Analytic evaluation of Coulomb integrals for one, two and three-electron distance operators, $R_{C1}^{-n}R_{D1}^{-m}$, $R_{C1}^{-n}r_{12}^{-m}$ and $r_{12}^{-n}r_{13}^{-m}$ with $n, m=0,1,2$

TL;DR

This paper derives new analytical expressions for complex Coulomb integrals involving multiple electrons and nuclei, which are essential for advanced quantum chemistry calculations, extending known solutions with novel Laplace transform techniques.

Contribution

It introduces new analytical formulas for Coulomb integrals with higher powers and mixed distances, expanding the toolkit for quantum chemical computations.

Findings

Derived analytical expressions for previously unsolved integrals.

Extended Laplace transformation methods to handle complex integrals.

Facilitated higher moment calculations in electron correlation studies.

Abstract

The state of the art for integral evaluation is that analytical solutions to integrals are far more useful than numerical solutions. We evaluate certain integrals analytically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the $(n,m)=(0,0)$ case is trivial, the $(n,m)=(1,0)$ and (0,1) cases are well known, fundamental milestone in integration and widely used in computation chemistry, as well as based on Laplace transformation with integrand exp(-$a^2t^2$). The rest of the cases are new and need the other Laplace transformation with integrand exp(-$a^2t$) also, as well as the necessity of a two dimensional version of Boys function comes up in case. These analytic expressions (up to Gaussian function integrand) are useful for manipulation with higher moments of inter-electronic distances, for example in correlation calculations.