Free-electron molecular orbital model description

The Kronig-Penney model, although rather crude, has been used extensively to generate a substantial amount of useful solid-state theory [73]. Simple free-electron models have likewise been used to provide logical descriptions of a variety of molecular systems, by a method known in modified form as the Hiickel Molecular Orbital (HMO) procedure [74]. 

We know that not all solids conduct electricity, and the simple free electron model discussed previously does not explain this. To understand semiconductors and insulators, we turn to another description of solids, molecular orbital theory. In the molecular orbital approach to bonding in solids, we regard solids as a very large collection of atoms bonded together and try to solve the Schrodinger equation for a periodically repeating system. For chemists, this has the advantage that solids are not treated as very different species from small molecules. 

The tight-binding quantum-chemical model with molecular orbitals that are a linear combination of atomic orbitals provides a reasonable description of the delocalization of the d valence electrons. The s,p valence electrons are well described as free electrons [11]. 

A simple description of electrons in a solid is the model of a free electron gas in the lattice of the ions as developed for the description of metals and metal clusters. The interaction of electrons and ions is restricted to Coulomb forces. This model is called a jellium model. Despite its simplicity, the model explains qualitatively several phenomena observed in the bulk and on the surface of metals. For a further development of the description of electrons in solids, the free electron gas can be treated by the rules of quantum mechanics. This treatment leads to the band model. Despite the complexity of the band model, Hoffmann presented a simple description of bands in solids based on the molecular orbital theory of organic molecules that will also be discussed below. 

---