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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

preprint
submitted on 25.10.2018 and posted on 26.10.2018 by Sandor Kristyan
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.

Funding

OTKA-K- 2015-115733 and 2016-119358

History

Email Address of Submitting Author

kristyan.sandor@ttk.mta.hu

Institution

Res. Centr. for Nat. Sci., Hung. Acad. Sci., Inst. Mater. and Env. Chem.

Country

Hungary

ORCID For Submitting Author

0000-0003-4169-2392

Declaration of Conflict of Interest

No

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