Nuclear motion in diatomic molecules

In Section 13.2, we shall show that the total energy E for an electronic state of a diatomic molecule is approximately the sum of electronic, vibrational, rotational, and translational energies, E iieiec + vib + rot + where the constant E [not to be confused with iiei in (13.7)] is given by iieiec = U Re). 

The approximation of separating electronic and nuclear motions is called the Born-Oppenheimer approximation and is basic to quantum chemistry. [The American physicist J. Robert Oppenheimer (1904-1967) was a graduate student of Born in 1927. During World War II, Oppenheimer directed the Los Alamos laboratory that developed the atomic bomb.] Born and Oppenheimer s mathematical treatment indicated that the true molecular wave function is adequately approximated as 

Bom and Oppenheimer s 1927 paper justifying the Bom-Oppenheimer approximation is seriously lacking in rigor. Subsequent work has better justified the Bora-Oppenheimer approximation, but significant questions stiU remain the problem of the coupling of nuclear and electronic motions is, at the moment, without a sensible solution and. .. is an area where much future woik can and must be done [B. T. Sutcliffe, J. Chem. Soc. Faraday Trans., 89, 2321 (1993) see also B. T. Sutcliffe and R. G. Woolley, Phys. Chem. Chem. Phys., 7, 3664 (2005), and Sutcliffe and Woolley, arxiv.org/abs/1206.4239]. 

Most of this chapter deals with the electronic Schrodinger equation for diatomic molecules, but this section examines nuclear motion in a bound electronic state of a diatomic molecule. From (13.10) and (13.11), the Schrodinger equation for nuclear motion in a diatomic-molecule bound electronic state is 

The potential energy U R) is a function of only the relative coordinates of the two nuclei, and the work of Section 6.3 shows that the two-particle Schrddinger equation (13.13) can be reduced to two separate one-particle Schrddinger equations—one for translational energy of the entire molecule and one for internal motion of the nuclei relative to each other. We have 

The application of the Bom-Oppenheimer and the adiabatic approximations to separate nuclear and electronic motions is best illustrated by treating the simplest example, a diatomic molecule in its electronic ground state. The diatomic molecule is sufficiently simple that we can also introduce center-of-mass coordinates and show explicitly how the translational motion of the molecule as a whole is separated from the internal motion of the nuclei and electrons. 

The total number of spatial coordinates for a molecule with Q nuclei and N electrons is 3(Q + N), because each particle requires three cartesian coordinates to specify its location. However, if the motion of each particle is referred to the center of mass of the molecule rather than to the external spaced-fixed coordinate axes, then the three translational coordinates that specify the location of the center of mass relative to the external axes may be separated out and eliminated from consideration. For a diatomic molecule (Q = 2) we are left with only three relative nuclear coordinates and with 3N relative electronic coordinates. For mathematical convenience, we select the center of mass of the nuclei as the reference point rather than the center of mass of the nuclei and electrons together. The difference is negligibly small. We designate the two nuclei as A and B, and introduce a new set of nuclear coordinates defined by 

The kinetic energy operator Tq for the two nuclei, as given by equation 

The laplacian operators in equation (10.23) refer to the spaced-fixed coordinates Qa with components Qxa, Qya, Qza, so that 

However, these operators change their form when the reference coordinate system is transformed from space fixed to center of mass. 

The translational energy levels can be taken as the energy levels (3.72) of a particle in a three-dimensional box whose dimensions are those of the container holding the gas of diatomic molecules. 

The most fundamental way to solve (13.19) is as follows (a) solve the electronic Schrodinger equation (13.7) at several values of R to obtain of the particular molecular electronic state one is interested in (b) add Z Z e IR to each E value to obtain U at these R values (c) devise a mathematical function U R) whose parameters are adjusted to give a good fit to the calculated U values (d) insert the function U R) found in (c) into the nuclear-motion radial Schrodinger equation (13.19) and solve 

A commonly used fitting procedure for step (c) is the method oi cubic splines, tor which computer programs exist (see Press et al. Chapter 3 Shoup, Chapter 6). 

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For more discussion of nuclear motion in diatomic molecules, see Section 13.2. For the rotational energies of polyatomic molecules, see Levine, Molecular Spectroscopy, Chapter 5. 

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