Reaction dynamics statistical theories

In the following chapters, we will consider an approach to the calculation of rate constants— transition-state theory—that do not take into account such details of the reaction dynamics. The theory will be based on the basic axioms of statistical mechanics where all partitionings of the total energy are equally likely, and it is assumed that all these partitionings are equally effective in promoting reaction. 

Intramolecular Dynamics Statistical Theories In sections A—7 we review theoretical and experimental work related to the internal dynamics in unimolecular reactions as opposed to collisional energy transfer. 

Temperature and Reaction Rates Statistical Theories of Rates Reaction Dynamics Reactions In the Gas Phase Reactions In Solution Reactions on Surfaces Composite Reaction Mechanisms Photochemical and Radiation-Chemical Reactions... 

Extensions of this statistical thermodynamical approach to estimating reaction rates include the RRK and RRKM theories of unimolecular decay rates, and the information theoretic formulation of reaction dynamics. These theories are remarkably successful, although generally more successful at interpreting experimental data and correlating results than at deriving results a priori. 

Classical trajectory calculations, performed on the PES1 and PESl(Br) potential energy surfaces described above, have provided a detailed picture of the microscopic dynamics of the Cl- + CH3Clb and Cl" + CH3Br SN2 nucleophilic substitution reactions.6,8,35-38 In the sections below, different aspects of these trajectory studies and their relation to experimental results and statistical theories are reviewed. 

The practice of physical chemistry came to include many subfields of research thermochemistry and thermodynamics, solution theory, phase equilibria, surface and transport phenomena, colloids, statistical mechanics, kinetics, spectroscopy, crystallography, photochemistry, and radiation. Here I concentrate only on three approaches within physical chemistry that had some promise for meeting the needs of organic chemists who wanted to explain affinity and reaction dynamics. 

Statistical theories present particularly useful approaches to the quantitative characterization of dynamical phenomena in chemical kinetics. On the one hand, they provide a shortcut to the overall rate of the reaction, avoiding the explicit treatment of the dynamics before and after reaching the reaction bot-... 

The production of species i (number of moles per unit volume and time) is the velocity of reaction,. In the same sense, one understands the molar flux, jh of particles / per unit cross section and unit time. In a linear theory, the rate and the deviation from equilibrium are proportional to each other. The factors of proportionality are called reaction rate constants and transport coefficients respectively. They are state properties and thus depend only on the (local) thermodynamic state variables and not on their derivatives. They can be rationalized by crystal dynamics and atomic kinetics with the help of statistical theories. Irreversible thermodynamics is the theory of the rates of chemical processes in both spatially homogeneous systems (homogeneous reactions) and inhomogeneous systems (transport processes). If transport processes occur in multiphase systems, one is dealing with heterogeneous reactions. Heterogeneous systems stop reacting once one or more of the reactants are consumed and the systems became nonvariant. 

Statistical theories, such as those just described, are currently the only practical approach for many ion-neutral reactions because the fine details of the collision process are unknown all the information concerning the dynamics of collision processes is, in principle, contained in the pertinent potential-energy surfaces. Although a number of theoretical groups are engaged in accurate ab initio calculations of potential surfaces (J. J. Kaufman, M. Krauss, R. N. Porter, H. F. Schaefer, I. Shavitt, A. C. Wahl, and others), this is an expensive and tedious task, and various approximate methods are also being applied. Some of these methods are listed in Table VI, for example, the diatomics-in-molecules method (DIM). 

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... 

Degenerate rearrangement of bicyclo[3.1.0]hex-2-ene (Chart 2) has a PES, in which four degenerate products are separated through four degenerate TSs with the common energy plateau on the surface.9 Here, four compounds are identical except for the position of deuterium. The rearrangement from 4-exo isomer (6x) is expected to afford 4-endo (6n), 6-exo (7x), and 6-endo (7n) isomers in equal amount if the reaction follows statistical reaction theory (TST). Thus, this reaction provides a situation previously presented by Carpenter to predict nonstatistical product distribution due to dynamics effect.1... 

Blomberg, M. R. A., Yi, S. S., Noll, R. J., Weisshaar, J. C., 1999, Gas Phase Ni+(2D5/2) + f -C4H10 Reaction Dynamics in Real Time Experiment and Statistical Modeling Based on Density Functional Theory , J. Phys. 

Since his arrival at McMaster in 1988, Randall Dumont has focused on statistical theories and their origin in quantum and classical mechanics. His interests include the development of Monte Carlo implementations of statistical theory wherein dynamical processes are simulated by random walks on potential energy surfaces. The breakdown of statistical theory and the appearance of nonexponential population decay are also topics of his ongoing investigations. Other questions of interest are the incorporation of quantum effects into statistical theory and the effects of collisions on reaction processes. He has a special interest in argon cluster evaporation in vacuum197 and in the description of simple isomerization reactions.198 His other interests include the semiclassical description of classically unallowed processes such as tunneling.199... 

TYoe, J. (1992). Statistical aspects of ion-molecule reactions, in State-Selected and State-to-State Ion-Molecule Reaction Dynamics, Part 2 Theory, ed. M. Baer and C.Y. Ng (Wiley, New York). 

In statistical theories of unimolecular reactions, the rate is determined from an approach that does not involve any explicit consideration of the reaction dynamics. 

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... 

Statistical calculations provide a relatively simple alternative to the solution of classical or quantum-mechanical reaction dynamics by replacing the detailed dynamical calculations of the progress of a reaction with probabilities of the possible outcomes. However, statistical theories are only an appropriate means of describing certain reactions and it is not generally possible to identify suitable candidates in advance of experimental measurements. There are many statistical methods available which are distinguished by various ways of describing the reaction intermediate or the possible states of the reagents or products. 

In the volumes to come, special attention will be devoted to the following subjects the quantum theory of closed states, particularly the electronic structure of atoms, molecules, and crystals the quantum theory of scattering states, dealing also with the theory of chemical reactions the quantum theory of time-dependent phenomena, including the problem of electron transfer and radiation theory molecular dynamics statistical mechanics and general quantum statistics condensed matter theory in general quantum biochemistry and quantum pharmacology the theory of numerical analysis and computational techniques. 

The statistical theories provide a relatively simple model of chemical reactions, as they bypass the complicated problem of detailed single-particle and quantum mechanical dynamics by introducing probabilistic assumptions. Their applicability is, however, connected with the collisional mechanism of the process in question, too. The statistical phase space theories, associated mostly with the work of Light (in Ref. 6) and Nikitin (see Ref. 17), contain the assumption of a long-lived complex formation and are thus best suited for the description of complex-mode processes. On the other hand, direct character of the process is an implicit dynamical assumption of the transition-state theory. 

---