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Numerical Evaluation of Coulomb Integrals For 1, 2 and 3-electron Distance Operators, RC1 -nRD1 -m , RC1 -n r12 -m and r12 -n r13 -m with Real (N, M) and The Descartes Product of 3 Dimension Common Density Functional Numerical Integration Scheme
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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.