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

Correlation energy for the homogeneous electron gas: exact Bethe-Salpeter solution and new approximate evaluation