Statistical mechanical modeling of chemical reactions

Cummings P T and Stell G 1984 Statistical mechanical models of chemical reactions analytic solution of models of A + S AS in the Percus-Yevick approximation Mol. Phys. 51 253... 

STATISTICAL MECHANICAL MODELING OF CHEMICAL REACTIONS IN CONDENSED PHASE SYSTEMS... 

There have been several other methods proposed for the statistical mechanical modeling of chemical reactions. We review these techniques and explain their relationship to RCMC in this section. These simulation efforts are distinct from the many quantum mechanical studies of chemical reactions. The goal of the statistical mechanical simulations is to find the equilibrium concentration of reactants and products for chemically reactive fluid systems, taking into account temperature, pressure, and solvent effects. The goals of the quantum mechanics computations are typically to find transition states, reaction barrier heights, and reaction pathways within chemical accuracy. The quantum studies are usually performed at absolute zero temperature in the gas phase. Quantum mechanical methods are confined to the study of very small systems, so are inappropriate for the assessment of solvent effects, for example. 

On a modest level of detail, kinetic studies aim at determining overall phenomenological rate laws. These may serve to discriminate between different mechanistic models. However, to it prove a compound reaction mechanism, it is necessary to determine the rate constant of each elementary step individually. Many kinetic experiments are devoted to the investigations of the temperature dependence of reaction rates. In addition to the obvious practical aspects, the temperature dependence of rate constants is also of great theoretical importance. Many statistical theories of chemical reactions are based on thermal equilibrium assumptions. Non-equilibrium effects are not only important for theories going beyond the classical transition-state picture. Eventually they might even be exploited to control chemical reactions [24]. This has led to the increased importance of energy or even quantum-state-resolved kinetic studies, which can be directly compared with detailed quantum-mechanical models of chemical reaction dynamics [25,26]. 

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. 

Beyond the clusters, to microscopically model a reaction in solution, we need to include a very big number of solvent molecules in the system to represent the bulk. The problem stems from the fact that it is computationally impossible, with our current capabilities, to locate the transition state structure of the reaction on the complete quantum mechanical potential energy hypersurface, if all the degrees of freedom are explicitly included. Moreover, the effect of thermal statistical averaging should be incorporated. Then, classical mechanical computer simulation techniques (Monte Carlo or Molecular Dynamics) appear to be the most suitable procedures to attack the above problems. In short, and applied to the computer simulation of chemical reactions in solution, the Monte Carlo [18-21] technique is a numerical method in the frame of the classical Statistical Mechanics, which allows to generate a set of system configurations... 

Simultaneously with the development of the Kinetic Model, the appKcation of statistical mechanics provided the basis for the Statistical Mechanics Model. Here a chemical reaction was viewed as the motion of a point in phase space, the co-ordinates of which were the distances between the molecules and their momentum. The expression for reaction rate was thus obtained from the passage of systems through the col point of the potential energy surface. 

The Transition State Model was an attempt to overcome the several shortcomings of the Thermodynamics, the Kinetics and the Statistical Mechanics Models, by forging a strong and precise link between thermodynamics and kinetic variables. By considering a chemical reaction as a process in which a system passed over the top of the energy barrier between the initial and the final states, this model proposed that rate could be calculated by focusing attention on the molecular complexes at the col of the surface. Moreover, the application of statistical methods made possible the development of equations that related the concentration of the species involved to the rate of a reaction. 

The book is therefore situated at the interface between physical chemistry (classical thermodynamics and statistical mechanics, chemical kinetics, transport phenomena) and the theory of reactors, themselves at the heart of chemical reaction engineering. It therefore possesses a marked pluridisciplinary character. However, in order to keep this book readable by newcomers to the fields both of GPTRs and the kinetic modelling of reactions, basic concepts, theories and laws of the underlying scientific disciplines are given. The main equations are illustrated by simple numerical applications in order to show how the data tables are used. 

As shown in Figure 26.1, the wide gap opens up between the particle and continuum paradigms. This gap cannot be spanned using statistical mechanical methods only. The existing theoretical models to be applied in the mesoscale are based on heuristics obtained via downscaling of macroscopic models and upscaling particle approach. Simphfied theoretical models of complex fluid flows, e.g., flows in porous media, non-Newtonian fluid dynamics, thin film behavior, flows in presence of chemical reactions, and hydrodynamic instabilities formation, involve not only vah-dation but should be supported by more accurate computational models as well. However, until now, there has not been any precisely defined computational model, which operates in the mesoscale, in the range from 10 A to tens of microns. 

Critics of first-principles models often point out that this AE corresponds to immobile atoms at zero Kelvin. For most catalytically interesting problems, the quantum mechanical AE is the largest contribution to a finite temperature reaction enthalpy or free energy. Statistical mechanical models can be used to determine and compute finite temperature and entropy contributions when necessary. In practice, the greater challenge is to solve the Schrodinger equation sufficiently accurately to give chemically useful values of AE. 

The theory of solvent-effects and some of its applications are overviewed. The generalized selfcon-sistent reaction field (SCRF)theory has been used to give a unified approach to quantum chemical calculations of subsystems embedded in a given milieu. The statistical mechanical theory of projected equations of motion has been briefly described. This theory underlies applications of molecular dynamics simulations to the study of solvent and thermal bath effects on carefully defined subsystem of interest. The relationship between different approaches used so far to calculate solvent effects and the general SCRF has been established. Recent work using the continuum approach to model the surrounding media is overviewed. Monte Carlo and molecular dynamics studies of solvent effects on molecular properties and chemical reactions together with simulations of solvent effects on protein structure and dynamics are reviewed. 

The alternative to analytical theories are simulations computer experiments that generate representative statistical samples of the system from which macroscopic properties can be derived. The theory used to generate simulations is much simpler than the statistical mechanical theories of liquids. Nevertheless the results can be far more accurate and less assumptions go into the derivation of the results. Computer simulations are thus often used to test the reliability of analytical theories, using model potentials. Using realistic potentials, computer simulations can be used to understand and predict properties of large systems of complex molecules. This is not restricted to simple molecular fluids but may extend to complicated macromolecules, liquid crystals, micels, surfaces and chemical reactions. 

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