  We develop the extended dynamical mean field theory (E-DMFT) with a view
towards realistic applications. {\bf 1)} We introduce an intuitive derivation
of the E-DMFT formalism. By identifying the Hartree contributions before the
E-DMFT treatment, it allows to handle systems in symmetry breaking phases
within a simple formalism. {\bf 2)} We make a new implementation of E-DMFT
through real Hubbard-Stratonovich transformation to decouple the non-local
two-particle interactions. We apply it to a 3D U-V model and investigate the
behavior of the various Green's functions, especially the density
susceptibility, as the density instability is approached. We obtain the phase
diagram at a finite temperature. {\bf 3)} We present a formalism incorporating
E-DMFT with Cellular DMFT. {\bf 4)} We suggest an improvement of the E-DMFT
approach by combining it with a generalized GW method. The method combines the
local self-energy from E-DMFT and the non-local ones from the perturbative
calculation of GW. We apply the method to a 1D U-V model with two sublattices
carrying different chemical potentials. By comparing with those from DMRG, we
show the results are shifted in the correct direction due to the GW
contributions. {\bf 5)} In order to handle the generic Coulomb repulsion within
E-DMFT, we describe a method to tailor E-DMFT so that proper momentum
dependence can be kept in general response functions.
