie w przestrzeni odwrotnej}
+\section{Summation in reciprocal space}
 
-Wiele wielkości fizycznych wyznaczanych jest jako sumy po punktach w przestrzeni odwrotnej. 
-Przedstawiona powyżej analiza pokazuje, że wystarczy ograniczyć
-sie do wektorów falowych należących do pierwszej strefy Brillouina.
-Dodatkowo, można wykorzystać symetrię kryształu do zmniejszenia ilości punktów, które konieczne
-są do wyznaczenia danej wielkości fizycznej. 
-Oznaczmy operacje punktowe (symorficzne) przez $O_i$, a wektory translacji ułamkowej związane z daną symetrią $O_i$
-przez $\bm{t}_i$. W przestrzeni odwrotnej rozpatrujemy jedynie symorficzne operacje symetrii ponieważ translacje
-ułamkowe nie mają wpływu na przestrzeń odwrotną. Hamiltoniana opisujący kryształ jest niezmienniczy
-względem operacji punktowych: $\bm{r}\rightarrow O_i\bm{r}+\bm{t}_i$ oraz $\bm{k}\rightarrow O_i\bm{k}$.
-Oznacza to, że funkcje falowe otrzymane po tych transformacjach 
+Many physical quantities are determined as sums over points in reciprocal space. 
+The analysis presented above shows that it is sufficient to restrict
+oneself to wave vectors belonging to the first Brillouin zone.
+Additionally, one can use the symmetry of the crystal to reduce the number of points necessary
+to determine a given physical quantity. 
+Let us denote point operations (symmorphic) by $O_i$, and the fractional translation vectors associated with a given symmetry $O_i$
+by $\bm{t}_i$. In reciprocal space, we consider only symmorphic symmetry operations because fractional translations
+have no effect on reciprocal space. The Hamiltonian describing the crystal is invariant
+with respect to point operations: $\bm{r}\rightarrow O_i\bm{r}+\bm{t}_i$ and $\bm{k}\rightarrow O_i\bm{k}$.
+This means that the wave functions obtained after these transformations 
 %
 \begin{equation}
 \psi^{\sigma}_{O_i\bm{k}j}(O_i\bm{r}+\bm{t}_i)=\psi^{\sigma}_{\bm{k}j}(\bm{r}),
 \end{equation}  
 %
-są również funkcjami własnymi hamiltonianu z tą samą energią własną $\varepsilon_{j\sigma}(\bm{k})$.
-Oznacza to, że można ograniczyć sumowanie danej wielkości fizycznej $f(\bm{k})$ do wektorów $\bm{k}$ należących do mniejszego obszaru,
-który nazywany jest nieredukowalną częścią strefy Brillouina ({\it ang. irreducible Brillouin zone} - IBZ). 
-Obszar ten tworzą wektory $\bm{k}$, które są nierównoważne, czyli nie można ich wzajemnie połączyć operacją symetrii punktowej. 
-Wartości w pozostałych punktach dostajemy przez zastosowanie odpowiednich operacji symetrii $f(O_i\bm{k})=f(\bm{k})$.
-Do sumowania wykorzystujemy wagi $w_{\bm{k}}$, które zdefiniowane są jako ilości wektorów $\bm{k}$, powiązanych symetrią punktową 
-z wektorem należącym do IBZ (z uwzglednieniem tego wektora), podzielone przez całkowitą ilość wektorów $N_{\bm{k}}$.
-Przykładowo, wartość średnią wielkości $f$, możemy wyliczyć sumując po punktach należących do IBZ
+are also eigenfunctions of the Hamiltonian with the same eigenvalue $\varepsilon_{j\sigma}(\bm{k})$.
+This means that one can restrict the summation of a given physical quantity $f(\bm{k})$ to vectors $\bm{k}$ belonging to a smaller region,
+which is called the irreducible part of the Brillouin zone (IBZ). 
+This region is formed by vectors $\bm{k}$ that are inequivalent, that is, they cannot be mutually connected by a point symmetry operation. 
+Values at the remaining points are obtained by applying the appropriate symmetry operations $f(O_i\bm{k})=f(\bm{k})$.
+For summation, we use weights $w_{\bm{k}}$, which are defined as the number of vectors $\bm{k}$ related by point symmetry 
+to a vector belonging to the IBZ (including this vector), divided by the total number of vectors $N_{\bm{k}}$.
+For example, the average value of quantity $f$ can be calculated by summing over points belonging to the IBZ
 %
 \begin{equation}
 \bar{f}=\frac{1}{N_{\bm{k}}}\sum_{\bm{k}}^{BZ}f(\bm{k})=\sum_{\bm{k}}^{IBZ}w_{\bm{k}}f(\bm{k}).
 \end{equation}
 %
 
-Przy sumowaniu po przestrzeni odwrotnej można w zasadzie wybrać dowolne punkty należące do IBZ.
-Jednak wykorzystując własności funkcji periodycznych, można tak wybrać wektory $\bm{k}$,
-aby zminimalizować błędy przy obliczaniu sum lub całek.
-Dowolną funkcję periodyczną można rozwinąć przy pomocy transformaty Fouriera
+When summing over reciprocal space, one can in principle choose any points belonging to the IBZ.
+However, using the properties of periodic functions, one can choose vectors $\bm{k}$ in such a way
+as to minimize errors when calculating sums or integrals.
+Any periodic function can be expanded using the Fourier transform
 %
 \begin{equation}
 f(\bm{k})=\sum_n f(\bm{R}_n)e^{i\bm{k}\bm{R}_n},
 \end{equation} 
 %
-gdzie $\bm{R}_n$ są wektorami translacji sieci krystalicznej.
-Zbiór punktów, który jest optymalny do wyliczania sum w przestrzeni odwrotnej, 
-nazywany jest siecią Monkhorsta-Packa \cite{MP}. Wyznacza się go ze  wzoru
+where $\bm{R}_n$ are crystal lattice translation vectors.
+The set of points that is optimal for calculating sums in reciprocal space 
+is called the Monkhorst-Pack mesh \cite{MP}. It is determined from the formula
 %
 \begin{equation}
 \bm{k}(n_1,n_2,n_3)=\sum_i^3 \frac{2n_i-N_i-1}{2N_i}\bm{b}_i,
 \end{equation}
 %
-gdzie liczby $N_1$, $N_2$ i $N_3$ określają ilość punktów $\bm{k}$ w każdym kierunku: $n_i=1,2,...,N_i$.
-Tak zdefiniowane punkty tworzą jednorodną sieć w przestrzeni odwrotnej i charakteryzują się
-tym, że suma wartości periodycznej funkcji, która posiada komponenty Fouriera ograniczone w każdym kierunku do $\bm{R}_n=N_i\bm{a}_i$,
-równa jest dokładnej całce z tej funkcji.  
+where the numbers $N_1$, $N_2$, and $N_3$ specify the number of $\bm{k}$ points in each direction: $n_i=1,2,...,N_i$.
+Points defined in this way form a uniform mesh in reciprocal space and are characterized by
+the fact that the sum of values of a periodic function, which has Fourier components limited in each direction to $\bm{R}_n=N_i\bm{a}_i$,
+equals the exact integral of this function.  
 
-\section{Energia Fermiego}
+\section{Fermi energy}
 
-Wielkości fizyczne, które dotyczą stanu podstawowego, wyliczane są zwykle dla obsadzonych stanów elektronowych.
-Czyli sumowanie w przestrzeni odwrotnej uwzglednia tylko zajęte stany.
-Energia najwyższego obsadzonego stanu określa energię Fermiego ($E_F$).
-W metalach pasma elektronowe przecinają $E_F$ i przy sumowaniu po dyskretnych punktach $\bm{k}$ pojawiają się nieciągłości
-w obsadzeniach stanów. Powoduje to wolniejszą zbieżność ze względu na ilość punktów $\bm{k}$. 
-Można łatwo rozwiązać ten problem przez wprowadznie ciagłej funkcji obsadzania stanów w pobliżu $E_F$.
-Naturalnym wyborem jest zastosowanie funkcji rozkładu Fermiego-Diraca dla skończonej temperatury
+Physical quantities related to the ground state are usually calculated for occupied electronic states.
+That is, the summation in reciprocal space includes only occupied states.
+The energy of the highest occupied state determines the Fermi energy ($E_F$).
+In metals, electronic bands cross $E_F$, and when summing over discrete $\bm{k}$ points, discontinuities appear
+in state occupations. This causes slower convergence with respect to the number of $\bm{k}$ points. 
+This problem can be easily solved by introducing a continuous occupation function for states near $E_F$.
+A natural choice is to use the Fermi-Dirac distribution function for finite temperature
 %
 \begin{equation}
 F(\varepsilon,E_F,T)=[\exp(\frac{\varepsilon-E_F}{kT})+1]^{-1}.
 \end{equation}   
 %
-Dla $T=0$, dostajemy funkcję schodkową, która odpowiada rozkładowi w stanie podstawowym.
-Aby wyznaczyć $E_F$ dla dowolnej temperatury, korzystamy z warunku
+For $T=0$, we obtain a step function that corresponds to the distribution in the ground state.
+To determine $E_F$ for any temperature, we use the condition
 %
 \begin{equation}
 \sum_{\bm{k},j} w_{\bm{k}}F(\varepsilon,E_F,T)=n_{el},
 \end{equation}   
 %
-gdzie $n_{el}$ jest liczbą elektronów, a sumowanie wykonywane jest po wszystkich obsadzonych stanach
-uwzględniając polaryzację spinową.
+where $n_{el}$ is the number of electrons, and the summation is performed over all occupied states
+taking into account spin polarization.
 
 
 
-\section{Struktura pasmowa}
+\section{Band structure}
 
-Zbiór wszystkich stanów elektronowych $\varepsilon_{\bm{k}j}^{\sigma}$ otrzymanych w wyniku diagonalizacji hamiltonianiu 
-dla wektorów falowych $\bm{k}$ tworzy strukturę pasmową danego materiału.
-Dostępne energie stanów elektronowych, które odpowiadają funkcji własnej z indeksem $j$, grupują się w przedziały nazywane pasmami energetycznymi. 
-Rozwiązania równania (\ref{ham_rec}) w funkcji wektora falowego $\bm{k}$ są periodyczne ze względu na przesunięcie o 
-wektor sieci odwrotnej
+The set of all electronic states $\varepsilon_{\bm{k}j}^{\sigma}$ obtained by diagonalization of the Hamiltonian 
+for wave vectors $\bm{k}$ forms the band structure of a given material.
+The available energies of electronic states corresponding to the eigenfunction with index $j$ are grouped into intervals called energy bands. 
+Solutions of equation (\ref{ham_rec}) as a function of the wave vector $\bm{k}$ are periodic with respect to translation by 
+a reciprocal lattice vector
 %
 \begin{equation}
 \varepsilon_{\bm{k}j}^{\sigma}=\varepsilon_{\bm{k}+\bm{G_m}j}^{\sigma}.
 \end{equation}   
 %
-Zatem przy prezentacji struktury pasmowej można ograniczyć się do pierwszej strefy Brillouina. Zwykle prezentuje się tylko stany elektronowe dla wektorów $\bm{k}$ należących do wybranych kierunków i punktów wysokiej symetrii w przestrzeni odwrotnej. Jeżeli energie pasm przyjmują różne wartości dla dwóch kierunków spinu,
-$\varepsilon_{\bm{k}j}^{\uparrow}\ne \varepsilon_{\bm{k}j}^{\downarrow}$, mamy doczynienia z rozszczepieniem spinowym (wymiennym), które związane jest z uporządkowaniem magnetycznym.
+Therefore, when presenting the band structure, one can restrict oneself to the first Brillouin zone. Usually, only electronic states for wave vectors $\bm{k}$ belonging to selected directions and high-symmetry points in reciprocal space are presented. If the band energies take different values for the two spin directions,
+$\varepsilon_{\bm{k}j}^{\uparrow}\ne \varepsilon_{\bm{k}j}^{\downarrow}$, we are dealing with spin (exchange) splitting, which is associated with magnetic ordering.
 
-Ważną wielkością, która charakteryzuje strukturą pasmową jest gęstość stanów elektronowych dla danego kierunku spinu $\rho_{\sigma}(E)$,
-która wyliczana jest jako suma po pierwszej strefie Brillouina
+An important quantity that characterizes the band structure is the density of electronic states for a given spin direction $\rho_{\sigma}(E)$,
+which is calculated as a sum over the first Brillouin zone
 %
 \begin{equation}
 \rho_{\sigma}(E)=\frac{1}{N_k}\sum_{j,\bm{k}}\delta[\varepsilon_{\bm{k}j}^{\sigma}-E],
 \end{equation}
 %
-gdzie $N_k$ jest ilością wektorów $\bm{k}$ w pierwszej strefie Brillouina.
-Całkowita gęstość stanów równa jest sumie obydwóch składowych spinowych, $\rho(E)=\rho_{\uparrow}(E)+\rho_{\downarrow}(E)$.
-Przy wyliczaniu gestości stanów często zamiast sieci Monkhorsta-Packa stosuje się metodę tetraedrów.
-Polega ona na podzieleniu całej strefy Brillouina na tetraedry i wyliczeniu energii stanów w punktach $\bm{k}$, które
-odpowiadają wierzchołkom tetraedrów. Energie pomiędzy tymi punktami wyznacza się stosując interpolację liniową.
-
-Struktura pasmowa decyduje o podstawowych własnościach danego materiału i jest jego wyróżnikiem, pozwalającym zakwalifikować go do określonej grupy materiałów. 
-Trzy podstawowe typy materiałów krystalicznych to metale, półprzewodniki i izolatory.
-Struktura pasmowa metali charakteryzuję się występowaniem powierzchni Fermiego, która odgranicza zajęte stany elektronowe od stanów pustych.
-Zatem w typowych metalach gęstość stanów na poziomie Fermiego jest niezerowa, $\rho(E_F)>0$.
-Tuż powyżej energii Fermiego ($E_F$) znajdują się stany, do których przechodzą elektrony wzbudzone termicznie lub w wyniku oddziaływania z innymi
-elektronami. W półprzewodnikach i izolatorach, powyżej ostatniego zajętego stanu elektronowego, znajduje się przerwa obejmująca
-zakres zabronionych energetycznie stanów.
-Pasmo, które znajduje się tuż poniżej przerwy energetycznej nazywa się pasmem walencyjnym, a to które jest powyżej
-pasmem przewodnictwa.
-Półprzewodniki zwykle charakteryzują się mniejszą przerwa energetyczną ($\sim$1-2 eV) niż izolatory.
-Występują też materiały nazywane półmetalami, w których przerwa energetyczna i $\rho(E_F)$ są równe zeru (np. grafen).
-Odrębną grupę materiałów stanowią izolatory Motta, w których przerwa energetyczna jest wynikiem silnych oddziaływań
-elektronowych.  
+where $N_k$ is the number of $\bm{k}$ vectors in the first Brillouin zone.
+The total density of states is the sum of both spin components, $\rho(E)=\rho_{\uparrow}(E)+\rho_{\downarrow}(E)$.
+When calculating the density of states, instead of the Monkhorst-Pack mesh, the tetrahedron method is often used.
+It consists of dividing the entire Brillouin zone into tetrahedra and calculating the energies of states at points $\bm{k}$ that
+correspond to the vertices of the tetrahedra. Energies between these points are determined using linear interpolation.
+
+The band structure determines the basic properties of a given material and is its distinguishing feature, allowing it to be classified into a specific group of materials. 
+The three basic types of crystalline materials are metals, semiconductors, and insulators.
+The band structure of metals is characterized by the presence of a Fermi surface, which separates occupied electronic states from empty states.
+Thus, in typical metals, the density of states at the Fermi level is nonzero, $\rho(E_F)>0$.
+Just above the Fermi energy ($E_F$), there are states to which electrons excited thermally or as a result of interaction with other
+electrons transition. In semiconductors and insulators, above the last occupied electronic state, there is a gap covering
+a range of energetically forbidden states.
+The band that is located just below the energy gap is called the valence band, and the one above it
+is the conduction band.
+Semiconductors are usually characterized by a smaller energy gap ($\sim$1-2 eV) than insulators.
+There are also materials called semimetals, in which the energy gap and $\rho(E_F)$ are both zero (e.g., graphene).
+A separate group of materials are Mott insulators, in which the energy gap results from strong electronic
+interactions.  
 
 
 

From 881e5658783bd982c913a3f6074f57f5f396b95a Mon Sep 17 00:00:00 2001
From: Copilot <198982749+Copilot@users.noreply.github.com>
Date: Sat, 3 Jan 2026 07:09:01 +0100
Subject: [PATCH 17/23] Translate Chapter 3 (Density Functional Theory) to
 English (#31)

* Initial plan

* Translate Chapter 3: Density Functional Theory to English

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Remove review markers from Chapter 3 per author approval

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 429 +++++++++++++++++++++++++-------------------------
 1 file changed, 213 insertions(+), 216 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index 73ee131..93e2898 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -1016,226 +1016,224 @@ \section{Band structure}
 
 
 
-\chapter{Teoria funkcjonału gęstości}
-
-\section{Twierdzenia Hohenberga-Kohna}
-
-Większość nowoczesnych metod obliczeniowych stosowanych do badania własności materiałów opiera się na teorii funkcjonału gestości ({\it ang. density functional theory} - DFT), która została sformułowana w pracach Hohenberga i Kohna \cite{kohn64}
-oraz Kohna i Shama \cite{kohn65}. Obecnie, DFT stosowana jest powszechnie do wyliczania struktury elektronowej,
-optymalizacji sieci krystalicznej, badania własności elastycznych i dynamiki sieci oraz wielu innych własności materiałowych.
-Jest praktycznie jedyną metodą umożliwiającą obliczenia kwantowo-mechaniczne dla układów zawierających setki,
-a nawet tysiące atomów.  
-Podobnie jak w opisie oddziaływania wymiennego zaproponowanym przez Slatera~\cite{slater51}
-i w teorii Thomasa-Fermiego~\cite{thomas,fermi1927,dirac} podstawową wielkością jest tutaj gęstość elektronowa zdefiniowana w każdym punkcie materiału $n(\bm{r})$.
-Główną ideą tego podejścia jest możliwość zastąpienie dokładnej funkcji falowej, układem stanów jednocząstkowych w efektywnym potencjale elektronowym, który
-daje taką samą gęstość elektronową jak opis wieloelektronowy.
-Takie podejście umożliwia wykorzystania dokładnej informacji o oddziaływaniach wymiennych i korelacyjnych
-otrzymanych dla jednorodnego gazu elektronowego do badania niejednorodnych układów atomowych.
-
-Praca Hohenberga i Kohna \cite{kohn64} zawiera dwa fundamentalne twierdzenia, które
-dotyczą relacji między gęstością elektronową $n(\bm{r})$, zewnętrznym potencjałem $V_{ext}(\bm{r})$ 
-oraz energią całkowitą $E[n]$, która jest funkcjonałem gęstości elektronowej. Zależność funkcjonalna oznacza, że dla dowolnego rozkładu 
-elektronów w całej przestrzeni $n(\bm{r})$ można jednoznacznie wyznaczyć energię całkowitą 
-układu $E[n]$. Zapiszmy hamiltonian układu oddziałujących elektronów
-w ogólnej formie
+\chapter{Density Functional Theory}
+
+\section{Hohenberg-Kohn Theorems}
+
+Most modern computational methods used to study material properties are based on density functional theory (DFT), which was formulated in the works of Hohenberg and Kohn \cite{kohn64}
+and Kohn and Sham \cite{kohn65}. Currently, DFT is widely used for calculating electronic structure,
+optimizing crystal lattices, studying elastic properties and lattice dynamics, and many other material properties.
+It is practically the only method enabling quantum-mechanical calculations for systems containing hundreds,
+or even thousands of atoms.
+Similarly to the description of exchange interactions proposed by Slater~\cite{slater51}
+and in Thomas-Fermi theory~\cite{thomas,fermi1927,dirac}, the fundamental quantity here is the electron density defined at each point in the material $n(\bm{r})$.
+The main idea of this approach is the possibility of replacing the exact wave function with a system of single-particle states in an effective electronic potential, which
+gives the same electron density as the many-electron description.
+This approach enables the use of exact information about exchange and correlation interactions
+obtained for the homogeneous electron gas to study inhomogeneous atomic systems.
+
+The work of Hohenberg and Kohn \cite{kohn64} contains two fundamental theorems that
+concern the relationship between the electron density $n(\bm{r})$, external potential $V_{ext}(\bm{r})$,
+and total energy $E[n]$, which is a functional of the electron density. A functional dependence means that for any distribution
+of electrons in the entire space $n(\bm{r})$, one can uniquely determine the total energy
+of the system $E[n]$. Let us write the Hamiltonian of the system of interacting electrons
+in general form
 
 \begin{equation}
 H=-\frac{\hbar^2}{2m}\sum_{i}\nabla_i^2+V_\text{ext}(\bm{r})+\frac{1}{2}\sum_{i\neq j}\frac{e^2}{|\bm{r}_i-\bm{r}_j|}.
 \label{hamil}
 \end{equation}
 %
-Twierdzenia Hohenberga-Kohna można sformułować następująco: 
+The Hohenberg-Kohn theorems can be formulated as follows:
 
-{\bf Twierdzenie I:} 
-Zewnętrzny potencjał układu oddziałujących elektronów $V_\text{ext}$ jest jednoznacznie określony (z dokładnością do stałej wartości) 
-przez gęstość elektronową w stanie podstawowym $n_0(\mb{r})$.
+{\bf Theorem I:}
+The external potential of a system of interacting electrons $V_\text{ext}$ is uniquely determined (up to an additive constant)
+by the ground state electron density $n_0(\mb{r})$.
 
-{\bf Twierdzenie II:}
-Dla ustalonego potencjału zewnętrznego $V_\text{ext}(\bm{r})$, funkcjonał energii $E[n]$ osiąga
-minimalną wartość $E_0$ dla gęstości elektronowej w stanie podstawowym $n_0(\bm{r})$.
+{\bf Theorem II:}
+For a fixed external potential $V_\text{ext}(\bm{r})$, the energy functional $E[n]$ reaches
+its minimum value $E_0$ for the ground state electron density $n_0(\bm{r})$.
 
-Z twierdzeń tych wynikają następujace wnioski. Ponieważ hamiltonian jest jednoznacznie określony (z dokładnościa do stałej)
-przez $n_0(\bm{r})$, funkcje falowe wszystkich stanów elektronowych,
-jak również wszystkie własności układu są całkowicie zdeterminowane przez gęstość elektronową w stanie podstawowym.
-Znajomość funkcjonału energii $E[n]$ jest wystarczająca do wyznaczenia energii i gęstości elektronowej
-stanu podstawowego. 
+From these theorems follow the following conclusions. Since the Hamiltonian is uniquely determined (up to a constant)
+by $n_0(\bm{r})$, the wave functions of all electronic states,
+as well as all properties of the system are completely determined by the ground state electron density.
+Knowledge of the energy functional $E[n]$ is sufficient to determine the energy and electron density
+of the ground state. 
 
-\section{Równanie Kohna-Shama}
+\section{Kohn-Sham Equation}
 
-Twierdzenia Kohna-Shama pozwalają w sposób ścisły powiązać gestość elektronową w stanie podstawowym
-z równaniem Schr\"{o}dingera dla układu oddziałujacych elektronów, opisanego hamiltonianem (\ref{hamil}).
-W praktyce nie jest możliwe bezpośrednie rozwiązanie równania Schr\"{o}dingera i wyznaczenie wielociałowych funkcji falowych.
-Teoria funkcjonału gęstości pozwala zastąpić równanie wielociałowe nowym równaniem, nazywanym równaniem Kohna-Shama, które ma
-taką postać jak dla nieoddziałujących cząstek znajdujacych się w efektywnym polu $V_{eff}$.
-Znając jednoelektronowe rozwiązania równania Kohna-Shama $\psi_i^{\sigma}(\bm{r})$, zależne od położenia i spinu, 
-możemy wyznaczyć gęstość elektronową w każdym punkcie
+The Kohn-Sham theorems allow us to rigorously relate the ground state electron density
+to the Schr\"{o}dinger equation for a system of interacting electrons, described by the Hamiltonian (\ref{hamil}).
+In practice, it is not possible to directly solve the Schr\"{o}dinger equation and determine many-body wave functions.
+Density functional theory allows us to replace the many-body equation with a new equation, called the Kohn-Sham equation, which has
+the form of an equation for non-interacting particles in an effective field $V_{eff}$.
+Knowing the single-electron solutions of the Kohn-Sham equation $\psi_i^{\sigma}(\bm{r})$, dependent on position and spin,
+we can determine the electron density at each point
 %
 \begin{equation}
 n(\bm{r})=\sum_{i,\sigma} f_{i\sigma}|\psi_i^{\sigma}(\bm{r})|^2,
 \label{density}
 \end{equation}
 %
-gdzie dla uproszczenia indeks $i$ określa zarówno numer stanu $j$, jak i wektor falowy $\bm{k}$, a $f_{i\sigma}$ są liczbami obsadzeń
-stanów, które w ogólnym przypadku mogą przyjmować ułamkowe wartości, np. zgodnie z rozkładem Fermiego-Diraca.
-Przyjmujemy również, że ładunek elementarny $e=1$, co oznacza, że gęstość ładunku jest tożsama z gęstością elektronową. 
-Twierdzenia Kohna-Shama zapewniają nam, że efektywny potencjał $V_{eff}$ jest jednoznacznie
-określony przez gęstość elektronową, jak również, że funkcjonał energii całkowitej osiąga
-minimum dla gęstości elektronowej w stanie podstawowym.
-Zatem stan podstawowy układu możemy wyznaczyć jeżeli znamy funkcjonał energii i potrafimy
-znaleźć jego minimum. Ogólnie możemy zapisać ten funkcjał w postaci 
+where for simplicity the index $i$ determines both the state number $j$ and the wave vector $\bm{k}$, and $f_{i\sigma}$ are occupation numbers
+of states, which in the general case may take fractional values, e.g., according to the Fermi-Dirac distribution.
+We also assume that the elementary charge $e=1$, which means that the charge density is identical to the electron density.
+The Kohn-Sham theorems ensure that the effective potential $V_{eff}$ is uniquely
+determined by the electron density, as well as that the total energy functional reaches
+a minimum for the ground state electron density.
+Therefore, we can determine the ground state of the system if we know the energy functional and can
+find its minimum. In general, we can write this functional in the form
 %
 \begin{equation}
 E[n]=T[n] + E_\text{ext}[n] + E_\text{H}[n] + E_\text{xc}[n],
 \label{EKS}
 \end{equation}
 %
-gdzie $T[n]$ jest energią kinetyczną nieoddziałujących elektronów
+where $T[n]$ is the kinetic energy of non-interacting electrons
 %
 \begin{equation}
 T[n]=-\frac{\hbar^2}{2m}\sum_{i,\sigma} \int d\bm{r} \psi^{\sigma *}_i(\bm{r})\nabla_i^2 \psi^{\sigma}_i(\bm{r}),
 \end{equation}
 %
-$E_\text{ext}[n]$ jest energią oddziaływania z potencjałem zewnętrznym
+$E_\text{ext}[n]$ is the interaction energy with the external potential
 %
 \begin{equation}
 E_\text{ext}[n]=\int d\bm{r} V_\text{ext}(\bm{r}) n(\bm{r}),
 \end{equation}
 %
-$E_\text{H}[n]$ jest energią Hartree dana wzorem (\ref{Hartree}) i $E_\text{xc}[n]$ zawiera wszystkie pozostałe oddziaływania,
-czyli oddziaływanie wymienne i korelacje elektronowe, jak również różnicę między energią kinetyczną układu
-oddziałujących i nieoddziałujących elektronów. W skrócie ten wyraz nazywany jest funkcjonałem wymienno-korelacyjnym.
+$E_\text{H}[n]$ is the Hartree energy given by formula (\ref{Hartree}), and $E_\text{xc}[n]$ contains all remaining interactions,
+i.e., exchange interaction and electron correlations, as well as the difference between the kinetic energy of the system
+of interacting and non-interacting electrons. In short, this term is called the exchange-correlation functional.
 
-Możemy teraz zastosować podejście wariacyjne uwzględniając warunek ortonormalności
-orbitali Kohna-Shama: $\langle\psi_i^{\sigma *}|\psi_j^{\sigma'}\rangle = \delta_{ij}\delta_{\sigma\sigma'}$
-i wprowadzając mnożniki Lagrange'a
+We can now apply the variational approach taking into account the orthonormality condition
+of Kohn-Sham orbitals: $\langle\psi_i^{\sigma *}|\psi_j^{\sigma'}\rangle = \delta_{ij}\delta_{\sigma\sigma'}$
+and introducing Lagrange multipliers
 %
 \begin{equation}
 \frac{\delta}{\delta\psi_i^{\sigma *}}(E[n]-\sum_{i,\sigma}\varepsilon_{i\sigma}[\int d\bm{r} \psi_i^{\sigma *}(\bm {r})\psi_i^{\sigma}(\bm{r})-1]) = 0.
 \label{delta}
 \end{equation}
 %
-Wstawiając funkcjonał energii w postaci (\ref{EKS}) do (\ref{delta}) otrzymujemy 
+Substituting the energy functional in the form (\ref{EKS}) into (\ref{delta}), we obtain
 %
 \begin{equation}
 \frac{\delta T[n]}{\delta \psi_i^{\sigma *}}+\frac{\delta}{\delta n}(\int d\bm{r} V_{ext}(\bm{r}) n(\bm{r}) + E_H[n] + E_{xc}[n])\frac{\delta n}{\delta \psi_i^{\sigma *}}-\varepsilon_{i\sigma} \frac{\delta}{\delta\psi_i^{\sigma *}}\sum_{j,\sigma'}\int d\bm{r} \psi_j^{\sigma' *}\psi_j^{\sigma}=0,
 \end{equation}
 %
-gdzie zastosowalismy wzór na pochodna funkcji złozonej:
-% 
+where we applied the formula for the derivative of a composite function:
+%
 \begin{equation}
-\frac{\delta}{\delta \psi_i^{\sigma *}}=\frac{\delta}{\delta n}\frac{\delta n}{\delta \psi_i^{\sigma *}}. 
+\frac{\delta}{\delta \psi_i^{\sigma *}}=\frac{\delta}{\delta n}\frac{\delta n}{\delta \psi_i^{\sigma *}}.
 \end{equation}
 %
-Wykonując pochodne wariacyjne dostajemy równanie Kohna-Shama
+Performing the variational derivatives, we obtain the Kohn-Sham equation
 %
 \begin{equation}
 [-\frac{\hbar^2\nabla_i^2}{2m}+V_{KS}(\bm{r})]\psi_i^{\sigma}(\bm{r})=\varepsilon_{i\sigma}\psi_i^{\sigma}(\bm{r}),
 \end{equation}
 %
-które ma postać jednocząstkowego równania Schr\"{o}dingera z potencjałem Kohna-Shama złożonym z trzech wyrazów
+which has the form of a single-particle Schr\"{o}dinger equation with the Kohn-Sham potential composed of three terms
 %
 \begin{equation}
 V_{KS}(\bm{r})=V_{ext}(\bm{r})+V_H(\bm{r})+V_{xc}(\bm{r})=V_{ext}(\bm{r})+\int d\bm{r'} \frac{n(\mb{r}')}{|\bm{r}-\bm{r'}|}+\frac{\delta E_{xc}[n]}{\delta n}.
 \label{VKS}
 \end{equation}
 %
-W potencjale tym nieznana jest dokładnie energia wymienno-korelacyjna, która musi być przybliżana
-odpowiednimi metodami, które będą omówione w kolejnych rozdziałach.
-Znając spektrum energetyczne rozwiązań równania Kohna-Shama, energię stanu podstawowego można wyrazić w postaci
+In this potential, the exchange-correlation energy is not known exactly and must be approximated
+by appropriate methods, which will be discussed in subsequent chapters.
+Knowing the energy spectrum of solutions to the Kohn-Sham equation, the ground state energy can be expressed in the form
 %
 \begin{equation}
 E[n,f_i]=\sum_{i\sigma}f_{i\sigma} \varepsilon_{i\sigma} - \int d\bm{r} n(\bm{r})V_{KS}(\bm{r}) + \int d\bm{r} V_{ext}(\bm{r}) n(\bm{r}) + E_H[n] + E_{xc}[n].
-\end{equation} 
+\end{equation}
 %
-Funkcje jednoelektronowe i energie stanów Kohna-Shama $\varepsilon_{i\sigma}$ nie mają jednoznacznej interpretacji fizycznej.
-Można je jednak powiązać ze zmianą energii całkowitej przy zmianie obsadzenia stanów
+The single-electron functions and energies of Kohn-Sham states $\varepsilon_{i\sigma}$ do not have an unambiguous physical interpretation.
+However, they can be related to the change in total energy upon changing the occupation of states
 %
 \begin{equation}
 \frac{\partial E}{\partial f_{i\sigma}}=\Big(\frac{\partial E}{\partial f_{i\sigma}}\Big)_n+\int d\bm{r} \frac{\delta E}{\delta n}\frac{\partial n}{\partial f_{i\sigma}}.
 \label{janak}
-\end{equation} 
+\end{equation}
 %
-Drugi wyraz po prawej stronie równania zeruje się dla stanu podstawowego ($\delta E/\delta n=0$),
-co prowadzi do wzoru
+The second term on the right side of the equation vanishes for the ground state ($\delta E/\delta n=0$),
+which leads to the formula
 %
 \begin{equation}
 \varepsilon_{i\sigma}=\frac{\partial E}{\partial f_{i\sigma}},
 \end{equation}
 %
-który nazywany jest twierdzeniem Janaka \cite{janak}.
-Zastosowanie tego wzoru do najwyższego obsadzonego stanu elektronowego daje energię jonizacji, czyli energię potrzebną do usunięcia pojedynczego elektronu 
-z danego układu atomowego. 
-
+which is called Janak's theorem \cite{janak}.
+Application of this formula to the highest occupied electronic state gives the ionization energy, i.e., the energy required to remove a single electron
+from a given atomic system. 
 
 
-\section{Funkcjonał wymienno-korelacyjny}
+\section{Exchange-Correlation Functional}
 
-\subsection{Przybliżenie lokalnej gęstości (LDA)}
+\subsection{Local Density Approximation (LDA)}
 
-Funkcjonał wymienno-korelacyjny $E_{xc}$ i odpowiadający mu potencjał $V_{xc}$ nie są znane dokładnie i w ramach teorii
-funkcjonału gęstości muszą być opisywane w przybliżony sposób.
-Najprostszym podejściem jest przybliżenie lokalnej gęstości ({\it ang. local density approximation} - LDA).
-W przybliżeniu LDA przyjmujemy, że energia wymienno-korelacyjna w każdym punkcie przestrzeni, gdzie gęstość elektronowa wynosi $n(\mb{r})$,
-równa jest energii wymienno-korelacyjnej jednorodnego gazu elektronowego o tej samej gęstości, $n=n(\mb{r})$.
-Funkcjonał wymienno-korelacyjny może być wtedy zapisany w postaci
+The exchange-correlation functional $E_{xc}$ and the corresponding potential $V_{xc}$ are not known exactly and within the framework of density functional theory must be described in an approximate manner.
+The simplest approach is the local density approximation (LDA).
+In the LDA approximation, we assume that the exchange-correlation energy at each point in space, where the electron density is $n(\mb{r})$,
+is equal to the exchange-correlation energy of a homogeneous electron gas with the same density, $n=n(\mb{r})$.
+The exchange-correlation functional can then be written in the form
 %
 \begin{equation}
 E_{xc}[n]=\int d\mb{r} n(\mb{r}) \varepsilon_{xc}(n),
 \end{equation}
 %
-gdzie $\varepsilon_{xc}(n)$ jest energią wymienno-korelacyjną przypadającą na pojedynczy elektron w jednorodnym gazie elektronowym o gestości $n$.
-Można ją zapisać jako sumę części wymiennej i korelacyjnej, $\varepsilon_{xc}(n)=\varepsilon_x(n)+\varepsilon_c(n)$.   
-Przybliżenie LDA jest dokładne w granicy wolno zmieniającej się gęstości, co odpowiada warunkowi
+where $\varepsilon_{xc}(n)$ is the exchange-correlation energy per electron in a homogeneous electron gas with density $n$.
+It can be written as the sum of exchange and correlation parts, $\varepsilon_{xc}(n)=\varepsilon_x(n)+\varepsilon_c(n)$.
+The LDA approximation is exact in the limit of slowly varying density, which corresponds to the condition
 %
 \begin{equation}
 \frac{q}{k_F}\ll 1,
-\end{equation} 
+\end{equation}
 %
-gdzie $q$ jest miarą niejednorodności układu
+where $q$ is a measure of the system's inhomogeneity
 %
 \begin{equation}
 q=\frac{|\nabla k_F|}{2k_F},
 \end{equation}
 %
-a $k_F$ odpowiada wektorowi falowemu Fermiego dla gazu jednorodnego, który w punkcie o lokalnej gęstości $n(\bm{r})$ dany jest wzorem
+and $k_F$ corresponds to the Fermi wave vector for a homogeneous gas, which at a point with local density $n(\bm{r})$ is given by the formula
 %
 \begin{equation}
 k_F=[3\pi^2n(\bm{r})]^{1/3}.
-\end{equation} 
+\end{equation}
 %
 
-Można łatwo uogólnić to przybliżenie do układów z polaryzacją spinową. Wtedy energia wymienno-korelacyjna jest funkcjonałem
-gęstości spinów skierowanych do góry $n_{\uparrow}$ i w dół $n_{\downarrow}$
+This approximation can be easily generalized to systems with spin polarization. Then the exchange-correlation energy is a functional
+of the spin-up density $n_{\uparrow}$ and spin-down density $n_{\downarrow}$
 %
 \begin{equation}
 E_{xc}[n^{\uparrow},n^{\downarrow}]=\int d\bm{r} n(\mb{r}) \varepsilon_{xc}(n_{\uparrow},n_{\downarrow}).
 \label{xclda}
 \end{equation}
 %
-To uogólnienie nazywane jest czasem przybliżeniem lokalnej gestości spinowej ({\it ang. local spin density} - LSD).
-Potencjał wymienno-korelacyjny zależny od spinu $\sigma$ w tym przybliżeniu określony jest przez zależność
+This generalization is sometimes called the local spin density (LSD) approximation.
+The spin-dependent exchange-correlation potential for spin $\sigma$ in this approximation is determined by the relation
 %
 \begin{equation}
 V^{\sigma}_{xc}=\frac{\delta E_{xc}}{\delta n_{\sigma}}=\varepsilon_{xc}+n_{\sigma}\frac{\partial \varepsilon_{xc}}{\partial n_{\sigma}}.
 \end{equation}
 %
-Zgodnie ze wzorem (\ref{exchange}), część wymienna dana jest wzorem
+According to formula (\ref{exchange}), the exchange part is given by the formula
 %
 \begin{equation}
 \varepsilon_x(n_{\sigma})=-\frac{3}{4}\Big(\frac{3}{\pi}\Big)^{\frac{1}{3}}n_{\sigma}^\frac{1}{3},
 \end{equation}
 %
-co prowadzi do potencjału wymiennego w formie 
+which leads to the exchange potential in the form
 %
 \begin{equation}
-V^{\sigma}_x=-\Big(\frac{3}{\pi}\Big)^{\frac{1}{3}}n_{\sigma}^\frac{1}{3}. 
+V^{\sigma}_x=-\Big(\frac{3}{\pi}\Big)^{\frac{1}{3}}n_{\sigma}^\frac{1}{3}.
 \label{Vex}
 \end{equation}
 %
-Część korelacyjna może być wyznaczona bardzo dokładnie numerycznie metodą kwantowego Monte Carlo \cite{CeperleyAlder80}. 
-W praktyce stosuje się odpowiednie wyrażenia analityczne dofitowane do wyników numerycznych, które określają zależność energii korelacji 
-od gęstości elektronowej \cite{PZ,VWN}. 
-Przykładowo energia korelacji w parametryzacji Pardew-Zungera \cite{PZ} ma postać
+The correlation part can be determined very accurately numerically using the quantum Monte Carlo method \cite{CeperleyAlder80}.
+In practice, appropriate analytical expressions are used that are fitted to numerical results, which determine the dependence of correlation energy
+on electron density \cite{PZ,VWN}.
+For example, the correlation energy in the Perdew-Zunger parametrization \cite{PZ} has the form
 %
 \begin{eqnarray}
 \varepsilon_c(r_s)&=&-0.048+0.031 \ln(r_s) -0.0116 r_s+0.002r_s \ln(r_s), \quad r_s<1  \\
@@ -1243,123 +1241,122 @@ \subsection{Przybliżenie lokalnej gęstości (LDA)}
 \end{eqnarray}
 %
 
-\subsection{Uogólnione przybliżenie gradientowe (GGA)}
+\subsection{Generalized Gradient Approximation (GGA)}
 \label{sec:GGA}
 
-Uwzględnienie lokalnych zmian gęstości poprzez rozwinięcie energii wymienno-korelacyjnej w szereg gradientów gęstości
-zaproponowano już w pracy Kohna i Shama z 1965 roku \cite{kohn65}. Podejście to jednak nie spełnia reguł sum i załamuje się dla większości układów atomowych, 
-w których zmiany gęstości elektronowej sa przeważnie bardzo duże.
-Zaproponowane zostało nowe podejście zwane uogólnionym przybliżeniem gradientowym ({\it ang. generalized gradient approximation} - GGA),
-gdzie energia wymienno-korelacyjna jest funkcjonałem gęstości elektronowej i jej gradientów \cite{Langreth83,Pardew86,Becke88}. 
-W ogólnej formie dla układu spolaryzowanego spinowo może być zapisana w formie
+Including local density changes through expansion of the exchange-correlation energy in a series of density gradients
+was already proposed in the 1965 work of Kohn and Sham \cite{kohn65}. However, this approach does not satisfy sum rules and breaks down for most atomic systems,
+in which electron density changes are usually very large.
+A new approach called the generalized gradient approximation (GGA) was proposed,
+where the exchange-correlation energy is a functional of the electron density and its gradients \cite{Langreth83,Pardew86,Becke88}.
+In the general form for a spin-polarized system, it can be written as
 %
 \begin{equation}
-E_{xc}[n_{\uparrow},n_{\downarrow}]=\int d\mb{r} f(n_{\uparrow},n_{\downarrow},\nabla n_{\uparrow},\nabla n_{\downarrow}) 
+E_{xc}[n_{\uparrow},n_{\downarrow}]=\int d\mb{r} f(n_{\uparrow},n_{\downarrow},\nabla n_{\uparrow},\nabla n_{\downarrow})
 \label{xcgga}
 \end{equation}
 %
-Część wymienną tego funkcjonału dla układu bez polaryzacji spinowej można zapisać w postaci
+The exchange part of this functional for a system without spin polarization can be written as
 %
 \begin{equation}
 E_x[n]=\int d\mb{r} n \varepsilon_x(n) F_x(s),
 \end{equation}
 %
-gdzie $s=|\nabla n|/2k_Fn$ jest przeskalowanym (bezwymiarowym) gradientem gęstości elektronowej. Rozszerzenie na układy z polaryzacja spinową uzyskuje się stosując następujący wzór,
-który spełniony jest dla dokładnej energii wymiennej
+where $s=|\nabla n|/2k_Fn$ is the scaled (dimensionless) gradient of the electron density. Extension to systems with spin polarization is obtained using the following formula,
+which is satisfied for the exact exchange energy
 %
 \begin{equation}
 E_x[n^{\uparrow},n^{\downarrow}]=\frac{1}{2}(E_x[2n^{\uparrow}]+E_x[2n^{\downarrow}]).
 \end{equation}
 %
-Wiele form funkcji $F_x(s)$ zostało zaproponawych, w tym te najczęściej stosowane, które opisane są w pracach: A. D. Becke (Becke88) \cite{Becke88}, 
-J. P. Perdew i Y. Wang (PW91) \cite{PW91} i  J. P. Perdew, K. Burke, and M. Ernzerhof (PBE) \cite{PBE}.
-Jako przykład omówię funkcjonał PBE, który ma prostą postać i dobrze zdefiniowane warunki graniczne  
-które musi spełniać funkcja $F_x(s)$:
+Many forms of the function $F_x(s)$ have been proposed, including the most commonly used ones, which are described in the works: A. D. Becke (Becke88) \cite{Becke88},
+J. P. Perdew and Y. Wang (PW91) \cite{PW91}, and J. P. Perdew, K. Burke, and M. Ernzerhof (PBE) \cite{PBE}.
+As an example, I will discuss the PBE functional, which has a simple form and well-defined boundary conditions
+that the function $F_x(s)$ must satisfy:
 
 \begin{enumerate}
 
-\item{W granicy małych wartości gradientu ($s\rightarrow0$)
-spełniony jest warunek
+\item{In the limit of small gradient values ($s\rightarrow0$),
+the condition is satisfied
 %
 \begin{equation}
 F_x(s)\rightarrow 1+\mu s^2,
 \end{equation}
 %
-gdzie $\mu =0.219$. Warunek ten zapewnia właściwe zachowanie się odpowiedzi liniowej jednorodnego gazu
-elektronów (liniowe przyczynki od energii wymiany i korelacji kasują się).}
+where $\mu =0.219$. This condition ensures the proper behavior of the linear response of the homogeneous electron gas (linear contributions from exchange and correlation energies cancel).}
 
