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Motion of a single electron in a disordered alloy and or interactingelectrons systems such as magnetic materials, strongly correlated systems andsuperconductors is replaced by motion of that in an effective medium which isdenoted by self-energy. The study of disordered alloy and interacting electronssystems based on single electron motion is an old challenge and an importantproblem in condensed matter physics. In this paper we introduce a real spaceapproximation beyond super cell approximation for the study of these systems tocapture multi-site effects. Average disordered alloy or interacting system isreplaced by a self-energy, $\Sigma(i,j,E)$. We divided self-energy in q-space$\Sigma({\bf q}; E)=\frac{1}{N}\sum_{ij}e^{i{\bf q}.{\bf r}_{ij}}\Sigma(i,j;E)$ into two parts $\Sigma({\bf q}; E)=\frac{1}{N_{c}}\sum_{IJ\in\; \mbox{\tinysame cluster}}e^{i{\bf q}.{\bf r}_{IJ}}\Sigma(I,J; E)+\frac{1}{N}\sum_{ij\notin\:\mbox{\tiny same cluster}}e^{i{\bf q}.{\bf r}_{IJ}}\Sigma(I,J,E)$ where$\{Lc_{1}, Lc_{2},Lc_{3}\}$ are dimensions of the super cell. We show thatneglecting the second term of q-space self-energy leads to super cellapproximation $e^{iq_{j} Lc_{j}}=1$, hence $ q_{j}$ determined by $ q_{j}Lc_{j}=2\pi n_{j}$. Then we kept this correction in the second step to add selfenergies of sites in different super cells which leads to fully q-dependentself energy in the first Brillouin zone (FBZ). Our self-energy in FBZ iscasual, fully q-dependent, and continuous with respect to ${\bf q}$. Itrecovers coherent potential approximation in the single site approximation andis exact when the number of sites in the super cell approaches to the totalnumber of lattice sites. We illustrate that this approximation undertakeselectrons localization for one and two dimensional alloy systems which isn'tobserved by previous multi site approximations.

Beyond real space super cell approximation, corrections to the real space cluster approximation