Tight binding model

In tight binding model, the crystal potential is assumed to be strong. When an electron is captured by an atomic core, it stays there for long time before tunnelling to the next atomic core. Most of the time, the electron is tightly bound to its atomic core.

In a periodic one-dimensional system, the wave function can be written as:

$$\psi_k(x) = \frac{1}{\sqrt{N}} \sum_{j=1}^N \exp(ikX_j) \phi_v(x - X_j)$$

Where j is the index for the ions, and v is for a specific orbital of that atom. $X_j$ is the coordinate of the j^th atom. $X_j = ja$, where a is the lattice constant.

Following our tight binding assumption, the function $\phi_v(x - X_j)$ is centered around the j^th atom, and decays rapidly away from that atom, there is little overlap between the orbitals of neighboring atoms.

Tight binding model is suitable description of low lying narrow bands. In such cases, the cell radius is much smaller than the lattice constant (or inter-atomic distance), e.g., the d-orbitals of transition metals.