Statistical theory of unimolecular reactions

In Section 2.2.4 we obtained expression (2.76) for the rate constant on the basis of the activated complex meAod. This expression is applicable for the calculation of equilibrium rate constants of unimolecular reactions, in particular, K (T) under the conditions when the temperature of reactants is constant (T = const). 

For the calculation of k(e) we apply the microscopic activated complex theory, which considers probabilities of the existence of a system with the energy between e and e + de in the phase space of the reactants T and on the critical surface 5. In this case, we should obtain a formula similar to (2.70). The latter consists of three co ctors VF, fdE. . and f(Piqi)dr. For the microcanonical 

After the insertion of these analogs into (2.70), we have A(e) = / p(e dEttansi / h p(e) 

If taking into account that e = e - Eq and J p(a ) d e = W(e is the number of quantum states of the activated complex with the energy e, we obtain 

On going from the active molecule to the activated complex, the quantum number J remains unchanged, and the moments of inertia change. This results in the situation when the difference of the energies Asj = Zj — ey is released on the internal degrees of fieedom of the activated complex, and Aej depends on J. Taking into account this fact results in e = e + Asj (J)— Eg depends on J. If the 

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In statistical theories of unimolecular reactions, the rate is determined from an approach that does not involve any explicit consideration of the reaction dynamics. 

As shown above, classical unimolecular reaction rate theory is based upon our knowledge of the qualitative nature of the classical dynamics. For example, it is essential to examine the rate of energy transport between different DOFs compared with the rate of crossing the intermolecular separatrix. This is also the case if one attempts to develop a quantum statistical theory of unimolecular reaction rate to replace exact quantum dynamics calculations that are usually too demanding, such as the quantum wave packet dynamics approach, the flux-flux autocorrelation formalism, and others. As such, understanding quantum dynamics in classically chaotic systems in general and quantization effects on chaotic transport in particular is extremely important. 

E.E.Nikitin, Parity conservation and statistical theories of unimolecular reactions, Khim. Fiz. 9, 723 (1990)... 

The important questions in the study of unimolecular reactions are (a) what is the initial state produced in the excitation step, (b) how fast does the system evolve toward products, (c) what are the reaction products, and (d) what are the product energy states Up until about 1975, the first and last questions could not be addressed experimentally. Most experiments were carried out with the reacting system specified in terms of a temperature with its attendant distribution of initial states. From the very beginning, it was recognized that a dissociation rate depends on the internal energy of the molecule (Hinshelwood, 1926). Thus, all detailed statistical theories of unimolecular reactions begin with the calculation of k(E), the rate constant as a function of the infernal energy, E. 

Before considering how the intramolecular dynamics determines the absolute value of the unimolecular reaction rate, and how van der Waals molecules can serve as a vehicle for the study of those intramolecular processes that compete with reaction, we ask if the characteristic features of the fragmentation reactions described in this section can be interpreted using perturbation theory. This approach is at the opposite end of the spectrum from the statistical theory of unimolecular reaction rate, since it focuses attention on state-to-state transitions. We shall see that such an analysis has some successes and some failures. 

It must be noted that the quantum mechanical version of the alternative statistical theory of unimolecular reaction rate remains to be developed. The difficulties to be surmounted are (i) the alternative rate theory makes extensive use of the detailed characteristics of trajectories in the nonlinear system, and there is no good quantum mechanical analogue of a classical trajectory (ii) there is very little understanding... 

Relative rate coefficients define relative product state (product channel and product energy) distributions. These can often be described by statistical theories of unimolecular reactions, such as the statistical adiabatic channel model, described in Statistical Adiabatic Channel Models. 

Selective excitation experiments on unimolecular reactions have two main aims. First, to test the proposition, inherent in the statistical theories of unimolecular reactions, that intramolecular energy transfer is extremely rapid and therefore the random lifetime assumption [see equation (1.52)] is valid. Secondly, to measure specific rate constants, k e and compare them with theoretical predictions. In the rest of this section, some of the experimental studies which have had greatest success in fulfilling these objectives are reviewed. 

With this brief overview of classical theories of unimolecular reaction rate, one might wonder why classical mechanics is so useful in treating molecular systems that are microscopic, and one might question when a classical statistical theory should be replaced by a corresponding quantum theory. These general questions bring up the important issue of quantum-classical correspondence in general and the field of quantum chaos [27-29] (i.e., the quantum dynamics of classically chaotic systems) in particular. For example, is it possible to translate the above classical concepts (e.g., phase space separatrix, NHIM, reactive islands) into quantum mechanics, and if yes, how What is the consequence of... 

