as a consequence, form bonding and anti-bonding states. Now with many atoms together, forming a crystalline lattice, the overlaps help the electron delocalise on the lattice by tunneling from one atom to the other. This tunneling potential is given by the overlap (inner product integral), denoted by $-t$, between the two states at different sites [@simon].

![Pictorial representation of a one dimensional tight-binding Hamiltonian.](fig4.pdf)

One can now write down a Hamiltonian describing a particle’s motion on a lattice structure - say, a one dimensional ring with $N$ such lattice points: $$\begin{aligned}
H & = & -\sum_{ij}t_{ij}\ket{i}\bra{j} \end{aligned}$$ Here $i,j$ represent the lattice site and $-t_{ij}$ is the overlap integral between the $i$th and $j$th site. The sum over $i,j$ runs independently over [*all*]{} $i,j$ pairs. Note since going to-and-fro between two lattice points is equivalent, we impose $t_{ij} = t_{ji}$ (this condition makes the Hamiltonian Hermitian) and $t_{ii} = 0$ for all site $i$. From quantum mechanics, we recall that the translation operator translates site $i$ to $i+1$ which we represent here as $T=\sum_i \ket{i+1}\bra{i}$. Now if we assume that the overlap integral only depends on the distance between the two sites $t_{i,i+n}\equiv t_n$ then we see that the tight 