The separation of nuclear and electronic motion

Bom and Oppenheimer expanded the molecular Hamiltonian in terms of a parameter k given by the ratio of a typical nuclear displacement to the internuclear distance R. Simple order-of-magnitude arguments showed that 

In this chapter we show how the separation of the quantum mechanical problem into translational, rotational, vibrational and electronic parts can be achieved. The basis of our approach is to define coordinates which describe the various motions and then attempt to express the wave function as a product of factors, each of which depends only on a small sub-set of coordinates, along the lines  

In this chapter we describe the various stages of the factorisation process. Following the separation of translational motion by reference of the particles coordinates to the molecular centre of mass, we separate off the rotational motion by referring coordinates to an axis system which rotates with the molecule (the so-called molecule-fixed axis system). Finally, we separate off the electronic motion to the best of our ability by invoking the Born-Oppenheimer approximation when the electronic wave function is obtained on the assumption that the nuclei are at a fixed separation R. Some empirical discussion of the involvement of electron spin, in either Hund s case (a) or (b), is also included. In conclusion we consider how the effects of external electric or magnetic fields are modified by the various transformations. 

The kinetic energy of the nuclei and electrons in field-free space may be written in the form 

In order to discuss the spectroscopic properties of diatomic molecules it is useful to transform the kinetic energy operators (2.5) or (2.6) so that the translational, rotational, vibrational, and electronic motions are separated, or at least partly separated. In this section we shall discuss transformations of the origin of the space-fixed axis system the following choices of origin have been discussed by various authors (see figure 2.1)  

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The separation of nuclear and electronic motion may be accomplished by expanding the total wave function in functions of the election coordinates, r, parametrically dependent on the nuclear coordinates... 

In chapter 2 we discussed at length the separation of nuclear and electronic coordinates in the solution of the Schrodinger equation. We described the Born-Oppenheimer approximation which allows us to solve the Schrodinger equation for the motion of the electrons in the electrostatic field produced by fixed nuclear charges. There are certain situations, particularly with polyatomic molecules, when the separation of nuclear and electronic motions cannot be made satisfactorily, but with most diatomic molecules the Born-Oppenheimer separation is acceptable. The discussion of molecular electronic wave functions presented in this chapter is therefore based upon the Born-Oppenheimer approximation. 

In such states, the molecule does not exhibit or exist in the full symmetry appropriate to the stationary states. Why molecules do this is the unanswered question. That they do it is simply not at issue. Given that they do, all we have to do is ask whether delocalization takes place on a time scale that rules out the ideas of separated motions and a rigid molecule. This is an empirical matter. For most chemical states, the separability of nuclear and electronic motions is valid. 

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