Hopping integral ( $t_{ij} = ⟨ i ∣ H ∣ j ⟩$ )

A hopping integral is an off-diagonal Hamiltonian matrix element between localized orbitals on different lattice sites. It has units of energy and controls the dispersion in a tight-binding model. Its sign and phase depend on the orbital phase convention; observable band energies do not. The sign of a real nearest-neighbour

t

E (k) = E_{0} + 2 t cos (ka)

determines the location of the minimum in that convention.

Table of contents
- Virtual hopping Hopping integral
- Projected hopping matrix element and sign convention Hopping integral

Virtual hopping ( $Δ E^{(2)} = \sum_{a \neq = 0} ∣ ⟨ a ∣ T ∣0 ⟩ ∣^{2} / (E_{0} - E_{a})$ )

 0  0

A hopping process through an intermediate configuration of different unperturbed energy, contributing to a low-energy effective description through perturbation theory. If an intermediate state costs

Δ > 0

, second-order nondegenerate perturbation theory gives a contribution

- ∣ ⟨ a ∣ T ∣0 ⟩ ∣^{2} /Δ

. Its amplitude is of order

t /Δ

and its energy correction of order

- t^{2} /Δ

; a degenerate low-energy sector requires the corresponding degenerate perturbation theory before identifying the ground state.

Projected hopping matrix element and sign convention ( $- t_{ij} = ⟨ ϕ_{i} ∣ \overset{p}{^}^{2} / (2 m^{*}) + V ∣ ϕ_{j} ⟩$ )

 0  0

If a tight-binding Hamiltonian is written with coefficient

- t_{ij}

multiplying

c_{i}^{†} c_{j}

, projecting the one-electron Hamiltonian gives the displayed relation. An alternative convention defines the coefficient itself as the hopping integral and has no leading minus. Orthogonality removes constant scalar potentials from off-diagonal matrix elements, but does not remove a spatially varying potential. Orbital rephasing changes both matrix elements together and cannot repair an algebraic mismatch between these two conventions in a fixed basis.