Statistical Molecular Model

The aim of statistical approach is to explain the macroscopic behavior of matter in terms of the behavior of the constituent molecules, that is, in terms of motions and interactions of a large number of particles. 

The motion of molecules can be expressed using Newton s second law as 

To obtain the actual dynamics of the problem, we have to solve 6N number of unknowns (Xj, Ci) using first-order differential equations and 6A initial conditions, that is, 

The one difficulty is to supply the initial data and Cf The other issue is that the detailed information may be unnecessary as we are interested in the average information, that is, the pressure exerted on a wall by a gas at a given density and temperature. 

In statistical approach, instead of having a definite position and velocity of a particle, we compute the probability of finding a molecule at a particular position and state. Once we solve the conservation equation for probability distribution, we can compute important statistical averaged quantities, that is, momentum and energy of the molecules. 

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One of the most popular and detailed statistical molecular models for the diffusion of simple penetrants in amorphous rubbery polymers was proposed in refs. (45,46). This model is based on features taken from those presented briefly above. Because it is frequently cited in the literature, it will be presented here in some detail. 

Molecules are usually represented as 2D formulas or 3D molecular models. WhOe the 3D coordinates of atoms in a molecule are sufficient to describe the spatial arrangement of atoms, they exhibit two major disadvantages as molecular descriptors they depend on the size of a molecule and they do not describe additional properties (e.g., atomic properties). The first feature is most important for computational analysis of data. Even a simple statistical function, e.g., a correlation, requires the information to be represented in equally sized vectors of a fixed dimension. The solution to this problem is a mathematical transformation of the Cartesian coordinates of a molecule into a vector of fixed length. The second point can... 

We shall rely heavily on models again in this chapter this time they are of two different types. We shall consider elasticity in terms of a molecular model in which the chains are described by random flight statistics. The phenomena of... 

It is not particularly difficult to introduce thermodynamic concepts into a discussion of elasticity. We shall not explore all of the implications of this development, but shall proceed only to the point of establishing the connection between elasticity and entropy. Then we shall go from phenomenological thermodynamics to statistical thermodynamics in pursuit of a molecular model to describe the elastic response of cross-linked networks. 

By combining random flight statistics from Chap. 1 with the statistical definition of entropy from the last section, we shall be able to develop a molecular model for the stress-strain relationship in a cross-linked network. It turns out to be more convenient to work with the ratio of stretched to unstretched lengths L/Lq than with y itself. Note the relationship between these variables ... 

The drawback of the statistical approach is that it depends on a model, and models are bound to oversimplify. Nevertheless, we can learn a great deal from the attempt to evaluate thermodynamic properties from molecular models, even if the effort falls short of quantitative success. 

Molecular modeling has evolved as a synthesis of techniques from a number of disciplines—organic chemistry, medicinal chemistry, physical chemistry, chemical physics, computer science, mathematics, and statistics. With the development of quantum mechanics (1,2) ia the early 1900s, the laws of physics necessary to relate molecular electronic stmcture to observable properties were defined. In a confluence of related developments, engineering and the national defense both played roles ia the development of computing machinery itself ia the United States (3). This evolution had a direct impact on computing ia chemistry, as the newly developed devices could be appHed to problems ia chemistry, permitting solutions to problems previously considered intractable. 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

At present, conformational searches provide for the most important application of computer molecular modeling in biology. In contrast, in statistical physics, from which MC and MD methods were originally borrowed, they are primarily used for studying... 

Tobolsky, A. V. and DuPre, D. B. Macro molecular Relaxation in the Damped Torsional Oscillator and Statistical Segment Models. VoL 6, pp. 103-127. 

The molecular models of rubber elasticity relate chain statistics and chain deformation to the deformation of the macroscopic material. The thermodynamic changes, including stress are derived from chain deformation. In this sense, the measurement of geometric changes is fundamental to the theory, constitutes a direct check of the model, and is an unambiguous measure of the mutual consistency of theory and experiment. 

The molecular modelling of systems consisting of many molecules is the field of statistical mechanics, sometimes called statistical thermodynamics [28,29], Basically, the idea is to go from a molecular model to partition functions, and then, from these, to predict thermodynamic observables and dynamic and structural quantities. As in classical thermodynamics, in statistical mechanics it is essential to define which state variables are fixed and which quantities are allowed to fluctuate, i.e. it is essential to specify the macroscopic boundary conditions. In the present context, there are a few types of molecular systems of interest, which are linked to so-called ensembles. 

Linusson A. Elofsson M. Andersson I.E. Dahlgren M.K. Statistical molecular design of balanced compound libraries for QSAR modeling. Current Medicinal Chemistry, 2010, 17 (19), 2001-2016. 

Self-consistent approaches in molecular modeling have to strike a balance of appropriate representation of the primary polymer chemistry, adequate treatment of molecular interactions, sufficient system size, and sufficient statistical sampling of structural configurations or elementary transport processes. They should account for nanoscale confinement and random network morphology and they should allow calculating thermodynamic properties and transport parameters. 

As discussed in Section 6.5.3, coarse-grained molecular modeling approaches offer the most viable route to the molecular modeling of hydrated ionomer membranes. The coarse-grained treatment implies simplification in interactions, which can be systematically improved with advanced forcematching procedures, but allows simulations of systems with sufficient size and sufficient statistical sampling. Structural correlations, thermodynamic properties, and transport parameters can be studied. 

Using the pitch, symmetry, monomer geometries and other stereochemical constraints, a number of types of molecular model can be constructed. Typical dilemmas are whether the molecular helix is left- or right-handed, whether the molecule is a single helix or coaxial double-helix (and in the later case whether the two chains in the duplex are parallel or antiparallel), or whether, if there are two or more molecules in the unit cell, the molecules are parallel or antiparallel. Solution of a structure therefore involves refinement and adjudication All candidate models are refined until the fit with the measured x-ray amplitudes or steric factors allows one model to be declared significantly superior to the others by some standard statistical test. 

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