  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.
