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