Tunneling through a parabolic barrier

Thus, these results indicated the involvement of heavy atom tunneling in the localized biradicals. The rates of decay for 19,20, and 9 could be fitted with Bell s simple model of tunneling through a parabolic barrier. Assuming log A (s ) = 8.0, and... 

Interestingly, this equation does not appear in this form in the oft-cited classic text by Bell (Ref. la), and refers to an earlier derivation. Bell derived a simpler expression, described in Ref. la, for tunneling through a parabolic barrier which affords approximately the same results. See Ref 2a for discussion. 

The temperature dependence of the large isotope effect for the 2,4,6-collidine is just as striking (see Chart 1 and Fig. 2). In place of the expected unit value of Ah/Aq, a value around 0.15 was found accompanied by an enormous isotopic difference in enthalpies of activation, equivalent to an isotope effect of 165. Both of these results had earlier been shown by Bell (as summarized by Caldin ) to be predicted by a onedimensional model for tunneling through a parabolic barrier. The outlines of Bell s treatment of tunneling are given in Chart 2, while Fig. 3 shows that the departure of the isotopic ratios of pre-exponential factors from unity and isotopic activation energy differences from the expected values are both predicted by the Bell approach. 

Calculate the tunneling probability of an electron and proton through a parabolic barrier of height 1 eV and width of 20 A. Assume particle = 1/2 barrier height and that a receptor state is available for the tunneling particle. (Sidik)... 

Consider a tunneling event through a parabolic barrier [Doll et al., 1972], The classical motion at energy E < V0 to the left of the barrier obeys the equation... 

A similar study by Schoofs et al. [43] of methane dissociation on the Pt(l 11) surface produced qualitatively similar results An exponential increase in the value of S0 with increasing normal energy and a weak dependence of S0 on surface temperature, Ts. Further, like Rettner et al. [40], Schoofs and coworkers find these trends consistent with a hydrogen tunneling mechanism through a one-dimensional parabolic-shaped activation barrier. 

One can ask why the electron transfer needs to he over the harrier. Can an electron not tunnel through it It can, hut just as flie case of over flie harrier, the solvent needs to adjust to the new parabolic potential. The propensity for doing so is maximal when the vibrational turning points in the two wells coincide. This occurs at the crossing distance r. There is however an exponentially small hut finite transfer probability also by tunneling under the barrier. For a discussion of solvent-mediated coupling see Zimmt and Waldeck (2003). 

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