Difference pair-correlations

For all the above reasons we have defined g(C) without reference to any hypothetical, independent-site system. One simply extracts both 1(C) and all from the experimental data, and then constructs the quantity g(C). When the sites are identical in a weak sense, i.e., all k = k, some of the correlations for a given / might differ. For example, four identical subunits arranged in a square will have only one intrinsic binding constant k, but two different pair correlation functions. For this particular example we have four nearest-neighbor pair correlations g (2), and two second-nearest-neighbor pair correlations gJJ)- The average correlation for this case is... 

In the square model, the sites are identical only in a weak sense. This means that there is only one (first) intrinsic binding constant, but we have two different pair correlations, which are denoted by g and for the nearest and next-nearest... 

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. 

We next turn to the indirect pair correlations. As noted before, we expect to have four different pair correlations that we denote by and... 

FIGURE 8.4 Two difference pair-correlation functions AG(rz). The dotted line shows AG(rz)(H), obtained from the difference between the vermiculite and its H-salt solution. The full line shows AG(rz)(D), obtained from the difference between the vermiculites in D- and H-salt solutions. 

FIGURE 13.5 Difference pair correlation functions AG(rz) between the atomic pair correlation functions of samples with deuterated and hydrogenous PEO. In the case of the 0.1 M salt concentration (solid line), the butylammonium chains were deuterated in both samples, whereas for the 0.03 M salt concentration (dashed line), both samples contained hydrogenous counterions. The volume fraction of PEO was 4% in all cases. 

The generalization to multicomponent systems is quite straightforward. Instead of one pair correlation g(X, X"), we shall have pair correlation functions for each pair of species a/ . For instance, if A and B are spherical particles, then we have three different pair correlation functions g,, (R), gAB(R) = 8ba(R) and gBB(R)- We shall describe these in more detail in section 2.9. 

Whereas the lEPA-scheme means physically that one treats one electron pair in the Hartree-Fock field of the other electrons, CEPA means that each pair is treated in the fields of the correlated other electrons which is somewhat more physical. Through the explicit consideration of the Bab block the interaction between the different pair correlations is accounted for. Though a rigorous justification of the CEPA-method is still lacking and probably not possible the practical applications have so far been very satisfactory. One main advantage of CEPA as compared with lEPA is that it can also be used with delocalized orbitals and the results are nearly invariant with respect to a unitary transformation of the occupied orbitals. 

Now consider what happens in a three-dimensional lattice. In the low-density limit, coupling terms between different pair correlations can be ignored, since they are always of higher order than the terms in Eq. (37). Therefore the total correlation energy can be obtained as a simple lattice sum [17]... 

In other words, the lEPA is accurate up to second order of perturbation theory based on the Hartree-Fock (HF) model as zeroth-order approximation. The exact third-order contribution involves—besides the terms which are properly included—the interaction between different pair correlation functions... 

Theories based on the solution to integral equations for the pair correlation fiinctions are now well developed and widely employed in numerical and analytic studies of simple fluids [6]. Furtlier improvements for simple fluids would require better approximations for the bridge fiinctions B(r). It has been suggested that these fiinctions can be scaled to the same fiinctional fomi for different potentials. The extension of integral equation theories to molecular fluids was first accomplished by Chandler and Andersen [30] through the introduction of the site-site direct correlation fiinction c r) between atoms in each molecule and a site-site Omstein-Zemike relation called the reference interaction site... 

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 simulations are repeated several times, starting from different matrix configurations. We have found that about 10 rephcas of the matrix usually assure good statistics for the determination of the local fluid density. However, the evaluation of the nonuniform pair distribution functions requires much longer runs at least 100 matrix replicas are needed to calculate the pair correlation functions for particles parallel to the pore walls. However, even as many as 500 replicas do not ensure the convergence of the simulation results for perpendicular configurations. 

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. 

Naively it may be expected that the correlation between pairs of electrons belonging to the same spatial MO would be the major part of the electron correlation. However, as the size of the molecule increases, the number of electron pairs belonging to different spatial MOs grows faster than those belonging to the same MO. Consider for example the valence orbitals for CH4. There are four intraorbital electron pairs of opposite spin, but there are 12 interorbital pairs of opposite spin, and 12 interorbital pairs of the same spin. A typical value for the intraorbital pair correlation of a single bond is 20kcal/ mol, while that of an interorbital pair (where the two MO are spatially close, as in CH4) is 1 kcal/mol. The interpair correlation is therefore often comparable to the intrapair contribution. 

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)]. 

The pair-correlation function for the segmental dynamics of a chain is observed if some protonated chains are dissolved in a deuterated matrix. The scattering experiment then observes the result of the interfering partial waves originating from the different monomers of the same chain. The lower part of Fig. 4 displays results of the pair-correlation function on a PDMS melt (Mw = 1.5 x 105, Mw/Mw = 1.1) containing 12% protonated polymers of the same molecular weight. Again, the data are plotted versus the Rouse variable. 

The result for the like and unlike partners definition can be obtained by very similar arguments and involves all three pair correlation functions. The various definitions and results can equally be applied to defects which are not ionic by merely substituting the words different kind for opposite charge and either kind for either charge in the definitions. 

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