  Self-consistent solutions of Hedin's diagrammatic theory equations (HE) for
the two-site Hubbard Model (HM) have been studied. They have been found for
three-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-consistent
iterative solving of HE and when QP approximation is not applied. The results
obtained with the exact vertex are directly related to the presently open
question - which approximation is more advantageous for future implementations
- GW+DMFT or QPGW+DMFT. It is shown that in the regime of strong correlations
only originally proposed GW+DMFT scheme is able to provide reliable results.
Vertex corrections based on Perturbation Theory (PT) systematically improve the
GW results when full self-consistency is applied. The application of the QP
self-consistency combined with PT vertex corrections shows similar problems to
the case when the exact vertex is applied combined with QP sc. The analysis of
Ward Identity violation is performed for all studied in this work
approximations and its relation to the general accuracy of the schemes used is
provided.
