Monte Carlo MC Methods

The Monte Carlo (MC) method starts with a particular arrangement of all the particles (solute and solvent molecules) in the system—a configuration. Then, a three-step procedure is applied. 

It is from step ii that the method derives its name—the process of choosing random numbers is as if dice were thrown at a casino. 

This problem can be circumvented by biasing the randomness of step ii, introducing importance sampling. This causes the method to favor good configurations over bad. The most important approach to importance sampling is the Metropolis method (Monte Carlo is a city, but Metropolis is a person s name). Steps i and ii are the same as above, followed by  

This approach biases the sampling toward low energy structures. It can be shovm that Metropolis sampling produces averages that are meaningful from a statistical mechanics viewpoint. Another sampling bias usually introduced is to favor moving solvent molecules that are closer to, rather than farther from, the solute molecule. 

With these approaches, the Monte Carlo method becomes a feasible, but still large, calculation. For example, to evaluate a simple solute like ethane in water, we might first evaluate 10 configurations just to let the system settle down (i.e., equilibrate). Then, we would average over 2-4 X 10 configurations to consider the solvation. 

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Prediction of Liquid Solubility with Molecular Dynamics (MD) and Monte Carlo (MC) Methods... 

An overreaching theme of the present chapter, besides broken ergodicity, has to do with the fact that most of the enhanced sampling methods that we shall discuss address situations in which one cannot clearly identify a reaction coordinate that can be conveniently used to describe the kinetic evolution of the system of interest. While methods for enhanced sampling are designed to yield accurate results faster than regular molecular dynamics or Monte Carlo (MC) methods, it is our belief that there is no perfect method, but that, rather, there are methods that perform better for particular applications. Moreover, it should be noted that, while in instances when a proper reaction coordinate can be identified methods described in other chapters are probably more efficient, they could still benefit by sampling in conformational directions perpendicular to the reaction coordinate. 

The Monte Carlo (MC) method can be used to efficiently calculate thermal equilibrium properhes (see Fig. 3.2). However, since it is an energy-barrier-based method, it will fail to generate dynamic features such as the precession of the spins, and will be able to generate the dynamic magnetizahon in the overdamped limit (X —> oo) only if an appropriate algorithm is used [35]. 

The method presented in this chapter serves as a link between molecular properties (e.g., cavities and their occupants as measured by diffraction and spectroscopy) and macroscopic properties (e.g., pressure, temperature, and density as measured by pressure guages, thermocouples, etc.) As such Section 5.3 includes a brief overview of molecular simulation [molecular dynamics (MD) and Monte Carlo (MC)] methods which enable calculation of macroscopic properties from microscopic parameters. Chapter 2 indicated some results of such methods for structural properties. In Section 5.3 molecular simulation is shown to predict qualitative trends (and in a few cases quantitative trends) in thermodynamic properties. Quantitative simulation of kinetic phenomena such as nucleation, while tenable in principle, is prevented by the capacity and speed of current computers however, trends may be observed. 

Monte Carlo (MC) methods can address the time gap problem of MD. The basis of MC methods is that the deterministic equations of the MD method are replaced by stochastic transitions for the slow processes in the system.3 MC methods are stochastic algorithms for exploring the system phase space although their implementation for equilibrium and non-equilibrium calculations presents some differences. 

Molecular dynamics (MD) and Monte Carlo (MC) methods have provided dynamic and atomic insights to understand complex biological systems. Thus, many techniques such as the A-dynamics and the chemical Monte Carlo/Molecular Dynamics (CMC/MD) method have been developed to improve their efficiencies.129... 

Molecular mechanics (MM), molecular dynamics (MD), and Monte-Carlo (MC) methods were employed to simulate the adsorption of methane, ethane, propane and isobutane on silicalite and HZSM-5. The silicalite was simulated using the same cluster-model adopted in the diffusion calculations. The H-ZMS-5 structure was constructed according to the procedure suggested by Vetrivel et al. [32], which consists in replacing one atom at the channel intersection by and protonating the oxygen atom bridging the Ta and Tg sites in order to preserve the lattice neutrality. 

