Direct-space analysis of the Hartree-Fock energy bands and density of states for metallic extended systems

The restricted Hartree-Fock equations for a crystal with at least one partially filled band are written in terms of Wannier functions. The orbital energy then contains lattice sums, whose convergence properties are of decisive importance for the analytic properties of the energy bands. In particular it is shown how the particular form of the restricted Hartree-Fock exchange operator, when applied to an extended system with Coulombic forces, leads to the well-known singularity at the Fermi energy. Our analysis makes it possible to trace the sources of this singularity for a realistic system, thus leading to a better understanding of the background for systematic improvements.