diff --git a/book/main.tex b/book/main.tex index 7b86b97..7561445 100644 --- a/book/main.tex +++ b/book/main.tex @@ -11,10 +11,12 @@ %\usepackage[polish]{babel} \usepackage{amsmath} \usepackage[urlcolor=blue,colorlinks=true,citecolor=blue,linkcolor=blue,pdfstartview={FitH},bookmarks=false]{hyperref} -\def\mb{\bm} +%\def\mb{\bm} % **[EDITORIAL]**: Macro removed - all instances of \mb replaced with \bm for consistency \usepackage[left=2cm,right=2cm,top=3cm,bottom=3cm]{geometry} \DeclareMathOperator{\Tr}{Tr} +% **[EDITORIAL]**: Mathematical notation - Applied change: all \mb replaced with \bm throughout document for consistency and standard LaTeX practice. + \begin{document} @@ -41,8 +43,10 @@ \chapter*{Introduction} \addcontentsline{toc}{chapter}{Introduction} +% **[EDITORIAL]**: Introduction - Overall structure: Excellent historical overview. Consider adding a roadmap paragraph at the end to outline the book's chapter structure and guide readers through the content progression. + 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}. +The application of the Schr\"{o}dinger equation~\cite{Schrodinger} to atoms and molecules enabled explanation of the basic 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. @@ -59,7 +63,7 @@ \chapter*{Introduction} 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}. +as a 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 parallel spins, effectively reducing the Coulomb repulsion between them. @@ -67,20 +71,20 @@ \chapter*{Introduction} 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. +Methods based on the Hartree-Fock approach, the so-called multi-determinant methods, correctly take into account electronic correlations. However, they require time-consuming computer calculations and 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. +with a 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. +into 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. +However, in the interstitial area, both the potential and wave functions are slowly changing, and plane waves are the natural basis. +Additionally, the condition of continuity at the boundaries between 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. @@ -96,16 +100,16 @@ \chapter*{Introduction} 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. +In practical applications, the electron density is determined from single-particle wave functions, calculated +self-consistently 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}. +the linearized augmented plane wave (LAPW) method \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. % **[REVIEW NEEDED]**: Technical terminology for DFT approximations and strongly correlated systems @@ -157,6 +161,12 @@ \chapter*{Introduction} 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. +% **[EDITORIAL]**: Roadmap paragraph added as requested +This book is structured to guide readers through the fundamental concepts and advanced methods of ab initio calculations in solid state physics. Chapter 1 introduces electron interactions and the Hartree-Fock approximation, providing the foundation for understanding many-body quantum systems. Chapter 2 explores electronic states in crystals, including Bloch's theorem and band structure methods. Chapter 3 presents the fundamental theorems of density functional theory (DFT), the cornerstone of modern computational materials science. Chapter 4 discusses various methods for determining electronic structure, including pseudopotentials and basis set approaches. Chapter 5 examines orbital-dependent functionals and potentials that address limitations of standard DFT. Chapters 6 and 7 focus on specialized topics: insulators and semiconductors, and electric polarization in crystals. Chapter 8 covers van der Waals interactions, essential for describing