Polymerization rate represents the instantaneous status of reaction locus, but the whole history of polymerization is engraved within the molecular weight distribution (MWD). Recently, a new simulation tool that uses the Monte Carlo (MC) method to estimate the whole reaction history, for both hnear [263-265] and nonlinear polymerization [266-273], has been proposed. So far, this technique has been applied to investigate the kinetic behavior after the nucleation period, where the overall picture of the kinetics is well imderstood. However, the versatility of the MC method could be used to solve the complex problems of nucleation kinetics. 

Like MD, Monte Carlo (MC) methods involve the generation of successive configurations of an ensemble of particles representing the system studied. However, unlike MD, there is no temporal connection between the different configurations. The aim of the technique is to generate a sufficient and representative number of configurations from which ensemble averages may then be calculated with acceptable accuracy. 

Molecular Dvnamics. In Monte Carlo (MC) methods a sequence of points in phase ... 

Experimental determination of excess molar quantities such as excess molar enthalpy and excess molar volume is very important for the discussion of solution properties of binary liquids. Recently, calculation of these thermodynamic quantities becomes possible by computer simulation of molecular dynamics (MD) and Monte Carlo (MC) methods. On the other hand, the integral equation theory has played an essential role in the statistical thermodynamics of solution. The simulation and the integral equation theory may be complementary but the integral equation theory has the great advantage over simulation that it is computationally easier to handle and it permits us to estimate the differential thermodynamic quantities. 

These methods starts from a given geometry, which typically is a (local) minimum, and new configurations are generated by adding a random kick to one or more atoms. In Monte Carlo (MC) methods the new geometry is accepted as a starting point for the next... 

There are two major techniques for generating an ensemble Monte Carlo and Molecular Dynamics. In Monte Carlo (MC) methods a sequence of points in phase space is generated from an initial geometry by adding a random kick to the... 

The concept of liquidlike clusters seems to have first emerged as a result of the explosion of simulations by the molecular dynamics (MD) method of McGinty, ° Cotterill et al., Damgaard Kristensen et al., and Briant and Burton, and by the Monte Carlo (MC) method of Lee et al. These were followed quickly by further MD simulations and MC simula-... 

Dynamical effects can also be defined in terms of the availability of special coherent motions. In this way, the dynamical proposal implies that enzymes activate special types of coherent motions, which are not available in the solution reaction. Now, the difference between the reaction in enzyme and in solution cannot be accounted for by evaluating the corresponding Ag using nondynamical Monte Carlo (MC) methods. In other words, if the results from MC and MD are identical, then we do not have dynamical contributions to catalysis. Careful and systematic studies (e.g., Refs. 4,129) have shown that the reactions in both enzymes and solutions involved large electrostatic fluctuations. However, these fluctuations follow the Boltzmann distribution, and thus, do not provide dynamical contributions to catalysis. 

In classical mechanics there exist, apart from the mean field theory, two popular methods to describe the dynamics of molecular systems, viz., the molecular dynamics (MD) method and the Monte Carlo (MC) method (Hansen and McDonald, 1976). In both methods the system is represented by a finite number, usually about 100 to 300, of molecules. In order to reduce boundary effects, this finite system is periodically repeated in all directions. 

The first molecular simulations were performed almost five decades ago by Metropolis et al. (1953) on a system of hard disks by the Monte Carlo (MC) method. Soon after, hard spheres (Rosenbluth and Rosenbluth, 1954) and Lennard-Jones (Wood and Parker, 1957) particles were also studied by both MC and molecular dynamics (MD). Over the years, the simulation techniques have evolved to deal with more complex systems by introducing different sampling or computational algorithms. Molecular simulation studies have been made of molecules ranging from simple systems (e.g., noble gases, small organic molecules) to complex molecules (e.g., polymers, biomolecules). 

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