Monte Carlo method dynamics approach

PBSA), the lambda dynamics approach and the chemical Monte Carlo/Molecular dynamics approach. Success using these methods will depend on their ability to accurately discriminate between structurally diverse compound series and thereby help prioritize the compound series for the medicinal chemists. 

The alternative simulation approaches are based on molecular dynamics calculations. This is conceptually simpler that the Monte Carlo method the equations of motion are solved for a system of A molecules, and periodic boundary conditions are again imposed. This method pennits both the equilibrium and transport properties of the system to be evaluated, essentially by numerically solvmg the equations of motion... 

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. 

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 ... 

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... 

Calculations on dynamics of solvation shells are still in their infancy. However, very recent papers on this subject, show that in most examples we cannot expect a realistic picture of solvent shells from a purely static approach. Most probably, molecular dynamics calculations and Monte Carlo methods will produce a variety of interesting data and will improve our knowledge on solvation of ions substantially. 

There are two basic approaches to the computer simulation of liquid crystals, the Monte Carlo method and the method known as molecular dynamics. We will first discuss the basis of the Monte Carlo method. As is the case with both these methods, a small number (of the order hundreds) of molecules is considered and the difficulties introduced by this restriction are, at least in part, removed by the use of artful boundary conditions which will be discussed below. This relatively small assembly of molecules is treated by a method based on the canonical partition function approach. That is to say, the energy which appears in the Boltzman factor is the total energy of the assembly and such factors are assumed summed over an ensemble of assemblies. The summation ranges over all the coordinates and momenta which describe the assemblies. As a classical approach is taken to the problem, the summation is replaced by an integration over all these coordinates though, in the final computation, a return to a summation has to be made. If one wishes to find the probable value of some particular physical quantity, A, which is a function of the coordinates just referred to, then statistical mechanics teaches that this quantity is given by... 

Simulations. In addition to analytical approaches to describe ion—solid interactions two different types of computer simulations are used Monte Carlo (MC) and molecular dynamics (MD). The Monte Carlo method relies on a binary collision model and molecular dynamics solves the many-body problem of Newtonian mechanics for many interacting particles. As the name Monte Carlo suggests, the results require averaging over many simulated particle trajectories. A review of the computer simulation of ion—solid interactions has been provided (43). 

Another and now more widely used computational approach to predicting the properties of ions in solution follows from the Monte Carlo method. Thus, in the latter, the particle is made mentally to move randomly in each experiment but only one question is posed Does the random move cause an increase or decrease of energy In molecular dynamic (MD) simulations, much more is asked and calculated. In fact, a random micromovement is subjected to all the questions that classical dynamics can ask and answer. By repeating calculations of momentum and energy exchanges... 

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. 

Yet another approach is to simulate adsorption by Monte Carlo methods or Molecular Dynamics. In particular with water as the solvent this is an... 

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. 

There are two other approaches that we want to mention here. The first is the collection of dynamic Monte Carlo methods that are defined using an algorithm. We include in this collection the method by Fichthorn and Weinberg. [34] The second approach consists of cellular automata (CA) in one form or another. 

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