Computational studies Monte Carlo method

Monte Carlo simulations are commonly used to compute the average thermodynamic properties of a molecule or a system of molecules, and have been employed extensively in the study of the structure and equilibrium properties of liquids and solutions. Monte Carlo methods have also been used to conduct conformational searches under non-equilibrium conditions. 

The study of surface chemical reaction processes using computer simulation techniques is quite an active field of research. Within this context the Monte Carlo method emerges as a powerful tool which contributes to the... 

Molecular dynamics and Monte Carlo simulations have been extensively applied to molten salts since 1968 to study structure, thermodynamic properties, and dynamic properties from a microscopic viewpoint. Several review papers have been published on computer simulation of molten salts. " Since the Monte Carlo method cannot yield dynamic properties, MD methods have been used to calculate dynamic properties. 

Sometimes the theoretical or computational approach to description of molecular structure, properties, and reactivity cannot be based on deterministic equations that can be solved by analytical or computational methods. The properties of a molecule or assembly of molecules may be known or describable only in a statistical sense. Molecules and assemblies of molecules exist in distributions of configuration, composition, momentum, and energy. Sometimes, this statistical character is best captured and studied by computer experiments molecular dynamics, Brownian dynamics, Stokesian dynamics, and Monte Carlo methods. Interaction potentials based on quantum mechanics, classical particle mechanics, continuum mechanics, or empiricism are specified and the evolution of the system is then followed in time by simulation of motions resulting from these direct... 

Monte Carlo Methods. Although several statistical mechanical ensembles may be studied using MC methods (2,12,14), the canonical ensemble has been the most frequently used ensemble for studies of interfacial systems. In the canonical ensemble, the number of molecules (N), cell volume (V) and temperature (T) are fixed. Hence, the canonical ensemble is denoted by the symbols NVT. The choice of ensemble determines which thermodynamic properties can be computed. 

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). 

A promising study of the lattice gas model is the computer statistical tests (by the Monte Carlo method). Such calculations have been carried out since the mid-1960s (see, for example, refs. 66 and 105). For calculations of gas adsorption on metals, see refs. 106-110. However, no systematic application of the Monte Carlo method to heterogeneous reactions has been carried out it is to be done in the future. 

Many computational studies of the permeation of small gas molecules through polymers have appeared, which were designed to analyze, on an atomic scale, diffusion mechanisms or to calculate the diffusion coefficient and the solubility parameters. Most of these studies have dealt with flexible polymer chains of relatively simple structure such as polyethylene, polypropylene, and poly-(isobutylene) [49,50,51,52,53], There are, however, a few reports on polymers consisting of stiff chains. For example, Mooney and MacElroy [54] studied the diffusion of small molecules in semicrystalline aromatic polymers and Cuthbert et al. [55] have calculated the Henry s law constant for a number of small molecules in polystyrene and studied the effect of box size on the calculated Henry s law constants. Most of these reports are limited to the calculation of solubility coefficients at a single temperature and in the zero-pressure limit. However, there are few reports on the calculation of solubilities at higher pressures, for example the reports by de Pablo et al. [56] on the calculation of solubilities of alkanes in polyethylene, by Abu-Shargh [53] on the calculation of solubility of propene in polypropylene, and by Lim et al. [47] on the sorption of methane and carbon dioxide in amorphous polyetherimide. In the former two cases, the authors have used Gibbs ensemble Monte Carlo method [41,57] to do the calculations, and in the latter case, the authors have used an equation-of-state method to describe the gas phase. 

The same computer revolution that started in the middle of the last century also plays an important, in fact crucial, role in the development of methods and algorithms to study solvation problems. Dealing, for instance, with a liquid system means the inclusion of explicit molecules, in different thermodynamic conditions. The number of possible arrangements of atoms or molecules is enormous, demanding the use of statistical mechanics. Here is where computer simulation, Monte Carlo (MC) or molecular dynamics (MD), makes its entry to treat liquid systems. Computer simulation is now an important, if not central, tool to study solvation phenomena. The last two decades have seen a remarkable development of methods, techniques and algorithms to study solvation problems. Most of the recent developments have focused on combining quantum mechanics and statistical mechanics using MC or... 

Followed by the rapid development of computer power, Monte Carlo (MC) and molecular dynamics (MD) simulation methods have been applied to many fields so as to connect the microscopic interaction model with the macroscopic properties, such as pVT relation, phase equilibria and so on [6]. They have also been used to analyze the adsorption characteristics of supercritical fluid [7-9] however, the simulation studies for adsorption phenomena in supercritical fluid mixtures are still limited. 

