Using a novel self-consistent implementation of Hedin's GW perturbation theory we calculate space and energy dependent self-energy for a number of materials. We find it to be local in real space and rapidly convergent on second-to third-nearest neighbors. Corrections beyond GW are evaluated and shown to be completely localized within a single unit cell. This can be viewed as a fully self consistent implementation of the dynamical mean field theory for electronic structure calculations of real solids using a perturbative impurity solver.PACS numbers: PACS numbers: 71.28.+d, 71.25.Pi, 75.30.Mb The construction of a controlled practical approximation to the many body problem of solid state physics is a long sought goal. Controlled approximations are important because the accuracy of the results can be improved in a systematic way. This goal has been achieved in quantum chemistry by the configuration interaction (CI) method. CI can be thought as a controlled approximation that becomes more accurate as two factors are increased: a) the number of configurations (i.e. Slater determinants) kept and b) the size of the basis used to represent the one-particle orbitals which are used to represent the configurations. Dynamical mean field theory (DMFT) and its cluster extensions (C-DMFT) [1][2] merge CI ideas with band structure methods. They allow us to tackle the problem of periodic infinite systems.The central goal is the computation of the one-electron Greens function, (its Fourier transform can be measured via photoemission and inverse photoemission spectroscopy), G(r, r ′ , ω), and the self-energy Σ(r, r ′ , ω). The interaction functional Φ[G, W ] is then expanded in a perturbative series. The first few graphs are shown in Fig 1(a) and corresponds to the Hartree, the GW, and the first correction beyond GW. Variations of Φ over G and W give us Σ and Π. For the self-energy these diagrams are given in Fig 1(b). To solve the corresponding Dyson equations numerically one introduces a basis set and corresponding expansions for the self energies polar- izations and effective interactions. Cluster DMFT ideas truncate the functional Φ, Σ and Π by setting its variables, i.e. the Greens functions, equal to zero beyond a given range R. When R is one lattice spacing we have the highly successful single site DMFT, as the range R increases the approximation converges to the solution of the full many-body problem. In this paper we address the central problem of determining the minimal range that is needed to obtain accurate results for various materials.There are three different parameters that need to be increased to achieve convergence, a) the size of the basis set L max , b) the order of the perturbation theory kept n max , c) the range of the graphs R max which needs to be kept to obtain accurate approximation. R max depends on L and n. We do not consider in this paper the important issue of convergence as a function of L max as well as the dependence of the range of the type of basis set chosen. Instead we make the choice of a...