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Self-consistent solutions of Hedin's diagrammatic theory equations (HE) forthe two-site Hubbard Model (HM) have been studied. They have been found forthree-point vertices of increasing complexity ($\Gamma=1$ (GW approximation),$\Gamma_{1}$ from the first order perturbation theory, and exact vertex$\Gamma_{E}$). The comparison is being made when an additional quasiparticle(QP) approximation for the Green function is applied during the self-consistentiterative solving of HE and when QP approximation is not applied. The resultsobtained with the exact vertex are directly related to the presently openquestion - which approximation is more advantageous for future implementations- GW+DMFT or QPGW+DMFT. It is shown that in the regime of strong correlationsonly originally proposed GW+DMFT scheme is able to provide reliable results.Vertex corrections based on Perturbation Theory (PT) systematically improve theGW results when full self-consistency is applied. The application of the QPself-consistency combined with PT vertex corrections shows similar problems tothe case when the exact vertex is applied combined with QP sc. The analysis ofWard Identity violation is performed for all studied in this workapproximations and its relation to the general accuracy of the schemes used isprovided.

Full self-consistency versus quasiparticle self-consistency in diagrammatic approaches: Exactly solvable two-site Hubbard Model