The LCAO method described in the previous chapter forms the basis for a number of empirical or qualitative models. In such models the LCAO matrix elements are treated as “fitting” parameters to be determined from experiment or in some empirical way. Such models have provided a great deal of physical insight into the electronic properties of molecules and solids.

One of the first and simplest LCAO models was used by Hückel [1] to discuss the general qualitative features of conjugated molecules. Later, Slater and Koster [2] introduced an LCAO method for the analysis of the energy bands of solids. The Slater–Koster LCAO model has been used extensively as an interpolation scheme.

The LCAO parameters are determined by choosing the model parameters to give results that approximate those of more accurate numerical energy band calculations at a few points in the Brillouin zone. Once the parameters are determined the LCAO model gives approximate energies at any point in the Brillouin zone.

LCAO models have been remarkably useful for ordered solids and molecules having a high degree of symmetry. The reason for this is that in many cases the electronic structure is qualitatively determined by symmetry or group theoretical considerations. The group theoretical properties of a system are preserved in LCAO models and therefore they are able to correctly represent the general features of the electronic states.