= Solution
The <tight-binding model> uses Bloch sums of localized atomic-like or Wannier orbitals,
$$
\phi_{\alpha\mathbf k}(\mathbf r)=\frac1{\sqrt N}\sum_{\mathbf R}e^{i\mathbf k\cdot\mathbf R}\varphi_\alpha(\mathbf r-\mathbf R-\boldsymbol\tau_\alpha),\qquad \psi_{b\mathbf k}=\sum_\alpha c_{b\alpha}(\mathbf k)\phi_{\alpha\mathbf k}.
$$
Including $\boldsymbol\tau_\alpha$ 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.
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