The Tight-Binding Method

The nearly-free-electron model starts from plane waves and adds a weak periodic potential. That works for the simple metals but fails wherever the electrons of interest are tightly bound to their atoms — the $d$ -electrons of a transition metal, the $π$ -electrons of a carbon sheet, the valence electrons of an insulator. For these the natural starting point is the isolated atom, whose sharp orbital levels broaden into bands only because neighbouring atoms overlap. The tight-binding method builds Bloch states directly from atomic orbitals and produces a band whose width is controlled by a single overlap-driven quantity, the hopping integral. It is the microscopic basis of the molecular-orbital picture extended from two atoms to an infinite lattice.

From atomic levels to bands

Consider $N$ identical atoms brought together from infinity onto a lattice. When they are far apart each atomic level is $N$ -fold degenerate — the same orbital on every atom, at the same energy. As the atoms approach, the tails of the orbitals on neighbouring sites overlap, the wavefunctions on different atoms are no longer independent, and the degeneracy lifts. The $N$ levels spread into a quasi -continuous band whose width grows as the spacing shrinks and the overlap increases. Deep core orbitals, which barely reach the next atom, stay razor -sharp; valence orbitals, which overlap substantially, broaden into wide bands. This broadening is the tight-binding picture of band formation, complementary to the gaps of the nearly-free-electron picture — the same bands seen from the atomic side.

N atomic levels, degenerate at large separation, broaden into bands as the interatomic spacing decreases and orbital overlap grows. Tightly bound core levels stay narrow; valence levels spread into wide bands that can eventually overlap.

The tight-binding Bloch sum

Let $ϕ (r)$ be a single atomic orbital (an $s$ -orbital, to begin) with isolated-atom energy $ε_{at}$ , satisfying $H_{at} ϕ = ε_{at} ϕ$ where $H_{at}$ is the single-atom Hamiltonian. Place one such orbital on every lattice site $R$ and form the linear combination that obeys Bloch's theorem,

ψ_{k} (r) = \frac{1}{N} R \sum e^{i k \cdot R} ϕ (r - R) .

Translating by a lattice vector $R^{'}$ shifts the sum's index and produces the factor $e^{i k \cdot R^{'}}$ , so this is a Bloch state of wavevector $k$ . The crystal Hamiltonian is $H = H_{at} + Δ U (r)$ , where $Δ U$ is the extra potential from all the other atoms. The band energy is the expectation value

E (k) = \frac{⟨ ψ _{k} ∣ H ∣ ψ _{k} ⟩}{⟨ ψ _{k} ∣ ψ _{k} ⟩} = \frac{\sum _{R} e ^{i k \cdot R} ⟨ ϕ _{0} ∣ H ∣ ϕ _{R} ⟩}{\sum _{R} e ^{i k \cdot R} ⟨ ϕ _{0} ∣ ϕ _{R} ⟩},

writing $ϕ_{R} (r) = ϕ (r - R)$ . Two simplifications make this tractable. First, atomic orbitals on different sites overlap weakly, so the denominator is dominated by its $R = 0$ term (normalization), and the small overlap integrals $⟨ ϕ_{0} ∣ ϕ_{R} ⟩$ for $R \neq = 0$ are dropped. Second, the matrix elements $⟨ ϕ_{0} ∣ H ∣ ϕ_{R} ⟩$ fall off rapidly with distance, so only nearest neighbours are kept. Define

ε = ⟨ ϕ_{0} ∣ H ∣ ϕ_{0} ⟩ = ε_{at} + ⟨ ϕ_{0} ∣ Δ U ∣ ϕ_{0} ⟩, - t = ⟨ ϕ_{0} ∣ H ∣ ϕ_{δ} ⟩,

the on-site energy $ε$ (the atomic level shifted by the crystal field) and the hopping integral $t$ between an orbital and its neighbour at $δ$ . The band energy becomes

E (k) = ε - t δ \sum e^{i k \cdot δ},

the sum running over the nearest-neighbour vectors.

The hopping integral t is the Hamiltonian matrix element between an orbital on one site and its neighbour; it is large where the orbital tails overlap strongly (shaded region) and decays quickly with separation.

