Using rectangular collocation with finite difference derivatives to solve electronic Schrodinger equation

TL;DR

This paper introduces a rectangular collocation method combined with finite difference derivatives for solving the electronic Schrödinger equation, demonstrating high accuracy and flexibility in basis function choice, applicable to atomic and molecular systems.

Contribution

The paper presents a novel collocation approach that allows flexible point distribution and basis functions, enabling the use of non-Gaussian and non-plane wave basis sets for electronic structure calculations.

Findings

Achieved millihartree accuracy in atomic and molecular solutions

Method is less sensitive to point set choice with better basis functions

Allows use of basis functions like Slater-type orbitals with collocation

Abstract

We show that a rectangular collocation method, equivalent to evaluating all matrix elements with a quadrature-like scheme and using more points than basis functions, is an effective approach for solving the electronic Schr\"odinger equation (ESE). We test the ideas by computing several solutions of the ESE for the H atom and the H2+ cation and several solutions of a Kohn-Sham equation for CO and H2O. In all cases, we achieve millihartree accuracy. Two key advantages of the collocation method we use are: 1) collocation points need not have a particular distribution or spacing and can be chosen to reduce the required number of points; 2) the better the basis, is the less sensitive are the results to the choice of the point set. The ideas of this paper make it possible to use any basis functions and thus open to the door to using basis functions that are not Gaussians or plane waves. We use basis functions that are similar to Slater type orbitals. They are rarely used with the variational method, but present no problems when used with collocation.