  Motion of a single electron in a disordered alloy and or interacting
electrons systems such as magnetic materials, strongly correlated systems and
superconductors is replaced by motion of that in an effective medium which is
denoted by self-energy. The study of disordered alloy and interacting electrons
systems based on single electron motion is an old challenge and an important
problem in condensed matter physics. In this paper we introduce a real space
approximation beyond super cell approximation for the study of these systems to
capture multi-site effects. Average disordered alloy or interacting system is
replaced 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 $\Sigma({\bf q}; E)=\frac{1}{N_{c}}\sum_{IJ\in\; \mbox{\tiny
same cluster}}e^{i{\bf q}.{\bf r}_{IJ}}\Sigma(I,J; E)+\frac{1}{N}\sum_{ij\notin
\:\mbox{\tiny same cluster}}e^{i{\bf q}.{\bf r}_{IJ}}\Sigma(I,J,E)$ where
$\{Lc_{1}, Lc_{2},Lc_{3}\}$ are dimensions of the super cell. We show that
neglecting the second term of q-space self-energy leads to super cell
approximation $e^{iq_{j} Lc_{j}}=1$, hence $ q_{j}$ determined by $ q_{j}
Lc_{j}=2\pi n_{j}$. Then we kept this correction in the second step to add self
energies of sites in different super cells which leads to fully q-dependent
self energy in the first Brillouin zone (FBZ). Our self-energy in FBZ is
casual, fully q-dependent, and continuous with respect to ${\bf q}$. It
recovers coherent potential approximation in the single site approximation and
is exact when the number of sites in the super cell approaches to the total
number of lattice sites. We illustrate that this approximation undertakes
electrons localization for one and two dimensional alloy systems which isn't
observed by previous multi site approximations.
