Correlation function indirect

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... 

In this book we examine various types of correlations that arise from (direct or indirect) communication between the ligands at different sites. We require that the correlation functions be unity whenever the two sites are physically independent. This excludes the type of correlation we found in Eq. (1.1.9). Yet, we wish to study systems with small values of m. This can be achieved by opening the system with respect to the ligands. We still keep m fixed, but now the ligands bound to the system are in equilibrium with a reservoir of ligands at a fixed chemical potential, or at a fixed density (see also Section 1.2). 

It is seen that the correlation function g(l, 1) is not simply related to the direct correlations and Clearly, this is not an average of the two direct correlations [see also Eq. (4.5.24) below]. In this section we wish to focus on the indirect correlation, Therefore, for the moment, we assume that the direct correlations are either negligible, i.e., 5 5 1, or that they are independent of the conformation, i.e.,S, = 5 , = 5. Hence, g(l, 1) may be written as... 

Perhaps the simplest two-site cooperative systems are small molecules having two binding sites for protons, such as dicarboxylic acids and diamines. Despite their molecular simplicity, most of these molecules do not conform with the modelistic assumptions made in this chapter. Therefore, their theoretical treatment is much more intricate. The main reasons for this are (1) there is, in general, a continuous range of macrostates (2) the direct and indirect correlations are both strong and intertwined, so that factorization of the correlation function is impossible. In addition, as with any real biochemical system, the solvent can have a major effect on the binding properties of these molecules. 

The molecular approach, adopted throughout this book, starts from the statistical mechanical formulation of the problem. The interaction free energies are identified as correlation functions in the probability sense. As such, there is no reason to assume that these correlations are either short-range or additive. The main difference between direct and indirect correlations is that the former depend only on the interactions between the ligands. The latter depend on the maimer in which ligands affect the partition function of the adsorbent molecule (and, in general, of the solvent as well). The argument is essentially the same as that for the difference between the intermolecular potential and the potential of the mean force in liquids. 

Fig. 6.86. Oxygen-oxygen pair correlation function obtained from molecular dynamic simulations on the adsorbed layer of a Pt(100) surface. Ax and Ay are the projections of the interatomic distances in the x- and indirections, respectively. They reflect the positions of the oxygen atoms on the top site of the platinum lattice, and the pronounced form of the peaks refers to their relatively small displacement. (Reprinted from E. Spohr, G. Toth, and K. Heinzinger, Electrochim. Acta 41 2131, copyright 1996, Fig. 10a, with peimission from Elsevier Science.)...

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

This estimate should be made more precise. To do it, let us use some results of the numerical solution of a set of the kinetic equations derived in the superposition approximation. The definition of the correlation length o in the linear approximation was based on an analysis of the time development of the correlation function Y(r,t) as it is noted in Section 5.1. Its solution is obtained neglecting the indirect mechanism of spatial correlation formation in a system of interacting particles, i.e., omitting integral terms in equations (5.1.14) to (5.1.16). Taking now into account such indirect interaction mechanism, the dissimilar correlation function, obtained as a solution of the complete set of equations in the superposition approximation... 

The performed analysis permits to assume that an indirect channel of the correlation emergence results in the correlation function Y(77) = z(rj) whose shape is a smoothed step, besides near 77 = 1 because of the continuity reasons one gets z (77) = 0 as 77 1... 

For the multipole interaction (4.1.44) the dissimilar correlation function could be also presented in a form of product (6.1.4), where Yo(r,t) = exp[—cr(r)f]. Neglecting indirect correlation mechanism, the dissimilar particle function Yo(r, t) — exp[— (r/ o)-m], with o defined by (4.1.45), is stationary in term of variable r)0 = r/ o- Indirect mechanism of the correlation formation, as follows from a solution of equations derived in the superposition approximation, results at long times in Y(r, t) z r) t),... 

From the many tools provided by statistical mechanics for determining the EOS [36, 173, 186-188] we consider first integral equation theories for the pair correlation function gxp(ra,rp) of spherical ions which relates the density of ion / at location rp to that of a at ra. In most theories gafi(ra,rp) enters in the form of the total correlation function hxp(rx,rp) = gxp(rx,rp) — 1. The Omstein-Zemike (OZ) equation splits up hap(rx,rp) into the direct correlation function cap(ra, rp) for pair interactions plus an indirect term that reflects these interactions mediated by all other particles y ... 

