  We discuss a generalization of the dynamical mean field theory (DMFT) for
strongly correlated systems close to a Mott transition based on a systematic
approximation of the fully irreducible four-point vertex. It is an atomic-limit
approximation of a functional of the one- and two-particle Green functions,
built with the second Legendre transform of the free energy with respect to the
two-particle Green function. This functional is represented diagrammatically by
four-particle irreducible (4PI) diagrams. Like the dynamical vertex
approximation (D$\Gamma$A), the fully irreducible vertex is computed from a
quantum impurity model whose bath is self-consistently determined by solving
the parquet equations. However, in contrast with D$\Gamma$A and DMFT, the
interaction term of the impurity model is also self-consistently determined.
The method interpolates between the parquet approximation at weak coupling and
the atomic limit, where it is exact. It is applicable to systems with
short-range and long-range interactions.
