Addressing the Coulomb potential singularity in exchange-correlation energy integrals with one-electron and two-electron basis sets Rogelio Cuevas-Saavedra, Paul W. Ayers ⇑ Department of Chemistry & Chemical Biology, McMaster University, Hamilton, Ontario, Canada article info Article history: Received 10 March 2012 In final form 19 April 2012 Available online 27 April 2012 abstract Nonlocal functionals for the exchange-correlation energy like the weighted density approximation require evaluating six-dimensional integrals with a Coulomb singularity. The convergence of a straight- forward grid-based approach is linear in the number of grid points, because grid points where the inte- grand’s magnitude exceeds a threshold must be neglected. This slow convergence makes extrapolation to the infinite-grid limit problematic. We introduce an alternative approach, based on basis-set expansion using either conventional three-dimensional basis functions or explicitly-correlated basis functions. The approach using explicitly-correlated GAUSSIAN geminal basis functions converges particularly rapidly. Ó 2012 Elsevier B.V. All rights reserved. 1. Motivation In quantum many-body theory, one often needs to evaluate six- dimensional integrals with the form, W½q¼ 1 2 ZZ qðrÞKðr; r 0 Þqðr 0 Þ jr  r 0 j drdr 0 ð1Þ where qðrÞ is the electron density. For many interesting choices of the correlation factor, Kðr; r 0 Þ, this integral cannot be performed analytically and numerical methods must be used. Six-dimensional numerical integration is challenging numerically, especially when the integrand is singular. If one uses the standard approach based on the direct product of two three-dimensional numerical integra- tion grids [1], or even a sparse-grid approach using a Smolyak-type construction [2], the integration grid will include points where the two particles’ coordinates are the same (r ¼ r 0 ). These points must be omitted from the numerical quadrature. Omitting these points, however, results in a quadrature that converges only with the slow rate of n 1 grid , where n grid is the number of grid points. There are other approaches to these integrals in the literature, including methods based on short-range/long-range decomposition [2,3], solution of the Poisson equation by eigenvector decomposition, Green’s theo- rem tricks, and center-of-mass transformation [4]. In this Letter we will discuss methods using basis sets, which we believe to be the most promising technique overall, partly because it makes con- tact with the computational machinery of conventional molecular electronic structure theory. Our interest in this type of integral stems from density func- tional theory (DFT), where integrals of this form appear in the expression for nonlocal exchange-correlation energy functionals, E xc ½q¼ 1 2 ZZ qðrÞ h xc ðr; r 0 Þqðr 0 Þ jr  r 0 j drdr 0 ð2Þ where h xc ðr; r 0 Þ¼ Z 1 0 h k xc ðr; r 0 Þdk ð3Þ is the exchange-correlation hole averaged over the constant-density adiabatic connection [5,6]. This type of two-point exchange-correla- tion functional is as old as the weighted density approximation [7–9] and has recently reappeared in the context of direct correla- tion function [10–13], exchange-correlation hole functionals [14–16] and nonlocal density functionals for dispersion [17–23]. 2. One-electron basis set approach We start with a density-fitting basis set, fg i ðrÞg N b i¼1 , containing N b basis functions. We define an associated potential-fitting basis set, fn i ðrÞg N b i¼1 , as the Coulomb potential of the density-fitting basis functions, n i ðrÞ¼ Z g i ðr 0 Þ jr  r 0 j dr 0 ð4Þ As long as the density-fitting basis functions are chosen so that Eq. (4) can be evaluated analytically, the singularity in the integral (1) can be subsumed in the potential-like basis functions. This is the key idea of this Letter. It is not new; similar ideas arise throughout the density-fitting approaches associated with DFT and ab initio 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.04.037 ⇑ Corresponding author. Fax: +1 905 972 2710. E-mail addresses: ayers@chemistry.mcmaster.ca, ayers@mcmaster.ca (P.W. Ayers). Chemical Physics Letters 539–540 (2012) 163–167 Contents lists available at SciVerse ScienceDirect Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett