Fitting the Coulomb potential variationally in linear-combination-of-atomic-orbitals density-functional calculations

A previously developed method for self-consistent-field density-functional calculations involving a variational fit to the charge density is generalized to the case in which the total (electronic plus nuclear) Coulomb potential is fit. Previously the total energy $E(\rho ,\tilde{\rho })$ was viewed as a functional of the exact $\rho$ and fitted $\tilde{\rho }$ charge density. The energy expression was modified to be correct when $\tilde{\rho }=\rho$ while at the same time allowing $\tilde{\rho }$ to be obtained variationally through $\frac{\partial E(\rho ,\tilde{\rho })}{\partial \tilde{\rho }}=0\mathrm{}$. Herein an expression for $E(\rho ,\tilde{V})$, where $\tilde{V}$ is the fit to the Coulomb potential, is derived with similar properties. In particular, $\tilde{V}$ can be determined variationally through $\frac{\partial E(\rho ,\tilde{V})}{\partial \tilde{V}}=0\mathrm{}$. In various linear-combination-of-atomic-orbitals calculations on atomic neon, the superiority of variational over conventional least-squares-fitting methods is demonstrated.