Quantum mechanics unimolecular reaction rate

CLASSICAL, SEMICLASSICAL, AND QUANTUM MECHANICAL UNIMOLECULAR REACTION RATE THEORY... 

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... 

Bourgin has recently attempted to apply the new quantum mechanics to the problem of unimolecular reaction rates. There are some features of his work which are perhaps less satisfactory from the chemical standpoint than might be desired, and it is intended to examine these in more detail. 

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... 

Isomerizations are important unimolecular reactions that result in the intramolecular rearrangement of atoms, and their rate parameters are of the same order of magnitude as other unimolecular reactions. Consequently, they can have significant impact on product distributions in high-temperature processes. A large number of different types of isomerization reactions seem to be possible, in which stable as well as radical species serve as reactants (Benson, 1976). Unfortunately, with the exception of cis-trans isomerizations, accurate kinetic information is scarce for many of these reactions. This is, in part, caused by experimental difficulties associated with the detection of isomers and with the presence of parallel reactions. However, with computational quantum mechanics theoretical estimations of barrier heights in isomerizations are now possible. 

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. 

RRKM theory represents the state of the art in understanding unimolecular reaction kinetics. However, because of the rigorous treatment of molecular energetics and quantum mechanics, it requires rather sophisticated numerical software to evaluate the rate constant. Computer programs to evaluate RRKM rate expressions are widely available examples are UNIMOL by Gilbert and Smith [143], and a program by Hase and Bunker [166]. 

Similarly, the p-fragmentation of tertiary alkoxyl radicals [reaction (2)] is a well-known process. Interestingly, this unimolecular decay is speeded up in a polar environment. For example, the decay of the ferf-butoxyl radical into acetone and a methyl radical proceeds in the gas phase at a rate of 103 s 1 (for kinetic details and quantum-mechanical calculations see Fittschen et al. 2000), increases with increasing solvent polarity (Walling and Wagner 1964), and in water it is faster than 106 s 1 (Gilbert et al. 1981 Table 7.2). 

In this chapter, we discussed the principle quantum mechanical effects inherent to the dynamics of unimolecular dissociation. The starting point of our analysis is the concept of discrete metastable states (resonances) in the dissociation continuum, introduced in Sect. 2 and then amply illustrated in Sects. 5 and 6. Resonances allow one to treat the spectroscopic and kinetic aspects of unimolecular dissociation on equal grounds — they are spectroscopically measurable states and, at the same time, the states in which a molecule can be temporally trapped so that it can be stabilized in collisions with bath particles. The main property of quantum state-resolved unimolecular dissociation is that the lifetimes and hence the dissociation rates strongly fluctuate from state to state — they are intimately related to the shape of the resonance wave functions in the potential well. These fluctuations are universal in that they are observed in mode-specific, statistical state-specific and mixed systems. Thus, the classical notion of an energy dependent reaction rate is not strictly valid in quantum mechanics Molecules activated with equal amounts of energy but in different resonance states can decay with drastically different rates. 

Modern unimolecular theory has its origins in the work of Rice, Ramsberger and Kassel [44] who investigated the rate of dissociation of a molecule as a function of energy. Marcus and Rice [44] subsequently extended the theory to take account of quantum mechanical features. This extended theory, referred to as RRKM theory, is currently the most widely used approach and is usually the point of departure for more sophisticated treatments of unimolecular reactions. The key result of RRKM theory is that the microcanonical rate coefficient can be expressed as... 

Vibrational predissociation (VP) of a van der Waals triatomic complex A..BC is an example of a unimolecular reaction the rate of which is controlled by the intramolecular vibrational energy redistribution (TVR) [1]. Within a rigorous quantum mechanical approach, the VP dynamics is completely characterized by the complex-valued energies E = - /T / 2 that lie above the dissociation threshold of A..BC into an atom A and... 

Our approach to the study of the departure from equilibrium in chemical reactions and of the "microscopic theory of chemical kinetics is a discrete quantum-mechanical analog of the Kramers-Brownian-motion model. It is most specifically applicable to a study of the energy-level distribution function and of the rate of activation in unimolecular (dissociation Reactions. Our model is an extension of one which we used in a discussion of the relaxation of vibrational nonequilibrium distributions.14 18 20... 

Theoretical chemistry is the discipline that uses quantum mechanics, classical mechanics, and statistical mechanics to explain the structures and dynamics of chemical systems and to correlate, understand, and predict their thermodynamic and kinetic properties. Modern theoretical chemistry may be roughly divided into the study of chemical structure and the study of chemical dynamics. The former includes studies of (1) electronic structure, potential energy surfaces, and force fields (2) vibrational-rotational motion and (3) equilibrium properties of condensed-phase systems and macromolecules. Chemical dynamics includes (1) bimolecular kinetics and the collision theory of reactions and energy transfer (2) unimolecular rate theory and metastable states and (3) condensed-phase and macromolecular aspects of dynamics. 

Consider again the potential given by equation (10). If its behaviour is examined at large p values, it is seen that it has three symmetric saddles at height (6X ) at p=X, and eventually goes to minus infinity. Therefore all the states which it supports are actually metastable, and they will eventually decay by quantum mechanical tunnelling in other words, they are typical quantum mechanical resonances, to which we may associate a width r and a lifetime =fi/r. This model has already been considered [35,36] for unimolecular reaction theory, where the resonance li fetime is most naturally related to the inverse of the unimolecular rate constants k=x... 

The pyrolysis mechanism of PPE and its derivatives given in Scheme 7.2 consist of bimolecular and unimolecular reactions. Applying transition state theory, we calculate the rate constants for the hydrogen abstraction reactions using Eq. (7.19) and the rate constants for reactions 1 and 3-5 using Eq. (7.21). The Wigner correction (Eq. (7.20)) is utilized to approximate quantum effects and the molecular partition functions are defined through Eqs (7.14), (7.15), (7.17), (7.18), and (7.28). 

In the early days of the transition-state theory, there was great hope that this theory would allow quantitative predictions of rate constants, although this requires adequate preexponential factors and energy barriers for the reactions. The theories previously discussed are essentially concerned with the pre-exponential factors of Arrhenius-type equations. Currently energy barriers for unimolecular reactions are estimated by ab initio methods or by density functional theory (DFT). Such quantum-mechanical methods are particularly useful for small systems, but the numerical solution they offer are not easily related to processes of a similar kind. 

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