Consistent treatment of charged systems within periodic boundary conditions: The projectoraugmented-wave and pseudopotential methods revisited

The ab initio calculation of charged defect properties in solids is not straightforward because of the delicate interplay between the long-range Coulomb interaction and the periodic boundary conditions. We derive the projector augmented-wave (PAW) energy and Hamiltonian with special care taken on the potentials from the Coulomb interaction. By explicitly treating the background compensation charge, we find additional terms in the total energy of the charged cells and in the potential. We show that these background terms are needed to accurately reproduce all-electron calculations of the formation energy of a charged defect. In particular, the previous PAW expressions were spuriously sensitive to the pseudization conditions and this artifact is removed by the background term. This PAW derivation also provides insights into the norm-conserving pseudopotential framework.We propose then an alternative definition for the total energy of charged cells and for the Kohn-Sham potential within this framework that better approximates the all-electron results.