General treatment of the singularities in Hartree-Fock and exact-exchange Kohn-Sham methods for solids

Abstract

We present a general scheme for treating the integrable singular terms within exact exchange (EXX) Kohn-Sham or Hartree-Fock (HF) methods for periodic solids. We show that the singularity corrections for treating these divergencies depend only on the total number and the positions of $\mathbf{k}$ points and on the lattice vectors, in particular, the unit cell volume, but not on the particular positions of atoms within the unit cell. The method proposed here to treat the singularities constitutes a stable, simple to implement, and general scheme that can be applied to systems with arbitrary lattice parameters within either the EXX Kohn-Sham or the HF formalism. We apply the singularity correction to a typical symmetric structure, diamond, and to a more general structure, trans-polyacetylene. We consider the effect of the singularity corrections on volume optimizations and $\mathbf{k}$-point convergence. While the singularity correction clearly depends on the total number of $\mathbf{k}$ points, it exhibits a remarkably small dependence upon the choice of the specific arrangement of the $\mathbf{k}$ points.