Figure 1 Double well potential for a generic conformational transition showing the regions of reactant and product states separated by the transition state surface.

to mount the barrier and cross from the reactant to the product state. Moreover, once the barrier is crossed and the system loses energy it will spend many oscillations in the product well before recrossing the barrier. When we examine the equilibrium probability distribution along such a reaction coordinate, we will note two macrostate maxima in the probability distribution separated by a probability minimum. The position of this minimum is often a good choice for the transition state dividing surface.

One possible definition of a transition state dividing surface would be to divide a single energy basin down the middle, recognizing that the reactants sit to the left of the minimum and the products to the right. This would be a bad idea. Such a choice of the ‘‘transition state’’ dividing surface would lead to fast oscillations between the reactant and product state populations as the system moves left to right and back again in the energy basin. Therefore, a condition is needed: There must be a separation in time scales between the fast, transient motion within a macrostate and the slow, activated dynamics of motion between macrostates.

B. Phenomenological Rate Equations

When it is possible to recognize distinct macrostates, we can write the phenomenological rate equations

where kPR is the rate of transition from reactant to product while kRP is the rate of reaction from product to reactant. It is these macroscopic population changes that are usually measured in chemical kinetics experiments.

These rate equations are easily solved. At long times, the chemical dynamics reaches a stationary ‘‘equilibrium’’ and the populations of reactants and products cease to change. The relative populations of reactants and products are given by the condition of ‘‘detailed balance,’’ where the rate of transition from products to reactants equals the rate of transition from reactants to products, or

This relation defines the equilibrium constant between reactants and products, Keq

NPeq/NeqR kPR/kRP.

At short times, there is a relaxation of the reactant and product state populations to their equilibrium values. For example, the deviation from the equilibrium concentration

where δNP(0) is a measure of the initial deviation from the equilibrium population of product. What is measured experimentally as the rate of change in the population of the product states is the kinetic rate constant k kPR kRP, which is the sum of the forward and backward rate constants.

The next step in understanding the chemical kinetics of this system is the calculation of the kinetic rate constant from a knowledge of the energetics of the reaction system.

II. TRANSITION STATE THEORY

The original microscopic rate theory is the transition state theory (TST) [10–12]. This theory is based on two fundamental assumptions about the system dynamics. (1) There is a transition state dividing surface that separates the short-time intrastate dynamics from the long-time interstate dynamics. (2) Once the reactant gains sufficient energy in its reaction coordinate and crosses the transition state the system will lose energy and become deactivated product. That is, the reaction dynamics is activated crossing of the barrier, and every activated state will successfully react to form product.

A. Building the TST Rate Constant

Given the foregoing assumptions, it is a simple matter to construct an expression for the transition state theory rate constant as the probability of (1) reaching the transition state dividing surface and (2) having a momentum along the reaction coordinate directed from reactant to product. Stated another way, kPRTST is the equilibrium flux of reactant states across

energy of the system is taken to be a Hamiltonian function of the positions and momenta of both the reactive (q, p) and nonreactive (Q, P) degrees of freedom as (Γ, γ) (P, p) (Q, q). dΓ dγ is an incremental volume of the total phase space of nonreactive dΓ dQ dP and reactive dγ dq dp degrees of freedom. Similarly we find that

where θ(q‡ q), the Heaviside step function, counts only those configurations in which the reaction coordinate is to the reactant side of the transition state dividing surface. Note that the nonreactive Q coordinates orthogonal to the reaction coordinate q have been averaged over to define the potential of mean force (q) as

where Zγ is a normalization constant. The probability of finding the system along the reaction coordinate q will be proportional to exp[ β (q)]. The potential (q) can be used to define the probability distribution of the reduced system. It is possible to compute an exact transition state theory rate constant using (q). However, if the dynamics of the reaction coordinate is studied on the effective potential (q), the assumption is that the degrees of freedom Q are averaged over during the time scale for the essential dynamics of the reaction coordinate q.

It is a remarkable fact that the microscopic rate constant of transition state theory depends only on the equilibrium properties of the system. No knowledge of the system dynamics is required to compute the transition state theory estimate of the reaction rate constant kTST.

B. Some Details

We counted the contribution of only those trajectories that have a positive momentum at the transition state. Trajectories with negative momentum at the transition state are moving from product to reactant. If any of those trajectories were deactivated as products, their contribution would need to be subtracted from the total. Why? Because those trajectories are ones that originated from the product state, crossed the transition state twice, and were deactivated in the product state. In the TST approximation, only those trajectories that originate in the reactant well are deactivated as product and contribute to the reactive flux. We return to this point later in discussing dynamic corrections to TST.

A few more comments are in order. The backward rate constant can be computed from the condition of detailed balance

Suppose that the reactant well can be approximated as harmonic and the activation energy is much larger than the thermal energy. In that case we can approximate the rate constant as