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 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 optimisations and k point convergence. While the singularity corrections clearly depends on the total number of k points, it exhibits a remarkably small dependence upon the choice of the specific arrangement of the k points.

Regularization of the Coulomb singularity in exact exchange by Wigner-Seitz truncated interactions: Towards chemical accuracy in nontrivial systems

“…Ultimately, each reciprocal-space method prescribes a kernelK( q) for use in (5), which translates to some K( r) in the real space version (10). The error in any method compared to the ideal case is governed by the discrepancy of K( r) from 1/r, given by…”

Communication: Singularity-free hybrid functional with a Gaussian-attenuating exact exchange in a plane-wave basis

“…14 It is well known that it is problematic to treat integrable singularities in exact exchange Kohn-Sham for periodic solids. There have been many challenges to remedy this problem: [15][16][17][18][19][20] Gygi and Baldereschi at first proposed a method to solve this problem based on an auxiliary function. 21 However, it cannot be denied that all the proposed methods inevitably make the algorithm complex.…”

Parallel implementation of γ‐point pseudopotential plane‐wave DFT with exact exchange

“…For CPU sizes beyond a 1,000 CPUs the overall performance of our hybrid-DFT calculations using Algorithm 3 was found to be fairly reasonable. We note that the parallel algorithms developed in this work can also be readily applied to other standard plane-wave formulations of exact exchange 29,30,32 , screened exchange 45 because the singularity at r r ′ = in Eq. A.1 is analytically integrated out.…”

“…As pointed out by Gygi and Balderechi 25-28 and others 8,[29][30][31][32] , this expression must be evaluated with some care especially for small Brillioun zone samplings and small unit cell size, because of the singularity at G -k -l = 0.…”