Reaction rate theory, classical unimolecular

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. 

The result just cited provides a formal justification of the assumption implicit in classical unimolecular reaction rate theory, namely that the reaction rate depends only on the equilibrium internal energy distribution of the activated molecule. 

There are certain general rules that are very helpful in constructing a mechanism, Laidler [10]. The initiation step can be considered from the viewpoint of classical unimolecular reaction rate theory and is first order if ... 

M. Zhao and S. A. Rice,/. Chem. Phys., 96,3542 (1992). Unimolecular Fragmentation Rate Theory Revisited An Improved Classical Theory. M. Zhao and S. A. Rice, /. Chem. Phys., 96, 6654 (1992). An Approximate Classical Unimolecular Reaction Rate Theory. M. Zhao and S. A. Rice, /. Chem. Phys., 98,2837 (1993). Comment on the Rate of Isomerization of 3-Phospholene. M. Zhao and S. A. Rice,/. Chem. Phys., 96,3542 (1992). Comment on the Classical Theory of Isomerization. S. Jang, M. Zhao, and S. A. Rice, /. Chem. Phys., 97,... 

III. AN ALTERNATIVE FORM FOR CLASSICAL UNIMOLECULAR REACTION RATE THEORY... 

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. 

In this chapter we have reviewed the development of unimolecular reaction rate theory for systems that exhibit deterministic chaos. Our attention is focused on a number of classical statistical theories developed in our group. These theories, applicable to two- or three-dimensional systems, have predicted reaction rate constants that are in good agreement with experimental data. We have also introduced some quantum and semiclassical approaches to unimolecular reaction rate theory and presented some interesting results on the quantum-classical difference in energy transport in classically chaotic systems. There exist numerous other studies that are not considered in this chapter but are of general interest to unimolecular reaction rate theory. 

Shalashilin and Thompson [46-48] developed a method based on classical diffusion theory for calculating unimolecular reaction rates in the IVR-limited regime. This method, which they referred to as intramolecular dynamics diffusion theory (IDDT) requires the calculation of short-time ( fs) classical trajectories to determine the rate of energy transfer from the bath modes of the molecule to the reaction coordinate modes. This method, in conjunction with MCVTST, spans the full energy range from the statistical to the dynamical limits. It in essence provides a means of accurately... 

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... 

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. 

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... 

The first of the theoretical chapters (Chapter 9) treats approaches to the calculation of thermal rate constants. The material is familiar—activated complex theory, RRKM theory of unimolecular reaction, Debye theory of diffusion-limited reaction—and emphasizes how much information can be correlated on the basis of quite limited models. In the final chapt, the dynamics of single-collision chemistry is analyzed within a highly simplified framework the model, based on classical mechanics, collinear collision geometries, and naive potential-energy surfaces, illuminates many of the features that account for chemical reactivity. 

It is worthwhile to first review several elementary concepts of reaction rates and transition state theory, since deviations from such classical behavior often signal tunneling in reactions. For a simple unimolecular reaction. A—>B, the rate of decrease of reactant concentration (equal to rate of product formation) can be described by the first-order rate equation (Eq. 10.1). 

The effects of QMT at cryogenic temperatures can be quite spectacular. At extremely low temperatures, even very small energy barriers can be prohibitive for classical overbarrier reactions. For example, if = Ikcal/mol and A has a conventional value of 10 s for a unimolecular reaction of a molecule, Arrhenius theory would predict k = 2 X 10 ° s , or a half-life of 114 years at lOK. But, many tunneling reactions of reactive intermediates have been observed to occur at measurable rates at this and lower temperatures, even when energy barriers are considerably higher. Reactive intermediates can, thus, still be quite elusive at extremely low temperatures if protected only by small and narrow energy barriers. 

Elements of classical dynamics of unimolecular reactions in particular, the Slater theory for indirect reactions, where the molecule is modeled as a set of uncoupled harmonic oscillators. Reaction is defined to occur when a particular bond length attains a critical value, and the rate constant is given as the frequency with which this occurs. 

Thermal unimolecular reactions usually exhibit first-order kinetics at high pressures. As pointed out originally by Lindemann [1], such behaviour is found because collisionally energised molecules require a finite time for decomposition at high pressures, collisional excitation and de-excitation are sufficiently rapid to maintain an equilibrium distribution of excited molecules. Rice and Ramsperger [2] and, independently, Kassel [3] (RRK), realised that a detailed theory must take account of the variation of decomposition rate of an excited molecule with its degree of internal excitation. Kassel s theory is still widely used and is valid for the chosen model of a set of coupled, classical, harmonic oscillators. 

The theoretical framework in the present discussion is transition state theory (TST), which yields the expression of the classical rate constant.9 For a unimolecular reaction, the forward rate constant is given below ... 

This effect of CO2 is at first somewhat surprising because it implies that the rate of dissociation of N2OB is being accelerated by CO2, which we would expect only if this dissociation were a unimolecular reaction below its high-pressure limit. N2O6 has 7 atoms and 15 internal vibrations (or their equivalent) and probably 2 active rotations. For a completely classical molecule (which N2O6 is not), the RRK theory (Table XI.2) would predict deviations from the high-pressure limit near 0.1 mm Hg, which is... 

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