### Abstract

In the title, where R stands for nucleus-electron and r for electron-electron distances in practice of computation chemistry or physics, the (n,m)=(0,0) case is trivial, the (n,m)=(1,0) and (0,1) cases are well known, fundamental milestone in integration and widely used, as well as based on Laplace transformation with integrand exp(-a^{2}t^{2}). The rest of the cases are new and need the other Laplace transformation with integrand exp(-a^{2}t) also, as well as the necessity of a two dimensional version of Boys function comes up in case. These analytic expressions (up to Gaussian function integrand) are useful for manipulation with higher moments of inter-electronic distances, for example in correlation calculations. The equations derived help to evaluate the important Coulomb integrals ∫ρ(r1)RC1-nRD1-mdr1,∫ρ(r1)ρ(r2)RC1-nr12-mdr1dr2,∫ρ(r1)ρ(r2)ρ(r3)r12-nr13-mdr1dr2dr3, where ρ(r_{i}), called one-electron density, is a linear combination of Gaussian functions of position vector variable r_{i}, capable to describe the electron clouds in molecules, solids or any media/ensemble of materials.

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

Title of host publication | International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017 |

Publisher | American Institute of Physics Inc. |

Volume | 1978 |

ISBN (Electronic) | 9780735416901 |

DOIs | |

Publication status | Published - Jul 10 2018 |

Event | International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017 - Thessaloniki, Greece Duration: Sep 25 2017 → Sep 30 2017 |

### Other

Other | International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017 |
---|---|

Country | Greece |

City | Thessaloniki |

Period | 9/25/17 → 9/30/17 |

### Fingerprint

### Keywords

- Analytic evaluation of Coulomb integrals for one, two and three-electron operators
- Higher moment Coulomb operators R R , R r and r r with n, m=0,1,2

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

_{C1}

^{-n}R

_{D1}

^{-m}, R

_{C1}

^{-n}r

_{12}

^{-m}and r

_{12}

^{-n}r

_{13}

^{-m}with n, m=0,1,2), useful in Coulomb integrals for one, two and three-electron operators. In

*International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017*(Vol. 1978). [470030] American Institute of Physics Inc.. https://doi.org/10.1063/1.5044100

**Analytic evaluation for integrals of product Gaussians with different moments of distance operators (R _{C1}
^{-n}R_{D1}
^{-m}, R_{C1}
^{-n}r_{12}
^{-m} and r_{12}
^{-n} r_{13}
^{-m} with n, m=0,1,2), useful in Coulomb integrals for one, two and three-electron operators.** / 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 n, m=0,1,2), useful in Coulomb integrals for one, two and three-electron operators. in

*International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017.*vol. 1978, 470030, American Institute of Physics Inc., International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017, Thessaloniki, Greece, 9/25/17. https://doi.org/10.1063/1.5044100

_{C1}

^{-n}R

_{D1}

^{-m}, R

_{C1}

^{-n}r

_{12}

^{-m}and r

_{12}

^{-n}r

_{13}

^{-m}with n, m=0,1,2), useful in Coulomb integrals for one, two and three-electron operators. In International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017. Vol. 1978. American Institute of Physics Inc. 2018. 470030 https://doi.org/10.1063/1.5044100

}

TY - GEN

T1 - Analytic evaluation for integrals of product Gaussians with different moments of distance operators (RC1 -nRD1 -m, RC1 -nr12 -m and r12 -n r13 -m with n, m=0,1,2), useful in Coulomb integrals for one, two and three-electron operators

AU - Kristyán, S.

PY - 2018/7/10

Y1 - 2018/7/10

N2 - In the title, where R stands for nucleus-electron and r for electron-electron distances in practice of computation chemistry or physics, the (n,m)=(0,0) case is trivial, the (n,m)=(1,0) and (0,1) cases are well known, fundamental milestone in integration and widely used, as well as based on Laplace transformation with integrand exp(-a2t2). The rest of the cases are new and need the other Laplace transformation with integrand exp(-a2t) also, as well as the necessity of a two dimensional version of Boys function comes up in case. These analytic expressions (up to Gaussian function integrand) are useful for manipulation with higher moments of inter-electronic distances, for example in correlation calculations. The equations derived help to evaluate the important Coulomb integrals ∫ρ(r1)RC1-nRD1-mdr1,∫ρ(r1)ρ(r2)RC1-nr12-mdr1dr2,∫ρ(r1)ρ(r2)ρ(r3)r12-nr13-mdr1dr2dr3, where ρ(ri), called one-electron density, is a linear combination of Gaussian functions of position vector variable ri, capable to describe the electron clouds in molecules, solids or any media/ensemble of materials.

AB - In the title, where R stands for nucleus-electron and r for electron-electron distances in practice of computation chemistry or physics, the (n,m)=(0,0) case is trivial, the (n,m)=(1,0) and (0,1) cases are well known, fundamental milestone in integration and widely used, as well as based on Laplace transformation with integrand exp(-a2t2). The rest of the cases are new and need the other Laplace transformation with integrand exp(-a2t) also, as well as the necessity of a two dimensional version of Boys function comes up in case. These analytic expressions (up to Gaussian function integrand) are useful for manipulation with higher moments of inter-electronic distances, for example in correlation calculations. The equations derived help to evaluate the important Coulomb integrals ∫ρ(r1)RC1-nRD1-mdr1,∫ρ(r1)ρ(r2)RC1-nr12-mdr1dr2,∫ρ(r1)ρ(r2)ρ(r3)r12-nr13-mdr1dr2dr3, where ρ(ri), called one-electron density, is a linear combination of Gaussian functions of position vector variable ri, capable to describe the electron clouds in molecules, solids or any media/ensemble of materials.

KW - Analytic evaluation of Coulomb integrals for one, two and three-electron operators

KW - Higher moment Coulomb operators R R , R r and r r with n, m=0,1,2

UR - http://www.scopus.com/inward/record.url?scp=85049950183&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049950183&partnerID=8YFLogxK

U2 - 10.1063/1.5044100

DO - 10.1063/1.5044100

M3 - Conference contribution

AN - SCOPUS:85049950183

VL - 1978

BT - International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017

PB - American Institute of Physics Inc.

ER -