Recent years have also witnessed exciting developments in the active control of unimolecular reactions [30,31]. Reactants can be prepared and their evolution interfered with on very short time scales, and coherent hght sources can be used to imprint information on molecular systems so as to produce more or less of specified products. Because a well-controlled unimolecular reaction is highly nonstatistical and presents an excellent example in which any statistical theory of the reaction dynamics would terribly fail, it is instmctive to comment on how to view the vast control possibihties, on the one hand, and various statistical theories of reaction rate, on the other hand. Note first that a controlled unimolecular reaction, most often subject to one or more external fields and manipulated within a very short time scale, undergoes nonequilibrium processes and is therefore not expected to be describable by any unimolecular reaction rate theory that assumes the existence of an equilibrium distribution of the internal energy of the molecule. Second, strong deviations Ifom statistical behavior in an uncontrolled unimolecular reaction can imply the existence of order in chaos and thus more possibilities for inexpensive active control of product formation. Third, most control scenarios rely on quantum interference effects that are neglected in classical reaction rate theory. Clearly, then, studies of controlled reaction dynamics and studies of statistical reaction rate theory complement each other. 

E.E.Nikitin, On the statistical theory of endothermic reactions. II. Unimolecular reactions, Teor. Eksp. Khim. 1, 144 (1965)... 

The results of simple statistical calculations based on the RRK and RRKM theories of unimolecular reactions agree well with the observed vibrational population distribution. At 293 K, the computed average CO vibrational energy is 9.8 kcal/mole according to RRK and 10.2 according to RRKM calculations. Both agree quantitatively with the experimental value, 9.9 0.5. 

As will be discussed in chapter 6, of fundamental importance in the theory of unimolecular reactions is the concept of a microcanonical ensemble, for which every zero-order state within an energy interval AE is populated with an equal probability. Thus, it is relevant to know the time required for an initially prepared zero-order state j) to relax to a microcanonical ensemble. Because of low resolution and/or a large number of states coupled to i), an experimental absorption spectrum may have a Lorentzian-like band envelope. However, as discussed in the preceding sections, this does not necessarily mean that all zero-order states are coupled to r) within the time scale given by the line width. Thus, it is somewhat unfortunate that the observation of a Lorentzian band envelope is called the statistical limit. In general, one expects a hierarchy of couplings between the zero-order states and it may be exceedingly difficult to identify from an absorption spectrum the time required for IVR to form a micro-canonical ensemble. 

The partition function and the sum or density of states are functions which are to statistical mechanics what the wave function is to quantum mechanics. Once they are known, all of the thermodynamic quantities of interest can be calculated. It is instructive to compare these two functions because they are closely related. Both provide a measure of the number of states in a system. The partition function is a quantity that is appropriate for thermal systems at a given temperature (canonical ensemble), whereas the sum and density of states are equivalent functions for systems at constant energy (microcanonical ensemble). In order to lay the groundwork for an understanding of these two functions as well as a number of other topics in the theory of unimolecular reactions, it is essential to review some basic ideas from classical and quantum statistical mechanics. 

Quantum Theory of Scattering and Unimolecnlar Breakdown.—From the theoretical viewpoint it would appear natural to compute lifetimes and cross-sections for unimolecular processes like equation (34) by one of the existing methods for the solution of the set of coupled equations of the scattering problem. There have been, however, hardly any calculations for experimental examples or at least realistic model systems. The present status of the quantum theory of unimolecular reactions is still rather in the domain of formal theories or hi y simplified models, which are not of immediate interest to the experimentalist. We shall, nevertheless, review some of the recent developments, because one may hope that in the future the detailed dynamical theories will provide a deeper understanding of unimolecular dynamics than the statistical theories presently do. 

Consequently, while I jump into continuous reactors in Chapter 3, I have tried to cover essentially aU of conventional chemical kinetics in this book. I have tried to include aU the kinetics material in any of the chemical kinetics texts designed for undergraduates, but these are placed within and at the end of chapters throughout the book. The descriptions of reactions and kinetics in Chapter 2 do not assume any previous exposure to chemical kinetics. The simplification of complex reactions (pseudosteady-state and equilibrium step approximations) are covered in Chapter 4, as are theories of unimolecular and bimolecular reactions. I mention the need for statistical mechanics and quantum mechanics in interpreting reaction rates but do not go into state-to-state dynamics of reactions. The kinetics with catalysts (Chapter 7), solids (Chapter 9), combustion (Chapter 10), polymerization (Chapter 11), and reactions between phases (Chapter 12) are all given sufficient treatment that their rate expressions can be justified and used in the appropriate reactor mass balances. 

Recently, Miller and co-workers have obtained a generalized form of the distribution of unimolecular decay rates for the case of coupled open channels contributing with unequal partial half-widths [139]. Further results have also recently been obtained in the statistical theory of reactions where the possibility of algebraic decay besides the RRKM exponential decay has been discussed [140]. ... 

Statistical methods represent a background for, e.g., the theory of the activated complex (239), the RRKM theory of unimolecular decay (240), the quasi-equilibrium theory of mass spectra (241), and the phase space theory of reaction kinetics (242). These theories yield results in terms of the total reaction cross-sections or detailed macroscopic rate constants. The RRKM and the phase space theory can be obtained as special cases of the single adiabatic channel model (SACM) developed by Quack and Troe (243). The SACM of unimolecular decay provides information on the distribution of the relative kinetic energy of the products released as well as on their angular distributions. 

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