Approximation of Atomic Interactions with Spherical Harmonics

We detail a framework to systematically derive polynomial basis functions for approximating isometry and permutation invariant functions, particularly with an eye to modelling properties of atomistic systems and analyzing the convergence to a high-fidelity reference model. Our presentation builds on (Drautz, Phys. Rev. B 99, 2019), extending the derivation in a way that yields immediate guarantees that a complete basis is indeed obtained. In addition, we discuss generalisations, a variety of related open challenges, particularly from a numerical analysis perspective, around basis optimisation and parameter estimation.