Statistical Treatment of Interacting Systems

At the moment there exist no quantum chemical method which simultaneously satisfies all demands of chemists. Some special demands with respect to treatment of macromolecular systems are, the inclusion of as many as possible electrons of various atoms, the fast optimization of geometry of large molecules, and the high reliability of all data obtained. To overcome the point 4 of the disadvantages, it is necessary to include the interaction of the molecule with its surroundings by means of statistical thermodynamical calculations and to consider solvent influence. 

Furthermore, applications in the case of neutral atoms are more difficult because of the lack of electrostatic moments in the atom for describing the interaction with the environment. A proper treatment of liquid systems should consider its statistical nature [6, 7] as there are many possible geometrical arrangements accessible to the system at nonzero temperature. Thus, liquid properties are best described by a statistical distribution [8-11], and all properties are obtained from statistical averaging over ensembles. Thus, in this direction, it is important to use statistical mechanics, with some sort of computer simulation of liquids [6, 7], combined with quantum mechanics to obtain the electronic property of interest. 

The results of statistical treatment of experimental data, suggesting the additivity of the properties with high coefficient of correlation, can conceal the true state of interactions between components at the phase boundary. The function =f(v) can then be convenient for a more detailed characterization of the system. 

In the RISM-SCF theory, the statistical solvent distribution around the solute is determined by the electronic structure of the solute, whereas the electronic strucmre of the solute is influenced by the surrounding solvent distribution. Therefore, the ab initio MO calculation and the RISM equation must be solved in a self-consistent manner. It is noted that SCF (self-consistent field) applies not only to the electronic structure calculation but to the whole system, e.g., a self-consistent treatment of electronic structure and solvent distribution. The MO part of the method can be readily extended to the more sophisticated levels beyond Hartree-Fock (HF), such as configuration interaction (Cl) and coupled cluster (CC). 

The lattice gas has been used as a model for a variety of physical and chemical systems. Its application to simple mixtures is routinely treated in textbooks on statistical mechanics, so it is natural to use it as a starting point for the modeling of liquid-liquid interfaces. In the simplest case the system contains two kinds of solvent particles that occupy positions on a lattice, and with an appropriate choice of the interaction parameters it separates into two phases. This simple version is mainly of didactical value [1], since molecular dynamics allows the study of much more realistic models of the interface between two pure liquids [2,3]. However, even with the fastest computers available today, molecular dynamics is limited to comparatively small ensembles, too small to contain more than a few ions, so that the space-charge regions cannot be included. In contrast, Monte Carlo simulations for the lattice gas can be performed with 10 to 10 particles, so that modeling of the space charge poses no problem. In addition, analytical methods such as the quasichemical approximation allow the treatment of infinite ensembles. 

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. 

Working with the density operator is a convenient alternative to using wavefunctions when dealing with a few-atom, isolated molecular system, insofar it suggests more efficient computational procedures or more consistent approximations, but it is not stricktly needed. The density operator is however essential in treatments of a many-atom system, when this interacts with a medium which constrains thermodynamical properties such as temperature or pressure, because the density operator incorporates statistical averages which would not be included in a treatment based on wavefunctions. 

A system of an ionic amphiphile contains in its simplest form three entities the solvent water, an amphiphilic ion, and a hydrophilic counterion. The properties of the total system can then be understood as the effects of the mutual interactions between these three species. A comprehensive treatment of all the interactions on a molecular level is not at present feasible. However, the development of methods for determining intermolecular potentials and for making statistical mechanical simulations292-294 should change this in the not too distant future. 

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