Norm-conserving Hartree-Fock pseudopotentials and their asymptotic behavior

We investigate the properties of norm-conserving pseudopotentials (effective core potentials) generated by inversion of the Hartree-Fock equations. In particular we investigate the asymptotic behaviour as $\mathbf{r} \to \infty$ and find that such pseudopotentials are non-local over all space, apart from a few special special cases such H and He. Such extreme non-locality leads to a lack of transferability and, within periodic boundary conditions, an undefined total energy. The extreme non-locality must therefore be removed, and we argue that the best way to accomplish this is a minor relaxation of the norm-conservation condition. This is implemented, and pseudopotentials for the atoms H$-$Ar are constructed and tested.