GW approximation with self-screening correction

The GW approximation takes into account electrostatic self-interaction contained in the Hartree potential through the exchange potential. However, it has been known for a long time that the approximation contains self-screening error, as is evident in the case of the hydrogen atom. When applied to the hydrogen atom, the GW approximation does not yield the exact result for the electron removal spectra because of the presence of self-screening: the hole left behind is erroneously screened by the only electron in the system that is no longer present. We present a scheme to take into account self-screening and show that the removal of self-screening is equivalent to including exchange diagrams, as far as self-screening is concerned. The scheme is tested on a model hydrogen dimer and it is shown that the scheme yields the exact result to second order in (U-0 - U-1)/2t, where U-0 and U-1 are, respectively, the on-site and off-site Hubbard interaction parameters and t is the hopping parameter.