  We explore the combination of the extended dynamical mean field theory
(EDMFT) with the GW approximation (GWA); the former sums the local
contributions to the self-energies to infinite order in closed form and the
latter handles the non-local ones to lowest order. We investigate the different
levels of self-consistency that can be implemented within this method by
comparing to the exact QMC solution of a finite-size model Hamiltonian. We find
that using the EDMFT solution for the local self-energies as input to the GWA
for the non-local self-energies gives the best result.
