Permutation-adapted atomic cluster expansions and applications

Defining physics-inspired basis sets for expanding physical properties in atomic systems often starts with base requirements of isometry and permutation invariance. Such mathematical mappings make machine learning tasks far more data efficient than physics-agnostic feature sets. The atomic cluster expansion (ACE) has recently been used to construct many different high-fidelity atomistic models. The success of the method may be partially attributed to the basis of physics-inspired ACE descriptors, which are rotation and permutation-invariant mappings of local atomic environments. Sets of rotation and permutation invariant ACE descriptors may be used to expand atomic properties such as the potential energy and may depend on variables such as atomic position, chemical labeling, and charge. Many interatomic potentials and models have been produced with ACE methodologies so far, but most of them rely on semi-numerically defined descriptor sets. This work outlines an all-analytic method for obtaining ACE descriptor sets without numerical steps and can be used for any system or ACE model. The utility of this approach is highlighted in some theoretical and practical cases. Specific cases highlighting the use of this all-analytic basis include machine-learned interatomic potential development, atomistic structure representations, and electronegativity models.