Barrier crossing rate model

In the previous sections a model of the frequency-dependent collisional friction has been derived. Because the zero-frequency friction for a spherical particle in a dense fluid is well modeled by the Stokes-Einstein result, even for particles of similar size as the bath particles, there has been considerable interest in generalizing the hydrodynamic approach used to derive this result into the frequency domain in order to derive a frequency-dependent friction that takes into account collective bath motions. The theory of Zwanzig and Bixon, corrected by Metiu, Oxtoby, and Freed, has been invoked to explain deviation from the Kramers theory for unimolec-ular chemical reactions. The hydrodynamic friction can be used as input in the Grote-Hynes theory [Eq. (2.35)] to determine the reactive frequency and hence the barrier crossing rate of the molecular reaction. However, the use of sharp boundary conditions leads to an unphysical nonzero high-frequency limit to Ib(s). which compromises its utility. 

Although, AE is not generally equal to AG° unless the frequencies of the reactant and product are the same, this assumption is almost universally made. With this assumption classical Marcus ET theory combined with a quantum mechanical (Landau-Zener) treatment of the barrier crossing also yields Eq. (4) [2,36-39]. This derivation of Eq. (4) is called semiclassical ET theory, and therefore in the rest of this paper Eq. (4) will also be referred to as the semiclassical rate expression or the semiclassical model. 

Here we allow for the possibility of diffusion control in the barrier-crossing process [cf) in contrast to the intertial rate constant (c) generally assumed for passage from activated reactants to products in the TST model [84, 87]. The activation a) or deactivation (A) rate constants may also be diffusive. The occurrence of k < 1 reflects diffusive recrossings of the system in the TS region. In the absence of diffusional bottlenecks cf = c h), k becomes unity (i.e., the TST limit). 

We would be remiss if we did not comment on isomerization dynamics in liquids, since they form a very important and widely studied class of reactions. There have been many theoretical models for isomerization reactions in liquids. In Section III.C we briefly outlined Kramers approach to this problem. Since that time much more extensive studies of such barrier-crossing problems have been carried out. These studies have been concerned mainly with obtaining expressions for the rates in the transition region between the low- and high-friction cases, or with effects arising from the nonlocality of the diffusion coefficient in the context of Smoluchowski equation descriptions, applications to polymeric systems, and so on." ... 

An earlier suggestion of Fixman of how to mimic flexibility has been discussed with respect to a system with two degrees of freedom. Harmonic stretching and bending of the bonds of butane in liquid CCI4 has subsequently been allowed transition state theory does not exactly apply as many barrier crossings are reflected by solvent collisions. In a modified molecular dynamics examination of conformational isomerizations in butane the effect of solvent was expressed with a stochastic model in which the Newtonian trajectory was modified by random impulses. The frequency of these impulses, which have a frictional effect upon the trajectory, reduced the value of the transmission coefficient by inducing oscillatory motion at the col. At the inner bonds of decane isomerization rates are less than in butane. 

The rate constant for barrier crossing is then calculated from the fluxes out of and into the reactant and product stable states. The rate constant Xqii obtaLlned with this model is the following ... 

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