The Mott-Hubbard gap and breakdown of the band model

The best approach to electron repulsion in solids is the Hubbard model, which assumes the only important repulsion effects are between electrons on the same atom. The repulsion between electrons on other atoms is not insignificant, but intra-atomic effects seem to be the main culprits for the breakdown of band theory. The Hubbard model therefore offers a useful picture for discussing electron localization.

Looking at a 1D array of atoms each with a single valence s orbital. If there is one electron per atom we would have a half full band, and band theory suggests the solid should be metallic. However, in a situation where there is very little orbital overlap, the electrons will be localized on each atom in the ground state. This is due to the extra electron repulsion encountered when you try to move an electron to another atom. (fig. 12)