Rate theories Monte Carlo methods

The Monte Carlo method as described so far is useful to evaluate equilibrium properties but says nothing about the time evolution of the system. However, it is in some cases possible to construct a Monte Carlo algorithm that allows the simulated system to evolve like a physical system. This is the case when the dynamics can be described as thermally activated processes, such as adsorption, desorption, and diffusion. Since these processes are particularly well defined in the case of lattice models, these are particularly well suited for this approach. The foundations of dynamical Monte Carlo (DMC) or kinetic Monte Carlo (KMC) simulations have been discussed by Eichthom and Weinberg (1991) in terms of the theory of Poisson processes. The main idea is that the rate of each process that may eventually occur on the surface can be described by an equation of the Arrhenius type ... 

Voter, A.F. A Monte Carlo method for determining free-energy differences and transition state theory rate constants. J. Chem. Phys. 1985, 82, 1890-9. 

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). 

Transition state theory can also be employed to calculate diffusion coefficients in hopping processes. Adsorbates prefer to reside at particular places in a zeolite and because an energy barrier is present between them, they do not transfer easily from one site to another. The possible adsorption sites are located via a Monte Carlo method, and the transition state via migration path analyses. A rate constant can be associated with jumps from site i to siteA surface can be defined that separates sites i and and contains the top of the energy... 

A. F. Voter, J. Chem. Phys., 82,1890 (1985). A Monte Carlo Method for Determining Free-Energy Differences and Transition State Theory Rate Constants. 

The collision probability is one of several possible formulations of integral transport theory. Three other formulations are the integral equations for the neutron flux, neutron birth-rate density, and fission neutron density. Oosterkamp (26) derived perturbation expressions for reactivity in the birth rate density formulation. The fission density formulation provides the basis for Monte Carlo methods for perturbation calculations (52, 55). 

In a staged multi-scale approach, the energetics and reaction rates obtained from these calculations can be used to develop coarse-grained models for simulating kinetics and thermodynamics of complex multi-step reactions on electrodes (for example see [25, 26, 27, 28, 29, 30]). Varying levels of complexity can be simulated on electrodes to introduce defects on electrode surfaces, composition of alloy electrodes, distribution of alloy electrode surfaces, particulate electrodes, etc. Monte Carlo methods can also be coupled with continuum transport/reaction models to correctly describe surfaces effects and provide accurate boundary conditions (for e.g. see Ref. [31]). In what follows, we briefly describe density functional theory calculations and kinetic Monte Carlo simulations to understand CO electro oxidation on Pt-based electrodes. 

More complex computational models using Monte Carlo methods have attempted to predict bubble size distributions for a combination of breakup and coalescence. These models typically treat bubble coalescence by analogy with the kinetic theory where bubbles are assumed to act as solid particles [18,19]. They use a binary collision rate (probability) and a collision efficiency factor to account for collisions that do not lead to coalescence. Since collision is assumed to be a random process in these models, turbulence of the same scale as the bubbles or smaller would increase collisions and, therefore, also increase the coalescence rate. 

An different kind of Monte Carlo method is the so-called Kinetic Monte Carlo method (sometimes also called Dynamic Monte Carlo) [21], in which the system is allowed to evolve dynamically from state to state, based on a catalog of transitions and associated rates. Each transition is accepted with a probability proportional to its rate. This, however, assumes that a complete catalog of possible transitions is known in advance (see [22] for an example of the importance of this). Alternatively, a catalog may be built on-the-fly, as proposed by Henkehnan et al. [23]. Similar to this technique is the transition state theory (TST)-based MC technique of Liu et al. [24]. 

Although the collision and transition state theories represent two important methods of attacking the theoretical calculation of reaction rates, they are not the only approaches available. Alternative methods include theories based on nonequilibrium statistical mechanics, stochastic theories, and Monte Carlo simulations of chemical dynamics. Consult the texts by Johnson (62), Laidler (60), and Benson (59) and the review by Wayne (63) for a further introduction to the theoretical aspects of reaction kinetics. 

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 lack of data to support claims for failure rates is an issue which is widely investigated by data uncertainty analyses. For example, Hauptmanns, 2008 compares the use of reliability data stemming from different sources on probabilistic safety calculations, and tends to prove that results do not differ substantially. Wang, 2004 discusses and identifies the inputs that may lead to SIL estimation changes. Propagation of error, Monte Carlo, and Bayesian methods (Guerin, 2003) are quite common. Fuzzy set theory is also often used to handle data uncertainties, especially into fault tree analyses (Tanaka, 1983, Singer, 1990). Other approaches are based on evidence, possibihty, and interval analyses (Helton, 2004). 

The simplest and most reliable method to determine the radioactivity will be the relative method in which the peak counting rate of a sample is compared quantitatively with that of the radioactive standard consisting of the same nuclide and geometry. Usually, the peak efficiency for each detector is prepared by means of experimental procedures using various kinds of radioactive standards. But theoretical methods based on Monte Carlo theory have also been designed. 

Rate constants can be estimated by means of transition-state theory. In principle all thermodynamic data can be deduced from the partion function. The molecular data necessary for the calculation of the partion function can be either obtained from quantum mechanical calculations or spectroscopic data. Many of those data can be found in tables (e.g. JANAF). A very powerful tool to study the kinetics of reactions in heterogeneous catalysis is the dynamic Monte-Carlo approach (DMC), sometimes called kinetic Monte-Carlo (KMC). Starting from a paper by Ziff et al. [16], several investigations were executed by this method. Lombardo and Bell [17] review many of these simulations. The solution of the problem of the relation between a Monte-Carlo step and real time has been advanced considerably by Jansen [18,19] and Lukkien et al. [20] (see also Jansen and Lukkien [21]). First principle quantum chemical methods have advanced to the stage where they can now offer quantitative predictions of structure and energetics for adsorbates on surfaces. Cluster and periodic density functional quantum chemical methods are used to analyze chemisorption and catalytic surface reactivity [see e.g. 24,25]. 

The selection process is the place where the optimization in EC methods really takes place. This is where methods such as dynamic or kinetic Monte Carlo (DMC) simulations become important. They are used to compute the properties of a system or process. These properties are then converted to a fitness value. This fitness value is for satisfaction of a particular requirement of performance which is then operated on by the EC methods. The conversion is different for each system and property and also determines how effective the selection is. Dynamic Monte Carlo simulation, as we have already discussed, is a method to simulate elementary processes along with the actual rate. The method uses each individual reaction as an elementary event, which means that timescales comparable to actual experiments can be simulated. The reaction rate constants that it needs as input can be calculated using quantum chemical methods such as density functional theory, which results in what has been termed ab initio kinetics (see Chapter 3.10.4). 

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