Transition state theory variable reaction coordinate

The idea in Kramers theory is to describe the motion in the reaction coordinate as that of a one-dimensional Brownian particle and in that way include the effects of the solvent on the rate constants. Above we have seen how the probability density for the velocity of a Brownian particle satisfies the Fokker-Planck equation that must be solved. Before we do that, it will be useful to generalize the equation slightly to include two variables explicitly, namely both the coordinate r and the velocity v, since both are needed in order to determine the rate constant in transition-state theory. 

Let us assume that a variable A(t) is coupled to the reaction coordinate and that (A) is its mean value. If a measurement of some property P depends on (A), but not on the particular details of the time dependence of A(t), then we will call it a statistical dependence. If the property P depends on particular details of the dynamics of A(t) we will call it a dynamical dependence. Note that in this definition it is not the mode A(t) alone that causes dynamical effects, but it also depends on the timescale of the measured property P. Promoting vibrations (to be discussed in Sections 2-4) are a dynamic effect in this sense, since their dynamics is coupled to the reaction coordinate and have similar timescales. Conformation fluctuations that enhance tunneling (to be discussed in Section 5) are a statistical effect the reaction rate is the sum of transition state theory (TST) rates for barriers corresponding to some configuration, weighted by the probability that the system reaches that configuration. This distinction between dynamic and statistical phenomena in proteins was first made in the classic paper of Agmon and Hopfield.4 We will discuss three kinds of motions ... 

In the context of association reactions, an algorithm in which the reaction coordinate definition is optimized along with the dividing surface along a one-parameter sequence of paths is called variable reaction coordinate (VRC) variational transition state theory... 

These SCTST expressions, in both the microcanonical (Eq. (27)) and canonical (Eq. (31)) forms, include coupling between all the degrees of freedom in a uniform manner. For example, even at the perturbative level, Eq. (23), there is an anharmonic coupling between modes of the activated complex x, , k and k < F -1) and between the reaction coordinate and modes of the activated complex (xkJ, < F — 1). This is not a dynamically exact theory, however, because these actions variables are in general only locally good. For energies too far above or below the barrier V0 they may fail to exist. This semiclassical theory is thus still a transition state theory (i.e., dynamical approximation). 

For radical-radical reactions, the full mode coupling and anharmonicity effects for the relative and overall rotational motions must be explicitly accounted for. We have derived a direct variable reaction coordinate transition state theory approach that appears to 3deld accurate rate coefficients for a number of alkyl radical reactions.This approach is analogous to that embodied in Eq. (4.10) for the long-range transition state, but includes variational optimizations of the form of the reaction coordinate and does not make the large orbital moment of inertia assumption. A detailed description of this approach was provided in some of our recent articles. 

S. Robertson, A. F. Wagner, and D. M. Wardlaw, /. Phys. Chem. A, 106, 2598 (2002). Flexible Transition State Theory for a Variable Reaction Coordinate Analytical Expressions and an Application. 

Y. Georgievskii and S. J. Klippenstein, /. Chem. Phys., 118, 5442 (2003). Variable Reaction Coordinate Transition State Theory Analytic Results and Application to the C2H3 -F H —> C2H4 Reaction. 

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