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We investigate various approximations to the correlation energy of a H$_2$molecule in the dissociation limit, where the ground state is poorly describedby a single Slater determinant. The correlation energies are derived from thedensity response function and it is shown that response functions derived fromHedin's equations (Random Phase Approximation (RPA), Time-dependentHartree-Fock (TDHF), Bethe-Salpeter equation (BSE), and Time-Dependent GW(TDGW)) all reproduce the correct dissociation limit. We also show that the BSEimproves the correlation energies obtained within RPA and TDHF significantlyfor intermediate binding distances. A Hubbard model for the dimer allow us toobtain exact analytical results for the various approximations, which isreadily compared with the exact diagonalization of the model. Moreover, themodel is shown to reproduce all the qualitative results from the \textit{abinitio} calculations and confirms that BSE greatly improves the RPA and TDHFresults despite the fact that the BSE excitation spectrum breaks down in thedissociation limit. In contrast, Second Order Screened Exchange (SOSEX) gives apoor description of the dissociation limit, which can be attributed to the factthat it cannot be derived from an irreducible response function.

Static correlation beyond the random phase approximation: Dissociating H2 with the Bethe-Salpeter equation and time-dependent GW