Permutation-adapted complete and independent basis for atomic cluster expansion descriptors

TL;DR

This paper introduces an analytical method to construct a complete, independent basis for atomic cluster expansion descriptors, reducing redundancy and improving the efficiency of modeling atomic environments.

Contribution

It derives linear relationships for cluster functions using recursion and permutation properties, enabling the selection of linearly independent basis functions analytically.

Findings

Derived linear relationships for cluster functions using recursion.

Proved that the new basis is complete and independent.

Demonstrated improved modeling accuracy with the new basis in a tantalum potential.

Abstract

Atomic cluster expansion (ACE) methods provide a systematic way to describe particle local environments of arbitrary body order. For practical applications it is often required that the basis of cluster functions be symmetrized with respect to rotations and permutations. Existing methodologies yield sets of symmetrized functions that are over-complete. These methodologies thus require an additional numerical procedure, such as singular value decomposition (SVD), to eliminate redundant functions. In this work, it is shown that analytical linear relationships for subsets of cluster functions may be derived using recursion and permutation properties of generalized Wigner symbols. From these relationships, subsets (blocks) of cluster functions can be selected such that, within each block, functions are guaranteed to be linearly independent. It is conjectured that this block-wise independent set of permutation-adapted rotation and permutation invariant (PA-RPI) functions forms a complete, independent basis for ACE. Along with the first analytical proofs of block-wise linear dependence of ACE cluster functions and other theoretical arguments, numerical results are offered to demonstrate this. The utility of the method is demonstrated in the development of an ACE interatomic potential for tantalum. Using the new basis functions in combination with Bayesian compressive sensing sparse regression, some high degree descriptors are observed to persist and help achieve high-accuracy models.