Non-RRKM lifetimes

For some systems qiiasiperiodic (or nearly qiiasiperiodic) motion exists above the unimoleciilar tlireshold, and intrinsic non-RRKM lifetime distributions result. This type of behaviour has been found for Hamiltonians with low uninioleciilar tliresholds, widely separated frequencies and/or disparate masses [12,, ]. Thus, classical trajectory simulations perfomied for realistic Hamiltonians predict that, for some molecules, the uninioleciilar rate constant may be strongly sensitive to the modes excited in the molecule, in agreement with the Slater theory. This property is called mode specificity and is discussed in the next section. 

F. Gabern, W. S. Koon, J. E. Marsden, and S. D. Ross, Theory and computation of non-RRKM lifetime distributions and rates in chemical systems with three or more degrees of freedom, Physica D 211, 391 (2005). 

That surfaces with a linear HCC equilibrium geometry give rise to intrinsic non-RRKM lifetimes is consistent with the fact that the coupling between the HCC bend and the HC and CC stretch vibrational momenta is zero for 0 = 180° and a maximum for 0 = 90°. This can be seen by writing the vibrational kinetic energy in internal coordinates using... 

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. 

A very good correspondence is found between the nature of the unimolecular lifetime distribution and the fraction of trajectories that are quasiperiodic. Surface VA, which has the most intrinsically non-RRKM lifetime distribution, also contains the largest fraction of quasiperiodic trajectories. The fraction of quasiperiodic trajectories is negligibly small for the surfaces with intrinsically RRKM lifetime distributions. A summary of our findings is given in Table 4. The A and B surfaces are the ones with the largest number of quasiperiodic trajectories, and these surface types are most similar to the ethyl radical potential energy surface. 

Figure A3.12.2. Relation of state oeeupation (sehematieally shown at eonstant energy) to lifetime distribution for the RRKM theory and for various aetiial situations. Dashed eiirves in lifetime distributions for (d) and (e) indieate RRKM behaviour, (a) RRKM model, (b) Physieal eounterpart of RRKM model, (e) Collisional state seleetion. (d) Chemieal aetivation. (e) Intrinsieally non-RRKM. (Adapted from [9].)...

In the above discussion it was assumed that the barriers are low for transitions between the different confonnations of the fluxional molecule, as depicted in figure A3.12.5 and therefore the transitions occur on a timescale much shorter than the RRKM lifetime. This is the rapid IVR assumption of RRKM theory discussed in section A3.12.2. Accordingly, an initial microcanonical ensemble over all the confonnations decays exponentially. However, for some fluxional molecules, transitions between the different confonnations may be slower than the RRKM rate, giving rise to bottlenecks in the unimolecular dissociation [4, ]. The ensuing lifetime distribution, equation (A3.12.7), will be non-exponential, as is the case for intrinsic non-RRKM dynamics, for an mitial microcanonical ensemble of molecular states. 

Definitive examples of intrinsic non-RRKM dynamics for molecules excited near their unimolecular tluesholds are rather limited. Calculations have shown that intrinsic non-RRKM dynamics becomes more pronounced at very high energies, where the RRKM lifetime becomes very short and dissociation begins to compete with IVR [119]. There is a need for establishing quantitative theories (i.e. not calculations) for identifying which molecules and energies lead to intrinsic non-RRKM dynamics. For example, at thenual... 

At this point it is worthwhile to review the possible failures of RRKM theory [9, 14] within a classical framework. First, the dynamics in some regions of phase space may not be ergodic. In this instance, which has been termed intrinsic non-RRKM behaviour [38], the use of the statistical distribution in Eq. (2.2) is inappropriate. In the extreme case of two disconnected regions of space, with one region nonreactive, the lifetime distribution is still random with an exponential decay of population to a non-zero value. However, the averaging of the flux must then be restricted to the reactive part of the phase space, and the rate coefficient is then increased by a factor equal to the reciprocal of the proportion of the phase space that is reactive. 

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... 

Fig. 15.2. Trajectory lifetime distributions for HCO, HNO, and HO2 dissociation, following microcanonical sampling. Extensive intrinsic non-RRKM dynamics are present in HCO and HNO decomposition. The smooth curves are obtained by fitting the data to an exponential function, with the initial fast decomposition times being excluded. Adapted from Ref. [51].

Classical trajectory studies of unimolecular decomposition have helped define what is meant by RRKM and non-RRKM behavior (Bunker, 1962, 1964 Bunker and Hase, 1973 Hase, 1976, 1981). RRKM theory assumes that the phase space density of a decomposing molecule is uniform. A microcanonical ensemble exists at t = 0 and rapid intramolecular processes maintain its existence during the decomposition [fig. 8.9(a), (b)]. The lifetime distribution, Eq. (8.35a), is then... 

It should be recognized that there is an ambiguity in the reported experimental isomerization rate constants. Following the discussion in section 20.3, the intrinsic non-RRKM isomerization lifetime distribution P t) and reactant population N i) are non-exponential with multiple rate constants. A possible model for the isomerization dynamics is that a microcanonical ensemble of reactant states has a P t) with an initial small component which has a rate constant kj much larger than the RRKM value and a much larger longer-time... 

If a wavepacket remains localized in the central-barrier region, it may then be possible to resolve its vibrational dynamics and spectrum. A direct probe of non-RRKM dynamics for the Cl ---CH3CI complex would involve measuring the lifetimes of its resonance states. 

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... 

State selective unimolecular chemistry will always exist for a short period of time after excitation as a result of apparent non-RRKM effects. However, a much more important question is whether state selective effects persist for times longer than the average RRKM lifetime of the molecule. This will be the case if there are long-lived vibrational states for highly vibrationally excited polyatomic molecules. Whether these states exist may be the most provocative question in molecular spectroscopy and dynamics at the present time. As described in the previous section highly excited molecules with long-lived vibrational states are expected to be intrinsically non-RRKM. 

It may seem inconsistent that statistical (RRKM) energy distributions can be observed at the critical configuration even though the lifetime distributions are apparent and/or intrinsic non-RRKM. However, there are two reasons why this is not the case. First, model master equation studies have shown that product energy distributions are insensitive to the rate of intramolecular vibrational energy relaxation.Second, RRKM lifetime distributions require... 

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