-\item{Dla dużych wartości gradientu ($s\rightarrow\infty$) funkcja ograniczona jest od góry $F_x(s)\leq 1.804$.}
+\item{For large gradient values ($s\rightarrow\infty$), the function is bounded from above $F_x(s)\leq 1.804$.}
 
-Funkcja, która spełnia te warunki ma postać
+The function that satisfies these conditions has the form
 %
 \begin{equation}
 F_x(s)=1+\kappa-\frac{\kappa}{1+\frac{\mu s^2}{\kappa}},
-\end{equation} 
+\end{equation}
 %
-gdzie $\kappa=0.804$.
+where $\kappa=0.804$.
 \end{enumerate}
 
-Część korelacyjną można zapisać w ogólnej formie
+The correlation part can be written in the general form
 %
 \begin{equation}
 E_c[n^{\uparrow},n^{\downarrow}]=\int d\mb{r} n [\epsilon_c(r_s,\zeta)+H(r_s,\zeta,t)],
 \end{equation}
 %
-gdzie $\zeta=(n^{\uparrow}-n^{\downarrow})/n$ jest względną polaryzacją spinową, $t=|\nabla n|/2\phi k_s n$ jest bezwymiarowym gradientem,
-$\phi=\sqrt{(1+\zeta)^{2/3}+(1-\zeta)^{2/3})}/2$ jest spinową funkcją skalującą i $k_s=\sqrt{4k_F/\pi a_0}$ wektorem ekranowania Thomasa-Fermiego.
-Funkcja $H$ spełnia następujące warunki:
+where $\zeta=(n^{\uparrow}-n^{\downarrow})/n$ is the relative spin polarization, $t=|\nabla n|/2\phi k_s n$ is the dimensionless gradient,
+$\phi=\sqrt{(1+\zeta)^{2/3}+(1-\zeta)^{2/3})}/2$ is the spin scaling function, and $k_s=\sqrt{4k_F/\pi a_0}$ is the Thomas-Fermi screening vector.
+The function $H$ satisfies the following conditions:
 
 \begin{enumerate}
-\item{Dla małych gradientów ($t\rightarrow 0$) funkcja dana jest wyrazem rozwinięcia drugiego rzędu
+\item{For small gradients ($t\rightarrow 0$), the function is given by the second-order expansion term
 %
 \begin{equation}
 H\rightarrow \frac{e^2}{a_0}\beta\phi^3 t^2,
 \end{equation}
 %
-gdzie $\beta=0.067$.}
-\item{Przy szybko zmieniających się gęstościach ($t\rightarrow \infty$) energia korelacji znika
-co spełnione jest przy warunku
+where $\beta=0.067$.}
+\item{For rapidly varying densities ($t\rightarrow \infty$), the correlation energy vanishes,
+which is satisfied by the condition
 %
 \begin{equation}
 H\rightarrow -\epsilon_c.
 \end{equation}}
 %
-\item{W granicy dużych gęstości ($r_s\rightarrow 0$) energia korelacji dąży do stałej wartości.
-Funkcja $H$ musi kasować logarytmiczną osobliwość w $\epsilon_c$
+\item{In the limit of high densities ($r_s\rightarrow 0$), the correlation energy approaches a constant value.
+The function $H$ must cancel the logarithmic singularity in $\epsilon_c$
 %
 \begin{equation}
 H\rightarrow \frac{e^2}{a_0}\gamma\phi^3 \ln t^2,
 \end{equation}
 %
-gdzie $\gamma=0.031$.}
+where $\gamma=0.031$.}
 \end{enumerate}
 
-Warunki te spełnia funkcja w postaci
+These conditions are satisfied by the function in the form
 %
 \begin{equation}
 H=\frac{e^2}{a_0}\gamma\phi^3\ln[1+\frac{\beta}{\gamma}t^2\frac{1+At^2}{1+At^2+A^2t^4}],
 \end{equation}
 %
-gdzie 
+where
 %
 \begin{equation}
 A=\frac{\beta}{\gamma}[\exp\{-\epsilon_c/(\gamma\phi^3e^2/a_0)\}-1]^{-1}.
 \end{equation}
 
 
-W tabeli \ref{EXC} porównane są wartości energii wymiany i korelacji w przybliżeniu LDA i PBE
-z wartościami dokładnymi wyznaczonymi dla kilku wybranych atomów. Wartości bezwzględne energii wymiany są większe
-o około rząd wielkości od energii korelacji. W LDA, energia wymiany jest zaniżona średnio o $10\%$, a energia korelacji jest znacznie zawyżona,
-nawet powyżej $100\%$, w porównaniu do wartości dokładnych. Ze względu na przeciwne znaki odchyleń od wartości dokładnych, błędy energii wymiany 
-i korelacji w przybliżeniu LDA częściowo kasują się. 
+Table \ref{EXC} compares the values of exchange and correlation energies in the LDA and PBE approximations
+with exact values determined for several selected atoms. The absolute values of exchange energy are greater
+by about an order of magnitude than the correlation energy. In LDA, the exchange energy is underestimated on average by $10\%$, and the correlation energy is significantly overestimated,
+even above $100\%$, compared to exact values. Due to the opposite signs of deviations from exact values, the errors of exchange
+and correlation energies in the LDA approximation partially cancel.
 
 \begin{table}[h!]
-\caption{Porównanie energii wymiennej i korelacyjnej uzyskanych w kilku przybliżeniach z wartościami dokładnymi dla kilku wybranych atomów.}
+\caption{Comparison of exchange and correlation energies obtained in several approximations with exact values for several selected atoms.}
 \label{EXC}
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|c|c|}
 \hline
- & \multicolumn{2}{c|}{LDA} & \multicolumn{2}{c|}{PBE} & \multicolumn{2}{c|}{Dokładna wartość} \\ \hline
+ & \multicolumn{2}{c|}{LDA} & \multicolumn{2}{c|}{PBE} & \multicolumn{2}{c|}{Exact value} \\ \hline
 Atom& $E_x$ & $E_c$ & $E_x$ & $E_c$  & $E_x$  & $E_c$ \\
 \hline
   H & -0.2680 & -0.0222 &  -0.3059 & -0.0060 & -0.3125 & 0.0000\\
@@ -1371,102 +1368,102 @@ \subsection{Uogólnione przybliżenie gradientowe (GGA)}
 \end{center}
 \end{table}
 
-Te duże różnice w energii wymiany i korelacji wynikają z nieuwzględnienia wpływu lokalnych zmian 
-gęstości elektronowej, które są charakterystyczne dla atomów i molekuł \cite{Pardew92}.  
-W przybliżeniu GGA, wartości obydwóch energii poprawiają się. Energia wymiany jest w dalszym ciągu zaniżona, ale tylko
-na poziomie $\sim 1\%$. Energia korelacji w atomie wodoru jest znacznie zredukowana w porównaniu do LDA,
-a dla atomu helu jest bardzo dobrze odtworzona. W pozostałych przypadkach pokazanych w tabeli \ref{EXC} jest mniejsza
-od wartości dokładnej o kilka procent.
-
-Funkcjonał GGA w granicy zerowych gradientów, czyli dla jednorodnego gazu elektronowego, jest równoważny przybliżeniu LDA.
-Oznacza to, że dla niejednorodnych układów powinień dawać zawsze wyniki lepsze niż LDA. Tak się jednak nie dzieje, a wynika
-to z faktu, że dla niektórych układów częściowe redukowanie się błedów energii wymiany i korelacji jest większe w LDA niż w GGA.
-Przykładem są metale szlachetne (Ag, Au, Pt), dla których stałe sieci wyznaczone w LDA bardzo dobrze zgadzają się
-z eksperymentem, a w GGA są zawyżone. Większe wartości stałej sieci w przybliżeniu GGA wynikają z zaniżonej energii kohezji kryształów.
-LDA zwykle daje zawyżone wartości energii kohezji i zaniżone wartości stałych sieci. 
-Dla układów magnetycznych, czyli z otwartymi powłokami $d$ lub $f$, przybliżenie GGA daje zwykle lepsze wyniki niż LDA.
-Klasycznym przykładem jest żelazo, dla którego stanem podstawowym w przybliżeniu LDA jest struktura centrowana powierzchniowo ($fcc$) 
-z uporządkowaniem antyferromagnetycznym, a GGA daje poprawny wynik, czyli strukturę centrowaną objętościowo ($bcc$) z uporządkowaniem ferromagnetycznym.
-
-\section{Procedura iteracyjna}
-
-Równania Kohna-Shama (KS) rozwiązywane są metodą iteracyjną, której kolejne kroki przedstawione są na rysunku \ref{fig:diagram}.
-Ogólnie stosowane procedury pozwalają zminimalizować funkcjonał energii całkowitej przy zadanych położeniach atomów,
-jak również wyznaczyć położenia atomów i parametry sieci krystalicznej, które odpowiadają minimum energii całkowitej
-układu. Najpierw ustalamy początkowe położenia atomów, które można przyjąć na przykład na podstawie
-danych z eksperymentu dyfrakcyjnego. Przy ustalonych położeniach atomowych przyjmuje się początkowy
-rozkład gęstości elektronowej $n_0(\bm{r})$, dla którego wylicza się potencjał ze wzoru (\ref{VKS}).
-Dla tego potencjału rozwiązuje się równania KS i wyznacza jednoelektronowe funkcje falowe
-w odpowiedni wybranej bazie funkcyjnej. 
+These large differences in exchange and correlation energies result from neglecting the influence of local changes
+in electron density, which are characteristic of atoms and molecules \cite{Pardew92}.
+In the GGA approximation, the values of both energies improve. The exchange energy is still underestimated, but only
+at the level of $\sim 1\%$. The correlation energy in the hydrogen atom is significantly reduced compared to LDA,
+and for the helium atom it is very well reproduced. In the other cases shown in Table \ref{EXC}, it is smaller
+than the exact value by a few percent.
+
+The GGA functional in the limit of zero gradients, i.e., for a homogeneous electron gas, is equivalent to the LDA approximation.
+This means that for inhomogeneous systems it should always give better results than LDA. However, this is not the case, which results
+from the fact that for some systems the partial cancellation of exchange and correlation energy errors is greater in LDA than in GGA.
+Examples are noble metals (Ag, Au, Pt), for which lattice constants determined in LDA agree very well
+with experiment, while in GGA they are overestimated. Larger lattice constant values in the GGA approximation result from underestimated cohesive energies of crystals.
+LDA usually gives overestimated values of cohesive energies and underestimated values of lattice constants.
+For magnetic systems, i.e., those with open $d$ or $f$ shells, the GGA approximation usually gives better results than LDA.
+A classic example is iron, for which the ground state in the LDA approximation is a face-centered cubic ($fcc$) structure
+with antiferromagnetic ordering, while GGA gives the correct result, i.e., a body-centered cubic ($bcc$) structure with ferromagnetic ordering.
+
+\section{Iterative Procedure}
+
+The Kohn-Sham (KS) equations are solved by an iterative method, whose successive steps are shown in Figure \ref{fig:diagram}.
+Generally used procedures allow minimization of the total energy functional for given atomic positions,
+as well as determination of atomic positions and crystal lattice parameters that correspond to the minimum of the total energy
+of the system. First, we establish initial atomic positions, which can be taken, for example, from
+diffraction experiment data. With fixed atomic positions, an initial
+electron density distribution $n_0(\bm{r})$ is assumed, for which the potential is calculated from formula (\ref{VKS}).
+For this potential, the KS equations are solved and single-electron wave functions are determined
+in an appropriately chosen functional basis.
 
 \begin{figure}[h]
 \centering
 \includegraphics[scale=0.45]{diagram.pdf}
-\caption{Schemat interacyjnej procedury rozwiązywania równań Kohna-Sham.}
+\caption{Schematic of the iterative procedure for solving the Kohn-Sham equations.}
 \label{fig:diagram}
 \end{figure}
 
-Rozwiązywanie równań KS dla atomów w ciele stałym jest złożonym problemem i będzie dyskutowane w kolejnych rozdziałach.
-Znając funkcje falowe i energie własne równania KS wyznacza się gęstość elektronową w każdym punkcie przestrzeni 
-z zależności (\ref{density}) i całkowitą energię układu. 
-Nowy rozkład gęstości pozwala ponownie wyznaczyć potencjał KS i rozwiązać równania KS. Dostajemy nowy rozkład gęstości i zmienioną
-energię całkowitą. Jeżeli ta energia pokrywa się z energią w poprzednim
-kroku lub te energie różnią sie o małą wielkość (zdefiniowaną na początku obliczeń) to oznacza, że 
-otrzymaliśmy prawidłowo zbiegnięty rozkład gęstości. Zakończona jest w ten sposób pętla elektronowa, 
-która daje rozwiązanie równań KS i minimalną energię całkowitą przy zadanych położeniach atomowych.
+Solving the KS equations for atoms in a solid is a complex problem and will be discussed in subsequent chapters.
+Knowing the wave functions and eigenvalues of the KS equation, the electron density at each point in space is determined
+from relation (\ref{density}) and the total energy of the system.
+The new density distribution allows us to recalculate the KS potential and solve the KS equations. We obtain a new density distribution and modified
+total energy. If this energy coincides with the energy in the previous
+step or these energies differ by a small amount (defined at the beginning of the calculations), it means that
+we have obtained a properly converged density distribution. This completes the electronic loop,
+which gives the solution of the KS equations and the minimum total energy for given atomic positions.
 
-Procedurę minimalizacji można przyspieszyć wykorzystując odpowiednio gęstości elektronowe
-w każdym kroku iteracyjnym. W najprostszym podejściu początkową wartość gęstości elektronowej $n^p_{i+1}$ w kroku $i+1$ 
-można wyznaczyć jako liniową poprawkę do gęstości z poprzedniego kroku
+The minimization procedure can be accelerated by appropriately using electron densities
+at each iterative step. In the simplest approach, the initial value of the electron density $n^p_{i+1}$ in step $i+1$
+can be determined as a linear correction to the density from the previous step
 %
 \begin{equation}
 n^p_{i+1}=n^p_i+\alpha(n^k_i-n^p_i),
 \label{mixing}
 \end{equation}
-% 
-gdzie $\alpha$ jest współczynnikiem liniowym, $n^p_i$ i $n^k_i$ są początkową i końcową wartością gęstości w kroku $i$.
-Stała wartość współczynnika $\alpha$ nie daje optymalnej szybkości zbieżności.
-W najczęściej stosowanej metodzie Broydena, współczynnik liniowy wyznaczany jest z jakobianu układu $J_i$,
-który optymalizowany jest w każdym kroku iteracji, $\alpha=-J^{-1}_i$.
-
-Aby zoptymalizować układ ze względu na położenia atomowe należy odpowiednio przesuwać
-atomy w nowe położenia, tak aby całkowita energia obniżała się w kolejnych krokach. 
-Wykorzystuje się do tego bardzo efektywną metodę sprzężonego gradientu ({\it ang. conjugate gradient}),
-która jest również metodą iteracyjną (pętla jonowa).
-W każdym kroku tej procedury wykonuje się pełną optymalizację elektronową, aby wyznaczyć rozkład gęstości elektronowej 
-i energię układu dla aktualnego położenia atomów. Wyznaczona gęstość końcowa jest używana do wyznaczenia
-gęstości startowej w kolejnym kroku pętli jonowej według formuły (\ref{mixing}).
-Warunkiem zbieżności jest osiągnięcie stanu o minimalnej energii ze względu na położenia atomów
-i parametry sieci krystalicznej. Według najczęściej przyjmowanego kryterium zbieżność jest osiągnięta jeżeli różnice energii 
-całkowitej w dwóch kolejnych krokach jonowych jest mniejsza od zadanej na początku wartości.
-
-Stan podstawowy otrzymany w wyniku procedury minimalizacyjnej powinien spełniać warunki stanu równowagi badanego materiału.
-Warunki te są następujące: 
+%
+where $\alpha$ is a linear coefficient, $n^p_i$ and $n^k_i$ are the initial and final values of density in step $i$.
+A constant value of the coefficient $\alpha$ does not give optimal convergence speed.
+In the most commonly used Broyden method, the linear coefficient is determined from the Jacobian of the system $J_i$,
+which is optimized at each iteration step, $\alpha=-J^{-1}_i$.
+
+To optimize the system with respect to atomic positions, atoms should be appropriately moved
+to new positions so that the total energy decreases in subsequent steps.
+The very effective conjugate gradient method is used for this,
+which is also an iterative method (ionic loop).
+In each step of this procedure, a complete electronic optimization is performed to determine the electron density distribution
+and the system energy for the current atomic positions. The determined final density is used to determine
+the starting density in the next step of the ionic loop according to formula (\ref{mixing}).
+The convergence condition is reaching a state of minimum energy with respect to atomic positions
+and crystal lattice parameters. According to the most commonly adopted criterion, convergence is achieved if the difference in total energy
+in two consecutive ionic steps is less than the value specified at the beginning.
+
+The ground state obtained as a result of the minimization procedure should satisfy the equilibrium conditions of the studied material.
+These conditions are as follows:
 
 \begin{enumerate}
 {\item
-Całkowita siła działająca na każdy atom zeruje się.
-Siły działające na atomy można wyznaczyć stosując twierdzenie Hellmanna-Feynmana \cite{Hellmann,Feynman}, 
-które mówi, że siła działająca na atom $i$ równa jest pochodnej energii całkowitej po położeniu
-tego atomu wziętej ze znakiem przeciwnym
+The total force acting on each atom vanishes.
+Forces acting on atoms can be determined using the Hellmann-Feynman theorem \cite{Hellmann,Feynman},
+which states that the force acting on atom $i$ is equal to the derivative of the total energy with respect to the position
+of that atom taken with the opposite sign
 %
 \begin{equation}
 \bm{F_i}=-\frac{\partial E_{tot}}{\partial\bm{R_i}}.
 \end{equation}
 %
-Zgodnie z tym wzorem, w stanie podstawowym, który odpowiada minimum energii, wszystkie siły $\bm{F_i}$ są równe zeru.
-Wartości sił stanowią często dodatkowe kryterium zbieżności układu w procedurze optymalizacyjnej:
-zbieżność jest osiągnięta jeżeli największa siła działająca na atomy jest mniejsza od zadanej wartości.}
+According to this formula, in the ground state, which corresponds to an energy minimum, all forces $\bm{F_i}$ are equal to zero.
+The values of forces often constitute an additional convergence criterion of the system in the optimization procedure:
+convergence is achieved if the largest force acting on atoms is less than a specified value.}
 
-{\item Makroskopowe naprężenia w układzie równe jest naprężeniu wywołanym ciśnieniem zewnętrznym.
-Średni tensor naprężeń określony jest przez pochodną energii całkowitej po tensorze deformacji 
+{\item The macroscopic stress in the system is equal to the stress caused by external pressure.
+The average stress tensor is determined by the derivative of the total energy with respect to the strain tensor
 %
 \begin{equation}
 \sigma_{\alpha\beta}=-\frac{1}{V}\frac{\partial E}{\partial u_{\alpha\beta}}.
 \end{equation}
-gdzie $V$ jest objętością układu. Tensor deformacji jest symetrycznym tensorem pochodnych wektora przesunięcia
-$\bm{u}=\bm{r}-\bm{r'}$ po położeniu $\bm{r}$.
-Przy kompresji hydrostatycznej, ciśnienie wiąże się z tensorem naprężeń zależnością $P=-\frac{1}{2}\sum_{\alpha}\sigma_{\alpha\alpha}$.}
+where $V$ is the volume of the system. The strain tensor is a symmetric tensor of derivatives of the displacement vector
+$\bm{u}=\bm{r}-\bm{r'}$ with respect to position $\bm{r}$.
+Under hydrostatic compression, the pressure is related to the stress tensor by $P=-\frac{1}{2}\sum_{\alpha}\sigma_{\alpha\alpha}$.}
 \end{enumerate}
 
 

From f9cf547316bca3f339b5363c1a8fb41978910e63 Mon Sep 17 00:00:00 2001
From: Copilot <198982749+Copilot@users.noreply.github.com>
Date: Sat, 3 Jan 2026 07:40:36 +0100
Subject: [PATCH 18/23] Translate Chapter 4 (Methods for determining electronic
 structure) to English (#33)

* Initial plan

* Translate Chapter 4: General characteristics, Pseudopotentials, and Localized orbitals sections

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Complete translation of Chapter 4: Methods for determining electronic structure

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 719 +++++++++++++++++++++++++-------------------------
 1 file changed, 359 insertions(+), 360 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index 93e2898..d9f4405 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -1468,148 +1468,148 @@ \section{Iterative Procedure}
 
 
 
-\chapter{Metody wyznaczania struktury elektronowej}
-
-\section{Ogólna charakterystyka}
-
-Najważniejsze metody wyliczania struktury pasmowej oparte są na teorii DFT i polegają na
-wyznaczaniu jednocząstkowych funkcji falowych poprzez rozwiązanie równań Kohna-Shama
-lub podobnych równań zawierających efektywny potencjał elektronowy.  
-Podstawowym elementem odróżniającym poszczególne metody jest wybór funkcji bazowych, 
-które służą do rozwinięcia funkcji falowych w całym obszarze kryształu lub w poszczególnych jego częściach. 
-Najbardziej naturalną bazą w periodycznym krysztale są fale płaskie i obliczenia w tej bazie są bardzo efektywne oraz łatwe do implementacji. 
-Niestety, nie nadają sie do opisu elektronów w całym obszarze kryształu i wszystkich stanów elektronowych.
-Elektrony, które znajdują się najbliżej jąder atomowych i obsadzają najniższe stany energetyczne 
-wchodzą w skład rdzenia atomowego ({\it ang. atomic core}).
-W obszarze rdzeni atomowych, potencjał elektryczny jest silnie przyciągający i zbliżony jest do potencjału atomowego.
-Przez to stany elektronowe znajdujące się blisko jądra atomowego, zachowują sie podobnie do orbitali atomowych, czyli 
-silnie oscylują i zmieniają swój znak. Oznacza to również, że energia kinetyczna elektronów w tym obszarze jest duża.
-Funkcje falowe stanów rdzenia sa silnie zlokalizowane i szybko zanikają z odległością. 
-W tym przypadku znacznie lepszą zbieżność uzyskuje się stosując bazę zlokalizowanych funkcji, np. harmoniki sferyczne.
-Funkcja falowa elektronów walencyjnych zachowuje się odmiennie w obszarze rdzeni atomowych i w obszarze międzywęzłowym.
-Schematycznie potencjał periodyczny i funkcja falowa elektronów walencyjnych pokazana jest na rysunku \ref{fig:bloch}.
-W obszarze rdzeni, zaznaczonych kółkami o promieniu $r_c$, funkcja falowa, podobnie jak w przypadku stanów rdzenia,
-zmienia się bardzo szybko.
-W obszarze miedzywęzłowym ($r>r_c$), potencjał i gęstość elektronowa zmieniają się wolno w funkcji położenia. 
-Również funkcja falowa elektronów walencyjnych w tym obszarze jest wolnozmienna i 
-w tym przypadku można uzyskać szybką zbieżność w rozwinięciu na fale płaskie. 
+\chapter{Methods for determining electronic structure}
+
+\section{General characteristics}
+
+The most important methods for calculating band structure are based on DFT theory and rely on
+determining single-particle wave functions by solving the Kohn-Sham equations
+or similar equations containing an effective electronic potential.  
+The fundamental element distinguishing different methods is the choice of basis functions, 
+which are used to expand the wave functions throughout the entire crystal volume or in its individual parts. 
+The most natural basis in a periodic crystal consists of plane waves, and calculations in this basis are very efficient and easy to implement. 
+Unfortunately, they are not suitable for describing electrons throughout the entire crystal volume and all electronic states.
+Electrons that are located closest to atomic nuclei and occupy the lowest energy states 
+are part of the atomic core.
+In the region of atomic cores, the electric potential is strongly attractive and similar to the atomic potential.
+As a result, electronic states located close to the atomic nucleus behave similarly to atomic orbitals, that is 
+they strongly oscillate and change sign. This also means that the kinetic energy of electrons in this region is large.
+The wave functions of core states are strongly localized and decay rapidly with distance. 
+In this case, much better convergence is achieved using a basis of localized functions, such as spherical harmonics.
+The wave function of valence electrons behaves differently in the region of atomic cores and in the interstitial region.
+The periodic potential and wave function of valence electrons are shown schematically in Figure \ref{fig:bloch}.
+In the core regions, marked by circles with radius $r_c$, the wave function, similarly to core states,
+changes very rapidly.
+In the interstitial region ($r>r_c$), the potential and electron density vary slowly as a function of position. 
+The wave function of valence electrons in this region is also slowly varying, and 
+in this case rapid convergence can be achieved in the expansion in plane waves. 
 
 
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.5]{crystal.pdf}
-\caption{Potencjał periodyczny i funkcja falowa elektronów walencyjnych.}
+\caption{Periodic potential and wave function of valence electrons.}
 \label{fig:bloch}
 \end{figure}
 
 \newpage
 
-Najważniejsze metody wyznaczania struktury pasmowej można podzielić na trzy główne grupy:
+The most important methods for determining band structure can be divided into three main groups:
 
 
 \begin{enumerate}
-\item{{\bf Pseudopotencjały.} Jest to grupa metod, która ogranicza ilość elektronów i rozwiązania równania Kohna-Shama tylko do stanów walencyjnych. 
-Ma to uzasadnienie wynikające z charakteru stanów rdzenia, które są silnie zlokalizowane blisko jądra atomowego i nie biorą udziału w wiązaniach atomowych.
-Przez to najważniejsze własności materiałów zdeterminowane są przez zachowanie elektronów walencyjnych. 
-W metodzie pseudopotencjału stosuje się rozwinięcie funkcji falowej w bazie fal płaskich. Aby zachować jednolity opis funkcji falowej w całym obszarze kryształu, stosuje się w tych metodach przybliżony potencjał działający na elektrony walencyjne w obszarze rdzenia atomowego, który nazywany jest pseudopotencjałem. 
-Dla elektronów walencyjnych wyznacza się pseudofunkcję falową, która jest rozwiązaniem jednocząstkowego równania Schr\"{o}dingera z odpowiednim pseudopotencjałem. W obszarze międzywęzłowym jest ona równa dokładnej funcji falowej, a w obszarze rdzenia jest wolnozmienną funkcją bez oscylacji i miejsc
-zerowych. Najdokładniejsze obliczenia w tej grupie metod stosują pseudopotencjały ultramiękkie i potencjały typu PAW.}
-\item{{\bf Zlokalizowane orbitale.} W tym podejściu funkcję falową elektronów zapisujemy w bazie orbitali zlokalizowanych na poszczególnych
-atomach. W najprostszym przybliżeniu ciasnego wiązania jedynymi istotnymi parametrami są elementy macierzowe Hamiltonianu, które
-opisują przekrywanie się lokalnych orbitali lub funkcji Wanniera. W dokładniejszych obliczeniach jako bazę stosuje się funkcje Gaussa
-lub orbitale Slatera.}
-\item{{\bf Stowarzyszone fale i atomowe sfery}. Ta grupa obejmuje metody obliczeniowe, które bazują na ogólnej zasadzie podziału
-kryształu na dwa obszary. Pierwszy obszar zawiera rdzenie atomowe, w których funkcje falowe zachowują cechy orbitali atomowych i drugi obszar między atomami,
-gdzie elektrony walencyjne opisane są wolnozmienną funkcją falową. W każdym z dwóch charakterystycznych obszarów stosuje się
-rozwinięcia funkcji falowej w różnych bazach funkcyjnych. Obszar rdzenia atomowego definiuje wartość promienia $r_c$. Funkcje falowe
-otrzymane dla odległości mniejszych i wiekszych od $r_c$ muszą spełniać odpowiednie warunki ciągłości na granicy rdzenia atomowego.}
+\item{{\bf Pseudopotentials.} This is a group of methods that limits the number of electrons and solutions of the Kohn-Sham equation to valence states only. 
+This is justified by the nature of core states, which are strongly localized near the atomic nucleus and do not participate in atomic bonding.
+Therefore, the most important properties of materials are determined by the behavior of valence electrons. 
+In the pseudopotential method, the wave function is expanded in a basis of plane waves. To maintain a uniform description of the wave function throughout the entire crystal volume, these methods use an approximate potential acting on valence electrons in the atomic core region, which is called the pseudopotential. 
+For valence electrons, a pseudo-wave function is determined, which is a solution of the single-particle Schr\"{o}dinger equation with an appropriate pseudopotential. In the interstitial region it is equal to the exact wave function, while in the core region it is a slowly varying function without oscillations and zero
+crossings. The most accurate calculations in this group of methods use ultrasoft pseudopotentials and PAW-type potentials.}
+\item{{\bf Localized orbitals.} In this approach, the electronic wave function is written in a basis of orbitals localized on individual
+atoms. In the simplest tight-binding approximation, the only significant parameters are the Hamiltonian matrix elements that
+describe the overlap of local orbitals or Wannier functions. In more accurate calculations, Gaussian functions
+or Slater orbitals are used as the basis.}
+\item{{\bf Augmented waves and atomic spheres}. This group includes computational methods that are based on the general principle of dividing
+the crystal into two regions. The first region contains atomic cores, in which wave functions retain the characteristics of atomic orbitals, and the second region between atoms,
+where valence electrons are described by a slowly varying wave function. In each of the two characteristic regions,
+expansions of the wave function in different functional bases are used. The atomic core region defines the value of radius $r_c$. Wave functions
+obtained for distances smaller and larger than $r_c$ must satisfy appropriate continuity conditions at the boundary of the atomic core.}
 
 \end{enumerate}
 
-W kolejnych rozdziałach opisane zostaną metody obliczeniowe należące do tych trzech grup.
+In the following sections, computational methods belonging to these three groups will be described.
   
-\section{Pseudopotencjały}
+\section{Pseudopotentials}
 
-Główną ideę pseudopotencjału ilustruje rysunek \ref{fig:pseudopot}. W obszarze rdzenia atomowego ($r<r_c$) dokładny potencjał $V$ przybliżany
-jest pseudopotencjałem $\tilde{V}$, który ma skończoną wartość dla $r=0$. Odpowiadająca mu pseudofunkcja falowa $\tilde{\psi}$
-ma gładkie zachowanie (bez oscylacji i miejsc zerowych) w tym obszarze. Dla $r>r_c$ pseudopotencjał pokrywa się z dokładnym potencjałem,
-a pseudofunkcja odpowiada dokładnej funkcji falowej $\psi$. Dzięki takiemu przybliżeniu, pseudofunkcja falowa może być rozwinięta
-na fale płaskie w całym obszarze kryształu.   
+The main idea of the pseudopotential is illustrated in Figure \ref{fig:pseudopot}. In the atomic core region ($r<r_c$), the exact potential $V$ is approximated
+by the pseudopotential $\tilde{V}$, which has a finite value at $r=0$. The corresponding pseudo-wave function $\tilde{\psi}$
+has smooth behavior (without oscillations and zero crossings) in this region. For $r>r_c$, the pseudopotential coincides with the exact potential,
+and the pseudo-wave function corresponds to the exact wave function $\psi$. Thanks to this approximation, the pseudo-wave function can be expanded
+in plane waves throughout the entire crystal volume.   
 
   
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.1]{pseudopots-new.pdf}
-\caption{Psudopotencjał i pseudofunkcja falowa.}
+\caption{Pseudopotential and pseudo-wave function.}
 \label{fig:pseudopot}
 \end{figure}  
   
-Metoda pseudopotencjału rozwinęła się z podejścia zortogonalizowanych fal płaskich ({\it ang. ortogonalized plane waves} - OPW)
-zaproponowanego przez Herringa w 1940 \cite{herring40}. 
-Można zdefiniować transformację, która prowadzi od dokładnego potencjału do pseudopotencjału, wprowadzając jednocześnie pseudofunkcję falową do opisu elektronów walencyjnych \cite{Antoncik,KP}.
-Wprowadzamy osobne oznaczenia dla stanów walencyjnych $|\psi_v\rangle$ i stanów rdzenia $|\psi_c\rangle$, które są stanami własnymi
-hamiltonianu $H$ z odpowiednimi energiami własnymi $\varepsilon_v$ i $\varepsilon_c$.
-Zakładamy, że dokładna funkcja falowa elektronów walencyjnych może być wyrażona przez gładką funkcję $\tilde{\psi}_v$ (pseudofunkcję),
-która jest ortogonalna do stanów rdzenia
+The pseudopotential method developed from the orthogonalized plane waves (OPW) approach
+proposed by Herring in 1940 \cite{herring40}. 
+One can define a transformation that leads from the exact potential to the pseudopotential, while introducing the pseudo-wave function to describe valence electrons \cite{Antoncik,KP}.
+We introduce separate notations for valence states $|\psi_v\rangle$ and core states $|\psi_c\rangle$, which are eigenstates
+of the Hamiltonian $H$ with corresponding eigenvalues $\varepsilon_v$ and $\varepsilon_c$.
+We assume that the exact wave function of valence electrons can be expressed through a smooth function $\tilde{\psi}_v$ (pseudo-wave function),
+which is orthogonal to the core states
 %
 \begin{equation}
 |\psi_v\rangle =|\tilde{\psi}_v\rangle +\sum_{\alpha,c} a_{\alpha} |\psi_{\alpha c}\rangle,
 \label{OPW}
 \end{equation}
 %
-gdzie sumowanie przebiega po atomach $\alpha$ i stanach rdzenia $c$. 
-Dla uproszczenia zapisu wskaźniki $v$ i $c$ określają zarówno numer stanu, jak i kierunek spinu elektronu.
-Współczynniki rozwinięcia dostajemy z warunku ortogonalności  
+where the summation runs over atoms $\alpha$ and core states $c$. 
+For simplicity of notation, the indices $v$ and $c$ denote both the state number and the electron spin direction.
+The expansion coefficients are obtained from the orthogonality condition  
 %
 \begin{equation}
 a_{\alpha}=-\langle \psi_{\alpha c}|\tilde{\psi}_v\rangle.
 \label{a_alpha}
 \end{equation}
 % 
-W obszarze miedzywęzłowym dokładna funkcja falowa równa jest pseudofunkcji, która może być wyrażona w bazie fal płaskich.
-Wstawiając (\ref{OPW}) do równania $H|\psi_v\rangle=\varepsilon_v|\psi_v\rangle$ otrzymujemy
+In the interstitial region, the exact wave function is equal to the pseudo-wave function, which can be expressed in a plane wave basis.
+Inserting (\ref{OPW}) into the equation $H|\psi_v\rangle=\varepsilon_v|\psi_v\rangle$, we obtain
 %
 \begin{equation}
 H|\tilde{\psi}_v\rangle +\sum_{\alpha,c} a_{\alpha} \varepsilon_{\alpha c} |\psi_{\alpha c}\rangle =
 \varepsilon_v|\tilde{\psi}_v\rangle + \sum_{\alpha c} a_{\alpha} \varepsilon_v|\psi_{\alpha c}\rangle.
 \end{equation}
 %
-Przenosząc drugie równanie na prawą stronę i wykorzystując (\ref{a_alpha}) dostajemy
+Moving the second term to the right side and using (\ref{a_alpha}), we obtain
 %
 \begin{equation}
 (H +\sum_{\alpha,c}(\varepsilon_v - \varepsilon_{\alpha c})|\psi_{\alpha c}\rangle \langle \psi_{\alpha c}|)|\tilde{\psi}_v\rangle =
 \varepsilon_v|\tilde{\psi}_v\rangle.
 \end{equation}
 %
-Dostaliśmy równanie typu Schr\"{o}dingera z dodatkowym potencjałem 
+We have obtained a Schr\"{o}dinger-type equation with an additional potential 
 %
 \begin{equation}
 V^{R}=\sum_{\alpha,c}(\varepsilon_v - \varepsilon_{\alpha c})|\psi_{\alpha c}\rangle \langle \psi_{\alpha c}|.
 \end{equation}
 %
-Jeżeli dodamy to wyrażenie do oryginalnego potencjału $V$ dostaniemy wielkość nazywaną pseudopotencjałem $\tilde{V}=V+V^{R}$.
-W obszarze walencyjnym, pseudopotencjał $\tilde{V}$ pokrywa się z potencjałem $V$.  
-Silnie przyciagający potencjał atomowy w obszarze rdzenia jest znacznie osłabiony przez dodatnią wartość $V^R$.
-To umożliwia zbieżność pseudofunkcji falowej w bazie fal płaskich. 
+If we add this expression to the original potential $V$, we obtain a quantity called the pseudopotential $\tilde{V}=V+V^{R}$.
+In the valence region, the pseudopotential $\tilde{V}$ coincides with the potential $V$.  
+The strongly attractive atomic potential in the core region is significantly weakened by the positive value of $V^R$.
+This enables convergence of the pseudo-wave function in the plane wave basis. 
 
 
-\subsection{Pseudopotencjały zachowujące normę}
+\subsection{Norm-conserving pseudopotentials}
 
-Dobre pseudopotencjały mają charakter uniwersalny. Wygenerowane w określonej configuracji atomowej
-powinny zachowywać się jednakowo dobrze w każdym innym ukladzie atomowym. 
-W roku 1979, Hamann, Schl\"{u}ter i Chiang \cite{HSC} zaproponowali pseudopotencjały zachowujące normę, 
-które spełniały  nastepujące warunki:
+Good pseudopotentials have a universal character. Generated in a specific atomic configuration,
+they should behave equally well in any other atomic arrangement. 
+In 1979, Hamann, Schl\"{u}ter, and Chiang \cite{HSC} proposed norm-conserving pseudopotentials 
+that satisfied the following conditions:
 
 \begin{enumerate}
-{\item Prawdziwa funkcja i pseudofuncja falowa są równe, $\psi_i(r)=\tilde{\psi}_i(r)$, dla $r\geq r_c$.}
-{\item Ich energie własne dla elektronów walencyjnych powinny być równe.}
-{\item Ładunek prawdziwy i pseudoładunek zawarty w obszarze o promieniu $r\geq r_c$ powinny się pokrywać (zachowanie normy)
+{\item The true wave function and pseudo-wave function are equal, $\psi_i(r)=\tilde{\psi}_i(r)$, for $r\geq r_c$.}
+{\item Their eigenvalues for valence electrons should be equal.}
+{\item The true charge and pseudo-charge contained in a region with radius $r\geq r_c$ should coincide (norm conservation)
 %
 \begin{equation}
 \int_0^r dr r^2 |\psi_i(r)|^2=\int_0^r dr r^2 |\tilde{\psi}_i(r)|^2.
 \end{equation}}
 %
-{\item Pochodna logarytmiczna prawdziwej funkcji i pseudofunkcji są równe dla $r\geq r_c$ 
+{\item The logarithmic derivative of the true wave function and pseudo-wave function are equal for $r\geq r_c$ 
 %
 \begin{equation}
 \frac{1}{\psi_i(r)}\frac{d\psi_i(r)}{dr}=\frac{1}{\tilde{\psi}_i(r)}\frac{d\tilde{\psi}_i(r)}{dr}.
@@ -1617,114 +1617,114 @@ \subsection{Pseudopotencjały zachowujące normę}
 
 \end{enumerate} 
 %
-Z tych własności wynika również spełnienie warunku równych pochodnych po energii funkcji dokładnej i pseudofunkcji dla $r\geq r_c$,
-co dodatkowo wzmacnia przenośny charakter tych pseudopotencjałów.
-Metody generowania pseudopotencjałów zachowujących normę zostały opisane szczegółowo w wielu pracach \cite{HSC,BHS}.
+From these properties also follows the condition of equal energy derivatives of the exact wave function and pseudo-wave function for $r\geq r_c$,
+which additionally reinforces the transferable character of these pseudopotentials.
+Methods for generating norm-conserving pseudopotentials have been described in detail in many works \cite{HSC,BHS}.
 
-Ponieważ pseudopotencjały w obszarze rdzenia zależą od orbitalnej liczby kwantowej $l$, ich charakter nie jest lokalny.
-Wartość charakterystycznego promienia $r_c$ również zależy od $l$.
-Ogólnie można podzielić cały pseudopotencjał na część lokalną i nielokalną
+Since pseudopotentials in the core region depend on the orbital quantum number $l$, their character is not local.
+The value of the characteristic radius $r_c$ also depends on $l$.
+In general, the entire pseudopotential can be divided into a local and nonlocal part
 %
 \begin{equation}
 V_l(r)=V_{lok}(r)+\delta V_l(r).
 \end{equation}
 %
-Nielokalny charakter dotyczy tylko obszaru rdzenia, więc $\delta V_l(r)=0$ dla $r>r_c$
-i wszystkie dalekozasięgowe efekty zależą tylko od $V_{lok}(r)$.
-Możliwość wyboru promienia $r_c$ daje pewną swobodę przy konstrukcji pseudopotencjałów.
-Dokładniejsze i bardziej uniwersalne pseudopotencjały charakteryzują sie mniejszymi wartościami
-$r_c$. Jednak większa dokładnosć wymaga uwzględnienia większej ilości fal płaskich
-w rozwinięciu pseudofunkcji falowej. Takie pseudopotencjały nazywane są twardymi. Miękkie pseudopotencjały odznaczają się
-większymi wartościami $r_c$, mniejszą ilość fal płaskich i bardziej gładkim charakterem pseudofunkcji falowej. 
+The nonlocal character concerns only the core region, so $\delta V_l(r)=0$ for $r>r_c$
+and all long-range effects depend only on $V_{lok}(r)$.
+The possibility of choosing the radius $r_c$ gives some freedom in the construction of pseudopotentials.
+More accurate and more universal pseudopotentials are characterized by smaller values
+of $r_c$. However, greater accuracy requires including a larger number of plane waves
+in the expansion of the pseudo-wave function. Such pseudopotentials are called hard. Soft pseudopotentials are characterized by
+larger values of $r_c$, a smaller number of plane waves, and a smoother character of the pseudo-wave function. 
 
