The method of addition and subtraction of electrons to calculate a gap energy within QMC produced only mixed quality results within VMC. Even with the use of the enhanced electron-electron interaction to reduce the finite size effects in the gap energies, the VMC results are only of broadly comparable accuracy to the LDA and if anything a little worse. In contrast to the LDA, the VMC results all overestimate the band gaps and this is attributable to the inferior quality of the trial wavefunctions for the systems with electrons.
Within DMC the results from addition and subtraction of electrons are much improved over VMC and represent a significant improvement over the LDA. It appears that contrary to our initial expectations, the results are not strongly sensitive to the choice of electron-electron interaction. In fact both the gap energy calculated using the Ewald interaction and the gap energy calculated using the new interaction agree with the experimental result to within error bars. This suggests that the finite size effects in the gap energies are a short range phenomena, not long range as suggested by Ref..
The method of promoting electrons to calculate excited state energies proved successful with DMC. The results show that for the relatively simple excitations performed here, a single determinantal product is sufficient to represent the excited state. As with the addition and subtraction method, the results appear relatively insensitive to the choice of electron-electron interaction. Also, it appears that when promoting electrons, there is no need to relax the single-particle orbitals within the LDA before constructing the Slater determinant. The additional simplification of using the one- and two-body functions from the ground state trial wavefunction in the guiding wavefunction for the excited state proved successful as all the DMC calculations were numerically stable and did not exhibit any large fluctuations in the population of walkers indicative of a poor quality guiding wavefunction.
When comparing the results of all the possible excitations from the top of the valence band to the bottom of the conduction band, accessible in a n=2 supercell, the DMC shows a significant improvement over the LDA for all the excitations. On average, the DMC reduced the the difference in the excitation energy between the LDA and experiment by a factor of 3.