  We have calculated the correlation energy of the homogeneous electron gas
(HEG) and the dissociation energy curves of molecules with covalent bonds from
a novel implementation of the adiabatic connection fluctuation dissipation
(ACFD) expression including the exact exchange (EXX) kernel. The EXX kernel is
defined from first order perturbation theory and used in the Dyson equation of
time-dependent density functional theory. Within this approximation (RPAx), the
correlation energies of the HEG are significantly improved with respect to the
RPA up to densities of the order of $r_s \approx 10$. However, beyond this
value, the RPAx response function exhibits an unphysical divergence and the
approximation breaks down. Total energies of molecules at equilibrium are also
highly accurate but we find a similar instability at stretched geometries.
Staying within an exact first order approximation to the response function we
use an alternative resummation of the higher order terms. This slight
redefinition of RPAx fixes the instability in total energy calculations without
compromising the overall accuracy of the approach.
