Kinetic Monte Carlo simulation dynamic processes

Contents 1. Introduction 176 2. Static NMR Spectra and the Description of Dynamic Exchange Processes 178 2.1. Simulation of static NMR spectra 178 2.2. Simulation of DNMR spectra with average density matrix method 180 3. Calculation of DNMR Spectra with the Kinetic Monte Carlo Method 182 3.1. Kinetic description of the exchange processes 183 3.2. Kinetic Monte Carlo simulation of DNMR spectra for uncoupled spin systems 188 3.3. Kinetic Monte Carlo simulation of coupled spin systems 196 3.4. The individual density matrix 198 3.5. Calculating the FID of a coupled spin system 200 3.6. Vector model and density matrix in case of dynamic processes 205 4. Summary 211 Acknowledgements 212 References 212... 

It is often stated that MC methods lack real time and results are usually reported in MC events or steps. While this is immaterial as far as equilibrium is concerned, following real dynamics is essential for comparison to solutions of partial differential equations and/or experimental data. It turns out that MC simulations follow the stochastic dynamics of a master equation, and with appropriate parameterization of the transition probabilities per unit time, they provide continuous time information as well. For example, Gillespie has laid down the time foundations of MC for chemical reactions in a spatially homogeneous system.f His approach is easily extendable to arbitrarily complex computational systems when individual events have a prescribed transition probability per unit time, and is often referred to as the kinetic Monte Carlo or dynamic Monte Carlo (DMC) method. The microscopic processes along with their corresponding transition probabilities per unit time can be obtained via either experiments such as field emission or fast scanning tunneling microscopy or shorter time scale DFT/MD simulations discussed earlier. The creation of a database/lookup table of transition... 

Statistical mechanical Monte Carlo as well as classical molecular dynamic methods can be used to simulate structure, sorption, and, in some cases, even diffusion in heterogeneous systems. Kinetic Monte Carlo simulation is characteristically different in that the simulations follow elementary kinetic surface processes which include adsorption, desorption, surface diffusion, and reactivity . The elementary rate constants for each of the elementary steps can be calculated from ab initio methods. Simulations then proceed event by event. The surface structure as well as the time are updated after each event. As such, the simulations map out the temporal changes in the atomic structure that occur over time or with respect to processing conditions. 

In this review we put less emphasis on the physics and chemistry of surface processes, for which we refer the reader to recent reviews of adsorption-desorption kinetics which are contained in two books [2,3] with chapters by the present authors where further references to earher work can be found. These articles also discuss relevant experimental techniques employed in the study of surface kinetics and appropriate methods of data analysis. Here we give details of how to set up models under basically two different kinetic conditions, namely (/) when the adsorbate remains in quasi-equihbrium during the relevant processes, in which case nonequilibrium thermodynamics provides the needed framework, and (n) when surface nonequilibrium effects become important and nonequilibrium statistical mechanics becomes the appropriate vehicle. For both approaches we will restrict ourselves to systems for which appropriate lattice gas models can be set up. Further associated theoretical reviews are by Lombardo and Bell [4] with emphasis on Monte Carlo simulations, by Brivio and Grimley [5] on dynamics, and by Persson [6] on the lattice gas model. 

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 ... 

In a few instances, quantum mechanical calculations on the stability and reactivity of adsorbates have been combined with Monte Carlo simulations of dynamic or kinetic processes. In one example, both the ordering of NO on Rh(lll) during adsorption and its TPD under UHV conditions were reproduced using a dynamic Monte Carlo model involving lateral interactions derived from DFT calculations and different adsorption... 

In the dynamic Monte Carlo simulations described earlier, we used a crystalline template to suppress supercooling (Sect. A.3). If this template is not present, there will be a kinetic interplay between polymer crystallization and liquid-liquid demixing during simulations of a cooling run. In this context, it is of particular interest to know how the crystallization process is affected by the vicinity of a region in the phase diagram where liquid-liquid demixing can occur. 

The dynamics of systems such as this one can be followed using a method called kinetic Monte Carlo (kMC) The idea behind this method is straightforward If we know the rates for all processes that can occur given the current configuration of our atoms, we can choose an event in a random way that is consistent with these rates. By repeating this process, the system s time evolution can be simulated. 

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. 

Investigation of the motion of adsorbed molecules, which give mechanisms and rates of re-orientation and diffusion, require alternative approaches. For systems that contain highly mobile species. Molecular Dynamics (MD) techniques are widely used. However, for many adsorbates the timescales of motion are much longer than can feasibly be simulated, so that MD is only relevant either for small molecules or at high temperatures. In order to simulate slower diffusion, the process must be considered in terms of rare events with significant activations. The activated processes are then usefully treated by transition state theory, and the associated processes treated over extended timescales and volumes by, for example, Kinetic Monte Carlo (KMQ techniques. 

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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