A perspective marking 20 years of using permutationally invariant polynomials for molecular potentials

This Perspective is focused on permutationally invariant polynomials (PIPs). Since their introduction in 2004 and first use in developing a fully permutationally invariant potential for the highly fluxional cation CH5+, PIPs have found widespread use in developing machine learned potentials (MLPs) for isolated molecules, chemical reactions, clusters, condensed phase, and materials. More than 100 potentials have been reported using PIPs. The popularity of PIPs for MLPs stems from their fundamental property of being invariant with respect to permutations of like atoms; this is a fundamental property of potential energy surfaces. This is achieved using global descriptors and, thus, without using an atom-centered approach (which is manifestly fully permutationally invariant). PIPs have been used directly for linear regression fitting of electronic energies and gradients for complex energy landscapes to chemical reactions with numerous product channels. PIPs have also been used as inputs to neural network and Gaussian process regression methods and in many-body (atom-centered, water monomer, etc.) applications, notably for gold standard potentials for water. Here, we focus on the progress and usage of PIPs since 2018, when the last review of PIPs was done by our group.