Numerical evaluation of Coulomb integrals for 1, 2 and 3-electron distance operators, RC1-Nrd1-M, RC1-Nr12-M and R12-Nr13-M with real (N, M) and the Descartes product of 3 dimension common density functional numerical integration scheme

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.

Original languageEnglish
Title of host publicationInternational Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2018
EditorsT.E. Simos, T.E. Simos, T.E. Simos, T.E. Simos, Ch. Tsitouras, T.E. Simos
PublisherAmerican Institute of Physics Inc.
ISBN (Electronic)9780735418547
DOIs
Publication statusPublished - Jul 24 2019
EventInternational Conference on Numerical Analysis and Applied Mathematics 2018, ICNAAM 2018 - Rhodes, Greece
Duration: Sep 13 2018Sep 18 2018

Publication series

NameAIP Conference Proceedings
Volume2116
ISSN (Print)0094-243X
ISSN (Electronic)1551-7616

Conference

ConferenceInternational Conference on Numerical Analysis and Applied Mathematics 2018, ICNAAM 2018
CountryGreece
CityRhodes
Period9/13/189/18/18

Fingerprint

functional integration
numerical integration
electrons
quantum chemistry
operators
electron
evaluation
seeds
products
analytical chemistry
seed
electronics
manipulators
chemistry
density functional theory
moments
nuclei
product
testing

Keywords

  • Generalization of 3 dimension Becke-Lebedev-Voronoi numerical integration scheme to 6 and 9 dimensions
  • Higher moment Coulomb distance operators RR
  • m≥0 and <0
  • Numerical evaluation of Coulomb integrals for one
  • Rr and rr with real n
  • two and three-electron distance operators

ASJC Scopus subject areas

  • Ecology, Evolution, Behavior and Systematics
  • Ecology
  • Plant Science
  • Physics and Astronomy(all)
  • Nature and Landscape Conservation

Cite this

Kristyán, S. (2019). Numerical evaluation of Coulomb integrals for 1, 2 and 3-electron distance operators, RC1-Nrd1-M, RC1-Nr12-M and R12-Nr13-M with real (N, M) and the Descartes product of 3 dimension common density functional numerical integration scheme. In T. E. Simos, T. E. Simos, T. E. Simos, T. E. Simos, C. Tsitouras, & T. E. Simos (Eds.), International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2018 [450029] (AIP Conference Proceedings; Vol. 2116). American Institute of Physics Inc.. https://doi.org/10.1063/1.5114496

Numerical evaluation of Coulomb integrals for 1, 2 and 3-electron distance operators, RC1-Nrd1-M, RC1-Nr12-M and R12-Nr13-M with real (N, M) and the Descartes product of 3 dimension common density functional numerical integration scheme. / Kristyán, S.

International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2018. ed. / T.E. Simos; T.E. Simos; T.E. Simos; T.E. Simos; Ch. Tsitouras; T.E. Simos. American Institute of Physics Inc., 2019. 450029 (AIP Conference Proceedings; Vol. 2116).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Kristyán, S 2019, Numerical evaluation of Coulomb integrals for 1, 2 and 3-electron distance operators, RC1-Nrd1-M, RC1-Nr12-M and R12-Nr13-M with real (N, M) and the Descartes product of 3 dimension common density functional numerical integration scheme. in TE Simos, TE Simos, TE Simos, TE Simos, C Tsitouras & TE Simos (eds), International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2018., 450029, AIP Conference Proceedings, vol. 2116, American Institute of Physics Inc., International Conference on Numerical Analysis and Applied Mathematics 2018, ICNAAM 2018, Rhodes, Greece, 9/13/18. https://doi.org/10.1063/1.5114496
Kristyán S. Numerical evaluation of Coulomb integrals for 1, 2 and 3-electron distance operators, RC1-Nrd1-M, RC1-Nr12-M and R12-Nr13-M with real (N, M) and the Descartes product of 3 dimension common density functional numerical integration scheme. In Simos TE, Simos TE, Simos TE, Simos TE, Tsitouras C, Simos TE, editors, International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2018. American Institute of Physics Inc. 2019. 450029. (AIP Conference Proceedings). https://doi.org/10.1063/1.5114496
Kristyán, S. / Numerical evaluation of Coulomb integrals for 1, 2 and 3-electron distance operators, RC1-Nrd1-M, RC1-Nr12-M and R12-Nr13-M with real (N, M) and the Descartes product of 3 dimension common density functional numerical integration scheme. International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2018. editor / T.E. Simos ; T.E. Simos ; T.E. Simos ; T.E. Simos ; Ch. Tsitouras ; T.E. Simos. American Institute of Physics Inc., 2019. (AIP Conference Proceedings).
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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.",
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