Vibrations in diatomic molecules

Consider a stable diatomic molecule with nuclei denoted as A and B. The Born-Oppenheimer potential V for such a molecule will depend on the internuclear distance rAB and will typically have the form shown in Fig. 3.1. The potential energy has a minimum at r0, which is often referred to as the equilibrium internuclear distance. As the distance rAB increases, the potential V increases and finally reaches a limiting value where the molecule is now better described as two separated atoms (or depending on the electronic state of the system, two separated atomic species one or both of which may be ions). The difference in energy between the two separated atoms and the minimum of the potential is the dissociation energy De of the molecule. As the internuclear distance of the diatomic molecule is decreased 

Wolfsberg et al., Isotope Effects in the Chemical, Geological, and Bio Sciences, 55 

the first derivative of the potential energy with respect to rAB vanishes (the potential is at a minimum) The second derivative of the potential with respect to distance is the first non-vanishing derivative and is referred to as a force constant, designated here as f. If one replaces the potential of Fig. 3.1 by a curve with a minimum at ro, a second derivative equal to f, and no higher derivatives, i.e. if one replaces the Born-Oppenheimer potential by the parabola 

As pointed out in Chapter 2, nuclear motion takes place on the Born-Oppenheimer potential surface. The motion of the center of mass (corresponding to translation) rigorously separates from the other motions of the atoms. Translational motion may be subject to a potential corresponding to the fact that the molecule 

Consider now the vibration of the AB diatomic molecule. The potential energy of vibration depends on the displacement of the diatomic molecule internuclear distance from its equilibrium value, S = r-re. The kinetic energy involves the velocity (time derivative), S = d(r-re)/dt, and is given by 

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