Rice-Ramsperger-Kassel—Marcus RRKM theory

In the statistical description of ununolecular kinetics, known as Rice-Ramsperger-Kassel-Marcus (RRKM) theory [4,7,8], it is assumed that complete IVR occurs on a timescale much shorter than that for the unimolecular reaction [9]. Furdiemiore, to identify states of the system as those for the reactant, a dividing surface [10], called a transition state, is placed at the potential energy barrier region of the potential energy surface. The assumption implicit m RRKM theory is described in the next section. 

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. 

Nowadays, the basic framework of our understanding of elementary processes is the transition state or activated complex theory. Formulations of this theory may be found in refs. 1—13. Recent achievements have been the Rice—Ramsperger—Kassel—Marcus (RRKM) theory of unimol-ecular reactions (see, for example, ref. 14 and Chap. 4 of this volume) and the so-called thermochemical kinetics developed by Benson and co-workers [15] for estimating thermodynamic and kinetic parameters of gas phase reactions. Computers are used in the theory of elementary processes for quantum mechanical and statistical mechanical computations. However, this theme will not be discussed further here. 

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. 

B. Rice-Ramsperger-Kassel-Marcus (RRKM) Theory... 

The most accepted modern activation theory for the outer electron transfer is that of Rudolph A. Marcus (Nobel Prize in Chemistry in 1992) [14], which is different from the transition state theory. His studies on unimolecular reactions and the transition and collision theories committed him to elaborate on the Rice-Ramsperger-Kassel-Marcus (RRKM) theory in 1952. This theory is an extension of the previous RRK theory proposed by Rice, Ramsperger, and Kassel between 1927 and 1928. Moreover, Hush and Marcus further extended the electron transfer theory of Marcus for inner electron transfers [15-17]. 

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... 

Perhaps the point to emphasise in discussing theories of translational energy release is that the quasiequilibrium theory (QET) neither predicts nor seeks to describe energy release [576, 720]. Neither does the Rice— Ramsperger—Kassel—Marcus (RRKM) theory, which for the purposes of this discussion is equivalent to QET. Additional assumptions are necessary before QET can provide a basis for prediction of energy release (see Sect. 8.1.1) and the nature of these assumptions is as fundamental as the assumption of energy randomisation (ergodic hypothesis) or that of separability of the transition state reaction coordinate (Sect. 2.1). The only exception arises, in a sense by definition, with the case of the loose transition state [Sect. 8.1.1(a)]. 

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... 

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]. 

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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