Ensembles for Molecular Simulation

The main idea of a lattice model is to assume that atomic or molecular entities constituting the system occupy well-defined lattice sites in space. This method is sometimes employed in simulations with the grand canonical ensemble for the simulation of surface electrochemical proceses. The Hamiltonians H of the lattice gas for one and two adsorbed species from which the ttansition probabilities 11 can be calculated have been discussed by Brown et al. (1999). We discuss in some detail MC lattice model simulations applied to the electrochemical double layer and electrochemical formation and growth two-dimensional phases not addressed in the latter review. MC lattice models have also been applied recently to the study the electrox-idation of CO on metals and alloys (Koper et al., 1999), but for reasons of space we do not discuss this topic here. 

Mitsutake, A. Sugita, Y. Okamoto, Y., Generalized-ensemble algorithms for molecular simulations of biopolymers, Biopolymers 2001, 60, 96-123... 

Another way to view MD simulation is as a technique to probe the atomic positions and momenta that are available to a molecular system under certain conditions. In other words, MD is a statistical mechanics method that can be used to obtain a set of configurations distributed according to a certain statistical ensemble. The natural ensemble for MD simulation is the microcanonical ensemble, where the total energy E, volume V, and amount of particles N (NVE) are constant. Modifications of the integration algorithm also allow for the sampling of other ensembles, such as the canonical ensemble (NVT) with constant temperature... 

In the last section we have assumed that we perform our simulation for a fixed number, N, of particles at constant temperature, T, and volume, V, the canonical ensemble. A major advantage of the Monte Carlo technique is that it can be easily adapted to the calculation of averages in other thermodynamic ensembles. Most real experiments are performed in the isobaric-isothermal (constant- ) ensemble, some in the grand-canonical (constant-pFT) ensemble, and even fewer in the canonical ensemble, the standard Monte Carlo ensemble, and near to none in the microcanonical (constant-NFE) ensemble, the standard ensemble for molecular-dynamics simulations. 

There are also many variations of the LJ 12-6 potential. One example is the computationally inexpensive tmncated and shifted Lennard-Jones potential (TSLJ), which is commonly used for molecular simulation studies in which large molecular ensembles are regarded, e.g., for investigating condensation processes [15, 16]. Another version of the LJ potential is the Kihara potential [17], which is a non-spherical generalization of the LJ model. 

To further illustrate how extended ensembles can be designed to conduct MD simulations under various macroscopic constraints, we discuss here the NTLxPyycr ensemble. NTL yyOzz is an appropriate statistical ensemble for the simulation of uniaxial tension experiments on solid polymers [12] or relaxation experiments in uniaxially oriented polymer melts [13]. This ensemble is illustrated in Fig. 2. The quantities that are kept constant during a molecular simulation in this ensemble are the following ... 

Equations (2) and (3) relate intermolecular interactions to measurable solution thermodynamic properties. Several features of these two relations are worth noting. The first is the test-particle method, an implementation of the potential distribution theorem now widely used in molecular simulations (Frenkel and Smit, 1996). In the test-particle method, the excess chemical potential of a solute is evaluated by generating an ensemble of microscopic configurations for the solvent molecules alone. The solute is then superposed onto each configuration and the solute-solvent interaction potential energy calculated to give the probability distribution, Po(AU/kT), illustrated in Figure 3. The excess... 

There are two important consequences of this equality for computer simulations of many-body systems. First, it means that statistically averaged properties of these systems are accessible from simulations that are aimed at generating trajectories -e.g., molecular dynamics, or ensemble averages such as Monte Carlo. Furthermore, for sufficiently long trajectories, the time-averaged properties become independent of the initial conditions. Stated differently, it means that for almost all values of qo, Po, the system will pass arbitrarily close to any point x, p, in phase space at some later time. 

One of the most powerful tools molecular simulation affords is that of measuring distribution functions and sampling probabilities. That is, we can easily measure the frequencies with which various macroscopic states of a system are visited at a given set of conditions - e.g., composition, temperature, density. We may, for example, be interested in the distribution of densities sampled by a liquid at fixed pressure or that of the end-to-end distance explored by a long polymer chain. Such investigations are concerned with fluctuations in the thermodynamic ensemble of interest, and are fundamentally connected with the underlying statistical-mechanical properties of a system. 

