The Mott-Hubbard gap and breakdown of the band model

The ionisation energy (I) is the energy needed to remove an electron from an orbital. The electron affinity (A) is the energy you get back when placing an electron on an already occupied site. Therefore the energy required to move an electron is;

U = I – A

This can be interpreted as the repulsion energy between two electrons on the same atom. For example, with hydrogen, U = 12.8eV (13.6 – 0.8)

The electron repulsion makes the half-filled band insulating when the interaction between atoms is small. The situation as the bandwidth increases is shown opposite. (fig. 13) On the left (where bandwidth, W, is zero) are the atomic energies. –I is the energy of the singly occupied orbitals. –A is the energy of an extra electron added to the solid, leading to a doubly occupied orbital. The gap, U, is the energy required to excite an electron to move to another orbital. The gap is different to the one that occurs in band theory and is a consequence of electron repulsion.

The sub-bands are the lower and upper levels, each holding one electron per atom and the band gap, U, is referred to as the Mott-Hubbard splitting. The sub-bands broaden with inter-atomic overlap and meet when W = U approximately. After this point, there is no energy gap and the solid is metallic. Therefore, for band theory to work we require W > U.
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