The Rate Constants for Many-Dimensional Systems

In considering the equivalent of eq. (2.8) for multidimensional systems we will start by defining the relevant reaction coordinate, X, and the probability, P(X), that the system will be at different points, along X. The reaction coordinate can be taken rather arbitrarily as any well-defined parameter fe.g., X = (r23 - r12) in Fig. 1.7]. Once X is selected we can obtain P(X) by dividing the coordinate space into subsets according to the specific value of X and evaluating the one-dimensional function. 

as before, the transmission factor F expresses the fraction of the trajectories which continue to the product state after arriving at X (see Fig. 

FIGURE 2.2. A schematic description of the evaluation of the transmission factor F. The figure describes three trajectories that reach the transition state region (in reality we will need many more trajectories for meaningful statistics). Two of our trajectories continue to the product region XP, while one trajectory crosses the line where X = X (the dashed line) but then bounces back to the reactants region XR. Thus, the transmission factor for this case is 2/3. 

CHEMICAL REACTIONS IN THE GAS PHASE AND IN SIMPLE SOLVENT MODELS 

When F is equal to unity, the equation reduces to the rate expression of the well-known transition state theory. In most of the cases considered in this book, we will deal with reactions in condensed phases where F is not much different from unity and the relation between k and Ag follows the qualitative role given in Table 2.1. 

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