  Self-consistent vertex corrected GW calculations have been performed to
evaluate the equilibrium volume and corresponding bulk modulus of 3d transition
metal vanadium. The study demonstrates the feasibility of this approach. The
accuracy of the results obtained for vanadium is considerably better as
compared to the accuracy of the results obtained with self consistent GW method
without vertex corrections. It is shown that vertex corrections are important
not only for the self energy, but also for the irreducible polarizability. The
contribution to the value of the $\Psi$-functional from the vertex corrections
is about 30 times smaller than the contribution from the GW part (correlation
only!), suggesting the fast convergence of the expansion in terms of the
screened interaction for this material. Consideration of the Ward-Takahashi
Identity shows that the first order vertex correction eliminates considerable
(about 80\%) part of the mismatch found in scGW calculations. The remaining
small mismatch in the Ward-Takahashi Identity leaves, however, a possibility of
the calculated low energy electronic structure being affected by higher order
diagrams (not included in this study).
