Permutationally invariant potential energy surfaces in high dimensionality

TLDR

The review situates permutationally invariant polynomial fitting within the broader landscape of potential energy surface methods. The paper surveys recent advances in constructing potential energy and dipole moment surfaces for polyatomic systems containing up to ten atoms. It describes a global linear least‑squares fitting scheme that uses a compact basis of permutationally invariant polynomials in Morse‑type variables, together with a review of the underlying mathematics, data acquisition, and a pedagogical symmetrization method. Illustrations of the approach are provided for the water dimer and acetaldehyde potential energy surfaces.

Abstract

We review recent progress in developing potential energy and dipole moment surfaces for polyatomic systems with up to 10 atoms. The emphasis is on global linear least squares fitting of tens of thousands of scattered ab initio energies using a special, compact fitting basis of permutationally invariant polynomials in Morse-type variables of all the internuclear distances. The computational mathematics underlying this approach is reviewed first, followed by a review of the practical approaches used to obtain the data for the fits. A straightforward symmetrization approach is also given, mainly for pedagogical purposes. The methods are illustrated for potential energy surfaces for , (H2O)2 and CH3CHO. The relationship of this approach to other approaches is also briefly reviewed.