= Tight binding
{wiki}
= Tight-binding model
{synonym}
= Tight-binding approximation
{synonym}
Tight binding approximates a quantum system using orbitals localized near sites and Hamiltonian matrix elements describing their energies and hopping. In a one-dimensional chain with one orbital per cell and nearest-neighbor hopping,
$$
H=\epsilon_0\sum_j|j\rangle\langle j|-t\sum_j(|j+1\rangle\langle j|+|j\rangle\langle j+1|).
$$
A <Bloch state> gives the dispersion $E(k)=\epsilon_0-2t\cos(ka)$ and bandwidth $4|t|$.
The finite bandwidth makes explicit how overlap between localized states produces delocalized bands. More orbitals, longer-range hopping, spin and magnetic phases give richer matrices. A localized-orbital basis is an approximation to the physical problem, and its parameters must be obtained from a microscopic calculation or fit. The hopping amplitude is a matrix element, not a probability of a completed stochastic hop.
Back to article page
