Application of the Monte Carlo Technique

Barker and Watts (1969) published a preliminary report on the computations of energy, heat capacity, and the radial distribution function for waterlike particles. The potential function used for these calculations is similar to the one discussed in Section 6.4 however, instead of a smooth switching function, they used a hard-sphere cutoff at 2 A so that the point charges could not approach each other to zero separation. 

This potential was devised by Rowlinson (1951) for the computation of the virial coefficients of water. 

It is difficult to trace the origin of these discrepancies between the computed and experimental radial distribution function. It is possible that the pair potential does not produce enough preference for the tetrahedral geometry (hence, the coordination number and the location of the second peak do not show the characteristic values as in water), or that the numerical procedure was not run with a sufficient number of particles and configurations. [Sixty four particles in three dimensions corresponds to four particles in one dimension, which is quite a small number. Furthermore, the number of configurations, of the order of 10 , is quite small to ensure convergence. See also the discussion in Section 6.11.] Some preliminary recent Monte Carlo results were reported by Popkie et ah (1973). 

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This section has illustrated a relatively simple application of the Monte Carlo technique for simulating atmospheric diffusion. With the availability of large-scale computing capacities, Monte Carlo methods can be envi-... 

The most straightforward application of the Monte Carlo technique arises where the probability of a parameter taking a particular value is constant over its entire range. For example, the initial angles between the BC inter-nuclear axis, and a line joining A to the center of mass of BC and in the plane containing A, B, and C, are distributed in this way, and a value can be selected by multiplying it by a random number between — 1 and 1. 

In this section we will discuss two applications of the Monte Carlo technique. 

For a more comprehensive introduction to Monte Carlo simulations we refer the interested reader to the excellent text Landau and Binder [126]. In Ref. 126 the authors discuss many applications of the Monte Carlo technique beyond the scope of the present book. 

Monte Carlo search methods are stochastic techniques based on the use of random numbers and probability statistics to sample conformational space. The name Monte Carlo was originally coined by Metropolis and Ulam [4] during the Manhattan Project of World War II because of the similarity of this simulation technique to games of chance. Today a variety of Monte Carlo (MC) simulation methods are routinely used in diverse fields such as atmospheric studies, nuclear physics, traffic flow, and, of course, biochemistry and biophysics. In this section we focus on the application of the Monte Carlo method for... 

The techniques and applications of the Monte Carlo trajectory method have been explained fully elsewhere. The object in the next few paragraphs is simply to highlight the results which are specially relevant to the subject of selective energy consumption. Trajectory studies that relate to particular systems are referred to later when the corresponding experimental data are reviewed. 

The acceptance or rejection of configurations with high potential energy is a special feature of modern Monte Carlo techniques [20, 21]. In the case of hard-sphere systems, this choice is particularly simple because U is equal to zero when spheres do not overlap, and infinity when they do. Details about application of this technique to more complex systems such as molecular fiuids involve some variations which are described in the original literature. 

This section illustrates application of the mass balance technique for two exploration prospects in the East Texas Basin. Gas volumes were computed using both the conventional reservoir engineering and mass balance approaches. Furthermore, uncertainty was incorporated into the calculations with a Monte Carlo simulation technique that generates probabilistic distributions of gas volumes. 

The analog or direct simulation ityproach to the solution of neutron transport problems makes the understanding of the application of die Monte Carlo method quite simple. Upon closer examination it is evident there is often need to supplement the analog method with techniques to reduce the computer time required to solve a particular problem. A summary of the commonly used techniques will be presented along with a brief justification of their use. 

In this review, we shall focus on the development and application of combined QM/MM potentials in condensed phase simulations where solute and solvent molecules are explicitly treated, in both aqueous and organic environments. The emphasis of this chapter is the use of molecular orbital (MO) theory in the QM treatment because it is well documented and familiar to chemists. Details of the Monte Carlo and molecular dynamics simulation techniques are available in an excellent book by Allen and Tildesley. In a recent paper, Aqvist and Warshel provided a number of additional details of the methodology, particularly on the use of the EVB method. We have also benefited both intellectually and technically from the thorough paper by Field, Bash, and Karplus, which gives additional insights on the implementation of combining molecular orbital and molecular mechanics programs. 

Finally, the concept of assigning probabilities to variables in order to quantify risk was discussed. The Monte-Carlo technique was introduced, and its application to simulate the cumulative distribution of net present values of a project was described. The interpretation of results from this technique was presented. Finally, the simulation of risk and the analysis of data using the CAPCOST package was illustrated by an example. 

For the equihbrium properties and for the kinetics under quasi-equilibrium conditions for the adsorbate, the transfer matrix technique is a convenient and accurate method to obtain not only the chemical potentials, as a function of coverage and temperature, but all other thermodynamic information, e.g., multiparticle correlators. We emphasize the economy of the computational effort required for the application of the technique. In particular, because it is based on an analytic method it does not suffer from the limitations of time and accuracy inherent in statistical methods such as Monte Carlo simulations. The task of variation of Hamiltonian parameters in the process of fitting a set of experimental data (thermodynamic and... 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

We have presented applications of a parameter estimation technique based on Monte Carlo simulation to problems in polymer science involving sequence distribution data. In comparison to approaches involving analytic functions, Monte Carlo simulation often leads to a simpler solution of a model particularly when the process being modelled involves a prominent stochastic coit onent. 

We first mentioned the applicability of optimization (minimization) methods in Section V.C of Chapter 1. Constraints pose no particular problem to many of these methods. It would seem that the deconvolution problem with object amplitude bounds should be a straightforward application. The most general case, however, deals with each sampled element om of the estimate as a parameter of the objective function and hence the solution. Excessive computation is then required. The likelihood is great that only local minima of the objective function O will be found. Nevertheless, the optimization idea may be teamed with a Monte Carlo technique and a decision rule to yield a method having some promise. 

The credit load for die computational chemistry laboratory course requires that the average student should be able to complete almost all of the work required for the course within die time constraint of one four-hour laboratory period per week. This constraint limits the material covered in the course. Four principal computational methods have been identified as being of primary importance in the practice of chemistry and thus in the education of chemistry students (1) Monte Carlo Methods, (2) Molecular Mechanics Methods, (3) Molecular Dynamics Simulations, and (4) Quantum Chemical Calculations. Clearly, other important topics could be added when time permits. These four methods are developed as separate units, in each case beginning with die fundamental principles including simple programming and visualization, and building to the sophisticated application of the technique to a chemical problem. 

The workhorse for the calculation of cross sections in full collisions is the so-called Monte Carlo technique (Schreider 1966 Porter and Raff 1976 Pattengill 1979). The application to photodissociation proceeds in an identical fashion. Within the Monte Carlo method an integral over a function f(x) is approximated by the average of the function over N values Xk randomly selected from a uniform distribution,... 

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