A composite ‘density fitting + numerical integration’ approximation for electron-repulsion integrals

Abstract

We present a new method for the evaluation of four-centre electron-repulsion integrals that combines density fitting (DF) with numerical integration by quadrature (QD). In this composite DF-QD approach, the orbital product density is fitted by a truncated spectral set of auxiliary functions and the fitting residual is expanded in a discrete space of functions localised at the eigenvalues of the coordinate operator. The new DF-QD approach has the advantage of DF of reducing computational effort and storage requirements of conventional quantum chemical algorithms, and allows for high accuracy and error control by means of QD. Systematic improvement of the accuracy is achieved for a chosen DF basis set by enlarging the number of grid points used for QD. Test calculations on a set of small molecules reveal that errors in absolute energy of the DF-QD approximation with less than 1000 and 3000 grid points per atom do not exceed one milli- and one microhartree, respectively, even if standard (non-optimised) AO basis sets are used for DF.