Correlation energy for the homogeneous electron gas: Exact Bethe-Salpeter solution and an approximate evaluation

The correlation energy of the homogeneous electron gas is evaluated by solving the Bethe-Salpeter equation (BSE) beyond the Tamm-Dancoff approximation for the electronic polarisation propagator. The BSE is expected to improve upon the random phase approximation, owing to the inclusion of exchange diagrams. For instance, since the BSE reduces in second order to M{\o}ller-Plesset perturbation theory, it is self-interaction free in second order. Results for the correlatione energy are compared with Quantum Monte Carlo benchmarks and excellent agreement is observed. For low densities, however, we find imaginary eigenmodes in the polarisation propagator. To avoid the occurence of imaginary eigenmodes, an approximation to the BSE kernel is proposed, which allows to completely remove this issue in the low electron density region. We refer to this approximation as the random phase approximation with screened exchange (RPAsX). We show that this approximation even slightly improves upon the standard BSE kernel.