Hybrid-functional calculations with plane-wave basis sets: The effect of the singularity correction on total energies, energy eigenvalues, and defect energy levels

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 $\mathrm{\Gamma }$-point, as often occurs in the study of surfaces, interfaces, and defects or in molecular dynamics simulations.