RRKM theory derivation

The above describes the fundamental assumption of RRKM theory regarding the intramolecular dynamics of A. The RRKM expression for k E) is now derived. 

In deriving the RRKM rate constant in section A3.12.3.1. it is assumed that the rate at which reactant molecules cross the transition state, in the direction of products, is the same rate at which the reactants fonn products. Thus, if any of the trajectories which cross the transition state in the product direction return to the reactant phase space, i.e. recross the transition state, the actual unimolecular rate constant will be smaller than that predicted by RRKM theory. This one-way crossing of the transition state, witii no recrossmg, is a fiindamental assumption of transition state theory [21]. Because it is incorporated in RRKM theory, this theory is also known as microcanonical transition state theory. 

When TRPD measurements are combined with PEPICO results, the dissociation rate-energy curve for styrene ion is known over perhaps the largest of any polyatomic ion s range. ° Simple RRKM theory gives an excellent fit (as does Klots thermodynamic formulation), and an extrapolated of 2.43 eV is derived. The thermochemistry for this dissociation to benzene ion plus acetylene [Equation (16)] is very well known from independent heats of formation, giving a calculated... 

Computationally, simple RRKM theory is easy to apply, and it has been used as the extrapolation tool in many studies. By simple RRKM we mean the rate derived from. 

Of course, in a thermal reaction, molecules of the reactant do not all have the same energy, and so application of RRKM theory to the evaluation of the overall unimolecular rate constant, k m, requires that one specify the distribution of energies. This distribution is usually derived from the Lindemann-Hinshelwood model, in which molecules A become activated to vibrationally and rotationally excited states A by collision with some other molecules in the system, M. In this picture, collisions between M and A are assumed to transfer energy in the other direction, that is, returning A to A ... 

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 derivation of RRKM theory given here is also deficient because it has assumed that there is a single unique reaction path between the reactants and the products. In fact there may be several such paths related by symmetry, each with its own transition state. For example in the dissociation of CH4 to CH3 + H there are obviously four equivalent paths. The correct way to allow for this reaction path degeneracy has been the subject of much controversy [25,27,29-34]. However this was finally resolved by Pechukas [35], and the resolution of the problem is relatively simple. 

The focus of this chapter is a review of the methodologies employed in a priori implementations of RRKM theory for the collisionless dissociation/ isomerization steps in gas-phase unimolecular reactions. Special attention will be paid to recent developments, particularly those that have proven their utility through substantive applications. With microscopic reversibility, RRKM treatments of the dissociation process are directly applicable to the reverse bimolecular associations. Furthermore, some of the more interesting illustrations are for bimolecular reactions and so we do not limit our discussion of RRKM theory to unimolecular reactions. However, one should bear in mind that TST was originally derived for bimolecular reactions and the specific term RRKM theory is really only applicable to the unimolecular direction. 

It is instructive to begin with a derivation of RRKM theory from the classical dynamical expression for the rate coefficient. The latter rate coefficient, classically, is generally time dependent, and may be expressed as... 

The determination of the microcanonical rate coefficient k E) is the subject of active research. A number of techniques have been proposed, and include RRKM theory (discussed in more detail in Section 2.4.4) and the derivatives of this such as Flexible Transition State theory. Phase Space Theory and the Statistical Adiabatic Channel Model. All of these techniques require a detailed knowledge of the potential energy surface (PES) on which the reaction takes place, which for most reactions is not known. As a consequence much effort has been devoted to more approximate techniques which depend only on specific PES features such as reaction threshold energies. These techniques often have a number of parameters whose values are determined by calibration with experimental data. Thus the analysis of the experimental data then becomes an exercise in the optimization of these parameters so as to reproduce the experimental data as closely as possible. One such technique is based on Inverse Laplace Transforms (ILT). 

A general expression taking into account the rotational energy was derived from RRKM theory.29 If the intermediate C is sufficiently short-lived (or the total pressure is sufficiently low) that it is not stabilized by collisions, the rate constant k for the formation of the product(s) can be written as... 

