Investigating exact density-functional theory of a model semiconductor

Using the diffusion quantum Monte Carlo method, we calculte the ground-state density and energy, and the quasiparticle band gap, of a model semiconductor. The exchange-correlation potential of density-functional theory (DFT), ${\mathit{V}}_{\mathrm{xc}}$(r), is obtained using optimization techniques. From this we calculate the DFT functionals ${\mathit{E}}_{\mathrm{sc}}$ and ${\mathit{T}}_{\mathit{s}}$ and the DFT band gap for various external potentials and compare the results with the local-density approximation (LDA). Whereas energies are found to be very accurate in the LDA, and the density reasonably good, we find large differences in the shape of ${\mathit{V}}_{\mathrm{xc}}$(r).