Microscopic-macroscopic Connection

In the very recent past, it has become possible under certain circumstances to observe single molecules in the laboratory. Nevertheless, the vast majority of chemical research concerns itself not with individual molecules, but instead witli macroscopic quantities of matter that are made up of unimaginably large numbers of molecules. The behavior of such ensembles of molecules is governed by the empirically determined laws of thermodynamics, and most chemical reactions and many chemical properties are defined in tenns of some of tlie fundamental variables of thermodynamics, such as enthalpy, entropy, free energy, and others. 

In this chapter, the most common procedures for augmenting electronic-structure calculations in order to convert single-molecule potential energies to ensemble thermodynamic variables will be detailed, and key potential ambiguities and pitfalls described. Within the context of certain assumptions, this connection can be established in a rigorous way. 

Note that the situation is less clear-cut for molecular mechanics calculations. As already discussed in Chapter 2, the strain energy from a typical MM calculation must be thought 

Essentials of Computational Chemistry. 2nd Edition Christopher J. Cramer 

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The Institute presented the subject of the liquid state and its electrical properties systematically. The treatment, both theoretical and experimental, started from the microscopic, quantum-statistical foundations of the liquid state and progressed to the macroscopic description of conduction in liquids, and the various applications that arise because of the electrical properties of liquids. Emphasis was placed on fundamental principles and physical points of view, and on achieving a microscopic-macroscopic connection in the description of the liquid state and its electrical properties. 

Fig. 1. The microscopic enlanglemenl slruciure, e.g, at an interface or in the bulk, is related to the measured macroscopic fracture energy G, via the VP theory of breaking connectivity in the embedded plastic zone (EPZ) at the crack tip. The VP theory determines

A microscopic description characterizes the structure of the pores. The objective of a pore-structure analysis is to provide a description that relates to the macroscopic or bulk flow properties. The major bulk properties that need to be correlated with pore description or characterization are the four basic parameters porosity, permeability, tortuosity and connectivity. In studying different samples of the same medium, it becomes apparent that the number of pore sizes, shapes, orientations and interconnections are enormous. Due to this complexity, pore-structure description is most often a statistical distribution of apparent pore sizes. This distribution is apparent because to convert measurements to pore sizes one must resort to models that provide average or model pore sizes. A common approach to defining a characteristic pore size distribution is to model the porous medium as a bundle of straight cylindrical or rectangular capillaries (refer to Figure 2). The diameters of the model capillaries are defined on the basis of a convenient distribution function. 

The linkage of microscopic and macroscopic properties is not without challenges, both theoretical and experimental. Statistical mechanics and thermodynamics provide the connection between molecular properties and the behavior of macroscopic matter. Coupled with statistical mechanics, computer simulation of the structure, properties, and dynamics of mesoscale models is now feasible and can handle the increase in length and time scales. 

In physical chemistry the most important application of the probability arguments developed above is in the area of statistical mechanics, and in particular, in statistical thermodynamics. This subject supplies the basic connection between a microscopic model of a system and its macroscopic description. The latter point of view is of course based on the results of experimental measurements (necessarily carried out in each experiment on a very large number of particle ) which provide the basis of classical thermodynamics. With the aid of a simple example, an effort now be made to establish a connection between the microscopic and macroscopic points of view. 

Corey has discussed the similarities and differences between synthetic and biological catalysts, coined the term chemzymes for the former, and discussed the connections between these microscopic catalysts and macroscopic robots. See Corey, E J. New Enantiose-lective Routes to Biologically Interesting Compounds Pure Appl. Chem 1990, 62,1209-1216. 

The most simple diblock copolymers are linear chains, in which one part of the chain consists of one type of monomer, say polystyrene (PS), and the other one of another type, say polybutadiene (PB), as illustrated in Figure 14. PS and PB usually phase separate at low temperatures however, because of their chemical connectivity, block copolymers cannot unmix on a macroscopic scale. They can only phase separate on a microscopic scale, the size of which is determined by the length of the polymers. 

A connection between microscopic quantities such as state-to-state cross sections, state-to-state rate constant and macroscopic quantities such as overall rate constant k(T) is summarized in Flow Chart 1. 

This relation provides a simple closure to the iGLE in which the microscopic dynamics is connected to the macroscopic behavior. Because of this closure, the microscopic dynamics are said to depend self-consistently on the macroscopic (averaged) trajectory. Formally, this construction is well-defined in the sense that if the true (R(t)) is known a priori, then the system of equations return to that of the iGLE with a known g(t). In practice, the simulations are performed either by iteration of (R(t)) in which a new trajectory is calculated at each step and (R(t)) is revised for the next step, propagation of a large number of trajectories with (R(t)) calculated on-the-fly, or some combination thereof. 

This alternative representation is entirely arbitrary of course, but its use is well established in the literature. It offers some advantage in discussing the connections between the macroscopic and microscopic descriptions of viscoelasticity. The parameters, t]0, J%, and G° can be expressed in terms of the moments of H(x) ... 

Time-dependent correlation functions are now widely used to provide concise statements of the miscroscopic meaning of a variety of experimental results. These connections between microscopically defined time-dependent correlation functions and macroscopic experiments are usually expressed through spectral densities, which are the Fourier transforms of correlation functions. For example, transport coefficients1 of electrical conductivity, diffusion, viscosity, and heat conductivity can be written as spectral densities of appropriate correlation functions. Likewise, spectral line shapes in absorption, Raman light scattering, neutron scattering, and nuclear jmagnetic resonance are related to appropriate microscopic spectral densities.2... 

It connects the macroscopic constant D with the microscopic jumps of the particle. 

In general, a particle migrates in a material by a series of thermally activated jumps between positions of local energy minima. Macroscopic diffusion is the result of all the migrations executed by a large ensemble of particles. The spread of the ensemble due to these migrations connects the macroscopic diffusivity to the microscopic particle jumping. 

The driving forces necessary to induce macroscopic fluxes were introduced in Chapter 3 and their connection to microscopic random walks and activated processes was discussed in Chapter 7. However, for diffusion to occur, it is necessary that kinetic mechanisms be available to permit atomic transitions between adjacent locations. These mechanisms are material-dependent. In this chapter, diffusion mechanisms in metallic and ionic crystals are addressed. In crystals that are free of line and planar defects, diffusion mechanisms often involve a point defect, which may be charged in the case of ionic crystals and will interact with electric fields. Additional diffusion mechanisms that occur in crystals with dislocations, free surfaces, and grain boundaries are treated in Chapter 9. 

The connection between the microscopic description of any system in terms of individual states and its macroscopic thermodynamical behavior was provided by Boltzmann through statistical mechanics. The key connection is that the entropy of a system is proportional to the natural logarithm of the number of levels available to the system, thus ... 

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