Double exponential transformation for computing three-center nuclear attraction integrals

ABSTRACT

Three-center nuclear attraction integrals, which arise in density functional and ab initio calculations, are one of the most time-consuming computations involved in molecular electronic structure calculations. Even for relatively small systems, millions of these laborious calculations need to be executed. Highly efficient and accurate methods for evaluating molecular integrals are therefore all the more vital in order to perform the calculations necessary for large systems. When using a basis set of B functions, an analytical expression for the three-center nuclear attraction integrals can be derived via the Fourier transform method. However, due to the presence of the highly oscillatory semi-infinite spherical Bessel integral, the analytical expression still remains problematic. By applying the S transformation, the spherical Bessel integral can be converted into a much more favorable sine integral. In the present work, we then apply two types of double exponential transformations to the resulting sine integral, which leads to a highly efficient and accurate quadrature formula. This method facilitates the approximation of the molecular integrals to a high predetermined accuracy, while still keeping the calculation times low. The fast convergence properties analyzed in the numerical section illustrate the advantages of the method.

GRAPHICAL ABSTRACT

Keywords:

• Molecular multi-center integrals
• exponential type orbitals
• spherical bessel integrals
• double exponential transformation
• trapezoidal rule

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The corresponding author acknowledges the financial support for this research by the Natural Sciences and Engineering Research Council of Canada (NSERC) - Grant RGPIN-2016-04317.