Parabolic barrier tunneling

Fig. 8. Arrhenius plot of dissipative tunneling rate in a cubic potential with Vq = Sficoo and r jlto = 0, 0.25 and 0.5 for curves 1-3, respectively. The cross-over temperatures are indicated by asterisks. The dashed line shows k(T) for the parabolic barrier with the same CO and Va-...

If the potential is parabolic, it seems credible that the inverted barrier frequency A should be substituted for the parabolic barrier transparency to give the dissipative tunneling rate as... 

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. 

In reality, as the barrier becomes narrower, it deviates from the square shape. One often used model is the parabolic barrier (dashed line in Fig. 1). When the barrier is composed of molecules, not only is the barrier shape difficult to predict, but the effective mass of the electron can deviate significantly from the free-electron mass. In order to take these differences into account, a more sophisticated treatment of the tunneling problem, based on the WKB method, can be used [21, 29-31]. Even if the metals are the same, differences in deposition methods, surface crystallographic orientation, and interaction with the active layer generally result in slightly different work functions on either side of the barrier. 

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. 

In Equation 21, T is the absolute temperature, h is Planck s constant, is Boltzmann constant, and AG is the free energy barrier height relative to infinitely-separated reactants. The temperature-dependent factor r(7) represents quantum mechanical tunneling and the Wigner approximation to tunneling through an inverted parabolic barrier ... 

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... 

Figure 3.2. Time contour (a) and real part (b) of the tunneling trajectory for a separable system of a parabolic barrier-harmonic oscillator (schematic). Curves E, E and E" are equipotentials. Vertical and horizontal dashed lines show the loci of vibrational and translational turning points. Points A, B, and C indicate the corresponding times and positions along trajectory. (From Altcorn and Schatz [1980]).

This expression is identical to the tunneling correction factor for a parabolic barrier. 

Tunneling has been suggested as the cause of these effects, and parabolic barrier widths of 1-3 A assigned (Caldin and Kasparian, 1965 Kreevoy, 1965a). Again the attribution seems reasonable, and no alternative is apparent. 

In the second section the calculation of the rate constant was discussed from the classical mechanics viewpoint. Voth, Chandler, and Miller derived a quantum mechanical expression for the rate constant based on a path integral formalism. Using this expression as a starting point, Voth and O Gormani derived an effective barrier model to allow the calculation of the barrier tunneling contribution to the quantum mechanical rate constant for reactions in dissipative baths. The spirit of their derivation is quite similar to that which treats Grote-Hynes theory o as transition state theory for a parabolic barrier in a harmonic bath. 

All above conclusions are involved as special cases in the general consequences of the collision theory rate equation (51j III) derived in Sec.7.III. The corresponding consequences from the statistical formulation (67.Ill) of the reaction rate theory were also discussed there. The current interpretations of kinetic isotope effects are based on transition state theory. The correction for proton tunneling is first taken into consideration by BELL et al./155/. More extensive work in this direction has been carried out by CALDIN et al. /I53/. In this treatment estimations of the tunneling correction are made using one-dimensional (parabolic) barrier by neglecting the coupling of the proton motion with other motions of reactants or solvent. 

The parabolic barrier approximation for the fluctuation-induced tunneling gives the following relationship in respect of the temperature dependence of conductivity [49, 50] ... 

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