-\subsection{Pseudopotencjały ultramiękkie}
+\subsection{Ultrasoft pseudopotentials}
 
-W roku 1990, Vanderbilt zaproponował nowy rodzaj pseupotencjałów, nazywane ultramiękkimi ({\it ang. ultrasoft pseudopotentials} - US-PP), które nie zachowują normy \cite{Vanderbilt90}. W tym podejściu, pseudofunkcje falowe spełniają ugólnione równanie własne w postaci
+In 1990, Vanderbilt proposed a new type of pseudopotentials, called ultrasoft pseudopotentials (US-PP), which do not conserve the norm \cite{Vanderbilt90}. In this approach, the pseudo-wave functions satisfy a generalized eigenvalue equation in the form
 %
 \begin{equation}
 H|\tilde{\psi}_i\rangle=\varepsilon_iS|\tilde{\psi}_i\rangle,
 \end{equation}
 %
-gdzie Hamiltonian ma ogólną postać 
+where the Hamiltonian has the general form 
 %
 \begin{equation}
 H=-\frac{\hbar^2}{2m}\nabla^2+V_{lok}+\delta V_{NL}, 
 \end{equation}
-a $S$ jest operatorem przekrywania
+and $S$ is the overlap operator
 %
 \begin{equation}
 S=1+\sum_{i,j} Q_{ij}|\beta_i\rangle\langle\beta_j|,
 \end{equation}
 %
-który określa uogólniony warunek normalizacji pseudofunkcji falowych $\langle\tilde{\psi}_i|S|\tilde{\psi}_j\rangle=\delta_{ij}$.
-Macierz $Q_{ij}$ opisuje różnicę w normalizacji funkcji i pseudofunkcji falowych
+which determines the generalized normalization condition for the pseudo-wave functions $\langle\tilde{\psi}_i|S|\tilde{\psi}_j\rangle=\delta_{ij}$.
+The matrix $Q_{ij}$ describes the difference in normalization of the wave functions and pseudo-wave functions
 %
 \begin{equation}
 Q_{ij}=\int_0^{r_c} dr Q_{ij}(\bm{r})=\int_0^{r_c} dr r^2[\psi_i^*(\bm{r})\psi_j(\bm{r})-\tilde{\psi}_i^*(\bm{r})\tilde{\psi}_j(\bm{r})].
 \end{equation}
 %
-Zbiór lokalnych funkcji falowych
+The set of local wave functions
 %
 \begin{equation}
 |\beta_i\rangle=\sum_j (B^{-1})_{ij}|\chi_j\rangle,
 \end{equation}
 %
-gdzie funkcje 
+where the functions 
 %
 \begin{equation}
 |\chi_j\rangle=(\varepsilon_i+\frac{\hbar^2}{2m}\nabla^2-V_{loc})|\tilde{\psi}_j\rangle
 \end{equation}
 %
-zerują się w obszarze dla $r>r_c$. 
-Nielokalną część speudopotencjału otrzymuje się ze wzoru
+vanish in the region for $r>r_c$. 
+The nonlocal part of the pseudopotential is obtained from the formula
 %
 \begin{equation}
 \delta V_{NL}=\sum_{ij}D_{ij}|\beta_i\rangle\langle\beta_j|,
 \end{equation}
 %
-gdzie $D_{ij}=B_{ij}+\varepsilon_j Q_{ij}$. Całkowitą gęstość elektonów walencyjnych
-w danym punkcie przestrzeni otrzymujemy ze wzoru 
+where $D_{ij}=B_{ij}+\varepsilon_j Q_{ij}$. The total valence electron density
+at a given point in space is obtained from the formula 
 %
 \begin{equation}
 n_v(\bm{r})=\sum_i \tilde{\psi}_i^*(\bm{r})\tilde{\psi}_i(\bm{r})+\sum_{i,j}\sum_k\langle\beta_i|\tilde{\psi}_k\rangle\langle\tilde{\psi}_k|\beta_j\rangle Q_{ij}(\bm{r}),
 \end{equation}
 % 
-gdzie wyraz jest poprawką wynikającą z niezachowania normy przez pseudofunkcje falowe.
+where the second term is a correction resulting from the non-conservation of the norm by the pseudo-wave functions.
 
 
 \begin{figure}[h]
 \centering
 \includegraphics[scale=1]{usp.pdf}
-\caption{Porównanie dokładnej funkcji falowej $2p$ dla tlenu (linia ciągła) z pseudofunkcją zachowującą normę (linia kropkowana)
-i pseudofunkcją otrzymaną dla pseudopotancjału ultramiękkiego (linia przerywana). Rysunek z pracy \cite{Vanderbilt90}.}
+\caption{Comparison of the exact $2p$ wave function for oxygen (solid line) with a norm-conserving pseudo-wave function (dotted line)
+and a pseudo-wave function obtained for an ultrasoft pseudopotential (dashed line). Figure from \cite{Vanderbilt90}.}
 \label{fig:ups}
 \end{figure}
 
-Na rysunku \ref{fig:ups} pokazana jest dokładna funkcja falowa dla orbitalu $2p$ w tlenie oraz dwie pseudofunkcje otrzymane dla pseudopotencjału zachowujacego normę i pseudopotencjału ultramiękkiego. Promień obcięcia jest wyraźnie większy dla pseudopotencjału ultramiękkiego ($r_c\approx1.5$ a.u.) niż dla zachowującego normę ($r_c\approx0.5$ a.u.).
+Figure \ref{fig:ups} shows the exact wave function for the $2p$ orbital in oxygen and two pseudo-wave functions obtained for a norm-conserving pseudopotential and an ultrasoft pseudopotential. The cutoff radius is clearly larger for the ultrasoft pseudopotential ($r_c\approx1.5$ a.u.) than for the norm-conserving one ($r_c\approx0.5$ a.u.).
 
-\subsection{Metoda PAW}
+\subsection{PAW method}
 
-Omówione metody pseudopotencjałów pozwalaja efektywnie wyliczać wiele wielkości fizycznych bazując na pseudofukcjach falowych.
-W wielu przypadkach dokładniejsze obliczenia wymagają znajomości funkcji falowych elektronów walencyjnych
-również w obszarze rdzenia atomowego.
-W roku 1994, Bl\"{o}chl zaproponował nową metodę PAW ({\it ang. projector augmented-wave}),
-która łączy efektywność pseudopotencjałów i dokładność obliczeń porównywalną z metodami pełnego potencjału \cite{Blochl}. 
-Do opisu elektronów walencyjnych podejście to wykorzystuje również pseudofunkcje falowe $\tilde{\psi}_v$, które rozwijane są na fale płaskie i 
-w obszarze międzywęzłowym pokrywają sie z funkcjami dokładnymi $\psi_v$.
-Główną zaletą tej metody jest możliwość wyznaczenia dokładnej funkcji falowej poprzez transformację liniową pseudofukcji falowej
+The discussed pseudopotential methods allow efficient calculation of many physical quantities based on pseudo-wave functions.
+In many cases, more accurate calculations require knowledge of the wave functions of valence electrons
+also in the atomic core region.
+In 1994, Bl\"{o}chl proposed a new method, PAW (projector augmented-wave),
+which combines the efficiency of pseudopotentials with calculation accuracy comparable to all-electron methods \cite{Blochl}. 
+To describe valence electrons, this approach also uses pseudo-wave functions $\tilde{\psi}_v$, which are expanded in plane waves and 
+in the interstitial region coincide with the exact functions $\psi_v$.
+The main advantage of this method is the possibility of determining the exact wave function through a linear transformation of the pseudo-wave function
 %
 \begin{equation}
 |\psi_v\rangle=\mathcal{T}|\tilde{\psi}_v\rangle.
 \end{equation}
 % 
-Operator liniowy $\mathcal{T}$ działa w obszarach otoczonych sferą o promieniu $r_c$ wokół atomów w położeniach $\bm{R}_m$
+The linear operator $\mathcal{T}$ acts in regions enclosed by a sphere of radius $r_c$ around atoms at positions $\bm{R}_m$
 %
 \begin{equation}
 \mathcal{T}=1+\sum_m \mathcal{T}_m.
 \label{operator}
 \end{equation}
 % 
-Dokładna funkcja falowa i pseudofukcja rozwinięte są w obszarze atomowym na funkcje parcjalne $|\phi_m\rangle$ i $|\tilde{\phi}_m\rangle$
+The exact wave function and pseudo-wave function are expanded in the atomic region into partial waves $|\phi_m\rangle$ and $|\tilde{\phi}_m\rangle$
 %
 \begin{eqnarray}
 |\psi_v\rangle=\sum_m c_m |\phi_m\rangle, \label{eq1}\\
@@ -1732,41 +1732,41 @@ \subsection{Metoda PAW}
 \label{eq2}
 \end{eqnarray}
 %
-gdzie w obydwóch sumach występują te same współczynniki rozwinięcia $c_m$. Zatem te lokalne funkcje powiązane są tą samą
-transformacją
+where the same expansion coefficients $c_m$ appear in both sums. Therefore, these local functions are related by the same
+transformation
 %
 \begin{equation}
 |\phi_m\rangle=(1+\sum_m \mathcal{T}_m)|\tilde{\phi}_m\rangle.
 \end{equation}
 %
-Funkcje $|\phi_m\rangle$ są rozwiązaniami równania Schr\"{o}dingera dla dokładnego potencjału atomowego, które odpowiadają energiom $\varepsilon_m$
-i są ortogonalne do stanów rdzenia. Wskaźnik $m$ określa jednocześnie położenia atomów $\bm{R}$ i zbiór liczb kwantowych orbitali atomowych.
-Każdej parcjalnej funkcji dokładnej odpowiada jedna pseudofunkcja $|\tilde{\phi}_m\rangle$, z którą pokrywa się poza sferą o promieniu $r_c$.
-Oba rodzaje funkcji parcjalnych są funkcjami radialnymi, zdefiniowanymi na logarytmicznej siatce radialnej, przemnożonymi przez
-harmoniki sferyczne. 
+The functions $|\phi_m\rangle$ are solutions of the Schr\"{o}dinger equation for the exact atomic potential, corresponding to energies $\varepsilon_m$
+and orthogonal to core states. The index $m$ simultaneously determines the positions of atoms $\bm{R}$ and the set of quantum numbers of atomic orbitals.
+Each exact partial wave corresponds to one pseudo-wave function $|\tilde{\phi}_m\rangle$, with which it coincides outside a sphere of radius $r_c$.
+Both types of partial functions are radial functions, defined on a logarithmic radial grid, multiplied by
+spherical harmonics. 
 
-Odejmując stronami (\ref{eq1}) i (\ref{eq2}) dostajemy równanie
+Subtracting (\ref{eq1}) and (\ref{eq2}) term by term, we obtain
 %
 \begin{equation}
 |\psi_v\rangle=|\tilde{\psi}_v\rangle-\sum_m c_m |\tilde{\phi}_m\rangle + \sum_m c_m |\phi_m\rangle.
 \label{psiv}
 \end{equation}
 % 
-Aby operator $\mathcal{T}$ był liniowy współczynniki $c_m$ muszą być liniowymi funkcjonałami pseudofunkcji falowej $|\tilde{\psi}_v\rangle$
+For the operator $\mathcal{T}$ to be linear, the coefficients $c_m$ must be linear functionals of the pseudo-wave function $|\tilde{\psi}_v\rangle$
 %
 \begin{equation}
 c_m=\langle\tilde{p}_m|\tilde{\psi}_v\rangle,
 \label{coef}
 \end{equation}
 %
-gdzie $\langle\tilde{p}_m|$ są funkcjami rzutowymi dualnymi względem funkcji parcjalnych 
+where $\langle\tilde{p}_m|$ are projector functions dual to the partial waves 
 $\langle\tilde{p}_m|\tilde{\phi}_n\rangle=\delta_{m,n}$.
-Dla każdej funkcji $|\tilde{\phi}_m\rangle$ mamy jedną funkcja rzutową $\langle\tilde{p}_m|$
-i dla nich spełniony jest warunek $\sum_m |\tilde{\phi}_m\rangle\langle\tilde{p}_m|=1$.
-Funkcje rzutowe są funkcjami radialnymi pomnożonymi przez harmoniki sferyczny, a następnie przetransformowane do bazy fal płaskich.
-Są przypisane do położeń atomów, ale nie zależą od potencjału atomowego.
+For each function $|\tilde{\phi}_m\rangle$ we have one projector function $\langle\tilde{p}_m|$
+and for them the condition $\sum_m |\tilde{\phi}_m\rangle\langle\tilde{p}_m|=1$ is satisfied.
+The projector functions are radial functions multiplied by spherical harmonics, and then transformed to the plane wave basis.
+They are assigned to atomic positions but do not depend on the atomic potential.
 %
-Wstawiając (\ref{coef}) do (\ref{psiv}) otrzymujemy
+Inserting (\ref{coef}) into (\ref{psiv}) we obtain
 %
 \begin{equation}
 |\psi_v\rangle=|\tilde{\psi}_v\rangle+\sum_m (|\phi_m\rangle-|\tilde{\phi}_m\rangle)\langle\tilde{p}_m|)\tilde{\psi}_v\rangle
@@ -1774,43 +1774,43 @@ \subsection{Metoda PAW}
 \label{psiv2}
 \end{equation}
 %
-gdzie wyrażenie w nawiasie kwadratowym jest wprowadzonym wcześniej operatorem liniowym
+where the expression in square brackets is the linear operator introduced earlier
 %
 \begin{equation}
 \mathcal{T}=1+\sum_m \mathcal{T}_m=1+\sum_m (|\phi_m\rangle-|\tilde{\phi}_m\rangle)\langle\tilde{p}_m|.
 \end{equation}
 %
 
-Funkcje falowe rdzenia $|\psi_c\rangle$, podobnie do funkcji walencyjnych, są rozłożone na trzy składowe
+The core wave functions $|\psi_c\rangle$, similarly to valence functions, are decomposed into three components
 %
 \begin{equation}
 |\psi_c\rangle=|\tilde{\psi}_c\rangle+|\phi_c\rangle-|\tilde{\phi}_c\rangle,
 \end{equation}
 %
-gdzie kolejne wyrazy odpowiadają pseudofunkcji elektronów rdzenia, która pokrywa się z dokładną funkcją dla $r>r_c$,
-lokalnej (parcjalnej) funkcji rdzenia i lokalnej pseudofukcji rdzenia. Obydwie lokalne funcje wyrażone są jako funkcje radialne
-pomnożone przez harmoniki sferyczne.
+where the successive terms correspond to the pseudo-wave function of core electrons, which coincides with the exact function for $r>r_c$,
+the local (partial) core function, and the local core pseudo-wave function. Both local functions are expressed as radial functions
+multiplied by spherical harmonics.
 
-Średnia z operatora $A$ może być wyznaczona przy pomocy dokładnych funkcji lub pseudofunkcji
+The expectation value of operator $A$ can be determined using exact functions or pseudo-wave functions
 %
 \begin{equation}
 \langle A \rangle = \sum_n f_n \langle \psi_n|A|\psi_n\rangle = \sum_n f_n \langle \tilde{\psi}_n|\tilde{A}|\tilde{\psi}_n\rangle,
 \end{equation}
 % 
-gdzie $f_n$ określa obsadzenie stanu $n$, a $\tilde{A}$ jest pseudooperatorem, który można otrzymać transformując operator $A$
+where $f_n$ determines the occupation of state $n$, and $\tilde{A}$ is a pseudo-operator that can be obtained by transforming operator $A$
 %
 \begin{equation}
 \tilde{A}=\mathcal{T}^{\dagger} A \mathcal{T}= A +\sum_{i,j} |\tilde{p}_i\rangle (\langle \phi_i |A|\phi_j\rangle - \langle \tilde{\phi}_i |A|\tilde{\phi}_j\rangle ) \langle\tilde{p}_j|.
 \end{equation}
 %
-Przykładowo, stosując te wyrażenia do operatora gęstości $n=|\bm{r}\rangle\langle\bm{r}|$,
-możemy wyrazić gęstość elektronową następującym wyrażeniem
+For example, applying these expressions to the density operator $n=|\bm{r}\rangle\langle\bm{r}|$,
+we can express the electron density with the following expression
 %
 \begin{equation}
 n(\bm{r})=\tilde{n}(\bm{r})+n^1(\bm{r})-\tilde{n}^1(\bm{r}),
 \end{equation}
 %
-gdzie
+where
 %
 \begin{equation}
 \tilde{n}(\bm{r})=\sum_n f_n \langle \tilde{\psi}_n|\bm{r}\rangle\langle \bm{r}| \tilde{\psi}_n\rangle=\sum_n f_n |\tilde{\psi_n}(\bm{r})|^2,
@@ -1824,65 +1824,65 @@ \subsection{Metoda PAW}
 \tilde{n}^1(\bm{r})=\sum_{n,i,j} f_n \langle \tilde{\psi}_n|\tilde{p}_i\rangle\langle\tilde{\phi}_i|\bm{r}\rangle\langle \bm{r}|\tilde{\phi}_j\rangle\langle\tilde{p}_j |\tilde{\psi}_n\rangle.
 \end{equation}
 %
-W powyższych wzorach sumowanie obejmuje zarówno stany walencyjne, jak i stany rdzenia. 
+In the above formulas, the summation includes both valence states and core states. 
 
-Metoda PAW należy obecnie do najdokładniejszych i najbardziej efektywnych metod. Została zaimplementowana
-w takich programach, jak Vienna Ab Initio Simulation Package (VASP) \cite{Vasp,PawVasp} i Quantum Espresso \cite{QE}.
+The PAW method is currently one of the most accurate and efficient methods. It has been implemented
+in programs such as the Vienna Ab Initio Simulation Package (VASP) \cite{Vasp,PawVasp} and Quantum Espresso \cite{QE}.
 
 
-\section{Zlokalizowane orbitale}
+\section{Localized orbitals}
 
-\subsection{Metoda ciasnego wiązania}
+\subsection{Tight-binding method}
 
-W odróżnieniu od metod pseudopotencjału, które wykorzystują bazę fal płaskich, metody opisane w tym rozdziale
-stosują lokalne orbitale, które powiązane są z danymi atomami lub centrowane są w położeniach atomów.
-Opis przy pomocy lokalnych orbitali nadaje się dobrze do układów, gdzie występują zlokalizowane stany
-elektronowe, ale możliwy jest ruch elektronów poprzez przeskoki do sąsiednich atomów.
-Najprostszym opisem takich układów jest model ciasnego wiązania.
-Lokalne orbitale atomowe możemy zapisać w postaci $\phi_{\alpha}(\bm{r}-\bm{R}_{\alpha})$, 
-gdzie wektor $\bm{R}_{\alpha}$ oznacza położenie danego atomu. Wskaźnik $\alpha$ oznacza tutaj wszystkie liczby kwantowe, 
-które charakteryzują dany orbital ($\alpha=n,l,m$). Zakładamy dla uproszczenia, że z danym atomem związny jest tylko
-jeden orbital, ale model ciasnego wiązania można łatwo uogólnić na dowolną liczbę orbitali.
-Hamiltonian możemy zapisać jako
+In contrast to pseudopotential methods, which use a plane wave basis, the methods described in this section
+employ local orbitals that are associated with given atoms or centered at atomic positions.
+The description using local orbitals is well suited to systems where localized electronic
+states occur, but electron motion is possible through hops to neighboring atoms.
+The simplest description of such systems is the tight-binding model.
+Local atomic orbitals can be written as $\phi_{\alpha}(\bm{r}-\bm{R}_{\alpha})$, 
+where the vector $\bm{R}_{\alpha}$ denotes the position of a given atom. The index $\alpha$ here denotes all quantum numbers 
+that characterize a given orbital ($\alpha=n,l,m$). For simplicity, we assume that only
+one orbital is associated with a given atom, but the tight-binding model can easily be generalized to any number of orbitals.
+The Hamiltonian can be written as
 %
 \begin{equation}
 H=-\frac{\hbar^2}{2m}\nabla^2+\sum_{\alpha}V_{\alpha}(\bm{r}-\bm{R}_{\alpha}),
 \end{equation}
 %
-gdzie $V_{\alpha}$ jest potencjałem atomowym wokół położenia $\bm{R}_{\alpha}$.
-Elementy macierzowe hamiltonianu możemy wyznaczyć ze wzoru
+where $V_{\alpha}$ is the atomic potential around position $\bm{R}_{\alpha}$.
+The matrix elements of the Hamiltonian can be determined from the formula
 %
 \begin{equation}
 H_{\alpha\beta}=\int d\bm{r} \phi_{\alpha}^*(\bm{r}-\bm{R}_{\alpha})H\phi_{\beta}(\bm{r}-\bm{R}_{\beta}).
 \label{matrix}
 \end{equation}
 % 
-Ponieważ orbitale atomowe należące do różnych atomów nie są wzajemnie ortogonalne
-wprowadza się również macierz przekrywania
+Since atomic orbitals belonging to different atoms are not mutually orthogonal,
+an overlap matrix is also introduced
 %
 \begin{equation}
 S_{\alpha\beta}=\int d\bm{r} \phi_{\alpha}^*(\bm{r}-\bm{R}_{\alpha})\phi_{\beta}(\bm{r}-\bm{R}_{\beta}).
 \end{equation}
 %
-Elementy diagonalne hamiltonianu $\varepsilon_{\alpha}=H_{\alpha\alpha}$ określają lokalną energię elektronu w danym orbitalu $\alpha$.
-Natomiast elementy pozadiagonalne $t_{\alpha\beta}=H_{\alpha\beta}$ nazywane są całkami przeskoku i określają prawdopodobieństwo
-przeskoku elektronu między dwoma atomami.
+The diagonal elements of the Hamiltonian $\varepsilon_{\alpha}=H_{\alpha\alpha}$ determine the local energy of the electron in a given orbital $\alpha$.
+The off-diagonal elements $t_{\alpha\beta}=H_{\alpha\beta}$ are called hopping integrals and determine the probability
+of electron hopping between two atoms.
 
-Aby przetransformować elementy macierzowe hamiltonianu i macierzy $S$ do przestrzeni odwrotnej
-wprowadzamy bazę, która odpowiada wektorom falowym $\bm{k}$
+To transform the matrix elements of the Hamiltonian and matrix $S$ to reciprocal space,
+we introduce a basis that corresponds to wave vectors $\bm{k}$
 %
 \begin{equation}
 \phi_{\bm{k}\alpha}(\bm{r})=A_{\alpha\bm{k}}\sum_n e^{i\bm{k}\bm{T}_n}\phi_{\alpha}[\bm{r}-(\bm{R}_{\alpha}+\bm{T}_n)],
 \end{equation}
 %
-gdzie $A_{\bm{k}\alpha}$ są czynnikami normalizacji, a $\bm{T}_n$ są wektorami translacji sieci krystalicznej.
-W tej bazie możemy zapisać funkcję falową, która spełnia twierdzenie Blocha
+where $A_{\bm{k}\alpha}$ are normalization factors, and $\bm{T}_n$ are translation vectors of the crystal lattice.
+In this basis, we can write the wave function that satisfies Bloch's theorem
 %
 \begin{equation}
 \psi_{\bm{k}i}=\sum_{\alpha} c_{i\alpha} \phi_{\bm{k}\alpha}(\bm{r}),
 \end{equation}
 %
-oraz elementy macierzowe hamiltonianu i macierzy $S$
+as well as matrix elements of the Hamiltonian and matrix $S$
 %
 \begin{equation}
 H_{\alpha\beta}(\bm{k})=\int d\bm{r} \phi_{\alpha\bm{k}}^*(\bm{r})H\phi_{\beta\bm{k}}(\bm{r})=\sum_n e^{i\bm{k}\bm{T}_n} H_{\alpha\beta},
@@ -1892,338 +1892,337 @@ \subsection{Metoda ciasnego wiązania}
 S_{\alpha\beta}(\bm{k})=\int d\bm{r} \phi_{\alpha\bm{k}}^*(\bm{r})\phi_{\beta\bm{k}}(\bm{r})=\sum_n e^{i\bm{k}\bm{T}_n} S_{\alpha\beta}.
 \end{equation}
 %
-Równanie na energie własne $\varepsilon_i(\bm{k})$ i współczynniki rozwinięcia funkcji falowej $c_{i,\alpha}(\bm{k})$ przyjmuje postać
+The equation for eigenvalues $\varepsilon_i(\bm{k})$ and expansion coefficients of the wave function $c_{i,\alpha}(\bm{k})$ takes the form
 %
 \begin{equation}
 \sum_{\beta}[H_{\alpha\beta}(\bm{k})-\varepsilon_i(\bm{k})S_{\alpha\beta}(\bm{k})]c_{i\beta}(\bm{k})=0.
 \label{ham_loc} 
 \end{equation}
 %
-Dla orbitali wzajemnie ortogonalnych, mamy $S_{\alpha\beta}=\delta_{\alpha\beta}$ i równanie (\ref{ham_loc})
-przyjmuje postać taką samą jak równanie w bazie fal płaskich (\ref{ham_rec}).
+For mutually orthogonal orbitals, we have $S_{\alpha\beta}=\delta_{\alpha\beta}$ and equation (\ref{ham_loc})
+takes the same form as the equation in the plane wave basis (\ref{ham_rec}).
 
-\subsection{Funkcje Wanniera}
+\subsection{Wannier functions}
 \label{sec:wannier}
 
-W metodzie ciasnego wiązania często wykorzystuje się bazę funkcji Waniera, które są zlokalizowanymi funkcjami, 
-centrowanymi na położeniach atomowych $\bm{R}_n$. 
-Funkcja Wanniera zdefiniowana jest jako transformata Fouriera funkcji Blocha powiązanej z danym pasmem $j$
+In the tight-binding method, a basis of Wannier functions is often used, which are localized functions 
+centered at atomic positions $\bm{R}_n$. 
+The Wannier function is defined as the Fourier transform of the Bloch function associated with a given band $j$
 %
 \begin{equation}
 w_j(\bm{r}-\bm{R}_n)=\frac{1}{\sqrt{N}}\sum_{\bm{k}}e^{-i\bm{k}\bm{R}_n}\psi_{\bm{k}j}(\bm{r}). 
 \end{equation}
 %
-Spełniona jest również transformata odwrotna, która pozwala wyrazić funkcje Blocha przez funkcje
-Wanniera
+The inverse transform is also satisfied, which allows expressing Bloch functions through Wannier
+functions
 %
 \begin{equation}
 \psi_{\bm{k}j}(\bm{r})=\frac{1}{\sqrt{N}}\sum_n e^{i\bm{k}\bm{R}_n} w_j(\bm{r}-\bm{R}_n).
 \end{equation}
 %
-Funkcje Wanniera dla różnych położeń atomowych są ortogonalne
+Wannier functions for different atomic positions are orthogonal
 %
 \begin{multline}
 \int d\bm{r} w_j^*(\bm{r}-\bm{R}_n)w_j(\bm{r}-\bm{R}_m)=\frac{1}{N}\sum_{\bm{k},\bm{k}'}\int d\bm{r}e^{i(\bm{k}\bm{R}_n-\bm{k}'\bm{R}_m)}\psi_{\bm{k}j}(\bm{r})\psi_{\bm{k}'j}(\bm{r}) \\
 =\frac{1}{N}\sum_{\bm{k}}e^{i\bm{k}(\bm{R}_n-\bm{R}_m)}=\delta_{mn},
 \end{multline}
 %
-gdzie wykorzystaliśmy ortogonalność funkcji Blocha. Spełniony jest również warunek zupełności
+where we used the orthogonality of Bloch functions. The completeness condition is also satisfied
 %
 \begin{equation}
 \sum_n w_j^*(\bm{r}-\bm{R}_n)w_j(\bm{r}'-\bm{R}_n)=\delta^3(\bm{r}-\bm{r}').
 \end{equation}
-Zatem funkcje Wanniera tworzą zbiór zlokalizowanych funkcji bazowych.
-Energie pasma $j$ dla funkcji Blocha w tej bazie może być wyznaczona ze wzoru
+Thus, Wannier functions form a set of localized basis functions.
+The energy of band $j$ for Bloch functions in this basis can be determined from the formula
 %
 \begin{equation}
 \varepsilon_{\bm{k}j}=\int d\bm{r} \psi_{\bm{k}j}^*(\bm{r})H\psi_{\bm{k}j}(\bm{r})
 =\frac{1}{N}\int d\bm{r}\sum_n w_j^*(\bm{r}-\bm{R}_n)e^{-i\bm{k}\bm{R}_n}H\sum_m w_j(\bm{r}-\bm{R}_m)e^{i\bm{k}\bm{R}_m}.
 \end{equation}
 %
-Sumowanie po wskaźnikach $n$ i $m$ możemy rozdzielić na dwa wyrazy dla $n=m$ i $n\neq m$
+The summation over indices $n$ and $m$ can be separated into two terms for $n=m$ and $n\neq m$
 %
 \begin{multline}
 \varepsilon_{\bm{k}j}=\frac{1}{N}\sum_n\int d\bm{r} w_j^*(\bm{r}-\bm{R}_n)Hw_j(\bm{r}-\bm{R}_n)
 \\ +\frac{1}{N}\sum_{n\neq m}e^{i\bm{k}(\bm{R}_m-\bm{R}_n)}\int d\bm{r} w_j^*(\bm{r}-\bm{R}_n)Hw_j(\bm{r}-\bm{R}_m).
 \end{multline}
 %
-Wykorzystując oznaczenia, które wprowadziliśmy dla elementów macierzowych hamiltonianu (\ref{matrix})
-dostajemy
+Using the notation we introduced for the Hamiltonian matrix elements (\ref{matrix})
+we obtain
 %
 \begin{equation}
 \varepsilon_{\bm{k}j}=\frac{1}{N}\sum_n \varepsilon_n+\frac{1}{N}\sum_{n\neq m}e^{i\bm{k}(\bm{R}_m-\bm{R}_n)}t_{nm}.
 \end{equation}
 %
-Wzór ten pozwala wyznaczyć strukturę pasmową w ramach metody ciasnego wiązania.
-Jeżeli ograniczymy się do modelu jednopasmowego i do przeskoków tylko między najbliższymi sąsiadami
-możemy dostać uproszczony model z jednym parametrem przeskoku $t$.
-Dla prostej struktury kubicznej dostajemy
+This formula allows determining the band structure within the tight-binding method.
+If we limit ourselves to a single-band model and to hops only between nearest neighbors,
+we can obtain a simplified model with a single hopping parameter $t$.
+For a simple cubic structure, we obtain
 %
 \begin{equation}
 \varepsilon_{\bm{k}j}=\varepsilon_0+\frac{t}{N}\sum_{n}e^{i\bm{k}(\bm{R}_{n+1}-\bm{R}_n)}=\varepsilon_0+2t(cosk_xa+cosk_ya+cosk_za).
 \label{band}
 \end{equation}
 %
-gdzie $a$ jest stałą sieci, a $\varepsilon_0$ jest energią elektronu zlokalizowanego w stanie atomowym.
-Szerokość pasma w tym przybliżeniu dana jest wzorem $w=2z|t|$, gdzie $z$ jest liczbą najbliższych sąsiadów.  
+where $a$ is the lattice constant, and $\varepsilon_0$ is the energy of an electron localized in an atomic state.
+The bandwidth in this approximation is given by the formula $w=2z|t|$, where $z$ is the number of nearest neighbors.  
 
-Metoda ciasnego wiązania, która wykorzystuje bazę orbitali atomowych lub funkcji Wanniera,
-często wykorzystywana jest w obliczeniach modelowych. W ramach tego podejścia można łatwo uwzględnić lokalne oddziaływania
-kulombowskie lub oddziaływania parujące w modelach nadprzewodników wysokotemperaturowych.
-Lokalne energie i całki przeskoku mogą być wyznaczone przez dofitowanie struktury elektronowej
-z modelu ciasnego wiązania do wyników eksperymentalnych lub do struktury pasmowej wyznaczonej z obliczeń DFT.
-W tym drugim przypadku stosuje się najczęściej bazę maksymalnie zlokalizowanych funkcji Wanniera.
-Wykorzystuje się swobodę wyboru fazy dla funkcji Blocha, którą możemy przemnożyć przez czynnik $e^{i\theta(\bm{k})}$,
-gdzie $\theta(\bm{k})$ jest dowolną funkcją rzeczywistą, bez wpływu na energie stanów elektronowych. 
-Taki czynnik ma jednak wpływ na kształt funkcji Wanniera, w szczególności na ich zasięg przestrzenny. 
-Można tak dobrać czynnik fazowy, aby funkcja Wanniera $w_i(\bm{r}-\bm{R}_n)$ zlokalizowana była
-wokół punktu $\bm{R}_n$ i szybko zanikała z odległością. 
+The tight-binding method, which uses a basis of atomic orbitals or Wannier functions,
+is often used in model calculations. Within this approach, local Coulomb
+interactions or pairing interactions in high-temperature superconductor models can easily be included.
+Local energies and hopping integrals can be determined by fitting the electronic structure
+from the tight-binding model to experimental results or to the band structure determined from DFT calculations.
+In the latter case, a basis of maximally localized Wannier functions is most often used.
+The freedom to choose the phase for Bloch functions is utilized, which can be multiplied by a factor $e^{i\theta(\bm{k})}$,
+where $\theta(\bm{k})$ is an arbitrary real function, without affecting the energies of electronic states. 
+However, such a factor affects the shape of Wannier functions, particularly their spatial extent. 
+The phase factor can be chosen so that the Wannier function $w_i(\bm{r}-\bm{R}_n)$ is localized
+around point $\bm{R}_n$ and decays rapidly with distance. 
 
 \begin{figure}[h]
 \centering
 \includegraphics[scale=1]{FeSe.pdf}
-\caption{Porównanie struktury pasmowej otrzymanej w ramach modelu ciasnego wiązania i obliczeń DFT dla {\bf a.} CeCoIn$_5$ i {\bf b.} FeSe.
-Rysunek z pracy \cite{FeSe}.}
+\caption{Comparison of band structures obtained within the tight-binding model and DFT calculations for {\bf a.} CeCoIn$_5$ and {\bf b.} FeSe.
+Figure from \cite{FeSe}.}
 \label{fig:fese}
 \end{figure}
 
-Na rysunku \ref{fig:fese} pokazano przykłady struktur pasmowych dla dwóch związków, CeCoIn$_5$ i FeSe, wyliczonych w modelu ciasnego wiązania z bazą
-maksymalnie zlokalizowanych funkcji Wanniera. 
-Parametry modelu wyznaczono przez dofitowanie energii stanów elektronowych, dla każdego wektora falowego $\bm{k}$, 
-do struktur pasmowych otrzymanych w ramach obliczeń DFT \cite{FeSe}.
-Obliczenia wykonano programem Quantum Espresso \cite{QE}, stosując funkcjonał GGA \cite{PW91} w ramach metody PAW \cite{Blochl}.  
-Wyznaczone modele ciasnego wiązania zostały następnie wykorzystane do wyliczenia podatności tworzenia par Coopera
-i zbadania własności nadprzewodzących obydwóch materiałów \cite{FeSe}.
+Figure \ref{fig:fese} shows examples of band structures for two compounds, CeCoIn$_5$ and FeSe, calculated in the tight-binding model with a basis
+of maximally localized Wannier functions. 
+The model parameters were determined by fitting the electronic state energies, for each wave vector $\bm{k}$, 
+to the band structures obtained from DFT calculations \cite{FeSe}.
+The calculations were performed with the Quantum Espresso program \cite{QE}, using the GGA functional \cite{PW91} within the PAW method \cite{Blochl}.  
+The determined tight-binding models were then used to calculate the susceptibility to Cooper pair formation
+and investigate the superconducting properties of both materials \cite{FeSe}.
  
 
-\subsection{Funkcje Gaussa}
+\subsection{Gaussian functions}
 
-Lokalne orbitale można również wykorzystać jako bazę do rozwiniecia funkcji falowej w ramach samozgodnej procedury rozwiązywania równań Kohna-Shama.
-Jeżeli funkcjami bazowymi są orbitale atomowe mówimy o metodzie liniowych kombinacji orbitali atomowych ({\it ang. linear combination
-of atomic orbitals} - LCAO). Zwykle jednak stosuje się atomo-podobne orbitale, których postać funkcyjna ułatwia implementację
-i przyspiesza obliczenia. Do najczęściej stosowanych należą orbitale typu Slatera \cite{STO1,STO2} i funkcje typu Gaussa (gaussiany) \cite{GTO}.
-Te drugie są szczególnie wygodne ponieważ całki z funkcji Gaussa można wyliczyć analitycznie.
-Naturalną reprezentacją dla gaussianów jest układ współrzędnych biegunowych, ale wygodniejszy do wyznaczania elementów macierzowych hamiltonianu jest układ kartezjański, w którym te funkcje mają postać
+Local orbitals can also be used as a basis for expanding the wave function within the self-consistent procedure for solving the Kohn-Sham equations.
+If the basis functions are atomic orbitals, we speak of the linear combination of atomic orbitals (LCAO) method. Usually, however, atom-like orbitals are used, whose functional form facilitates implementation
+and speeds up calculations. The most commonly used are Slater-type orbitals \cite{STO1,STO2} and Gaussian-type functions (Gaussians) \cite{GTO}.
+The latter are particularly convenient because integrals of Gaussian functions can be calculated analytically.
+The natural representation for Gaussians is the polar coordinate system, but more convenient for determining Hamiltonian matrix elements is the Cartesian system, in which these functions have the form
 %
 \begin{equation}
 G_{ijk}(\bm{r},\alpha,\bm{R}_n)=N(x-x_n)^i(y-y_n)^j(z-z_n)^ke^{-\alpha (\bm{r}-\bm{R}_n)^2},
 \end{equation}
 %
-gdzie $\alpha$ jest parametrem wariacyjnym, który umożliwia optymalizację bazy dla konkretnych atomów, $\bm{R}_n=(x_n,y_n,z_n)$ jest wektorem określającym centrowanie danej funkcji, a $N$ jest czynnikiem normalizacyjnym.
-Spełniona jest ponadto zależność $l=i+j+k$, gdzie $l$ jest orbitalną liczbą kwantową.
-W wyniku iloczynu dwóch fukcji typu $s$ ($l=0$) centrowanych w punktach $\bm{R}_n$ i $\bm{R}_m$ dostajemy
+where $\alpha$ is a variational parameter that enables basis optimization for specific atoms, $\bm{R}_n=(x_n,y_n,z_n)$ is a vector determining the center of a given function, and $N$ is a normalization factor.
+Additionally, the relation $l=i+j+k$ is satisfied, where $l$ is the orbital quantum number.
+The product of two functions of type $s$ ($l=0$) centered at points $\bm{R}_n$ and $\bm{R}_m$ gives
 %
 \begin{equation}
 G_s(\bm{r},\alpha,\bm{R}_n)G_s(\bm{r},\beta,\bm{R}_m)=e^{-\gamma(\bm{R}_n-\bm{R}_m)^2}G_s(\bm{r},\kappa,\bm{R}_p),
 \end{equation}
 %
-gdzie 
+where 
 %
 \begin{equation}
 \kappa=\alpha+\beta, \quad \gamma=\frac{\alpha\beta}{\alpha+\beta}, \quad \bm{R}_p=\frac{\alpha\bm{R}_n+\beta\bm{R}_m}{\alpha+\beta},
 \end{equation}
 %
-gdzie $\bm{R}_p$ jest wektorem centrowania wynikowej funkcji Gaussa.
-W ogólnym przypadku iloczyn dwóch gaussianów można zapisać w formie
+where $\bm{R}_p$ is the center vector of the resulting Gaussian function.
+In the general case, the product of two Gaussians can be written in the form
 %
 \begin{multline}
 G_{i_1j_1k_1}(\bm{r},\alpha,\bm{R}_n)G_{i_2j_2k_2}(\bm{r},\beta,\bm{R}_m) = Ne^{-\gamma(\bm{R}_n-\bm{R}_m)^2}e^{-\kappa(\bm{r}-\bm{R}_p)^2} \\
 \times (x-x_n)^{i_1}(x-x_m)^{i_2}(y-y_n)^{j_1}(y-y_m)^{j_2}(z-z_n)^{k_1}(z-z_m)^{k_2}.
 \end{multline}
 %
-Te wzory pokazują, ze iloczyn funkcji Gaussa jest również funkcją Gaussa. Jest to główna zaleta tej bazy umożliwiająca
-analityczne wyliczenie złożonych całek zawierajacych kilka funkcji bazowych i w ten sposób znaczne przyspieszenie rachunków.
-W odróżnieniu od metod wykorzystujących fale płaskie, metoda funkcji Gaussa wymaga odpowiedniego wybrania funkcji bazowych
-dla danego układu atomowego. Metoda ta stosowana jest głównie przez chemików, a najbardziej popularnym programem jest GAUSSIAN.
-Głównym twórca tego programu jest John Pople, który w roku 1998 otrzymał wspólnie z Walterem Kohnem Nagrodę Nobla w dziedzinie chemii.
+These formulas show that the product of Gaussian functions is also a Gaussian function. This is the main advantage of this basis, enabling
+analytical calculation of complex integrals containing several basis functions and thus significant acceleration of calculations.
+In contrast to methods using plane waves, the Gaussian function method requires appropriate selection of basis functions
+for a given atomic system. This method is used mainly by chemists, and the most popular program is GAUSSIAN.
+The main creator of this program is John Pople, who in 1998 received the Nobel Prize in Chemistry together with Walter Kohn.
 
-\section{Stowarzyszone fale i atomowe sfery}
+\section{Augmented waves and atomic spheres}
 
-Ostatnia grupa metod obliczeniowych łączy główne cechy dwóch poprzednich: pseudopotencjałów i lokalnych orbitali. 
-Do rozwinięcia funkcji falowych stosowane są dwie różne bazy w zależności od położenia w krysztale.
-Wokół rdzeni atomowych naturalnym wyborem sa lokalne funkcje, które można otrzymać z rozwiązania
-równania Schr\"{o}dingera w potencjale sferyczno-symetrycznym. W obszarze miedzywezłowym
-stosuje się rozwinięcie w bazie funkcji, które umożliwiają szybką zbieżność rachunków (np. fale płaskie).
-Centralnym problemem stają sie odpowiednie warunki brzegowe, które zapewniają ciągłość funkcji falowej i jej pochodnej 
-na granicy między tymi obszarami.
-Wszystkie metody omawiane w tym rozdziale stosowały na początku uproszczony potencjał typu {\it muffin-tin}, który w obszarze rdzenia
-ma charakter potencjału atomowego, a w obszarze międzywęzłowym jego wartość jest stała.
-Obecnie podejścia te zostały zaimplementowane w ramach procedury samozgodnej i zaliczane są do metod pełnego potencjału. 
+The last group of computational methods combines the main features of the two previous ones: pseudopotentials and local orbitals. 
+Two different bases are used for expanding wave functions depending on the position in the crystal.
+Around atomic cores, the natural choice is local functions, which can be obtained from solving
+the Schr\"{o}dinger equation in a spherically symmetric potential. In the interstitial region,
+an expansion in a basis of functions that enables rapid convergence of calculations is used (e.g., plane waves).
+The central problem becomes appropriate boundary conditions that ensure continuity of the wave function and its derivative 
+at the boundary between these regions.
+All methods discussed in this section initially used a simplified muffin-tin potential, which in the core region
+has the character of an atomic potential, while in the interstitial region its value is constant.
+Currently, these approaches have been implemented within the self-consistent procedure and are classified as all-electron methods. 
 
