Linear optical response in silicon and germanium including self-energy effects

We calculate the optical response of silicon, ${\epsilon }_1$(ω), below the optical-absorption threshold and the static dielectric constant of germanium. The time-dependent local-density approximation (LDA) is modified by the addition of a self-energy term taken to be a ‘‘scissors operator,’’ $P_{c\mathrm{k}}$ ${\Delta }_{\mathrm{k}}$. This form leads to a Ward identity replacement p→(${\epsilon }_{n\mathrm{k}}$-$H_{\mathrm{k}}^{\mathrm{LDA}{)}^{-1}}$ (${\epsilon }_{n\mathrm{k}}$-$H_{\mathrm{k}}$)p. For silicon, we obtain 11.3 for ${\epsilon }_1$(ω=0), compared to 11.7 for experiment, 13.1 in our LDA calculation, and 8.4 in a naive self-energy corrected theory (i.e., simply modifying the eigenvalues without modifying the momentum operator). For germanium, the corresonding values as 16.5, 15.8, 21.3 and 10.4