Pictorial descriptions of the phase difference between bound and continuum vibrational wavefunctions

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)  

Alternatively, the level shift for E Ec can be understood in terms of a noncrossing curve (adiabatic) representation. The outer- (inner-) crossing case corresponds to an adiabatic curve that is wider (narrower) below Ec than the diabatic curve. Broadening (narrowing) the potential curve has the effect of shifting each v, J-level to lower (higher) energy. 

The difference between the behavior of T at inner versus outer wall crossings can be understood by examining Eq. (7.6.11). For an outer crossing, the phase difference, f (E), varies rapidly with v and the sin2 4 (E) function will oscillate rapidly. For an inner crossing, the phase difference does not change very much because the potential curves are nearly parallel. This is evident from the definition of f (E), 

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Figure 7.19 Pictorial descriptions of the phase difference between bound and continuum vibrational wavefunctions. The top part of the figure shows the crossing bound and repulsive potential curves and the two paths between which the phase shift is to be determined. The lower part of the figure represents the classical phase-space trajectories for motion on Vj (ellipse) and V2 (parabola). The shaded area is the phase difference between the two paths, (o) Outer wall crossing. Path I (single arrows) is the most direct dissociation path ai to Rc on Vj, Rc to oo on V2. Path II (double arrows) is the shortest indirect path 01 to i to Rc on Vi, Rc to a2 to oo on V2. (6) Inner wall crossing. The phase difference is between the shortest ( i to Rc on V, Rc to a2 to oo on V2) and next longer ( i to 01 to Rc on Vi,Rc to oo on V2) path.

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