RRKM theory classical limit

Intrinsic non-RRKM behavior occurs when an initial microcanonical ensemble decays nonexponentially or exponentially with a rate constant different from that of RRKM theory. The former occurs when there is a bottleneck (or bottlenecks) in the classical phase space so that transitions between different regions of phase space are less probable than that for crossing the transition state [fig. 8.9(e)]. Thus, a micro-canonical ensemble is not maintained during the unimolecular decomposition. A limiting case for intrinsic non-RRKM behavior occurs when the reactant molecule s phase space is metrically decomposable into two parts, for example, one part consisting of chaotic trajectories which can decompose and the other of quasiperiodic trajectories which are trapped in the reactant phase space (Hase et al., 1983). If the chaotic motion gives rise to a uniform distribution in the chaotic part of phase space, the unimolecular decay will be exponential with a rate constant k given by... 

The numerator is the sum of states for the transition state with energies from 0 to - Eq. Only in the high energy limit does classical mechanics give accurate sums and density of state. Thus, in general, quantum RRKM theory is applied where quantum statistical mechanics is used to determine the density of... 

The first of the theoretical chapters (Chapter 9) treats approaches to the calculation of thermal rate constants. The material is familiar—activated complex theory, RRKM theory of unimolecular reaction, Debye theory of diffusion-limited reaction—and emphasizes how much information can be correlated on the basis of quite limited models. In the final chapt, the dynamics of single-collision chemistry is analyzed within a highly simplified framework the model, based on classical mechanics, collinear collision geometries, and naive potential-energy surfaces, illuminates many of the features that account for chemical reactivity. 

RRKM theory is also at the basis of localization of loose transition states in the PES. Another assumption of the theory is that a critical configuration exists (commonly called transition state or activated complex) which separates internal states of the reactant from those of the products. In classical dynamics this is what is represented by a dividing surface separating reactant and product phase spaces. Furthermore, RRKM theory makes use of the transition state theory assumption once the system has passed this barrier it never comes back. Here we do not want to discuss the limits of this assumption (this was done extensively for the liquid phase [155] but less in the gas phase for large molecules we can have a situation similar to systems in a dynamical solvent, where the non-reacting sub-system plays the role... 

Apart from the analysis of the energy transfer properties as expressed by Zl. and Pc, the foregoing discussion of feo represents an elaboration of the low-pressure limit of RRKM theory (Robinson and Holbrook, 1972). Since calculation of the various factors of Eq. (3.6) is time consuming, it is often avoided in favor of the simpler RRK theory, in which all of the factors of Eq. (3.6) are given the form prescribed by classical statistical mechanics, i.e., Eq. (3.5). This simplification can give quite unsatisfactory results. Inspection... 

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