Orthogonal tight-binding approximation

In the orthogonal tight-binding approximation, localized basis states satisfy $⟨i|j⟩={\delta }_{ij}$ while the Hamiltonian retains nonzero hopping integrals. These are independent matrix elements: orthogonal states need not have vanishing Hamiltonian coupling. One can construct localized orthonormal orbitals from overlapping atomic orbitals; truncating the resulting Hamiltonian to nearest neighbours is a further approximation. An overlap matrix must instead be retained for raw nonorthogonal orbitals.

For one orthonormal localized orbital per simple-cubic site and real equal nearest-neighbour hopping integral $t$, the six neighbours give the displayed energy band. Each cosine ranges from minus one to one, so the positive bandwidth is $12|t|$, independently of the sign convention for $t$. Near the zone centre, $E=E_0+6t-t{a}^2{k}^2+O(k^4)$ and the band effective mass is $-{\hslash }^2/(2t{a}^2)$. It is positive there for $t<0$; for $t>0$ the zone centre is a maximum instead.