Monte Carlo methods, reaction dynamics

Initially computational chemistry mainly referred to the more applied aspects of quantum chemistry. Computational chemistry now encompasses a wide variety of areas, which include quantum chemistry, molecular mechanics, molecular dynamics, Monte Carlo methods. Brownian dynamics, continuum electrostatics, reaction dynamics, numerical analysis methods, artificial intelligence, chemometrics and others. This chapter deals mainly with the first three of these areas. We focus on these areas for reasons of space, personal interest, and expertise, and because two of these (quantum mechanics and molecular mechanics) are areas that have received attention in the Journal of Chemical Education. We do not cover aspects related to computational polymer chemistry or computational materials science. 

Specific solute-solvent interactions involving the first solvation shell only can be treated in detail by discrete solvent models. The various approaches like point charge models, siipennoleciilar calculations, quantum theories of reactions in solution, and their implementations in Monte Carlo methods and molecular dynamics simulations like the Car-Parrinello method are discussed elsewhere in this encyclopedia. Here only some points will be briefly mentioned that seem of relevance for later sections. 

Molecular mechanics methods have been used particularly for simulating surface-liquid interactions. Molecular mechanics calculations are called effective potential function calculations in the solid-state literature. Monte Carlo methods are useful for determining what orientation the solvent will take near a surface. Molecular dynamics can be used to model surface reactions and adsorption if the force held is parameterized correctly. 

The calculation of the potential of mean force, AF(z), along the reaction coordinate z, requires statistical sampling by Monte Carlo or molecular dynamics simulations that incorporate nuclear quantum effects employing an adequate potential energy function. In our approach, we use combined QM/MM methods to describe the potential energy function and Feynman path integral approaches to model nuclear quantum effects. 

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... 

GH Theory was originally developed to describe chemical reactions in solution involving a classical nuclear solute reactive coordinate x. The identity of x will depend of course on the reaction type, i.e., it will be a separation coordinate in an SnI unimolecular ionization and an asymmetric stretch in anSN2 displacement reaction. To begin our considerations, we can picture a reaction free energy profile in the solute reactive coordinate x calculated via the potential of mean force Geq(x) -the system free energy when the system is equilibrated at each fixed value of x, which would be the output of e.g. equilibrium Monte Carlo or Molecular Dynamics calculations [25] or equilibrium integral equation methods [26], Attention then focusses on the barrier top in this profile, located at x. ... 

Interestingly, in the experiments devoted solely to computational chemistry, molecular dynamics calculations had the highest representation (96-98). The method was used in simulations of simple liquids, (96), in simulations of chemical reactions (97), and in studies of molecular clusters (98). One experiment was devoted to the use of Monte Carlo methods to distinguish between first and second-order kinetic rate laws (99). One experiment used DFT theory to study two isomerization reactions (100). 

The force controls the remarkably persistent coherence in products, a feature that was unexpected, especially in view of the fact that all trajectory calculations are normally averaged (by Monte Carlo methods) without such coherences. Only recently has theory addressed this point and emphasized the importance of the transverse force, that is, the degree of anharmonicity perpendicular to the reaction coordinate. The same type of coherence along the reaction coordinate, first observed in 1987 by our group, was found for reactions in solutions, in clusters, and in solids, offering a new opportunity for examining solvent effects on reaction dynamics in the transition-state region. 

Bifurcation and global stability in surface catalyzed reactions using the Monte Carlo method (with D.G. Vlachos and L.D. Schmidt). In H. Swinney, R. Aris, and D. Aronson (eds.), Patterns and Dynamics in Reactive Media, (Vol. 37, pp. 187-206). New York Springer-Verlag, 1991. 

The Monte Carlo method is a very powerful numerical technique used to evaluate multidimensional integrals in statistical mechanics and other branches of physics and chemistry. It is also used when initial conditions are chosen in classical reaction dynamics calculations, as we have discussed in Chapter 4. It will therefore be appropriate here to give a brief introduction to the method and to the ideas behind the method. 

