Molecular Ornstein-Zernike

As is briefly described in the Introduction, an exact equation referred to as the Ornstein-Zernike equation, which relates h(r, r ) with another correlation function called the direct correlation function c(r, r/), can be derived from the grand canonical partition function by means of the functional derivatives. Our theory to describe the molecular recognition starts from the Ornstein-Zernike equation generalized to a solution of polyatomic molecules, or the molecular Ornstein-Zernike (MOZ) equation [12],... 

In principle, the three-dimensional RISM (3D-RISM) theory described in Chapter 4 is significantly more accurate than the RISM theory employed so far which can be distinguished from the 3D-RISM theory by calling it the one-dimensional RISM (ID-RISM) theory. This is because the 3D-RISM theory, in contrast to the ID-RISM theory, takes orientational average of the molecular Ornstein-Zernike (OZ) equation for solvent molecules only, keeping full description of the shape and orientation of the solute molecule. In reality, a solvent site cannot access to a completely buried atom in the solute molecule. Even if is... 

Richardi J, Fries PH, Fischer R, Rast S, Krienke H Liquid acetone and chloroform a comparison between Monte Carlo simulation, molecular Ornstein-Zernike theory, and site-site Omstein-Zemike theory. Mol Phys 93(6) 925—938, 1998. 

Richardi J, MiUot C, Fries PH A molecular Ornstein-Zernike study of popular models for water and methanol, f Chem Phys 110(2) 1138—1147, 1999. 

From the preceding sections, it seems evident that a real description of ion specificities in solutions can only be done if the geometry and the properties of water molecules are explicitly taken into account. Such models are called non-primitive or Born-Oppenheimer models. In the 1970s and 1980s, they were developed in two different directions. In particular, integral equation theories, such as the hypernetted chain (HNC) approach, were extended to include angle-dependent interaction potentials. The site-site Ornstein-Zernike equation with a HNC-like closure and the molecular Ornstein-Zernike equation are examples. For more information, see Ref. 17. 

A PCF is important not only as a mathematical expression but also as a measurable quantity in scattering or diffraction experiments. It is also possible to obtain a PCF by simply using computers. By employing molecular simulations such as Monte Carlo or molecular dynamics, a PCF can be calculated directly. Now, we have another way to obtain a PCF. The central equation of this strategy is the Ornstein-Zernike (OZ) equation given below [1,2]... 

The PRISM (Polymer-Reference-Interaction-Site model) theory is an extension of the Ornstein-Zernike equation to molecular systems [20-22]. It connects the total correlation function h(r)=g(r) 1, where g(r) is the pair correlation function, with the direct correlation function c(r) and intramolecular correlation functions (co r)). For a primitive model of a polyelectrolyte solution with polymer chains and counterions only, there are three different relevant correlation functions the monomer-monomer, the counterion-counterion, and the monomer-counterion correlation function [23, 24]. Neglecting chain end effects and considering all monomers as equivalent, we obtain the following three PRISM equations for a homogeneous and isotropic system in Fourier space ... 

This point of interest is brought forward by the RISM approach to the structure of molecular liquids, and a RISM model with HNC closure supports a similar result for the excess chemical potential in terms of atom-atom correlations (Singer and Chandler, 1985 Hirata, 1998). RISM - reference interaction site model - is an acronym that refers to a class of theories for the joint two-atom distributions in molecular liquids. The most basic decision of RISM models is that theories of molecular liquids should focus first on the atom-atom distributions extracted from X-ray and neutron scattering data rather than more complex possibilities this highly practical point was not so obvious in an earlier epoch when models of molecular liquids were scarcely realistic on an atomic scale. That basic decision was encapsulated by invention of a site-site (or atom-atom) Ornstein-Zernike (SSOZ) (Cummings and Stell, 1982) equation that involved intramolecular atom-atom correlations. The original suggestions (Chandler and Andersen, 1972) were sufficiently successful as to support subsequent flamboyant developments, and to be substantially impervious to more fundamental improvements (Chandler et al, 1982). For these reasons a full discussion of the RISM models wouldn t fit here. Fortunately, a devoted exposition of current RISM work is already available (Hirata, 1998). 

In some respects, this approach is very attractive since, if the spherical harmonic expansions of the correlation functions are sufficiently rapidly convergent, the approximate solution of the Ornstein-Zernike equation for a molecular fluid can be placed upon essentially the same footing as that for a simple atomic fluid. The question of convergence of the spherical harmonic expansions turns out to be the key issue in determining the efficacy of the approach, so it is worthwhile to review briefly the available evidence on this question. Most of the work on this problem has concerned the spherical harmonic expansion of (1,2) for linear molecules. This work was pioneered by Streett and Tildesley, who showed how it was possible to write the spherical harmonic expansion coefficients as ensemble averages obtainable from a Monte Carlo or molecular dynamics simulation via... 

This follows immediately from some elementary algebra, when (4.33) is rewritten in terms of w, fi, and c. Chandler and his colleagues refer to (4.31) as an Ornstein-Zernike-like equation we see in fact that if (4.34) is used to define c, (4.31) is nothing but the usual OZ equation for the mixture of atoms that make up the molecules. The w is the contribution to C from the m/ra molecular correlations, whereas c is the contribution from correlations between atoms in different molecules, that is, direct inter-molecular correlations. 

It is the purpose of this chapter to sketch a statistical approach of wave-number-dependent fluctuations within the background of the STL, which goes beyond the correlation length concept, by incorporating in particular high order moments of the pair direct correlation function, at the level of description of molecular liquids. This approach permits one to unify the Teubner-Strey (TS) approach of microemulsions (Teubner and Strey 1987) to the usual Ornstein-Zernike approach of molecular emulsions in general. 

A third approach is to inject particles based on a grand canonical ensemble distribution. At each predetermined molecular dynamics time step, the probability to create or destroy a particle is calculated and a random number is used to determine whether the update is accepted (the probability for both the creation and the destruction of a particle must be equal to ensure reversibility). The probability function depends on the excess chemical potential and must be calculated in a way that is consistent with the microscopic model used to describe the system. In the work of Im et al., a primitive water model is used, and the chemical potential is determined through an analytic solution to the Ornstein-Zernike equation using the hypemetted chain as a closure relation. This method is very accurate from the physical viewpoint, but it has a poorer CPU performance compared with simpler schemes based on... 

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