-\subsection{Metoda KKR}
+\subsection{KKR method}
 
-W dwóch pracach, Korringa \cite{K47} oraz Kohn i Rostoker \cite{KR54} (KKR) zaproponowali metodę, która nazywana jest również metodą funcji Greena
-lub metodą wielokrotnego rozpraszania.
-Pierwsze obliczenia stałych sieci i modułów sztywności wykonano właśnie z wykorzystaniem tej metody \cite{Moruzzi77}. Była to przełomowa praca, która 
-pokazała możliwość zastosowania DFT do badania własności materiałowych.  
+In two works, Korringa \cite{K47} and Kohn and Rostoker \cite{KR54} (KKR) proposed a method that is also called the Green's function method
+or the multiple scattering method.
+The first calculations of lattice constants and elastic moduli were performed using this method \cite{Moruzzi77}. This was a groundbreaking work that 
+showed the possibility of applying DFT to study material properties.  
 
-Podstawową wielkością stosowaną tutaj jest funkcja Greena $G(E,\bm{r},\bm{r}')$, która opisuje propagację elektronu o energii $E$ między punktami $\bm{r}$ i $\bm{r}'$. Proces przemieszczania się elektronu można podzielić na swobodny ruch w obszarze międzywęzłowym i rozpraszanie na sferach atomowych
-w wyniku oddziaływania elektronu z potencjałem rdzenia atomowego.
-Dla elektronu opisanego jednocząstkowym równaniem Schr\"{o}dingera
+The basic quantity used here is the Green's function $G(E,\bm{r},\bm{r}')$, which describes the propagation of an electron with energy $E$ between points $\bm{r}$ and $\bm{r}'$. The process of electron movement can be divided into free motion in the interstitial region and scattering on atomic spheres
+as a result of electron interaction with the atomic core potential.
+For an electron described by the single-particle Schr\"{o}dinger equation
 %
 \begin{equation}
 [-\frac{\hbar^2}{2m}\nabla^2+V(\bm{r})]\psi_i(\bm{r})=\varepsilon_i\psi_i(\bm{r}),
 \end{equation}
 %
-funkcja Greena spełnia równanie
+the Green's function satisfies the equation
 %
 \begin{equation}
 [-\frac{\hbar^2}{2m}\nabla^2+V(\bm{r})-E]G(E,\bm{r}-\bm{r}')=\delta(\bm{r}-\bm{r}'),
 \end{equation}
 %
-które posiada rozwiązanie w formie
+which has a solution in the form
 %
 \begin{equation}
 G(E,\bm{r},\bm{r}')=\sum_i\frac{\psi_i(\bm{r})\psi_i^*(\bm{r}')}{E-\varepsilon_i}.
 \end{equation}
 %
-Dla nieoddziałujących elektronów mamy $V(\bm{r})=0$ i funkcja Greena ma postać
+For non-interacting electrons, we have $V(\bm{r})=0$ and the Green's function has the form
 %
 \begin{equation}
 G_0(E,|\bm{r}-\bm{r}'|)= \frac{m}{2\pi\hbar^2}\frac{e^{ik_0|\bm{r}-\bm{r}'|}}{|\bm{r}-\bm{r}'|},
 \end{equation}
 %
-gdzie $k_0=\frac{\sqrt{2mE}}{\hbar}$.
+where $k_0=\frac{\sqrt{2mE}}{\hbar}$.
 
-Funkcja Greena dla elektronu oddziałującego z siecią atomową może być zapisana w postaci rozwinięcia
+The Green's function for an electron interacting with the atomic lattice can be written as an expansion
 %
 \begin{equation}
 G=G_0+G_0tG_0+G_0tG_0tG_0+...=G_0+G_0tG,
 \end{equation}
 %
-które prowadzi do zależności
+which leads to the relation
 %
 \begin{equation}
 G=(G_0^{-1}-t)^{-1},
 \end{equation}
 %
-gdzie $t$ jest macierzą rozpraszania elektronów na pojedynczych atomach sieci krystalicznej.
-Można wprowadzić również macierz całkowitego, wielokrotnego rozpraszania elektronów $T$, która zdefiniowana jest wzorem
+where $t$ is the scattering matrix for electrons on individual atoms of the crystal lattice.
+One can also introduce the total multiple scattering matrix for electrons $T$, which is defined by the formula
 %
 \begin{equation}
 G=G_0+G_0TG_0=G_0+G_0(t+tG_0T)G_0,
 \end{equation}
 %
-z którego dostajemy
+from which we obtain
 %
 \begin{equation}
 T=(t^{-1}-G_0)^{-1}.
 \end{equation}
 %
-Stany stacjonarne, które odpowiadają rozwiązaniom równania Schr\'{o}dingera, są biegunami macierzy $T$
-i mogą być wyznaczone jako miejsca zerowe wyznacznika
+Stationary states, which correspond to solutions of the Schr\'{o}dinger equation, are poles of the matrix $T$
+and can be determined as zeros of the determinant
 %
 \begin{equation}
 \det(t^{-1}-G_0)=0.
 \end{equation}
 %
-Równanie to dostarcza rozwiązania problemu wielokrotnego rozpraszania elektronów w przybliżeniu potencjału {\it muffin-tin}.
-Pełna informacja o rozpraszaniu elektronu na pojedynczej sferze atomowej zawarta jest w macierzy rozpraszania $t_l(E)$, 
-która zależy tylko od energii stanu elektronowego $E$ i orbitalnego krętu $l$.
-Korzystając z teorii rozpraszania, można wyrazić macierz rozpraszania jako funkcję przesunięcia fazy funkcji falowej $\eta_l(E)$
+This equation provides a solution to the problem of multiple scattering of electrons in the muffin-tin potential approximation.
+Complete information about electron scattering on a single atomic sphere is contained in the scattering matrix $t_l(E)$, 
+which depends only on the energy of the electronic state $E$ and the orbital angular momentum $l$.
+Using scattering theory, one can express the scattering matrix as a function of the phase shift of the wave function $\eta_l(E)$
 %
 \begin{equation}
 t_l(E)=-\frac{1}{k_0}e^{i\eta_l(E)}\sin(\eta_l(E)).
 \label{fshift}
 \end{equation} 
 %
-Funkcja Greena $G_0$ zależy tylko od geometrii sieci krystalicznej i energii $E$.
-Wykorzystując rozwinięcie fal płaskich w bazie harmonik sferycznych, funcję Greena
-możemy zapisać w formie
+The Green's function $G_0$ depends only on the crystal lattice geometry and energy $E$.
+Using the expansion of plane waves in the basis of spherical harmonics, the Green's function
+can be written in the form
 %
 \begin{equation}
 G_0(E,|\bm{r}-\bm{r}'|)=\sum_{L,L'} i^lj_l(k_0r)Y_L(\hat{\bm{r}})B_{LL'}(E,\bm{R}-\bm{R}')(-i)^{l'}j_{l'}(k_0r')Y^*_{L'}(\hat{\bm{r}}'),
 \end{equation} 
 %
-gdzie $j_l(k_0r)$ są funkcjami Bessela, $B_{LL'}$ są stałymi struktury KKR i $L=\{l,m\}$.
-Wektory położenia $\bm{r}$ i $\bm{r}'$ odnoszą się do dwóch sfer atomowych, których środki
-określają wektory $\bm{R}$ i $\bm{R}'$. Przy założeniu, że rozpraszanie w każdym węźle atomowym jest takie samo,
-co odpowiada sieci złożonej z jednego rodzaju atomów, możemy wprowadzić zależność stałych struktury od wektora falowego
+where $j_l(k_0r)$ are Bessel functions, $B_{LL'}$ are the KKR structure constants, and $L=\{l,m\}$.
+The position vectors $\bm{r}$ and $\bm{r}'$ refer to two atomic spheres whose centers
+are determined by vectors $\bm{R}$ and $\bm{R}'$. Assuming that scattering at each atomic site is the same,
+which corresponds to a lattice composed of one type of atom, we can introduce the dependence of structure constants on the wave vector
 % 
 \begin{equation}
 B_{LL'}(E_{\bm{k}},\bm{k})=\sum_{m} B_{LL'}(E_{\bm{k}},\bm{T}_m)e^{-i\bm{k}\bm{T}_m},
 \end{equation}
 %
-gdzie $\bm{T}_m$ są wektorami translacji sieci krystalicznej, a energie $E_{\bm{k}}$
-są rozwiązaniami równania
+where $\bm{T}_m$ are translation vectors of the crystal lattice, and the energies $E_{\bm{k}}$
+are solutions of the equation
 %
 \begin{equation}
 \det[t_l^{-1}(E_{\bm{k}})\delta_{LL'}-B_{LL'}(E_{\bm{k}},\bm{k})]=0.
 \end{equation}
 %
-Korzystając z wyrażenia (\ref{fshift}) dostajemy podstawowe równania na strukturę pasmową w metodzie KKR
+Using expression (\ref{fshift}), we obtain the basic equation for band structure in the KKR method
 %
 \begin{equation}
 \sum_{L'}[B_{LL'}(E_{\bm{k}},\bm{k})+k_0\cot(\eta_l(E_{\bm{k}}))\delta_{LL'}]a_{L'}(\bm{k})=0.
 \label{KKR}
 \end{equation}
 %
-Rozwiązania tego równania pozwalają nam wyznaczyć funkcje falowe wewnątrz sfer atomowych
+Solutions of this equation allow us to determine wave functions inside atomic spheres
 %
 \begin{equation}
 \psi_{\bm{k}}(\bm{r})=\sum_{L'}a_{L'}(\bm{k})u_l(r,E_{\bm{k}})Y_L(\hat{\bm{r}}),
 \end{equation}
 %
-gdzie $u_l(r,E_{\bm{k}})$ są rozwiązaniami równania radialnego wewnątrz sfery, umówionego dokładnie w następnym rozdziale.
-Funkcje falowe w obszarze międzywęzłowym są rozwiązaniami następującego równania całkowego
+where $u_l(r,E_{\bm{k}})$ are solutions of the radial equation inside the sphere, discussed in detail in the next section.
+Wave functions in the interstitial region are solutions of the following integral equation
 %
 \begin{equation}
 \psi_{\bm{k}}(\bm{r})=-\int d\bm{r}'G_0(E_{\bm{k}},|\bm{r}-\bm{r}'|)V(\bm{r}')\psi_{\bm{k}}(\bm{r}').
 \end{equation} 
 %
-Funkcje falowe w obydwóch obszarach muszą spełniać warunki brzegowe na każdej sferze atomowej. 
+Wave functions in both regions must satisfy boundary conditions on each atomic sphere. 
 
-W ogólnym przypadku macierz rozpraszania zależy od położenia atomu $t_l(E,\bm{R})$
-i równanie na funkcje Greena przyjmuje postać
+In the general case, the scattering matrix depends on the atomic position $t_l(E,\bm{R})$
+and the equation for the Green's function takes the form
 %
 \begin{equation}
 [G_{LL'}(E,\bm{R},\bm{R}')]^{-1}=\Big[[B_{LL'}(E,\bm{R}-\bm{R}')]^{-1}-t_l(E,\bm{R})\delta_{\bm{R}\bm{R}'}\delta_{LL'}]\Big],
 \end{equation}
 %
-a stany stacjonarne otrzymujemy z warunku
+and stationary states are obtained from the condition
 %
 \begin{equation}
 \det\Big[t_l^{-1}(E,\bm{R})\delta_{\bm{R}\bm{R}'}\delta_{LL'}-B_{LL'}(E,\bm{R}-\bm{R}')\Big]=0.
 \end{equation}
 
-Funkcja Greena pozwala wyznaczyć różne wielkości fizycznych. Na przykład, część urojona funkcji Greena zwiazana jest z lokalną gęstością stanów 
+The Green's function allows determining various physical quantities. For example, the imaginary part of the Green's function is related to the local density of states 
 %
 \begin{equation}
 n_L(E,\bm{R})=-\frac{1}{\pi}\operatorname{Im}G_{LL}(E+i\delta,\bm{R}).
 \end{equation}
 %
-Podobnie część urojona transformaty Fouriera funkcji Green daje nam gęstość spektralną,
-czyli gęstość stanów elektronowych dla danej energii i wektora falowego
+Similarly, the imaginary part of the Fourier transform of the Green's function gives us the spectral density,
+i.e., the density of electronic states for a given energy and wave vector
 %
 \begin{equation}
 n_L(E,\bm{k})=-\frac{1}{\pi}\operatorname{Im}G_{LL}(E+i\delta,\bm{k}).
 \end{equation}
 %
 
-Metoda funkcji Greena umożliwia wprowadzenie efektywnego (koherentnego) potencjału, który łączy własności dwóch różnych atomów.
-Jest to przybliżenie koherentnego potencjału ({\it ang. coherent potential approximation} - CPA), które stosowane jest do badania
-stopów metalicznych lub układów domieszkowanych \cite{CPA1,CPA2}.
-Metoda polega na uśrednieniu własności rozpraszania dwóch atomów umieszczonych w pobliżu efektywnego potencjału.
-Warunek równoważności średniej ważonej po obydwóch atomach i efektywnego potencjału prowadzi do zespolonego,
-zależnego od energii potencjału CPA. 
+The Green's function method enables the introduction of an effective (coherent) potential that combines the properties of two different atoms.
+This is the coherent potential approximation (CPA), which is used to study
+metallic alloys or doped systems \cite{CPA1,CPA2}.
+The method consists of averaging the scattering properties of two atoms placed near an effective potential.
+The condition of equivalence between the weighted average over both atoms and the effective potential leads to a complex,
+energy-dependent CPA potential. 
 
 
-\subsection{Metoda LAPW}
+\subsection{LAPW method}
 
-Metoda zlinearyzowanych stowarzyszonych fal płaskich ({\it ang. linearized augmented plane wave} - LAPW)
-jest modyfikacją oryginalnego podejścia rozszerzonych fal płaskich (APW) zaproponowanego przez Slatera w roku 1937 \cite{slater37}.
-Podobnie jak w metodzie pseudopotencjału, cały kryształ podzielony jest na obszary wewnątrz sfer otaczających atomy ($r<r_c$)
-i obszar międzywęzłowy ($r>r_c$). Funkcje falowe w otoczeniu jądra atomowego charakteryzują sie dużą zmiennością i
-w przybliżeniu maja kształt sferyczny. Z tego względu naturalnym wyborem bazy sa funkcje radialne, będące rozwiązaniami równania Schr\"{o}dingera z
-potencjałem sferycznie symetrycznym. W obszarze więdzywęzłowym, gdzie potencjał zmienia sie bardzo wolno, rozwiązania
-funkcji falowej rozwijane są w bazie fal płaskich. Funkcję falową w metodzie APW można
-zapisać w postaci
+The linearized augmented plane wave (LAPW) method
+is a modification of the original augmented plane wave (APW) approach proposed by Slater in 1937 \cite{slater37}.
+Similarly to the pseudopotential method, the entire crystal is divided into regions inside spheres surrounding atoms ($r<r_c$)
+and the interstitial region ($r>r_c$). Wave functions in the vicinity of the atomic nucleus are characterized by high variability and
+approximately have a spherical shape. For this reason, the natural choice of basis is radial functions, which are solutions of the Schr\"{o}dinger equation with
+a spherically symmetric potential. In the interstitial region, where the potential changes very slowly, solutions
+of the wave function are expanded in a plane wave basis. The wave function in the APW method can be
+written as
 %
 \begin{equation}
 \psi_{\bm{k}j}(\bm{r})=\sum_{\bm{G}}c_{\bm{G}j}\phi_{\bm{k}+\bm{G}}(\bm{r}),
 \end{equation}
 %
-gdzie $\bm{G}$ oznaczają wektory sieci odwrotnej. Funkcje bazowe mogą być zapisane w formie 
+where $\bm{G}$ denotes reciprocal lattice vectors. The basis functions can be written in the form 
 %
 \begin{equation} 
 \phi_{\bm{k}+\bm{G}}(\bm{r})=\begin{cases}
@@ -2233,56 +2232,56 @@ \subsection{Metoda LAPW}
 \label{APWbase}                      
 \end{equation}
 %
-gdzie $A_{lm}$ są współczynnikami rozwinięcia, $Y_{lm}(\theta,\varphi)$ są harmonikami sferycznymi i $u_l(r,E_l)$ są rozwiązaniami równania
+where $A_{lm}$ are expansion coefficients, $Y_{lm}(\theta,\varphi)$ are spherical harmonics, and $u_l(r,E_l)$ are solutions of the equation
 %
 \begin{equation}
 [-\frac{d^2}{dx^2}+\frac{l(l+1)}{r^2}+V(r)-E_l]ru_l(r,E_l)=0,
 \label{radial}
 \end{equation}
 %
-gdzie $V(r)$ jest potencjałem wewnątrz sfery, a $E_l$ pełni rolę parametru i niekoniecznie jest to energia własna. 
-W przybliżeniu potencjału {\it muffin-tin} (MT), w obszarze atomowym jest on sferycznie
-symetryczny ($V(\bm{r})=V(r)$), a w obszarze międzywezłowym ma wartość stałą ($V(\bm{r})=V_0$) lub zero. Wtedy funkcje falowe $\psi(\bm{r})$ mogą być wyrażone przez rozwiazania równania Schr\"{o}dingera w poszczególnych obszarach kryształu, odpowiednio przez harmoniki sferyczne wokół atomów i fale płaskie między atomami. 
-Współczynniki $A_{lm}$ wyznacza się tak, aby spełniony był warunek ciągłości funkcji falowej na granicy sfery (warunek ciągłości pierwszych pochodnych nie jest spełniony). w ten sposób współczynniki $A_{lm}$ stają się zależne od vectora falowego.  
-Współczynniki $c_{\bm{G}}$ i liczby $E_l$ są parametrami wariacyjnymi metody APW. Rozwiązania, które numerowane są wektorami $\bm{G}$, składające sie z pojedynczych fal płaskich w obszarze miedzywęzłowym i dopasowane do nich rozwiązania radialne wewnątrz sfery nazywane są stowarzyszonymi falami płaskimi.
+where $V(r)$ is the potential inside the sphere, and $E_l$ plays the role of a parameter and is not necessarily an eigenvalue. 
+In the muffin-tin (MT) potential approximation, in the atomic region it is spherically
+symmetric ($V(\bm{r})=V(r)$), while in the interstitial region it has a constant value ($V(\bm{r})=V_0$) or zero. Then wave functions $\psi(\bm{r})$ can be expressed through solutions of the Schr\"{o}dinger equation in individual regions of the crystal, respectively through spherical harmonics around atoms and plane waves between atoms. 
+The coefficients $A_{lm}$ are determined so that the continuity condition of the wave function at the sphere boundary is satisfied (the continuity condition for first derivatives is not satisfied). In this way, the coefficients $A_{lm}$ become dependent on the wave vector.  
+The coefficients $c_{\bm{G}}$ and numbers $E_l$ are variational parameters of the APW method. Solutions, which are numbered by vectors $\bm{G}$, consisting of single plane waves in the interstitial region and radial solutions matched to them inside the sphere, are called augmented plane waves.
 
-Funkcje bazowe wewnątrz sfery zależą od energii $E_l$, co powoduje, że odpowiednie równanie własne na funkcje falowe jest nieliniowe. Jest to główny problem
-podejścia APW, uniemożliwiający łatwe rozszerzenie stosowalności tej metody na dowolne potencjały. W szczególności potencjały wyznaczane metodą samozgodną w ramach teorii funkcjonału gęstość.
-Rozwiązaniem tego problemu była zaproponowana przez Andersena w roku 1975 linearyzacja równań APW, która prowadzi do metody LAPW \cite{Andersen75}.
-W tym podejściu, modyfikuje się funkcje bazowe wewnątrz sfery do postaci
+The basis functions inside the sphere depend on energy $E_l$, which causes the corresponding eigenvalue equation for wave functions to be nonlinear. This is the main problem
+of the APW approach, preventing easy extension of the applicability of this method to arbitrary potentials. In particular, potentials determined by the self-consistent method within density functional theory.
+The solution to this problem was the linearization of APW equations proposed by Andersen in 1975, which leads to the LAPW method \cite{Andersen75}.
+In this approach, the basis functions inside the sphere are modified to the form
 %
 \begin{equation}
 \phi_{\bm{k}+\bm{G}}(\bm{r})=\sum_{lm}[A_{lm}(\bm{k}+\bm{G})u_l(r,E_l)Y_{lm}(\theta,\varphi)+B_{lm}(\bm{k}+\bm{G})\dot{u}_l(r,E_l)Y_{lm}(\theta,\varphi)],
 \end{equation}    
 %
-gdzie współczynniki rozwinięcia $A_{lm}$ i $B_{lm}$ wyznacza się z warunku ciągłości funkcji falowej i jej pochodnej na sferze, a pochodna $\dot{u}_l(r,E_l)=\partial u_l/\partial E$ spełnia równanie
+where the expansion coefficients $A_{lm}$ and $B_{lm}$ are determined from the continuity condition of the wave function and its derivative on the sphere, and the derivative $\dot{u}_l(r,E_l)=\partial u_l/\partial E$ satisfies the equation
 %
 \begin{equation}
 [-\frac{d^2}{dx^2}+\frac{l(l+1)}{r^2}+V(r)-E_l]r\dot{u}_l(r,E_l)=ru_l(r,E_l).
 \end{equation}
-Dodatkowo narzuca sie warunek normalizacji funkcji $u_l$
+Additionally, the normalization condition for the function $u_l$ is imposed
 %
 \begin{equation}
 \int_0^{r_c}dr[ru_l(r,E_l)]^2=1,
 \end{equation}
 %
-oraz ortogonalności funkcji $u_l$ i $\dot{u}_l$
+as well as the orthogonality of functions $u_l$ and $\dot{u}_l$
 %
 \begin{equation}
 \int_0^{r_c}drr^2u_l(r,E_l)\dot{u}_l(r.E_l)=0.
 \end{equation}
 
-Znajomość gradientu $\dot{u}_l$ umożliwia wyznaczenie funkcji radialnej dla danej energii pasma $\varepsilon$, 
-uwzlędniając poprawkę liniową
+Knowledge of the gradient $\dot{u}_l$ enables determination of the radial function for a given band energy $\varepsilon$, 
+taking into account the linear correction
 %
 \begin{equation}
 u_l(r,\varepsilon)=u_l(r,E_l)+(\varepsilon-E_l)\dot{u}_l(r,E_l)+O((\varepsilon-E_l)^2),
 \end{equation}
 %
-gdzie $O((\varepsilon-E_l)^2)$ oznacza błąd, który jest rzędu kwadratu różnicy tych dwóch energii.
+where $O((\varepsilon-E_l)^2)$ denotes an error that is of the order of the square of the difference between these two energies.
 
-W ramach procedury samozgodnej, pełny potencjał wyliczany jest w obydwóch obszarach kryształu,
-stosując rozwinięcia w odpowiednich bazach
+Within the self-consistent procedure, the full potential is calculated in both regions of the crystal,
+using expansions in appropriate bases
 %
 \begin{equation} 
 V(\bm{r})=\begin{cases}
@@ -2292,51 +2291,51 @@ \subsection{Metoda LAPW}
 \label{fullpot}                       
 \end{equation}
 %
-Podobnie wylicza się gęstość ładunku wewnątrz sfery korzystając ze współczynników rozwinięcia funkcji falowych
-w bazie harmonic sferycznych oraz w obszarze międzywęzłowym w bazie fal płaskich.
+Similarly, the charge density is calculated inside the sphere using the expansion coefficients of wave functions
+in the spherical harmonics basis and in the interstitial region in the plane wave basis.
 
-W odróżnieniu od metody pseudopetcjału, gdzie uwzględniane są tylko elektrony walencyjne, w metodzie LAPW wyznaczane są
-zarówno stany walencyjne, jak i stany rdzenia.
-Ponieważ funkcje falowe stanów rdzenia sa zlokalizowane blisko jądra atomowego, wylicza się je stosując tylko 
-potencjał centrosymetryczny. Jeżeli zasięg funkcji falowej stanu rdzenia przekracza granicę strefy atomowej,
-stosuje się gładkie przedłużenie potencjału centrosymetrycznego poza tę granice. 
-Do wyznaczenia stanów rdzenia stosuje sie podejście w pełni relatywistyczne,
-czyli rozwiązuje się równanie Diraca, zamiast równania Schr\"{o}dingera (\ref{radial}). 
-Stany walencyjne wyznaczone są dla pełnego potencjału w całym obszarze kryształu.
+In contrast to the pseudopotential method, where only valence electrons are considered, in the LAPW method both
+valence states and core states are determined.
+Since the wave functions of core states are localized close to the atomic nucleus, they are calculated using only 
+the centrally symmetric potential. If the range of the core state wave function exceeds the boundary of the atomic zone,
+a smooth extension of the centrally symmetric potential beyond this boundary is used. 
+To determine core states, a fully relativistic approach is used,
+i.e., the Dirac equation is solved instead of the Schr\"{o}dinger equation (\ref{radial}). 
+Valence states are determined for the full potential throughout the entire crystal volume.
 
-W niektórych materiałach występują również stany elektronowe, mające wspólne cechy stanów rdzenia i walencyjnych, które nazywane są
-pseudordzeniowymi. Przykładem są wysoko położone i wzlędnie rozległe stany $5d$ w metalach ziem rzadkich.
-Występowanie tych stanów wymaga modyfikacji metody LAPW. W standardowej wersji, baza konstruowana
-jest tak, aby jak najdokładniej opisać stany elektronowe bliskie energii $E_l$
-i zwykle wybiera się tę wartość blisko środka pasma walencyjnego.
-Jeżeli występują dodatkowe stany w innym zakresie energii, wybór wartości $E_l$ nie jest oczywisty. 
-Najlepszym podejściem do opisu stanów pseudordzeniowych jest rozszerzenie bazy przez wprowadzenie dodatkowych 
-lokalnych orbitali (LO), które mieszczą się wewnątrz atomowej sfery. Ta modyfikacja nazywa się metodą APW+LO \cite{singh91}. 
-Ponieważ istnieje swoboda wyboru bazy, pochodną funkcji radialnej $\dot{u}_l$ usuwa sie z głównej bazy i dołacza do LO.
-Zatem całkowita baza w tym podejściu składa się z oryginalnej bazy APW (\ref{APWbase})  
-i dodatkowych orbitali w formie
+In some materials, there also exist electronic states having common features of core and valence states, which are called
+semi-core states. An example is the high-lying and relatively extended $5d$ states in rare earth metals.
+The occurrence of these states requires modification of the LAPW method. In the standard version, the basis is constructed
+to describe as accurately as possible electronic states close to energy $E_l$,
+and usually this value is chosen close to the center of the valence band.
+If additional states occur in a different energy range, the choice of $E_l$ value is not obvious. 
+The best approach to describe semi-core states is to extend the basis by introducing additional 
+local orbitals (LO) that fit within the atomic sphere. This modification is called the APW+LO method \cite{singh91}. 
+Since there is freedom in choosing the basis, the derivative of the radial function $\dot{u}_l$ is removed from the main basis and attached to LO.
+Thus, the total basis in this approach consists of the original APW basis (\ref{APWbase})  
+and additional orbitals in the form
 %
 \begin{equation}
 \chi^{lo}_L(\bm{r})=[a^{lo}_{lm}u_l(r,E_l)+b^{lo}_{lm}\dot{u}_l(r,E_l)]Y(\theta,\varphi),
 \end{equation}
 %
-gdzie $r<r_c$, a współczynniki $a^{lo}_{lm}$ i $b^{lo}_{lm}$ są tak dobrane, aby LO
-znikały na sferze. Liczba orbitalna ograniczona jest w tym przypadku do wartości, które odpowiadają 
-fizycznym stanom kwantowym ($l\le 3$). W odróżnieniu od LAPW, w metodzie APW+LO nie ma ograniczenia na
-wartości pochodnych funkcji bazowych na sferze atomowej. 
+where $r<r_c$, and the coefficients $a^{lo}_{lm}$ and $b^{lo}_{lm}$ are chosen so that the LO
+vanishes on the sphere. The orbital quantum number is limited in this case to values corresponding to 
+physical quantum states ($l\le 3$). In contrast to LAPW, in the APW+LO method there is no restriction on
+the values of derivatives of basis functions on the atomic sphere. 
   
 
-\subsection{Metoda LMTO}
+\subsection{LMTO method}
 
-W metodzie zlinearyzowanych orbitali {\it muffin-tin} ({\it ang. linearized muffin-tin orbitals} - LMTO)
-również wprowadza się podział kryształu na obszary wewnątrz sfer atomowych i obszar miedzywęzłowy.
-Funkcja falowa odpowiadająca energii $\varepsilon$ może być zapisana w ogólnej formie 
+In the linearized muffin-tin orbitals (LMTO) method,
+the crystal is also divided into regions inside atomic spheres and the interstitial region.
+The wave function corresponding to energy $\varepsilon$ can be written in the general form 
 %
 \begin{equation}
 \psi_{\bm{k}}(\varepsilon,\bm{r})=\sum_{lm} a_{lm}(\bm{k}) \phi_{lm}(\varepsilon,\kappa,\bm{r}),
 \end{equation}
 %
-gdzie funkcje bazowe w poszczególnych obszarach mają postać
+where the basis functions in individual regions have the form
 %
 \begin{equation}
 \phi_{lm}(\varepsilon,\kappa,\bm{r})=i^lY_{lm}(\theta,\phi)\begin{cases}
@@ -2345,35 +2344,35 @@ \subsection{Metoda LMTO}
              \end{cases}    
 \end{equation}
 %
-Funkcja $J_l(\kappa,r)$ jest tak skonstruowana, aby funkcje bazowe wewnątrz sfery nie zależały od energii, 
-czyli spełnione było równanie
+The function $J_l(\kappa,r)$ is constructed so that the basis functions inside the sphere do not depend on energy, 
+i.e., the equation is satisfied
 %
 \begin{equation}
 \frac{d}{d\varepsilon}\phi_{lm}(\varepsilon,\kappa,\bm{r})=i^lY_{lm}[\dot{u}_l(\varepsilon,r) +\kappa\frac{d}{d\varepsilon} cot(\eta_l(\varepsilon))J_l(\kappa,r)]=0
 \end{equation}
 %
-dla energii $\varepsilon=E_{\nu}$ odpowiadającej średniej energii danego orbitalu LMTO.
-Ten warunek prowadzi do wzoru
+for energy $\varepsilon=E_{\nu}$ corresponding to the average energy of a given LMTO orbital.
+This condition leads to the formula
 %
 \begin{equation}
 J_l(\kappa,r)=-\frac{\dot{u}_l(E_{\nu},r)}{\kappa\frac{d}{d\varepsilon}cot(\eta_l(E_{\nu}))}.
 \end{equation}
 %
-W obszarze międzywęzłowym funkcje bazowe centrowane w położeniu danej sfery $\bm{R}$ otrzymuje się przez rozwinięcie na funkcji $J_l$
-powiązane z sąsiednimi sferami w położeniach $\bm{R}'$
+In the interstitial region, basis functions centered at the position of a given sphere $\bm{R}$ are obtained by expansion in functions $J_l$
+associated with neighboring spheres at positions $\bm{R}'$
 %
 \begin{equation}
 N_l(\kappa,\bm{r}-\bm{R})=4\pi\sum_{l',l''}C_{l',l'',l'''}n^*_{l''}(\kappa,\bm{R}-\bm{R'})J_{l'}(\kappa,\bm{r}-\bm{R}').
 \end{equation}
 %
-Taka konstrukcja powoduje, że funkcje LMTO są kombinacją liniową funkcji $u_l$ i $\dot{u}_l$ wewnątrz danej sfery
-i są gładko przedłużane do obszaru międzywęzłowego, łącząc się w sposób ciagły z funkcjami $\dot{u}_l$ z każdej
-sąsiedniej sfery.
+This construction causes LMTO functions to be a linear combination of functions $u_l$ and $\dot{u}_l$ inside a given sphere
+and to be smoothly extended to the interstitial region, joining continuously with functions $\dot{u}_l$ from each
+neighboring sphere.
 
-Metodę LMTO można uprościć poprzez pominięcie obszaru międzywęzłowego i ograniczenie się do optymalizacji funkcji falowych tylko wewnątrz sfer.
-W podejściu nazywanym przybliżeniem atomowych sfer ({\it ang. atomic sphere approximation} - ASA)
-stosuje się sfery, które częściowo na siebie nachodzą i ich sumaryczna objętość równa się całkowitej objetości kryształu. 
-Zastoswanie takiego podejścia ma uzasadnienie jedynie dla kryształów z gęstym upakowaniem atomów.
+The LMTO method can be simplified by omitting the interstitial region and limiting oneself to optimization of wave functions only inside spheres.
+In the approach called the atomic sphere approximation (ASA),
+spheres are used that partially overlap and their total volume equals the total volume of the crystal. 
+Application of such an approach is justified only for crystals with close-packed atomic structures.
 
 \chapter{Funkcjonały i potencjały zależne od orbitali}
 

From 4a22d6cd51b12c6890c4bb9cc3f9361b46095531 Mon Sep 17 00:00:00 2001
From: Copilot <198982749+Copilot@users.noreply.github.com>
Date: Sat, 3 Jan 2026 08:58:00 +0100
Subject: [PATCH 19/23] Translate Chapter 5: Orbital-dependent functionals and
 potentials (#34)

* Initial plan

* Translate Chapter 5 sections 5.1-5.4 (through hybrid functionals)

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Complete translation of Chapter 5 (Orbital-dependent functionals and potentials)

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Address review feedback: fix parameter range and remove resolved review markers

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 584 +++++++++++++++++++++++++-------------------------
 1 file changed, 293 insertions(+), 291 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index d9f4405..c83c789 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -2374,655 +2374,657 @@ \subsection{LMTO method}
 spheres are used that partially overlap and their total volume equals the total volume of the crystal. 
 Application of such an approach is justified only for crystals with close-packed atomic structures.
 
-\chapter{Funkcjonały i potencjały zależne od orbitali}
+\chapter{Orbital-dependent functionals and potentials}
 
-\section{Problemy lokalnych funkcjonałów}
+\section{Problems with local functionals}
 \label{sec:mgga}
 
-Teoria funkcjonału gęstości odniosła wiele spektakularnych sukcesów w badaniach własności ciał stałych.
-Posiada jednak braki, które nie pozwalają opisać poprawnie struktury elektronowej wielu materiałów.
-Te problemy wynikają zasadniczo z przybliżeń stosowanych do wyliczenia energii wymienno-korelacyjnej, 
-której dokładna postać funkcjonalna nie jest znana.  
-Główną zaletą przybliżeń stosowanych w DFT, pozwalająca na obliczenia nawet dla dużych układów, jest lokalny lub kwazilokalny 
-charakter potencjału wymienno-korelacyjnego. Dokładny potencjał jest zwykle nielokalny i zależny jest od funkcji falowych (orbitali), a nie tylko gęstości elektronowej. Przykładem jest dokładna energia oddziaływania wymiennego, która wyliczana jest z orbitali jednocząstkowych w przybliżeniu Hartree-Focka. 
-Standardowe funkcjonały wymienno-korelacyjnych stosowane w DFT nie zależą od rodzaju obsadzonych orbitali. 
-Mówiąc ściślej, nie odróżniają stanów o różnej liczbie kwantowej $m$.
-Wszystkie orbitale traktowane są jednakowo i nie ma możliwości wyróżnienia cech, które decydują o nierównomiernym ich obsadzeniu.
-Taka możliwość często decyduje o charakterze stanu podstawowego i jest szczególnie istotna w układach silnie skorelowanych. 
-
-W przybliżeniach LDA i GGA występują dwa główne problemy, które są ze sobą ściśle powiązane.
-Pierwszym z nich jest brak nieciągłości pochodnej funkcjonalnej energii całkowitej po gęstości elektronowej, która jest równa potencjałowi chemicznemu. 
-Ta nieciągłość jest cechą charakterystyczną funkcjonału wymienno-korelacyjnym, ale występuje również w wyrazie opisującym energię kinetyczną elektronów. 
-Ponieważ nieciągłość potencjału chemicznego powiązana jest z przerwą energetyczną, to jej brak skutkuje zaniżonymi wartościami, a nawet
-zerowaniem się tej wielkości w izolatorach i półprzewodnikach.
-Problem przerwy energetycznej będzie dokładniej omawiany w rozdziale~(\ref{sec:insulators}).
-
-Drugim problemem jest występowanie samoodziaływania elektronów, czyli niezerowej energii oddziaływania elektronu 
-z jego własnym polem kulombowskim. Jest to efekt niefizyczny, który wynika z niedokładnego kasowania się występujących w hamiltonianie 
-przyczynków do oddziaływania elektronu z jego własnym polem elektrycznym.   
-W dokładnym opisie ujemny potencjał wymienno-korelacyjny wytwarzany przez każdy elektron powinien dokładnie kasować jego dodatni potencjał Hartree.
-W przypadku przybliżenia Hartree-Focka samoodziaływanie nie występuje, ponieważ oddziaływanie wymienne i Hartree wzajemnie kasują się.
-W przypadku DFT, potencjał Hartree wyliczany jest dokładnie, natomiast potencjał wymienno-korelacyjny jest przybliżony,
-stając się źródłem samooddziaływania. 
-Nawet w przypadku układu jednoelektronowego, energia Hartree nie kasuje się z energią wymienną wyznaczoną w przybliżeniu LDA lub GGA.
-Na przykładzie atomu wodoru, dyskutowanym w rozdziale \ref{sec:GGA}, dobrze widać efekt jaki powodują różnice między wartościami dokładnymi i przybliżonymi.
-Ponieważ wartość bezwględna energii wymiany jest mniejsza niż wartość dokładna, nie kasuje całkowicie dodatniej energii Hartree. 
-Wprawdzie częściowo różnica ta redukowana jest przez niezerową i ujemną energię korelacji, ale pozostaje skończona. 
-Ta dodatkowa energia oddziaływania elektronu z własnym polem elektrycznym może powodować duże błędy w obliczeniach.
-W zależności od rodzaju materiału, samoodziaływanie przyjmuje bardzo różne wartości. Jest ono zaniedbywalnie małe dla rozległych stanów elektronowych,
-zdelokalizowanych w dużym obszarze danego materiału. Dlatego wpływ samoodziaływania na stany elektronowe w prostych metalach, 
-gdzie dominują stany walencyjne typu $s$ i $p$, może być pominięty. 
-W materiałach, w których występują stany elektronowe mające tendencję do lokalizacji (np. stany {\it 3d} i {\it 4f}),
-jest ono szczególnie silne i często prowadzi do wyników jakościowo niezgodnych z eksperymentem.
-W stanach zlokalizowanych, oddziaływanie elektronu z jego własnym polem kulombowskim, które ma charakter odpychający,
-może prowadzić do znacznego wzrostu energii. Zredukowanie tej energie jest możliwe tylko poprzez zwiększenie delokalizacji
-ładunku i poszerzenie pasm elektronowych. Prowadzi to efektywnie do zmniejszenia przerwy energetycznej
-otrzymywanej w obliczeniach dla półprzewodników i izolatorów. W skrajnym przypadku błędy samoodziaływania powodują delokalizację ładunków w stanach $3d$ 
-i zamknięcie przerwy energetycznej, co skutkuje niewłaściwym stanem metalicznym w materiałach, które w rzeczywistości są izolatorami Motta. 
-
-Problemy te mogą być wyeliminowane lub częściowo zredukowane przy zastosowaniu funkcjonałów zależnych od orbitali elektronowych~\cite{kummel}.
-W kolejnych rozdziałach omawiane są metody, które uwzlędniają zależność orbitalną w funkcjonale wymienno-korelacyjnym
-lub wprowadzają poprawki do energii całkowitej zależne od obsadzenia orbitali.  
-
-\section{Funkcjonały OEP}
+Density functional theory has achieved many spectacular successes in the study of solid state properties.
+However, it has deficiencies that prevent the correct description of the electronic structure of many materials.
+These problems arise essentially from the approximations used to calculate the exchange-correlation energy,
+whose exact functional form is not known.
+The main advantage of the approximations used in DFT, allowing calculations even for large systems, is the local or quasilocal
+character of the exchange-correlation potential. The exact potential is usually nonlocal and depends on the wave functions (orbitals), not just the electron density. An example is the exact exchange interaction energy, which is calculated from single-particle orbitals in the Hartree-Fock approximation.
+Standard exchange-correlation functionals used in DFT do not depend on the type of occupied orbitals.
+More precisely, they do not distinguish states with different quantum number $m$.
+All orbitals are treated equally and there is no possibility to distinguish features that determine their unequal occupation.
+This possibility often determines the character of the ground state and is particularly important in strongly correlated systems. 
+
+In the LDA and GGA approximations there are two main problems that are closely related.
+The first is the lack of discontinuity in the derivative of the total energy functional with respect to electron density, which is equal to the chemical potential.
+This discontinuity is a characteristic feature of the exchange-correlation functional, but also appears in the term describing the kinetic energy of electrons.
+Since the discontinuity of the chemical potential is associated with the energy gap, its absence results in underestimated values, and even
+vanishing of this quantity in insulators and semiconductors.
+The problem of the energy gap will be discussed in more detail in chapter~(\ref{sec:insulators}).
+
+The second problem is the occurrence of electron self-interaction, i.e., non-zero interaction energy of an electron
+with its own Coulomb field. This is a non-physical effect that results from the imperfect cancellation of the contributions in the Hamiltonian
+to the interaction of an electron with its own electric field.
+In an exact description, the negative exchange-correlation potential produced by each electron should exactly cancel its positive Hartree potential.
+In the case of the Hartree-Fock approximation, self-interaction does not occur because the exchange and Hartree interactions mutually cancel.
+In the case of DFT, the Hartree potential is calculated exactly, while the exchange-correlation potential is approximate,
+becoming the source of self-interaction.
+Even in the case of a one-electron system, the Hartree energy does not cancel with the exchange energy determined in the LDA or GGA approximation.
+Using the example of the hydrogen atom, discussed in chapter \ref{sec:GGA}, one can clearly see the effect caused by differences between exact and approximate values.
+Since the absolute value of the exchange energy is smaller than the exact value, it does not completely cancel the positive Hartree energy.
+Although this difference is partially reduced by the non-zero and negative correlation energy, it remains finite.
+This additional energy of electron interaction with its own electric field can cause large errors in calculations.
+% **[REVIEW NEEDED]**: The term "self-interaction" is consistently translated; verify terminology matches conventions in English DFT literature.
+Depending on the type of material, self-interaction takes very different values. It is negligibly small for extended electronic states,
+delocalized over a large region of a given material. Therefore, the influence of self-interaction on electronic states in simple metals,
+where valence states of type $s$ and $p$ dominate, can be neglected.
+In materials where electronic states tend to localize (e.g., {\it 3d} and {\it 4f} states),
+it is particularly strong and often leads to results qualitatively inconsistent with experiment.
+In localized states, the interaction of an electron with its own Coulomb field, which is repulsive,
+can lead to a significant energy increase. Reduction of this energy is only possible by increasing the delocalization
+of the charge and broadening of the electronic bands. This effectively leads to a reduction of the energy gap
+obtained in calculations for semiconductors and insulators. In extreme cases, self-interaction errors cause charge delocalization in $3d$ states
+and closure of the energy gap, resulting in an incorrect metallic state in materials that are actually Mott insulators.
+
+These problems can be eliminated or partially reduced by using functionals dependent on electronic orbitals~\cite{kummel}.
+In the following sections, methods are discussed that take orbital dependence into account in the exchange-correlation functional
+or introduce corrections to the total energy dependent on orbital occupation.  
+
+\section{OEP functionals}
 \label{sec:mgga}
 
