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

A Diagonally Updated Limited-Memory Quasi-Newton Method for the Weighted Density Approximation

We propose a limited-memory quasi-Newton method using the bad Broyden update and apply it to the nonlinear equations that must be solved to determine the effective Fermi momentum in the weighted density approximation for the exchange energy density functional. This algorithm has advantages for nonlinear systems of equations with diagonally dominant Jacobians, because it is easy to generalize the method to allow for periodic updates of the diagonal of the Jacobian. Systematic tests of the method for atoms show that one can determine the effective Fermi momentum at thousands of points in less than fifteen iterations.