Semiclassical calculation of vibrational overlap integrals

The accuracy of the semiclassical approach to vi vj (typically better than 10%) is not limited to the case Ei Vi = E2 Vj. This approach is particularly valuable for displaying the general qualitative form of versus E (see 

When an electronic state is known only through its perturbations of a better known state, frequently only the energy and rotational constant of one or two vibrational levels of unknown absolute vibrational numbering can be determined. If the information available is insufficient to generate a realistic potential energy curve, then one has no choice but to adopt a model potential and exploit relationships between Dunham (l)m) and other derived constants (vibrational overlaps), which are rigorously valid for the model potential and approximately valid for general potentials. 

The two most useful primitive model potentials are the harmonic and Morse oscillators, 

1 When Ei,Vi E2,Vj, it is necessary before applying Eq. (5.1.12) to vertically shift one of the potentials to ensure strict energy degeneracy. The new curve crossing point obtained in this way is called a pseudo-crossing point (Miller, 1968). 

In order to construct a harmonic or Morse potential from spectroscopic data, respectively, two (Re and k) or three (De,Re, and j3) independent constants must be determined. The usual routes to these constants, through Be,uje, and wexe, are often impassable and it is necessary to exploit special relationships, two of the most useful of which are due to Kratzer (Kratzer, 1920), 

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Figure 5.5 Semiclassical calculation of vibrational overlap integrals between successive vibrational levels of the N2 c 1E J Rydberg state and the bound region of the b Sj valence state where the vibrational energy in the b -state is treated as a continuous variable. Both states are represented by Morse potentials. The zero of energy is taken as v = 0, J = 0 of b 1 ). For the E values corresponding to the bound vibrational levels of the b Sj state, the values calculated using Eq. (5.1.12) reproduce the exact values (Table V of Lefebvre-Brion, 1969) to two decimal places. (Courtesy R. Lefebvre.)...

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