On the electrostatic potential in crystalline systems where the charge density is expanded in Gaussian functions

Methods for the evaluation of the electrostatic potential in systems which are periodic in three dimensions, where the electronic charge distribution is expanded as a linear combination of Gaussian functions of arbitrary quantum number, are described. An ‘exact’ method based on the Ewald potential function is derived, and an approximate scheme using a distributed point multipole representation of the charge distribution is then presented. Recursion formulae enable Cartesian and thence spherical derivatives (of any order) of the potential to be computed. Related procedures for the evaluation of the Fock matrix elements and the coulombic contribution to the total energy per unit cell are described. Tests of the exact and approximate procedures establish their relative accuracy and cost in the MgO, Si and Si₁₂O₂₄ systems in the rock-salt, diamond and chabazite structures, respectively. Applications include a study of the electric field gradient at the C, N and O sites in urea (crystalline and molecular), and contour mapping of the electrostatic potential in MgO and Si₁₂O₂₄.