Tight binding

A tight-binding model expands a particle's state in localized orbitals and represents tunnelling between sites by off-diagonal Hamiltonian matrix elements.

For equivalent minima with nearest-neighbour hopping magnitude $\Delta$, paths with $r$ right steps and $s$ left steps have $r-s=n-m$. Summing $(\Delta \tau /\hslash {)}^{r+s}/(r!s!)$ and applying the Fourier representation of a Kronecker delta gives the displayed modified Bessel function kernel in a normalized localized-site basis. Its Fourier exponent gives the tight-binding model energy $E(\theta )=E_{\mathrm{well}}-2\Delta \cos \theta$. Position-endpoint kernels can have an additional common local-wavefunction prefactor.

Near a nondegenerate minimum $x_0$, a smooth potential satisfiesThe localized levels are therefore approximately those of a harmonic oscillator of frequency $\sqrt{V^{\prime \prime }(x_0)/m}$; tunnelling broadens each level into a narrow band.

For $N$ identical sites of spacing $a$ with periodic boundary conditions, translation eigenstates have coefficientsThus $k=2\pi j/(Na)$ modulo the reciprocal-lattice period $2\pi /a$.

For hopping amplitude $-\lambda$ from each lattice site to its neighbours at $\pm a_1$ and $\pm a_2$, a Bloch state has energyThe band ranges from $E_0-4|\lambda |$ to $E_0+4|\lambda |$ and therefore has width $8|\lambda |$.