weakly bonded systems. Finally, Chapter 9 introduces many-body effects and excited states, including Green's functions and the GW method, providing a bridge to advanced many-body techniques. + + +% **[EDITORIAL]**: Transition paragraph added to connect introduction with Chapter 1 +Having established the historical development and current landscape of ab initio methods, we now turn to a detailed examination of the fundamental physics underlying these approaches, beginning with the quantum mechanical description of electron interactions. \chapter{Electron interactions} @@ -215,7 +225,7 @@ \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} % -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, +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 $n$ labels a specific 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. Let us write the wave function of a system of $N$ electrons, at fixed positions of atomic nuclei, in the form $\Phi(\bm{r}_1,\bm{r}_2,...,\bm{r}_N)$. @@ -272,7 +282,8 @@ \section{Basic properties} \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, +% **[EDITORIAL]**: Line 285 - Typo in formula: "\phi_B" should be "\psi_B" (consistency with other terms) +N_{-}[\psi_A(\bm{r}_1)\psi_B(\bm{r}_2)-\psi_B(\bm{r}_1)\psi_A(\bm{r}_2)]|\downarrow\downarrow\rangle, \end{cases} \end{equation} % @@ -281,7 +292,7 @@ \section{Basic properties} 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. +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 section. 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} @@ -289,7 +300,7 @@ \section{Basic properties} \end{equation} % 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}. +Minimizing the system energy for $M=13$ yields values $E_d=4.70$ eV and $r_{AB}=0.74$ \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. @@ -297,6 +308,8 @@ \section{Basic properties} \section{Hartree-Fock equation} \label{sec:HF} +% **[EDITORIAL]**: Section organization - This section introduces a complex mathematical formalism. Consider adding a brief introductory paragraph before Eq. (\ref{slater}) that explains the physical motivation for the Slater determinant form and its advantages over the simpler Hartree approximation. + % **[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 % @@ -338,7 +351,7 @@ \section{Hartree-Fock equation} 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}) +\langle\Phi|H|\Phi\rangle=\sum_{i,\sigma}\int d\bm{r} \psi_i^{\sigma*}(\bm{r})[-\frac{\hbar^2}{2m}\nabla_i^2+V_Z(\bm{r})]\psi_i^{\sigma}(\bm{r}) +E_H+E_x, \label{E_HF} \end{equation} @@ -348,20 +361,20 @@ \section{Hartree-Fock equation} % \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}'|}. +\int d\bm{r} \int d\bm{r}' \frac{\psi_i^{\sigma*}(\bm{r})\psi_j^{\sigma'*}(\bm{r}')\psi_i^{\sigma}(\bm{r})\psi_j^{\sigma'}(\bm{r}')}{|\bm{r}-\bm{r}'|}. \end{equation} % -Introducing the charge density at point $\mb{r}$ +Introducing the charge density at point $\bm{r}$ % \begin{equation} -n(\mb{r})=e\sum_{i,\sigma} |\psi_i^{\sigma}(\mb{r})|^2, +n(\bm{r})=e\sum_{i,\sigma} |\psi_i^{\sigma}(\bm{r})|^2, \label{dens} \end{equation} % 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}'|}, +E_H=\frac{1}{2} \int d\bm{r} \int d\bm{r}' \frac{n(\bm{r})n(\bm{r}')}{|\bm{r}-\bm{r}'|}, \label{Hartree} \end{equation} % @@ -371,7 +384,7 @@ \section{Hartree-Fock equation} % \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}'|}. +\int d\bm{r} \int d\bm{r}' \frac{\psi_i^{\sigma*}(\bm{r})\psi_j^{\sigma*}(\bm{r}')\psi_i^{\sigma}(\bm{r}')\psi_j^{\sigma}(\bm{r})}{|\bm{r}-\bm{r}'|}. \label{exc} \end{equation} % @@ -390,42 +403,42 @@ \section{Hartree-Fock equation} 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}) --\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}), +[-\frac{\hbar^2}{2m}\nabla_i^2+V_Z(\bm{r})+V_H(\bm{r})]\psi_i^{\sigma}(\bm{r}) +-\frac{e^2}{2}\sum_{j,\sigma}\int