Partial correlation function

In this equation Exc is the exchange correlation functional [46], is the partial charge of an atom in the classical region, Z, is the nuclear charge of an atom in the quantum region, is the distance between an electron and quantum atom q, r, is the distance between an electron and a classical atom c is the distance between two quantum nuclei, and r is the coordinate of a second electron. Once the Kohn-Sham equations have been solved, the various energy terms of the DF-MM method are evaluated as... 

The multidensity Ornstein-Zernike equation (70) and the self-consistency relation (71) actually describe a nonuniform system. To solve these equations numerically for inhomogeneous fluids one needs only an appropriate generalization of the Lowett-Mou-Buff-Wertheim equation (14). Such a generalization, employing the concept of the partial correlation function has been considered in Refs. 34,35. 

Eq. (101) is the multidensity Ornstein-Zernike equation for the bulk, one-component dimerizing fluid. Eqs. (102) and (103) are the associative analog of the singlet equation (31). The last equation of the set, Eq. (104), describes the correlations between two giant particles and may be important for theories of colloid dispersions. The partial correlation functions yield three... 

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. 

In spite of its simplicity and the visual similarity of this equation to Eq. (7), we would like to note that Eq. (11) leads to a nontrivial thermodynamics of a partially quenched system in terms of correlation functions, see, e.g.. Ref. 25 for detailed discussion. Evidently, the principal route for and to the virial theorem is to exploit the thermodynamics of the replicated system. However, special care must be taken then, because the V and s derivatives do not commute. Moreover, the presence of two different temperatures, Pq and P, requires attention in taking temperature derivatives, setting those temperatures equal, if appropriate, only at the end of the calculations. 

This matrix must be diagonalized to obtain the largest eigenvalue and its eigenvector, which allows the partial coverages and the correlation functions to be obtained. This is trivial for no interactions (yi = yi = 3 12 = 1) and gives... 

Figure. 3 (a) Partial pair correlation function.s gij(B.) in liquid K-Sb alloys, (b) Total, partial, and local electronic densities of states in liquid Ko.soSbo.so- Cf. text. 

According to Eq. (2.13), the spectra we are interested in are given by a Fourier transform of the orientational correlation functions of the corresponding order Similarly to what was done in Chapter 3, the correlation functions for linear and spherical molecules may be represented as a superposition of the partial (marginal) components... 

A closed kinetic equation may be derived not for correlation function K t) directly, but for its partial (marginal) value... 

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 following equation shows the relationship between the partial structure factors, Sa/iiq), and the atom—atom correlation functions, ga/iir), that refer to the distance, r, between atoms a and /3 ... 

Ec = E c - Ex have been employed. On the one hand, LDA and GGA type correlation functionals have been used [14], However, the success of the LDA (and, to a lesser extent, also the GGA) partially depends on an error cancellation between the exchange and correlation contributions, which is lost as soon as the exact Ex is used. On the other hand, the semiempirical orbital-dependent Colle-Salvetti functional [22] has been investigated [15]. Although the corresponding atomic correlation energies compare well [15] with the exact data extracted from experiment [23], the Colle-Salvetti correlation potential deviates substantially from the exact t)c = 8Ecl5n [24] in the case of closed subshell atoms [25]. 

Three doubly spin-labelled [2]catenanes with different sizes were studied by 4-pulse DEER.52 The experimental distribution of interspin distances was compared with a theoretical pair-correlation function computed based on geometrical constraints. In chloroform solution the medium and large catenanes were close to fully expanded, but in glassy o-terphenyl they were partially collapsed. For the smaller catenane there was a higher population of shorter interspin distances, which was attributed to interactions between unsaturated sections of the molecule. 

The computer-reconstructed catalyst is represented by a discrete volume phase function in the form of 3D matrix containing information about the phase in each volume element. Another 3D matrix defines the distribution of active catalytic sites. Macroporosity, sizes of supporting articles and the correlation function describing the macropore size distribution are evaluated from the SEM images of porous catalyst (Koci et al., 2006 Kosek et al., 2005). Spatially 3D reaction-diffusion system with low concentrations of reactants and products can be described by mass balances in the form of the following partial differential equations (Koci et al., 2006, 2007a). For gaseous components ... 

It is convenient to divide a set of fluctuation-controlled kinetic equations into two basic components equations for time development of the order parameter n (concentration dynamics) and the complementary set of the partial differential equations for the joint correlation functions x(r, t) (correlation dynamics). Many-particle effects under study arise due to interplay of these two kinds of dynamics. It is important to note that equations for the concentration dynamics coincide formally with those known in the standard kinetics... 

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