  We propose a novel variational method to calculate the two-hole propagators
relevant for Auger spectroscopy in transition metal oxides. This method can be
thought of as an intermediary step between the full solution (which is
difficult to generalize to systems with partially filled bands) and the
impurity approximation. Like the former, our solution has full translational
invariance, and like the latter, it can be generalized to certain types of
systems with partially filled bands. Here we compare both our variational
approximation and the impurity approximation against the exact solution for a
simple one-dimensional model with filled bands. We show that when the energies
of the eigenstates residing primarily on the transition metal ions do not
overlap with those of the eigenstates residing primarily on Oxygen ions, both
approximations are valid but the variational approach is superior.
