The Born–Oppenheimer approximation is an essential element without which the very notion of a potential energy surface would not exist (1). It also provides an example of how different coordinates can often be treated independently as a first approximation. This approach has far-reaching consequences, since it greatly simplifies the construction of partition functions in statistical mechanics. The approximation involves neglect of terms that couple together the electronic and nuclear degrees of freedom. The nuclear motion is then governed entirely by a single PES for each electronic state because the Schrödinger equation can be separated into independent nuclear and electronic parts. The simplest approach to the nuclear dynamics then leads to the normal mode approximation via successive coordinate transformations. These developments are treated in some detail, partly because the results are used extensively in subsequent chapters, and partly because they highlight important general principles, which can easily be extended to other situations. The consequences of breakdown in the Born–Oppenheimer approximation, and treatments of dynamics beyond the normal mode approach, are discussed in Section 2.4 and Section 2.5, respectively.

Independent degrees of freedom

The Schrödinger equation that we normally wish to solve in order to identify wavefunctions and energy levels is a partial differential equation if more than one coordinate is involved. The most common method of solution for such equations involves separation of variables (2).