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Self-consistent vertex corrected GW calculations have been performed toevaluate the equilibrium volume and corresponding bulk modulus of 3d transitionmetal vanadium. The study demonstrates the feasibility of this approach. Theaccuracy of the results obtained for vanadium is considerably better ascompared to the accuracy of the results obtained with self consistent GW methodwithout vertex corrections. It is shown that vertex corrections are importantnot only for the self energy, but also for the irreducible polarizability. Thecontribution to the value of the $\Psi$-functional from the vertex correctionsis about 30 times smaller than the contribution from the GW part (correlationonly!), suggesting the fast convergence of the expansion in terms of thescreened interaction for this material. Consideration of the Ward-TakahashiIdentity shows that the first order vertex correction eliminates considerable(about 80\%) part of the mismatch found in scGW calculations. The remainingsmall mismatch in the Ward-Takahashi Identity leaves, however, a possibility ofthe calculated low energy electronic structure being affected by higher orderdiagrams (not included in this study).

Vertex corrections in self-consistent GW$\Gamma$ calculations: ground state properties of vanadium