Efficient Algorithm for Two-Center Coulomb and Exchange Integrals of Electronic Prolate Spheroidal Orbitals

TL;DR

This paper introduces a fast, accurate algorithm for calculating Coulomb and exchange integrals of prolate spheroidal orbitals, enhancing quantum chemistry computations for diatomic molecules.

Contribution

The paper presents a novel algorithm that symbolically solves integrals and uses Taylor expansion for efficiency, avoiding numerical integration and enabling reuse of coefficients.

Findings

High accuracy due to exponential convergence

Application to O2 and CO molecules demonstrates effectiveness

Coefficients are independent of specific wavefunctions

Abstract

We present a fast algorithm to calculate Coulomb/exchange integrals of prolate spheroidal electronic orbitals, which are the exact solutions of the single-electron, two-center Schr\"odinger equation for diatomic molecules. Our approach employs Neumann's expansion of the Coulomb repulsion 1/|x-y|, solves the resulting integrals symbolically in closed form and subsequently performs a numeric Taylor expansion for efficiency. Thanks to the general form of the integrals, the obtained coefficients are independent of the particular wavefunctions and can thus be reused later. Key features of our algorithm include complete avoidance of numeric integration, drafting of the individual steps as fast matrix operations and high accuracy due to the exponential convergence of the expansions. Application to the diatomic molecules O2 and CO exemplifies the developed methods, which can be relevant for a quantitative understanding of chemical bonds in general.