Direct correlation function cluster expansion

The direct correlation function c is the sum of all graphs in h with no nodal points. The cluster expansions for the correlation functions were first obtained and analyzed in detail by Madden and Glandt [15,16]. However, the exact equations for the correlation functions, which have been called the replica Ornstein-Zernike (ROZ) equations, have been derived by Given and Stell [17-19]. These equations, for a one-component fluid in a one-component matrix, have the following form... 

The equlibrium between the bulk fluid and fluid adsorbed in disordered porous media must be discussed at fixed chemical potential. Evaluation of the chemical potential for adsorbed fluid is a key issue for the adsorption isotherms, in studying the phase diagram of adsorbed fluid, and for performing comparisons of the structure of a fluid in media of different microporosity. At present, one of the popular tools to obtain the chemical potentials is an approach proposed by Ford and Glandt [23]. From the detailed analysis of the cluster expansions, these authors have concluded that the derivative of the excess chemical potential with respect to the fluid density equals the connected part of the fluid-fluid direct correlation function (dcf). Then, it follows that the chemical potential of a fluid adsorbed in a disordered matrix, p ), is... 

The first contribution to h(r) is the direct correlation function c(r) that represents the correlation between a particle of a pair with its closest neighbor separated by a distance r. The second contribution is the indirect correlation function y(r), which represents the correlation between the selected particle of the pair with the rest of the fluid constituents. The total and direct correlation functions are amenable to an analysis in terms of configurational integrals clusters of particles, known as diagrammatic expansions. Providing a brief resume of the diagrammatic approach of the liquid state theory is beyond the scope of this chapter. The reader is invited to refer to appropriate textbooks on this approach [7, 9, 18, 26]. 

That means, as for g(r), the direct correlation function c(r) can be, in principle, derived from experiment. As can be seen, the OZ relation deals with a priori two unknown functions h(r) and c(r). The cluster expansion provides an important exact feature for c(r) with respect to the potential interaction at large distances... 

In the mean spherical approximation (MSA) treatment of the ion association in aqueous solutions, the linearity of the relative permittivity and of the hydrated cation diameters with the electrolyte concentration was taken into account and a good fit of the experimental activity and osmotic coefficient was obtained [72-75]. The MSA model was elaborated on the basis of cluster expansion considerations involving the direct correlation function the treatment can deal with the many-body interaction term and with a screening parameter and proved expedient for the interpretation of experimental results concerning inorganic electrolyte solutions [67,75-77]. 

They refer to Eq. (3.4.3) as the proper integral equation in view of the fact that the direct correlation function so defined does correspond to the sum of the nodeless diagrams in the interaction site cluster expansion of the total correlation function. Following Lupkowski and Monson, we shall refer to Eq. (3.4.3) as the Chandler-Silbey-Ladanyi (CSL) equation. Interestingly, the component functions have a simple physical interpretation. The elements of Ho correspond to the total correlation functions for pairs of sites at infinite dilution in the molecular solvent. The elements of the sum of Hq and //, (or H ) matrices are the solute-solvent site-site total correlation functions for sites at infinite dilution in the molecular solvent. 

When used in connection with integral equations, it is considered the definition for the direct correlation function Cij. It can be shown by cluster expansion that the direct correlation function has the following closed analytical expression ... 

An analysis of clusters expansion to higher order (as compared to PY equation) leads to the hypernetted-chain (HNC) approximation [44—46]. In other words, directly solving the OZ relation in conjunction with Eq. (28) is possible, under a drastic assumption on B(r). The total correlation function is given simply by... 

A first contribution to J-p is caused by the bonding between two atoms which constitute one molecule. This direct intra-molecular interaction between the atoms is taken into considerations as a f -bond. In addition there are intra-molecular interactions of indirect nature between the both atoms of a molecule. These atoms affect each other indirectly by n point interactions with all remaining atoms and combinations of atoms. The so-called intra-molecular pair cavity function y (ryw) expresses the ensemble of all indirect interactions which appear between the atoms of a molecule in f-bonds [13] and establishes the searched correlation function for all indirect interactions between the atoms inside a molecule. TTie molecular DFT approach evaluates the cluster expansion to calculate y (rjvr) using TPT. This approximation takes into account only presentations with vertices n <— 2, for what reason it is called the single chain approximation (TPT1)[12]. [7,8]... 

In this section, we review some of the important formal results in the statistical mechanics of interaction site fluids. These results provide the basis for many of the approximate theories that will be described in Section III, and the calculation of correlation functions to describe the microscopic structure of fluids. We begin with a short review of the theory of the pair correlation function based upon cluster expansions. Although this material is featured in a number of other review articles, we have chosen to include a short account here so that the present article can be reasonably self-contained. Cluster expansion techniques have played an important part in the development of theories of interaction site fluids, and in order to fully grasp the significance of these developments, it is necessary to make contact with the results derived earlier for simple fluids. We will first describe the general cluster expansion theory for fluids, which is directly applicable to rigid nonspherical molecules by a simple addition of orientational coordinates. Next we will focus on the site-site correlation functions and describe the interaction site cluster expansion. After this, we review the calculation of thermodynamic properties from the correlation functions, and then we consider the calculation of the dielectric constant and the Kirkwood orientational correlation parameters. 

The CASSCF method itself is not very useful for anything else than systems with few electrons unless an effective method to treat dynamical correlation effects could be developed. The Multi-Reference Cl (MRCI) method was available but was limited due to the steep increase of the size of the Cl expansion as a function of the number of correlated electrons, the basis set, and the number of active orbitals in the reference function. The direct MRCI formulation by P. Siegbahn helped but the limits still prevented applications to larger systems with many valence electrons [20], The method is still used with some success due to recent technological developments [21], Another drawback with the MRCI approach is the lack of size-extensivity, even if methods are available that can approximately correct the energies. Multi-reference coupled-cluster methods are studied but have not yet reached a state where real applications are possible. 

---