= Nonorthogonal nearest-neighbour tight-binding band
For one localized orbital per site in a chain, take on-site matrix elements $H_{jj}=E_0$, nearest-neighbour matrix elements $H_{j,j\pm1}=h$, overlaps $S_{jj}=1$ and $S_{j,j\pm1}=s$, with other terms neglected and $h,s$ real. A <Bloch state> reduces the <generalized eigenvalue problem> $Hc=ESc$ to
$$
E_0+2h\cos(ka)=E[1+2s\cos(ka)],\qquad
E(k)=\frac{E_0+2h\cos(ka)}{1+2s\cos(ka)}.
$$
The 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>.
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