Rate constants microcanonical

RRKM fit to microcanonical rate constants of isolated tran.s-stilbene and the solid curve a fit that uses a reaction barrier height reduced by solute-solvent interaction [46],... 

As a result of possible recrossings of the transition state, the classical RRKM lc(E) is an upper bound to the correct classical microcanonical rate constant. The transition state should serve as a bottleneck between reactants and products, and in variational RRKM theory [22] the position of the transition state along q is varied to minimize k E). This minimum k E) is expected to be the closest to the truth. The quantity actually minimized is N (E - E ) in equation (A3.12.15). so the operational equation in variational RRKM theory is... 

The most sophisticated and computationally demanding of the variational models is microcanonical VTST. In this approach one allows the optimum location of the transition state to be energy dependent. So for each k(E) one finds the position of the transition state that makes dk(E)/dq = 0. Then one Boltzmann weights each of these microcanonical rate constants and sums the result to find fc ni- There is general agreement that this is the most reliable of the statistical kinetic models, but it is also the one that is most computationally intensive. It is most frequently necessary for calculations on reactions with small barriers occurring at very high temperatures, for example, in combustion reactions. 

RRKM theory, an approach to the calculation of the rate constant of indirect reactions that, essentially, is equivalent to transition-state theory. The reaction coordinate is identified as being the coordinate associated with the decay of an activated complex. It is a statistical theory based on the assumption that every state, within a narrow energy range of the activated complex, is populated with the same probability prior to the unimolecular reaction. The microcanonical rate constant k(E) is given by an expression that contains the ratio of the sum of states for the activated complex (with the reaction coordinate omitted) and the total density of states of the reactant. The canonical k(T) unimolecular rate constant is given by an expression that is similar to the transition-state theory expression of bimolecular reactions. 

The relation between the key quantities (the rate constant k(T), the microcanonical rate constant k(E), and the reaction probability P) and various approaches to the description of the nuclear dynamics is illustrated below. 

In dynamical theories, one solves the equation of motion for the individual nuclei, subject to the potential energy surface. This is the exact approach, provided one starts with the Schrodinger equation. The aim is to calculate k(E) and kn(hi/), the microcanonical rate constants associated with, respectively, indirect (apparent or true) unimolecular reactions and true (photo-activated) unimolecular reactions. 

This is an approach for the calculation of the microcanonical rate constant k(E) for indirect unimolecular reactions that is based on several approximations. The molecule is represented by a collection of s uncoupled harmonic oscillators. According to Appendix E, such a representation is exact close to a stationary point on the potential energy surface. Furthermore, the dynamics is described by classical mechanics. 

The microcanonical rate constant takes, accordingly, the form... 

First, we want to derive an expression for the microcanonical rate constant k(E) when the total internal energy of the reactant is in the range E to E + dE. From Eq. (7.43), the rate of reaction is given by the rate of disappearance of A or, equivalently, by the rate at which activated complexes A pass over the barrier, i.e., the flow through the saddle-point region. The essential assumptions of RRKM theory are equivalent to the assumptions underlying transition-state theory. 

The dimension of the factor IIf=1z/j/IIlz v is that of a frequency. If the frequencies of the reactant and the activated complex are not too different, this frequency is roughly a typical vibrational frequency vr (typically in the range 1013 to 1014 s 4). Since the energy-dependent factor is less than one, we have that the microcanonical rate constant k(E) < i/r, i.e., it is less than a typical vibrational frequency. The energy dependence as a function of the number of vibrational degrees of freedom was illustrated in Fig. 7.3.2, and as shown previously in Eq. (7.38) it can be interpreted as the probability that the energy in one out of s vibrational modes exceeds the energy threshold Eq for the reaction. Note that if we make the identification vr n =1z/i/n 11i/ , we have recovered RRK theory, Eq. (7.39), from RRKM theory. 

In the RRKM theory, the microcanonical rate constant k(E, J) at a given E and total angular momentum quantum number / is given by [62, 68],... 

The first applications (230-234) concerned a study of the dependence of the tunneling effect on the curvature of the reaction path, of an energy transfer among different vibrational modes in the course of a reaction, and of the evaluation of microcanonical rate constants for study-case reactions, HCN - CNH, H2CO -> H2 + CO, H2CC -> HC=CH, and proton transfer in malonaldehyde. 

Briefly, the ARRKM theory represents the microcanonical rate constant in the form... 

The microcanonical rate constant at energy E can be expressed using the RRKM theory [48],... 

The discovery, the linear surprisal, due to Kinsey, Bernstein, and Levine is about a rule on microcanonical rate constants ( /( /)) or the associated product distribution (p/(s/)) experimentally observed in a chemical reaction, in which a final state, for instance, in a vibrational level of a given energy Ej is specified. A statistically estimated product distribution pj (s ) corresponding to Pj(Ej) is called the prior distribution, which is usually evaluated in terms of the volume of a relevant classical phase space and is frequently represented in terms of energy parameters. Their remarkable finding [2-5] is an exponential form... 

Another system where accurate microcanonical rate constants have been calculated is Li + HF - LiF + H with 7 = 0 (172). This reaction has variational transition states in the exit valley. Variational transition state theory agrees very well with accurate quantum dynamical calculations up to about 0.15 eV above threshold. After that, deviations are observed, increasing to about a factor of 2 about 0.3 eV above threshold. These deviations were attributed to effective barriers in the entrance valley these are supernumerary transition states. After Gaussian convolution of the accurate results, only a hint of step structure due to the variational transition states remains. Densities of reactive states, which would make the transition state spectrum more visible, were not published (172). 

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. 

It is useful for some purposes to define the microcanonical rate constant k(E),... 

Figure 12 Microcanonical rate constant, k(E), obtained theoretically (solid line) compare to the experimental results of Ref. 33.

The connection between k(T), often called the canonical rate constant, and k E), the microcanonical rate constant, involves averaging k E) over the energy distribution... 

This section begins with a simple derivation of the microcanonical rate constant k(E) in which rotations are ignored and in which the location of the transition state is assumed to be fixed at a saddle point and is thus independent of the energy in the system. Methods for including rotations and for treating the transition state for reactions with no saddle points will be discussed in the following chapter. 

The conversion of the microcanonical rate constant to a rate expression appropriate for a constant temperature system requires an averaging over the distribution of internal energies at the temperature T. Suppose that this distribution function is given by P(E, T) ... 

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