Permutation-Adapted ACE

To use this module add this directory to your pythonpath:

export PYTHONPATH=$PYTHONPATH:/path/to/PA_ACE

Descriptor labels and linear independence

ACE descriptors

The way the labels for the N-bond ACE descriptors are defined are chemical index of the central atom, u0, then chemical indices for other atom in the N bonds, radial indices for the N bonds, angular indices for the N bonds, and then intermediate angular indices (when applicable). There are N-2 intermediate angular indices for descriptors with rank N>=3. For ranks >=3, one will find these intermediate indices after the N angular indices for the bonds. In general, the descriptor labels will have the form: ` u0, u1, u2, ... uN, n1,n2,... nN, l1,l2,... lN, { ... L(N-3), L(N-2) }

For a rank (N=3) example with mu_i=0 , n_i =1, l_i = 0: 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0 u0, u1, u2, u3, n1, n2, n3, l1, l2, l3, L1

For a non-trivial rank 4 example: 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0 u0, u1, u2, u3, u4, n1, n2, n3, n4, l1, l2, l3, l4, L1, L2

Where, in the above rank 4 example, only one of two sets of intermediates { (00) , (22) } yields an independent ACE descriptor. This may be shown analytically using recursion relationships derived from ladder operations. `

The key functionality of this package is to supply users with complete, linearly independent ACE descriptors analytically as above.

The labels for linearly independent ACE descriptors may be generated using the following functions from pa_gen.py:

from pa_gen import * pa_labels_raw(rank=4,nmax=4,lmax=4,mumax=1)

where rank is the number of bonds for the features, nmax is the maximum quantum number for the radial basis, lmax is the maximum angular quantum number for the spherical harmonic basis, and mumax is the number of chemical types. This function will exhaustively generate labels that form a complete, linearly-independent set of ACE basis functions or descriptors.

Coupling coefficients

Both the generalized Wigner symbols and the generalized Clebsch-gordan coefficients may be generated in this package. The corresponding functions to generate these are given in wigner_couple.py and clebsch_couple.py, respectively and pedagogical implmementations for calculating them are given in wigner_tree.py and clebsch_tree.py.

Pythonized ACE calculator

A stand-alone ACE calculator outside of LAMMPS is provided for testing purposes, but is not intended for production usage in simulations. Examples of this may be found in all_examples and the corresponding functions/classes are given in equivariant_calc.

LAMMPS usage

This code supplies .yace coupling_coefficient and potential files that may be ran in LAMMPS. More examples may be found in the FitSNAP repo: https://github.com/FitSNAP/FitSNAP.

NOTES

This version of the library is out of date, but a new version will be provided here or in the FitSNAP repo, https://github.com/FitSNAP/FitSNAP.