Functions total correlation function

The actual solution of Eqs. (15.1) or (15.4) can be obtained if the relation between direct correlation functions, total correlation functions, and site-site potentials, is known. Several such closure relations are... 

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... 

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...

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 ... 

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. 

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,... 

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... 

Having defined the different Interactions occurlng In [3.6.1], we now need to specify the probability of finding an Ion a at some position r. The one-particle (singlet) density p fr jls defined In sec. I.3.9d as the number of particles per volume at position r. Now we apply the definition to Ions. The radial distribution function g (r)and the ion-wall total correlation function h (r) follow from (1.3.9.22 and 23] as... 

Here, I is the identity matrix, hag are the Fourier transforms of the matrix-matrix total correlation function, and the elements of the remaining matrices are the following [1 % = hddm - cddm, [C ]kl = cddm, [rd ]fc = hd -cd 0 and likewise for Cd , being cdd, cd the Fourier transform of the dipole-dipole and dipole-charge direct correlation functions respectively, pd is the... 

---