### 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 |
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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 |
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Volume | 2116 |

ISSN (Print) | 0094-243X |

ISSN (Electronic) | 1551-7616 |

### Conference

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

City | Rhodes |

Period | 9/13/18 → 9/18/18 |

### Keywords

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

### ASJC Scopus subject areas

- Physics and Astronomy(all)

## Fingerprint Dive into the research topics of 'Numerical evaluation of Coulomb integrals for 1, 2 and 3-electron distance operators, R<sub>C1</sub><sup>-N</sup>r<sub>d1</sub><sup>-M</sup>, R<sub>C1</sub><sup>-N</sup>r<sub>12</sub><sup>-M</sup> and R<sub>12</sub><sup>-N</sup>r<sub>13</sub><sup>-M</sup> with real (N, M) and the Descartes product of 3 dimension common density functional numerical integration scheme'. Together they form a unique fingerprint.

## 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