d\bm{r}' \frac{\psi_j^{\sigma*}(\bm{r}')\psi_i^{\sigma}(\bm{r}')}{|\bm{r}-\bm{r}'|}\psi_j^{\sigma}(\bm{r})=E_i\psi_i^{\sigma}(\bm{r}), \label{HF} \end{equation} % 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}'|}. +V_H(\bm{r})=\int d\bm{r}' \frac{n(\bm{r}')}{|\bm{r}-\bm{r}'|}. \end{equation} % 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 +By 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}) --\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}), +[-\frac{\hbar^2}{2m}\nabla_i^2+V_Z(\bm{r})+V_H(\bm{r}) +-\frac{e}{2}\int d\bm{r}' \frac{n(\bm{r},\bm{r}')}{|\bm{r}-\bm{r}'|}]\psi_i^{\sigma}(\bm{r})=E_i\psi_i^{\sigma}(\bm{r}), \label{HFS} \end{equation} % -where $n(\mb{r},\mb{r}')$ can be interpreted as the exchange charge density +where $n(\bm{r},\bm{r}')$ 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})}, +n(\bm{r},\bm{r}')= e\sum_{j,\sigma}\frac{\psi_j^{\sigma*}(\bm{r}')\psi_i^{\sigma}(\bm{r}')\psi_j^{\sigma}(\bm{r})\psi_i^{\sigma*}(\bm{r})}{\psi_i^{\sigma*}(\bm{r})\psi_i^{\sigma}(\bm{r})}, \label{EC} \end{equation} % -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 +which is a function of two positions $\bm{r}$ and $\bm{r}'$ 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 $\bm{r}'$ 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. +q=e\sum_{j,\sigma}[\int d\bm{r}'\psi_j^{\sigma*}(\bm{r}')\psi_i^{\sigma}(\bm{r}')]\frac{\psi_j^{\sigma}(\bm{r})}{\psi_i^{\sigma}(\bm{r})}= +e\sum_{j,\sigma}\delta_{ij}\frac{\psi_j^{\sigma}(\bm{r})}{\psi_i^{\sigma}(\bm{r})}=e. \end{equation} % % **[REVIEW NEEDED]**: Self-interaction cancellation and Koopmans' theorem @@ -435,7 +448,7 @@ \section{Hartree-Fock equation} 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 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}. +$\psi_i^{\sigma}(\bm{r})$~\cite{Koopman}. \section{Self-consistent field method} @@ -443,7 +456,7 @@ \section{Self-consistent field method} 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 $V(\mb{r})$ +For these wave functions, we calculate the electron density $n(\bm{r})$ and potential $V(\bm{r})$ 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 @@ -537,7 +550,7 @@ \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_y}{L},\frac{2\pi n_z}{L}), +\bm{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. @@ -548,13 +561,13 @@ \section{Electron gas} \end{equation} % \begin{equation} -\psi_k(\mb{r})=\frac{1}{\sqrt{V}}e^{i\mb{kr}}, +\psi_k(\bm{r})=\frac{1}{\sqrt{V}}e^{i\bm{kr}}, \label{pw} \end{equation} % We assume that the system is not magnetically polarized and the number of electrons with spins pointing 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 +According to the Pauli exclusion principle, in a quantum state with a given wave vector $\bm{k}$, there can be a maximum of two electrons with opposite spins. At temperature $T=0$ K, electrons occupy consecutive states from the lowest to the maximum value $E_F$ called the Fermi energy. @@ -613,14 +626,14 @@ \section{Electron gas} 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}, +\frac{1}{|\bm{r}-\bm{r}'|}=4\pi\int \frac{d\bm{q}}{(2\pi)^3}\frac{e^{i\bm{q}(\bm{r}-\bm{r}')}}{q^2}, \end{equation} % 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} +E_k=\frac{\hbar^2 k^2}{2m} - \int_{\bm{k'}<\bm{k_F}} \frac{d\bm{k'}}{(2\pi)^3} \frac{4\pi}{|\bm{k}-\bm{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} % @@ -675,6 +688,8 @@ \section{Electron gas} by the quantum Monte Carlo (QMC) method \cite{CeperleyAlder80}. +% **[EDITORIAL]**: Chapter 2 transition - Good organization moving from electron interactions to crystal structures. The chapter opening is clear and direct. Consider briefly connecting to Chapter 1 by mentioning how periodic structures affect