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Energy bands in graphene: Comparison between the tight-binding model and {\it ab initio} calculations

We compare the classification of the electron bands in graphene, obtained by group theory algebra in the framework of tight-binding model (TBM), with that calculated in the density-functional theory (DFT) framework. Identification in the DFT band-structure of all eight energy bands (four valence and four conduction bands) corresponding to the TBM-derived energy bands is performed and corresponding analysis is presented. The four occupied (three $σ$- and one $π$-like) and three unoccupied (two $σ$- and one $π$-like) bands given by DFT closely correspond to those predicted by TBM, both by their symmetry and their dispersion law. However, the two lowest lying at the $Γ$-point unoccupied bands (one of them of a $σ$-like type and the other of a $π$-like one), are not of TBM type. According both to their symmetry and to the electron density these bands are plane waves orthogonal to the TBM valence bands; dispersion of these states can be determined unambiguously up to the Brillouin zone borders. On the other hand, the fourth unoccupied band given by the TBM, can be identified among those given by the DFT band calculations; it is situated rather high with respect to energy. The interaction of this band with the free-electron states is so strong, that it exists only in a part of $k$-space.