Some remarks on the tight-binding method

$\bullet$ The characteristic feature of the tight-binding energy bands is that hte bandwidth is determined by the small overlap integral $\gamma$. Thus, the tight-binsing bands are narrow bands, and the smaller the overlap, the narrower the bands are. In the limit of vanishing overlap, the bandwidth also vanishes and the states become $N$-fold degenerate. This would correspond to core electrond residing near the nucleus, resembling $N$ isolated atoms, or atoms that are pulled very far apart.

$\bullet$ Interestingly, although commonly associated to the kinetic energy, the integral $\gamma$ -also called hopping integral- is purely generated by the potential energy, and how it hybridizes neighboring orbitals.

$\bullet$ Near the bottom of the bands, the energy is quadratic in $k$, and the constant-energy surfaces are spherical.

$\bullet$ The slope of the energy curve is zero when crossing perpendicular to one of the faces of the Brillouin zone.

$\bullet$ In solids that are not monoatomic Bravais lattices, i.e they are decorated lattices with more than one atom species, the tight-binding calculation becomes more complicated. if we have more than one atom per unit cell, we can write:

$$\vert i\rangle = \sum_{\alpha p} c_{\alpha p}\vert i \alpha p\rangle$$ (204)

where $\alpha$ denotes the different atoms in the unit cell. Then, we need to generalize the equations to obtain the matrix elements $H_{\alpha p, \beta q}$, and $S_{\alpha p, \beta q}$.