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.
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