Direct correlation functions in terms of

2 Direct correlation functions in terms of geometry and electrostatics 

We consider an arbitrary mixture of charged hard spheres. The general solution of the MSA [1] yields simple expressions for the thermodynamics and pair correlation functions. The dcf was obtained by Hiroike [2] For a system of hard spheres of radius Hi = Oi/2 charges Zi and number density ft = Ni/V, the dcf Ci (r) can be written 

An important step in constructing a free energy model for the hard sphere mixture begins by casting the known in geometric form [3] 

Our first step is to rewrite the dcf of the MSA as written by Hiroike in terms of the geometric and/or electrostatic forms, for the charge part We shall 

After some manipulations Hiroike s dcf can be cast in the form 

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For a planar interface the interpolation (19)-(20) of the inhomogeneous site-site direct correlation function in terms of the homogeneous ones simplifies the SS-LMBW equation (18) to the form... 

Typically, scaling approaches are employed to explain the behavior in the semidilute regime. By examining static correlations near the temperature, Daoud and Jannick( ) have expressed the density-density correlation function in terms of a correlation length that is inversely proportional to concentration. Since the diffusion coefficient is inversely proportional to the correlation length it is directly proportional to the concentration. 

This another popular closure [43] deals with spherical particles fluids that interact through an infinite repulsive potential at short range u(r) = +oo for r mean-spherical approximation (MSA) is formulated in terms of an ansatz for the direct correlation function. In this approach, c(r) is supposed to be... 

The parameter r can be optimized for thermodynamic consistency between the virial and compressibility routes to the free energy of the system. As discussed in Section 5.3.2, the KHM closure (7) brings about an additional long-range term in the direct correlation functions in the critical regime. 

Electron correlation in atoms and molecules finds a very natural expression in terms of the reduced density matrices within the formalism of second quantization. The fundamental relations of this theory lead here to a discussion of the correlation effects and their connexion with two properties of the group functions frequently studied by Prof. Ede Kapuy strong orthogonality and the concept of independence. The other aim of this paper is to examine the direct expression of the correlation effects in terms of reduced density matrices. 

As we have expressed the compressibility equation in terms of the integral over the direct correlation function in (C.16), one can write the KB theory in terms of Cy instead of G / the two are equivalent formulations. O Connel (1971) has expressed the view that the formulation in terms of Cy might be more useful for numerical work since the direct correlation function is considered to be shorter range than the pair correlation function. For further applications of this approach, see O Connel (1971), Perry and O Connel (1984), and Hamad et al. (1987, 1989, 1990a, b, 1993, 1997, 1998). 

Ihe so-called vertex functions r (x,...,x ) are directly related to the single-chain correlation fiinctions. In terms of wave vectors this becomes... 

The pair correlation functions can be expressed directly in terms of the computed coefficients from Eq. (61) in particular, the number-number pair distribution function gN ir) and the number-number structure factor SNN k). Thus,... 

Let us begin our discussion from the model of Cummings and Stell for heterogeneous dimerization a + P ap described in some detail above. In the case of singlet-level equations, HNCl or PYl, the direct correlation function of the bulk fluid c (r) represents the only input necessary to obtain the density profiles from the HNCl and PYl equations see Eqs. (6) and (7) in Sec. II A. It is worth noting that the transformation of a square-well, short-range attraction, see Eq. (36), into a 6-type associative interaction, see Eq. (39), is unnecessary unless one seeks an analytic solution. The 6-type term must be treated analytically while solving the HNCl... 

The definition of the terms can be found in Refs. 91, 92. The excess chemical potential, computed from the direct correlation function of the... 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

There are several ways of obtaining functionals for nonideal systems. In most cases the free energy functional is expressed as the sum of an ideal gas term, a hard-sphere term, and a term due to attractive forces. Below, I present a scheme by which approximate expression for the free energy functional may be obtained. This approach relies on the relationship between the free energy functional and the direct correlation function. Because the direct correlation functions are defined through functional derivatives of the excess free energy functional, that is,... 

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