Barrier Bending

The curves follow the linear + cubic barrier-bending form expected for tunneling of electrons across an insulating layer [21,22]. 

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

In the light of the path-integral representation, the density matrix p Q-,Q-,p) may be semi-classically represented as oc exp[ —Si(Q )], where Si(Q ) is the Eucledian action on the -periodic trajectory that starts and ends at the point Q and visits the potential minimum Q = 0 for r = 0. The one-dimensional tunneling rate, in turn, is proportional to exp[ —S2(Q-)], where S2 is the action in the barrier for the closed straight trajectory which goes along the line with constant Q. The integral in (4.32) may be evaluated by the method of steepest descents, which leads to an optimum value of Q- = Q. This amounts to minimization of the total action Si -i- S2 over the positions of the bend point Q. ... 

The multiple minima nature of the bending energy and the low barriers for interconversion resemble the torsional energy for organic molecules. An expansion of bend in tenns of cosine or sine functions of the angle is therefore more natural than a... 

Some of the empirical evidence, such as the small effect of changing from H to F, Cl, Br on one end and the apparently small effect on the barrier of bending the C—H bonds, would be understandable in terms of a noncylindrical C—C bond. Further, the crudity of available quantum mechanical methods may be such that this effect is perhaps still a possible one. However, there is one serious... 

Similarly, examples of barriers arising largely from simple steric hindrance can be found, as for instance in the hindered diphenyls.35 On the other hand there are many arguments suggesting that this is not the important force in ethane and similar molecules. It would be difficult to understand the relatively slow fall in barrier from ethane to methyl silane to methyl germane on a van der Waals repulsion basis. Furthermore, the small effect of substituting F, Cl, or Br on one end would also seem mysterious. The equilibrium orientation in propylene is opposite to the predictions of one of the quantitative van der Waals theories. Finally, the apparently small effect of bending back the C—H bonds is not in accord with either the electrostatic or van der Waals pictures. 

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