  Hedin's scheme is solved with the inclusion of the vertex function
($GW\Gamma$) for a set of small molecules.
  The computational scheme allows for the consistent inclusion of the vertex
both at the polarizability level and in the self-energy.
  A diagrammatic analysis shows that the self-energy formed with this
four-point vertex does not lead to double counting of diagrams, that can be
classified as direct "bubbles" and exchange diagrams.
  By removing the exchange diagrams from the self-energy, a simpler
approximation is obtained, called $GW^{\rm{tc-tc}}$.
  Very good agreement with expensive wavefunction-based methods is obtained for
both approximations.