the electron interactions discussed previously. + \chapter{Electronic states in crystals} \section{Crystal lattice} @@ -706,6 +721,8 @@ \section{Crystal lattice} 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. +% **[EDITORIAL]**: Figure references - Ensure all figures are properly referenced and that captions are sufficiently descriptive for standalone understanding. Consider adding more detailed figure captions that explain the physical significance of what is shown. + \begin{figure}[h] \centering \includegraphics[scale=0.4]{lattice.pdf} @@ -727,12 +744,14 @@ \section{Crystal lattice} System & Lattice constants & Angles & Bravais lattices\\ \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 +% **[EDITORIAL]**: Line 745 - Inconsistent use of "ne" - should use proper \neq symbol throughout for consistency; also missing space in some table cells +tetragonal & $a=b\neq c$ & $\alpha=\beta=\gamma=90^{\degree}$ & P, I \\ +% **[EDITORIAL]**: Continuing table consistency - use \neq throughout for clarity +orthorhombic & $a\neq b\neq c$ & $\alpha=\beta=\gamma=90^{\degree}$ & P, F, C, I \\ +hexagonal & $a=b\neq c$ & $\alpha=\beta=90^{\degree}$, $\gamma=120^{\degree}$ & P \\ +trigonal & $a=b\neq c$ & $\alpha=\beta=90^{\degree}$, $\gamma=120^{\degree}$ & P \\ +monoclinic & $a=b\neq c$ & $\alpha=\gamma=90^{\degree}$, $\beta\neq 90^{\degree}$ & P, C \\ +triclinic & $a\neq b\neq c$ & $\alpha\neq \beta\neq \gamma\neq 90^{\degree}$ & P \\ \hline \end{tabular} \end{center} \end{table} @@ -1048,7 +1067,7 @@ \section{Hohenberg-Kohn Theorems} {\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 $n_0(\mb{r})$. +by the ground state electron density $n_0(\bm{r})$. {\bf Theorem II:} For a fixed external potential $V_\text{ext}(\bm{r})$, the energy functional $E[n]$ reaches @@ -1062,7 +1081,8 @@ \section{Hohenberg-Kohn Theorems} \section{Kohn-Sham Equation} -The Kohn-Sham theorems allow us to rigorously relate the ground state electron density +% **[EDITORIAL]**: Line 1082 - "Kohn-Sham theorems" should be "Hohenberg-Kohn theorems" (the theorems were established by Hohenberg & Kohn; Kohn-Sham developed the equations) +The Hohenberg-Kohn 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 @@ -1078,7 +1098,8 @@ \section{Kohn-Sham Equation} 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 +% **[EDITORIAL]**: Line 1098 - Clarify reference: "Kohn-Sham theorems" should be "Hohenberg-Kohn theorems" (the theorems establishing uniqueness; Kohn-Sham equations come later) +The Hohenberg-Kohn 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 @@ -1135,7 +1156,7 @@ \section{Kohn-Sham Equation} 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}. +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(\bm{r}')}{|\bm{r}-\bm{r'}|}+\frac{\delta E_{xc}[n]}{\delta n}. \label{VKS} \end{equation} % @@ -1173,12 +1194,12 @@ \subsection{Local Density Approximation (LDA)} 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, $n=n(\mb{r})$. +In the LDA approximation, we assume that the exchange-correlation energy at each point in space, where the electron density is $n(\bm{r})$, +is equal to the exchange-correlation energy of a homogeneous electron gas with the same density, $n=n(\bm{r})$. 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), +E_{xc}[n]=\int d\bm{r} n(\bm{r}) \varepsilon_{xc}(n), \end{equation} % where $\varepsilon_{xc}(n)$ is the exchange-correlation energy per electron in a homogeneous electron gas with density $n$. @@ -1206,7 +1227,7 @@ \subsection{Local Density Approximation (LDA)} 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}). +E_{xc}[n^{\uparrow},n^{\downarrow}]=\int d\bm{r} n(\bm{r}) \varepsilon_{xc}(n_{\uparrow},n_{\downarrow}). \label{xclda} \end{equation} % @@ -1252,14 +1273,14 @@ \subsection{Generalized Gradient Approximation (GGA)} 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\bm{r} f(n_{\uparrow},n_{\downarrow},\nabla n_{\uparrow},\nabla n_{\downarrow}) \label{xcgga} \end{equation} % 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), +E_x[n]=\int d\bm{r} n \varepsilon_x(n) F_x(s), \end{equation} % 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, @@ -1299,7 +1320,7 @@ \subsection{Generalized Gradient Approximation (GGA)} 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)], +E_c[n^{\uparrow},n^{\downarrow}]=\int d\bm{r} n [\epsilon_c(r_s,\zeta)+H(r_s,\zeta,t)], \end{equation} % where $\zeta=(n^{\uparrow}-n^{\downarrow})/n$ is the relative spin polarization, $t=|\nabla n|/2\phi k_s n$ is the dimensionless gradient, @@ -2791,11 +2812,11 @@ \subsection{Becke-Johnson potential} 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}'|}, +V_{\text{x}\sigma}^\text{Slater}(\bm{r})=-\frac{e}{2}\int d\bm{r}' \frac{n(\bm{r},\bm{r}')}{|\bm{r}-\bm{r}'|}, \label{SlatPot} \end{equation} % -where the exchange charge $n(\mb{r},\mb{r}')$ is described by formula (\ref{EC}). The BJ potential significantly improves the exchange interaction energy +where the exchange charge $n(\bm{r},\bm{r}')$ 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} % @@ -3656,7 +3677,8 @@ \section{Green's functions and self-energy} 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), +% **[EDITORIAL]**: Line 3678 - Formula error: The interaction term should have annihilation operators \psi (not \psi^\dagger) in the rightmost two positions; correct form should be: \psi^\dagger(\bm{r},t)\psi^\dagger(\bm{r}',t)v(\bm{r}-\bm{r}') \psi(\bm{r}',t)\psi(\bm{r},t) +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(\bm{r}',t)\psi(\bm{r},t), \end{equation} % where $h_0(x)$ describes the kinetic energy and the interaction of electrons with an external field, $v(\bm{r}-\bm{r}')$ is @@ -3938,12 +3960,14 @@ \section{The GW method} -%\section{Teoria dynamicznego średniego pola (DMFT)} +%\section{Dynamical mean-field theory (DMFT)} % Original Polish: Teoria dynamicznego średniego pola (DMFT) %\label{sec:dmft} -%\section{Kwantowe Monte Carlo (QMC)} +%\section{Quantum Monte Carlo (QMC)} % Original Polish: Kwantowe Monte Carlo (QMC) %\label{sec:qmc} +% **[EDITORIAL]**: Commented sections translated - Section titles translated from Polish. Sections remain commented out pending decision on whether to develop content or remove placeholders. + %\bibliographystyle{acm} %\bibliography{biblio.bib} @@ -4226,6 +4250,8 @@ \section{The GW method} \bibitem{martin2} R. M. Martin, L. Reining, and D. M. Ceperley, {\it Interacting Electrons. Theory and Computational Approaches}, Cambridge University Press, 2016. \end{document} + +% **[EDITORIAL]**: Overall manuscript assessment - This is a comprehensive and well-structured academic textbook on ab initio methods in solid state physics. The translation from Polish to English is generally accurate with good technical terminology. Key strengths: (1) Clear logical progression from basic quantum mechanics to advanced many-body methods, (2) Appropriate mathematical rigor for PhD-level students, (3) Good integration of historical context with modern computational approaches, (4) Comprehensive coverage spanning DFT, pseudopotentials, and advanced functionals. Areas for enhancement: (1) Add transitional paragraphs between major sections, (2) Expand figure captions for standalone clarity, (3) Consider adding worked examples or problem sets at chapter ends, (4) Review mathematical notation consistency (\mb vs \bm), (5) Complete or remove commented placeholder sections (DMFT, QMC). The manuscript successfully balances theoretical foundations with practical computational methods, making it suitable for its intended audience of computational physics PhD candidates.