  A few self-consistent schemes to solve the Hedin equations are presented.
They include vertex corrections of different complexity. Commonly used
quasiparticle approximation for the Green function and static approximation for
the screened interaction are avoided altogether. Using alkali metals Na and K
as well as semiconductor Si and wide gap insulator LiF as examples, it is shown
that both the vertex corrections in the polarizability P and in the self energy
$\Sigma$ are important. Particularly, vertex corrections in $\Sigma$ with
proper treatment of frequency dependence of the screened interaction always
reduce calculated band widths/gaps, improving the agreement with experiment.
The complexity of the vertex included in P and in $\Sigma$ can be different.
Whereas in the case of polarizability one generally has to solve the
Bethe-Salpeter equation for the corresponding vertex function, it is enough
(for the materials in this study) to include the vertex of the first order in
the self energy. The calculations with appropriate vertices show remarkable
improvement in the calculated band widths and band gaps as compared to the
self-consistent GW approximation as well as to the self-consistent
quasiparticle GW approximation.
