Molecular pair correlation functions

They refer to Eq. (3.4.3) as the proper integral equation in view of the fact that the direct correlation function so defined does correspond to the sum of the nodeless diagrams in the interaction site cluster expansion of the total correlation function. Following Lupkowski and Monson, we shall refer to Eq. (3.4.3) as the Chandler-Silbey-Ladanyi (CSL) equation. Interestingly, the component functions have a simple physical interpretation. The elements of Ho correspond to the total correlation functions for pairs of sites at infinite dilution in the molecular solvent. The elements of the sum of Hq and //, (or H ) matrices are the solute-solvent site-site total correlation functions for sites at infinite dilution in the molecular solvent. 

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. 

Owing to the sensitivity of the chemical source term to the shape of the composition PDF, the application of the second approach to model molecular mixing models in Section 6.6, a successful model for desirable properties. In addition, the Lagrangian correlation functions for each pair of scalars (( (fO fe) ) should agree with available DNS data.130 Some of these requirements (e.g., desirable property (ii)) require models that control the shape of /, and for these reasons the development of stochastic differential equations for micromixing is particularly difficult. 

The function So (s) describes the neutron scattering from randomly oriented D2O molecules with the positional correlation between molecular centers specified by the correlation function hoo R). We expect the orientational correlation between pairs of water molecules to be of much shorter range than the positional correlation between molecular centers, and hence the functions 3o (s) and 3 (s) should be nearly equal for values of s <2 A-1. 

Fig. 29. The OO pair correlation function for the STa potential from molecular dynamics simulation of liquid water (from Ref. 3>)...

The interfacial pair correlation functions are difficult to compute using statistical mechanical theories, and what is usually done is to assume that they are equal to the bulk correlation function times the singlet densities (the Kirkwood superposition approximation). This can be then used to determine the singlet densities (the density and the orientational profile). Molecular dynamics computer simulations can in... 

The symmetry requirements and the need to very effectively describe the correlation effects have been the main motivations that have turned our attention to explicitly correlated Gaussian functions as the choice for the basis set in the atomic and molecular non-BO calculations. These functions have been used previously in Born-Oppenheimer calculations to describe the electron correlation in molecular systems using the perturbation theory approach [35 2], While in those calculations, Gaussian pair functions (geminals), each dependent only on a single interelectron distance in the exponential factor, exp( pr ), were used, in the non-BO calculations each basis function needs to depend on distances between aU pairs of particles forming the system. 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

Fig. 6.86. Oxygen-oxygen pair correlation function obtained from molecular dynamic simulations on the adsorbed layer of a Pt(100) surface. Ax and Ay are the projections of the interatomic distances in the x- and indirections, respectively. They reflect the positions of the oxygen atoms on the top site of the platinum lattice, and the pronounced form of the peaks refers to their relatively small displacement. (Reprinted from E. Spohr, G. Toth, and K. Heinzinger, Electrochim. Acta 41 2131, copyright 1996, Fig. 10a, with peimission from Elsevier Science.)...

The pair correlation function g describes the distribution of molecular centers in the solution. In concentrated systems at rest, gxl. Flow alters g, and it is this change which gives rise to the drag forces. For sufficiently slow shearing flows,... 

It appears that the formal theories are not sufficiently sensitive to structure to be of much help in dealing with linear viscoelastic response Williams analysis is the most complete theory available, and yet even here a dimensional analysis is required to find a form for the pair correlation function. Moreover, molecular weight dependence in the resulting viscosity expression [Eq. (6.11)] is much too weak to represent behavior even at moderate concentrations. Williams suggests that the combination of variables in Eq. (6.11) may furnish theoretical support correlations of the form tj0 = f c rjj) at moderate concentrations (cf. Section 5). However the weakness of the predicted dependence compared to experiment and the somewhat arbitrary nature of the dimensional analysis makes the suggestion rather questionable. 

Pair correlation function in molecular theory of liquids (Part 6). 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

It will be useful now to review some elementary facts regarding the structure of liquids at equilibrium. When a crystalline solid melts to form a liquid, the long range order of the crystal is destroyed. However, a residue of local order persists in the liquid state with a range of several molecular diameters. The local order characteristic of the liquid state is described in terms of a pair correlation function, g-i(R)> defined as the ratio of the average molecular density, p(R), at a distance R from an arbitrary molecule to the mean bulk density, p, of the liquid... 

This equals the negative of the internal (configurational) energy of the solvent, Econb which is related to its pair potential u(r) (see Eq. (3.18)) and pair correlation function g(r) on the molecular level (Marcus... 

The pair correlation function is a short range quantity in liquids, decaying to unity after a few molecular diameters, the correlation length However, in supercritical fluids g(r) has a much longer range and t, becomes considerably larger than the mean inter-molecular separation at the critical density. For instance, for carbon dioxide , = 5.5 nm at Tc compared to the mean intermolecular separation of 0.55 nm (Eckert, Knutson and Debenedetti 1996). 

Here, H and C are symmetric matrices whose elements are the partial total hap(r) and direct cap,(r) pair correlation functions a,ft = A,B) W is the matrix of intramolecular correlation functions wap r) that characterize the conformation of a macromolecule and its sequence distribution and p is the average number density of units in the system. Equation 17 is complemented by the closure relation corresponding to the so-called molecular Percus-... 

The persistence of the fluctuating local fields before being averaged out by molecular motion, and hence their effectiveness in causing relaxation, is described by a time-correlation function (TCF). Because the TCF embodies all the information about mechanisms and rates of motion, obtaining this function is the crucial point for a quantitative interpretation of relaxation data. As will be seen later, the spectral-density and time-correlation functions are Fourier-transform pairs, interrelating motional frequencies (spectral density, frequency domain) and motional rates (TCF, time domain). 

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 ... 

Figure 18. Pair correlation function g(r), at T = 297.6 K, for p = 0.95, 1.37, 1.69, and 1.93 nm 3 (from the bottom to the top), calculated by the ODS integral equation with both the two-body interactions (dotted lines), and the two- plus three-body interactions (solid lines). The curves for p= 1.37, 1.69, and 1.93 nm-3 are shifted upward by 0.5, 1, and 1.5, respectively. The comparison is made with molecular dynamics simulation (open circles). Taken from Ref. [129].

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