Metropolis Monte-Carlo

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. 

The main difference between the force-bias and the smart Monte Carlo methods is that the latter does not impose any limit on the displacement that m atom may undergo. The displacement in the force-bias method is limited to a cube of the appropriate size centred on the atom. However, in practice the two methods are very similar and there is often little to choose between them. In suitable cases they can be much more efficient at covering phase space and are better able to avoid bottlenecks in phase space than the conventional Metropolis Monte Carlo algorithm. The methods significantly enhance the acceptance rate of trial moves, thereby enabling Icirger moves to be made as well as simultaneous moves of more than one particle. However, the need to calculate the forces makes the methods much more elaborate, and comparable in complexity to molecular dynamics. 

By far the most common methods of studying aqueous interfaces by simulations are the Metropolis Monte Carlo (MC) technique and the classical molecular dynamics (MD) techniques. They will not be described here in detail, because several excellent textbooks and proceedings volumes (e.g., [2-8]) on the subject are available. In brief, the stochastic MC technique generates microscopic configurations of the system in the canonical (NYT) ensemble the deterministic MD method solves Newton s equations of motion and generates a time-correlated sequence of configurations in the microcanonical (NVE) ensemble. Structural and thermodynamic properties are accessible by both methods the MD method provides additional information about the microscopic dynamics of the system. 

We can now take one of two approaches (1) construct a probabilistic CA along lines with the Metropolis Monte Carlo algorithm outlined above (see section 7.1.3.1), or (2) define a deterministic but reversible rule consistent with the microcanonical prescription. As we shall immediately see, however, neither approach yields the expected results. 

Figure 3.2 Cyclic voltammograms for H adsorption on Pt(lll) and Pt(lOO). Two different methods have been applied. In (a) and (b), the H particles were assumed not to interact in the expression for the configurational entropy. In (c) and (d), the more elaborate model involving Metropolis Monte Carlo was applied. As can be seen, for these homogenous surfaces, the simple method suffices. The figure is adopted from [Karlberg et al., 2007a], where the full details of the calculations can also be found.

Miller, M.A. Amon, L.M. Reinhardt, W.P., Should one adjust the maximum step size in a Metropolis Monte Carlo simulation Chem. Phys. Lett. 2000, 331, 278-284... 

Instead of using MD, the X variables may also be sampled stochastically. In the hybrid CMC/MD approach, Metropolis Monte Carlo69 is used to evolve the X variables and molecular dynamics is used to evolve the atomic coordinates. The Metropolis Monte Carlo criteria leads to the generation of a canonical ensemble of the ligands when the following transition probability is used... 

Metropolis Monte Carlo (MMC) method, 26 1035-1036 Met-spar, 4 577 Metsulfuron-methyl, 13 322 Mettler dropping point, 10 827 Mevacor, 5 142... 

Fig. 7. The tunneling paths in a double minimum potential like that of Fig. 5 may be classified as having one or more (odd) numbers of instantons, or tunneling segments. Three such traverses of the classically forbidden region are shown above. The classification of all paths according to the number N of instantons is the basis for evaluating the path integral as the sum in Eq. 29 note however that the therein is constructed to include non-harmonic (beyond semiclassical) fluctuations around the minimum action instanton paths, which are evaluable by Metropolis Monte Carlo...

Metropolis Monte Carlo method sorption on zeolites, 42 62, 66 MgO... 

As the appropriate Boltzmann weights are included in the Metropolis Monte Carlo sampling technique, the average value of the polarizability, or any other property calculated from the MC data, is given as a simple average over all the values calculated for each configuration. 

Podtelezhnikov, A.A., Wild, D.L. Exhaustive Metropolis Monte Carlo sampling and analysis of polyalanine conformations adopted under the influence of hydrogen bonds. Protein. Struct. Funct. Genet. 2005, 61, 94—104. 

A more interesting problem is that the Metropolis Monte Carlo studies used a different (physically simplified) kinetic rate law for atomic motion than the KMC work. That is, the rules governing the rate at which atoms jump from one configuration to the next were fundamentally different. This can have serious implications for such dynamic phenomena as step fluctuations, adatom mobility, etc. In this paper, we describe the physical differences between the rate laws used in the previous work, and then present results using just one of the simulation techniques, namely KMC, but comparing both kinds of rate laws. 

In Zeolite A. An extensive series of papers concerned with the sorption location and isotherms of Xe in zeolite A have been published (118-122). The locations of sorbates and their structures were investigated by using Metropolis Monte Carlo simulations of zeolite A models (118, 119). Initially, an idealized truncated cuboctahedron was used, with Si and Al atoms occupying vertices and O atoms occupying the midpoints of line segments (118). Subsequent calculations were based on the positions of atoms in... 

Yashonath etal. (46) used a Metropolis Monte Carlo method to simulate the infinite-dilution adsorption of methane in NaY zeolite. The lattice had a Si/Al ratio of 3.0 and was treated as rigid, whereas methane was modeled... 

In Silicalite. A variety of papers are concerned with sorption of methane in the all-silica pentasil, silicalite. June et al. (87) used a Metropolis Monte Carlo method and MC integration of configuration integrals to determine low-occupancy sorption information for methane. The predicted heat of adsorption (18 kJ/mol) is within the range of experimental values (18-21 kJ/ mol) (145-150), as is the Henry s law coefficient as a function of temperature (141, 142). Furthermore, the center of mass distribution for methane in silicalite at 400 K shows that the molecule is delocalized over most of the total pore volume (Fig. 9). Even in the case of such a small sorbate, the channel intersections are unfavorable locations. 

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