Addressing the Coulomb potential singularity in exchange-correlation energy integrals with one-electron and two-electron basis sets

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 straightforward grid-based approach is linear in the number of grid points, because grid points where the integrand’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.