  We present an approach to calculate the electronic structure for a range of
materials using the quasiparticle self-consistent GW method with vertex
corrections included in the screened Coulomb interaction W. This is achieved by
solving the Bethe-Salpeter equation for the polarization matrix at all k-points
in the Brillouin zone. We refer to this method as QSGW^. We show that including
ladder diagrams in W can greatly reduce the band gap overestimation of
RPA-based QSGW. The resultant discrepency of the calculated band gap in this
method is then attributed mostly to the fact that electron-phonon contributions
to W are neglected; which would allow one to then obtain an estimate for the
size of this effect. We present results for a range of systems from simple sp
semiconductors to the strongly correlated systems NiO and CoO. Results for
systems where the RPA-based QSGW band gap is larger than expected are
investigated, and an estimate for the Frolich contribution to the gap is
included in a few polar compounds where QSGW can overestimate the gap by as
much as 2 eV. The improvement over QSGW for the dielectric constants is also
presented
