Intrinsic non-RRKM behavior

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... 

Figure 8.15 illustrates the presence of an intramolecular bottleneck in the interaction region of phase space. The transition rate through the turnstile in this bottleneck can be calculated using concepts described in section 4.3.1. An intramolecular bottleneck, such as the one depicted in figure 8.15, is expected to give rise to intrinsic non-RRKM behavior. 

In the following parts of this section, the selection of two different types of initial conditions for a unimolecular reactant are described. Selecting a microcanonical ensemble of states is described first. These initial conditions are never realized in an actual experiment, but are important for identifying intrinsic non-RRKM behavior and studying a molecule s intramolecular dynamics. The last procedure described is the selection of initial conditions for the nonrandom excitation of initial states for a molecule, as occurs in actual experiments. 

Bunker and Hase have suggested that statistical lifetime distributions are never quite attained for real molecules.They defined two different categories of non-RRKM (nonstatistical) behavior, apparent and intrinsic. Intrinsic non-RRKM behavior occurs when transitions between vibrational states of the excited molecule are slower than transitions leading to products. If a molecule which exhibits intrinsic non-RRKM behavior is excited randomly the intercept of its lifetime distribution at t = 0 gives k(E), which equals the RRKM unimolecular rate constant. However, for t 7 0 the unimolecular rate constant, given by equation (7), will not be a constant and its change with time will reflect the transition rates between the vibrational states. For intrinsic non-RRKM behavior P(E,t) can be represented by a sum of exponentials... 

Both apparent and intrinsic non-RRKM behavior have been observed in classical trajectory calculations. A list of the various unimole-cular reactions which have been studied by classical trajectories is given in Table 1.10,33-35, f9,51,80-10 this brief overview no... 

With respect to our understanding of the fundamental intramolecular dynamics of highly vibrationally excited molecules, intrincsic non-RRKM behavior is more significant than is apparent non-RRKM behavior. Intrinsic non-RRKM behavior exists when there are vibrational states weakly coupled to the reaction coordinate so that uni-molecular decomposition does not have a random probability. Closely related to intrinsic non-RRKM behavior is the presence of long-lived vibrational states in highly excited molecules. This is discussed in the next section. 

From the results of classical trajectory calculations intrinsic non-RRKM behavior has been predicted for ethane dissociation, ethyl radical dissociation,and methyl isocyanide isomerization. These predictions are supported by classical trajectory calculations for model H-C-C -> H + C=C dissociation. To generalize, classical trajectory calculations have predicted intrinsic non-RRKM behavior for molecules with isolated high frequency modes [e.g, CH3NC, clusters like Li (H20)j, and van der Waals molecules], molecules like acetylene with linear geometries for which bending and stretching motions are nearly separable, and molecules with tight activated complexes. 

As initially proposed by Rice and further described by Bunker and Hase, an intrinsic non-RRKM molecule should have at least one false high-pressure limit in its unimolecular fall-off plot of k versus pressure. The true unimolecular rate constant is observed as p -> CO where p is the pressure. The rate constants at the false high-pressure limit or limits are related to the transition rates between vibrational states. Thus, it should be possible to detect intrinsic non-RRKM behavior by measuring k i over the complete pressure range. However, there are several difficulties associated with this procedure. It is very difficult to perform unimolecular experiments at high pressure, and, as recently noted by Thiele et unimole-... 

Though unimolecular fall-off curves are important historically, their general insensitivity to intramolecular dynamics suggests that other types of experiments are more useful for detecting intrinsic non-RRKM behavior. Such an experiment is the one by Reddy and Berry on allyl isocyanide isomerization ... 

Each one of the eleven model H C C H + C=C potential energy surfaces was tested for intrinsic" non-RRKM behavior. This was done by choosing points at random from the HCC molecular phase space and comparing the trajectory lifetime distribution with that predicted by equation (14). The trajectory P(t) is given in histogram form by... 