In response to a question by pliVa, schaefer indicated that the calculation of one geometry of HNC or HCN (with 6343 configurations included) required about 20 min on an IBM-360/195. Calculations were made for about 10 geometries each on HCN and HNC therefore, the total problem required about 7 h of computer time, pliva also inquired if second and higher derivatives of the energy with respect to intemuclear distance had been obtained, schaefer indicated these had been calculated for HCN and HNC since this information was needed to test the rrkm theory for the unimolecular reaction. 

The modification to the RRKM theory that makes possible accurate modeling of loose transition states is variational transition state theory (Pechukas, 1981 Miller, 1983 Forst, 1991 Wardlaw and Marcus, 1984, 1985, 1988 Hase, 1983, 1987). In this approach the rate constant k E, J) is calculated as a function of the reaction coordinate, R. The location of the minimum flux is found by setting the derivative of the sum of states equal to zero and solving for / . Thus, we evaluate... 

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]. 

Extensions of this statistical thermodynamical approach to estimating reaction rates include the RRK and RRKM theories of unimolecular decay rates, and the information theoretic formulation of reaction dynamics. These theories are remarkably successful, although generally more successful at interpreting experimental data and correlating results than at deriving results a priori. 

For a detailed derivation of the RRKM theory, please refer to textbooks on chemical kinetics. 

RRKM theory is the well-known and consolidated statistical theory for unimolecular dissociation. It was developed in the late 1920s by Rice and Ramsperger [141, 142] and Kassel [143], who treated a system as an assembly of s identical harmonic oscillators. One oscillator is truncated at the activation energy Eq. The theory disregards any quantum effect and the approximation of having all identical is too cmde, such that the derived equation for micro canonical rate constant, k(E),... 

Around a fixed energy E, the average reaction rate is given by the famous RRKM formula, which can be derived from both quasiclassical and quantal considerations [71, 72]. In the context of the Wigner matrix theory [133], the rate is given by the sum of the half-widths of all the open channels. The rate is thus the product of the number v(E) of open channels and the rate per channel fcchannei(E) = l/hnav(E), where h is the Planck constant. The average reaction rate is obtained as [134, 135]... 

Since an early stage of the history of ab initio MD study, many cases have been observed in which the calculated trajectories do not support expectation derived from traditional reaction theories, such as RRKM and TST, and thus the applicability or suitability of these theories has been a matter of argument. In this section examples of one of those dynamics-derived phenomena are shown, namely nonstatistical barrier recrossing. 

In general, the structure and frequencies of the transition complex are not known for unimolecular reactions and, consequently, neither transition state theory nor detailed RRKM calculations can be tested. However, provided a physically plausible choice is made which will match the koc over the range of measured temperatures, the derived ft (e) are only slightly dependent on the particular model selected. Details of these procedures are available [11—13] and an excellent discussion is given by Robinson and Holbrook [11]. Readers should also refer to the detailed methods used by Schneider and Rabinovitch [14] for the CH3NC isomerisation. The following brief comments are intended to complete this introductory outline of the basic theory and to show how it may be applied. 

The work is divided into several parts. Part A in Sec. II briefly sets out the relevant aspects of the RRKM formulation14 for unimolecular reactions. Rather than repeat the derivation of the equations, emphasis is placed upon the present status of the theory, and the best techniques for carrying out computations simply. In Sec. II-B, characteristics of various model hydrocarbon-type molecular species are outlined and are used in Sec. II-C for theoretical calculations that illustrate various aspects of the theory. Some related aspects of kinetic isotope effects are... 

Table XXIII displays the rate constants obtained from the MRRKM theory and the reaction path analysis. It is seen that the former are about a factor of two smaller, and the latter about a factor of two larger, than those derived from direct trajectory calculations. We infer that, since both the RRKM and the MRRKM calculated rate constants are smaller than that calculated from trajectory calculations, there is a nonstatistical contribution to the isomerization rate that is not captured by the MRRKM theory.

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