Rate calculations, quantum statistical

Theoretical rate calculations. Statistical mechanics permits one in principle to compute reaction-rate expressions from first principles if one knows the potential energy surface over which the reaction occurs, and quantum mechanics permits one to calculate this potential energy surface. In Chapter 4 we consider briefly the theory of reaction rates from which reaction rates would be calculated. In practice, these are seldom simple calculations to perform, and one needs to find a colleague who is an accomplished statistical mechanic or quantum mechanic to do these calculations, and even then considerable computer time and costs are usually involved. 

The most accurate theories of reaction rates come from statistical mechanics. These theories allow one to write the partition function for molecules and thus to formulate a quantitative description of rates. Rate expressions for many homogeneous elementary reaction steps come from these calculations, which use quantum mechanics to calculate the energy levels of molecules and potential energy surfaces over which molecules travel in the transition between reactants and products. These theories give... 

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. 

Statistical models are widely exploited in kinetic studies and their usefulness certainly does not need any further proof. Nevertheless, the good agreement between average quantum mechanical and statistical rates, calculated for real molecules, as opposed to model systems, and for the same potential energy surfaces, found in the studies over the past decade may convince even the few non-believers. 

Quantitative estimates of E are obtained the same way as for the collision theory, from measurements, or from quantum mechanical calculations, or by comparison with known systems. Quantitative estimates of the A factor require the use of statistical mechanics, the subject that provides the link between thermodynamic properties, such as heat capacities and entropy, and molecular properties (bond lengths, vibrational frequencies, etc.). The transition state theory was originally formulated using statistical mechanics. The following treatment of this advanced subject indicates how such estimates of rate constants are made. For more detailed discussion, see Steinfeld et al. (1989). 

While details of the solution of the quantum mechanical eigenvalue problem for specific molecules will not be explicitly considered in this book, we will introduce various conventions that are used in making quantum calculations of molecular energy levels. It is important to note that knowledge of energy levels will make it possible to calculate thermal properties of molecules using the methods of statistical mechanics (for examples, see Chapter4). Within approximation procedures to be discussed in later chapters, a similar statement applies to the rates of chemical reactions. 

The complexity and importance of combustion reactions have resulted in active research in computational chemistry. It is now possible to determine reaction rate coefficients from quantum mechanics and statistical mechanics using the ideas of reaction mechanisms as discussed in Chapter 4. These rate coefficient data are then used in large computer programs that calculate reactor performance in complex chain reaction systems. These computations can sometimes be done more economically than to carry out the relevant experiments. This is especially important for reactions that may be dangerous to carry out experimentally, because no one is hurt if a computer program blows up. On the other hand, errors in calculations can lead to inaccurate predictions, which can also be dangerous. 

The purpose of this chapter is a detailed comparison of these systems and the elucidation of the transition from regular to irregular dynamics or from mode-specific to statistical behavior. The main focus will be the intimate relationship between the multidimensional PES on one hand and observables like dissociation rate and final-state distributions on the other. Another important question is the rigorous test of statistical methods for these systems, in comparison to quantum mechanical as well as classical calculations. The chapter is organized in the following way The three potential-energy surfaces and the quantum mechanical dynamics calculations are briefly described in Sections II and III, respectively. The results for HCO, DCO, HNO, and H02 are discussed in Sections IV-VII, and the overview ends with a short summary in Section VIII. 

Figure 7. Dissociation rates k as extracted from the quantum mechanical calculations (open circles). The statistical rates are represented by the step functions and the filled circles represent the classical rate constants as obtained from elaborate classical trajectory calculations. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)...

A quantum dynamical study of the Cl- + CH3 Br 5k2 reaction has been made.78 The calculations are described in detail and the resulting value of the rate constant is in much better agreement with experiment than is that derived from statistical theory, hi related work on the same reaction, a reaction path Hamiltonian analysis of the dynamics is presented.79 The same research group has used statistical theory to calculate the rate constant for the 5n2 reaction... 

Sn2 reactions of methyl halides with anionic nucleophiles are one of the reactions most frequently studied with computational methods, since they are typical group-transfer reactions whose reaction profiles are simple. Back in 1986, Basilevski and Ryaboy have carried out quantum dynamical calculations for Sn2 reactions of X + CH3Y (X = H, F, OH) with the collinear collision approximation, in which only a pair of vibrations of the three-center system X-CH3-Y were considered as dynamical degrees of freedom and the CH3 fragment was treated as a structureless particle [Equation (11)].30 They observed low efficiency of the gas-phase reactions. The results indicated that the decay rate constants of the reactant complex in the product direction and in the reactant direction did not represent statistical values. This constitutes a... 

The last decade has witnessed an intense interest in the theory of radiative association rate coefficients because of the possible importance of the reactions in the interstellar medium and because of the difficulty of measuring these reactions in the laboratory. Several theories have been proposed these are all directed toward systems of at least three or four atoms and utilize statistical approximations to the exact quantum mechanical treatment. The utility of these treatments can be partially gauged by using them to calculate three body rate coefficients which can be compared with laboratory measurements. In order to explain these theories briefly, it would be helpful to write down equations for the mechanism of association reactions. Consider two species A+ and B that come together with bimolecular rate coefficient kj to form a complex AB+ which can then be stabilized radiatively with rate coefficient kr, be stabilized collisionally with helium with rate coefficient kcoll, or redissociate with rate coefficient k j ... 

At low temperature the classical approximation fails, but a quantum generalization of the long-range-force-law collision theories has been provided by Clary (1984,1985,1990). His capture-rate approximation (called adiabatic capture centrifugal sudden approximation or ACCSA) is closely related to the statistical adiabatic channel model of Quack and Troe (1975). Both theories calculate the capture rate from vibrationally and rotationally adiabatic potentials, but these are obtained by interpolation in the earlier work (Quack and Troe 1975) and by quantum mechanical sudden approximations in the later work (Clary 1984, 1985). 

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