At the present time, we only have a qualitative understanding of the features that lead to intrinsic non-RRKM behavior for the A and B surfaces. In studying the potential energy contour maps in the r,R plane (Fig. 2), one sees that the C and D surfaces become strongly enharmonic with negative curvature as the HC bond is extended. This anharmonicity is expected to make trajectories which sample this part of the surface separate exponentially in time instead of linearly.Exponential separation of trajectories results in stochastic behavior which should give rise to RRKM dissociation probabilities. The absence of this anharmonicity on the A and B surfaces is an explanation for their intrinsic non-RRKM lifetime distributions. 

Classical dynamics of a micro-canonical ensemble intrinsic RRKM and non-RRKM behavior... 

Initiated by the work of Bunker [323,324], extensive trajectory simulations have been performed to determine whether molecular Hamiltonians exhibit intrinsic RRKM or non-RRKM behavior. Both types have been observed and in Fig. 43 we depict two examples, i.e., classical lifetime distributions for NO2 [271] and O3. While Pd t) for NO2 is well described by a single-exponential function — in contrast to the experimental and quantum mechanical decay curves in Fig. 31 —, the distribution for ozone shows clear deviations from an exponential decay. The classical dynamics of NO2 is chaotic, whereas for O3 the phase space is not completely mixed. This is in accord with the observation that the quantum mechanical wave... 

It is not immediately obvious, by simply looking at a molecule s Hamiltonian and/or its PES, whether the unimolecular dynamics will be intrinsic RRKM or not and computer simulations as outlined here are required. Intrinsic non-RRKM dynamics is indicative of mode-specific decomposition, since different regions of phase space are not strongly coupled and a micro-canonical ensemble is not maintained during the fragmentation. The phase space structures, which give rise to intrinsic RRKM or non-RRKM behavior, are discussed in the next section. 

Intrinsic RRKM behavior is defined by Eq. (3), where an initial microcanonical ensemble of states decomposes exponentially with the RRKM rate constant [56]. Such dynamics can be investigated by computational chemical dynamics simulations. Therefore, an intrinsic non-RRKM molecule is one for which the intercept in P(t) is k(E), as a result of the initial microcanonical ensemble, but whose decomposition probability versus time is not described by k E). For such a molecule there is a bottleneck (or bottlenecks) restricting energy flow into the dissociating coordinate. Intrinsic RRKM and non-RRKM dynamics are illustrated in Fig. 15.3(a), (b), and (e). 

Trajectory calculations have been used to study the intrinsic RRKM and apparent non-RRKM dynamics of ethyl radical dissociation, i.e. C2H5 — H - - C2H4 [61,62]. When C2H5 is excited randomly, with a microcanonical distribution of states, it dissociates with the exponential P t) of RRKM theory [61], i.e. it is an intrinsic RRKM molecule. However, apparent non-RRKM behavior is present in a trajectory simulation of C2H5... 

The irregular trajectories in Fig. 15.6 display the type of motion expected by RRKM theory. These trajectories moves chaotically throughout the coordinate space, not restricted to any particular type of motion. RRKM theory requires this type of irregular motion for all of the trajectories so that the intramolecular dynamics is ergodic [1]. In addition, for RRKM behavior the rate of intramolecular relaxation associated with the ergodicity must be sufficiently rapid so that a microcanonical ensemble is maintained as the molecule decomposes [1]. This assures the RRKM rate constant k E) for each time interval f —> f + df. If the ergodic intramolecular relaxation is slower than l/k(E), the unimolecular dynamics will be intrinsically non-RRKM. 

Figure 8.9 Relation of state occupation (schematically shown at constant energy) to lifetime distribution for the RRKM theory and for various actual situations. Dashed lines in lifetime distributions for (d) and (e) indicate RRKM behavior, (a) RRKM model, (b) Physical counterpart of RRKM model, (c) Collisional state selection, (d) Chemical activation, (e) Intrinsically non-RRKM (Bunker and Hase, 1973).

As shown in Figure 20.4, the nature of the spectrum, near the unimolecular threshold, indicates whether the unimolecular dynamics will be intrinsically RRKM or non-RRKM. RRKM theory assumes irregular chaotic dynamics and, if this is the nature of the spectrum near the dissociation threshold, intrinsic RRKM behavior is expected. In contrast, if an appreciable fraction of the spectrum is regular near the dissociation threshold, the dynamics should be intrinsically non-RRKM. 

---