Solvent effects can significantly influence the function and reactivity of organic molecules.1 Because of the complexity and size of the molecular system, it presents a great challenge in theoretical chemistry to accurately calculate the rates for complex reactions in solution. Although continuum solvation models that treat the solvent as a structureless medium with a characteristic dielectric constant have been successfully used for studying solvent effects,2,3 these methods do not provide detailed information on specific intermolecular interactions. An alternative approach is to use statistical mechanical Monte Carlo and molecular dynamics simulation to model solute-solvent interactions explicitly.4 8 In this article, we review a combined quantum mechanical and molecular mechanical (QM/MM) method that couples molecular orbital and valence bond theories, called the MOVB method, to determine the free energy reaction profiles, or potentials of mean force (PMF), for chemical reactions in solution. We apply the combined QM-MOVB/MM method to... 

The following five chapters deal with problems associated with solid phases, in some cases involving surface and interfacial problems. In Chapter 14, Steele presents a review of physical adsorption investigated by MD techniques. Jiang and Belak describe in Chapter 15 the simulated behavior of thin films confined between walls under the effect of shear. Chapter 16 contains a review by Benjamin of the MD equilibrium and non-equilibrium simulations applied to the study of chemical reactions at interfaces. Chapter 17 by Alper and Politzer presents simulations of solid copper, and methodological differences of these simulations compared to those in the liquid phase are presented. In Chapter 18 Gelten, van Santen, and Jansen discuss the application of a dynamic Monte Carlo method for the treatment of chemical reactions on surfaces with emphasis on catalysis problems. Khakhar in... 

The next section gives a brief overview of the main computational techniques currently applied to catalytic problems. These techniques include ab initio electronic structure calculations, (ab initio) molecular dynamics, and Monte Carlo methods. The next three sections are devoted to particular applications of these techniques to catalytic and electrocatalytic issues. We focus on the interaction of CO and hydrogen with metal and alloy surfaces, both from quantum-chemical and statistical-mechanical points of view, as these processes play an important role in fuel-cell catalysis. We also demonstrate the role of the solvent in electrocatalytic bondbreaking reactions, using molecular dynamics simulations as well as extensive electronic structure and ab initio molecular dynamics calculations. Monte Carlo simulations illustrate the importance of lateral interactions, mixing, and surface diffusion in obtaining a correct kinetic description of catalytic processes. Finally, we summarize the main conclusions and give an outlook of the role of computational chemistry in catalysis and electrocatalysis. 

Dynamic or kinetic Monte Carlo methods have been used to simulate the catalytic surface chemistry for various different reaction systems. The vapor-phase oxidation of CO to form CO2, however, has been the most widely studied due to its simplicity as well as its general applicability. Pioneering work by Ziff [82] and Zhdanov [83] shows the formations of interesting phase transitions as a function of the kinetics and lateral interactions. Many subsequent studies by various other groups extend the basic models to cover more general features. 

Exact determination of entropy effects in enzymatic reactions is not an easy task even nowadays when sophisticated Monte Carlo and molecular dynamics methods are available for calculations (Warshel, 1991 Aqvist and Warshel, 1993). One way to examine the importance of entropy is to analyse the configuration space available to the system in its ground and transition states, both in the enzyme and in solution. The entropic contribution to the catalytic effect, relative to the uncatalysed solution can be expressed as... 

The reported results for equilibrium properties were obtained by means of the standard Monte Carlo (MC), molecular dynamics (MD), and Gibbs ensemble (GE) simulation methods [23, 24], For the trial systems of a finite range the simple spherical cutoff was used, whereas in simulations of the full systems either the Ewald summation or the reaction field method were used. For further technical details we refer the reader to the original papers. 

In the modeling of catalytic reactions at the molecular level, the stochastic approach is also fruitful along with the simulation based on equations (the deterministic approach). Stochastic simulations (the dynamic Monte Carlo method) makes it possible to penetrate into the microlevel and monitor detailed changes in the adsorption layer, and explain the observed phenomena. 

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