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

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

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(-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.

Original languageEnglish
Title of host publicationInternational Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017
PublisherAmerican Institute of Physics Inc.
Volume1978
ISBN (Electronic)9780735416901
DOIs
Publication statusPublished - Jul 10 2018
EventInternational Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017 - Thessaloniki, Greece
Duration: Sep 25 2017Sep 30 2017

Other

OtherInternational Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017
CountryGreece
CityThessaloniki
Period9/25/179/30/17

Fingerprint

Laplace transformation
moments
operators
evaluation
products
electron clouds
electrons
manipulators
chemistry
physics
nuclei
electronics
molecules

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

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. / Kristyán, S.

International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2017. Vol. 1978 American Institute of Physics Inc., 2018. 470030.

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

Kristyán, S 2018, 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. 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
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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(-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.",
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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.

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