Statistical state specificity

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

In this chapter we elucidate the state-specific perspective of unimolec-ular decomposition of real polyatomic molecules. We will emphasize the quantum mechanical approach and the interpretation of the results of state-of-the-art experiments and calculations in terms of the quantum dynamics of the dissociating molecule. The basis of our discussion is the resonance formulation of unimolecular decay (Sect. 2). Summaries of experimental and numerical methods appropriate for investigating resonances and their decay are the subjects of Sects. 3 and 4, respectively. Sections 5 and 6 are the main parts of the chapter here, the dissociation rates for several prototype systems are contrasted. In Sect. 5 we shall discuss the mode-specific dissociation of HCO and HOCl, while Sect. 6 concentrates on statistical state-specific dissociation represented by D2CO and NO2. Vibrational and rotational product state distributions and the information they carry about the fragmentation step will be discussed in Sect. 7. Our description would be incomplete without alluding to the dissociation dynamics of larger molecules. For them, the only available dynamical method is the use of classical trajectories (Sect. 8). The conclusions and outlook are summarized in Sect. 9. 

Irregular quantum systems, which dissociate in a statistical state-specific manner, cannot be analyzed in terms of progressions and the adiabatic picture becomes irrelevant. Nevertheless, the fluctuations have the same physical origin For each resonance state there is a unique distribution of the total excitation energy among the internal degrees of freedom and. 

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. 

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. 

In this chapter, we discussed the principle quantum mechanical effects inherent to the dynamics of unimolecular dissociation. The starting point of our analysis is the concept of discrete metastable states (resonances) in the dissociation continuum, introduced in Sect. 2 and then amply illustrated in Sects. 5 and 6. Resonances allow one to treat the spectroscopic and kinetic aspects of unimolecular dissociation on equal grounds — they are spectroscopically measurable states and, at the same time, the states in which a molecule can be temporally trapped so that it can be stabilized in collisions with bath particles. The main property of quantum state-resolved unimolecular dissociation is that the lifetimes and hence the dissociation rates strongly fluctuate from state to state — they are intimately related to the shape of the resonance wave functions in the potential well. These fluctuations are universal in that they are observed in mode-specific, statistical state-specific and mixed systems. Thus, the classical notion of an energy dependent reaction rate is not strictly valid in quantum mechanics Molecules activated with equal amounts of energy but in different resonance states can decay with drastically different rates. 

The classical unimolecular dynamics is ergodic for molecules like NO2 and D2CO, whose resonance states are highly mixed and unassignable. As described above, their unimolecular dynamics is identified as statistical state specific. The classical dynamics for these molecules are intrinsically RRKM and a microcanonical ensemble of phase space points decays exponentially in accord with Eq. (3). The correspondence found between statistical state specific quantum dynamics and quantum RRKM theory is that the average of the N resonance rate constants fe,) in an energy window E + AE approximates the quantum RRKM rate constant k E) [27,90]. Because of the state specificity of the resonance rates, the decomposition of an ensemble of the A resonances is non-exponential, i.e. 

The conclusion one reaches is that quantum RRKM theory is an incomplete model for unimolecular decomposition. It does not describe fluctuations in state-specific resonance rates, which arise from the nature of the couplings between the resonance states and the continuum. It also predicts steps in k E), which appear to be inconsistent with the actual quantum dynamics as determined from computational chemistry. However, for molecules whose classical unimolecular dynamics is ergodic and intrinsically RRKM and/or whose resonance rates are statistical state specific (see Section 15.2.4), the quantum RRKM k E) gives an accurate average rate constant for an energy interval E E + AE [47]. 

Possible forms of P k) have not been established for groups of resonance states which are mode-specific or groups which contain both mode specific and non-mode-specific resonances. More work is required to determine P k) for systems that are not statistical-state-specific. However, as discussed below, the P k) in Eq. (8.17) may also fit systems which are not statistical-state-specific. 

The connection between the Porter-Thomas nonexponential N(r, E) distribution and RRKM theory is made through the parameters k and v. The average of the statistical state-specific rate constants k is expected to be similar to the RRKM rate constant k(E). This can be illustrated (Waite and Miller, 1980) by considering a separable (uncoupled) two-dimensional Hamilton H = + Hy whose decomposition path is... 

Levine (1988) has found that the ABA resonances studied by Manz and coworkers (Bisseling et al., 1985, 1987) can be fit by Eq. (8.17) with v = 1.8. These ABA resonances include a large number of mode-specific states and the ABA system is certainly not statistical state specific. The inferences to be made, in light of this result, is that the ability to fit a collection of resonance widths to Eq. (8.17) does not prove the system is statistical-state-specific. As discussed above, the evidence for statistical state specificity is the absence of any patterns in the positions of the resonances in the spectrum so that all the resonance states are intrinsically unassignable. This will be the case when the expansion coefficients, for the resonance wave functions i j , are Gaussian random variables for any zero-order basis set (Polik et al., 1990b). 

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. 

It is also interesting to consider the classical/quantal correspondence in the number of energized molecules versus time N(/, E), Eq. (8.22), for a microcanonical ensemble of chaotic trajectories. Because of the above zero-point energy effect and the improper treatment of resonances by chaotic classical trajectories, the classical and quantal I l( , t) are not expected to agree. For example, if the classical motion is sufficiently chaotic so that a microcanonical ensemble is maintained during the decomposition process, the classical N(/, E) will be exponential with a rate constant equal to the classical (not quantal) RRKM value. However, the quantal decay is expected to be statistical state specific, where the random 4i s give rise to statistical fluctuations in the k and a nonexponential N(r, E). This distinction between classical and quantum mechanics for Hamiltonians, with classical f (/, E) which agree with classical RRKM theory, is expected to be evident for numerous systems. 

The other limiting case for i/r is the situation for which they are intrinsically unassignable. As a result, the rate constant k for may not be related to any particular mode excitation of the unimolecular reactant. What is found is that the distribution of k for j/ in an energy interval E E + AE, is as statistical as possible. Such unimolecular dynamics is referred to as statistical state specific . 

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