### 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(-|r_{1}|^{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(-|r_{1}|) as well.

Original language | English |
---|---|

Title of host publication | International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2018 |

Editors | T.E. Simos, T.E. Simos, T.E. Simos, T.E. Simos, Ch. Tsitouras, T.E. Simos |

Publisher | American Institute of Physics Inc. |

ISBN (Electronic) | 9780735418547 |

DOIs | |

Publication status | Published - Jul 24 2019 |

Event | International Conference on Numerical Analysis and Applied Mathematics 2018, ICNAAM 2018 - Rhodes, Greece Duration: Sep 13 2018 → Sep 18 2018 |

### Publication series

Name | AIP Conference Proceedings |
---|---|

Volume | 2116 |

ISSN (Print) | 0094-243X |

ISSN (Electronic) | 1551-7616 |

### Conference

Conference | International Conference on Numerical Analysis and Applied Mathematics 2018, ICNAAM 2018 |
---|---|

Country | Greece |

City | Rhodes |

Period | 9/13/18 → 9/18/18 |

### Fingerprint

### 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

_{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 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, 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 dimension common density functional numerical integration scheme.** / Kristyán, S.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

_{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 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

_{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 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

}

TY - GEN

T1 - 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

AU - Kristyán, S.

PY - 2019/7/24

Y1 - 2019/7/24

N2 - 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.

AB - 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.

KW - Generalization of 3 dimension Becke-Lebedev-Voronoi numerical integration scheme to 6 and 9 dimensions

KW - Higher moment Coulomb distance operators RR

KW - m≥0 and <0

KW - Numerical evaluation of Coulomb integrals for one

KW - Rr and rr with real n

KW - two and three-electron distance operators

UR - http://www.scopus.com/inward/record.url?scp=85069961755&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85069961755&partnerID=8YFLogxK

U2 - 10.1063/1.5114496

DO - 10.1063/1.5114496

M3 - Conference contribution

T3 - AIP Conference Proceedings

BT - International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2018

A2 - Simos, T.E.

A2 - Simos, T.E.

A2 - Simos, T.E.

A2 - Simos, T.E.

A2 - Tsitouras, Ch.

A2 - Simos, T.E.

PB - American Institute of Physics Inc.

ER -