Closed clusters approach to graphene

The Closed Cluster method (CC method) is applied to find solutions for various calculation problems of the energy band structure of graphene. The essence of the CC method consists in the addition of closing bonds between edge atoms to the usual cluster method in order to eliminate the dangling bonds on the edges of the cluster. We study the cases of an infinite layer of graphene as well as nanoribbons, nanotubes and bilayer graphene. Results for these cases are in agreement to that what was obtained by means of other methods (tight binding approximation and others). By means of the CC method we also study the problem of point defects in graphene and obtain the distortion of the energy spectrum. The energy spectrum of the layer C$_{1-x}$ Si$_{x}$ $(0\leq x \leq 1)$ is found as well as the dependence of the energy gap on the concentration of silicon. We show that the energy band structure of C$_{1-x}$ Si$_{x}$ looks like a tunnel transition. Wave functions of graphene in the symmetry points of Brillouin zone are also obtained.