The s-band dispersion

On a one-dimensional chain of spacing $a$ each site has two neighbours at $δ = \pm a \overset{x}{^}$ , and the sum gives

E (k) = ε - t (e^{ik a} + e^{- ik a}) = ε - 2 t cos k a .

The band is a cosine of full width $4 t$ , centred on the shifted atomic level $ε$ . Its minimum $ε - 2 t$ sits at $k = 0$ (all orbitals in phase, the fully bonding combination) and its maximum $ε + 2 t$ at the zone boundary $k = \pm π / a$ (alternating signs, the fully antibonding combination) — the infinite-lattice generalization of the bonding/antibonding splitting of a diatomic molecule. Near the band bottom the cosine expands to $E \approx (ε - 2 t) + t a^{2} k^{2}$ , a parabola with an effective mass $m^{*} = ℏ^{2} /2 t a^{2}$ : a narrow band (small $t$ ) gives a heavy effective mass, a wide band a light one.

On a simple cubic lattice the six neighbours at $\pm a \overset{x}{^}, \pm a \overset{y}{^}, \pm a \overset{z}{^}$ give the separable form

E (k) = ε - 2 t (cos k_{x} a + cos k_{y} a + cos k_{z} a),

a band of width $12 t$ running from $ε - 6 t$ at the zone centre $Γ$ to $ε + 6 t$ at the zone corner. The bandwidth counts the neighbours: the more atoms an orbital can hop to, the wider the band.

The s-band cosine dispersion along a line of the Brillouin zone. The band runs from its minimum at the zone centre (bonding, all orbitals in phase) to its maximum at the boundary (antibonding, alternating phase); the full width is 4t in one dimension.

Multiple orbitals and p-bands

An atom contributes more than one valence orbital, and each generates its own band. When several orbitals are close in energy — the three $p$ -orbitals, or an $s$ and a $p$ — they must be treated together: the Bloch sum becomes a vector of coefficients, one per orbital, and $E (k)$ is found by diagonalizing a small matrix at each $k$ , exactly as the two-atom molecular-orbital problem diagonalizes a two-by-two secular determinant. The hopping now depends on orientation. Two $p$ -orbitals pointing along the bond ( $σ$ overlap) hop strongly; two pointing across the bond ( $π$ overlap) hop weakly, because their lobes overlap with opposite signs on the two sides. The $σ$ -band is therefore wider than the $π$ -band from the same orbitals. This directional hopping, tabulated as the Slater–Koster two-centre integrals, is what makes the $p$ -bands of a real crystal anisotropic and is the mechanism behind the linear Dirac bands of graphene, where the honeycomb $π$ -band touches at the zone corners.

Orientation dependence of p-orbital hopping. End-on (sigma) overlap along the bond is strong and gives a wide band; side-on (pi) overlap across the bond is weak, with the lobes meeting at opposite signs, giving a narrow band.

Wannier functions

The Bloch states $ψ_{n k}$ are delocalized: each spreads over the whole crystal with equal amplitude on every atom. Their localized counterparts are the Wannier functions, obtained by a Fourier sum over the zone,

w_{n} (r - R) = \frac{1}{N} k \sum e^{- i k \cdot R} ψ_{n k} (r) .

One Wannier function is centred on each site $R$ ; they are mutually orthogonal, span the same subspace as the Bloch states of band $n$ , and — for an isolated band — decay rapidly away from their centre. In the tight-binding limit the Wannier function is essentially the atomic orbital $ϕ (r - R)$ itself, which is why the method works: the atomic orbital is a good approximation to the exact localized basis. Wannier functions make the connection between the band and the bond explicit, and they are the natural language for anything local in a crystal — impurities, surfaces, the real-space hopping models used in correlated-electron theory.

A Bloch state (top) has uniform amplitude across the crystal, its envelope modulated only by the plane-wave phase. The corresponding Wannier function (bottom) is localized on a single site, decaying rapidly to the neighbours; the two describe the same band.

Tight binding and nearly-free electrons are the two analytic limits of band theory. A real band structure — computed today by density-functional methods that this course leaves to numerics — lies between them, wide and free-electron -like where the orbitals overlap strongly, narrow and atomic where they do not. With the bands in hand, the next lesson fills them with electrons and reads off which crystals conduct.