Statistical mechanical averages, dynamics

Statistical mechanical averages in a molecular dynamics run are obtained by simply averaging an energetic or structural value over time steps. Thus if the values (xj, i are being computed in a trajectory. the statistical mechanical average is jiist... 

A molecular dynamics (MD) calculation collects statistical information as it progresses. So, for example, if the calculated position vector of atom A at times ti,t2, -, tn is rA(f2), .., t-A(tn), then the statistical-mechanical average is... 

Osted, A., J. Kongsted, K. V. Mikkelson, P.-O. Astrand, and 0. Christiansen, Statistical mechanically averaged molecular properties of liquid water calculated using the combined coupled cluster/molecular dynamics method. Chem. Phys., 2006.124 124503-124516. 

Molecular level computer simulations based on molecular dynamics and Monte Carlo methods have become widely used techniques in the study and modeling of aqueous systems. These simulations of water involve a few hundred to a few thousand water molecules at liquid density. Because one can form statistical mechanical averages with arbitrary precision from the generated coordinates, it is possible to calculate an exact answer. The value of a given simulation depends on the potential functions contained in the Hamiltonian for the model. The potential describing the interaction between water molecules is thus an essential component of all molecular level models of aqueous systems. 

As reactants transfonn to products in a chemical reaction, reactant bonds are broken and refomied for the products. Different theoretical models are used to describe this process ranging from time-dependent classical or quantum dynamics [1,2], in which the motions of individual atoms are propagated, to models based on the postidates of statistical mechanics [3], The validity of the latter models depends on whether statistical mechanical treatments represent the actual nature of the atomic motions during the chemical reaction. Such a statistical mechanical description has been widely used in imimolecular kinetics [4] and appears to be an accurate model for many reactions. It is particularly instructive to discuss statistical models for unimolecular reactions, since the model may be fomuilated at the elementary microcanonical level and then averaged to obtain the canonical model. 

Beyond the clusters, to microscopically model a reaction in solution, we need to include a very big number of solvent molecules in the system to represent the bulk. The problem stems from the fact that it is computationally impossible, with our current capabilities, to locate the transition state structure of the reaction on the complete quantum mechanical potential energy hypersurface, if all the degrees of freedom are explicitly included. Moreover, the effect of thermal statistical averaging should be incorporated. Then, classical mechanical computer simulation techniques (Monte Carlo or Molecular Dynamics) appear to be the most suitable procedures to attack the above problems. In short, and applied to the computer simulation of chemical reactions in solution, the Monte Carlo [18-21] technique is a numerical method in the frame of the classical Statistical Mechanics, which allows to generate a set of system configurations... 

The theory of statistical mechanics provides the formalism to obtain observables as ensemble averages from the microscopic configurations generated by such a simulation. From both the MC and MD trajectories, ensemble averages can be formed as simple averages of the properties over the set of configurations. From the time-ordered properties of the MD trajectory, additional dynamic information can be calculated via the time correlation function formalism. An autocorrelation function Caa( = (a(r) a(t + r)) is the ensemble average of the product of some function a at time r and at a later time t + r. 

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. 

Kinetic Theory. In the kinetic theory and nonequilibrium statistical mechanics, fluid properties are associated with averages of pruperlies of microscopic entities. Density, for example, is the average number of molecules per unit volume, times the mass per molecule. While much of the molecular theory in fluid dynamics aims to interpret processes already adequately described by the continuum approach, additional properties and processes are presented. The distribution of molecular velocities (i.e., how many molecules have each particular velocity), time-dependent adjustments of internal molecular motions, and momentum and energy transfer processes at boundaries are examples. 

Surface tension is one of the most basic thermodynamic properties of the system, and its calculation has been used as a standard test for the accuracy of the intermolecular potential used in the simulation. It is defined as the derivative of the system s free energy with respect to the area of the interface[30]. It can be expressed using several different statistical mechanical ensemble averages[30], and thus we can use the molecular dynamics simulations to directly compute it. An example for such an expression is ... 

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