The Monte Carlo method has been applied more extensively to the study of solvation phenomena. Clementi has looked at solvent distribution around a large variety of amino acids and nucleotides. This work is summarized by Clementi in detail in this volume[l7e]. It should be noted that all this work on solvent using these powerful simulation techniques has appeared within the last five years and most of it even more recently. This is a direct result of the advances in computer technology as well as the adaptation of the... 

The effect of introduction of regularly disposed cyclic fragments into the macromolecular backbone on conformational and hydrodynamic parameters is also studied [30], For this purpose, copolymers 3 and 4 (Table 1) were fractionated into twelve fractions from the benzene (solvent) - methanol (precipitator) system. The influence of cyclic groups, introduced into the polymer backbone, on rigi-dity parameters was determined by direct computer simulation of macromolecular coil with the help of the Monte-Carlo method. 

It has already been seen in Seetion 2.17 that computer simulation of structures in aqueous solution can give rise to calculations of some static (e.g coordination numbers) and dynamic (e.g., diffusion coefficients) properties of ions in aqueous and nonaqueous solutions. One such computer approach is the Monte Carlo method. In this method, imaginary movements of the particles present are studied, but only those movements that /ower the potential energy. Another technique is molecular dynamics. In this method, one takes a manageable number of atoms (only a few hundred because of the expense of the computer time) and works out their movements at femtosecond intervals by applying Newtonian mechanics to the particles under force laws in which it is imagined that only pairwise interactions count. The parameters needed to compute these movements numerically are obtained by assuming that the calculations are correct and that one needs to find the parameters that fit. 

The numerical results reviewed above were obtained for infinite lattices. How do the various quantities of interest behave near the percolation threshold in a large but finite lattice This problem has been studied by renormalization methods, which are essentially equivalent to finite-size scaling. For finite lattices the percolation transition is smeared out over a range of p, and one must expect a similar trend in other functions, including the conductivity. Computer simulations by the Monte Carlo method have been carried out for bond percolation on a three-dimensional simple cubic lattice by Kirkpatrick (1979). Five such experimental curves are shown in Fig. 40, each of which corresponds to a cube of size b, containing bonds. In Fig. 40 the vertical axis gives the fraction p of such samples that percolate (i.e., have opposite faces con-... 

Two methods are in common use for simulating molecular liquids the Monte Carlo method (MC) and molecular dynamics calculations (MD). Both depend on the availability of reasonably accurate potential energy surfaces and both are based on statistical classical mechanics, taking no account of quantum effects. In the past 10-15 years quantum Monte Carlo methods (QMC) have been developed that allow intramolecular degrees of freedom to be studied, but because of the computational complexity of this approach results have only been reported for water clusters. 

The Intention of this volume is to give a flavour of the types of problems in biochemistry that theoretical calculations can solve at present, and to illustrate the tremendous predictive power these approaches possess. With these aspects in mind, I have tried to gather some of the leading scientists in the field of theoretical/computational biochemistry and let them present their work. You will hence find a wide range of computational approaches, from classical MD and Monte Carlo methods, via semi-empirical and DFT approaches on isolated model systems, to Car-Parrinello QM-MD and novel hybrid QM/MM studies. The systems investigated also cover a broad range from membrane-bound proteins to various types of enzymatic reactions as well as inhibitor studies, cofactor properties, solvent effects, transcription and radiation damage to DNA. 

Molecular dynamics uses classical mechanics to study the evolution of a system in time. At each point in time the classical equations of motion are solved for a system of particles (atoms), interacting via a set of predefined potential functions (force field), after which the solution obtained is applied to predict positions and velocities of the particles for a (short) step in time. This step-by-step process moves the system along a trajectory in phase space. Assuming that the trajectory has sampled a sufficiently large part of phase space and the ergodicity principle is obeyed, all properties of interest can then be computed by averaging along the trajectory. In contrast to the Monte Carlo method (see below), the MD method allows one to calculate both the structural and time-dependent characteristics of the system. An interested reader can find a comprehensive description of the MD method in the books by Allen and Tildesley or Frenkel and Smit. ... 

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. 

Finally, we stress that the quantum chemical method presented here has the advantage over DFT-based techniques that it also furnishes wavefunctions that can be used to perform computations of spectra, and therefore have a better contact with the experiment. Another advantage of this approach is that, unlike the diffusion Monte-Carlo method, it can coherently be applied to studies of fermion and mixed boson/fermion doped clusters. An example can be found in our recent work on the Raman spectra of (He)w-Br2(X) clusters [27,28]. 

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