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]. 

The reader has to notice that ( (r ) = h(r12 — c(r12), which is the indirect correlation function y(r12). Unfortunately, even if the types of diagrams are known, the resultant series cannot be transformed to any tractable analytical formula or evaluated numerically with a good accuracy. Recently, from the bridge diagram series, simple phenomenological forms for B r) haven been proposed for the LJ fluid [29]. They present the advantage of bringing some... 

As compared to Eq. (32), the long-range part of the potential has been added to the indirect correlation function. For convenience, the term y(r) — [Smlr (r) is noted j (r), which is the so-called renormalized indirect correlation function, whose importance is crucial (see the next paragraph). When r > rm, this closure reduces to the MSA approximation c(r) = ( t/(r). In the case of a purely... 

The previous basic observations have suggested to adopt a closure that allows us to interpolate continuously between two existing theories. In this framework, the famous HNC-SMSA(HMSA) closure [22] has been proposed for the Lennard-Jones fluid. It interpolates between HNC and SMSA. The HMSA has strong theoretical basis since it can be derived from Percus functional expansion formalism and its bridge function expresses as a functional of the remormalized indirect correlation function y (r) = y(r) — Pmi,r(/") so that... 

As seen, compared to (38) the parameter a has been replaced by a more complicated term involving itself the indirect correlation function y (r). Even if reputed to be accurate, this approximation suffers from a slight thermodynamic inconsistency. As recognized by the authors, the compressibility obtained from the virial route differs little from that obtained directly from the compressibility equation . Nevertheless, this approximation is here considered as a SCIET. 

As seen above, most of the recent SCIETs are involved with the function y (r). The use of this renormalized indirect correlation function in the diagrams expansions has a simple and practical justification The h(r) diagrams for the bridge function contain an exp[co(r) — factor in the integrand expressions,... 

Thus, another approach consists in selecting some boundary conditions and properties. It is obvious that all exact correlation functions must satisfy and incorporate them in the closure expressions at the outset, so that the resulting correlations and properties are consistent with these criteria. These criteria have to include the class of Zero-Separation Theorems (ZSTs) [71,72] on the cavity function v(r), the indirect correlation function y(r) and the bridge function B(r) at zero separation (r = 0). As will be seen, this concept is necessary to treat various problems for open systems, such as phase equilibria. For example, the calculation of the excess chemical potential fi(iex is much more difficult to achieve than the calculation of usual thermodynamic properties since one of the constraints it has to satisfy is the Gibbs-Duhem relation... 

As far as the LJ fluid is concerned, things are less straightforward and require, as it has been said above, a renormalization of the indirect correlation function y (r) that has to be accompanied by an optimized division scheme of the potential. As will be seen, the WCA [51] splitting is not sufficient to get accurate results for the correlation functions inside the core region, while these values are crucial for phase-equilibria treatment. 

In this framework, we present the repercussions on the physical properties of a renormalized indirect correlation function y (r) conjugated with an optimized division scheme. All the units are expressed in terms of the LJ parameters, that is, reduced temperature T = kBT/e and reduced density p = pa3. In order to examine the consequences of a renormalization scheme, the direct correlation function c(r) calculated from ZSEP conjugated with DHH splitting is compared in Fig. 7 to those obtained with the WCA separation. For high densities, the differences arise mainly in the core region for y(r) and c(r) [77]. These calculated quantities are in excellent agreement with simulation data. The reader has to note that similar results have been obtained with the ODS scheme (see Ref [80]). Since the acuracy of c(r) can be affected by the choice of a division scheme, the isothermal compressibility is affected too, as can be seen in Table III for the pkBTxT quantity. As compared to the values obtained with... 

The second assumption is the unique functionality [72] of the bridge function, meaning that B = B [y] is a simple function of the indirect correlation function, as expressed in the majority of the approximations for the bridge function. Changing the integration variable from X to y, and integrating by parts yields... 

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]... 

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