Start from the bonding and antibonding orbitals of H2 and add atoms one at a time. With N atoms there are N energy levels — yet every level stays inside the window α ± 2|β| and just gets more crowded, so for N → ∞ they form a continuous band of width 4|β|. Closing the chain into a ring and solving again brings out Bloch’s theorem and E(k); the density of states piles up at the band edges (van Hove singularities); and the moment you alternate the atoms or the bonds, a band gap opens. Finally we move on to the two-dimensional square lattice and see how the band shapes of s, p, d, and f orbitals differ, with real space and reciprocal space linked together.
Mochizuki Research Group, Tokyo University of Science / Tight-binding (Hückel) model. The derivation, from the 2×2 matrix of H2 to the eigenvalues of an N-atom chain, Bloch’s theorem, the density of states, and finally the origin of the gap, is given on the explanation page (in Japanese).