Efficient evaluation of the geometrical first derivatives of three-center Coulomb integrals

Gyula Samu, M. Kállay

Research output: Contribution to journalArticle

Abstract

The calculation of the geometrical derivatives of three-center electron repulsion integrals (ERIs) over contracted spherical harmonic Gaussians has been optimized. We compared various methods based on the Obara-Saika, McMurchie-Davidson, Gill-Head-Gordon-Pople, and Rys polynomial algorithms using Cartesian, Hermite, and mixed Gaussian integrals for each scheme. The latter ERIs contain both Hermite and Cartesian Gaussians, and they combine the advantageous properties of both types of basis functions. Furthermore, prescreening of the ERI derivatives is discussed, and an efficient approximation of the Cauchy-Schwarz bound for first derivatives is presented. Based on the estimated operation counts, the most promising schemes were implemented by automated code generation, and their relative performances were evaluated. We analyzed the benefits of computing all of the derivatives of a shell triplet simultaneously compared to calculating them just for one degree of freedom at a time, and it was found that the former scheme offers a speedup close to an order of magnitude with a triple-zeta quality basis when appropriate prescreening is applied. In these cases, the Obara-Saika method with Cartesian Gaussians proved to be the best approach, but when derivatives for one degree of freedom are required at a time the mixed Gaussian Obara-Saika and Gill-Head-Gordon-Pople algorithms are predicted to be the best performing ones.

Original languageEnglish
Article number124101
JournalJournal of Chemical Physics
Volume149
Issue number12
DOIs
Publication statusPublished - Sep 28 2018

Fingerprint

Derivatives
evaluation
degrees of freedom
Electrons
electrons
spherical harmonics
polynomials
Polynomials
approximation

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Physical and Theoretical Chemistry

Cite this

Efficient evaluation of the geometrical first derivatives of three-center Coulomb integrals. / Samu, Gyula; Kállay, M.

In: Journal of Chemical Physics, Vol. 149, No. 12, 124101, 28.09.2018.

Research output: Contribution to journalArticle

@article{1089232574e64a6dbbf152084d2f120d,
title = "Efficient evaluation of the geometrical first derivatives of three-center Coulomb integrals",
abstract = "The calculation of the geometrical derivatives of three-center electron repulsion integrals (ERIs) over contracted spherical harmonic Gaussians has been optimized. We compared various methods based on the Obara-Saika, McMurchie-Davidson, Gill-Head-Gordon-Pople, and Rys polynomial algorithms using Cartesian, Hermite, and mixed Gaussian integrals for each scheme. The latter ERIs contain both Hermite and Cartesian Gaussians, and they combine the advantageous properties of both types of basis functions. Furthermore, prescreening of the ERI derivatives is discussed, and an efficient approximation of the Cauchy-Schwarz bound for first derivatives is presented. Based on the estimated operation counts, the most promising schemes were implemented by automated code generation, and their relative performances were evaluated. We analyzed the benefits of computing all of the derivatives of a shell triplet simultaneously compared to calculating them just for one degree of freedom at a time, and it was found that the former scheme offers a speedup close to an order of magnitude with a triple-zeta quality basis when appropriate prescreening is applied. In these cases, the Obara-Saika method with Cartesian Gaussians proved to be the best approach, but when derivatives for one degree of freedom are required at a time the mixed Gaussian Obara-Saika and Gill-Head-Gordon-Pople algorithms are predicted to be the best performing ones.",
author = "Gyula Samu and M. K{\'a}llay",
year = "2018",
month = "9",
day = "28",
doi = "10.1063/1.5049529",
language = "English",
volume = "149",
journal = "Journal of Chemical Physics",
issn = "0021-9606",
publisher = "American Institute of Physics Publising LLC",
number = "12",

}

TY - JOUR

T1 - Efficient evaluation of the geometrical first derivatives of three-center Coulomb integrals

AU - Samu, Gyula

AU - Kállay, M.

PY - 2018/9/28

Y1 - 2018/9/28

N2 - The calculation of the geometrical derivatives of three-center electron repulsion integrals (ERIs) over contracted spherical harmonic Gaussians has been optimized. We compared various methods based on the Obara-Saika, McMurchie-Davidson, Gill-Head-Gordon-Pople, and Rys polynomial algorithms using Cartesian, Hermite, and mixed Gaussian integrals for each scheme. The latter ERIs contain both Hermite and Cartesian Gaussians, and they combine the advantageous properties of both types of basis functions. Furthermore, prescreening of the ERI derivatives is discussed, and an efficient approximation of the Cauchy-Schwarz bound for first derivatives is presented. Based on the estimated operation counts, the most promising schemes were implemented by automated code generation, and their relative performances were evaluated. We analyzed the benefits of computing all of the derivatives of a shell triplet simultaneously compared to calculating them just for one degree of freedom at a time, and it was found that the former scheme offers a speedup close to an order of magnitude with a triple-zeta quality basis when appropriate prescreening is applied. In these cases, the Obara-Saika method with Cartesian Gaussians proved to be the best approach, but when derivatives for one degree of freedom are required at a time the mixed Gaussian Obara-Saika and Gill-Head-Gordon-Pople algorithms are predicted to be the best performing ones.

AB - The calculation of the geometrical derivatives of three-center electron repulsion integrals (ERIs) over contracted spherical harmonic Gaussians has been optimized. We compared various methods based on the Obara-Saika, McMurchie-Davidson, Gill-Head-Gordon-Pople, and Rys polynomial algorithms using Cartesian, Hermite, and mixed Gaussian integrals for each scheme. The latter ERIs contain both Hermite and Cartesian Gaussians, and they combine the advantageous properties of both types of basis functions. Furthermore, prescreening of the ERI derivatives is discussed, and an efficient approximation of the Cauchy-Schwarz bound for first derivatives is presented. Based on the estimated operation counts, the most promising schemes were implemented by automated code generation, and their relative performances were evaluated. We analyzed the benefits of computing all of the derivatives of a shell triplet simultaneously compared to calculating them just for one degree of freedom at a time, and it was found that the former scheme offers a speedup close to an order of magnitude with a triple-zeta quality basis when appropriate prescreening is applied. In these cases, the Obara-Saika method with Cartesian Gaussians proved to be the best approach, but when derivatives for one degree of freedom are required at a time the mixed Gaussian Obara-Saika and Gill-Head-Gordon-Pople algorithms are predicted to be the best performing ones.

UR - http://www.scopus.com/inward/record.url?scp=85053893923&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053893923&partnerID=8YFLogxK

U2 - 10.1063/1.5049529

DO - 10.1063/1.5049529

M3 - Article

C2 - 30278674

AN - SCOPUS:85053893923

VL - 149

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 12

M1 - 124101

ER -