  We develop an extension of the Gutzwiller Approximation (GA) formalism that
includes the effects of Coulomb interactions of arbitrary range (including
density density, exchange, pair hopping and Coulomb assisted hopping terms).
This formalism reduces to the ordinary GA formalism for the multi-band Hubbard
models in the presence of only local interactions. This is accomplished by
combining the 1/z expansion -- where z is the coordination number and only the
leading order terms contribute in the limit of infinite dimensions -- with a
P^2-I expansion, where P is the Gutzwiller projector. The method is
conveniently formulated in terms of a Gutzwiller Lagrange function. We apply
our theory to the extended single band Hubbard model. Similarly to the usual
Brinkman-Rice mechanism we find a Mott transition. A valence skipping
transition is observed, where the occupation of the empty and doubly occupied
states for the Gutzwiller wavefunction is enhanced with respect to the
uncorrelated Slater determinant wavefunction.