-Podejście nazywane metodą zoptymalizowanego potencjału efektywnego ({\it ang. optimized effective potential} - OEP)  
-zostało zaproponowane prez Sharpa i Hortona w roku 1953, jeszcze zanim powstała teoria funkcjonału gęstości~\cite{Sharp}. 
-Ta metoda zakłada funkcjonalną zależność energii wymienno-korelacyjnej od orbitali $\phi_{j\sigma}$, przy zachowanym lokalnym charakterze potencjału $V^{OEP}_{\sigma}$. Równanie OEP zostało rozwiązane numerycznie po raz pierwszy przez Talmana i Shadwicka~\cite{Talman}. 
-Energię całkowitą Kohna-Shama możemy zapisać jako funkcjonał orbitali
+The approach called the optimized effective potential (OEP) method
+was proposed by Sharp and Horton in 1953, even before density functional theory was developed~\cite{Sharp}.
+% **[REVIEW NEEDED]**: Verify historical accuracy of Sharp-Horton contribution and year.
+This method assumes a functional dependence of the exchange-correlation energy on orbitals $\phi_{j\sigma}$, while maintaining the local character of the potential $V^{OEP}_{\sigma}$. The OEP equation was first solved numerically by Talman and Shadwick~\cite{Talman}.
+The Kohn-Sham total energy can be written as a functional of orbitals
 %
 \begin{equation}
 E[\{\phi_{j\sigma}\}]=T[\{\phi_{j\sigma}\}]+\int d\bm{r} V_{ext}(\bm{r})n(\bm{r})+\frac{1}{2}\int \int d\bm{r} d\bm{r}' \frac{n(\bm{r})n(\bm{r}')}{|\bm{r}-\bm{r}'|}+E_{xc}[\{\phi_{j\sigma}\}].
 \end{equation}
 %
-Metoda OEP ma charakter wariacyjny i opiera się na dwóch równaniach. Pierwsze określa warunek minium (stacjonarności) energii całkowitej
-względem efektywnego potencjału
+The OEP method has a variational character and is based on two equations. The first defines the condition of minimum (stationarity) of the total energy
+with respect to the effective potential
 %
 \begin{equation}
 \frac{\delta E[\{\phi_{j\sigma}\}]}{\delta V^{OEP}_{\sigma}}=0,
 \end{equation} 
 %
-który jest równoważny zasadzie wariacyjne Kohna-Shama $\delta E_\text{tot}/\delta n=0$~\cite{Sahni}.
-Drugie równanie ma postać równania Kohna-Shama z efektywnym potencjałem zależnym od orbitali
+which is equivalent to the Kohn-Sham variational principle $\delta E_\text{tot}/\delta n=0$~\cite{Sahni}.
+The second equation has the form of the Kohn-Sham equation with an effective potential dependent on orbitals
 %
 \begin{equation}
 \Big[-\frac{\hbar^2}{2m}\nabla^2+V^{OEP}_{\sigma}[\{\phi_{j\sigma}\}](\bm{r})\Big]\phi_{j\sigma}(\bm{r})=\varepsilon_{j\sigma}\phi_{j\sigma}(\bm{r}).
 \end{equation}
 %
-Część potencjału opisującą oddziaływanie wymienno-korelacyjne można zapisać w postaci podwójnej całki wykorzystując własności pochodnych funkcjonalnych
+The part of the potential describing the exchange-correlation interaction can be written as a double integral using properties of functional derivatives
 %
 \begin{equation}
 V^{OEP}_{xc\sigma}(\bm{r})=\frac{\delta E_{xc}[\{\phi_{j\sigma}\}]}{\delta n_{\sigma}(\bm{r})}=\sum_{\nu\mu\i}\int\int d\bm{r}' d\bm{r}''\frac{\delta E_{xc}}{\delta \phi_{i\nu}(\bm{r}')}\frac{\delta \phi_{i\nu}(\bm{r}')}{\delta V^{OEP}_{\mu}(\bm{r}'')}\frac{\delta V^{OEP}_{\mu}(\bm{r}'')}{\delta n_{\sigma}(\bm{r})}.
 \label{voep}
 \end{equation}
 %
-Drugą z pochodnych funkcjonalnych pod całką można wyrazić w pierwszym rzędzie rozwinięcia pertubacyjnego 
+The second of the functional derivatives under the integral can be expressed in the first order of perturbation expansion
 %
 \begin{equation}
 \frac{\delta \phi_{i\nu}(\bm{r}')}{\delta V^{OEP}_{\mu}(\bm{r}'')}=\delta_{\nu\mu}\sum_{k\neq i}\frac{\phi^*_{k\mu}(\bm{r}')\phi_{k\mu}(\bm{r}'')}{\varepsilon_{i\mu}-\varepsilon_{k\mu}}\phi_{i\mu}(\bm{r}'')=\delta_{\nu\mu}G^R_{i\mu}(\bm{r}',\bm{r}'')\phi_{i\mu}(\bm{r}''),
 \end{equation}
 %
-gdzie wprowadzono funkcję Greena dla nieodziałujących elektronów
+where we introduced the Green's function for non-interacting electrons
 %
 \begin{equation}
 G^R_{i\mu}(\bm{r}',\bm{r}'')=\sum_{k\neq i}\frac{\phi^*_{k\mu}(\bm{r}')\phi_{k\mu}(\bm{r}'')}{\varepsilon_{i\mu}-\varepsilon_{k\mu}}.
 \label{fgreena}
 \end{equation} 
 %
-Trzecia z pochodnych jest odwrotnością liniowej funkcji odpowiedzi, którą również możemy wyrazić przez funkcję Greena
+The third of the derivatives is the inverse of the linear response function, which we can also express through the Green's function
 %
 \begin{equation}
 \chi_{\sigma}(\bm{r},\bm{r}'')=\delta_{\sigma\mu}\frac{\delta n_{\sigma}(\bm{r})}{\delta V^{OEP}_{\mu}(\bm{r}'')}=\delta_{\sigma\mu}\sum_i G^R_{i\sigma}(\bm{r}'',\bm{r})\phi_{i\sigma}(\bm{r}'')\phi^*_{i\sigma}(\bm{r}).
 \label{vchi}
 \end{equation}
 %
-Mnożąc równanie (\ref{voep}) przez $\chi_{\sigma}(\bm{r}',\bm{r})$, całkując wzlędem $\bm{r}'$ i wykorzystując (\ref{vchi}) dostajemy 
-równanie całkowe metody OEP
+Multiplying equation (\ref{voep}) by $\chi_{\sigma}(\bm{r}',\bm{r})$, integrating with respect to $\bm{r}'$ and using (\ref{vchi}) we obtain
+the integral equation of the OEP method
 %
 \begin{equation}
 \sum_i\int d\bm{r}' \phi^*_{i\sigma}(\bm{r})[V^{OEP}_{xc\sigma}(\bm{r})-u^{OEP}_{xci\sigma}(\bm{r})]G^R_{i\sigma}(\bm{r}',\bm{r})\phi_{i\sigma}(\bm{r}) + c.c.=0,
 \label{oepint}
 \end{equation}
 %
-gdzie
+where
 %
 \begin{equation}
 u^{OEP}_{xci\sigma}(\bm{r})=\frac{1}{\phi^*_{i\sigma}(\bm{r})}\frac{\delta E_{xc}[\{\phi_{j\sigma}\}]}{\delta \phi_{i\sigma}(\bm{r})}.
 \end{equation}
 %
-Jeżeli funkcjonał wymienno-korelacyjny zależy tylko od gęstości, wtedy $u^{OEP}_{xci\sigma}(\bm{r})=V^{OEP}_{xc\sigma}(\bm{r})$ i równanie (\ref{oepint})
-jest automatycznie spełnione.
-W ramach metody OEP, część wymienną można traktować dokładnie, tak jak w przybliżeniu Hartree-Focka, i wtedy potencjał zależny od orbitali
-ma postać
+If the exchange-correlation functional depends only on density, then $u^{OEP}_{xci\sigma}(\bm{r})=V^{OEP}_{xc\sigma}(\bm{r})$ and equation (\ref{oepint})
+is automatically satisfied.
+Within the OEP method, the exchange part can be treated exactly, as in the Hartree-Fock approximation, and then the orbital-dependent potential
+has the form
 %
 \begin{equation}
 u^{OEP}_{xi\sigma}=-\frac{1}{\phi^*_{i\sigma}(\bm{r})}\sum_j \phi^*_{j\sigma}\int d\bm{r}'\frac{\phi^*_{i\sigma}(\bm{r}')\phi_{j\sigma}(\bm{r}')}{|\bm{r}-\bm{r}'|}.
 \end{equation} 
 %
-Ze względu na całkowy charakter równań OEP, wyznaczenie samozgodnych rozwiązań, nawet przy zastosowaniu dodatkowych przybliżeniach, wymaga bardzo czasochłonnych obliczeń~\cite{kummel}. Z tego powodu potencjały OEP rzadko stosowane są w praktyce.
-Znacznie efektywniejsze są potencjały zależne od orbitali omówione w kolejnych rozdziałach. 
+Due to the integral character of the OEP equations, determining self-consistent solutions, even with additional approximations, requires very time-consuming calculations~\cite{kummel}. For this reason, OEP potentials are rarely used in practice.
+Significantly more effective are orbital-dependent potentials discussed in the following sections. 
 
-\section{Korekcja samoodziaływania (SIC)}
+\section{Self-interaction correction (SIC)}
 \label{sec:sic}
 
-W roku 1981, Pardew i Zunger zaproponowali metodę korekcji samoodziaływania ({\it ang. self-interaction correction} - SIC) \cite{PZ}.
-Metoda ta dotyczy stanów zlokalizowanych, więc rozpatrujemy zbiór orbitali $\phi_{\alpha\sigma}(\bm{r})$, gdzie $\alpha$ oznacza numer orbitalu,
-a $\sigma$ kierunek spinu. Oznaczamy gęstość elektronową pojedynczego orbitalu przez 
+In 1981, Perdew and Zunger proposed the self-interaction correction (SIC) method \cite{PZ}.
+This method concerns localized states, so we consider a set of orbitals $\phi_{\alpha\sigma}(\bm{r})$, where $\alpha$ denotes the orbital number,
+and $\sigma$ the spin direction. We denote the electron density of a single orbital by
 %
 \begin{equation}
 n_{\alpha\sigma}(\bm{r})=f_{\alpha\sigma}|\phi_{\alpha\sigma}(\bm{r})|^2,
 \end{equation}
 %
-gdzie liczba $f_{\alpha\sigma}$ określa obsadzenie danego orbitalu. 
-Warunkiem braku samoodziaływania w każdym orbitalu jest kompletne wzajemne kasowanie się energii Hartree danej wzorem (\ref{Hartree}) 
-i energii wymienno-korelacyjnej
+where the number $f_{\alpha\sigma}$ determines the occupation of a given orbital.
+The condition for the absence of self-interaction in each orbital is the complete mutual cancellation of the Hartree energy given by formula (\ref{Hartree})
+and the exchange-correlation energy
 %
 \begin{equation}
 E_H[n_{\alpha\sigma}]+E_{xc}[n_{\alpha\sigma}]=0,
 \label{nosic}
 \end{equation}
 %
-gdzie obydwie energie są funkcjonałami gęstości pojedynczego orbitalu.
-Dla pojedynczego elektronu spełniony jest również warunek kasowania
-się energii Hartree i dokładnej energii wymiany wylicznej w podejściu Hartree-Focka
+where both energies are functionals of the single orbital density.
+For a single electron, the condition of cancellation
+of the Hartree energy and the exact exchange energy calculated in the Hartree-Fock approach is also satisfied
 %
 \begin{equation}
 E_H[n_{\alpha\sigma}]+E_{x}[n_{\alpha\sigma}]=0.
 \end{equation}
 %
-Ponieważ energia korelacji $E_c=E_{xc}-E_{x}$, z tych dwóch warunków wynika, że energia korelacji dla pojedynczego orbitalu musi się zerować
+Since the correlation energy $E_c=E_{xc}-E_{x}$, from these two conditions it follows that the correlation energy for a single orbital must vanish
 %
 \begin{equation}
 E_c[n_{\alpha\sigma}]=0.
 \end{equation}
 %
-Warunki te nie są spełnione dla energii wymienno-korelacyjnej danej wzorami w przybliżeniu LDA (\ref{xclda})
-lub GGA (\ref{xcgga}), co prowadzi do samoodziaływania elektronów. Oznaczmy ogólnie przez $E_{xc}^{DFT}[n_{\uparrow},n_{\downarrow}]$ funkcjonał wymienno-korelacyjny wyznaczonony w jednym z tych przybliżeń. Wtedy wzór na poprawioną energię wymienno-korelacyjną można zapisać w formie
+These conditions are not satisfied for the exchange-correlation energy given by the formulas in the LDA approximation (\ref{xclda})
+or GGA (\ref{xcgga}), which leads to electron self-interaction. Let us denote generally by $E_{xc}^{DFT}[n_{\uparrow},n_{\downarrow}]$ the exchange-correlation functional determined in one of these approximations. Then the formula for the corrected exchange-correlation energy can be written in the form
 %
 \begin{equation}
 E_{xc}^{SIC}[n_{\alpha\sigma}]=E_{xc}^{DFT}[n_{\uparrow},n_{\downarrow}]-\sum_{\alpha\sigma}\delta_{\alpha\sigma},
 \label{xcsic}
 \end{equation}
 %
-gdzie
+where
 %
 \begin{equation}
 \delta_{\alpha\sigma}=E_H[n_{\alpha\sigma}]+E_{xc}^{DFT}[n_{\alpha\sigma}],
 \end{equation}
 %
-a sumowanie jest tylko po zajętych orbitalach $\alpha$.
-Dla dokładnej wartości energii wymienno-korelacyjnej mielibyśmy $\delta_{\alpha\sigma}=0$,
-czyli spełniony byłby warunek (\ref{nosic}). 
-Energia wymienno-korelacyjna zdefiniowana wzorem (\ref{xcsic}) nie jest funkcjonałem gęstości elektronowej,
-tylko funkcjonałem orbitali atomowych. Powoduje to, że również funkcjonał energii całkowitej (\ref{EKS}) staje się zależny od orbitali
+and the summation is only over occupied orbitals $\alpha$.
+For the exact value of the exchange-correlation energy we would have $\delta_{\alpha\sigma}=0$,
+i.e., condition (\ref{nosic}) would be satisfied.
+The exchange-correlation energy defined by formula (\ref{xcsic}) is not a functional of electron density,
+but a functional of atomic orbitals. This causes the total energy functional (\ref{EKS}) to also become dependent on orbitals
 % 
 \begin{equation}
 E^{SIC}[n_{\alpha\sigma}]=\sum_{\alpha\sigma}\langle\phi_{\alpha\sigma}|-\frac{\hbar^2}{2m}\nabla^2|\phi_{\alpha\sigma}\rangle+E_{ext}[n]+E_H[n]+E_{xc}^{SIC}[n_{\alpha\sigma}].
 \label{esic}
 \end{equation}
 % 
-Podobnie, jak w przypadku funkcjonału Kohna-Shama, możemy zastosować metodę wariacyjną do zmninimalizowania tej energię względem
-zbioru orbitali $\phi_{\alpha\sigma}$, przy założonym warunku ortonormalności 
+Similarly, as in the case of the Kohn-Sham functional, we can apply the variational method to minimize this energy with respect to
+the set of orbitals $\phi_{\alpha\sigma}$, assuming the orthonormality condition
 %
 \begin{equation}
 \langle\phi_{\alpha\sigma}|\phi_{\alpha'\sigma'}\rangle=\delta_{\alpha\alpha'}\delta_{\sigma\sigma'}.
 \end{equation}
 %
-Dostajemy wtedy równania jednocząstkowe analogiczne do równań Kohna-Shama
+We then obtain single-particle equations analogous to the Kohn-Sham equations
 %
 \begin{equation}
 [-\frac{\hbar^2}{2m}\nabla^2+V_{eff}(n_{\alpha\sigma},\bm{r})]\phi_{\alpha\sigma}(\bm{r})=\varepsilon_{\alpha\sigma}^{SIC}\phi_{\alpha\sigma}(\bm{r}),
 \label{eqsic}
 \end{equation} 
 %
-z efektywnym potencjałem danym wzorem
+with an effective potential given by the formula
 %
 \begin{equation}
 V_{eff}(n_{\alpha\sigma},\bm{r})=V^{DFT}(\bm{r})-V_H(n_{\alpha\sigma},\bm{r})-V_{xc}(n_{\alpha\sigma},\bm{r}),
 \end{equation}
 %
-gdzie ostatnie dwa wyrazy to korekcja samoodziaływania do efektywnego potencjału elektronowego.
-Podobnie jak w przypadku stanów Kohna-Shama spełniona jest zależność
+where the last two terms are the self-interaction correction to the effective electron potential.
+Similarly to the Kohn-Sham states, the following relation is satisfied
 %
 \begin{equation}
 \varepsilon_{\alpha\sigma}^{SIC}=\frac{\partial E^{SIC}}{\partial f_{\alpha\sigma}}.
 \end{equation}
 
-Równanie (\ref{eqsic}) pozwala wyznaczyć orbitale i energie własne zlokalizowanych stanów elektronowych. 
-Jednak w rzeczywistych materiałach, mamy często doczynienia z współistnieniem stanów zlokalizowanych i zdelokalizowanych, co powoduje, że rozwiązania tego równania nie muszą odpowiadać globalnemu minimum. Pełna optymalizacja funkcji falowych powinna 
-uwzlędniać zarówno stany zlokalizowane (typu Wanniera) i zdelokalizowane (typu Blocha).
-W praktyce polega to na wyznaczeniu różnych możliwych konfiguracji stanów zlokalizowanych i zdelokalizowanych,
-a następnie wybraniu tej o najniższej energii.    
+Equation (\ref{eqsic}) allows determination of orbitals and eigenvalues of localized electronic states.
+However, in real materials, we often deal with coexistence of localized and delocalized states, which means that solutions to this equation may not correspond to the global minimum. Full optimization of wave functions should
+take into account both localized (Wannier-type) and delocalized (Bloch-type) states.
+In practice, this involves determining different possible configurations of localized and delocalized states,
+and then selecting the one with the lowest energy.    
 
 \begin{figure}[h]
 \centering
 \includegraphics[scale=1.2]{rare-earths.pdf}
-\caption{Różnice energii całkowitej dla walencyjności II i III wyliczone dla ziemiach rzadkich (o), porównane z eksperymentem (linia przerywana)
-i wartościami dla związków ziem rzadkich z siarką (+). Rysunek z pracy \cite{RE}.}
+\caption{Total energy differences for valence II and III calculated for rare earths (o), compared with experiment (dashed line)
+and values for rare earth sulfur compounds (+). Figure from ref. \cite{RE}.}
 \label{fig:rare}
 \end{figure}
 
 
-Dobrym przykładem są metale ziem rzadkich, w których występują zarówno stany zlokalizowane ($4f$),
-jak i zdelokalizowane ($5d$ i $6s$). Dodatkowo występują dwa rodzaje elektronów $4f$. Cześć z nich jest zlokalizowana,
-tak jak elektrony rdzenia, mając decydujący wpływ na moment magnetyczny i walencyjność danego atomu. 
-Pozostałe, w wyniku hybrydyzacji ze stanami $5d$ i $6s$, tworzą pasmo elektronowe i biorą udział w wiązaniu metalicznym.
-W ramach tradycyjnych przybliżeń energii wymienno-korelacyjnej nie można opisać prawidłowo tak złożonej
-struktury elektronowej ziem rzadkich. W pracy \cite{RE} zastosowano metodę SIC do wyliczenia obsadzeń
-stanów $4f$ i walencyjności we wszystkich atomach ziem rzadkich (rysunek \ref{fig:rare}).
-Walencyjność zdefiniowana jest jako ilość stanów zdelokalizowanych przypadająca na jeden atom i wyliczana jest ze wzoru
+A good example is rare earth metals, where both localized ($4f$) states occur,
+as well as delocalized ($5d$ and $6s$) states. Additionally, there are two types of $4f$ electrons. Part of them is localized,
+like core electrons, having a decisive influence on the magnetic moment and valence of a given atom.
+The remaining ones, as a result of hybridization with $5d$ and $6s$ states, form an electronic band and participate in metallic bonding.
+Within the framework of traditional approximations of exchange-correlation energy, one cannot correctly describe such a complex
+electronic structure of rare earths. In ref. \cite{RE}, the SIC method was applied to calculate the occupation
+of $4f$ states and valence in all rare earth atoms (figure \ref{fig:rare}).
+Valence is defined as the number of delocalized states per atom and is calculated from the formula
 %
 \begin{equation}
 N_{val}=Z-N_{core}-N_{SIC},
 \end{equation}
 %
-gdzie $Z$ jest liczbą atomową, $N_{core}$ ilością elektronów w rdzeniu i $N_{SIC}$ liczbą stanów zlokalizowanych $4f$,
-dla których odjęto energię samoodziaływania. Dla ziem rzadkich walencyjność przyjmuje dwie możliwe wartości:
-$N_{val}=2$ (atomy dwuwartościowe, II) lub $N_{val}=3$ (atomy trójwartościowe, III).
-Dla tych dwóch przypadków wyliczone różnice ich energii całkowitych $E_{II}-E_{III}$ pokazane są na rysunku \ref{fig:rare}.
-Otrzymane wartości bardzo dobrze zgadzają się z danymi eksperymentalnymi. 
-Jak widać, oprócz Eu i Yb, które są dwuwartościowe, wszystkie pozostałe ziemie rzadkie są trójwartościowe. 
-Podobnie dobrą zgodność z eksperymentem otrzymano dla promieni Wignera-Seitza i stałych sieci kryształów ziem rzadkich \cite{RE}.
+where $Z$ is the atomic number, $N_{core}$ the number of electrons in the core and $N_{SIC}$ the number of localized $4f$ states,
+for which the self-interaction energy was subtracted. For rare earths, valence takes two possible values:
+$N_{val}=2$ (divalent atoms, II) or $N_{val}=3$ (trivalent atoms, III).
+For these two cases, the calculated differences of their total energies $E_{II}-E_{III}$ are shown in figure \ref{fig:rare}.
+The obtained values agree very well with experimental data.
+As can be seen, except for Eu and Yb, which are divalent, all other rare earths are trivalent.
+Similarly good agreement with experiment was obtained for Wigner-Seitz radii and lattice constants of rare earth crystals \cite{RE}.
 
-Ze względu na jawną zależność od orbitali, funkcjonały SIC wykazują nieciągłość pochodnej po gęstości elektronowej.
-Dzięki temu metoda ta znacznie poprawia wartości przerwy energetycznej i momentów magnetycznych
-w układach silnie skorelowanych~\cite{svane1990,temmerman2001}. 
+Due to the explicit dependence on orbitals, SIC functionals exhibit discontinuity of the derivative with respect to electron density.
+Thanks to this, the method significantly improves the values of energy gap and magnetic moments
+in strongly correlated systems~\cite{svane1990,temmerman2001}.
 On the other hand, the SIC does not provide the systamic improvement in description of molecular
 systems~\cite{kummel}.
 
 
-\section{Funkcjonały hybrydowe}
+\section{Hybrid functionals}
 \label{sec:hybrid}
 
-Problem samoodziaływania nie występuje w przybliżeniu Hartree-Focka, gdzie dla każdego elektronu
-jego własne oddziaływanie kulombowskie kasowane jest przez oddziaływanie wymienne.
-Zatem częściowe uwzlędnienie dokładnej wartości energii wymiany w funkcjonale wymienno-korelacyjnym powinno
-zmniejszyć efekt samoodziaływania i w ten sposób poprawić całkowitą energię układu. 
-Takie funkcjonały, w których energia wymiany otrzymana w przybliżeniu LDA lub GGA jest częściowo zastąpiona
-dokładną wartością otrzymaną z metody Hartree-Focka, nazywane są funkcjonałami hybrydowymi.  
-Pierwszy funkcjonał hybrydowych zaproponował Becke w roku 1993 \cite{becke93}, uzasadniając jego postać
-na podstawie tzw. formuły adiabatycznego połączenia ({\it ang. adiabatic connection formula}).
-Rozważmy ogólny hamiltoniana w postaci
+The self-interaction problem does not occur in the Hartree-Fock approximation, where for each electron
+its own Coulomb interaction is canceled by the exchange interaction.
+Therefore, partial inclusion of the exact value of exchange energy in the exchange-correlation functional should
+reduce the self-interaction effect and thus improve the total energy of the system.
+Such functionals, in which the exchange energy obtained in the LDA or GGA approximation is partially replaced by
+the exact value obtained from the Hartree-Fock method, are called hybrid functionals.
+The first hybrid functional was proposed by Becke in 1993 \cite{becke93}, justifying its form
+based on the so-called adiabatic connection formula.
+Let us consider a general Hamiltonian in the form
 %
 \begin{equation}
 H_{\lambda}=T+\lambda V,
 \end{equation}
 %
-gdzie $T$ jest operatorem energii kinetycznej, $V$ jest potencjałem oddziaływania między elektronami, a $\lambda$ jest parametrem, który
-określa siłę tego ddziaływania ($0\leq\lambda\leq 1$). 
-Funkcje falowe $\psi_{\lambda}$ i energie własne $E_{\lambda}$ tego hamiltonianu są zależne od parametru $\lambda$.
-Dla nich spełnione jest równanie
+where $T$ is the kinetic energy operator, $V$ is the interaction potential between electrons, and $\lambda$ is a parameter that
+determines the strength of this interaction ($0\leq\lambda\leq 1$).
+The wave functions $\psi_{\lambda}$ and eigenvalues $E_{\lambda}$ of this Hamiltonian depend on the parameter $\lambda$.
+For them the following equation is satisfied
 %
 \begin{equation}
 \frac{dE_{\lambda}}{d\lambda}=\langle\psi_{\lambda}|V|\psi_{\lambda}\rangle,
 \end{equation} 
 %
-z którego dostajemy
+from which we get
 %
 \begin{equation}
 E_{\lambda=1}=E_{\lambda=0}+\int_0^1 d\lambda \langle\psi_{\lambda}|V|\psi_{\lambda}\rangle.
 \end{equation}
 %
-Energia dla $\lambda=0$ odpowiada energii kinetycznej, natomiast $\lambda=1$ daje energie z pełnym oddziaływaniem elektronowym $V$.
-Ta formuła określa właśnie adiabatyczne połączenie między układem nieoddziałujących i oddziałujących elektronów.
-Po zastosowaniu tej formuły do funkcjonału energii (z pominięciem potencjału zewnętrznego $V_{ext}$) dostajemy 
+The energy for $\lambda=0$ corresponds to the kinetic energy, while $\lambda=1$ gives the energy with full electron interaction $V$.
+This formula defines precisely the adiabatic connection between the system of non-interacting and interacting electrons.
+After applying this formula to the energy functional (omitting the external potential $V_{ext}$) we get
 %
 \begin{equation}
 E[n]=T[n]+\int_0^1 d\lambda \langle\psi_{\lambda}|V|\psi_{\lambda}\rangle.
 \end{equation}
 %
-Biorąc pod uwagę funkcjonał Kohna-Shama (\ref{EKS}), możemy napisać wzór na energię wymienno-korelacyjną w postaci
+Taking into account the Kohn-Sham functional (\ref{EKS}), we can write the formula for exchange-correlation energy in the form
 %
 \begin{equation}
 E_{xc}[n]=\int_0^1 d\lambda \langle\psi_{\lambda}|V|\psi_{\lambda}\rangle-E_H[n]=\int_0^1d\lambda E_{xc,\lambda}[n].
 \label{xcn}
 \end{equation}
 %
-Powyższą całkę można przybliżyć, rozpatrując jedynie wartości końcowe. Dla $\lambda=0$, dostajemy energie wymiany
-w przybliżeniu Hartree-Focka, $E_{xc,0}=E_x^{HF}$. Becke zaproponował, aby dla $\lambda=1$ zastosować jedno z przybliżeń stosowanych w DFT (LDA lub GGA),
-$E_{xc,1}=E_{xc}^{DFT}$, a dla pośrednich wartości $\lambda$ zastosować interpolację liniową
+The above integral can be approximated by considering only the end values. For $\lambda=0$, we get the exchange energy
+in the Hartree-Fock approximation, $E_{xc,0}=E_x^{HF}$. Becke proposed that for $\lambda=1$ one should apply one of the approximations used in DFT (LDA or GGA),
+$E_{xc,1}=E_{xc}^{DFT}$, and for intermediate values of $\lambda$ use linear interpolation
 %  
 \begin{equation}
 E_{xc,\lambda}[n]=(1-\lambda)E_x^{HF}[n]+\lambda E_{xc}^{DFT}[n].
 \end{equation}
 %
-Po wstawieniu tej zależności do (\ref{xcn}) i wyliczeniu całki dostajemy funkcjonał hybrydowy 
+After substituting this dependence into (\ref{xcn}) and calculating the integral we get the hybrid functional
 %
 \begin{equation}
 E_{xc}^{hyb}[n]=\frac{1}{2}E_x^{HF}+\frac{1}{2}E_{xc}^{DFT}[n].
 \label{becke}
 \end{equation} 
 %
-Funkcjonał ten można ulepszyć poprawiając przybliżone całkowanie $E_{xc,\lambda}[n]$.
-Dla dowolnego układu, można znaleźć parametr $b$, który najlepiej opisuje
-energię wymienną
+This functional can be improved by refining the approximate integration of $E_{xc,\lambda}[n]$.
+For an arbitrary system, one can find a parameter $b$ that best describes the
+exchange energy
 %
 \begin{equation}
 E_{xc}^{hyb}[n]=bE_x^{HF}+(1-b)E_{xc}^{DFT}[n].
 \end{equation}
-W praktyce stosuje się funkcjonały, w których liniowo miksowane są tylko dokładna i przybliżona część
-energii wymiennej, a część korelacyjna jest przybliżona w ramach LDA lub GGA.
-Parametr $b$ można wybrać empirycznie przez dofitowanie różnych wielkości, na przykład energii
-atomizacji, do eksperymentalnych wartości dla dużej grupy materiałów.
-Optymalne zgodności otrzymuje się zwykle dla wartości z przedziału $b=[0.15,30]$.
+In practice, functionals are used in which only the exact and approximate parts
+of the exchange energy are linearly mixed, and the correlation part is approximated within LDA or GGA.
+The parameter $b$ can be chosen empirically by fitting different quantities, for example atomization
+energies, to experimental values for a large group of materials.
+Optimal agreement is usually obtained for values from the range $b=[0.15,0.30]$.
 
-Wybór parametru $b$ można dodatkowo uzasadnić stosująć podejście, które zaproponowali Pardew, Ernzerhof i Burke w roku 1996~\cite{PBE0}.  
-Zapisując zależność energii wymienno-korelacyjnej od $\lambda$ w formie rozwinięcia perturbacyjnego Mollera-Plesseta, 
-dostaje się zależność
+The choice of parameter $b$ can be additionally justified using the approach proposed by Perdew, Ernzerhof and Burke in 1996~\cite{PBE0}.
+Writing the dependence of exchange-correlation energy on $\lambda$ in the form of Møller-Plesset perturbation expansion,
+one obtains the dependence
 %
 \begin{equation}
 E_{xc,\lambda}[n]=E_{xc}^{DFT}[n]+(E_x^{HF}[n]-E_x^{DFT}[n])(1-\lambda)^{m-1},
 \end{equation}
 %
-która prowadzi do funkcjonału hybrydowego w postaci
+which leads to the hybrid functional in the form
 %
 \begin{equation}
 E_{xc}^{hyb}[n]=E_{xc}^{DFT}[n]+\frac{1}{m}(E_x^{HF}[n]-E_x^{DFT}[n]).
 \end{equation}
 %
-Dla $m=1$, energia wymiany jest równa dokładnej wartości $E_x^{HF}$. W tym przypadku dodanie przybliżonej wartość energii korelacji $E_c^{DFT}$ 
-poprawia wyniki w porównaniu do przybliżenia Hartree-Focka. Wyniki są jednak gorsze niż w przybliżeniach LDA lub GGA, w których błędy energii wymiany i korelacji mają przeciwne znaki, co powoduję ich częściowe kasowanie się. Przypadek $m=2$ odpowiada funkcjonałowi, który dany jest wzorem~(\ref{becke}). 
-Za optymalny uznano wybór $m=4$~\cite{PBE0}, wskazując na bardzo dobrą zgodnością wyników otrzymanych w ramach teorii perturbacyjnej Mollera-Plesseta czwartego rzędu z danymi eksperymentalnymi \cite{pople89}. Co więcej, w tym przypadku zarówno wartości $E_{xc,\lambda}$ i $E_{xc}^{DFT}$, jak również ich pierwsze oraz drugie pochodne wzajemnie pokrywają się dla $\lambda=1$. Parametr $b$ związany jest z wykładnikiem $m$ prostą zależnością $b=\frac{1}{m}$, co daje $b=0.25$, bliskie wartościom empirycznym.
+For $m=1$, the exchange energy equals the exact value $E_x^{HF}$. In this case, adding the approximate value of correlation energy $E_c^{DFT}$
+improves results compared to the Hartree-Fock approximation. However, the results are worse than in the LDA or GGA approximations, where the errors of exchange and correlation energy have opposite signs, which causes their partial cancellation. The case $m=2$ corresponds to the functional given by formula~(\ref{becke}).
+The optimal choice was recognized as $m=4$~\cite{PBE0}, indicating very good agreement of results obtained within the fourth-order Møller-Plesset perturbation theory with experimental data \cite{pople89}. Moreover, in this case both the values of $E_{xc,\lambda}$ and $E_{xc}^{DFT}$, as well as their first and second derivatives mutually coincide for $\lambda=1$. The parameter $b$ is related to the exponent $m$ by the simple dependence $b=\frac{1}{m}$, which gives $b=0.25$, close to empirical values.
 
-Funkcjonał hybrydowy jest specjalnym przypadkiem funkcjonału zależnego od orbitali, w którym 
-potencjał Kohna-Shama składa się z części lokalnej (kwazilokalnej) i części
-potencjału Focka opisującego dokładną energię wymiany. 
-Oddziaływanie wymienne ma charakter krótkozasięgowy i można uwzględnić wkład od dokładnej wartości jedynie
-w pewnym zakresie odległości. Uwzględniając ten efekt Heyd, Scuseria i Enzerhof (HSE) \cite{HSE} zaproponowali nowy funkcjonał
-w postaci
+The hybrid functional is a special case of an orbital-dependent functional, in which
+the Kohn-Sham potential consists of a local (quasilocal) part and a part
+of the Fock potential describing the exact exchange energy.
+Exchange interaction has a short-range character and one can include the contribution from the exact value only
+within a certain distance range. Taking this effect into account, Heyd, Scuseria and Ernzerhof (HSE) \cite{HSE} proposed a new functional
+in the form
 %
 \begin{equation}
 E_{xc}^{hyb}(\mu) = (1-b)E_{x}^{GGA,SR}(\mu)+bE_x^{HF,SR}(\mu) + E_x^{GGA,LR} + E_c^{GGA},
 \end{equation}
 %
-gdzie $E_x^{GGA,SR}$ i $E_x^{GGA,LR}$ odpowiadają krótkozasięgowemu i długozasięgowemu oddziaływaniu wymiennemu w przybliżeniu GGA,
-$E_x^{HF,SR}$ jest częścią krótkozasięgowego dokladnego oddziaływania wymiennego, a parametr $\mu$
-określa zasięg oddziaływania krótkozasięgowego. .
+where $E_x^{GGA,SR}$ and $E_x^{GGA,LR}$ correspond to short-range and long-range exchange interaction in the GGA approximation,
+$E_x^{HF,SR}$ is the part of the short-range exact exchange interaction, and the parameter $\mu$
+determines the range of short-range interaction.
 
 
-\section{Funkcjonały i potencjały meta-GGA}
+\section{meta-GGA functionals and potentials}
 \label{mgga}
 
 
-Dwa podstawowe przybliżenia stosowane do opisu funkcjonału wymienno-korelacyjnego uwzlędniają tylko gęstość ładunku w danym punkcie
-przestrzeni (LDA) lub dodatkowo jego gradient (GGA). 
-Kolejnym krokiem pozwalającym poprawić ten funkcjonał jest uwzlędnienie jego zależności od gradientu orbitali, czyli 
-części kinetycznej całkowitej energii. 
+The two basic approximations used to describe the exchange-correlation functional only take into account the charge density at a given point
+in space (LDA) or additionally its gradient (GGA).
+The next step to improve this functional is to take into account its dependence on the gradient of orbitals, i.e.,
+the kinetic part of the total energy.
 
 %
 \begin{figure}[h]
 \centering
 \includegraphics[scale=1]{Jacob.jpg}
-\caption{Drabina Jakuba opisująca kolejne poziomy dokładności opisu funkcjonału wymienno-korelacyjnego w DFT.}
+\caption{Jacob's ladder describing successive levels of accuracy in describing the exchange-correlation functional in DFT.}
 \label{fig:jacob}
 \end{figure}
 %
 
-Takie funkcjonały noszą nazwę meta-GGA i zapisywane są w ogólnej postaci
+Such functionals are called meta-GGA and are written in the general form
 %
 \begin{equation}
 E_{xc}[n_{\uparrow},n_{\downarrow}]=\int d\bm{r} n(\bm{r}) \varepsilon_{xc}(n_{\uparrow},n_{\downarrow},\nabla n_{\uparrow},\nabla n_{\downarrow},\tau_{\uparrow},\tau_{\downarrow}),
 \label{metaGGA}
 \end{equation}
 %
-gdzie $\tau_{\sigma}$ jest gęstością energii kinetycznej wyznaczoną dla obsadzonych stanów Kohna-Shama 
+where $\tau_{\sigma}$ is the kinetic energy density determined for occupied Kohn-Sham states
 %
 \begin{equation}
 \tau_{\sigma}(\bm{r})= \frac{1}{2}\sum_i |\nabla \psi_{i\sigma}(\bm{r})|^2,
 \end{equation}
 %
-w jednostkach $\hbar=m_\text{e}=1$. Uzasadnieniem włączenia $\tau_{\sigma}$ do funkcjonału jest występowanie tej wielkości w rozwinięciu Taylora
-uśrednionego sferycznie ładunku wymiennego wokół elektronu~\cite{Becke1998}. Drugą korzystną cecha jest możliwość zredukowania samoodziaływania
-w części korelacyjnej funkcjonału.
+in units $\hbar=m_\text{e}=1$. The justification for including $\tau_{\sigma}$ in the functional is the occurrence of this quantity in the Taylor expansion
+of the spherically averaged exchange charge around an electron~\cite{Becke1998}. The second beneficial feature is the possibility of reducing self-interaction
+in the correlation part of the functional.
 
 
-\subsection{Potencjał Becke-Johnsona}
+\subsection{Becke-Johnson potential}
 
-Becke and Johnson (BJ) zastosowali rozszerzenie potencjału oddziaływania wymiennego Slatera w postaci~\cite{Becke2006}
+Becke and Johnson (BJ) applied an extension of the Slater exchange potential in the form~\cite{Becke2006}
 %
 \begin{equation}
 V^\text{BJ}_{\text{x}\sigma}(\bm{r})=V_{\text{x}\sigma}^\text{Slater}(\bm{r})+\frac{1}{\pi}\sqrt{\frac{5}{12}}\sqrt{\frac{2\tau_{\sigma}(\bm{r})}{n_{\sigma},(\bm{r})}}.
 \label{BJ}
 \end{equation} 
 %
-gdzie omawiany w rozdziale (\ref{sec:HF}) potencjał Slatera dany jest wzorem
+where the Slater potential discussed in chapter (\ref{sec:HF}) is given by the formula
 %
 \begin{equation}
 V_{\text{x}\sigma}^\text{Slater}(\bm{r})=-\frac{e}{2}\int d\mb{r}' \frac{n(\mb{r},\mb{r}')}{|\mb{r}-\mb{r}'|}, 
 \label{SlatPot}
 \end{equation}
 % 
-w którym występuje ładunek wymienny $n(\mb{r},\mb{r}')$ opisany wzorem (\ref{EC}). Potencjał BJ znacznie poprawia energie odziaływania wymiennego
-w porównaniu do przybliżenia LDA i daje wyniki porównywalne z potencjałem OEP~\cite{Becke2006}.  
-Tran i Blaha zaproponowali modyfikację potencjału wymiennego BJ w postaci~\cite{Tran2009}
+where the exchange charge $n(\mb{r},\mb{r}')$ is described by formula (\ref{EC}). The BJ potential significantly improves the exchange interaction energy
+compared to the LDA approximation and gives results comparable to the OEP potential~\cite{Becke2006}.
+Tran and Blaha proposed a modification of the BJ exchange potential in the form~\cite{Tran2009}
 %
 \begin{equation}
 V^\text{mBJ}_{\text{x}\sigma}(\bm{r})=cV^\text{BR}_{\text{x}\sigma}(\bm{r})+(3c-2)\frac{1}{\pi}\sqrt{\frac{5}{12}}\sqrt{\frac{2\tau_{\sigma}(\bm{r})}{n_{\sigma}(\bm{r})}},
 \label{mBJ}
 \end{equation}
 %
-gdzie zamiast potencjału Slatera występuje potencjał Becke-Roussel'a
+where instead of the Slater potential, the Becke-Roussel potential appears
 %
 \begin{equation}
 V^\text{BR}_{\text{x}\sigma}(\bm{r})=-\frac{1}{b_{\sigma}(\bm{r})}(1-e^{-x_{\sigma}(\bm{r})}-\frac{1}{2}x_{\sigma}(\bm{r})e^{-x_{\sigma}(\bm{r})}).
 \end{equation}
 %  
-Wielkość $x_{\sigma}(\bm{r})$ zależy od gęstości elektronowej i jej gradientu oraz od $\tau_{\sigma}(\bm{r})$, natomiast
+The quantity $x_{\sigma}(\bm{r})$ depends on the electron density and its gradient as well as on $\tau_{\sigma}(\bm{r})$, while
 %
 \begin{equation}
 b_{\sigma}(\bm{r})=[\frac{x^3_{\sigma}(\bm{r})e^{-x_{\sigma}(\bm{r})}}{8\pi n_{\sigma}(\bm{r})}]^{\frac{1}{3}}.       
 \end{equation}
 %
-Parametr $c$ wyraża się następującym wzorem
+The parameter $c$ is expressed by the following formula
 %
 \begin{equation}
 c=\alpha+\beta[\frac{1}{V_\text{cell}}\int d\bm{r}'\frac{|\nabla n(\bm{r}')|}{n(\bm{r}')}]^{\frac{1}{2}},
 \end{equation}
 %
-gdzie $\alpha$ i $\beta$ wyznaczone są tak, aby zminimalizować odstępstwa wyliczanych przerw energetycznych
-od wartości eksperymentalnych dla dużej grupy materiałów.
-Potencjał mBJ (\ref{mBJ}) przechodzi w potencjał BJ (\ref{BJ}) dla $c=1$ oraz dobrze
-przybliża potencjał wymienny LDA dla stałej gęstości elektronowej dany wzorem (\ref{Vex}),
-niezależnie od wartości parametru $c$.
-W porównaniu do przybliżeń LDA lub GGA, obliczenia z potencjałem mBJ znacznie poprawiają wartości 
-przerwy energetycznej w gazach szlachetnych, półprzewodnikach i izolatorach~\cite{Tran2009}.
-Wartości te są porównywalne z wynikami otrzymanymi przy zastosowaniu funkcjonałów hybrydowych lub metody GW,
-które wymagają bardzo czasochłonnych obliczeń.
-Ponieważ w tym podejściu poprawiany jest potencjał wymienny, a nie funkcjonał wymienny, nie można go zastosować
-do optymalizacji położeń atomowych i stałych sieci krystalicznej.  
+where $\alpha$ and $\beta$ are determined to minimize deviations of calculated energy gaps
+from experimental values for a large group of materials.
+The mBJ potential (\ref{mBJ}) reduces to the BJ potential (\ref{BJ}) for $c=1$ and approximates well
+the LDA exchange potential for constant electron density given by formula (\ref{Vex}),
+independently of the value of parameter $c$.
+Compared to the LDA or GGA approximations, calculations with the mBJ potential significantly improve the values
+of the energy gap in noble gases, semiconductors and insulators~\cite{Tran2009}.
+These values are comparable to results obtained using hybrid functionals or the GW method,
+which require very time-consuming calculations.
+Since in this approach the exchange potential is improved, not the exchange functional, it cannot be applied
+to optimize atomic positions and crystal lattice constants.  
   
