Pair-correlation model

T introduces true n-particle correlation, and products like T T etc., arising out of the expansion eq.(3.2 ), generate simultaneous presence of k-particle amd m-particle correlations in a (k+m)-fold excited determinants etc. The truncation of T == T then corresponds to the pair—correlation model of Sinanoglu, while incorporating higher excited states with several disjoint pair excitations induced through the powers T - The amplitudes for T may be called linked or connected clusters for n electrons. The difficulty of a linear variation method such as Cl lies in its inability to realize the cluster expansion structure eq.(3.2),in a simple and practicable manner. 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

Figure 2 Pair correlation functions of 0-0 and O-H at ( puted with the parameters of the SPC water model.

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

To Anally complete the model a formula for the contact value of the pair correlation function g must be given. We choose the Carnahan formula... 

Local Average Density Model (LADM) of Transt)ort. In the spirit of the Flscher-Methfessel local average density model. Equation 4, for the pair correlation function of Inhomogeneous fluid, a local average density model (LADM) of transport coefficients has been proposed ( ) whereby the local value of the transport coefficient, X(r), Is approximated by... 

In the case of molten salts, no obvious model based on statistical mechanics is available because the absence of solvent results in very strong pair correlation effects. It will be shown that the fundamental properties of these liquids can be described by quasi-chemical models or, alternatively, by computer simulation of molecular dynamics (MD). 

The prerequisite for an experimental test of a molecular model by quasi-elastic neutron scattering is the calculation of the dynamic structure factors resulting from it. As outlined in Section 2 two different correlation functions may be determined by means of neutron scattering. In the case of coherent scattering, all partial waves emanating from different scattering centers are capable of interference the Fourier transform of the pair-correlation function is measured Eq. (4a). In contrast, incoherent scattering, where the interferences from partial waves of different scatterers are destructive, measures the self-correlation function [Eq. (4b)]. 

In the case of coherent scattering, which observes the pair-correlation function, interference from scattering waves emanating from various segments complicates the scattering function. Here, we shall explicitly calculate S(Q,t) for the Rouse model for the limiting cases (1) QRe -4 1 and (2) QRe > 1 where R2 = /2N is the end-to-end distance of the polymer chain. 

In summary, the chain dynamics for short times, where entanglement effects do not yet play a role, are excellently described by the picture of Langevin dynamics with entropic restoring forces. The Rouse model quantitatively describes (1) the Q-dependence of the characteristic relaxation rate, (2) the spectral form of both the self- and the pair correlation, and (3) it establishes the correct relation to the macroscopic viscosity. 

The results of calculations based on the central force model are very satisfying. As shown in Figs. C, D and E the pair correlation functions gHH(R), g(m(R)... 

Fig. D. The pair correlation function g0H and the running coordination number oh predicted by the central force model...

Fig. 53. Fit to the pair correlation function goo for the model consisting of a local pentamer plus continuum (from Ref. 5>).-------neutron diifraction data,-------model calculation...

These configurational changes will affect the total pair correlation in the model compounds. In addition, in the real molecules we have also a large effect due to changes in proton-proton distances (as well as differences in solvent effects and intramolecular hydrogen bonding). 

In the tetrahedral model, which possesses the highest symmetry, all the sites are identical in the strict sense. This means that there is only one (first) intrinsic binding constant, only one pair correlation, one triplet and one quadruplet correlation. 

In the linear model, which possesses the lowest symmetry, we find two different binding constants, denoted by iP and iP for binding to the first (or fourth) and second (or third) subunits, respectively. (Clearly, =k when there are only direct correlations in the system see Section 6.3.) We have four different pair correlations denoted by g g and g and two different triple correlations denoted by and Note that in Table 6.1 we assigned a direct correlation factor S only for nearest-neighbor pairs and assumed that S is independent of the state of the subunits. 

The first model is that discussed in Section 4.3. This may be called the one-macrostate approximation. In this model the adsorbent molecule has only one state, and the binding process does not induce any conformational changes. Hence, the ligand-ligand pair correlation is due only to the direct ligand-ligand interaction... 

The model is that discussed in Section 4.5, for which the ligand-ligand pair correlation has the form... 

We shall now examine the effect of size on the cooperativity. We use the model of Section 4.5, for which we found the formal expression for the ligand-ligand pair correlation 1, 1) in Eq. (9.3.16). The solvent effect enters this expression via three factors, which we shall examine separately. 

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