There are several commercial packages that realise the above strategy for molecularly realistic systems. It is useful to discuss some of the limitations. Ideally, one would like to do simulations on macroscopic systems. However, it is impossible to use a computer to deal with numbers of degrees of freedom on the order of /Vav. In lipid systems, where the computations of all the interactions in the system are expensive, a typical system can contain of the order of tens of thousands of particles. Recently, massive systems with up to a million particles have been considered [33], Even for these large simulations, this still means that the system size is limited to the order of 10 nm. Because of this small size, one refers to this volume as a box, although the system boundaries are typically not box-like. Usually the box has periodic boundary conditions. This implies that molecules that move out of the box on one side will enter the box on the opposite side. In such a way, finite size effects are minimised. In sophisticated simulations, i.e. (N, p, y, Tj-ensembles, there are rules defined which allow the box size and shape to vary in such a way that the intensive parameters (p, y) can assume a preset value. 

Once the boundary conditions have been implemented, the calculation of solution molecular dynamics proceeds in essentially the same manner as do vacuum calculations. While the total energy and volume in a microcanonical ensemble calculation remain constant, the temperature and pressure need not remain fixed. A variant of the periodic boundary condition calculation method keeps the system pressure constant by adjusting the box length of the primary box at each step by the amount necessary to keep the pressure calculated from the system second virial at a fixed value (46). Such a procedure may be necessary in simulations of processes which involve large volume changes or fluctuations. Techniques are also available, by coupling the system to a Brownian heat bath, for performing simulations directly in the canonical, or constant T,N, and V, ensemble (2,46). 

Principal component analysis (PCA) is another tool that has been used extensively to analyze molecular simulations. The technique, which attempts to extract the large-scale characteristic motions from a structural ensemble, was first applied to biomolecular simulations by Garcia [28], although an analogous technique was used by Levy et al. [30]. The first step is the construction of the 3N x 3N (for an N-atom system) fluctuation correlation matrix... 

We will delay a more detailed discussion of ensemble thermodynamics until Chapter 10 indeed, in this chapter we will make use of ensembles designed to render the operative equations as transparent as possible without much discussion of extensions to other ensembles. The point to be re-emphasized here is that the vast majority of experimental techniques measure molecular properties as averages - either time averages or ensemble averages or, most typically, both. Thus, we seek computational techniques capable of accurately reproducing these aspects of molecular behavior. In this chapter, we will consider Monte Carlo (MC) and molecular dynamics (MD) techniques for the simulation of real systems. Prior to discussing the details of computational algorithms, however, we need to briefly review some basic concepts from statistical mechanics. 

A series of articles were published by Ennari et al. on MD simulation of transport processes in Poly(Ethylene Oxide) and sulfonic acid-based polymer electrolyte.136,137 The work was started by the determination of the parameters for the ions missing from the PCFF forcefield made by MSI (Molecular Simulations Inc.), to create a new forcefield, NJPCFF. In the models, the proton is represented as a hard ball with a positive charge. Zhou et al. used the similar approach to model Nation.138 The repeating unit of Nafion (Fig. 17) was optimized using ab initio VAMP scheme. The protons were modeled with hydronium ions. Three unit cell or molecular models were used for the MD simulation. The unit cell contains 5000 atoms 20 pendent side chains, and branched Nafion backbone created with the repeating unit. Their water uptakes or water contents were 3, 13, or 22 IEO/SO3, which correspond to the room temperature water uptakes at 50% relative humidity (RH), at 100% RH, and in liquid water respectively.18 The temperature was initially set at a value between 298.15 and 423.15 K under NVE ensemble with constant particle number, constant volume (1 bar), and constant energy. 

The most obvious way to achieve a theoretical description of molecules in the liquid state is the path from single molecules, through small and large clusters of molecules, to very large ensembles of molecules. Even at times when realistic quantum chemical simulations were impossible for molecular sizes of... 

---