Quantum unimolecular theory

More recently, Mies and Kraus have presented a quantum mechanical theory of the unimolecular decay of activated molecules.13 Because of the similarity between this process and autoionization they used the Fano theory of resonant scattering.2 Their theory provides a detailed description of the relationships between level widths, matrix elements coupling discrete levels to the translational continuum, and the rate of fragmentation of the molecule. 

Rigorous Quantum Rate Theory Versus the Quantized ARRKM Theory A Semiclassical Approximation to the Rigorous Quantum Rate Theory Effective Hamiltonian Approach to Unimolecular Dissociation Wave Packet Dynamics Approach VII. Quantum Transport in Classically Chaotic Systems... 

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. 

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... 

The classical unimolecular dynamics is ergodic for molecules like NO2 and D2CO, whose resonance states are highly mixed and unassignable. As described above, their unimolecular dynamics is identified as statistical state specific. The classical dynamics for these molecules are intrinsically RRKM and a microcanonical ensemble of phase space points decays exponentially in accord with Eq. (3). The correspondence found between statistical state specific quantum dynamics and quantum RRKM theory is that the average of the N resonance rate constants fe,) in an energy window E + AE approximates the quantum RRKM rate constant k E) [27,90]. Because of the state specificity of the resonance rates, the decomposition of an ensemble of the A resonances is non-exponential, i.e. 

The conclusion one reaches is that quantum RRKM theory is an incomplete model for unimolecular decomposition. It does not describe fluctuations in state-specific resonance rates, which arise from the nature of the couplings between the resonance states and the continuum. It also predicts steps in k E), which appear to be inconsistent with the actual quantum dynamics as determined from computational chemistry. However, for molecules whose classical unimolecular dynamics is ergodic and intrinsically RRKM and/or whose resonance rates are statistical state specific (see Section 15.2.4), the quantum RRKM k E) gives an accurate average rate constant for an energy interval E E + AE [47]. 

S. Nordholm and S. A. Rice, A quantum ergodic theory approach to unimolecular fragmentation, J. Chem. Phys. 62 157 (1975). 

Stumpf M, Dobbyn A J, Keller H-M, Hase W L and Schinke R 1995 Quantum mechanical study of the unimolecular dissociation of HO2 a rigorous test of RRKM theory J. Chem. Phys. 102 5867-70... 

Pritchard, H. O. The quantum theory of unimolecular reactions, Cambridge University Press, Cambridge, 1984... 

Letter from G. N. Lewis to Paul Ehrenfest, undated but probably 1925, G. N. Lewis Correspondence, BL.UCB. G. N. Lewis and D. F. Smith promised in their paper, "The Theory of Reaction Rate," JACS 47 (1925) 15081520, to publish a demonstration that a range of frequencies of radiation affecting degrees of freedom in a molecule is responsible for chemical reaction. This paper was the subject of the letter, with anonymous referee s report, from Arthur B. Lamb to G. N. Lewis, 28 February 1925, G. N. Lewis Papers, BL.UCB. The referee said "No real unimolecular reaction has actually been observed they have been shown to be merely catalytic the idea that a unimolecular reaction is due to collision between a quantum and a molecule is not original with Lewis."... 

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. 

Rice, Ramsperger, and Kassel [206,333,334] developed further refinements in the theory of unimolecular reactions in what is known as RRK theory. Kassel extended the model to account for quantum effects [207] this treatment is known as QRRK theory. 

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]. 

We now turn to a more detailed analysis of the theory of control of quantum many-body dynamics, focusing attention on the particular case of control of product formation in a photoinduced unimolecular reaction. 

Laval University was one of the first Canadian universities to hire a theoretical chemist. Wendell Forst arrived in 1961 and developed a research program based on the theory of unimolecular reactions75 and quantum chemistry. He maintained ties with experimental physical chemistry through a strong interest in mass spectrometry and gas phase kinetics. In many of his papers he sought analytical solutions to fundamental problems.76 In 1986, after a quarter-century at Laval, he moved to the University of Nancy in France. 

The absorption of a quantum of light by a molecule in the gas phase may initiate a unimolecular decomposition or rearrangement process. The potentially intimate relationship between photochemistry and nonequilibrium unimolecular reaction theory has yet to be realized, since most of these photoprocesses take place on electronically excited poten-... 

Theoretical Expressions. The Rice-Ramsperger-Kassel-Marcus (RRKM14) theory of unimolecular reactions employs a quantum statis-... 

Extending the theory to interpret or predict the rovibrational state distribution of the products of the unimolecular dissociation, requires some postulate about the nature of the motion after the unimolecularly dissociating system leaves the TS on its way to form products. For systems with no potential energy maximum in the exit channel, the higher frequency vibrations will tend to remain in the same vibrational quantum state after leaving the TS. That is, the reaction is expected to be vibrationally adiabatic for those coordinates in the exit channel (we return to vibrational adiabaticity in Section 1.2.9). The hindered rotations and the translation along the reaction coordinate were assumed to be in statistical equilibrium in the exit channel after leaving the TS until an outer TS, the PST TS , is reached. With these assumptions, the products quantum state distribution was calculated. (After the system leaves the PST TS, there can be no further dynamical interactions, by definition.)... 

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... 

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. 

This hnding concerning quantum transport in classically chaotic systems sheds new light on quantum effects in unimolecular reaction dynamics. For example, one expects that intramolecular bottlenecks associated with canton, if treated quantum mechanically, would be more effective than in a classical statistical theory even when nh is smaller than the reaction flux crossing the intramolecular dividing surface. Clearly, it would be interesting to examine realistic molecular systems in a similar fashion. 

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