Exponential supercell convergence of the exact exchange energy via truncated coulomb potentials

Hybrid density functionals have become increasingly popular as a solution to mitigate the self-interaction error in semi-local density functionals, but widespread application to periodic systems has been limited by computational cost. This cost is exacerbated by poor $k$-point convergence due to the $G\to0$ singularity in the exact exchange energy, in spite of several singularity correction methods such as auxilliary function integration,\footnote{P. Carrier, S. Rohra and A. G\"orling, {\it Phys. Rev. B} {\bf 75}, 205126 (2007)}$^{,}$\footnote{I. Duchemin and F. Gygi, {\it Comp. Phys. Comm} {\bf 181}, 855 (2010)} image subtraction,\footnote{J. Paier et al., {\it J. Chem. Phys.} {\bf 122}, 234102 (2005)} and spherical truncation of the coulomb potential.\footnote{J. Spencer and A. Alavi, {\it Phys. Rev. B} {\bf 77}, 193110 (2008)} We analyze these rather disparate methods in an intuitive formalism based on Wannier function localization, which naturally suggests the truncation of the Coulomb potential on the superlattice Wigner-Seitz cell. We demonstrate that this scheme systematically exhibits the best $k$-point convergence, comparable to that of semi-local functionals, even for low-symmetry and reduced-periodicity systems where previous methods fail.