The Vibrational Factor

Spectroscopically determined potential energy curves for bound electronic states may be extrapolated into the repulsive region by an expression V2 (R) — aR n+b where n = 12 is typically chosen. (A repulsive wall with n 12 would cause the overlap factor to be even smaller than for n = 12 because the amplitude 

As long as the two parameters A and b or A and n are chosen so that the repulsive curve crosses the bound curve at the same point, Rc, and with the same slope, overlap factors obtained from either form of the potential are nearly identical (Julienne and Krauss, 1975). In practice, the two parameters of Eqs. (7.6.1) or (7.6.2) axe varied until optimal agreement with the experimental vibration-rotation dependence of T (or r) is obtained. 

Analytic formulas can be very useful if linewidths have been measured for many vibrational levels (for example, Child, 1974). It is convenient to represent Xv,J and xe,j in a uniform semiclassical approximation (Section 5.1.1). As for the bound bound case, the overlap integral between bound and continuum wavefunctions can be expressed as an Airy function (taking into account the proper normalization factor). The linewidth is then 

If V and the form of p Ev) are known, the repulsive curve Vi can be determined by an RKR-like method that computes individual turning points (Child, 1973, 1974). This method is useful for obtaining an initial approximation for the repulsive potential. However, if only a few experimental r -values are known, it is difficult to identify unambiguously the oscillatory frequency of T versus v. For example, in Fig. 7.20 the number of vibrational levels sampled is insufficient to determine the actual shape of r . 

The predissociation level shift, 5Ev,j, for Ev J Ec has also been treated semiclassically (Bandrauk and Child, 1970)  

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The isotope effects of reactions of HD + ions with He, Ne, Ar, and Kr over an energy range from 3 to 20 e.v. are discussed. The results are interpreted in terms of a stripping model for ion-molecule reactions. The technique of wave vector analysis, which has been successful in nuclear stripping reactions, is used. The method is primarily classical, but it incorporates the vibrational and rotational properties of molecule-ions which may be important. Preliminary calculations indicate that this model is relatively insensitive to the vibrational factors of the molecule-ion but depends strongly on rotational parameters. 

Thus for a diatomic molecule, the vibrational factor is always symmetric with respect to E and so does not play any part in the symmetry classification scheme. 

The vibrational factors f0, gi and g j of (4.5) can be expressed in terms of creation operators at, of SSANMV operating on the ground state function of noninteracting phonons namely... 

The electronic factor can also provide an explanation for a nonperturbation and a test of a vibrational assignment (Robbe, et al., 1981). However, this factor can only be computed ab initio or, sometimes, estimated semiempiri-cally. This dependence on theoreticians for the electronic factor is in contrast to the situation for the vibrational factor, which is routinely calculable by the... 

As this interaction occurs between states that converge to the same electronic state of the ion (but to different N levels), their vibrational functions are nearly identical. Thus, the vibrational factor is almost unity for Av = 0 and zero for Av O. 

We can show that the vibrational factor equals 1 by expanding the... 

We can determine the effect of these operations on each factor of the wave function. The translational factor depends only on the coordinates of the center of mass of the molecule, and is unaffected. The vibrational factor is unaffected because it depends only on TAB, which is a positive scalar quantity that remains unchanged under inversion, rotation, or reflection. The rotational factor of the wave function of a diatomic molecule is a spherical harmonic function. For even values of the rotational quantum number J the spherical harmonic functions are eigenfunctions of the inversion operator with... 

We assume that the eleetronic factors in the partition fimetions of HHBrt and HBr can be approximated by l.(X)0. The ground state of the hydrogen atom has a degeneracy of 2, so we assume that its eleetronic factor can be approximated by 2.000. We also assume that the vibrational factor in the HBr partition funetion can he approximated hy 1.000. The partition function of the H atom is... 

The relationship between the classical and quantum partition functions is not so simple for the vibrational factor. If we divide the classical vibrational factor for a diatomic molecule by Planck s constant, we obtain... 

As previously discussed, one of the vibrational factors in the partition function corresponds to a very loose vibration expressed usually by k TIhv, where v is its vibrational frequency. If we define a new partition function that lacks the contribution of such loose vibration, <2 = G (k TIh), the rate of the reaction is given by... 

The difficulties noted above can be avoided by comparing the data presented in Tables 7.6 and 7.7. The nearly invariant Wnr listed in Table 7.6 and the calculated vibronic overlaps reported in Table 7.7 indicate that neither the electronic nor the vibrational factors vary essentially within the group of investigated DAPs. The former can be estimated by means of Equation 7.42, using the data in Tables 7.6 and 7.7. 

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