Statistical state specific behavior

PSD s. They are reflections of the underlying wave functions, their nodal structures, and the dynamics in the exit channel. As outlined in detail in Ref. 20 (Chapters 9 and 10), in many cases the wave function at the TS defines the starting conditions for the final step of the fragmentation process If the system shows mode specificity, the PSD s also will show qualitative behaviors which are typical for excitation of particular modes. However, if the dissociation rates show statistical state-specific behavior, it does not necessarily follow that the PSD s have a statistical dependence on the quantum numbers of the fragments. An illuminating example is the dissociation of H2CO to be discussed in 7.3. 

If all the resonance states which form a microcanonical ensemble have random as defined by Eqs. (8.15) and (8.16), and are thus intrinsically unassignable, a situation arises which we will refer to as statistical state-specific behavior. Since the wave function coefficients of the il are Gaussian random variables when projected onto the basis functions for any zero-order representation (Polik et al., 1990b), the distribution of the state-specific rate constants k will be as statistical as possible. If these k within the energy interval E— E + dE form a continuous distributions, Levine (1987) has argued that the probability of a particular k is given by the Porter-Thomas (1956) distribution... 

Comparisons between state-specific quantum mechanical and classical calculations have been made for four systems, HO2 [60], NO2 [271], HNO [39], and HCO [51]. For the first three systems the quantum dynamics is statistical state-specific and the classical dynamics is in essence irregular above the dissociation threshold HCO is an example of mode-specific quantum mechanical behavior and the classical phase space is certainly not completely chaotic. 

Thus, in the high-pressure limit k(state-specific rate constants within the energy interval E—>E + dE, while /c(co, E) for the low-pressure limit is one divided by the average of the inverse of the state-specific rate constants. If all the k( are equal, (oo, E) - k(0, E) and normal RRKM behavior is observed. However, for statistical state specificity, where there are random fluctuations in the k , k(u>, E) will be pressure dependent. 

We see here then the beginnings of statistical behavior at the level of the individual quantum eigenstates. Similar results have been reported for the HO2 dissociation (Dobbyn et al., 1995). It is only in molecules where such state-specific experiments can be carried out. In most molecules, the density of states is so great that individual quantum state excitation is not possible because they are overlapped. 

The purpose of this chapter is a detailed comparison of these systems and the elucidation of the transition from regular to irregular dynamics or from mode-specific to statistical behavior. The main focus will be the intimate relationship between the multidimensional PES on one hand and observables like dissociation rate and final-state distributions on the other. Another important question is the rigorous test of statistical methods for these systems, in comparison to quantum mechanical as well as classical calculations. The chapter is organized in the following way The three potential-energy surfaces and the quantum mechanical dynamics calculations are briefly described in Sections II and III, respectively. The results for HCO, DCO, HNO, and H02 are discussed in Sections IV-VII, and the overview ends with a short summary in Section VIII. 

The essential nature of this relationship is clear statistical theories are based on a number of simplifying assumptions consistent with chaotic behavior. Specifically,2 any such theory must satisfy microscopic reversibility and the condition of zero relevance. The latter condition requires that the final state be independent of all initial conditions other than conserved quantities, that is, from the viewpoint of classical mechanics, that the system display the relaxation characteristic of chaotic motion. We note, for reference, that this minimal set of requirements allows for the construction of a large number of theories,3 the most prominant of which are the RRK.M theory of uni-molecular dissociation4 and the phase space theory of bimolecular reactions.5 Such theories have analogues, and in some cases their origins are in other areas such as nuclear physics.6... 

Consideration of bound-state dynamics affords one advantage not shared by systems undergoing reaction or decay. Specifically, since formal ergodic conditions require a compact phase space, ideal chaotic systems exist for bound systems but not for bimolecular collisions or unimolecular decay. Studies of these ideal bound systems therefore provide a route for analyzing statistical behavior in circumstances where the system is fully characterized. Furthermore, these ideal system results can be compared with the behavior of model molecular systems to assess the degree to which realistic systems display chaotic relaxation. 

The second direction has been statistical-mechanical. This usually follows the kind of reasoning described above for virial equations of state (equation 15.34). The interactions of all particles are calculated by summing interactions of pairs, triples, quadruples, and so on. In fact, what results is another regression equation with an appropriate shape and adjustable parameters with possible physical significance. Such equations are fit to real data at the present time it is not feasible to calculate the parameters theoretically and then predict the behavior of a specific gas (except for highly idealized systems). With the advent of very fast computers it has been possible... 

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