Barrier curvature

The key quantity in barrier crossing processes in tiiis respect is the barrier curvature Mg which sets the time window for possible influences of the dynamic solvent response. A sharp barrier entails short barrier passage times during which the memory of the solvent environment may be partially maintained. This non-Markov situation may be expressed by a generalized Langevin equation including a time-dependent friction kernel y(t) [ ]... 

We can now use the typical value of the barrier curvature from our tunneling argument in Section IIIA (see Fig. 10) to estimate the typical frequency of motion at the tunneling barrier top. We now express the barrier profile F(7/) as a function of the droplet s radius r = a(3Al/47t) and obtain... 

As an exercise, let us compare the spontaneous fission half-lives of two nuclei with barrier heights of 5 and 6 MeV, respectively, and barrier curvatures of 0.5 MeV. One quickly calculates that the spontaneous fission half-lives of these two nuclei differ by a factor of 3 x 105. The barrier heights and curvatures in this example are relevant for the actinides and illustrate the difficulty that a 1-MeV uncertainty in the fission barrier height corresponds to a factor of 105 in the spontaneous fission half-life. 

An elegant explanation for the unusual viscosity dependence was provided by the non-Markovian rate theory (NMRT) of Grote and Hynes [149] which incorporates the idea of frequency dependence of the friction. According to this theory the friction experienced by the reactive motion is not the zero frequency macroscopic friction (related to viscosity) but the friction at a finite frequency which itself depends on the barrier curvature. The rate is obtained by a self-consistent calculation involving the frequency-dependent friction. 

The potential, U(x), in the barrier region is approximated to an inverted parabola with a frequency Wf> related to the barrier curvature (tu , = [/x 1 d2U 0)/d2x Y 2). Actually, Kramers solves the steady-state Fokker-Planck equation associated with eq.(19) to find the following rate constant ... 

Fig. 14.2 A one-dimensional model for a barrier crossing process. The potential barrier is characterized by the well and barrier curvatures which determine the frequencies (Uq and (dq, and by the barrier height E-q.

In order to fit the data, it was however necessary to adjust the barrier height as well as the barrier curvature. This was done in order to fit the data for normal ethyl-chloride (C2H5CH). The ab initio and the experimentally determined barrier heights and curvatures are shown in table 7.6. 

Barrier curvature parameter. Dots experimentai data crosses curvature of the inner barrier line caicuiated curvature of the iiquid drop barrier inserts representative barrier shapes for an actinide and a transactinide nucieus... 

Here hcofis the barrier curvature parameter and the number of barrier assaults, = 2.5 x 10 ° O Figure 19.14 shows the experimental barrier curvatures plotted against tbe fissiiity parameter. The liquid drop barrier (Nix 1967) is indicated as aline. Tbe actinides show weakly curved thick liquid-drop dominated barriers with a curvature of 0.5 MeV. The transactinide barriers have curvatures close to 1 MeV, compatible with the curvature of the inner barrier measured for the light actinides (Bjornholm and Lynn 1980). 

On the other hand generalization from this behaviour would be rash. Comparison of calculations for the H -I- H2 hydrogen-exchange reactions with the best available ab initio results suggest that both BEBO and LEPS overestimate their barrier curvatures (vj ), as is shown in Table 8. Moreover, the relatively high ab initio value of... 

In view of the high values of Oil and the substantial barrier curvatures they imply, even in the case of the modified ab initio value, it may be questioned whether tunnelling corrections can be neglected. However, in what is probably the best available theoretical estimate,... 

Sterically-hindered transition states for hydrogen transfers have especially large tunnel corrections. This arises both from solvent exclusion and from the increase in the energy of the transition state and in the interaction force constant, and hence in the barrier curvature. 

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