Direct correlation function intramolecular

However, it is known that the direct correlation functions have an exact long-range asymptotic form, arising due to intramolecular correlations in clusters formed via the association mechanism. This asymptotics is not included in the Percus-Yevick approximation. Other common liquid state approximations also do not provide correct asymptotic behavior of Ca ir). 

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 ... 

Defining the intramolecular site-site direct correlation function as... 

This is our new equation for the single-site density distributions which generalizes the LMBW equation (2) to polyatomic fluids, below called a site-site LMBW (SS-LMBW) equation [47]. As distinct from the site-site DFT approaches [20-24], the SS-LMBW equation properly treats the short- and long-range correlations coupled in the site-site direct correlation function Cy (r2, rs). The SS-LMBW theory also differs from the RISM approach to inhomogeneous polyatomic fluids derived within the density functional formalism by Chandler et al. [25,26]. In Equations (3.7) and (3.13) of Ref. [26] for the site density profiles p (r), the orientational averaging with the intramolecular matrix (r 12)... 

Given the graphical expansion for the site-site total correlation function h y r) it is possible to generate on Omstein-Zernike like equation. We will adopt the physical approach that the total correlation function is the sum of all possible direct site-site correlations, both intramolecular correlations via an s -bond, or intermolecular correlations via a c, (r). This defines a site-site direct correlation function c y r). From the expansion (2.2.5), it follows that at most one s -bond is connected to each circle. 

As we discussed in Section II.B, site-site correlation functions provide a very useful formalism for describing the structure of fluids modeled with interaction site potentials. In this formalism, information equivalent to g l,2) is obtained from the set of site-site correlation functions and intramolecular correlation functions. For this reason, a great deal of effort has been put into the development of integral equation theories for these correlation functions. The seminal contribution in this area was made by Chandler and Andersen, who sought to write an integral equation of the Ornstein-Zernike form in which the set of site-site total correlation functions were related to a set of site-site direct correlation functions. Their equation has the form... 

Another exciting development gets rid of the limitation (i) from the start by basing the approach on theoretical developments in the theory of fluids [44-49, 52, 224-225]. These theories all are based on the concept of the direct correlation function [226] related to the direct interaction between monomers the total correlation function is expressed approximately in terms of the intramolecular structure factor and this direct correlation function, which needs a suitable decoupling approximation, which can be motivated by suitable... 

The polymer RISM theory relates the set of total correlation functions (r) = g(r)-l to the set of site-site direct correlation functions, (c(r), and the set of intramolecular distribution functions, w(r), via the nonlinear site-site Ornstein-Zemike-like (SSOZ) integral matrix equation [41]... 

This result can be deduced from a number of perspectives such as renormalized perturbation theory [93], polymer density fimctional theory [95,96], and Per-cus functional expansion methods [97]. The medium-induced pair potential is determined by the direct correlation function and collective density fluctuations which are both functionally related to the intramolecular pair correlations via the PRISM equation. Hence, a coupled intramolecular/intermolecular theory is obtained. 

A second, entirely different class of new polymer integral equation theories have been developed by Lipson and co-workers, Eu and Gan, " and Attard based on the site-site version of the Born-Green-Yvon (BGY) equation. The earliest work in this direction was apparently by Whittington and Dunfield, but they addressed only a special aspect of the isolated polymer problem (dilute solution). The central quantity in the BGY approaches is the formally exact expressions that relate two and three (or more) intramolecular and intermolecular distribution functions. The generalized site-site Ornstein-Zernike equations and direct correlation functions do not enter. In the BGY schemes the closure approximation(s) enter as approximate relations between the two- and three-body distribution functions supplemented with exact normalization and asymptotic conditions. In the recent BGY work of Taylor and Lipson a four-point distribution function also enters. 

Since 1986, theories have been developed which employ the concept of direct correlation functions. The total correlation is factorized into a contribution from intermolecular correlations, Ceff(q), and intramolecular correlations, P(q), the form factor of the polyion. Benmouna etal. [48,49] presented a generalization of the single contact approximation due to the Zimm [50] to express the direct correlation function as... 

Here, H and C are symmetric matrices whose elements are the partial total hap(r) and direct cap,(r) pair correlation functions a,ft = A,B) W is the matrix of intramolecular correlation functions wap r) that characterize the conformation of a macromolecule and its sequence distribution and p is the average number density of units in the system. Equation 17 is complemented by the closure relation corresponding to the so-called molecular Percus-... 

Intramolecular pair correlation function e( r I of directly connected monomers within a sphere of radius r of a given monomer, for randomly branched polymers with N monomers made from ulcanizing linear chains with degree of polymerization Nq. Both axes have logarithmic scales. 

A central problem in liquid state theory is the calculation of these pair correlation functions. One phenomenological way to address the problem is based on a mechanistic view which pictures the pair correlations between atoms on different molecules as arising from all possible simple linear sequences of intramolecular and direct intermolecular pair correlations. We will use the notation c ( r-r ) to denote this direct correlation or coupling function. With diagrams, the idea is expressed as... 

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