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A general formulation for the efficient evaluation of *n*-electron integrals over products of Gaussian charge distributions with Gaussian geminal functions

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10.1063/1.3600745

### Abstract

In this work, we present a general formulation for the evaluation of many-electron integrals which arise when traditional one particle expansions are augmented with explicitly correlated Gaussian geminal functions. The integrand is expressed as a product of charge distributions, one for each electron, multiplied by one or more Gaussian geminal factors. Our formulation begins by focusing on the quadratic form that arises in the general n-electron integral. Using the Rys polynomial method for the evaluation of potential energy integrals, we derive a general formula for the evaluation of any n-electron integral. This general expression contains four parameters ω, θ, v, and h, which can be evaluated by an examination of the general quadratic form. Our analysis contains general expressions for any n-electron integral over s-type functions as well as the recursion needed to build up arbitrary angular momentum. The general recursion relation requires at most n + 1 terms for any n-electron integral. To illustrate the general method, we develop explicit expressions for the evaluation of two, three, and four particle electron repulsion integrals as well as two and three particle overlap and nuclear attraction integrals. We conclude our exposition with a discussion of a preliminary computational implementation as well as general computational requirements. Implementation on parallel computers is briefly discussed.

© 2011 American Institute of Physics

Received 24 March 2011
Accepted 25 May 2011
Published online 29 June 2011
Publisher error corrected 05 July 2011

Acknowledgments: We would like to sincerely acknowledge the help and support of several colleagues in the course of this work. Professor Walter Thiel provided several very useful comments after reading an early draft of this paper. Dr. Sergey Varganov provided some very useful comments after very thorough review of this paper. Finally, we would like to acknowledge the moral support of Professor Peter Taylor who has seen this work from inception to completion.

Article outline:

I. INTRODUCTION

II. PRELIMINARY CONCEPTS

A. Orbitals, geminals, and charge distributions

B. Rys quadrature and Gaussian transform

C. The *G* _{ x } integral

III. OVERLAP INTEGRALS

A. Two-electron overlap integrals

B. Three-electron overlap integrals

IV. ELECTRON REPULSION INTEGRALS, CONCEPTS, AND METHODS

V. TWO-ELECTRON REPULSION INTEGRALS

VI. THREE-ELECTRON REPULSION INTEGRALS

A. Three-electron repulsion: Type I

B. Three-electron repulsion: Type II

VII. FOUR-ELECTRON REPULSION INTEGRALS

VIII. NUCLEAR ATTRACTION INTEGRALS

IX. COMPUTATIONAL DISCUSSION

A. Rys roots and weights

B. Outline of three-electron evaluation

X. CONCLUSIONS

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2011-06-29

2014-04-25

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