Calculation of the Kohn-Sham potential and its discontinuity for a model-semiconductor

Energy band gaps cannot be interpreted solely as differences in eigenvalues of the density-functional equation for the N-particle ground state. The deviation has recently been attributed to a discontinuity in the functional derivative of the exchange-correlation potential $V_{\mathrm{xc}}$ on adding holes or electrons to the ground state. The present work shows how this discontinuity can be calculated within the framework of density-functional theory. Use is made of the random-phase approximation to calculate the electronic self-energy. The frequency-dependent inverse dielectric matrix is determined with the help of moment techniques which allow us to define accurate generalizations of the plasma pole approximation. A complete calculation is performed on a two-plane-wave model, yielding not only the discontinuity in $V_{\mathrm{xc}}$ across the band gap but also information concerning the validity of the local-density approximation for the ground state.