Carlo Method

The Metropolis Monte Carlo method attempts to sample a representative set of equilibrium states in a manner that facilitates the calculation of meaningful averages for properties of the system. This method is discussed in the following subsections Basic Aspects of the Metropolis Monte Carlo Method, Monte Carlo Moves, and General Pointers for Conducting Monte Carlo Simulations. 

Basic Aspects of the Metropolis Monte Carlo Method 

The probability of an arrangement (labeled a) of N indistinguishable water molecules, in the canonical ensemble, is given by 

A Metropolis Monte Carlo simulation starts with a collection of molecules in a known configuration. The simulation consists of a large number of steps, each of which is an attempt to introduce an acceptable change in the collection of molecules. This change is either accepted or rejected based on a simple set of rules that ensure consistency of the results with the desired ensemble. If a change is accepted, the new state is used to generate the next step of the simulation otherwise, the unmodified configuration is used. Translational and rotational moves in the canonical ensemble are described below, followed by 

The condition most commonly used to reject or accept the new state b is based on the concept of detailed balance, requiring that for a sufficiently long 

An excellent example of a Monte Carlo approach in molecular modeling is provided 

Similar Monte Carlo approaches have been successfully used to characterize the sorptive properties of zeolites for alkanes (Smit and Siepmann 1994 Smit 1995 Nascimento 1999 Suzuki et al. 2000), for aromatic organic compounds (Bremard et al. 1997 Klemm et al. 1998), for water (Channon et al. 1998), and for the sorption and 

Fignre 12. Perspective view of zeolite A showing one of the low energy configurations for the distribution of twelve Na ions within the structure as determined using a Monte Carlo sampling approach. The label on each Na ion represents the size of the Si-Al ring stmctuie that the cation is associated with. 

Applications of molecular dynamics in mineralogy and geochemistry are often associated with the simulation of the structure and transport properties of fluids and melts due to the relatively rapid dynamics of the species in these systems. Melt and glass 

In contrast to the deterministic nature of the classical equations of the MD method, the Monte Carlo method makes use of a probabilistic picture of atomic motion. According to statistical mechanics, - an observable can be calculated from 

In MC methods the integrand in Eq. [66] is randomly sampled and the integral is approximated by a sum 

Importance sampling is central to Monte Carlo applications in statistical physics. In the subsections that follow, we describe the concepts of the schemes that are most frequently used in zeolite modeling. 

If At/ 0, generate a random number from a uniform distribution on the 

Steps 4 and 5 are often summarized by the expression Paccept = min[l, exp -AU/k T)]. Step 2 is usually carried out in such way that about 50% of the trial moves are accepted. The Metropolis method allows calculation of the structural and energetic properties of a system. 

Only when polymer chains form part of a crystal lattice can a precise structure be specified for the sample. In such a case the system can be defined by copying the contents of the unit cell along directions parallel to the lattice vectors. But polymers by their nature depart drastically from this ideal. Motions in noncrystalline media are subject to so many random or indeterminable factors that they cannot be treated by methods that require specifying a precise structure. Nevertheless we include a brief account of how randomness can be introduced into the simulation of polymer systems, since results have been shown to produce useful descriptions of noncrystalline polymer states. For more details about the methods, we refer the reader to books and specialist review articles and descriptions of polymer simulation codes.  

A random-number-based algorithm is then applied in a sequence of steps of a Markov chain, attempting at each step to change the conformation of one section of the chain. The algorithm cx)ntains conditions with which to test the feasibility of the new structure at each step. In the simplest test, the move is rejected if it has placed two atoms on the same site. Alternatively if the energy change associated with the move is 5U, calculated by using a force field as described in Section 1.3, then the move is accepted if the factor exceeds a random number drawn 

A static assembly of chains resulting from a selection of random walks through conformation space is taken to constitute a time-averaged structure. Since Monte Carlo methods are quite demanding on computing time, simulating a highly complex polymer system is beyond the power of current methods, and simulated chains have to be much shorter than those of a true polymer. Despite these caveats the method can still provide a useful description of a polymer sample and permit the evaluation of such structural parameters as the mean square end-to-end distance 

The MC method considers the configuration space of a model and generates a discrete-time random walk through configuration space following a master equation41,51 

