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For one localized orbital per site in a chain, take on-site matrix elements $H_{jj}=E_0$, nearest-neighbour matrix elements $H_{j,j\pm 1}=h$, overlaps $S_{jj}=1$ and $S_{j,j\pm 1}=s$, with other terms neglected and $h,s$ real. A Bloch state reduces the generalized eigenvalue problem $Hc=ESc$ toThe overlap metric must be positive for the retained model; $|s|<1/2$ is sufficient for all $k$. Dropping it yields the cosine band of the 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.

Adopt the usual orthogonal tight-binding approximation, so the three localized spatial states are orthonormal. Each site has two neighbors, with the same real hopping matrix element. For coefficients $c_n$, the Schrödinger equation at site $n$ isThe constant-coefficient state has $c_1=c_2=c_3$, giving energy $E_0+2t$. Its normalization therefore yieldsTo verify that it is the ground state, the three discrete ring momenta are $\theta =0,\pm 2\pi /3$, with $c_n\propto e^{i\theta (n-1)}$ and energies $E_0+2t\cos \theta$. The other two states both have energy $E_0-t$. Since $t<0$, $E_0+2t<E_0-t$ and the gap is $3|t|$. This is the three-site tight-binding ring spectrum.

The spatial ground state is unique and has equal probability $1/3$ on every site; its common overall phase is arbitrary. Spin would supply a separate twofold degeneracy when no spin-dependent terms are included, but does not change the requested spatial state. If the localized functions were instead nonorthogonal raw atomic orbitals, their overlap matrix would be additional data needed for normalization and a generalized eigenvalue problem.

Use one orthonormalized localized atomic orbital $|j⟩$ at each position $ja$. In the nearest-neighbour tight-binding model, translation invariance gives the on-site energy and real hopping integralThe Bloch state isSince $H|j⟩=E_0|j⟩+t|j-1⟩+t|j+1⟩$, shifting the two summation indices givesThe hopping integral is an energy, and its sign sets whether $k=0$ is a minimum or maximum. $E_0$ is the diagonal matrix element in the crystal Hamiltonian, which can differ from an isolated-atom energy.

The orthogonal tight-binding approximation matters. If raw neighbouring orbitals retain a nonzero orbital overlap $s$ in their normalization, projection instead gives a generalized eigenvalue problem and the nonorthogonal nearest-neighbour tight-binding band $E(k)=[E_0+2h\cos (ka)]/[1+2s\cos (ka)]$. The stated cosine band follows after orthonormalization and nearest-neighbour truncation, or after neglecting that overlap metric while retaining the hopping it motivates.