Microcanonical dissociation rates

In this section we summarize methods for solution of the master equation, which couples the collisional relaxation of the highly excited unimolecular species with the microcanonical dissociation rates to determine, for a given temperature and pressure, the non-equilibrium probability distribution for the molecular population over energies and angular momenta, and thence the thermal rate coefficient k(T, P). The separability of molecular interactions in the gas phase into unimolecular events and bimolecular events enables the overall thermal dissociation process to be modeled by the two-dimensional master equation, expressed in continuum notation as... 

It is important to realize that the average dissociation rate coefficient for a microcanonical ensemble may or may not accurately eflect the phenomenological behavior of a given molecular system. The assumption that it will accurately reflect the kinetics is of course known as the ergodic assumption. The conditions governing the validity of the ergodic assumption have been thoroughly discussed and documented elsewhere hence we shall not repeat the discussion in any detail here. However, one or two comments are in order. 

In Section 2.2, the significance of VTST in relation to the important class of reactions involving no barrier to recombination was outlined. The classical tracing methods described above find particularly fruitful application within the context of E- and 7-resolved microcanonical VTST [ VTST( , 7)] calculations of the microcanonical dissociation or association rate coefficients in such reaction systems. In this context, the crucial degrees of freedom which control the kinetics of the reaction are the fragment rotations which turn into hindered rotations and ultimately vibrations as the fragments recombine to form a metastable molecule. These degrees of freedom are referred to as the transitional modes, and a precise count of the sum of states J) is essential for accurate predic-... 

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. 

Modern unimolecular theory has its origins in the work of Rice, Ramsberger and Kassel [44] who investigated the rate of dissociation of a molecule as a function of energy. Marcus and Rice [44] subsequently extended the theory to take account of quantum mechanical features. This extended theory, referred to as RRKM theory, is currently the most widely used approach and is usually the point of departure for more sophisticated treatments of unimolecular reactions. The key result of RRKM theory is that the microcanonical rate coefficient can be expressed as... 

State specific experiments can now test unimolecular rate theories by probing microcanonical rate coefficients. Moore and coworkers [45] have studied the dissociation of ketene close to the reaction threshold in an attempt to test RRKM theory. 

These remarks apply equally to the complementary unimolecular reaction and it is helpful to look at the unimolecular reaction to begin with, always bearing in mind that association and dissociation are connected via the equilibrium constant. In Section 2.4.4 it was shown that for the RRKM model, the microcanonical rate coefficient is proportional to the sum of states, G, at the transition state, which is a function of the energy, E. Application of the minimum flux criterion means that G must be altered... 

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). 

The classical anharmonic RRKM rate constant for a fluxional molecule may be calculated from classical trajectories by following the initial decay of a microcanonical ensemble of states for the unimolecular reactant, as given by equation 1A3.12.41. Such a calculation has been performed for dissociation of the Alg and A1j3 clusters using a model analytic potential energy function written as a sum of Lennard-Jones and Axelrod-Teller potentials [30]. Stmctures of some of the Alg minima, for the potential function, are shown in figure A3.12.6. The deepest potential minimum has... 

Another experimental system where steplike structure possibly related to quantized transition states was observed is the work of Wittig and co-workers (186-188) on N02 dissociation these experiments have been further analyzed by Klippenstein and Radi-voyevitch (189) and Katagiri and Kato (190), both of whose studies indicate that the interpretation may be more complicated. Wittig and co-workers concluded (187) that the steps observed in the experimental microcanonical rate constants may correspond to overlapping of vibrationally adiabatic thresholds. 

RRKM theory is a microcanonical transition state theory and as such, it gives the connection between statistical unimolecular rate theory and the transition state theory of thermal chemical reaction rates. Isomerization or dissociation of an energized molecule A is assumed in RRKM theory to occur via the mechanism... 

Microcanonical rate coefficient versus energy above dissociation for the H00H 20H reaction with J=Q computed using three methods. The red curve is the semiclassical adiabatic model, the blue curve is the HO-RR method, and the purple curve shows the result of the separable hindered rotor approximation. 

Figure 6.14 Microcanonical rate coefficient versus energy above dissociation calculated using the semiclassical adiabatic model evaluated at five values of the total angular momentum.

If the dissociating ion is characterized by a wellde-fined internal energy, i.e. if P E) reduces to a single value, then each ion of the ionized sample can be considered to be one of the replicas of a microcanonical ensemble. For this reason, k E) is often referred to as the microcanonical rate constant . If, however, P E)... 

One of the most widespread approaches to model the experimental microcanonical rate constant is the RRKM-QET statistical theory, which postulates that a rapid internal energy randomization takes place before dissociation and that a transition state can be defined. This leads to the following well-known equation ... 

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