Rice-Ramsperger-Kassel-Marcus rate

In more detail, our approach can be briefly summarized as follows gas-phase reactions, surface structures, and gas-surface reactions are treated at an ab initio level, using either cluster or periodic (plane-wave) calculations for surface structures, when appropriate. The results of these calculations are used to calculate reaction rate constants within the transition state (TS) or Rice-Ramsperger-Kassel-Marcus (RRKM) theory for bimolecular gas-phase reactions or unimolecular and surface reactions, respectively. The structure and energy characteristics of various surface groups can also be extracted from the results of ab initio calculations. Based on these results, a chemical mechanism can be constructed for both gas-phase reactions and surface growth. The film growth process is modeled within the kinetic Monte Carlo (KMC) approach, which provides an effective separation of fast and slow processes on an atomistic scale. The results of Monte Carlo (MC) simulations can be used in kinetic modeling based on formal chemical kinetics. 

More sophisticated treatments of Lindemann s scheme by Lindemann— Hinshelwood, Rice—Ramsperger—Kassel (RRK) and finally Rice— Ramsperger—Kassel—Marcus (RRKM) have essentially been aimed at re-interpreting rate coefficients of the Lindemann scheme. RRK(M) theories are extensively used for interpreting very-low-pressure pyrolysis experiments [62, 63]. 

This standard mechanistic analysis has a long successful history. Organic chemistry textbooks are filled with PESs and discussions of the implication of single-step versus multiple-step mechanisms, concerted TSs, and so on. - Transition state theory (TST) and Rice-Ramsperger-Kassel-Marcus (RRKM) theory provide tools for predicting rates based upon simple assumptions built upon the notion of reaction on the PES following the reaction coordinate. " ... 

Another advantage of the quantum calculations is that they provide a rigorous test of approximate methods for calculating dissociation rates, namely classical trajectories and statistical models. Two commonly used statistical theories are the Rice-Ramsperger-Kassel-Marcus (RRKM) theory and the statistical adiabatic channel model (SACM). The first one is thoroughly discussed in Chapter 2, while the second one is briefly reviewed in the Introduction. Moreover, the quantum mechanical approach is indispensable in analyzing the reaction mechanisms. A resonance state is characterized not only by its position, width and the distribution of product states, but also by an individual wave function. Analysis of the nodal structure of resonance wave functions gives direct access to the mechanisms of state- and mode-selectivity. 

Suppose we have a simple unimolecular dissociation embedded in a microcano-nical ensemble in phase space, in which only one dissociating channel is available. The Rice-Ramsperger-Kassel-Marcus (RRKM) rate constant is given as [9]... 

To use the master equation, one needs a general formula for the rate constant, kj, out of minimum j through transition state f. In the micro-canonical ensemble this relation is provided by Rice-Ramsperger-Kassel-Marcus (RRKM) theory [166] ... 

The rate constants were calculated with the transition state theory (TST) for direct abstraction reactions and the Rice-Ramsperger-Kassel-Marcus (RRKM) theory for reactions occuring via long-lived intermediates. For reactions taking place without well-defined TS s, the Variflex [35] code and the ChemRate [36] code were used for one-well and multi-well systems, respectively, based on the variational transition-state theory approach... 

Using Rice-Ramsperger-Kassel-Marcus (RRKM) theory (11, 12), we can model the rates of these reactions as a function of the energy difference separating the two transition states. The result of the analysis is an estimate... 

A discussion of recent published results on the cyclobutane reaction illustrates the kind of excited molecule dynamical information available from recoil techniques. The primary assumption in the approach reported is that the RRKM (Rice-Ramsperger-Kassel-Marcus) method for describing statistical energy redistribution and calculating decomposition rate constants is valid. TTius, deviations from expected RRKM behavior... 

Rf(Ei) denotes the rate of formation of KUlzCFs molecules containing sufficient internal energy to populate the ith level ksy the energy-dependent Rice-Ramsperger-Kassel-Marcus (RRKM) rate constant the... 

In this spirit, we will also briefly describe the basis for some of the microscopic kinetic theories of unimolecular reaction rates that have arisen from nonlinear dynamics. Unlike the classical versions of Rice-Ramsperger-Kassel-Marcus (RRKM) theory and transition state theory, these theories explicitly take into account nonstatistical dynamical effects such as barrier recrossing, quasiperiodic trapping (both of which generally slow down the reaction rate), and other interesting effects. The implications for quantum dynamics are currently an active area of investigation. 

The recombination reaction PHa + H + M PH3 + M was also proposed for the PH3 photolysis [15,16] and PH3 + H reaction [17]. A high-pressure limit of the recombination rate constant, Kec = 3.7 x1exp(-340/T) cm molecule" s", was derived [20] from the Rice-Ramsperger-Kassel-Marcus (RRKM) theory of the activated complex, modified in [21]. 

Rate constants for unimolecular homogeneous PH3 decomposition were calculated by the Rice-Ramsperger-Kassel-Marcus (RRKM) theory and by the use of estimated values for the activation energies. Rate constants at the high-pressure limit for reaction (5), log(k/s)= 14.18-11 610/T [5] or 14.00-12610/T [4], include activation energies of 222 or 241 kJ/mol, respectively. Calculated rate constants for reaction (6) are log(k/s)=15.74-18 040/T with an activation energy of 345 kJ/mol. At 900 K PH formation is thus predicted to exceed PH2 formation by a factor -10. Calculated fall-off pressures for both reactions which indicate the onset of second-order decomposition, are quite high, about 10 Torr in an H2 bath gas [5]. 

By the end of 1972, a second cornerstone of the transition state approach was beginning to crumble significantly, for it was now quite evident that widely different transition states could be assumed for a given reaction, but the Rice-Ramsperger-Kassel-Marcus (RRKM) procedure would give the same result for the shape of the fall-off curve [72.N 72.R 74.F 79.A1]. This, as is now well known, arises through the adjustment of the model after the transition state has been chosen so as to force it to be consistent with the observed high pressure rate constant [72.R 80.P1], Perhaps it should have sounded the knell for the RRKM theory, much as the unsymmetric isotopic replacement experiments did for the Slater theory a decade earlier, but there was no other substitute available. 

Here, most quantities are defined above and k(e + Ei) = k(E ) is the unimolecular dissociation rate constant, evaluated using modern statistical theories, such as Rice-Ramsperger-Kassel-Marcus (RRKM) theory. Note that Equation (8) combines the distribution of deposited energies (Equation (5)) with the probability that the complex dissociates in time r (term in square brackets), and a summation over the internal energy available to the reactants. Importantly, the integration recovers Equation (2) when the dissociation rate, A ( ), is faster than the experimental time scale, such that the term in brackets is unity. 

The QET is formally identical to the Rice-Ramsperger-Kassel-Marcus (RRKM) theory of unimolecular decay, in which the rate constant for dissociation to reaction products of an energized species with total angular momentum J and internal energy E over a barrier of Eq is given by the following relation ... 

Langevin collision theory. For the calculation of the dissociation rate constant, k, statistical models like Rice-Ramsperger-Kassel-Marcus (RRKM) theory [14, 15] are used. The Langevin and RRKM theories will be presented briefly in the next section and detailed in Chap. 3. 

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