Morse curves, intersecting

A Simple Picture of the Symmetry Factor. In order to employ simple geometry, one now ignores the curvature of the Morse curves and considers that the potential energy barrier near the intersection point is made up of straight lines (Fig. 9.11). This simplifying analogue of the barrier is useful for a first-base discussion of the symmetry factor p. 

Hine (1966a) also addressed the question of the physical origin of least motion effects. He considered them to arise from bond stretches and deformations in the substrate molecule, and treated the energetic consequences of these stretches and deformations in terms of intersecting Morse curves which could be approximated by parabolas. PLNM could thus be recast in terms of energetics. The energy required for a molecular deformation was considered to vary as the square of the displacement from a stable equilibrium geometry, be it reactant or product. As a consequence, reactions with very early or very late transition states were anticipated to show only small least motion effects, which should be most pronounced in reactions with central transition states. 

It was shown [8] that in the case of initial-state continuous spectra this temperature for parabolic barrier is the same as for a discrete spectrum corresponding to the intersection of two Morse curves with similar parameters and shifted equilibrium positions. 

It is well known [6, 14a, 27] that the Marcus relation may be derived from a model of intersecting parabolas for the reactant and product energy curves (Fig. 19.2). A parabola provides an unrealistic description of the energy profile for extension of an A-H bond far from its equilibrium length and towards complete dissociation a Morse curve offers a better description. It is perhaps not too widely appreciated [28, 29] that transformation of a Morse function (Eq. (19.7)) from bond length (r) to bond order (n) coordinates (Eq. (19.8)) yields a parabola (Eq. (19.9)). 

The problem lies in the model representing the forces on the proton in its transfer and two other models give results qualitatively similar to the Marcus theory. One of these involves intersecting Morse curves and the other the proton moving in an electrostatic force field between two negative charges a fixed distance apart [25]. 

With the inclusion of the electrophihcity index, the transition state given by the intersection of the Morse curves in eq. (6.61), can be obtained from... 

Now let us idealize the intersection region of the overlapping Morse potential energy curves as shown in Figure 2.15. In the nonequilibrium situation, we arbitrarily define a and b so that the change in barrier height for the reverse reaction is... 

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