MCs+ method

The Boltzmaim weight appears implicitly in the way the states are chosen. The fomi of the above equation is like a time average as calculated in MD. The MC method involves designing a stochastic algorithm for stepping from one state of the system to the next, generating a trajectory. This will take the fomi of a Markov chain, specified by transition probabilities which are independent of the prior history of the system. 

For a multicomponent system, it is possible to simulate at constant pressure rather than constant volume, as separation into phases of different compositions is still allowed. The method allows one to study straightforwardly phase equilibria in confined systems such as pores [166]. Configuration-biased MC methods can be used in combination with the Gibbs ensemble. An impressive demonstration of this has been the detennination by Siepmaim et al [167] and Smit et al [168] of liquid-vapour coexistence curves for n-alkane chain molecules as long as 48 atoms. 

We have implemented the generalized Monte Carlo algorithm using a hybrid MD/MC method composed of the following steps. 

Monte Carlo (MC) techniques for molecular simulations have a long and rich history, and have been used to a great extent in studying the chemical physics of polymers. The majority of molecular modeling studies today do not involve the use of MC methods however, the sampling capabiUty provided by MC methods has gained some popularity among computational chemists as a result of various studies (95—97). Relevant concepts of MC are summarized herein. 

As stated above, MC simulations are popular in many diverse fields. Their popularity is due mainly to their ease of use and their good convergence properties. Nonetheless, straightforward and application of MC methods to biomolecules is often problematic due... 

Among the brilliant mathematicians who developed the minutiae of the MC method, major disputes broke out concerning basic issues, particularly the question whether any (determinate) computer-based method is in principle capable of... 

The MC method can be implemented by a modification of the classic Metropolis scheme [25,67]. The Markov chain is generated by a three-step sequence. The first step is identical to the classic Metropolis algorithm a randomly selected molecule i is displaced within a small cube of side length 26r centered on its original position... 

While static Monte Carlo methods generate a sequence of statistically independent configurations, dynamic MC methods are always based on some stochastic Markov process, where subsequent configurations X of the system are generated from the previous configuration X —X —X" — > with some transition probability IF(X —> X ). Since to a large extent the choice of the basic move X —X is arbitrary, various methods differ in the choice of the basic unit of motion . Also, the choice of transition probability IF(X — > X ) is not unique the only requirement is that the principle... 

Most of the simulational results described in this chapter have been obtained by using a very efficient dynamic MC method [61,27], so below we shall describe it in more detail. As mentioned earlier, one is not aiming at the explanation of the properties of particular polymers (such as... 

Both MD and MC methods employ a temperature as a guiding parameter for generating new geometries. At sufficiently high temperature and long run time, all the... 

Prediction of Liquid Solubility with Molecular Dynamics (MD) and Monte Carlo (MC) Methods... 

As a first attempt to modify the code to be able to run simulations on SiH4-H2 discharges, a hybrid PlC/MC-fluid code was developed [264, 265]. It turned out in the simulations of the silane-hydrogen discharge that the PIC/MC method is computationally too expensive to allow for extensive parameter scans. The hybrid code combines the PIC/MC method and the fluid method. The electrons in the discharge were handled by the fluid method, and the ions by the PIC/MC method. In this way a large gain in computational effort is achieved, whereas kinetic information of the ions is still obtained. 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

For one specific set of discharge parameters, in a comparison between the hybrid approach and a full PIC/MC method, the spectra and the ion densities of the hybrid model showed some deviations from those of the full particle simulation. Nevertheless, due to its computational advantages, the hybrid model is appropri-... 

Most liquid phase molecular simulations with explicit atomic polarizabilities are performed with MD rather than MC techniques. This is due to the fact that, despite its general computational simplicity, MC with explicit polarization [173, 174] requires that Eq. (9-21) be solved every MC step, when even one molecule in the system is moved, and the number of configurations in an average Monte Carlo computation is orders of magnitude greater than in a MD simulation. For nonpolarizable, pairwise-additive models, MC methods can be efficient because only the... 

Applications of the fluctuating charge model have relied on iterative methods to determine the converged charges [52, 159, 164, 196] and for very large-scale systems, multilevel methods have also been developed [197, 198], MC methods have also been used with fluctuating-charge models [116, 194],... 

An overreaching theme of the present chapter, besides broken ergodicity, has to do with the fact that most of the enhanced sampling methods that we shall discuss address situations in which one cannot clearly identify a reaction coordinate that can be conveniently used to describe the kinetic evolution of the system of interest. While methods for enhanced sampling are designed to yield accurate results faster than regular molecular dynamics or Monte Carlo (MC) methods, it is our belief that there is no perfect method, but that, rather, there are methods that perform better for particular applications. Moreover, it should be noted that, while in instances when a proper reaction coordinate can be identified methods described in other chapters are probably more efficient, they could still benefit by sampling in conformational directions perpendicular to the reaction coordinate. 

Other variations on these basic free energy methods have been published, although for various reasons they have not yet been widely adopted. These methods include MD/MC methods,38 the acceptance ratio method,39, 40 the weighted histogram method,41 the particle insertion method,42 43 and the energy distribution method.39 The reader is referred to the original publications for additional discussion of these approaches. 

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... 

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. 

In MC methods the ultimate objective is to evaluate macroscopic properties from information about molecular positions generated over phase space. To evaluate average macroscopic properties, p, in the canonical ensemble from statistical mechanics, the following expression is used ... 

Molecular Dynamics Methods. In contrast to the MC method, both kinetic and structural properties of a molecular system can be evaluated from MD studies. These properties are evaluated as averages over configurations generated during time. In microcanonical ensemble studies with the MD method, the properties which are controlled... 

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