 
-\subsection{Funkcja lokalizacji elektronów (ELF)}
+\subsection{Electron localization function (ELF)}
 
-Funkcjonały meta-GGA używają bezwymiarowego parametru 
+meta-GGA functionals use the dimensionless parameter
 %
 \begin{equation}
 \alpha=\frac{\tau-\tau^w}{\tau^{u}},
 \end{equation}
 %
-gdzie $\tau^w$ jest gęstością energii kinetycznej Weizsäckera
+where $\tau^w$ is the Weizsäcker kinetic energy density
 %
 \begin{equation}
 \tau^w=\frac{|\nabla n|^2}{8n},
 \end{equation}
 %
-która określa dokładną wartość $\tau$ dla pojedynczego orbitalu, a $\tau^u$ jest jego wartością dla jednorodnego gazu
+which determines the exact value of $\tau$ for a single orbital, and $\tau^u$ is its value for a homogeneous gas
 %
 \begin{equation}
 \tau^u=\frac{3}{10}\Big{(}\frac{3}{\pi^2}\Big{)}^{\frac{2}{3}}n^{\frac{5}{3}}.
 \end{equation}
 %
-Parametr $\alpha$ wykorzystywany jest do charakteryzacji rozkładu gęstości elektronowej i pozwala odróżnić
-układy z wolnozmienną gestością elektronową, typową dla metali ($\alpha\approx 1$), 
-materiały z wiązaniami kowalencyjnymi między pojedynczymi orbitalami ($\alpha=0$) oraz słabymi wiązaniami niekowalencyjnymi między
-zamkniętymi powłokami atomowymi ($\alpha\rightarrow\infty$).
-Jest on bezpośrednio powiązany z funkcją lokalizacji elektronów ({\it ang. electron localization function} - ELF)~\cite{ELF1}
+The parameter $\alpha$ is used to characterize the electron density distribution and allows distinguishing
+systems with slowly varying electron density, typical for metals ($\alpha\approx 1$),
+materials with covalent bonds between single orbitals ($\alpha=0$) and weak non-covalent bonds between
+closed atomic shells ($\alpha\rightarrow\infty$).
+It is directly related to the electron localization function (ELF)~\cite{ELF1}
 %
 \begin{equation}
 \textrm{ELF} = \frac{1}{1+\alpha^2},
 \end{equation}     
 %
-która przyjmuje wartości z przedziału $[0,1]$ i jest używana do opisu wiązań chemicznych~\cite{ELF2,ELF3}.
-$\text{ELF}=1$ odpowiada idealnej lokalizacji elektronu, a $\text{ELF}=\frac{1}{2}$ jest wartością dla gazu elektronowego.  
+which takes values from the range $[0,1]$ and is used to describe chemical bonds~\cite{ELF2,ELF3}.
+$\text{ELF}=1$ corresponds to ideal electron localization, and $\text{ELF}=\frac{1}{2}$ is the value for an electron gas.  
 
 
-\subsection{Funkcjonał SCAN}
+\subsection{SCAN functional}
 
-W omówionych potencjałach BJ i mBJ zmiania się tylko część opisująca oddziaływanie wymienne, natomiast
-część korelacyjna pozostaje bez zmiany i może być wyznaczona w ramach przybliżeń LDA lub GGA.
-Funkcjonały meta-GGA, których ogólna postać wyraża się wzorem (\ref{metaGGA}), uwzględniają poprawki w części
-wymiennej i korelacyjnej. Zaproponowano kilka wersji funkcjonałów meta-GGA, których nazwy określane są skrótami
-PKZB~\cite{metaGGA}, TPSS~\cite{TPSS}, revTPSS~\cite{revTPSS} lub SCAN~\cite{SCAN}.
-Tutaj omówimy krótko funkcjonał SCAN ({\it ang. strongly contrained and appropriately normed}), który 
-spełnia 17 znanych warunków i norm dla funkcjonału wymienno-korelacyjnego.
-Jednym z nich jest warunek dla współczynnika określającego stosunek energii wymiany do odpowiedniej wartości w LDA, $F_x=E_x/E^{LDA}_x$, który
-nie może przekraczać wartości 1.174. Poprawnie opisuje również układy, dla których dokładne wartości są znane, np. jednorodny gaz elektronowy.  
+In the discussed BJ and mBJ potentials, only the part describing the exchange interaction is changed, while
+the correlation part remains unchanged and can be determined within the LDA or GGA approximations.
+meta-GGA functionals, whose general form is expressed by formula (\ref{metaGGA}), include corrections in both the
+exchange and correlation parts. Several versions of meta-GGA functionals have been proposed, whose names are determined by the acronyms
+PKZB~\cite{metaGGA}, TPSS~\cite{TPSS}, revTPSS~\cite{revTPSS} or SCAN~\cite{SCAN}.
+Here we will briefly discuss the SCAN functional (strongly constrained and appropriately normed), which
+satisfies 17 known conditions and norms for the exchange-correlation functional.
+One of them is the condition for the coefficient determining the ratio of exchange energy to the corresponding value in LDA, $F_x=E_x/E^{LDA}_x$, which
+cannot exceed the value 1.174. It also correctly describes systems for which exact values are known, e.g., the homogeneous electron gas.
 
-Część wymienną funkcjonału można zapisać w postaci
+The exchange part of the functional can be written in the form
 %
 \begin{equation}
 E_x[n]=\int d\bm{r} n(\bm{r}) \varepsilon_x(n)F_x(s,\alpha),
 \end{equation}
 %
-gdzie $\varepsilon_x(n)$ jest energią wymiany dla jednorodnego gazu przypadającą na pojedynczy elektron,
-$\alpha$ jest parametrem zdefiniowanym w poprzednim rozdziale, a $s$ jest bezwymiarowym gradientem gęstości
+where $\varepsilon_x(n)$ is the exchange energy for a homogeneous gas per single electron,
+$\alpha$ is the parameter defined in the previous section, and $s$ is the dimensionless density gradient
 %
 \begin{equation}
 s=\frac{|\nabla n|}{2(3\pi^2)^{1/3}n^{4/3}}.
 \end{equation}
 %
-Rysunek~\ref{fig:Fxs} pokazuje zależność współczynnika $F_x$ od $s$ dla trzech charakterystycznych wartości $\alpha$.
-$F_x$ spełnia odpowiednie warunki zarówno dla małych $s\rightarrow0$, jak i dużych wartości $s\rightarrow\infty$.  
-Dla $\alpha=1$ i małych $s$, $F_x$ pokrywa się z wartościami dla funkcjonału PBE.
-Wartości $F_x$ dla innych wartości $\alpha$ uzyskuje się przez odpowiednią interpolację między $\alpha=0$ i $\alpha=1$ oraz ekstrapolację
-do $\alpha\rightarrow\infty$.
+Figure~\ref{fig:Fxs} shows the dependence of the coefficient $F_x$ on $s$ for three characteristic values of $\alpha$.
+$F_x$ satisfies the appropriate conditions both for small $s\rightarrow0$ and large values $s\rightarrow\infty$.
+For $\alpha=1$ and small $s$, $F_x$ coincides with the values for the PBE functional.
+The values of $F_x$ for other values of $\alpha$ are obtained by appropriate interpolation between $\alpha=0$ and $\alpha=1$ and extrapolation
+to $\alpha\rightarrow\infty$.
 %
 \begin{figure}[h]
 \centering
 \includegraphics[scale=0.5]{Fxs.png}
-\caption{Współczynnik $F_x$ w funkcji $s$ dla różnych wartości $\alpha$. Rysunek z pracy \cite{SCAN}.}
+\caption{Coefficient $F_x$ as a function of $s$ for different values of $\alpha$. Figure from ref. \cite{SCAN}.}
 \label{fig:Fxs}
 \end{figure}
 %
 
-Podobnie skonstruowany jest całkowity fukcjonał wymienno-korelacyjny $F_{xc}=F_{x}+F_{c}$, który w granicy
-dużych gęstości niespolaryzowanego gazu przyjmuje wartość energii wymiennej $F_x$.
-Dla małych gęstości funkcjonał spełnia warunek ograniczenia Lieba-Oxforda, $F_{xc}\leq 2.215$.
+Similarly constructed is the total exchange-correlation functional $F_{xc}=F_{x}+F_{c}$, which in the limit
+of high densities of unpolarized gas takes the value of exchange energy $F_x$.
+For low densities the functional satisfies the Lieb-Oxford bound, $F_{xc}\leq 2.215$.
  
 
-\section{Metoda LDA+U}
+\section{LDA+U method}
 \label{sec:ldau}
 
-Metoda LDA+U zaproponowana została przez Anisimova, Zaanena i Andersena w roku 1991~\cite{anisimov}.
-Jednym z pierwszych jej zastosowań było wyliczenie struktury elektronowej dla nadprzewodnika wysokotemperaturowego La$_2$CuO$_4$~\cite{czyzyk}.
-Metoda ta polega na połączeniu podejścia DFT z modelem Hubbarda, który stosowany jest od lat sześćdziesiątych ubiegłego wieku do opisu układów silnie skorelowanych elektronów~\cite{hubbard}. Zacznę od omówienia tego modelu. 
-
-W najbardziej tradycyjnym podejściu, stany elektronowe w krysztale dzielimy na dwie grupy: stany zlokalizowane, które znajdują się w obrębie rdzenia atomowego i posiadają cechy orbitali atomowych oraz stany zdelokalizowane, które rozciągają się w całej przestrzeni kryształu.
-O własnościach transportowych danego materiału decydują głównie elektrony należące do drugiej grupy.
-Natomiast jeżeli chodzi o własności magnetyczne, to zarówno elektrony zlokalizowane, jak i wędrowne mogą oddziaływać wymiennie
-i decydować o rodzaju uporządkowania magnetycznego. W pierwszym przypadku mówimy o zlokalizowanych momentach magnetycznych, 
-których oddziaływanie można opisać w ramach modelu Heisenberga. W drugim mamy doczynienia z magnetyzmem pasmowym, który wynika
-z oddziaływania między mobilnymi elektronami walencyjnymi, które można opisać przy pomocy funkcji Blocha.
-
-Występują również stany elektronowe, które posiadają cechy obydwu tych grup. Stany te tworzą pasma, zachowując jednocześnie
-cechy orbitali atomowych. Ruch elektronów w takiej sytuacji polega na przeskokach między lokalnymi orbitali i odpowiadajace im pasma są znacznie węższe od
-typowych pasm metalicznych. Żródłem takiego zachowania są efekty korelacyjne, które uniemożliwiają swobodny przepływ elektronów w krysztale.
-Najlepszym przykładem są stany należące do niezapełnionej powłoki $3d$ w metalach przejściowych.
-Zgodnie z zakazem Pauliego, w każdym orbitalu mogą znajdować się tylko dwa elektrony z przeciwnymi kierunkami spinu. 
-Jeżeli występuje silne, lokalne oddziaływanie kulombowskie między ładunkami, to stany kwantowe, w których dwa elektrony   
-znajdują się w tym samym orbitalu są niekorzystne energetycznie. Zatem ruch elektronu jest bardzo utrudniony i dla odpowiednio dużego oddziaływania kulombowskiego
-elektrony zostają zlokalizowane. Materiały, w których występuje taki mechanizm lokalizacji to izolatory Motta.
-
-Hubbard zaproponował model do opisu stanów elektronowych, które posiadają cechy zlokalizowanych orbitali atomowych,
-wynikające z korelacji elektronowych~\cite{hubbard}.
-Jest to przykład  modelu ciasnego wiązania z orbitalami Wanniera, który opisany jest w rozdziale~\ref{sec:wannier}. 
-W najprostszym przypadku rozważamy pojedyncze orbitale typu $s$, zlokalizowane w węzłach atomowych. Silne korelacje elektronów znajdujących
-się w tym samym orbitalu można uwzględnić dodają do hamiltonianu lokalne oddziaływanie kulombowskie o ustalonej energii $U$. 
-Hamiltonian modelu Hubbarda zapisujemy stosując formalizm drugiego kwantowania
+The LDA+U method was proposed by Anisimov, Zaanen and Andersen in 1991~\cite{anisimov}.
+One of its first applications was the calculation of electronic structure for the high-temperature superconductor La$_2$CuO$_4$~\cite{czyzyk}.
+This method consists of combining the DFT approach with the Hubbard model, which has been used since the 1960s to describe strongly correlated electron systems~\cite{hubbard}. I will start by discussing this model.
+
+In the most traditional approach, electronic states in a crystal are divided into two groups: localized states, which are located within the atomic core and possess features of atomic orbitals, and delocalized states, which extend throughout the entire crystal space.
+The transport properties of a given material are mainly determined by electrons belonging to the second group.
+As for magnetic properties, both localized and itinerant electrons can interact by exchange
+and determine the type of magnetic ordering. In the first case we speak of localized magnetic moments,
+whose interaction can be described within the framework of the Heisenberg model. In the second case we deal with band magnetism, which results
+from the interaction between mobile valence electrons, which can be described using Bloch functions.
+
+There are also electronic states that possess features of both these groups. These states form bands while maintaining
+features of atomic orbitals. Electron motion in such a situation consists of hopping between local orbitals and the corresponding bands are much narrower than
+typical metallic bands. The source of such behavior is correlation effects that prevent free flow of electrons in the crystal.
+The best example is states belonging to the unfilled $3d$ shell in transition metals.
+According to the Pauli exclusion principle, each orbital can contain only two electrons with opposite spin directions.
+If there is strong, local Coulomb interaction between charges, then quantum states in which two electrons
+are in the same orbital are energetically unfavorable. Therefore, electron motion is very difficult and for sufficiently large Coulomb interaction
+electrons become localized. Materials in which such a localization mechanism occurs are Mott insulators.
+
+Hubbard proposed a model to describe electronic states that possess features of localized atomic orbitals,
+resulting from electron correlations~\cite{hubbard}.
+This is an example of a tight-binding model with Wannier orbitals, which is described in chapter~\ref{sec:wannier}.
+In the simplest case, we consider single $s$-type orbitals, localized at atomic sites. Strong correlations of electrons located
+in the same orbital can be accounted for by adding to the Hamiltonian a local Coulomb interaction of fixed energy $U$.
+The Hubbard model Hamiltonian is written using second quantization formalism
 %
 \begin{equation}
 H=\sum_{i,j,\sigma}t_{ij}c^{\dagger}_{i\sigma}c_{j\sigma}+U\sum_i n_{i\uparrow}n_{i\downarrow},
 \label{hubbard}
 \end{equation}
 %
-gdzie $c^{\dagger}_{i\sigma}$ i $c_{i\sigma}$ to operatory kreacji i anihilacji elektronu ze spinem $\sigma$ w zlokalizowanym orbitalu Wanniera $w(\bm{r}-\bm{R}_i)$,
-a całki przeskoku dane są wzorem
+where $c^{\dagger}_{i\sigma}$ and $c_{i\sigma}$ are creation and annihilation operators of an electron with spin $\sigma$ in the localized Wannier orbital $w(\bm{r}-\bm{R}_i)$,
+and the hopping integrals are given by the formula
 %
 \begin{equation}
 t_{ij}=\int d\bm{r} w^*(\bm{r}-\bm{R}_i)H_0w(\bm{r}-\bm{R}_j).
 \end{equation}
 %
-$H_0$ jest jednocząstkowym hamiltonianem złożonym z energii kinetycznej i efektywnego potencjału atomowego. 
-Najczęście uwzlędnia się tylko całki przeskoku między najbliższymi sąsiadami. 
-Operator liczby cząstek ze spinem $\sigma$ na węźle $i$ dany jest wyrażeniem $n_{i\sigma}=c^{\dagger}_{i\sigma}c_{i\sigma}$.
-Energię oddziaływania kulombowskiego dwóch elektronów w tym samym orbitalu można wyliczyć ze wzoru
+$H_0$ is the single-particle Hamiltonian composed of kinetic energy and effective atomic potential.
+Most often only hopping integrals between nearest neighbors are taken into account.
+The particle number operator with spin $\sigma$ at site $i$ is given by the expression $n_{i\sigma}=c^{\dagger}_{i\sigma}c_{i\sigma}$.
+The Coulomb interaction energy of two electrons in the same orbital can be calculated from the formula
 %
 \begin{equation}
 U=\int d\bm{r}d\bm{r}' |w(\bm{r}-\bm{R}_i)|^2\frac{e^2}{|\bm{r}-\bm{r}'|}|w(\bm{r}-\bm{R}_i)|^2.
 \end{equation}
 %
-Jeżeli wykonamy transformatę Fouriera operatorów kreacji i anihilacji,
-pierwszy wyraz hamiltonianu (\ref{hubbard}) przyjmie postać $\sum_{\bm{k}\sigma}\varepsilon_{\bm{k}}n_{\bm{k}\sigma}$,
-gdzie $\varepsilon_{\bm{k}}$ opisuje strukturę pasmową w modelu ciasnego wiązania,
-a $n_{\bm{k}\sigma}=c^{\dagger}_{\bm{k}\sigma}c_{\bm{k}\sigma}$ jest operatorem liczby obsadzeń w stanie $\bm{k}$ i spinie $\sigma$.
+If we perform the Fourier transform of the creation and annihilation operators,
+the first term of the Hamiltonian (\ref{hubbard}) takes the form $\sum_{\bm{k}\sigma}\varepsilon_{\bm{k}}n_{\bm{k}\sigma}$,
+where $\varepsilon_{\bm{k}}$ describes the band structure in the tight-binding model,
+and $n_{\bm{k}\sigma}=c^{\dagger}_{\bm{k}\sigma}c_{\bm{k}\sigma}$ is the occupation number operator in state $\bm{k}$ and spin $\sigma$.
 
-Zaproponowano kilka sformułowań metody LDA+U \cite{anisimov,orbital1,dudarev}, które różnią się opisem lokalnych 
-oddziaływań eletronowych. Ogólny schemat używany do wyliczenia całkowitej energii ma postać
+Several formulations of the LDA+U method have been proposed \cite{anisimov,orbital1,dudarev}, which differ in the description of local
+electronic interactions. The general scheme used to calculate the total energy has the form
 %
 \begin{equation}
 E_\text{tot}=E_\text{DFT}+E_U-E_\text{dc},
 \label{eldau}
 \end{equation}
 %
-gdzie $E_\text{DFT}$ jest energią układu otrzymaną w ramach metody DFT (w przybliżeniu LDA lub GGA), $E_U$ jest energią lokalnych oddziaływań elektronowych,
-a $E_\text{dc}$ odpowiada energii oddziaływań kulombowskich w przybliżeniu średniego pola.
-Ten ostatni wyraz jest konieczny, aby odjąć przybliżoną energię oddziaływań elektronowych, która uwzlędniona jest w $E_\text{DFT}$.
-W pracy \cite{orbital1} zaproponowano uogólnioną postać energii $E_U$, która uwzglądnia orbitalną zależność oddziaływań elektronowych
+where $E_\text{DFT}$ is the energy of the system obtained within the DFT method (in the LDA or GGA approximation), $E_U$ is the energy of local electronic interactions,
+and $E_\text{dc}$ corresponds to the Coulomb interaction energy in the mean field approximation.
+This last term is necessary to subtract the approximate electron interaction energy, which is included in $E_\text{DFT}$.
+In ref. \cite{orbital1}, a generalized form of energy $E_U$ was proposed, which takes into account the orbital dependence of electronic interactions
 %
 \begin{multline}
 E_U=\frac{1}{2}\sum_{i,\{m\},\sigma}[\langle m,m''|V_{ee}|m',m'''\rangle n_{i\sigma}^{mm'}n_{i-\sigma}^{m''m'''}
 \\+(\langle m,m''|V_{ee}|m',m'''\rangle - \langle m,m''|V_{ee}|m''',m'\rangle  )n_{i\sigma}^{mm'}n_{i\sigma}^{m''m'''}],
 \end{multline}  
 %
-gdzie $i$ numeruje węzły sieci,  $\{m\}=(m, m',m'', m''')$ są magnetycznymi liczbami kwantowymi,
-a elementy macierzowe oddziaływania kulombowskiego dane są wzorem
+where $i$ numbers the lattice sites, $\{m\}=(m, m',m'', m''')$ are magnetic quantum numbers,
+and the matrix elements of the Coulomb interaction are given by the formula
 %
 \begin{equation}
 \langle m,m''|V_{ee}|m',m'''\rangle=\int d\bm{r} \int \bm{r}' \phi_{lm}^*(\bm{r})\phi_{lm'}(\bm{r})\frac{e^2}{|\bm{r}-\bm{r}'|}\phi_{lm''}^*(\bm{r}')\phi_{lm'''}(\bm{r}').
 \label{integral}
 \end{equation}
 %
-Występujące pod całką atomowe funkcje falowe określone są dla orbitalnej liczby kwantowej $l$, która definiują zakres magnetycznych liczb kwantowych $-l\le \{m\} \le l$. Liczby obsadzeń orbitali atomowych tworzą tensor drugiego rzędu, którego elementy wyznaczane są przez rzutowanie funkcji falowych Kohna-Shama $\psi_{\bm{k}j}^{\sigma}$ na orbitale atomowe
+The atomic wave functions appearing under the integral are determined for the orbital quantum number $l$, which defines the range of magnetic quantum numbers $-l\le \{m\} \le l$. The occupation numbers of atomic orbitals form a second-rank tensor, whose elements are determined by projecting the Kohn-Sham wave functions $\psi_{\bm{k}j}^{\sigma}$ onto atomic orbitals
 %
 \begin{equation}
 n_{i\sigma}^{mm'}=\sum_{\bm{k},j} f_{\bm{k}j}^{\sigma}\langle\psi_{\bm{k}j}^{\sigma}|\phi_{lm'}\rangle\langle\phi_{lm}|\psi_{\bm{k}j}\rangle, 
 \end{equation}
 %
-gdzie współczynniki $f_{\bm{k}j}^{\sigma}$ okreśkaja obsadzenia stanów $j$ z wektorem falowym $\bm{k}$ i spinie $\sigma$.
-Całkę (\ref{integral}) można przedstawić w postaci sumy iloczynów części kątowej i radialnej 
+where the coefficients $f_{\bm{k}j}^{\sigma}$ determine the occupation of states $j$ with wave vector $\bm{k}$ and spin $\sigma$.
+The integral (\ref{integral}) can be represented as a sum of products of angular and radial parts
 %
 \begin{equation}
 \langle m,m''|V_{ee}|m',m'''\rangle=\sum_p a_p(m,m',m'',m''')F^p,
 \label{vee}
 \end{equation}
 %
-gdzie $p$ jest liczbą parzystą z przedziału $0\le p \le 2l$. Część kątowa $a_p$ wyliczana jest przy pomocy iloczynów współczynników Clebsha-Gordana 
+where $p$ is an even number from the range $0\le p \le 2l$. The angular part $a_p$ is calculated using products of Clebsch-Gordan coefficients
 %
 \begin{equation}
 a_p(m,m',m'',m''')=\frac{4\pi}{2p+1}\sum_{q=-p}^p \langle lm|Y_{pq}|lm'\rangle\langle lm''|Y_{pq}^*|lm'''\rangle.
 \end{equation}
 %
-$F^p$ nazywane są całkami Slatera i wyznaczane są ze wzoru
+$F^p$ are called Slater integrals and are determined from the formula
 %
 \begin{equation}
 F^p=e^2\int d\bm{r}\int d\bm{r}' r^2 r'^2 R_{nl}^2(\bm{r})\frac{r_1^p}{r_2^{p+1}} R_{nl}^2(\bm{r}'),
 \end{equation}
 %
-gdzie $R_{nl}(\bm{r})$ są częścią radialną atomu funkcji falowej. 
-Dla stanów $d$ potrzebne są trzy całki Slatera $F^0$, $F^2$ i $F^4$, a dla stanów $f$ dodatkowo $F^6$, aby wyznaczyć elementy macierzowe 
-potencjału kulombowskiego (\ref{vee}).
-Efektywne parametry, które określają lokalne oddziaływanie kulombowskie ($U$) i wymienne ($J$), można wyrazić przy pomocy całek Slatera
+where $R_{nl}(\bm{r})$ is the radial part of the atomic wave function.
+For $d$ states, three Slater integrals $F^0$, $F^2$ and $F^4$ are needed, and for $f$ states additionally $F^6$, to determine the matrix elements
+of the Coulomb potential (\ref{vee}).
+Effective parameters that determine the local Coulomb ($U$) and exchange ($J$) interactions can be expressed using Slater integrals
 %
 \begin{eqnarray}
 U&=&\frac{1}{(2l+1)^2}\sum_{m.m'} \langle m,m'|V_{ee}|m,m'\rangle=F^0,\\
@@ -3030,18 +3032,18 @@ \section{Metoda LDA+U}
 \label{HundJ}
 \end{eqnarray} 
 %
-Wykorzystując te parametry, możemy zapisać ostatni wyraz we wzorze (\ref{eldau}) w następującej formie
+Using these parameters, we can write the last term in formula (\ref{eldau}) in the following form
 %
 \begin{equation}
 E_{dc}=\frac{1}{2}\Big[\sum_i Un_i(n_i-1)-J[n_{i\uparrow}(n_{i\uparrow}-1)+n_{i\downarrow}(n_{i\downarrow}-1)]\Big],
 \end{equation}
 %
-gdzie $n_{i\sigma}=\Tr(n_{i\sigma}^{mm'})$ i $n_i=n_{i\uparrow}+n_{i\downarrow}$ jest całkowitym obsadzeniem orbitali w węźle $i$.
-Tak wyznaczone parametry $U$ i $J$ odnoszą sie do izolowanych atomów. Efektywne wartości parametru kulombowskiego $U$,
-stosowane dla danego rodzaju atomu, uwzględniaja efekty ekranowania elektronowego i zależą od rodzaju materiału.
-Można go wyznaczyć w ramach metody odpowiedzi liniowej poprzez wyliczenie zmiany obsadzenia stanów elektronowych na wybranym atomie
-pod wpływem przyłożonego lokalnego potencjału \cite{coco1,coco2}.
-Drugi z parametrów $J$, nazywany energią wymiany Hunda, znacznie słabiej zależy od rodzaju materiału i często stosowana jest jego wartość atomowa (\ref{HundJ}). 
+where $n_{i\sigma}=\Tr(n_{i\sigma}^{mm'})$ and $n_i=n_{i\uparrow}+n_{i\downarrow}$ is the total occupation of orbitals at site $i$.
+The parameters $U$ and $J$ determined in this way refer to isolated atoms. Effective values of the Coulomb parameter $U$,
+used for a given type of atom, take into account electron screening effects and depend on the type of material.
+It can be determined within the linear response method by calculating the change in electronic state occupation at a selected atom
+under the influence of an applied local potential \cite{coco1,coco2}.
+The second parameter $J$, called the Hund exchange energy, depends much less weakly on the type of material and often its atomic value (\ref{HundJ}) is used. 
 
 
 \chapter{Izolatory i półprzewodniki}

From 3be9d756f2f4ee8542fae4851698b93cba43acb3 Mon Sep 17 00:00:00 2001
From: Copilot <198982749+Copilot@users.noreply.github.com>
Date: Sat, 3 Jan 2026 09:29:22 +0100
Subject: [PATCH 20/23] Translate Chapter 6: Insulators and semiconductors to
 English (#35)

* Initial plan

* Translate Chapter 6: Insulators and semiconductors

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Fix Perdew name misspelling in bibliography entries

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 154 +++++++++++++++++++++++++-------------------------
 1 file changed, 77 insertions(+), 77 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index c83c789..b368e3a 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -3046,14 +3046,14 @@ \section{LDA+U method}
 The second parameter $J$, called the Hund exchange energy, depends much less weakly on the type of material and often its atomic value (\ref{HundJ}) is used. 
 
 
-\chapter{Izolatory i półprzewodniki}
+\chapter{Insulators and semiconductors}
 \label{sec:insulators}
 
-\section{Przerwa energetyczna}
+\section{Energy gap}
 \label{sec:bandgap}
 
-Fundamentalna przerwa energetyczna zdefiniowana jest jako różnica między energią jonizacji ($I$), czyli procesu usunięcia elektronu z danego materiału, 
-a powinowactwem elektronowym ($A$), czyli energią uzyskaną z dodania elektronu
+The fundamental energy gap is defined as the difference between the ionization energy ($I$), i.e., the process of removing an electron from a given material, 
+and the electron affinity ($A$), i.e., the energy obtained from adding an electron
 %
 \begin{eqnarray}
 E_g=I-A&=&[E(N-1)-E(N)]-[E(N)-E(N+1)] \nonumber \\ 
@@ -3061,89 +3061,89 @@ \section{Przerwa energetyczna}
 \label{energygap}       
 \end{eqnarray}
 %
-gdzie $E(N)$, $E(N-1)$ i $E(N+1)$ to energie całkowite wyznaczone kolejno dla układu neutralnego składającego się z $N$ elektronów oraz
-układów z odjętym i dodanym elektronem. Każda z tych energii może być wyznaczona jako energia stanu podstawowego układu z odpowiednią ilością
-elektronów. Wartość przerwy możemy również zapisać używając energii stanów Kohna-Shama 
+where $E(N)$, $E(N-1)$, and $E(N+1)$ are the total energies determined respectively for the neutral system consisting of $N$ electrons and
+systems with a removed and added electron. Each of these energies can be determined as the ground state energy of the system with the appropriate number
+of electrons. The gap value can also be expressed using Kohn-Sham state energies 
 %
 \begin{equation}
 E_g=E(N+1)-E(N)-[E(N)-E(N-1)]=\varepsilon_{N+1}(N+1)-\varepsilon_N(N),
 \label{truegap}
 \end{equation}
 %
-gdzie $\varepsilon_N(N)$ jest energią najwyższego poziomu pasma walencyjnego w układzie z $N$ elektronami,
-a $\varepsilon_{N+1}(N+1)$ energią najniższego stanu pasma przewodnictwa w układzie z $N+1$ elektronami.
-Czyli wartość przerwy energetycznej jest teoretycznie osiągalna w ramach DFT, 
-pod warunkiem, że znana jest dokladna zależność $E(N)$. Stosując funkcjonały LDA i GGA możemy otrzymać ze wzoru (\ref{truegap}) jedynie przybliżone wartości przerwy fundamentalnej. 
+where $\varepsilon_N(N)$ is the energy of the highest level of the valence band in the system with $N$ electrons,
+and $\varepsilon_{N+1}(N+1)$ is the energy of the lowest state of the conduction band in the system with $N+1$ electrons.
+Thus, the value of the energy gap is theoretically attainable within DFT, 
+provided that the exact dependence $E(N)$ is known. Using the LDA and GGA functionals, we can obtain from formula (\ref{truegap}) only approximate values of the fundamental gap. 
    
-Zwykle wykonujemy obliczenia dla układu z ustaloną liczbą elektronów $N$ i otrzymujemy przerwę w spektrum energetycznym Kohna-Shama, 
-która wynosi
+Usually, we perform calculations for a system with a fixed number of electrons $N$ and obtain a gap in the Kohn-Sham energy spectrum, 
+which is
 %
 \begin{equation}
 \Delta_{KS}=\varepsilon_{N+1}(N)-\varepsilon_N(N).
 \label{ksgap}
 \end{equation}
 %
-Porównując wzory (\ref{truegap}) i (\ref{ksgap}) dostajemy związek między tymi dwiema przerwami
+Comparing formulas (\ref{truegap}) and (\ref{ksgap}), we obtain the relationship between these two gaps
 %
 \begin{equation}
 E_g=\Delta_{KS}+\varepsilon_{N+1}(N+1)-\varepsilon_{N+1}(N)=\Delta_{KS}+\Delta_{xc},
 \end{equation}
 %
-gdzie różnica między nimi $\Delta_{xc}$ wynika ze zmiany nachylenia liniowej funkcji $E(N)$ przy całkowitej liczbie elektronów \cite{pardew82}.
-Ta nieciągłość ma swoje źródło w potencjale wymienno-korelacyjnym, którego pochodna zmienia się skokowo przy zmianie ilości elektronów
+where the difference between them $\Delta_{xc}$ results from the change in the slope of the linear function $E(N)$ at integer electron numbers \cite{pardew82}.
+This discontinuity has its origin in the exchange-correlation potential, whose derivative changes abruptly when the number of electrons changes
 %
 \begin{equation}
 \Delta_{xc}=\frac{\delta E_{xc}[n]}{\delta n(\bm{r})}|_{N+\delta}-\frac{\delta E_{xc}[n]}{\delta n(\bm{r})}|_{N-\delta},
 \label{deltaxc}
 \end{equation}
 %
-gdzie pochodne funkcjonalne wyznaczone są dla gęstości elektronów $n(\bm{r})$, których całki po całej przestrzeni równe są $N+\delta$ i $N-\delta$, w granicy $\delta\rightarrow 0$. 
+where the functional derivatives are determined for electron densities $n(\bm{r})$, whose integrals over the entire space are equal to $N+\delta$ and $N-\delta$, in the limit $\delta\rightarrow 0$. 
 
-Występowanie tej nieciągłości jest podstawową cechą DFT, ale w stosowanych przybliżeniach LDA i GGA, które charakteryzują się analityczną zależnością
-energii od gęstości elektronowej, te nieciągłości nie występują. Przybliżony charakter energii stanów jednocząstkowych,
-z których wyznaczana jest wartość $\Delta_{KS}$, w połączeniu z brakiem nieciągłości powoduje, że przerwy energetyczne wyznaczone w przybliżeniu LDA są średnio zaniżone o $40\%$. W GGA te wartości są zwykle poprawione, ale również znacznie odbiegają od wartości eksperymentalnych. 
+The occurrence of this discontinuity is a fundamental feature of DFT, but in the LDA and GGA approximations, which are characterized by an analytical dependence
+of energy on electron density, these discontinuities do not occur. The approximate character of the single-particle state energies,
+from which the value of $\Delta_{KS}$ is determined, combined with the lack of discontinuity, causes the energy gaps determined in the LDA approximation to be on average underestimated by $40\%$. In GGA, these values are usually corrected, but also significantly deviate from experimental values. 
 
-\section{Izolatory pasmowe i półprzewodniki}
+\section{Band insulators and semiconductors}
 
-W większości półprzewodników i izolatorów przerwa energetyczna wynika z istnienia całkowicie zapełnionych pasm walencyjnych
-i pustych pasm przewodzących. Rozróżnienie między półprzewodnikami i izolatorami nie jest jednoznacznie określone.
-Zwykle półprzewodniki mają mniejszą przerwę energetyczną i ich przewodnictwo elektryczne wykazuje znaczną zależność
-od temperatury w wyniku występowania indukowanych termicznie nośników prądu w paśmie przewodnictwa.
-W tych materiałach korelacje elektronowe zwykle nie są silne, ale mimo tego wyznaczana w ramach standardowych przybliżeń DFT 
-przerwa energetyczna jest znacznie zaniżona. Wynika to z braku uwzglednienia efektów wielociałowych, które występują
-przy wzbudzaniu elektronu z pasma walencyjnego do pasma przewodnictwa. Przykładem jest oddziaływanie elektron-dziura
-wystepujace dla stanów wzbudzonych.
-W przypadku tych materiałów, metoda LDA+U nie jest skuteczna i lepiej nadają się metody SIC i funkcjonały hybrydowe,
-jak również metody wielociałowe typu GW. 
-W metodzie SIC, energia całkowita zależy od obsadzenia orbitali atomowych, co daje możliwość odtworzenie nieciągłości
-pochodnych funkcjonału wymienno-korelacyjnego (\ref{deltaxc}) i poprawienia wartości przerwy energetycznej.
-SIC daje dobre wyniki dla izolatorów z dużą przerwa takich, jak kryształy gazów szlachetnych \cite{PZ}, czy kryształy jonowe. Przykładowo dla NaCl przerwa energetyczna wynosi $E_g^{SIC}=9.2$~eV \cite{norman83}, co dobrze zgadza się z wartością eksperymentalną $E_g^{exp}=8.97$~eV.  
+In most semiconductors and insulators, the energy gap results from the existence of completely filled valence bands
+and empty conduction bands. The distinction between semiconductors and insulators is not unambiguously defined.
+Usually, semiconductors have a smaller energy gap and their electrical conductivity shows a significant dependence
+on temperature due to the occurrence of thermally induced current carriers in the conduction band.
+In these materials, electronic correlations are usually not strong, but despite this, the energy gap determined within standard DFT approximations
+is significantly underestimated. This results from the lack of inclusion of many-body effects that occur
+when exciting an electron from the valence band to the conduction band. An example is the electron-hole interaction
+occurring for excited states.
+In the case of these materials, the LDA+U method is not effective and SIC methods and hybrid functionals are more suitable,
+as well as many-body methods of the GW type.
+In the SIC method, the total energy depends on the occupation of atomic orbitals, which provides the possibility of reproducing the discontinuity
+of the derivatives of the exchange-correlation functional (\ref{deltaxc}) and improving the value of the energy gap.
+SIC gives good results for insulators with a large gap such as noble gas crystals \cite{PZ} or ionic crystals. For example, for NaCl the energy gap is $E_g^{SIC}=9.2$~eV \cite{norman83}, which agrees well with the experimental value $E_g^{exp}=8.97$~eV.  
 
-Funkcjonały hybrydowe, które częściowo uwzlędniają dokladną wartość oddziaływania wymiennego i przez to redukują błąd samooddziaływania elektronów, 
-również poprawiają wartość przerwy energetycznej. Na rysunku~\ref{fig:gaphyb} porównano eksperymentalne wartości przerwy energetycznej
-z wyliczonymi w ramach czterech funkcjonałów hybrydowych dla wybranej grupy półprzewodników i tlenków metali przejściowych (FeO, CoO, NiO, MnO, and VO$_2$)~\cite{Gap-hyb}. Dwa z tych funkcjonałów zawierają 20 \% dokładnej wartości energii wymiany (B3PW91 i B3LYP), a dwa pozostałe 25 \% (PBE0 i HSE). Dla wiekszości materiałów 
-z przerwą mniejszą niź 5 eV, zgodność z eksperymentem jest bardzo dobra dla wszystkich funkcjonałów. Jednocześnie widzimy, że wybrane funkcjonały znacznie zaniżają wartości przerwy dla trzech półprzewodników o dużej przerwie (NaCl, LiCl i LiF).
-Dokładna analiza błędów pokazała, że najlepszą zgodność uzyskano dla funkcjonału HSE, który ma tendencję do niewielkiego zaniżania wartości przerwy (średnio o -0.24 eV). Dwa funkcjonały B3PW91 i B3LYP lekko zawyżają przerwę (średnie błedy wynoszą 0.14 eV i 0.13 eV). Największe odstępstwo obserwujemy dla funkcjonału PBE0, który zawyża wartość przerwy średnio o 0.43 eV.
+Hybrid functionals, which partially include the exact value of exchange interaction and thereby reduce the electron self-interaction error, 
+also improve the value of the energy gap. Figure~\ref{fig:gaphyb} compares experimental energy gap values
+with those calculated using four hybrid functionals for a selected group of semiconductors and transition metal oxides (FeO, CoO, NiO, MnO, and VO$_2$)~\cite{Gap-hyb}. Two of these functionals contain 20\% of the exact exchange energy value (B3PW91 and B3LYP), and the other two 25\% (PBE0 and HSE). For most materials 
+with a gap smaller than 5 eV, the agreement with experiment is very good for all functionals. At the same time, we see that the selected functionals significantly underestimate the gap values for three semiconductors with a large gap (NaCl, LiCl, and LiF).
+Detailed error analysis showed that the best agreement was obtained for the HSE functional, which has a tendency to slightly underestimate the gap value (on average by -0.24 eV). Two functionals B3PW91 and B3LYP slightly overestimate the gap (average errors are 0.14 eV and 0.13 eV). The largest deviation is observed for the PBE0 functional, which overestimates the gap value by an average of 0.43 eV.
 
 \begin{figure}[t!]
 \centering
 \includegraphics[scale=0.5]{gap-hyb.pdf}
-\caption{Porównanie eksperymentalnych wartości przerwy energetycznej z obliczonymi dla różnych funkcjonałów hybrydowych. Rysunek z pracy \cite{Gap-hyb}.}
+\caption{Comparison of experimental energy gap values with those calculated for different hybrid functionals. Figure from \cite{Gap-hyb}.}
 \label{fig:gaphyb}
 \end{figure}
 
-W tabeli~\ref{gap} zebrane są wartości przerwy energetycznej dla wybranych materiałów, wyliczone różnymi metodami
-i porównane z danymi eksperymentalnymi. LDA i GGA we wszystkich przypadkach znacznie zaniżają wartość przerwy.
-Dotyczy to zarówno półprzewodników, szczególnie germanu, gdzie przerwa jest równa zeru, jak i izolatorów.
-Duże rozbieżności w porównaniu do eksperymentu widzimy dla trzech ostatnich materiałów, zaliczanych do izolatorów Motta.
-Zastosowanie funkcjonału hybrydowego HSE poprawia wartości przerwy i dla większości półprzewodników zgodność z eksperymentem jest bardzo dobra.
-Dla izolatorów, ta zgodność jest lepsza lub gorsza w zależności od materiału.
-W tabeli~\ref{gap} pokazano również wyniki otrzymane w ramach metody GW i jej uproszczonej wersji G$_0$W$_0$, które również dają
-lepsze wartości przerwy. Metoda GW, która uwzlędnia oddziaływania wielociałowe w ramach rozwinięcia perturbacyjnego, będzie tematem jednego z
-kolejnych rozdziałów. 
+Table~\ref{gap} presents energy gap values for selected materials, calculated using different methods
+and compared with experimental data. LDA and GGA in all cases significantly underestimate the gap value.
+This applies to both semiconductors, especially germanium, where the gap is equal to zero, and insulators.
+Large discrepancies compared to experiment are seen for the last three materials, classified as Mott insulators.
+Application of the HSE hybrid functional improves the gap values and for most semiconductors the agreement with experiment is very good.
+For insulators, this agreement is better or worse depending on the material.
+Table~\ref{gap} also shows results obtained using the GW method and its simplified version G$_0$W$_0$, which also give
+better gap values. The GW method, which includes many-body interactions within a perturbative expansion, will be the subject of one of
+the following chapters. 
 
 \begin{table}[h!]
-\caption{Wartości przerwy energetycznej wyliczone różnymi metodami i porównane z danymi eksperymentalnymi.}
+\caption{Energy gap values calculated using different methods and compared with experimental data.}
 \label{gap}
 \begin{center}
 \begin{tabular}{|c|c|c|c|c|c|c|}
@@ -3168,51 +3168,51 @@ \section{Izolatory pasmowe i półprzewodniki}
 \end{center}
 \end{table}
 
-\section{Izolatory Motta}
+\section{Mott insulators}
 
-Szczególną klasę materiałów stanowią izolatory Motta, których własności elektronowe wynikają z silnych oddziaływaniań kulombowskich miedzy elektronami.
-W tym przypadku wielkość charakteryzująca nieciągłość wyliczana w ramach standardowych przybliżeń $\Delta_{KS}$ jest równe zeru 
-lub jest bardzo małą wielkością. Przerwa energetyczna wynika zasadniczo z nieciągłości funkcjonału energii ($\Delta_{xc}>0$), powstającej przy zmianie obsadzenia stanów orbitalnych. 
-Ta nieciągła zmiana energii jest proporcjonalna do energii oddziaływania kulombowskiego $U$. Dla stanów $d$ w metalach przejściowych parametr $U$ można zdefiniować
-jako energię związaną z transferem elektronu między dwoma atomami z początkowym obsadzeniem $d^n$,
-który prowadzi do zwiększenia ilości elektronów na jednym atomie $d^{n+1}$ i zmniejszenia na drugim $d^{n-1}$
+A special class of materials are Mott insulators, whose electronic properties result from strong Coulomb interactions between electrons.
+In this case, the characteristic quantity describing the discontinuity calculated within standard approximations $\Delta_{KS}$ is equal to zero 
+or is a very small quantity. The energy gap essentially results from the discontinuity of the energy functional ($\Delta_{xc}>0$), arising from changes in orbital state occupation. 
+This discontinuous energy change is proportional to the Coulomb interaction energy $U$. For $d$ states in transition metals, the parameter $U$ can be defined
+as the energy associated with electron transfer between two atoms with initial occupation $d^n$,
+which leads to an increase in the number of electrons on one atom $d^{n+1}$ and a decrease on the other $d^{n-1}$
 %
 \begin{equation}
 U=E(d^{n+1})+E(d^{n-1})-2E(d^n),
 \end{equation} 
 %
-gdzie $E(d^n)$ jest całkowitą energią układu, w którym pojedynczy atom posiada obsadzenie $d^n$.
-Ta relacja oznacza, że sama przerwa energetyczna dana wzorem (\ref{energygap}) powinna być liniową funkcją $U$.
-Potwierdzają to obliczenia przeprowadzone w ramach metody LDA+U.
+where $E(d^n)$ is the total energy of the system in which a single atom has occupation $d^n$.
+This relation means that the energy gap itself given by formula (\ref{energygap}) should be a linear function of $U$.
+This is confirmed by calculations performed using the LDA+U method.
 
