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

We obtain an accurate density-functional exchange-correlation potential, $V_{\mathrm{xc}}\left(\mathrm{r}\right)$, for silicon, from calculations of the self-energy $\Sigma (\mathrm{r},{\mathrm{r}}^{\prime },\omega )$. 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 $\Sigma$. The 50% band-gap error found in LDA calculations is therefore not caused by the LDA but by the discontinuity, $\Delta$, in the exact $V_{\mathrm{xc}}$ on addition of an electron.