Accurate Exchange-Correlation Potential for Silicon and Its Discontinuity on Addition of an Electron

Abstract

We obtain an accurate density-functional exchange-correlation potential, ${V}_{\mathrm{xc}}(\mathrm{r})$, for silicon, from calculations of the self-energy $\ensuremath{\Sigma}(\mathrm{r},{\mathrm{r}}^{\ensuremath{'}},\ensuremath{\omega})$. No local-density approximation (LDA) is used for ${V}_{\mathrm{xc}}$. The band structure with this ${V}_{\mathrm{xc}}$ is in remarkably close agreement with that obtained with the LDA, while both differ significantly from the quasiparticle spectrum of $\ensuremath{\Sigma}$. The 50% band-gap error found in LDA calculations is therefore not caused by the LDA but by the discontinuity, $\ensuremath{\Delta}$, in the exact ${V}_{\mathrm{xc}}$ on addition of an electron.