-Jako przykład omówię wyniki obliczeń gestości stanów elektronowych dla izolatora Motta Fe$_2$SiO$_4$, otrzymane metodą GGA+U \cite{Mariana}.
-Na rysunku 6.2 pokazane sa wyniki dla stanu antyferromagnetycznego (AFM).
-Dla $U=0$ dostajemy nieprawidłowy stan metaliczny ($\Delta_{KS}=0$) z energią Fermiego przecinającą pasmo $t_{2g}$. 
-Pasmo to obsadzone jest przez elektrony z mniejszościowym kierunkiem spinów (kierunek przeciwny do momentu magnetycznego danego atomu). 
-Obliczenia dla $U=4.5$ eV dają prawidłowy stan elektronowy z przerwą energetyczną równą około 2 eV,
-która rozdziela pasmo $t_{2g}$ na dwie części: obsadzone dolne pasmo Hubbarda i nieobsadzone górne pasmo Hubbarda.
+As an example, I will discuss the results of calculations of electronic density of states for the Mott insulator Fe$_2$SiO$_4$, obtained by the GGA+U method \cite{Mariana}.
+Figure 6.2 shows the results for the antiferromagnetic (AFM) state.
+For $U=0$ we get an incorrect metallic state ($\Delta_{KS}=0$) with the Fermi energy crossing the $t_{2g}$ band. 
+This band is occupied by electrons with the minority spin direction (direction opposite to the magnetic moment of the given atom). 
+Calculations for $U=4.5$ eV give the correct electronic state with an energy gap equal to about 2 eV,
+which separates the $t_{2g}$ band into two parts: the occupied lower Hubbard band and the unoccupied upper Hubbard band.
 
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.21]{Fe2SiO4-2.pdf}
-\caption{Gęstość stanów elektronowych w Fe$_2$SiO$_4$ dla $U=0$ (górny panel) i dla $U=4.5$~eV (dolny panel). Rysunek z pracy \cite{Mariana}.}
+\caption{Electronic density of states in Fe$_2$SiO$_4$ for $U=0$ (upper panel) and for $U=4.5$~eV (lower panel). Figure from \cite{Mariana}.}
 \label{fig:mott}
 \end{figure}
 
 \begin{figure}[t!]
 \centering
 \includegraphics[scale=0.18]{Fe2SiO4-1.pdf}
-\caption{Zależność przerwy energetycznej od parametru oddziaływania kulombowskiego $U$ w Fe$_2$SiO$_4$ dla stanu antyferromagnetycznego (czerwone kółka)
-i ferromagnetycznego (niebieskie kwadraty). Rysunek z pracy \cite{Mariana}.}
+\caption{Dependence of the energy gap on the Coulomb interaction parameter $U$ in Fe$_2$SiO$_4$ for the antiferromagnetic state (red circles)
+and ferromagnetic state (blue squares). Figure from \cite{Mariana}.}
 \label{fig:gap}
 \end{figure}
 
-Wartość przerwy energetycznej zależy od dokładnej wartości parametru $U$, co pokazano na rysunku 6.3.
-Dla małych wartości $U\leq 2$ eV, przerwa energetyczna równa jest zeru. Wynika to z kryterium, które mówi, że w izolatorach Motta, 
-niezerowa przerwa tworzy się dla wartości energii oddziaływania kulombowskiego większych od szerokości
-pasma elektronowego, $U>W$. Rzeczywiście, z rysunku 6.2 wynika, że szerokość pasma $t_{2g}$ jest równa z dobrym przybliżeniem $W=2$ eV.
-Gdy energia oddziaływania kulombowskiego $U$ przewyższa $W$, przerwa energetyczna rośnie praktycznie liniowo z jej wartością. 
-Jak widać z rysunku 6.3 wartości przerwy zależą od rodzaju uporządkowania magnetycznego i dla stanu AFM są większe niż dla stanu FM.
+The value of the energy gap depends on the exact value of the parameter $U$, as shown in Figure 6.3.
+For small values $U\leq 2$ eV, the energy gap is equal to zero. This follows from the criterion that states that in Mott insulators, 
+a nonzero gap forms for values of Coulomb interaction energy greater than the width
+of the electronic band, $U>W$. Indeed, from Figure 6.2 it follows that the width of the $t_{2g}$ band is equal with good approximation to $W=2$ eV.
+When the Coulomb interaction energy $U$ exceeds $W$, the energy gap grows practically linearly with its value. 
+As can be seen from Figure 6.3, the gap values depend on the type of magnetic ordering and for the AFM state are larger than for the FM state.
 
 \chapter{Polaryzacja elektryczna}
 
@@ -4010,7 +4010,7 @@ \section{Metoda GW}
 \bibitem{CeperleyAlder80} D. M. Ceperley and B. J. Alder, {\it Ground state of the electron gas by a stochastic method},
 Phys. Rev. Lett. {\bf 45}, 566 (1980).
 
-\bibitem{PZ} J. P. Pardew and A. Zunger, {\it Self-interaction correction to density-functional approximations for many-electron systems},
+\bibitem{PZ} J. P. Perdew and A. Zunger, {\it Self-interaction correction to density-functional approximations for many-electron systems},
 Phys. Rev. B {\bf 23}, 5048 (1981).
 
 \bibitem{VWN} S. Vosko, L. Wilk, and M. Nusair, {\it Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis}, Can. J. Phys. {\bf 58}, 1200 (1983).
@@ -4182,7 +4182,7 @@ \section{Metoda GW}
 
 \bibitem{pardew82} J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz, {Density-functional theory for fractional particle number: derivative discontinuities of the energy}, Phys. Rev. Lett. {\bf 49}, 1691 (1982).
                    
-\bibitem{norman83} M. R. Norman and J. P. Pardew, {\it Simplified self-interaction correction applied to the energy bands of neon and sodium chloride},
+\bibitem{norman83} M. R. Norman and J. P. Perdew, {\it Simplified self-interaction correction applied to the energy bands of neon and sodium chloride},
 Phys. Rev. B {\bf 28}, 2135 (1983).                   
 
 \bibitem{Gap-hyb} A. J. Garza and G. E. Scuseria, {\it Predicting band gaps with hybrid density functionals}, J. Phys. Chem. Lett. {\bf  7}, 4165 (2016).

From 2979ae66ccf72a9a597931f9f6543185b3886ced Mon Sep 17 00:00:00 2001
From: Copilot <198982749+Copilot@users.noreply.github.com>
Date: Sat, 3 Jan 2026 11:56:38 +0100
Subject: [PATCH 21/23] Translate Chapter 7 (Electric polarization) from Polish
 to English (#36)

* Initial plan

* Translate Chapter 7 (Electric polarization) from Polish to English

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Fix missing summation index m in equation (line 3268)

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Change "extreme cases" to "limiting cases" per author feedback

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Remove review marker for "structural phase transition" terminology

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 225 +++++++++++++++++++++++++-------------------------
 1 file changed, 112 insertions(+), 113 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index b368e3a..30340a8 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -3214,88 +3214,88 @@ \section{Mott insulators}
 When the Coulomb interaction energy $U$ exceeds $W$, the energy gap grows practically linearly with its value. 
 As can be seen from Figure 6.3, the gap values depend on the type of magnetic ordering and for the AFM state are larger than for the FM state.
 
-\chapter{Polaryzacja elektryczna}
+\chapter{Electric polarization}
 
-\section{Teoria fazy Berry'ego}
+\section{Berry phase theory}
 
-Polaryzacja elektryczna odgrywa ważną rolę w wielu zjawiskach fizycznych występujacych w dielektrykach. 
-Wyznaczenie wartości polaryzacji elektrycznej w materiałach periodycznych jakimi są kryształy stanowiło duży problem teoretyczny. 
-W obliczeniach z pierwszych zasad, dzięki periodycznym warunkom brzegowym, takie układy traktowane sa jako nieskończone
-i zwykle nie bierze sie pod uwagę wpływu powierzchni materiału. 
-W układzie skończonym można wyznaczyć polaryzację elektryczną obliczając całkowity moment dipolowy złożony z części jonowej
-i elektronowej
+Electric polarization plays an important role in many physical phenomena occurring in dielectrics. 
+Determining the value of electric polarization in periodic materials such as crystals posed a major theoretical problem. 
+In first-principles calculations, due to periodic boundary conditions, such systems are treated as infinite
+and the influence of the material surface is usually not taken into account. 
+In a finite system, electric polarization can be determined by calculating the total dipole moment composed of ionic
+and electronic parts
 %
 \begin{equation}
 \bm{P}=\frac{e}{\Omega}\sum_jZ_j\bm{R}_j+\frac{1}{\Omega}\int d\bm{r} n(\bm{r}) \bm{r},
 \label{polaryzacja}
 \end{equation}
 % 
-gdzie zarówno sumowanie po wszystkich jonach o ładunku $Z_je$, jak i całkowanie gestości elektronowej $n(\bm{r})$ 
-odbywa się po całej objetości $\Omega$. Z tego samego wzoru można wyliczyć zmianę polaryzacji $\Delta \bm{P}$
-wywołaną zmianą rozkładu gęstości elektronowej $\Delta n(\bm{r})$, np. na skutek przemieszczenia się atomów. 
+where both the summation over all ions with charge $Z_je$ and the integration of electron density $n(\bm{r})$ 
+is performed over the entire volume $\Omega$. From this same formula, one can calculate the change in polarization $\Delta \bm{P}$
+induced by a change in the electron density distribution $\Delta n(\bm{r})$, for example, as a result of atomic displacements. 
 
-W układach periodycznych własności całego kryształu zależą od rozmieszczenia atomów w komórce prymitywnej.
-Wydaje się, że całkowity moment dipolowy kryształu i polaryzację elektryczną można wyznaczyć stosując wzór (\ref{polaryzacja}),
-do pojedynczej komórki. Jednak takie podejście możliwe jest tylko w skrajnych przepadkach kryształów jonowych
-lub molekularnych i odpowiada klasycznemu opisowi typu Clausiusa-Mossottiego, w którym występują izolowane dipole elektryczne.
-Gęstość elektronowa jest wielkością ciągłą i nie jest możliwe podzielenia całego obszaru kryształu w sposób jednoznaczny.  
-Co więcej, wiązania kowalencyjne, ktore występuja w ferroelektrykach, mają naturę kwantową i poprawny opis polaryzacji elektrycznej
-wymaga formalizmu kwantowego. 
+In periodic systems, the properties of the entire crystal depend on the arrangement of atoms in the primitive cell.
+It seems that the total dipole moment of the crystal and electric polarization can be determined by applying formula (\ref{polaryzacja})
+to a single cell. However, such an approach is only possible in limiting cases of ionic
+or molecular crystals and corresponds to the classical Clausius-Mossotti description, in which isolated electric dipoles occur.
+Electron density is a continuous quantity and it is not possible to divide the entire crystal region in a unique way.  
+Moreover, covalent bonds that occur in ferroelectrics have a quantum nature and a proper description of electric polarization
+requires a quantum formalism. 
 
-Punktem startowym, który umożliwia wyznaczenie polaryzacji w układach periodycznych jest 
-fundamentalna zależność między lokalną wartością polaryzacji elektronowej i przepływającym prądem
+The starting point that enables the determination of polarization in periodic systems is 
+the fundamental relationship between the local value of electronic polarization and the flowing current
 %
 \begin{equation}
 \bm{P}_e(\bm{r},t)=\int_{t_0}^t dt' j_e(\bm{r},t').
 \label{delP}
 \end{equation}
 %
-Wzór ten umożliwia również pomiar zmiany polaryzacji $\Delta \bm{P}$ przez wyznaczenie przepływającego prądu.
-Przykładowo, mierząc zmiany polaryzacji w funkcji deformacji kryształu można wyznaczyć stałą piezoelektryczną.
-Przełomem w teorii były prace, w których pokazano związek polaryzacji elektrycznej z geometryczną fazą Berry'ego~\cite{berry1,berry2}. 
-W obliczeniach zamiast czasu wprowadza się parametr $\lambda\in [0,1]$, który  
-opisuje adiabatyczną transformację układu generującą zmianę polaryzacji
+This formula also enables the measurement of the change in polarization $\Delta \bm{P}$ by determining the flowing current.
+For example, by measuring changes in polarization as a function of crystal deformation, one can determine the piezoelectric constant.
+The breakthrough in theory came with works showing the relationship of electric polarization with the geometric Berry phase~\cite{berry1,berry2}. 
+In calculations, instead of time, a parameter $\lambda\in [0,1]$ is introduced, which  
+describes the adiabatic transformation of the system generating the change in polarization
 %
 \begin{equation}
 \Delta\bm{P}_e = \int_0^1 d\lambda \frac{\partial\bm{P}_e}{\partial\lambda}. 
 \end{equation}
 %
-Przykładowo parametr $\lambda$ może być współrzędną, która określa położenie atomu w sieci krystalicznej.  
-Stosując wyrażenie z rachunku zaburzeń można powiązać pochodną polaryzacji ze strukturą elektronową 
+For example, the parameter $\lambda$ can be a coordinate that determines the position of an atom in the crystal lattice.  
+Using an expression from perturbation theory, one can relate the derivative of polarization to the electronic structure 
 %
 \begin{equation}
-\frac{\partial\bm{P}_e}{\partial\lambda}=-i\frac{e\hbar}{\Omega m_e}\sum_{\bm{k}}\sum_{n=1}^{occ}\sum_{=1}^{emp}\frac{\langle\psi^{\lambda}_{\bm{k}n}|\bm{p}
+\frac{\partial\bm{P}_e}{\partial\lambda}=-i\frac{e\hbar}{\Omega m_e}\sum_{\bm{k}}\sum_{n=1}^{occ}\sum_{m=1}^{emp}\frac{\langle\psi^{\lambda}_{\bm{k}n}|\bm{p}
 |\psi^{\lambda}_{\bm{k}m}\rangle\langle\psi^{\lambda}_{\bm{k}m}|\partial V^{\lambda}_{\text{KS}}/\partial\lambda|\psi^{\lambda}_{\bm{k}n}\rangle}{(\varepsilon^{\lambda}_{\bm{k}n}-\varepsilon^{\lambda}_{\bm{k}m})^2}+c.c.,
 \label{partialP}
 \end{equation}
 %
-gdzie sumowanie wykonywane jest po wszystkich stanach zajętych i pustych z pierwszej strefy Brillouina. 
-Elementy maciorzowe operatora pędu można wyrazić zależnością
+where the summation is performed over all occupied and empty states from the first Brillouin zone. 
+Matrix elements of the momentum operator can be expressed by the relation
 %
 \begin{equation}
 \langle\psi^{\lambda}_{\bm{k}n}|\bm{p}|\psi^{\lambda}_{\bm{k}m}\rangle=\frac{m_e}{\hbar}\langle u^\lambda_{\bm{k}n}|[\nabla_{\bm{k}},H^\lambda_{\bm{k}}]|u^\lambda_{\bm{k}m}\rangle,
 \end{equation}
 %
-w której $u^\lambda_{\bm{k}m}$ jest częścią periodyczną funkcji Blocha, a hamiltonian dany jest wyrażeniem
+in which $u^\lambda_{\bm{k}m}$ is the periodic part of the Bloch function, and the Hamiltonian is given by the expression
 %
 \begin{equation}
 H^\lambda_{\bm{k}}=\frac{1}{2m_e}(-i\hbar\nabla+\hbar\bm{k})^2+V^\lambda_\text{KS}.
 \end{equation}
 %
-Podobnie można wyrazić elementy macierzowe pochodnej potencjału Kohna-Shama
+Similarly, one can express the matrix elements of the derivative of the Kohn-Sham potential
 %
 \begin{equation}
 \langle\psi^{\lambda}_{\bm{k}n}|\frac{\partial V^{\lambda}_{\text{KS}}}{\partial\lambda}|\psi^{\lambda}_{\bm{k}m}\rangle=\frac{m_e}{\hbar}\langle u^\lambda_{\bm{k}n}|[\frac{\partial}{\partial\lambda},H^\lambda_{\bm{k}}]|u^\lambda_{\bm{k}m}\rangle.
 \end{equation}
 %
-Po wstawieniu elementów macierzowych do (\ref{partialP}) otrzymujemy
+After inserting the matrix elements into (\ref{partialP}), we obtain
 %
 \begin{equation}
 \Delta \bm{P}_e=\frac{-ie}{8\pi^3}\sum_{n=1}^{occ}\int_\text{BZ}d\bm{k}\int_0^1d\lambda[\langle\nabla u^\lambda_{\bm{k}n}|                  
 \frac{\partial u^\lambda_{\bm{k}n}}{\partial\lambda}\rangle-\langle\frac{\partial u^\lambda_{\bm{k}n}}{\partial\lambda}|                  
 \nabla_{\bm{k}} u^\lambda_{\bm{k}n}\rangle].
 \end{equation}
-Wykonując całkowanie przez części i wykorzystując periodyczność funkcji $u_{\bm{k}n}$ dostajemy
+Performing integration by parts and using the periodicity of the function $u_{\bm{k}n}$, we obtain
 %
 \begin{equation}
 \Delta \bm{P}_e=\bm{P}^{\lambda=1}_e-\bm{P}^{\lambda=0}_e,
@@ -3309,154 +3309,153 @@ \section{Teoria fazy Berry'ego}
 \label{pol2}
 \end{equation}
 %
-Występująca w tym wzorze całka jest bezpośrednio związana z fazą Berry'ego stanów elektronowych~\cite{Zak}.
-Wielkość wektorowa $\bm{A}(\bm{k})=i\langle u^\lambda_{\bm{k}n}|\nabla_{\bm{k}}|u^\lambda_{\bm {k}n}\rangle$
-nazywa się potencjałem cechowania lub połączeniem Berry'ego, a całka tej wielkość
-po zamkniętym obszarze jest fazą Berry'ego, którą dla danego pasma $n$ i kierunku $j$ można zapisać w formie
+The integral appearing in this formula is directly related to the Berry phase of electronic states~\cite{Zak}.
+The vector quantity $\bm{A}(\bm{k})=i\langle u^\lambda_{\bm{k}n}|\nabla_{\bm{k}}|u^\lambda_{\bm {k}n}\rangle$
+is called the gauge potential or Berry connection, and the integral of this quantity
+over a closed region is the Berry phase, which for a given band $n$ and direction $j$ can be written in the form
 %
 \begin{equation}
 \phi^\lambda_{nj}=\frac{i}{\Omega_\text{BZ}}\int_\text{BZ} d\bm{k}\langle u^\lambda_{\bm{k}n}|\bm{G}_j\nabla_{\bm{k}}|u^\lambda_{\bm {k}n}\rangle,
 \end{equation}
 %
-gdzie $\bm{G}_j$ jest wektorem sieci odwrotnej. 
-Polaryzacja związana z pasmem $n$ wyraża się wzorem
+where $\bm{G}_j$ is a reciprocal lattice vector. 
+The polarization associated with band $n$ is expressed by the formula
 %
 \begin{equation}
 \bm{P}^\lambda_n=\frac{e}{2\pi\Omega}\sum_j\phi^\lambda_{nj}\bm{R}_j.
 \end{equation}
 %
-Otrzymany wynik, który pokazuje związek polaryzacji elektrycznej z fazą funkcji falowej nie jest zaskoczeniem.
-W mechanice kwantowej, faza funkcji falowej jest bezpośrednio związana z prądem elektrycznym
-i zgodnie ze wzorem (\ref{delP}) określa również polaryzację elektryczną. 
-Tak jak każda faza, również polaryzacja elektryczna nie jest wartością zdefiniowaną jednoznaczne, a jedynie określoną {\it modulo}
-stała wartość. Żeby określić tę wartość zakładamy, że stan początkowy ($\lambda=0$) i końcowy ($\lambda=1$) opisane
-są tym samym hamiltonianem. Wtedy funkcje falowe dla tych stanów mogą różnić się tylko wartością fazy
+The obtained result, which shows the relationship of electric polarization with the phase of the wave function, is not surprising.
+In quantum mechanics, the phase of the wave function is directly related to the electric current
+and according to formula (\ref{delP}) also determines the electric polarization. 
+% **[REVIEW NEEDED]**: Consider the precise wording of "modulo a constant value" in this context.
+Like any phase, electric polarization is also not a uniquely defined value, but only defined {\it modulo}
+a constant value. To determine this value, we assume that the initial state ($\lambda=0$) and final state ($\lambda=1$) are described
+by the same Hamiltonian. Then the wave functions for these states can differ only by the value of the phase
 %
 \begin{equation}
 u^{\lambda=1}_{\bm{k}n}(\bm{r})=e^{i\theta_{\bm{k}n}}u^{\lambda=0}_{\bm{\bm{k}n}}(\bm{r}).
 \end{equation}
 %
-Zmianę fazy można zapisać w ogólnej formie
+The change in phase can be written in the general form
 %
 \begin{equation}
 \theta_{\bm{k}n}=\beta_{\bm{k}n}+\bm{k}\bm{R}_n,
 \end{equation}
 %
-gdzie $\beta_{\bm{k}n}$ jest funkcją periodyczną w $\bm{k}$. Zgodnie ze wzorem (\ref{pol2}), zmiana polaryzacji w tym przypadku wynosi
+where $\beta_{\bm{k}n}$ is a periodic function in $\bm{k}$. According to formula (\ref{pol2}), the change in polarization in this case is
 %
 \begin{equation}
 \Delta\bm{P}_e=-\frac{e}{8\pi^3}\sum_{n=1}^{occ}\int_\text{BZ}d\bm{k}\nabla_{\bm{k}}\theta_{\bm{k}n}=\frac{e}{\Omega}\sum_{n=1}^{occ}\bm{R}_n=\frac{e}{\Omega}\bm{R}.
 \end{equation}
 %
-Dla każdej sieci krystalicznej, można określić najmniejszą wartość $e\bm{R}_1/\Omega$, 
-która definiuje kwant polaryzacji, związany z symetrią translacyjną potencjału elektronowego, $V(\bm{r})=V(\bm{r}-\bm{R}_1)$.
-Ze względu na periodyczność kryształu wartość bezwględna polaryzacji nie ma większego sensu i istotna jest zmiana polaryzacji między dwoma stanami.
-Przesunięcia atomów, które zwykle są brane pod uwagę, są znacznie mniejsze niż odległości między atomami i wyliczana zmiana polaryzacji
-jest dużo mniejsza niż kwant polaryzacji, $\Delta\bm{P}_e\ll e\bm{R}_1/\Omega$. Pozwala to wyeliminować niejednoznaczność wartości polaryzacji
-w praktycznych zastosowaniach.  
+For each crystal lattice, one can determine the smallest value $e\bm{R}_1/\Omega$, 
+which defines the polarization quantum, associated with the translational symmetry of the electronic potential, $V(\bm{r})=V(\bm{r}-\bm{R}_1)$.
+Due to the periodicity of the crystal, the absolute value of polarization does not make much sense and what matters is the change in polarization between two states.
+Atomic displacements that are typically considered are much smaller than distances between atoms, and the calculated change in polarization
+is much smaller than the polarization quantum, $\Delta\bm{P}_e\ll e\bm{R}_1/\Omega$. This allows one to eliminate the ambiguity in the value of polarization
+in practical applications.  
 
-Polaryzację elektryczna można wyrazić również w reprezentacji funkcji Wanniera,
-które sa transformatami Fouriera funkcji Blocha
+Electric polarization can also be expressed in the representation of Wannier functions,
+which are Fourier transforms of Bloch functions
 %
 \begin{equation}
 \psi^{\lambda}_{\bm{k}j}=e^{i\bm{k}{r}}u^\lambda_{\bm{k}n}(\bm{r})=\frac{1}{\sqrt{N}}\sum_je^{i\bm{k}\bm{R}_j}w^\lambda_n(\bm{r}-\bm{R}_j),
 \end{equation}
 %
-gdzie $\bm{R}_j$ są wektorami sieci krystalicznej. Wyliczają z tej zależności $u^\lambda_{\bm{k}n}$ i wstawiając do wzoru (\ref{pol2}) otrzymujemy 
+where $\bm{R}_j$ are crystal lattice vectors. Calculating $u^\lambda_{\bm{k}n}$ from this relation and inserting it into formula (\ref{pol2}), we obtain 
 %
 \begin{equation}
 \Delta\bm{P}_e=-\frac{e}{\Omega}\sum_{n=1}^{occ} [\int d\bm{r} \bm{r}|w^{\lambda=1}_n(\bm{r})|^2-\int d\bm{r} \bm{r}|w^{\lambda=0}_n(\bm{r})|^2].
 \end{equation}
 %
-Ten wzór pokazuje, że zmiana polaryzacji jest proporcjonalna do przesunięcia środka ciężkości rozkładu gestości ładunku,
-zlokalizowanego na orbitalach Wanniera, wywołanego adiabatyczną zmianą potencjału elektronowego.
+This formula shows that the change in polarization is proportional to the shift of the center of gravity of the charge density distribution,
+localized on Wannier orbitals, induced by an adiabatic change in the electronic potential.
 
 
-\section{Ładunki efektywne}
+\section{Effective charges}
 
-Zgodnie ze wzorem (\ref{polaryzacja}), całkowita zmiana polaryzacji wzdłuż kierunku $\alpha$
-składa się z części jonowej i elektronowej
+According to formula (\ref{polaryzacja}), the total change in polarization along direction $\alpha$
+consists of ionic and electronic parts
 %
 \begin{equation}
 (\Delta\bm{P})_\alpha=(\Delta\bm{P}_\text{ion})_\alpha+(\Delta\bm{P}_e)_\alpha.
 \label{pol3}
 \end{equation}
 % 
-Część elektronowa polaryzacji dana jest wyrażeniami (\ref{pol1}) i (\ref{pol2}), natomiast część jonowa wyliczana jest ze wzoru
+The electronic part of polarization is given by expressions (\ref{pol1}) and (\ref{pol2}), while the ionic part is calculated from the formula
 %
 \begin{equation}
 (\Delta\bm{P}_\text{ion})_\alpha=\frac{e}{\Omega}Z_\text{ion}u_\alpha,
 \label{pol4}
 \end{equation}
 %
-gdzie $Z_{ion}$ jest ładunkiem rdzenia atomowego, a $u_\alpha$ wartością przesunięcia atomów.
-Zmianę całkowitej polaryzacji można zapisać w formie
+where $Z_{ion}$ is the atomic core charge, and $u_\alpha$ is the value of atomic displacement.
+The change in total polarization can be written in the form
 %
 \begin{equation}
 (\Delta\bm{P})_\alpha = \frac{\partial P_\alpha}{\partial u_\beta}u_\beta = \frac{e}{\Omega}Z^*_{\alpha\beta} u_\beta,
 \label{pol5}
 \end{equation}
 %
-gdzie $Z^*_{\alpha\beta}$ jest tensorem ładunków efektywnych Borna.
-Liczba niezależnych i niezerowych składowych tensora zależy od symetrii kryształu i ilości nierównowaznych położeń atomowych. 
-Wszystkie składowe tensora $Z^*_{\alpha\beta}$ można wyznaczyć korzystając ze wzoru (\ref{pol5}), wyliczając
-zmianę polaryzacji dla wychyleń atomów w różnych kierunkach. 
+where $Z^*_{\alpha\beta}$ is the Born effective charge tensor.
+The number of independent and nonzero components of the tensor depends on the crystal symmetry and the number of inequivalent atomic positions. 
+All components of the tensor $Z^*_{\alpha\beta}$ can be determined using formula (\ref{pol5}), by calculating
+the change in polarization for atomic displacements in different directions. 
 
-Ładunki efektywne przyjmują niezerowe wartości we wszystkich izolatorach i odgrywają ważną rolę przy opisie
-dynamiki sieci. Przesunięcia jonów związane z podłużnymi modami optycznymi (LO) w centrum strefy Brillouina ($\bm{k}=0$), które aktywne są w podczerwieni ({\it ang. infrared modes}), generują makroskopową polaryzację elektryczną i w ten sposób modyfikują siły międzyatomowe.
-Te dodatkowe siły powodują, że energia modów LO blisko punktu $\Gamma$ jest większa niż poprzecznych modów optycznych (TO).
-Powoduje to rozszczepienie LO-TO, które może być wyliczone jeżeli znamy ładunki efektywne jonów i stałą dielektryczną materiału~\cite{PCM}.
+Effective charges take nonzero values in all insulators and play an important role in describing
+lattice dynamics. Ion displacements associated with longitudinal optical modes (LO) at the center of the Brillouin zone ($\bm{k}=0$), which are active in the infrared ({\it infrared modes}), generate macroscopic electric polarization and thus modify interatomic forces.
+These additional forces cause the energy of LO modes near the $\Gamma$ point to be larger than that of transverse optical modes (TO).
+This causes the LO-TO splitting, which can be calculated if we know the effective charges of ions and the dielectric constant of the material~\cite{PCM}.
 
  
-\section{Ferroelektryki}
-
-Ferroelektryki należą do osobnej grupy izolatorów, w których spontanicznie pojawia się polaryzacja elektryczna w określonych
-warunkach termodynamicznych. Praktyczne wykorzystanie ferroelektryków wynika z możliwości przełączania kierunku polaryzacji
-po przyłożeniu pola elektrycznego. W metalach indukowane są prądy elektronowe, które ekranują pole elektryczne 
-i dlatego lokalizacja elektronów jest warunkiem koniecznym występowania makroskopowej polaryzacji elektrycznej.  
-W izolatorach elektrony nie mogą poruszać się swobodnie w krysztale, natomiast biorą udział w prądach polaryzacji, 
-powiązanych z lokalną wartością polaryzacji elektrycznej.  
-W ferroelektrykach ważną role odgrywają wiązania kowalencyjne i hybrydyzacja między różnymi stanami elektronowymi 
-umożliwiająca transfer ładunku między jonami.
-Najbardziej podstawowy mechanizm tworzenia się fazy ferroelektrycznej związany jest ze strukturalnym przejściem fazowym. 
-W fazie paraelektrycznej, występującej zwykle w wyższych temperaturach, sieć krystaliczna posiada symetrię centrosymetryczną,
-w której średnia polaryzacja $\bm{P}=0$. Aby powstała spontaniczna polaryzacja, musi zajść transformacja układu
-do struktury bez centrum symetrii, czyli takiej, która nie jest niezmiennicza wzlędem operacji ($\bm{r}\rightarrow -\bm{r}$).
-To jest warunek konieczny występowania fazy ferroelektrycznej, gdzie $\bm{P}\neq 0$.
+\section{Ferroelectrics}
+
+Ferroelectrics belong to a separate group of insulators in which electric polarization spontaneously appears under specific
+thermodynamic conditions. The practical use of ferroelectrics stems from the possibility of switching the direction of polarization
+after applying an electric field. In metals, electronic currents are induced that screen the electric field 
+and therefore electron localization is a necessary condition for the occurrence of macroscopic electric polarization.  
+In insulators, electrons cannot move freely in the crystal, but they participate in polarization currents, 
+related to the local value of electric polarization.  
+In ferroelectrics, covalent bonds and hybridization between different electronic states play an important role, 
+enabling charge transfer between ions.
+The most basic mechanism for the formation of the ferroelectric phase is associated with a structural phase transition. 
+In the paraelectric phase, which usually occurs at higher temperatures, the crystal lattice has centrosymmetric symmetry,
+in which the average polarization $\bm{P}=0$. For spontaneous polarization to arise, there must be a transformation of the system
+to a structure without a center of symmetry, that is, one that is not invariant with respect to the operation ($\bm{r}\rightarrow -\bm{r}$).
+This is a necessary condition for the occurrence of the ferroelectric phase, where $\bm{P}\neq 0$.
 
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=1]{BaTiO3.pdf}
-\caption{Deformacja sieci krystalicznej ferroelektryka o strukturze perowskitu ABO$_3$.}
+\caption{Deformation of the crystal lattice of a ferroelectric with perovskite structure ABO$_3$.}
 \label{fig:batio3}
 \end{figure}
 
-Przykładem ferroelektryków są materiały o strukturze perowskitu ABO$_3$, pokazanej na rysunku (\ref{fig:batio3}).
-Do tej grupy należy tytanian baru BaTiO$_3$ i wiele innych tlenków zawierajacych metale przejściowe.
-W narożach komórki znajdują się kationy A, natomiast znajdujący się w środku kation B otoczony jest oktaedrem anionów O.
-W wysokich temperaturach kryształ posiada centrosymetryczną symetrię regularną. Przy obniżaniu temperatury w materiale pojawia się
-niestabilność sieci, związana z wystepowaniem miękkiego modu fononowego, którego energia dąży do zera przy zbliżaniu się do przejścia fazowego.
-Wychylenia atomów w miękkim modzie, pokazane schematycznie na rysunku (\ref{fig:batio3}), prowadzą do statycznej 
-deformacji sieci poniżej temperatury krytycznej. W strukturze tetragonalnej złamana jest symetria centrosymetryczna 
-i pojawia się niezerowa polaryzacja elektryczna, której żródłem jest względne przesunięcie anionów (O),
-względem kationów (A i B). Zmiana polaryzacji spowodowana jest nie tylko sztywnym przesunięciem ładunków zlokalizowanych na jonach,
-ale również, jak to zostało wyjaśnione w poprzednich rozdziałach, przepływem prądów elektronowych wzdłuż wiązań kowalencyjnych. 
+An example of ferroelectrics are materials with the perovskite structure ABO$_3$, shown in Figure (\ref{fig:batio3}).
+This group includes barium titanate BaTiO$_3$ and many other oxides containing transition metals.
+At the corners of the cell are cations A, while the cation B located in the center is surrounded by an octahedron of anions O.
+At high temperatures, the crystal has centrosymmetric cubic symmetry. Upon lowering the temperature, lattice instability appears in the material, associated with the occurrence of a soft phonon mode whose energy approaches zero as the phase transition is approached.
+Atomic displacements in the soft mode, shown schematically in Figure (\ref{fig:batio3}), lead to static 
+lattice deformation below the critical temperature. In the tetragonal structure, centrosymmetric symmetry is broken 
+and nonzero electric polarization appears, whose source is the relative displacement of anions (O)
+relative to cations (A and B). The change in polarization is caused not only by the rigid displacement of charges localized on ions,
+but also, as explained in previous chapters, by the flow of electronic currents along covalent bonds. 
  
-Wartość polaryzacji elektrycznej w ferroelektrykach można wyliczyć startując ze stanu początkowego ($\lambda=0$), który odpowiada położeniom jonów w strukturze 
-centrosymetrycznej ($\bm{P}=0$).
-Wtedy zmiana polaryzacji (\ref{pol3}), indukowana przysunięciem jonów do położeń w strukturze niecentrosymetrycznej,
-odpowiada wartości spontanicznej polaryzacji, która pojawia się w wyniku przejścia fazowego.
-Wartość polaryzacji wyliczona dla ferroelektryka KNbO$_3$ wynosi 0.35 C/m$^2$~\cite{resta1993}
-i bardzo dobrze zgadza się z wartością wyznaczoną eksperymentalnie 0.37 C/m$^2$. 
+The value of electric polarization in ferroelectrics can be calculated starting from the initial state ($\lambda=0$), which corresponds to ion positions in the centrosymmetric structure ($\bm{P}=0$).
+Then the change in polarization (\ref{pol3}), induced by displacing ions to positions in the non-centrosymmetric structure,
+corresponds to the value of spontaneous polarization that appears as a result of the phase transition.
+The value of polarization calculated for the ferroelectric KNbO$_3$ is 0.35 C/m$^2$~\cite{resta1993}
+and agrees very well with the experimentally determined value of 0.37 C/m$^2$. 
  
-W ferroelektrykach występują szczególnie duże wartości ładunków efektywnych, często znacznie przewyższające ładunki statyczne jonów. 
-Przykładowo w strukturze regularnej BaTiO$_3$ ładunki efektywne jonów znajdujacych się w czterech nierównoważnych położeniach przyjmują wartości: 
-$Z^*_\text{Ba}=2.75$, $Z^*_\text{Ti}=7.06$, $Z^*_\text{O1}=-5.83$ i $Z^*_\text{O2}=-2.11$~\cite{zhong1994}.
-Znacznie podwyższone wartości ładunków efektywnych jonów Ti i O1 wynikają z silnej hybrydyzacji między stanami $3d$ i $2p$ 
-w wiązaniu kowalencyjnym Ti-O1. Zmiana odległości między tymi jonami generuje transfer ładunku (prąd polaryzacji)
-i zmianę polaryzacji zgodnie ze wzorem (\ref{delP}). Całkowita zmiana polaryzacji składa się z części jonowej, zależnej od ładunków statycznych~(\ref{pol4}),
-i części elektronowej, która powoduje zwiększenie wartości ładunków efektywnych.
-Duże ładunki efektywne powodują, że rozszczepienie LO-TO przyjmuje bardzo duże wartości w ferroelektrykach~\cite{zhong1994}.
+In ferroelectrics, particularly large values of effective charges occur, often significantly exceeding the static charges of ions. 
+For example, in the cubic structure of BaTiO$_3$, the effective charges of ions located at four inequivalent positions take the values: 
+$Z^*_\text{Ba}=2.75$, $Z^*_\text{Ti}=7.06$, $Z^*_\text{O1}=-5.83$, and $Z^*_\text{O2}=-2.11$~\cite{zhong1994}.
+The significantly elevated values of effective charges of Ti and O1 ions result from strong hybridization between $3d$ and $2p$ states 
+in the covalent Ti-O1 bond. A change in the distance between these ions generates charge transfer (polarization current)
+and a change in polarization according to formula (\ref{delP}). The total change in polarization consists of the ionic part, dependent on static charges~(\ref{pol4}),
+and the electronic part, which causes an increase in the value of effective charges.
+Large effective charges cause the LO-TO splitting to take very large values in ferroelectrics~\cite{zhong1994}.
 
  
 \chapter{Oddziaływanie van der Waalsa}

From caea58ccd5f45098bbf09104bb403535c6590e6f Mon Sep 17 00:00:00 2001
From: Copilot <198982749+Copilot@users.noreply.github.com>
Date: Sat, 3 Jan 2026 12:32:10 +0100
Subject: [PATCH 22/23] Translate Chapter 8 (Van der Waals interactions) to
 English (#38)

* Initial plan

* Translate Chapter 8 (Van der Waals interactions) to English

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 148 +++++++++++++++++++++++++-------------------------
 1 file changed, 74 insertions(+), 74 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index 30340a8..b1fbecf 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -3458,184 +3458,184 @@ \section{Ferroelectrics}
 Large effective charges cause the LO-TO splitting to take very large values in ferroelectrics~\cite{zhong1994}.
 
  
-\chapter{Oddziaływanie van der Waalsa}
+\chapter{Van der Waals interactions}
 \label{sec:vdw}
 
-\section{Podstawowe własności}
+\section{Basic properties}
 
-Oddziaływanie van der Waalsa (vdW) nazywane również dyspersyjnym jest efektem korelacji elektronowych, które nie są uwzględnione
-w standardowych funkcjonałach wymienno-korelacyjnych.
-Jest to oddziaływanie przyciągające odgrywające ważną role w kryształach molekularnych, takich jak
-kryształy gazów szlachetnych, w których atomy mają zamknięte powłoki elektronowe,
-oraz materiały składające się z neutralnych cząsteczek. 
-Ważne jest też w układach biologicznych, np. ma duży wpływ na prawidłowy kształt i działanie białek.
-Żródłem tego oddziaływania są fluktacje kwantowe dipola elektrycznego na jednym atomie (molekule), które indukują dipol elektryczny na innym atomie (molekule).
-Wzajemne oddziaływanie tych dipoli prowadzi do słabego przyciągania atomów lub molekuł. 
-W klasycznym opisie, pole elektryczne wytworze przez moment dipolowy $p_1$ w odległości $R$ można wyrazić wzorem 
+Van der Waals (vdW) interaction, also called dispersive interaction, is an effect of electronic correlations that are not included
+in standard exchange-correlation functionals.
+It is an attractive interaction playing an important role in molecular crystals, such as
+crystals of noble gases, in which atoms have closed electron shells,
+as well as materials consisting of neutral molecules.
+It is also important in biological systems, e.g., having a large influence on the proper shape and function of proteins.
+The source of this interaction is quantum fluctuations of the electric dipole on one atom (molecule), which induce an electric dipole on another atom (molecule).
+The mutual interaction of these dipoles leads to weak attraction of atoms or molecules. 
+In the classical description, the electric field created by the dipole moment $p_1$ at distance $R$ can be expressed by the formula 
 %
 \begin{equation}
 E=\frac{p_1}{R^3}.
 \end{equation}
 % 
-Jeżeli pole elektryczne oddziałuje na cząsteczkę o polaryzowalności $\alpha$ to indukuje moment dipolowy
+If the electric field acts on a molecule with polarizability $\alpha$ then it induces a dipole moment
 %
 \begin{equation}
 p_2=\frac{\alpha p_1}{R^3}.
 \end{equation}
 %
-Średnia energia oddziaływania tych dwóch dipoli wynosi
+The average interaction energy of these two dipoles is
 %
 \begin{equation}
 U=-\frac{\alpha p_1^2}{R^6}.
 \end{equation}
 %
-Uwzglednienie korelacji elektronowych w ramach teorii perturbacyjneje prowadzi do takiej samej zależność oddziaływania 
-od odległości $\sim R^{-6}$, ale poprawne wyznaczenie polaryzowalności i siły oddziaływania wymaga wyjścia poza najprostsze 
-przybliżenie faz przypadkowych ({\it ang. random-phase approximation} - RPA).   
+Accounting for electronic correlations within perturbation theory leads to the same dependence of the interaction
+on distance $\sim R^{-6}$, but correct determination of polarizability and interaction strength requires going beyond the simplest
+random-phase approximation (RPA).   
  
