Intramolecular correlations

However, it is known that the direct correlation functions have an exact long-range asymptotic form, arising due to intramolecular correlations in clusters formed via the association mechanism. This asymptotics is not included in the Percus-Yevick approximation. Other common liquid state approximations also do not provide correct asymptotic behavior of Ca ir). 

The equations that describe a blend or mixture of homopolymers is similar to that shown for multisite systems above. The only difference is that the matrix of intramolecular correlation functions is different and is given by... 

We can see from Figure 7 that for momentum transfers larger than about 3 A-1 in PB, i.e., starting around the second maximum, one observes only intramolecular correlations in the melt structure factor112-114 when one considers only scattering from the united atom centers. The melt structure factor can always be decomposed into a chain contribution (Sck(q)) and a contribution that captures the correlations between distinct melt chains (S,nj(c])). 

The intramolecular correlations of O—H at r= 1.00 A and of H—H at r = 1.56 A are identical, and the second peaks show small differences, perhaps arising from interactions between water molecules and sulfonate groups, although it would seem that water coordination around Ni would have to be considered, as well. From this, the authors concluded that the structure of water in the clusters is essentially that of water in the bulk state. This would seem reasonable considering that the water uptake of these samples corresponded to an average of 21 H2O molecules per —SOa" group. 

Catcd, quantities Total intramolecular correlation factor <7intra d (In gintra) I d T... 

The temperature and frequency dependence of the complex dielectric permittivity a for both 2-chlorocydohexyi isobutyrate (CCHI) and poly(2-chlorocyclohexyl acrylate) (PCCHA) is reported. The analysis of the dielectric results in terms of the electric modulus suggests that whereas the conductive processes in CCHI are produced only by free charges, the conductivity observed in PCCHA involves both free charges and interfacial phenomena. The 4x4 RIS scheme presented which accounts for two rotational states about the CH-CO bonds of the side group reproduces the intramolecular correlation coefficient of the polymer. 

Combining the above results, we write the intramolecular correlation function as... 

K. Szalewicz and B. Jeziorski. Symmetry adapted, double perturbation analysis of intramolecular correlation effects in weak intermolecular interactions. Molec. Phys., 38 191, 1979. 

We will use the basis vectors (1) where > i2 and apply equation (4) when needed. For two tlu electrons, our basis (1) consists of 15 different state vectors I/) (for two holes, the fivefold hu degeneracy leads to 45 states). In the following we will study the intramolecular correlations of electrons (holes) within a multipole expansion of the two-body Coulomb potential V(r, f) = 1/lr —1 (charge e = unity). In terms of real spherical harmonics YJ, where r stands for m = 0,... 

Structural, energetic, and electronic aspects of a series of porphyrins and related compounds in their neutral, anion, and dianion electronic states were studied by means of DFT calculations [B3LYP/6-31+G(d,p)]. The analysis of the electron density put forward a number of different kinds of interactions between the inner atoms. Intramolecular correlations similar to the ones found in intermolecular N-H—N interactions have been found [214],... 

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 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 ... 

The new function w appearing in the derivation of (10.11) is called the intramolecular correlation function, which is defined for a pair of atoms a and 7 in a molecule by... 

Calculations of a similar nature have demonstrated that replacement of both hydrogens of water, yielding dimethyl ether, also has only a minor effect upon the nature of the H-bond in the water dimer. With their polarized basis set, and with inclusion of corrections for BSSE, dispersion, and intramolecular correlation effects, these authors found the first methyl substitution raises the binding energy by 0.5 kcal/mol and the second by 0.6. The authors cautioned that an unpolarized basis set would fail to pick up these small effects, which they attribute to Coulomb and dispersion components of the interaction. 

Here ujxx (k) is the Fourier transform of the solute intramolecular correlation function, and... 

Given the graphical expansion for the site-site total correlation function h y r) it is possible to generate on Omstein-Zernike like equation. We will adopt the physical approach that the total correlation function is the sum of all possible direct site-site correlations, both intramolecular correlations via an s -bond, or intermolecular correlations via a c, (r). This defines a site-site direct correlation function c y r). From the expansion (2.2.5), it follows that at most one s -bond is connected to each circle. 

As we discussed in Section II.B, site-site correlation functions provide a very useful formalism for describing the structure of fluids modeled with interaction site potentials. In this formalism, information equivalent to g l,2) is obtained from the set of site-site correlation functions and intramolecular correlation functions. For this reason, a great deal of effort has been put into the development of integral equation theories for these correlation functions. The seminal contribution in this area was made by Chandler and Andersen, who sought to write an integral equation of the Ornstein-Zernike form in which the set of site-site total correlation functions were related to a set of site-site direct correlation functions. Their equation has the form... 

---