Effect of coupling of forward- and backward-going electron-hole pairs on the static local-field factor of jellium

We investigate the effect of the coupling of the forward- and backward-going electron-hole pairs on the static local-field factor of jellium with and without a gap. We calculate the static local-field factor for the two types of jellium as a function of q using the Bethe-Salpeter equation. We assume that a particle and a hole interact via a statically screened Coulomb interaction. The polarization diagrams that include this interaction are summed to infinite order, leading to a matrix equation for the inverse of the polarization propagator. Employing a string of manipulations we convert the matrices in convenient forms, and we then invert the resulting matrices iteratively. This allows us to use matrices of very large size, something that would have not been feasible with straightforward inversion, and therefore achieve a very high level of convergence of results with respect to sampling of electron states in the Fermi sea with relative computational ease. For the calculation of the static local-field factor, we find that the coupling of both kinds of pairs gives qualitatively different results compared to the case when there is no coupling, and that the coupling of both kinds of electron-hole pairs is necessary for obtaining the most accurate results. We compare our results with recent calculations and point out the similarities and differences.