Much of what we understand about the chemistry and optical properties of molecules has come from theoretical studies of very simple, empirical models. In most cases the theoretical models employ such drastic approximations that one may wonder why the results have any relevance at all to actual molecular systems. The success of these models may be attributed almost entirely to their use of group-theoretical concepts. In many cases symmetry is the dominant factor determining the electronic structure of a molecule. While the models are crude approximations, the general structure imposed by symmetry is usually exact and often independent of the details of the model employed. As a result many of the features have a much deeper truth than the model from which they are derived.

In this chapter we discuss the use of the LCAO method (linear combinations of atomic orbitals) to analyze the electronic structure of molecules. The term “atomic orbitals” is used loosely to mean one-electron orbitals whose angular functions are the spherical harmonics. The precise specifications of the radial parts of the orbitals are not needed for our discussion.

N-electron systems

It is generally assumed that the electronic states of a molecule or solid can be calculated for fixed positions of the nuclei. The electron's velocity is very large compared with the speed of vibratory motion, so that in effect the electrons instantly readjust to any motion of the nuclei. This assumption is referred to as the Born–Oppenheimer approximation.