  Hybrid density functionals show great promise for chemically-accurate first
principles calculations, but their high computational cost limits their
application in non-trivial studies, such as exploration of reaction pathways of
adsorbents on periodic surfaces. One factor responsible for their increased
cost is the dense Brillouin-zone sampling necessary to accurately resolve an
integrable singularity in the exact exchange energy. We analyze this
singularity within an intuitive formalism based on Wannier-function
localization and analytically prove Wigner-Seitz truncation to be the ideal
method for regularizing the Coulomb potential in the exchange kernel. We show
that this method is limited only by Brillouin-zone discretization errors in the
Kohn-Sham orbitals, and hence converges the exchange energy exponentially with
the number of k-points used to sample the Brillouin zone for all but
zero-temperature metallic systems. To facilitate the implementation of this
method, we develop a general construction for the plane-wave Coulomb kernel
truncated on the Wigner-Seitz cell in one, two or three lattice directions. We
compare several regularization methods for the exchange kernel in a variety of
real systems including low-symmetry crystals and low-dimensional materials. We
find that our Wigner-Seitz truncation systematically yields the best k-point
convergence for the exchange energy of all these systems and delivers an
accuracy to hybrid functionals comparable to semi-local and screened-exchange
functionals at identical k-point sets.
