On the approximation of electronic wavefunctions by anisotropic Gauss and Gauss-Hermite functions

TL;DR

This paper demonstrates that complex electronic wavefunctions can be effectively approximated using anisotropic Gauss-Hermite functions, achieving high accuracy with relatively few terms despite their non-smooth features.

Contribution

It shows that singular electronic wavefunctions can be approximated efficiently by anisotropic Gauss-Hermite functions, reaching arbitrary convergence orders with minimal residual error.

Findings

Wavefunctions can be approximated with negligible residual error.

Approximation achieves high accuracy with few terms.

Non-smooth cusps have little impact on convergence.

Abstract

The electronic Schr\"odinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wavefunctions, depend on 3N variables, three spatial dimensions for each electron. We study the approximability of these wavefunctions by linear combinations of anisotropic Gauss functions, or more precisely Gauss-Hermite functions, products of polynomials and anisotropic Gauss functions in the narrow sense. We show that the original, singular wavefunctions can up to given accuracy and a negligibly small residual error be approximated with only insignificantly more such terms than their convolution with a Gaussian kernel of sufficiently small width and that basically arbitrary orders of convergence can be reached. This is a fairly surprising result, since it essentially means that by this type of approximation, the intricate hierarchies of non-smooth cusps in electronic wavefunctions have almost no impact on the convergence, once the global structure is resolved.