Total correlation functions

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... 

We apply the singlet theory for the density profile by using Eqs. (101) and (103) to describe the behavior of associating fluids close to a crystalline surface [120-122], First, we solve the multidensity OZ equation with the Percus-Yevick closure for the bulk partial correlation functions, and next calculate the total correlation function via Eq. (68) and the direct correlation function from Eq. (69). The bulk total direct correlation function is used next as an input to the singlet Percus-Yevick or singlet hypernetted chain equation, (6) or (7), to obtain the density profiles. The same approach can be used to study adsorption on crystalline surfaces as well as in pores with walls of crystalline symmetry. 

The first crystal-independent structural order metric that we will explore is the translational parameter l w an integral measure of the amplitude of the material s total correlation function h(r),... 

In the absence of a correlation between the local dynamics and the overall rotational diffusion of the protein, as assumed in the model-free approach, the total correlation function that determines the 15N spin-relaxation properties (Eqs. (1-5)) can be deconvolved (Tfast, Tslow < Tc) ... 

One can easily adjust the values of the dielectric constants D(, and Dj to obtain the experimental values of W, as in Table 4.4. With a choice of = 19.6 and Dj. = 51.0 for water, and D. = 12.5 and Dj. = 31.8 for 50% water-ethanol, we obtain the experimental values of W. We now compute the total correlation function for the two-state model for succinic acid. Here the correlation cannot be computed as an average correlation of the two configurations (see Section 4.5). The total correlation of the equilibrated two-state model is... 

Figure 4.1-2 Total correlation functions for (a) LiSCN/AICIj and (b) LiSCN/AICb- The bold lines are the experimental neutron data ( ), the fit (-), the Gaussian functions for each of the atomic pairs used to fit the data (-) and...

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 matrix elements of the total correlation function, h, are related to all pairs of atoms. The intramolecular correlation function, to, introduced here represents the shape of the molecule. 8(r) in the diagonal element is the Dirac delta function and represents the position of an atom. The function appearing in the off-diagonal element is given by,... 

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 ... 

The correlation functions play an essential role in the static description of homogenous classical liquids whose particles are taken to interact through an effective pair potential. The starting point of the liquid-state theory, in terms of correlation functions, is the well-known Ornstein-Zernike equation [25]. The total correlation function h r) defined in Section II is actually a sum of two contributions that is illustrated by the following relationship... 

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... 

The function F(r) is left unchanged by a change of sign of a. We note, however, that only the total correlation function, and not /q (r), possesses symmetry under a change of sign of a. This assertion is evident, for example, in Fig. 11, which shows the dependence of /31 (a) on a as an illustration. 

The relaxation dynamics (W7 in Fig. 38) is the response of the environment around Trp7 to its sudden shift in charge distribution from the ground state to the excited state. Under this perturbation, the response can result from both the surrounding water molecules and the protein. We separately calculated the linear-response correlation functions of indole-water, indole-protein, and the sum of the two. The results for isomer 1, relative to the time-zero values, are shown in Fig. 42a. The linear response correlation function is accumulated from a 6-ns interval indicated in Fig. 41a during which the protein was clearly in the isomer 1 substate. All three correlation functions show a significant ultrafast component 63% for the total response, 50% for indole-water, and nearly 100% for indole-protein. A fit to the total correlation function beyond the ultrafast inertial decrease requires two exponential decays 1.4 ps (3.6kJ/mol) and 23 ps (2.0kJ/mol). Despite the 6-ns simulation window for isomer 1, the 23-ps long component is not well determined on account of the noise apparent in the linear response correlation function (Fig. 42a) between 30 and 140 ps. The slow dynamics are mainly observed in the indole-water relaxation and the overall indole-protein interactions apparently make nearly no contributions to the slowest relaxation component. 

Assuming the pair potential known, the radial distribution function for two-dimensional systems can be calculated using the two-dimensional version of the Ornstein-Zernike equation, Eq. (22), and one of the closure relations. Although Eq. (22) does not relate one to one the radial distribution function with the pair potential, one might attempt to invert the procedure to get u(r) from the experimental values for g(r). Thus, by taking the Fourier-Bessel (FB) transform [43,44] of Eq. (22) an expression for c(k) is obtained in terms of the FB transform of the measured total correlation function, i.e. 

Most modern theories are based on the equation of Ornstein and Zernike (OZ). This equation was developed eighty years ago to describe light scattering in a fluid. Its utility in the theory of fluids was realized much later, about thirty-five years ago. The OZ equation is obtained by defining the total correlation function,... 

One more relation is required to achieve closure, i.e., to determine the two types of correlation functions. The most commonly used relations are the Percus-Yevick (PY) and the hypernetted chain (HNC) approximations [47-49]. From graph or diagram expansion of the total correlation function in powers of the density n(r) and resummation, an exact relation between the total and direct correlation functions is obtained, namely... 

Its Hankel transform has no singularity at p -> 0, and so the expansion of the DCF at p = 0 keeps the analytic form (44). Accordingly, the total correlation function keeps the asymptotics (43). However, the matrices of the expansion coefficients Co, C2, C4,... in (44) have other, modified values. Through Equation (40) this, in general, changes the profile p and hence results in a modified inverse decay length appearing in the asymptotics (42), (43) and (46). 

To calculate the ampUtudes we have used a procedure like that in ref. 51, assuming as the initial state [ 0) = 1] the state of the system having all the dipoles aligned and distributed in the various environments according to the binomial distribution. The total correlation function is therefore... 

After solving this problem we should have reduced our model to a five-states model for water, each state with its g (i),) and ft(rij) factor, that is, to a model like those considered before by other authors. In this case we can write the total correlation function 4>(/) as... 

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