RRKM theory quantum

In addition to experiments, a range of theoretical techniques are available to calculate thermochemical information and reaction rates for homogeneous gas-phase reactions. These techniques include ab initio electronic structure calculations and semi-empirical approximations, transition state theory, RRKM theory, quantum mechanical reactive scattering, and the classical trajectory approach. Although still computationally intensive, such techniques have proved themselves useful in calculating gas-phase reaction energies, pathways, and rates. Some of the same approaches have been applied to surface kinetics and thermochemistry but with necessarily much less rigor. 

Figure A3,12.2(a) illnstrates the lifetime distribution of RRKM theory and shows random transitions among all states at some energy high enongh for eventual reaction (toward the right). In reality, transitions between quantum states (though coupled) are not equally probable some are more likely than others. Therefore, transitions between states mnst be snfficiently rapid and disorderly for the RRKM assumption to be mimicked, as qualitatively depicted in figure A3.12.2(b). The situation depicted in these figures, where a microcanonical ensemble exists at t = 0 and rapid IVR maintains its existence during the decomposition, is called intrinsic RRKM behaviour [9].

RRKM theory allows some modes to be uncoupled and not exchange energy with the remaining modes [16]. In quantum RRKM theory, these uncoupled modes are not active, but are adiabatic and stay in fixed quantum states n during the reaction. For this situation, equation (A3.12.15) becomes... 

The parameter v in equation (A3.12.59) has also been related to RRKM theory. Polik et al [ ] have shown that for decomposition by quantum mechanical tiumelling... 

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... 

RRKM theory uses the actual vibrational frequencies of the molecule. The density of molecular states (i.e., the number of quantum states per unit energy range) is obtained using direct counting techniques. Modem high-speed computing and efficient algorithms make this aspect of the theory quite accurate [33,375,430]. 

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]. 

The vibrationally excited precursor AB/s/(fs) can decay not only via energy transfer to the bulk but also via a chemical transformation (desorption of B and reaction with the formation of D and C/s/). These chemical processes can be characterized by the chemical lifetime Tch, which can be estimated in the framework of the statistical RRKM theory (see, e.g., Refs. [50, 51]) using the reaction parameters of reagents B and A/s/, precursor AB/s/, and transition complexes determined based on the results of quantum-chemical calculations. Such estimates were performed for many reactions of interest for the growth of metal oxide films [20]. It appeared that in the wide temperature range... 

Nowadays, the basic framework of our understanding of elementary processes is the transition state or activated complex theory. Formulations of this theory may be found in refs. 1—13. Recent achievements have been the Rice—Ramsperger—Kassel—Marcus (RRKM) theory of unimol-ecular reactions (see, for example, ref. 14 and Chap. 4 of this volume) and the so-called thermochemical kinetics developed by Benson and co-workers [15] for estimating thermodynamic and kinetic parameters of gas phase reactions. Computers are used in the theory of elementary processes for quantum mechanical and statistical mechanical computations. However, this theme will not be discussed further here. 

In the RRKM theory, the microcanonical rate constant k(E, J) at a given E and total angular momentum quantum number / is given by [62, 68],... 

A comprehensive quantum mechanical model for the effect has been developed by Marcus and his colleagues at the California Institute of Technology. The Gao and Marcus (2001, 2002) model accounts for many of the experimental observations and utilizes classical quantum mechanical RRKM theory in its development. Statistical RRKM theory quantitatively describes the energetics of gas phase atom-molecule encounters and the relevant parameters which lead to either stabilization and product formation or re-dissociation to atomic and molecular species. This is a well-developed theory and will not be described in detail here. An important application of this theory is that it determines... 

Because of its classical nature, RRKM theory has the some of the same defects as the classical RRK theory, discussed in section 3 above. Thus, when the density of states and the sum of states are calculated it is necessary to take account of the fact that these are actually quantum states and not a continuum. 

Although the use of the correct energy levels for calculating the density of states is strictly a quantum correction of the classical RRKM theory, there are two other effects that are much more fundamental to the theory quantum mechanical tunnelling and fluctuations. The first of these is dealt with in Chapter 2, and the second is the main subject of Chapter 3. 

The main difficulty in applying the SACM is that an adiabatic channel must be calculated for each quantum state of the reactant. This level of detail is only possible for small molecules. For larger molecules the RRKM theory is more practical. 

With this replacement of the strong collider assumption now commonplace, the term RRKM theory has become largely synonymous with quantum TST for unimolecular reactions, and we use this terminology here. The foundations of RRKM theory have been tested in depth with a wide variety of inventive theoretical and experimental studies [9]. While these tests have occasionally indicated certain limitations in its applicability, for example to timescales of a picosecond or longer, the primary conclusion remains that RRKM theory is quantitatively valid for the vast majority of conditions of importance to chemical kinetics. The H + O2 HO2 OH + O reaction is an example of an important reaction where deviations from RRKM predictions are significant [10, 11]. The foundations of RRKM theory and TST have been aptly reviewed in various places [7, 9, 12-15]. Thus, the present chapter begins with only a brief... 

In some instances, a quantitative understanding of anharmonic effects may be required to acheive a priori accuracies of better than a factor of two. Procedures for incorporating one-dimensional corrections, particularly for hindered rotors are well developed and commonly employed. Increased quantum chemical and computational capabilities should now allow for studies of the fully coupled nonrigid anharmonic state densities and/or partition functions via direct Monte Carlo sampling. Such accurate state density studies are a necessary prerequisite to furthering our understanding of the accuracy limits of both quantum chemical estimates and of RRKM theory itself. 

Another advantage of the quantum calculations is that they provide a rigorous test of approximate methods for calculating dissociation rates, namely classical trajectories and statistical models. Two commonly used statistical theories are the Rice-Ramsperger-Kassel-Marcus (RRKM) theory and the statistical adiabatic channel model (SACM). The first one is thoroughly discussed in Chapter 2, while the second one is briefly reviewed in the Introduction. Moreover, the quantum mechanical approach is indispensable in analyzing the reaction mechanisms. A resonance state is characterized not only by its position, width and the distribution of product states, but also by an individual wave function. Analysis of the nodal structure of resonance wave functions gives direct access to the mechanisms of state- and mode-selectivity. 

By choosing the initial conditions for an ensemble of trajectories to represent a quantum mechanical state, trajectories may be used to investigate state-specific dynamics and some of the early studies actually probed the possibility of state specificity in unimolecular decay [330]. However, an initial condition studied by many classical trajectory simulations, but not realized in any experiment is that of a micro-canonical ensemble [331] which assumes each state of the energized reactant is populated statistically with an equal probability. The classical dynamics of this ensemble is of fundamental interest, because RRKM unimolecular rate theory assumes this ensemble is maintained for the reactant [6,332] as it decomposes. As a result, RRKM theory rules-out the possibility of state-specific unimolecular decomposition. The relationship between the classical dynamics of a micro-canonical ensemble and RRKM theory is the first topic considered here. 

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