Numerical evaluation of Coulomb integrals for 1, 2 and 3-electron distance operators, $R_{C1}^{-n}R_{D1}^{-m}$, $R_{C1}^{-n}r_{12}^{-m}$ and $r_{12}^{-n}r_{13}^{-m}$ with real $(n,m$) and the Descartes product of 3 dim. common density functional numerical integration scheme

TL;DR

This paper develops a numerical method to evaluate complex Coulomb integrals involving multiple electron and nucleus distances, extending existing schemes to higher dimensions for use in advanced quantum chemistry calculations.

Contribution

It generalizes the Becke-Lebedev-Voronoi numerical integration scheme to 6 and 9 dimensions for Coulomb integrals with real exponents, enabling calculations beyond analytical solutions.

Findings

Numerical scheme successfully evaluates higher-dimensional Coulomb integrals.

Method works with Gaussian and Slater-type functions.

Applicable to correlation and higher moment calculations in quantum chemistry.

Abstract

Analytical solutions to integrals are far more useful than numeric, however, the former is not available in many cases. We evaluate integrals indicated in the title numerically 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 case is trivial, the (n, m)= (1,0) or (0,1) cases are well known, a fundamental milestone in the integration and widely used in computational quantum chemistry, as well as analytical integration is possible if Gaussian functions are used. For the rest of the cases the analytical solutions are restricted, but worked out for some, e.g. for n, m= 0,1,2 with Gaussians. In this work we generalize the Becke-Lebedev-Voronoi 3 dimensions numerical integration scheme (commonly used in density functional theory) to 6 and 9 dimensions via Descartes product to evaluate integrals indicated in the title, and test it. This numerical recipe (up to Gaussian integrands with seed exp(-|r1|2), as well as positive and negative real n and m values) is useful for manipulation with higher moments of inter-electronic distances, for example, in correlation calculations; more, our numerical scheme works for Slaterian type functions with seed exp(-|r1|) as well.