Construction of the two-electron contribution to the Fock matrix by numerical integration

Abstract

A novel method to numerically calculate the Fock matrix is presented. The Coulomb operator is re-expressed as an integral identity, which is discretized. The discretization of the auxiliary t dimension separates the x, y, and z dependencies transforming the two-electron Coulomb integrals of Gaussian-type orbitals (GTO) to a linear sum of products of two-dimensional integrals. The s-type integrals are calculated analytically and integrals of the higher angular-momentum functions are obtained using recursion formulae. The contributions to the two-body Coulomb integrals obtained for each discrete t value can be evaluated independently. The two-body Fock matrix elements can be integrated numerically, using common sets of quadrature points and weights. The aim is to calculate Fock matrices of enough accuracy for electronic structure calculations. Preliminary calculations indicate that it is possible to achieve an overall accuracy of at least 10⁻¹² E _h using the numerical approach.

Keywords:

• Fock matrix
• two-electron integrals
• numerical integration

Acknowledgements

This research has been supported by the Academy of Finland through its Computational Science Research Programme (LASTU) and within project 137460 and the Human Frontier Science Program through the grant RGP00391/2008. CSC – the Finnish IT Center for Science is thanked for computer time. We also acknowledge the Magnus Ehrnrooth Foundation for financial support. The authors would like to thank Jussi Lehtola for his helpful suggestions.