Correlation energy within exact-exchange ACFD theory: systematic development and simple approximations

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.