Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 5 / b / Solution

The tight-binding model uses Bloch sums of localized atomic-like or Wannier orbitals,Including ${\mathbf{\tau }}_{\alpha }$ in the phase is an equivalent basis convention. Overlap and hopping between these localized orbitals create dispersive bands. Narrow transition-metal $d$ bands and the bonding/antibonding bands of covalent solids are natural examples; graphene's $p_z$ bands are another. The model can be accurate beyond weak hopping when enough orbitals and hopping integrals are retained.