Local-density approximation for dynamical correlation corrections to single-particle excitations in insulators

This paper presents a general approach to the calculation of single-particle excitations in insulating crystals, with application to silicon and diamond. The method includes the energy dependence of the self-energy which is evaluated in the Green's-function—screened-Coulomb-interaction ("GW") approximation of Hedin [Phys. Rev. 139, A796 (1965)], which is applied in the local-density approximation of Sham and Kohn. This self-energy describes changes in exchange-correlation effects beyond those described by the ground-state exchange-correlation potential. The essential ingredient is the model of a homogeneous insulating electron gas, built on the ideas of Penn and of Levine and Louie. It is shown that this quasiparticle local-density-approximation (QPLDA) selfenergy for an insulator with gap $E_g$ is topologically distinct from its metallic counterpart due to a nonanalytic contribution to the self-energy proportional to $E_g(k-k_F)\ln |k-k_F|$. This gives a discontinuity in the energy derivative of the dynamic self-energy, which, in turn, leads to an increase in band gaps over the ground-state values. Both effects can be related directly to the gap between occupied and unoccupied states; charge inhomogeneities play only a minor role in this approach. Direct and indirect band gaps in both silicon and diamond are in much better agreement with experiment than are the results from the ground-state theory. The zone-boundary optical gaps and the valence-band width in Si appear to be predicted very accurately by the QPLDA, while a 0.24-eV discrepancy remains in the indirect band gap. In diamond the valence-band width and fundamental direct and indirect band gaps are within 5% of the experimental values. To provide a basis for an intuitive grasp of exchange-correlation corrections to excitation energies, the full nonlocal interaction for the model homogeneous system is presented and analyzed in some detail.