Ornstein-Zernike OZ Equations

It is the Ornstein-Zernike equation that has been used most popularly to describe the density fluctuations in liquids [37], [35]. The equation by itself is nothing more than the defining of a correlation function called the direct correlation function c(r, r ) in terms of the total correlation function h r,r ) defined in Eq. (1.7). 

The OZ equation can be formulated from the grand partition function by functional differentiations in a spirit of the density functional theory. Here, we give just a brief sketch of the formulation. 

In the present section, we confine ourselves with a simple liquid. Let us define the potential energy of the liquid system by 

We define the single particle and pair correlation functions of density, respectively, by 

We now define the direct correlation function by an inverse relation of Eq. (1.20), 

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Most integral equations are based on the Ornstein-Zernike (OZ) equation [3-5]. The idea behind the OZ equation is to divide the total correlation function h ri2) iiito a direct correlation function (DCF) c r 12) that describes the fact that molecules 1 and 2 can be directly correlated, and an indirect correlation function 7( 12), that describes the correlation of molecule 1 with the other molecules that are also correlated with molecule 2. At low densities, when only direct correlations are possible, 7(r) = 0. At higher densities, where only triplet correlations are possible, we can write... 

The structure of the bulk associating fluid in the framework of the model in question can be determined by solving the common Ornstein-Zernike (OZ) equation... 

The starting point for the calculation of g(r) is the Ornstein-Zernike (OZ) equation, which, for a one-component system of liquids interacting via spherically symmetric potentials (e.g. Argon), is [89]... 

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]... 

This is the Ornstein-Zernike (OZ) equation from which the function c .y(r,/) acquires its name. 

Different techniques can be applied to calculate the effective interaction in a form of the potential of mean force. Because we are looking for W (r) which in the following will be used in computer simulations, the mostly suitable is the integral equation theory (IET) approach. The advantage of IET lies in a fact that in many most important applications this approach leads to an analytical equation for W (r). The IET technique is based on the Ornstein-Zernike (OZ) equation [33]... 

At the core of any integral equation approach we have the (exact) Ornstein-Zernike (OZ) equation [300] relating the total correlation function(s) of a given fluid to the so-called direct correlation function(s). For the replicated system at hand, the OZ equation is that of a multicomponent mixture [30],... 

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... 

All measured SAXS data were analyzed by the generalized indirect Fourier transformation (GIFT) technique with the Boltzmann simplex simulated annealing (BSSA) algorithm [34, 35]. The GIFT calculation is based on the analytical or numerical solution of the Ornstein-Zernike (OZ) equation that describes the interplay between the total (h(r)) and direct (c(r)) correlation functions ... 

Equation (6.14) can be derived for a much more general class of distributions and is know under the name of Ornstein-Zernike (OZ) equation. 

The behavior of fluids and solutions would be more easily understood if the distribution functions could be calculated exactly for a simple but realistic model potential at a moderately high concentration (e.g., l.OM). This is unfortunately not the case, and it is necessary to devise accurate but computationally convenient approximations for the distribution functions. They are more easily formulated in terms of the direct correlation function, c j(r), which is defined by the Ornstein-Zernike (OZ) equation (1914) ... 

I have already mentioned the limit X = (cr/L) /h with L < a/2 when the system should consist of dipolar dumbbells. In the absence of a solvent, the asymptotic form of the direct correlation function [defined through the Ornstein-Zernike (OZ) equation] for this system is given by (Rasaiah and Lee, 1985a)... 

Simulation results were compared with the predictions of the Ornstein-Zernike (OZ) equation with the hypernetted chain (HNC) closure approximation and the non-linear Poisson-Boltzmann equation, both augmented by pertinent Lifshitz NES potentials. We show in Fig. 1 that there is very good agreement between modified Poisson-Boltzmann theory, MC simulations, and HNC calculations when the counterions and co-ions are monovalent. There is also good agreement between the different approaches with divalent co-ions (not shown here). However, the results from MPBE cannot account for ion correlation effects that occur in Fig. 2 when the counterions are divalent. The reason is simply that the... 

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