  In solid-state physics, energies of crystals are usually computed with a
plane-wave discretization of Kohn-Sham equations. However the presence of
Coulomb singularities requires the use of large plane-wave cut-offs to produce
accurate numerical results. In this paper, an analysis of the plane-wave
convergence of the eigenvalues of periodic linear Hamiltonians with Coulomb
potentials using the variational projector-augmented wave (VPAW) method is
presented. In the VPAW method, an invertible transformation is applied to the
original eigenvalue problem, acting locally in balls centered at the
singularities. In this setting, a generalized eigenvalue problem needs to be
solved using plane-waves. We show that cusps of the eigenfunctions of the VPAW
eigenvalue problem at the positions of the nuclei are significantly reduced.
These eigenfunctions have however a higher-order derivative discontinuity at
the spheres centered at the nuclei. By balancing both sources of error, we show
that the VPAW method can drastically improve the plane-wave convergence of the
eigenvalues with a minor additional computational cost. Numerical tests are
provided confirming the efficiency of the method to treat Coulomb
singularities.
