Abstract
The numerical evaluations of the four-center two-electron Coulomb integrals are among the most time-consuming computations involved in molecular electronic structure calculations. In the present paper we extend the double exponential (DE) transform method, previously developed for the numerical evaluation of threecenter one-electron molecular integrals [J. Lovrod, H. Safouhi, Molecular Physics (2019) DOI:10.1030/0026867.2019.1619854], to four-center two-electron integrals. The fast convergence properties analyzed in the numerical section illustrate the advantages of the new approach.
Highlights
The double exponential (DE) transformation method for the numerical evaluation of three-center oneelectron molecular integrals [1] is extended in the present paper for the evaluation of the notoriously difficult four-center molecular integrals
The need for a higher value M for the transformation φ1(τ) in comparison with the lower value M for the transformation φ2(τ) is a feature similar to the results reported in [1] for the three-center molecular integrals
The computation of the multicenter two-electron Coulomb integrals involves the highest degree of difficulty in the molecular electronic structure calculations
Summary
The double exponential (DE) transformation method for the numerical evaluation of three-center oneelectron molecular integrals [1] is extended in the present paper for the evaluation of the notoriously difficult four-center molecular integrals. The analytical expression of a four-center molecular integral, which can be derived via the Fourier transform method, contains a semi-infinite integral J(s, t), the integrand of which is slowly decaying and involves the spherical Bessel function jλ(vx). When λ and/or v are large, the oscillations of the integrand can become huge, complicating the accurate approximation of the integral by any of the existing numerical programming platforms. By applying the S transformation [2] followed by the DE transformation [1], we obtain a bi-infinite integral, which can be approximated by a trapezoidal rule. Few collocation points were required in order to obtain accurate approximations
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