  When described through a plane-wave basis set, the inclusion of exact
nonlocal exchange in hybrid functionals gives rise to a singularity, which
slows down the convergence with the density of sampled $k$ points in reciprocal
space. In this work, we investigate to what extent the treatment of the
singularity through the use of an auxiliary function is effective for $k$-point
samplings of limited density, in comparison to analogous calculations performed
with semilocal density functionals. Our analysis applies for instance to
calculations in which the Brillouin zone is sampled at the sole $\Gamma$-point,
as often occurs in the study of surfaces, interfaces, and defects or in
molecular dynamics simulations.