Here x,x denote two configurations of the system (specified, for instance, by the set of coordinates of all atoms r or the position of one chain end for all chains and all bond lengths, bond angles, and torsion angles rf, If, 0a, where a = 1. M runs over all chains and the indices i,j, k run over all internal degrees of freedom of one chain). The transition rates W(x — x ) are chosen to fulfill the detailed balance condition 

Equation [19] ensures that the thermodynamic equilibrium distribution of Eq. [20] is the stationary (long-time) limit of the Markov chain generated by Eq. [18]. It does not specify the transition rates uniquely, however. Let us write them in the following way  

The two main sources for slow relaxation in polymers are entanglement effects and the glass transition. The first is entropic in origin, whereas the second—at least in chemically realistic polymer models—is primarily enthalpic. We write the largest relaxation time in the melt as 

In the double-bridging algorithm, an inner monomer of a chain attacks an inner monomer of another chain (or the same chain) and tries to form a 

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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. 

Mbiler-Krumbhaar H and Binder K 1973 Dynamic properties of the Monte-Carlo method in statistical mechanics J. Stat. Phys. 8 1-24... 

Binder K (ed) 1995 The Monte Carlo Method in Condensed Matter Physics vol 71 Topics in Applied Physics 2nd edn (Berlin Springer)... 

Newman M E J and Barkema G T 1999 Monte Carlo Methods in Statistical Physics (Oxford Clarendon)... 

Salsburg Z W, Jacobson J D, Fickett W and Wood W W 1959 Application of the Monte Carlo method to the lattice gas model. Two dimensional triangular lattice J. Chem. Phys. 30 65-72... 

McDonald I R and Singer K 1967 Calculation of thermodynamic properties of liquid argon from Lennard-Jones parameters by a Monte Carlo method Discuss. Faraday Soc. 43 40-9... 

Heermann D W and Burkitt A N 1995 Parallel algorithms for statistical physics problems The Monte Carlo Method In Condensed Matter Physios vol 71 Toplos In Applied Physios ed K Binder (Berlin Springer) pp 53-74... 

Bruce A D, Wilding N B and Ackland G J 1997 Free energies of crystalline solids a lattice-switch Monte-Carlo method Phys. Rev. Lett. 79 3002-5... 

Demidov A A 1999 Use of Monte-Carlo method in the problem of energy migration in molecular complexes Resonance Energy Transfer e6 D L Andrews and A A Demidov (New York Wiley) pp 435-65... 

Agranovich V M, Efremov N A and Kirsanov V V 1980 Computer simulation of kinetics of excitation bimolecular quenching by Monte-Carlo method Fiz. Tverd. Tela 22 2118-27... 

The basic scheme of this algorithm is similar to cell-to-cell mapping techniques [14] but differs substantially In one important aspect If applied to larger problems, a direct cell-to-cell approach quickly leads to tremendous computational effort. Only a proper exploitation of the multi-level structure of the subdivision algorithm (also for the eigenvalue problem) may allow for application to molecules of real chemical interest. But even this more sophisticated approach suffers from combinatorial explosion already for moderate size molecules. In a next stage of development [19] this restriction will be circumvented using certain hybrid Monte-Carlo methods. 

Ch. Schiitte, A. Fischer, W. Huisinga, P. Deuflhard. A Hybrid Monte-Carlo Method for Essential Molecular Dynamics. Preprint, Preprint SC 98-04, Konrad Zuse Zentrum, Berlin (1998)... 

Treatment of Multiple Ionizations by a Monte Carlo Method... 

Monte Carlo Methods for Pure Tsallis Statistics... 

Alternatively, one may use a phase space Monte Carlo method with uniform random trial moves and an acceptance probability... 

A similar algorithm has been used to sample the equilibrium distribution [p,(r )] in the conformational optimization of a tetrapeptide[5] and atomic clusters at low temperature.[6] It was found that when g > 1 the search of conformational space was greatly enhanced over standard Metropolis Monte Carlo methods. In this form, the velocity distribution can be thought to be Maxwellian. 

B. Mehlig, D. W. Heermann, and B. M. Forrest. Hybrid Monte Carlo method for condensed-matter systems. Phys. Rev. B, 45 679-685, 1992. 

In this chapter we shall discuss some of the general principles involved in the two most common simulation techniques used in molecular modelling the molecular dynamics and the Monte Carlo methods. We shall also discuss several concepts that are common to both of these methods. A more detailed discussion of the two simulation methods can be found in Chapters 7 and 8. 

Differences Between the Molecular Dynamics and Monte Carlo Methods... 

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