Extending the Gutzwiller approximation to intersite interactions

We develop an extension of the Gutzwiller Approximation (GA) formalism which 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 on a site R. The method is conveniently formulated in terms of a Gutzwiller Lagrange function. We apply our theory to the single band t-J model and the extended single band Hubbard model. We find a Mott transition but for the extended Hubbard model. An enhanced valence fluctuations transition is also discovered.