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The tight-binding model constructs crystal electronic states by starting from isolated atomic orbitals and introducing hopping between neighboring atoms. An electron in atomic orbital phi(r - R_i) at site R_i can tunnel to a neighboring site R_j with amplitude t (the hopping or transfer integral). The resulting Bloch states have energies E(k) = epsilon_0 - t sum_delta eik·delta, where the sum runs over nearest-neighbor vectors delta. For a simple cubic lattice with one orbital per site, this gives E(k) = epsilon_0 - 2t(cos k_x a + cos k_y a + cos k_z a) — a cosine band whose width is 12t. The tight-binding approach naturally produces narrow bands from localized orbitals and is the complement of the nearly free electron model.
The tight-binding model approaches band theory from the atomic limit: start with isolated atoms, each with well-defined atomic orbitals, then bring them together to form a crystal and see how the discrete atomic energy levels broaden into bands. This is essentially the LCAO (linear combination of atomic orbitals) method applied to an infinite periodic system. The key parameter is the hopping integral t, which measures the quantum mechanical amplitude for an electron to tunnel from an orbital on one atom to an orbital on a neighboring atom.
For a one-dimensional chain with one orbital per atom, the Bloch states are psi_k(r) = (1/sqrt(N)) sum_n eikna phi(r - na), and the energy eigenvalue is E(k) = epsilon_0 - 2t cos(ka), where epsilon_0 is the on-site atomic energy. The cosine dispersion has a bandwidth of 4t: the bonding state at k = 0 (all orbitals in phase) has the lowest energy, and the antibonding state at k = pi/a (alternating phases) has the highest. In three dimensions on a simple cubic lattice, the sum over three directions gives E(k) = epsilon_0 - 2t(cos k_x a + cos k_y a + cos k_z a) with bandwidth 12t.
The physical content of the model is that bandwidth measures delocalization. Larger orbital overlap means larger t and wider bands — the electron is more "free" to move through the crystal. Smaller overlap means narrower bands and more localized behavior. This is why s and p electrons, which extend far from the nucleus, form wide bands and behave nearly free-electron-like, while d and f electrons form narrow bands where strong correlation effects (magnetism, Mott insulating behavior, heavy fermion physics) become important. The tight-binding model quantifies this intuition precisely.
In practice, realistic tight-binding models include multiple orbitals per site, different hopping amplitudes for sigma and pi bonding, next-nearest-neighbor hopping, and spin-orbit coupling. The method is computationally efficient because the Hamiltonian is sparse (only neighboring sites are coupled), and it provides excellent physical intuition about how band structure arises from chemistry. It is also the natural language for many modern topics: the Hubbard model adds on-site electron-electron repulsion to tight-binding, graphene's band structure is a two-orbital tight-binding model on the honeycomb lattice, and topological insulator models are often formulated in tight-binding language.