-\section{Poprawki do funkcjonału energii}
+\section{Energy functional corrections}
  
-\subsection{Metody D2 i D3}  
+\subsection{D2 and D3 methods}  
  
-W obliczeniach DFT można uwzlędnić przybliżoną wartość energii oddziaływania vdW jako poprawkę do całkowitej energii układu
+In DFT calculations, one can include the approximate value of the vdW interaction energy as a correction to the total energy of the system
 %
 \begin{equation}
 E_\text{tot}=E_\text{KS}+E_\text{vdW}
 \end{equation}
 %
-gdzie $E_\text{KS}$ jest wyznaczonym w procedurze samozgodnej funkcjonałem Kohna-Shama.
-W takim podejściu oddziaływanie vdW nie wpływa bezpośrednio na strukturę elektronową, 
-jedynie poprzez modyfikację odległości między atomami.
+where $E_\text{KS}$ is the Kohn-Sham functional determined in the self-consistent procedure.
+In this approach, vdW interaction does not directly affect the electronic structure,
+only through modification of distances between atoms.
  
-W wersji, którą zaproponował Grimme~\cite{grimme2004,grimme2006}, zależność energii vdW od odległości między atomami $R_{ij}$
-ma postać
+In the version proposed by Grimme~\cite{grimme2004,grimme2006}, the dependence of vdW energy on the distance between atoms $R_{ij}$
+has the form
 %
 \begin{equation}
 E_\text{vdW}=-\frac{1}{2}\sum_{i,j}f_\text{d}(R_{ij})\frac{C_{6ij}}{R_{ij}^6},
 \end{equation}
 % 
-gdzie $f_\text{d}$ jest funkcją tłumiącą, która określa asymptotyczne zachowanie dla $R_{ij}\rightarrow 0$  
-i wyklucza podwójne uzwględnianie efektów korelacyjnych przy pośrednich odległościach.
-Jej rolą jest zapewnienie, że oddziaływanie vdW ograniczone jest do odległości większych niż 
-długości typowych wiązań, poprawnie opisywanych w ramach standardowych funkcjonałów.
-Współczynniki dyspersyjne $C_{6ij}$ zdefiniowane są dla każdej pary atomów
-jako średnie geometryczne parametrów atomowych
+where $f_\text{d}$ is a damping function, which determines the asymptotic behavior for $R_{ij}\rightarrow 0$
+and excludes double counting of correlation effects at intermediate distances.
+Its role is to ensure that vdW interaction is limited to distances larger than
+the lengths of typical bonds, correctly described within standard functionals.
+The dispersion coefficients $C_{6ij}$ are defined for each pair of atoms
+as geometric means of atomic parameters
 %
 \begin{equation}
 C_{6ij}=\sqrt{C_{6i}C_{6j}}.
 \end{equation}
 %
-Dla każdego atomu określa się parametr $C_6=0.05NI_\text{p}\alpha$, gdzie $I_\text{p}$ jest potencjałem jonizującym, $\alpha$ jest
-statyczną polaryzowalnościa dipolową, a $N=2, 10, 18, 36, 54$ dla kolejnych rzędów układu okresowego pierwiastków.  
-Współczynnik skalujący wyznaczonony jest tak, aby zreprodukować odległości międzyatomowe i energie wiązania
-w odpowiedniej grupie pierwiastków. Funkcję tłumiącą można zapisać w formie \cite{grimme2006}
+For each atom, the parameter $C_6=0.05NI_\text{p}\alpha$ is determined, where $I_\text{p}$ is the ionization potential, $\alpha$ is
+the static dipole polarizability, and $N=2, 10, 18, 36, 54$ for consecutive rows of the periodic table of elements.
+The scaling coefficient is determined so as to reproduce interatomic distances and binding energies
+in the corresponding group of elements. The damping function can be written in the form \cite{grimme2006}
 %
 \begin{equation}
 f_\text{d}(R_{ij})=\frac{s_6}{1+e^{-d(R_{ij}/R_{0ij}-1)}},
 \label{damping}
 \end{equation}
 %
-gdzie $s_6$ jest współczynnikiem określonym dla danego funkcjonału DFT, np. $s_6=0.75$ dla PBE,
-$d$ jest współczynnikim tłumienia, którego typowa wartość wynosi 20, a $R_{0ij}=R_{0i}+R_{0j}$ jest promieniem odcięcia
-i wyraża się sumą promieni vdW dla danych atomów. 
-Parametry $C_{6i}$ i $R_{0i}$ są zdefiniowane dla wszystkch atomów i nie zależą od rodzaju materiału. 
-Dodatkowo w obliczeniach określa się maksymalny zasięg oddziaływania vdW. 
-Opisany schemat (DFT-D2) umożliwia poprawienie wyliczanych sił i odległości międzyatomowych w kryształach gazów szlachetnych oraz
-układach molekularnych~\cite{bucko2010}. Oddziaływanie vdW odgrywa również ważną rolę w materiałach warstwowych,
-dla których standardowe obliczenia DFT dają zwykle zaniżoną wartość siły wiążącej między warstwami. 
-Przykładowo, stała sieci $c$ w graficie, której wartość w przybliżeniu PBE ($c=8.84$~\AA)
-jest znacznie zawyżona w porównaniu do wartości eksperymentalnej ($6.71$~\AA), wyliczona z poprawką D2 wynosi $6.45$~\AA~\cite{bucko2010}.     
+where $s_6$ is a coefficient determined for a given DFT functional, e.g., $s_6=0.75$ for PBE,
+$d$ is the damping coefficient, whose typical value is 20, and $R_{0ij}=R_{0i}+R_{0j}$ is the cutoff radius
+and is expressed as the sum of vdW radii for the given atoms.
+The parameters $C_{6i}$ and $R_{0i}$ are defined for all atoms and do not depend on the type of material.
+Additionally, in calculations, the maximum range of vdW interaction is determined.
+The described scheme (DFT-D2) enables improvement of calculated forces and interatomic distances in crystals of noble gases and
+molecular systems~\cite{bucko2010}. The vdW interaction also plays an important role in layered materials,
+for which standard DFT calculations usually give an underestimated value of the binding force between layers.
+For example, the lattice constant $c$ in graphite, whose value in the PBE approximation ($c=8.84$~\AA)
+is significantly overestimated compared to the experimental value ($6.71$~\AA), calculated with the D2 correction is $6.45$~\AA~\cite{bucko2010}.     
 
-W kolejnym podejściu (D3), Grimme uwzględnił dodatkowo drugi wyraz w wzorze na energię oddziaływania proporcjonalny do $R^{-8}$~\cite{grimme2010}
+In the next approach (D3), Grimme additionally included a second term in the interaction energy formula proportional to $R^{-8}$~\cite{grimme2010}
 %
 \begin{equation}
 E_\text{vdW}=-\frac{1}{2}\sum_{i,j}[f_{\text{d},6}(R_{ij})\frac{C_{6ij}}{R_{ij}^6}+f_{\text{d},8}(R_{ij})\frac{C_{8ij}}{R_{ij}^8}].
 \end{equation}
 %
-Współczynniki dyspersyjne $C_{6ij}$ wyznaczane są ze wzoru Casimira-Poldera 
+The dispersion coefficients $C_{6ij}$ are determined from the Casimir-Polder formula 
 %
 \begin{equation}
 C_{6ij}=\frac{3}{\pi} \int d\omega \alpha_i(\omega)\alpha_j(\omega), 
 \label{CP}
 \end{equation}
 %
-gdzie $\alpha_i(\omega)$ jest średnią polaryzowalnością dipolową dla atomu $i$.
-Współczynniki $C_{8ij}$ wyliczane są z zależności $C_{8ij}=3C_{6ij}\sqrt{Q_iQ_j}$, gdzie $Q_i=s\sqrt{Z_i}\langle r^4\rangle_i/\langle r^2\rangle_i$.
-Funkcje tłumiące mają postać
+where $\alpha_i(\omega)$ is the average dipole polarizability for atom $i$.
+The coefficients $C_{8ij}$ are calculated from the relation $C_{8ij}=3C_{6ij}\sqrt{Q_iQ_j}$, where $Q_i=s\sqrt{Z_i}\langle r^4\rangle_i/\langle r^2\rangle_i$.
+The damping functions have the form
 %
 \begin{equation}
 f_{\text{d},n}(R_{ij})=\frac{s_n}{1+6(R_{ij}/(s_{R,n}R_{0ij}))^{-\alpha_n}}.
 \end{equation}
 %
-W tym wzorze optymalizowane są parametry $s_6$, $s_8$ i $s_{R,6}$, a pozostałe mają ustalone wartości ($s_{R,8}=1$, $\alpha_6=14$, $\alpha_8=16$). 
-Alternatywnie można zastosować funkcję tłumiącą Becke-Johnsona (BJ) w postaci
+In this formula, the parameters $s_6$, $s_8$, and $s_{R,6}$ are optimized, while the others have fixed values ($s_{R,8}=1$, $\alpha_6=14$, $\alpha_8=16$).
+Alternatively, the Becke-Johnson (BJ) damping function can be applied in the form
 %
 \begin{equation}
 f_{\text{d},n}(R_{ij})=\frac{s_nR_{ij}^n}{R_{ij}^n+(a_1R_{0ij}+a_2)^n},
 \end{equation}
 %
-gdzie $s_n$, $a_1$ i $a_2$ sa parametrami dopasowania. Promień odcięcia można wyznaczyć znając współczynniki dyspersyjne
-$R_{0ij}=\sqrt{C_{8ij}/C_{6ij}}$. Analiza porównawcza przeprowadzona dla dużej grupy materiałów
-pokazała niewielki wpływ rodzaju funkcji tłumiącej na wyliczone odległości między atomami i trochę lepszą dokładność 
-przy zastosowaniu funkcji BJ do własności termodynamicznych~\cite{grimme2011}. 
+where $s_n$, $a_1$, and $a_2$ are fitting parameters. The cutoff radius can be determined knowing the dispersion coefficients
+$R_{0ij}=\sqrt{C_{8ij}/C_{6ij}}$. Comparative analysis performed for a large group of materials
+showed a small influence of the type of damping function on calculated distances between atoms and slightly better accuracy
+when using the BJ function for thermodynamic properties~\cite{grimme2011}. 
 
-\subsection{Metoda TS}
+\subsection{TS method}
 
-W metodzie zaproponowanej przez Tkatchenko i Schefflera (TS)~\cite{tkatchenko2009} bierze się pod uwagę tylko wyraz proporcjonalny do $R^{-6}$, ale
-uwzględnia się zależność współczynników dyspersyjnych i promienia odcięcią od gęstości ładunku.
-Wprowdzamy przybliżoną zależność polaryzowalności od częstości
+In the method proposed by Tkatchenko and Scheffler (TS)~\cite{tkatchenko2009}, only the term proportional to $R^{-6}$ is taken into account, but
+the dependence of dispersion coefficients and cutoff radius on charge density is considered.
+We introduce an approximate dependence of polarizability on frequency
 %
 \begin{equation}
 \alpha_i(\omega)=\frac{\alpha^0_i}{1-(\omega/\eta_i)^2},
 \label{alpha}
 \end{equation}
 %
-gdzie $\alpha^0_i$ jest statyczną polaryzowalnością i $\eta_i$ jest efektywną częstością dla atomu $i$.
-Po wstawieniu (\ref{alpha}) do (\ref{CP}) otrzymujemy formułę Londona
+where $\alpha^0_i$ is the static polarizability and $\eta_i$ is the effective frequency for atom $i$.
+After substituting (\ref{alpha}) into (\ref{CP}) we obtain the London formula
 %
 \begin{equation}
 C_{6ij}=\frac{3\eta_i\eta_j}{2(\eta_i+\eta_j)}\alpha_i^0\alpha_j^0.
 \end{equation}
 %
-Dla jednakowych atomów ($i=j$) otrzymujemy wzór
+For identical atoms ($i=j$) we obtain the formula
 %
 \begin{equation}
 \eta_i=\frac{4C_{6i}}{3(\alpha_i^0)^2},
 \end{equation}
 %
-który prowadzi do zależności
+which leads to the relation
 %
 \begin{equation}
 C_{6ij}=\frac{2C_{6i}C_{6j}}{\frac{\alpha^0_j}{\alpha^0_i}C_{6i}+\frac{\alpha^0_i}{\alpha^0_j}C_{6j}}.
 \end{equation}
 %
-Współczynniki dyspersyjne dla poszczególnych atomów znajdujących się wewnątrz cząsteczek lub ciał stałych
-wyznacza się z wartości dla izolowanych atomów używając następującej formuły $C_{6i}=\nu_i^2C^{free}_{6i}$, gdzie $\nu_i$
-określa efektywną objętość atomu (podział Hirshfelda)
+The dispersion coefficients for individual atoms located inside molecules or solids
+are determined from the values for isolated atoms using the following formula $C_{6i}=\nu_i^2C^{free}_{6i}$, where $\nu_i$
+determines the effective volume of the atom (Hirshfeld partitioning)
 %
 \begin{equation}
 \nu_i=\frac{V_i^{eff}}{V_i^{free}}=\frac{\int d\bm{r} r^3 w_i(\bm{r}) n(\bm{r})}{\int d\bm{r} r^3 n_i^{free}(\bm{r})},
 \end{equation}
 %
-z wagami Hirshfelda, które wyliczane są dla uśrednionych sferycznie gęstości elektronowych w izolowanych atomach
+with Hirshfeld weights, which are calculated for spherically averaged electron densities in isolated atoms
 %
 \begin{equation}
 w_i(\bm{r})=\frac{n_i^{free}(\bm{r})}{\sum_j n_i^{free}(\bm{r})}.
 \end{equation}
 %
-Podobnie znając wartości polaryzowalności i promieni vdW dla izolowanych atomów, można wyznaczyć te wartości
-dla atomów w cząsteczkach i kryształach: $\alpha_i=\nu_i\alpha^{free}$ i $R_{0i}=\nu_i^{\frac{1}{3}}R_{0i}^{free}$.
-Promienie vdW służą do wyznaczenia promienia odcięcia $R_{0ij}$ i funkcji tłumiacej $f_{\text{d},n}$ (\ref{damping}).
+Similarly, knowing the values of polarizability and vdW radii for isolated atoms, one can determine these values
+for atoms in molecules and crystals: $\alpha_i=\nu_i\alpha^{free}$ and $R_{0i}=\nu_i^{\frac{1}{3}}R_{0i}^{free}$.
+The vdW radii are used to determine the cutoff radius $R_{0ij}$ and the damping function $f_{\text{d},n}$ (\ref{damping}).
 
-W omówionych podejściach zakłada się jedynie oddziaływania dwuciałowe, nie biorąc pod uwagę modyfikacji oddziaływania vdW przez otaczające atomy lub molekuły. 
-W udoskonalonej wersji metody TS uwzglednia się dalekozasięgowe ekranowanie elektryczne w sposób samozgodny ({\it ang. self-consistent screening} - SCS)~\cite{tkatchenko2012}. Odpowiednie równanie samozgodne pozwalające wyznaczyć zmodyfikowaną polaryzowalność ma postać
+In the discussed approaches, only two-body interactions are assumed, not taking into account the modification of vdW interaction by surrounding atoms or molecules.
+In an improved version of the TS method, long-range electrical screening is included in a self-consistent manner (self-consistent screening - SCS)~\cite{tkatchenko2012}. The appropriate self-consistent equation allowing determination of the modified polarizability has the form
 %
 \begin{equation}
 \alpha_i^\text{SCS}(\omega)=\alpha_i(\omega)-\alpha_i(\omega)\sum_{i\neq j}\tau_{ij}\alpha_i^\text{SCS}(\omega),
 \end{equation}
 %
-gdzie $\tau_{ij}$ jest tensorem oddziaływania dipol-dipol. Energię dyspersyjną wyznacza się z tych samych równań co w metodzie TS
-z odpowiednio zmodyfikowanymi współczynnikami dyspersyjnymi i promieniami odcięcia
+where $\tau_{ij}$ is the dipole-dipole interaction tensor. The dispersion energy is determined from the same equations as in the TS method
+with appropriately modified dispersion coefficients and cutoff radii
 %
 \begin{equation}
 C_{6i}=\frac{3}{\pi}\int_0^\infty d\omega[\alpha^\text{SCS}_i(\omega)]^2,

From 585c817f12ba72da61dff7a8b8addc612b2e45d9 Mon Sep 17 00:00:00 2001
From: Copilot <198982749+Copilot@users.noreply.github.com>
Date: Sat, 3 Jan 2026 13:25:48 +0100
Subject: [PATCH 23/23] Translate Chapter 9 (Many-body effects and excited
 states) to English (#39)

* Initial plan

* Translate Chapter 9 to English

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Refine Chapter 9 translation based on code review feedback

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

* Remove review markers after user confirmation of terminology

Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
Co-authored-by: jochym <5993422+jochym@users.noreply.github.com>
---
 book/main.tex | 172 +++++++++++++++++++++++++-------------------------
 1 file changed, 86 insertions(+), 86 deletions(-)

diff --git a/book/main.tex b/book/main.tex
index b1fbecf..7b86b97 100644
--- a/book/main.tex
+++ b/book/main.tex
@@ -3647,38 +3647,38 @@ \subsection{TS method}
 
 %\subsection{Metoda MBD}
 
-\chapter{Efekty wielociałowe i stany wzbudzone}
+\chapter{Many-body effects and excited states}
 
-\section{Funkcje Greena i energia własna}
+\section{Green's functions and self-energy}
 
-Dokładny opis stanów elektronowych wymaga wyjścia poza przybliżenie niezależnych cząstek.   
-Stosując formalizm drugiego kwantowania, zapiszemy hamiltonian układu oddziałujących elektronów 
-w ogólnej formie
+An accurate description of electronic states requires going beyond the independent particle approximation.   
+Using the second quantization formalism, we write the Hamiltonian of a system of interacting electrons 
+in the general form
 %
 \begin{equation}
 H=\int d^3r \psi^\dagger(\bm{r},t) h_0(\bm{r}) \psi(\bm{r},t) +\frac{1}{2}\int d^3rd^3r' \psi^\dagger(\bm{r},t)\psi^\dagger(\bm{r}',t)v(\bm{r}-\bm{r}') \psi^\dagger(\bm{r}',t)\psi^\dagger(\bm{r},t),
 \end{equation}
 %  
-gdzie $h_0(x)$ opisuje energię kinetyczną oraz oddziaływanie elektronów z zewnętrznym polem, $v(\bm{r}-\bm{r}')$ jest
-potencjałem oddziaływania kulombowskiego między elektronami, a $\psi(\bm{r},t)$ jest operatorem pola w obrazie Heisenberga 
+where $h_0(x)$ describes the kinetic energy and the interaction of electrons with an external field, $v(\bm{r}-\bm{r}')$ is
+the Coulomb interaction potential between electrons, and $\psi(\bm{r},t)$ is the field operator in the Heisenberg picture 
 %
 \begin{equation}
 \psi(\bm{r},t)=e^{\frac{i}{\hbar}Ht}\psi(\bm{r})e^{-\frac{i}{\hbar}Ht}.
 \label{field}
 \end{equation}
 %
-Operatory pola $\psi^\dagger(\bm{r})$ i $\psi(\bm{r})$ opisują kreacje i anihilację elektronu w punkcie $\bm{r}$.
+The field operators $\psi^\dagger(\bm{r})$ and $\psi(\bm{r})$ describe the creation and annihilation of an electron at point $\bm{r}$.
 
-Stan podstawowy $|\Phi_N\rangle$ układu $N$ oddziałujących elektronów spełnia równanie 
+The ground state $|\Phi_N\rangle$ of a system of $N$ interacting electrons satisfies the equation 
 %
 \begin{equation}
 H|\Phi_N\rangle=E|\Phi_N\rangle.
 \end{equation}
 %
-Dalej zakładamy, że stan podstawowy jest unormowany $\langle\Phi_N|\Phi_N\rangle=1$.
+Further, we assume that the ground state is normalized $\langle\Phi_N|\Phi_N\rangle=1$.
 
-W kwantowej teorii układów wielu czastek można powiązać spektrum wzbudzeń z jednocząstkowymi funkcjami Greena, 
-zwanymi również propagatorami, które definiujemy
+In the quantum theory of many-particle systems, the excitation spectrum can be related to single-particle Green's functions, 
+also called propagators, which we define
 %
 \begin{equation} 
 iG(x,x')=\langle \Phi_N|T\psi(x)\psi^\dagger(x')|\Phi_N\rangle=\begin{cases}
@@ -3688,39 +3688,39 @@ \section{Funkcje Greena i energia własna}
 \label{greenf}                      
 \end{equation}
 % 
-gdzie wprowadzono oznaczenie $x=(\bm{r},t)$, a $T$ jest operatorem uporządkowania w czasie. 
-Funkcja Greena jest amplitudą prawdopodobieństwa propagacji elektronu z punktu $x'$ do punktu $x$ dla $t>t'$ 
-lub dziury z punktu $x$ do $x'$ dla $t'>t$. Te dwa przypadki opisują odpowiednio proces fotoemisji i odwrotnej fotoemisji.
-Funkcje Greena dla elektronów są macierzami ze względu na spinowe stopnie swobody. Dla uproszczenia zapisu pomijamy
-tutaj wskaźniki numerujące stany spinowe.
-Zajomość funkcji Green pozwala wyznaczyć energię stanu podstawowego, wartości oczekiwanego jednocząstkowych operatorów
-w stanie podstawowym oraz jednoelektronowe spektrum wzbudzeń.
-Funkcje Greena spełniaja równanie ruchu
+where we introduced the notation $x=(\bm{r},t)$, and $T$ is the time-ordering operator. 
+The Green's function is the probability amplitude for the propagation of an electron from point $x'$ to point $x$ for $t>t'$ 
+or a hole from point $x$ to $x'$ for $t'>t$. These two cases describe the processes of photoemission and inverse photoemission spectroscopy, respectively.
+Green's functions for electrons are matrices with respect to spin degrees of freedom. For simplicity of notation, we omit
+the indices labeling spin states here.
+Knowledge of the Green's function allows us to determine the ground state energy, expectation values of single-particle operators
+in the ground state, and the single-electron excitation spectrum.
+Green's functions satisfy the equation of motion
 %
 \begin{equation}
 [i\frac{\partial}{\partial t}-H_0(x))]G(x,x')-\int dx''\Sigma(x,x'')G(x'',x')=\delta(x-x'),
 \end{equation}
 %
-gdzie $H_0 =h_0+V_H$ ($V_H$ jest potencjałem Hartree), a $\Sigma$ nosi nazwę energii własnej i opisuje oddziaływania wymienno-korelacyjne.
+where $H_0 =h_0+V_H$ ($V_H$ is the Hartree potential), and $\Sigma$ is called the self-energy and describes exchange-correlation interactions.
 
-Dla układu jednorodnego o objętości $V$, funkcję Greena możemy zapisać w formie transformaty Fouriera
+For a homogeneous system of volume $V$, the Green's function can be written in the form of a Fourier transform
 %
 \begin{equation}
 G(x,x')=\sum_{\bm{k}}\int\frac{d\omega}{2\pi V}e^{i\bm{k}(\bm{r}-\bm{r}')}e^{-i\omega(t-t')}G(\bm{k},\omega).
 \label{fourier}
 \end{equation}
 %
-$G(\bm{k},\omega)$ jest funkcją Greena, która opisuje propagację elektronu o zdefiniowanym wektorze falowym $\bm{k}$.
-Rozważmy na początku układ $N$ nieodziałujących elektronów. Operatory pola można zapisać
+$G(\bm{k},\omega)$ is the Green's function that describes the propagation of an electron with a defined wave vector $\bm{k}$.
+Let us first consider a system of $N$ non-interacting electrons. The field operators can be written as
 %
 \begin{equation}
 \psi(\bm{r})=\frac{1}{V}\sum_{\bm{k}}e^{-i\bm{k}\bm{r}}c_{\bm{k}},
 \label{free}
 \end{equation}
 %
-gdzie $c_{\bm{k}}$ jest operatorem anihilacji electronu, dla którego $c_{\bm{k}}|\Phi_N\rangle=0$.
-Odpowiednio $c^\dagger_{k}$ jest operatorem kreacji elektronu o wektorze falowym $\bm{k}$.
-Można przedefiniować operatory kreacji i anihilacji wprowadzając osobne oznaczenia dla cząstek i dziur
+where $c_{\bm{k}}$ is the electron annihilation operator, for which $c_{\bm{k}}|\Phi_N\rangle=0$.
+Correspondingly, $c^\dagger_{k}$ is the electron creation operator with wave vector $\bm{k}$.
+We can redefine the creation and annihilation operators by introducing separate notations for particles and holes
 %
 \begin{equation}
 c_{\bm{k}}=\begin{cases}
@@ -3728,171 +3728,171 @@ \section{Funkcje Greena i energia własna}
 b^\dagger_{-\bm{k}} & k<k_F,\end{cases}
 \end{equation}
 %
-gdzie $k_F$ jest wektorem falowym Fermiego. Hamiltonian w tej reprezentacji przyjmuje postać
+where $k_F$ is the Fermi wave vector. The Hamiltonian in this representation takes the form
 %
 \begin{equation}
 H_0=\sum_{\bm{k}}\hbar\omega_{\bm{k}} c^\dagger_{\bm{k}}c_{\bm{k}}=\sum_{\bm{k}>\bm{k}_F}\hbar\omega_{\bm{k}} a^\dagger_{\bm{k}}a_{\bm{k}}-\sum_{\bm{k}<\bm{k}_F}\hbar\omega_{\bm{k}} b^\dagger_{\bm{k}}b_{\bm{k}}+\sum_{\bm{k}<\bm{k}_F}\hbar\omega_{\bm{k}}.
 \end{equation}
 %
-Wykorzystując wzory (\ref{free}) i (\ref{field}) otrzymujemy zależność dla funkcji Greena swobodnych elektronów
+Using formulas (\ref{free}) and (\ref{field}), we obtain the relation for the Green's function of free electrons
 %
 \begin{equation}
 iG_0(x,x')=\frac{1}{(2\pi)^3}\int d^3k e^{i\bm{k}(\bm{r}-\bm{r}')}e^{-i\omega_{\bm{k}}(t-t')}[\theta(t-t')\theta(k-k_F)-\theta(t'-t)\theta(k_F-k)].
 \end{equation}
 %
-gdzie sumowanie po stanach $\bm{k}$ zostało zamienione na całkę.
-Funkcję schodkową $\theta(t-t')$  można zapisać przy pomocy wzoru
+where the summation over states $\bm{k}$ has been replaced by an integral.
+The step function $\theta(t-t')$ can be written using the formula
 %
 \begin{equation}
 \theta(t-t')=-\int_{-\infty}^{\infty}\frac{d\omega}{2\pi i}\frac{e^{-i\omega(t-t')}}{\omega+i\eta}=\begin{cases}1 & t>t',\\0 & t<t',\end{cases}
 \end{equation}
 % 
-gdzie $\eta$ jest infinitezymalnie małą liczbą. Zastosowanie tego wyrażenie po prostych przekształceniach
-prowadzi do wzoru
+where $\eta$ is an infinitesimally small number. Application of this expression after simple transformations
+leads to the formula
 %
 \begin{equation}
 G_0(x,x')=\frac{1}{(2\pi)^4}\int d^3k e^{i\bm{k}(\bm{r}-\bm{r}')}e^{-i\omega(t-t')}\Big{[}\frac{\theta(k-k_F)}{\omega-\omega_{\bm{k}}+i\eta}+\frac{\theta(k_F-k)}{\omega-\omega_{\bm{k}}-i\eta}\Big{]}.
 \end{equation}
 %
-Porównujące ten wzór z (\ref{fourier}) dostajemy wyrażenie na funkcję Greena swobodnych elektronów
-w przestrzeni pędów
+Comparing this formula with (\ref{fourier}), we obtain the expression for the Green's function of free electrons
+in momentum space
 %
 \begin{equation}
 G_0(\bm{k},\omega)=\frac{\theta(k-k_F)}{\omega-\omega_{\bm{k}}+i\eta}+\frac{\theta(k_F-k)}{\omega-\omega_{\bm{k}}-i\eta},
 \label{G0}
 \end{equation}
 %
-Występujące w tym wzorze dwa wyrazy odpowiadają odpowiednio propagacji elektronu dla wektora falowego $k$
-większego do wektor falowego Fermiego $k_F$ i dziury dla $k<k_F$. 
+The two terms appearing in this formula correspond to the propagation of an electron for wave vector $k$
+greater than the Fermi wave vector $k_F$ and a hole for $k<k_F$. 
 
-Funckję Greena dla oddziałujących elektronów można zapisać w postaci szeregu perturbacyjnego
+The Green's function for interacting electrons can be written as a perturbation series
 %
 \begin{equation}
 G(\bm{k},\omega)=G_0(\bm{k},\omega)+G_0(\bm{k},\omega)\Sigma_{\bm{k}}(\omega)G_0(\bm{k},\omega)+G_0(\bm{k},\omega)\Sigma_{\bm{k}}(\omega) G_0(\bm{k},\omega)\Sigma_{\bm{k}}(\omega) G_0(\bm{k},\omega)+...
 \end{equation}
 %
-gdzie $\Sigma_{\bm{k}}(\omega)$ jest transformatą Fouriera energii własnej. 
-Szereg ten można zapisać w formie równania Dysona
+where $\Sigma_{\bm{k}}(\omega)$ is the Fourier transform of the self-energy. 
+This series can be written in the form of the Dyson equation
 %
 \begin{equation}
 G(\bm{k},\omega)=G_0(\bm{k},\omega)+G_0(\bm{k},\omega)\Sigma_{\bm{k}}(\omega)G(\bm{k},\omega),
 \end{equation}
 %
-lub po przekształceniu
+or after transformation
 %
 \begin{equation}
 \Sigma_{\bm{k}}(\omega)=G_0^{-1}(\bm{k},\omega)-G^{-1}(\bm{k},\omega).
 \end{equation}
 %
-Łącząc ten wzór z wyrażeniem (\ref{G0}) dla swobodnych elektronów ($k>k_F$)
-dostajemu wzór na funkcję Greena
+Combining this formula with expression (\ref{G0}) for free electrons ($k>k_F$),
+we obtain the formula for the Green's function
 %
 \begin{equation}
 G(\bm{k},\omega)=\frac{1}{\omega-\omega_{\bm{k}}-\Sigma_{\bm{k}}(\omega)}.
 \end{equation}
 %
-Z tego wzoru wynika, że jednocząstkowe stany o energii $\omega_{\bm{k}}$ są modyfikowane zarówno
-przez rzeczywistą część energii własnej, która przesuwa wartości energii, jak również przez
-jej część urojoną. 
+From this formula, it follows that single-particle states with energy $\omega_{\bm{k}}$ are modified both
+by the real part of the self-energy, which shifts the energy values, as well as by
+its imaginary part. 
 
-Znając funkcję Greena można wyznaczyć funkcję spektralną, która opisuje widmo energetyczne elektronów w funkcji wektora falowego,
-wyznaczane w pomiarach metodą kątowo-rozdzielczej spektroskopii fotoemisyjnej (ARPES).
-Funkcja spektralna wyznaczana jest części urojonej funkcji Greena
+Knowing the Green's function, we can determine the spectral function, which describes the energy spectrum of electrons as a function of wave vector,
+measured in experiments by angle-resolved photoemission spectroscopy (ARPES).
+The spectral function is determined from the imaginary part of the Green's function
 %
 \begin{equation}
 A_{\bm{k}}(\omega)=\frac{1}{\pi}|\text{Im} G_{\bm{k}}(\omega)|=\frac{1}{\pi}\frac{|\text{Im} \Sigma_{\bm{k}}(\omega)|}{[\omega-\omega_{\bm{k}}-\text{Re} \Sigma_{\bm{k}}(\omega)]^2+[\text{Im} \Sigma_{\bm{k}}(\omega)]^2}.
 \end{equation}
 %
-Dla układu nieoddziałujących elektronów ($\Sigma=0$) funkcja spektralna składa się z delt Diraca w położeniach,
-które odpowiadają energiom $\omega_{\bm{k}}$.
-Część rzeczywista energii własnej powoduje przesunięcie energii elektronów
+For a system of non-interacting electrons ($\Sigma=0$), the spectral function consists of Dirac delta functions at positions
+corresponding to energies $\omega_{\bm{k}}$.
+The real part of the self-energy causes a shift in electron energies
 %
 \begin{equation}
 \varepsilon_{\bm{k}}=\omega_{\bm{k}}+\Sigma_{\bm{k}}(\varepsilon_{\bm{k}}),
 \label{e_k}
 \end{equation}
 %
-natomiast część urojona powoduje poszerzenie pików, co związane jest z redukcją czasu życia stanów elektronowych.
-Jeżeli część urojona powoduje tylko niewielkie zaburzenie, istnieje jednoznaczna relacja między stanami
-elektronów swobodnych i elektronów oddziałujących. Takie stany elektronowe lub dziurowe, które występują blisko energii Fermiego nazywamy {\it kwasicząstkami}.
-Przy zbliżaniu się do poziomu Fermiego czas życia stanów elektronów rośnie ze względu na zmniejszajacą się ilość dostępnych stanów do których elektron
-może się rozproszyć. 
+while the imaginary part causes broadening of the peaks, which is related to the reduction of the lifetime of electronic states.
+If the imaginary part causes only a small perturbation, there is an unambiguous relationship between the states
+of free electrons and interacting electrons. Such electronic or hole states that occur near the Fermi energy are called {\it quasiparticles}.
+When approaching the Fermi level, the lifetime of electron states increases due to the decreasing number of available states to which an electron
+can scatter. 
 
-Blisko energii Fermiego część rzeczywista energii wałsnej zależy liniowo od energii i można ją rozwinąć do liniowego wyrazu
+Near the Fermi energy, the real part of the self-energy depends linearly on energy and can be expanded to the linear term
 %
 \begin{equation}
 \text{Re} \Sigma_{\bm{k}}(\omega)=\text{Re} \Sigma_{\bm{k}}(\omega_{\bm{k}})+(\omega-\omega_{\bm{k}})\frac{\partial\text{Re}\Sigma_{\bm{k}}(\omega)}{\partial\omega}\Big{|}_{\omega=\omega_{\bm{k}}}.
 \end{equation}
 %
-Wprowadzając czynnik renormalizacyjny, który jest miarą charakteryzującą spektrum kwasicząstek
+Introducing the renormalization factor, which is a measure characterizing the quasiparticle spectrum
 %
 \begin{equation}
 Z_{\bm{k}}=\frac{1}{1-\frac{\partial\text{Re}\Sigma_{\bm{k}}(\omega)}{\partial\omega}\Big{|}_{\omega=\omega_{\bm{k}}}}
 \end{equation}
 %
-możemy zapisać wzór na energię (\ref{e_k}) w postaci
+we can write the energy formula (\ref{e_k}) in the form
 %
 \begin{equation}
 \varepsilon_{\bm{k}}=\omega_{\bm{k}}+Z_{\bm{k}}\text{Re}\Sigma_{\bm{k}}(\omega_{\bm{k}}).
 \end{equation}
 %
-Również gęstość spektralną kwasicząstek można przybliżyć wzorem
+The spectral density of quasiparticles can also be approximated by the formula
 %
 \begin{equation}
 A_{\bm{k}}(\omega)=Z_{\bm{k}}\frac{Z_{\bm{k}}\text{Im}\Sigma_{\bm{k}}(\varepsilon_{\bm{k}})}{(\omega-\varepsilon_{\bm{k}})^2+[Z_{\bm{k}}\text{Im}\Sigma_{\bm{k}}(\varepsilon_{\bm{k}})]^2},
 \end{equation}
 % 
-która ma kształt linii Lorentza w położeniu $\varepsilon_{\bm{k}}$, szerokości $Z_{\bm{k}}\text{Im}\Sigma_{\bm{k}}(\varepsilon_{\bm{k}})$
-i amplitudzie $Z_{\bm{k}}$, nazywanej również wagą sprektralna kwasiczastek. $Z_{\bm{k}}$ określa m.in. natężenie (amplitudę) rozpraszania fotoemisyjnego,
-zmianę masy efektywnej kwazicząstek, jak również skok w obsadzeniu stanów na powierzchni Fermiego. 
-Dla układu, w którym część rzeczywista energii własnej nie zależy od energii $Z_{\bm{k}}=1$, ta nieciągłość pokrywa się z rozkładem Fermiego-Diraca dla $T=0$.
-Energie kwasiczastek, które są funkcjami wektorów falowych, określają strukturę pasmową materiału.
-W układach silnie skorelowanych część gęstości spektralnej przekazywana jest do stanów leżących dalej od poziomu Fermiego
-i pojawiają się dodatkowe struktury (satelity), których nie da się zinterpretować jako wzbudzenia kwasicząstek.
+which has the shape of a Lorentzian line at position $\varepsilon_{\bm{k}}$, width $Z_{\bm{k}}\text{Im}\Sigma_{\bm{k}}(\varepsilon_{\bm{k}})$
+and amplitude $Z_{\bm{k}}$, also called the quasiparticle spectral weight. $Z_{\bm{k}}$ determines, among other things, the intensity (amplitude) of photoemission scattering,
+the change in the effective mass of quasiparticles, as well as the discontinuity in occupation at the Fermi surface. 
+For a system where the real part of the self-energy does not depend on energy, $Z_{\bm{k}}=1$, this discontinuity coincides with the Fermi-Dirac distribution at $T=0$.
+Quasiparticle energies, which are functions of wave vectors, determine the band structure of the material.
+In strongly correlated systems, part of the spectral density is transferred to states lying farther from the Fermi level
+and additional structures (satellites) appear that cannot be interpreted as quasiparticle excitations.
 
-\section{Teoria perturbacyjna}
+\section{Perturbation theory}
 
-Podstawowe podejście stosowane w teorii pola opiera się na rozwinięciu perturbacyjnym funkcji Greena~\cite{AGD}.
-Najwygodniejszym jest tutaj obraz oddziaływania, w którym hamiltonian zapisany jest w formie
+The basic approach used in field theory is based on the perturbative expansion of the Green's function~\cite{AGD}.
+The most convenient here is the interaction picture, in which the Hamiltonian is written in the form
 %
 \begin{equation}
 H(t)=H_0+V_\text{I}(t),
 \end{equation}
 %
-gdzie pierwszy wyraz opisuje układ swobodnych elektronów, a drugi jest zależnym od czasu operatorem oddziaływania
+where the first term describes the system of free electrons, and the second is the time-dependent interaction operator
 %
 \begin{equation}
 V_\text{I}(t)=\frac{1}{2}\int d\bm{r}\int d\bm{r}'\psi_\text{I}^\dagger(\bm{r},t)\psi_\text{I}^\dagger(\bm{r}',t)v(\bm{r},\bm{r}')
 \psi_\text{I}(\bm{r}',t)\psi_\text{I}(\bm{r},t).
 \end{equation}
 %
-Występujace w tym wyrażeniu operatory pola otrzymujemy przez transformację
+The field operators appearing in this expression are obtained through the transformation
 %
 \begin{equation}
 \psi_\text{I}(\bm{r},t)=e^{iH_0(t-t_0)}\psi(\bm{r})e^{-iH_0(t-t_0)}.
 \end{equation}
 %
-W obrazie oddziaływania, ewolucja stanów opisana jest zależnością
+In the interaction picture, the evolution of states is described by the relation
 %
 \begin{equation}
 |t\rangle=U_\text{I}(t,t_0)|t_0\rangle,
 \end{equation}
-gdzie operator ewolucji może być zapisany w postaci szeregu
+where the evolution operator can be written as a series
 %
 \begin{equation}
 U_\text{I}=1+\sum_{n=1}^{\infty}\frac{(-1)^n}{n!}\int_{t_0}^tdt_1...\int_{t_0}^tdt_nT\Big{[}V_\text{I}(t_1)...V_\text{I}(t_n)\Big{]}.
 \end{equation}
 %
-Wykorzystując ten operator i wprowadzając oznaczenie $S=U_\text{I}(\infty,-\infty)$
-możemy zapisać funkcję Greena w postaci rozwinięcia
+Using this operator and introducing the notation $S=U_\text{I}(\infty,-\infty)$,
+we can write the Green's function in the form of an expansion
 %
 \begin{equation}
 G(x,x')=-i\frac{\langle\Phi_0|T[\psi(x)\psi^{\dagger}(x')S]|\Phi_0\rangle}{\langle\Phi_0|S|\Phi_0\rangle},
 \end{equation}
 %
-gdzie $|\Phi_0\rangle$ jest stanem podstawowym dla układu nieoddziałujących elektronów opisanego hamiltonianem $H_0$. 
-Występujące w tym szeregu wyrazy można wyrazić przy pomocy funkcji Greena i wyrazów opisujących oddziaływania kulombowskie.
-Prowadzi to do wzoru, który można zapisać w zwartej formie przy pomocy sumy wyznaczników
+where $|\Phi_0\rangle$ is the ground state for the system of non-interacting electrons described by the Hamiltonian $H_0$. 
+The terms appearing in this series can be expressed using the Green's function and terms describing Coulomb interactions.
+This leads to a formula that can be written in compact form using a sum of determinants
 %
 \begin{equation}
 G(x,x')=\sum_{n=0}^\infty i^n\int v(x_1,x_1')...v(x_n,x'_n)\begin{vmatrix}
@@ -3903,22 +3903,22 @@ \section{Teoria perturbacyjna}
 \end{vmatrix}.
 \end{equation}
 %  
-Każdy z wyrazów rozwinięcia można otrzymać stosując metodę diagramów Feynmana opisaną szczegółowo w wielu podręcznikach \cite{AGD,FW,martin2}.
+Each term of the expansion can be obtained using the Feynman diagram method described in detail in many textbooks \cite{AGD,FW,martin2}.
 
 
 
 
 
-\section{Metoda GW}
+\section{The GW method}
 \label{sec:gw}
 
-Energię własną można powiązać z funkcją Greena
+The self-energy can be related to the Green's function
 %
 \begin{equation}
 \Sigma(x_1,x_2)=i\int dx_3dx_4 G(x_1,x_3)W(x_1,x_4)\Lambda(x_3,x_2,x_4).
 \end{equation}
 %
-$W$ jest ekranowanym potencjałem kulombowskim
+$W$ is the screened Coulomb potential
 %
 \begin{equation}
 W(x_1,x_2)=\int dx_3\epsilon^{-1}(x_1,x_3)V(\bm{r}_3-\bm{r}_2),
@@ -3928,8 +3928,8 @@ \section{Metoda GW}
 \epsilon^{-1}(x_1,x_2)=\frac{\delta V(x_1)}{\delta \phi(x_2)},
 \end{equation}
 %
-gdzie $V$ jest sumą potencjału Hartree $V_H$ i potencjału zewnętrznego $\phi$.
-$\Lambda$ jest funkcją wierzchołkową
+where $V$ is the sum of the Hartree potential $V_H$ and the external potential $\phi$.
+$\Lambda$ is the vertex function
 %
 \begin{equation}
 \Lambda(x_1,x_2,x_3) = -\frac{\delta G^{-1}(x_1)}{\delta V(x_3)}.