Integral equation theory

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... 

Sohweizer K S and Curro J G 1997 Integral equation theories of the struoture, thermodynamios and phase transitions of polymer fluids Adv. Chem. Phys. 98 1... 

One important class of integral equation theories is based on the reference interaction site model (RISM) proposed by Chandler [77]. These RISM theories have been used to smdy the confonnation of small peptides in liquid water [78-80]. However, the approach is not appropriate for large molecular solutes such as proteins and nucleic acids. Because RISM is based on a reduction to site-site, solute-solvent radially symmetrical distribution functions, there is a loss of infonnation about the tliree-dimensional spatial organization of the solvent density around a macromolecular solute of irregular shape. To circumvent this limitation, extensions of RISM-like theories for tliree-dimensional space (3d-RISM) have been proposed [81,82],... 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

M. Lozada-Cassou, E. Diaz-Herrera. Three-point extension for hypernetted chain and other integral equation theories numerical results. J Chem Phys 92 1194-1210, 1990. 

The subjeet of previous seetions has been to review and diseuss some theo-retieal approaehes and eomputer algorithms used to deseribe nonuniform assoeiating fluids. However, many studies have been earried out by using eomputer simulation teehniques. Computer simulation provides a powerful tool enabling a detailed study of the inner strueture and thermodynamie properties of ehemieally assoeiating fluids. In partieular, it may be used to determine several struetural properties sueh as, for example, the angular distributions of eomplexes formed due to assoeiation, that are not aeeessible from elassieal integral equation theories. 

D. Henderson. Integral equation theories for inhomogeneous fluids. In D. Henderson, ed. Fundamentals of Inhomogeneous Fluids. New York Marcel Dekker, 1992. 

Saa = -esB = e was chosen, and an effective coordination number Zcir = 2.5 occurred ) Thus one finds that the coefficient in the relation T N ,r is somewhat smaller than expected from Eq. (3), and in addition there is a "chain end"-correction. However, an integral equation theoryyielded Tc cc Vn, and th simulations showed that this result clearly is incorrect. More recent improved versions of integral equation theories now yield Tc N as well but still the theoretical understanding of the prefactor in this relation is limited. 

The role of electrolyte is critical in these nanoscopic interfaces, but is difficult to predict and quantify. For sufficiently large rigid interfacial structures, one can apply the model of electrolyte interaction with a single charged surface in Figure 1(a). The double-layer theories or the recent integral-equation theories have been applied. Reviews of this subject are available in the literature [4,5]. For electrolytes in a nanostructure, the double layers from two surfaces overlap and behave differently from the case of a single surface. Ad-... 

In the theoretical approaches of Poisson-Boltzmann, modified Gouy-Chapman (MGC), and integral equation theories such as HNC/MSA, concentration or density profiles of counterions and coions are calculated with consideration of the ion-waU and ion-ion in-... 

Heine, D. R., Grest, G. S. and Curro,J. G. Structure of Polymer Melts and Blends Comparison of Integral Equation theory and Computer Sumulation. Vol. 173, pp. 209-249. 

Integral equation theories are widely used in the theoretical study of liquids. There are two broad classes of integral equation theories those based on the Bom-Green-Yvon (BGY) hierarchy and those based on the Omstein-Zemike (OZ) equation [88]. Although the formalism is exact in both classes, it is generally easier to fashion approximations in the case of the OZ-equation-based approach, and this type of theory has therefore been more popular. Surprisingly, the BGY approach has never been implemented for nonuniform polymers, and this section is therefore restricted to a discussion of the OZ-equation-based approach. 

The integral equation theory is a simple means of studying the density profiles of dense polymer melts at surfaces where the structure is dominated by... 

Note added in -proof. The application of the usual integral equation theories of the liquid state 2> to water has not been successful.1) A recent study by H. C. Andersen [J. Chem. Phys. 61, 4985 (1974)] promises to change this situation. Briefly, Andersen reformulates the well known Mayer cluster expansion of the distribution function 2> by consistently taking into account the saturation of interaction characteristic of hydrogen bonding. Approximations are selected which satisfy this saturation condition at each step of the analysis. Preliminary calculations (H. C. Andersen, private communication) indicate that even low order approximations that preserve the saturation condition lead to qualitative be-... 

Tbe results for water are mainly available from computer simulations, although some results of integral equation theories have been reported in recent years. Despite the varied nature of the intermolecular potential energy functions used, most of these studies give a qualitatively similar picture. As an example, we consida in some detail the results obtained for the potential energy function given above (for the interface between water and Pt). 

From the many tools provided by statistical mechanics for determining the EOS [36, 173, 186-188] we consider first integral equation theories for the pair correlation function gxp(ra,rp) of spherical ions which relates the density of ion / at location rp to that of a at ra. In most theories gafi(ra,rp) enters in the form of the total correlation function hxp(rx,rp) = gxp(rx,rp) — 1. The Omstein-Zemike (OZ) equation splits up hap(rx,rp) into the direct correlation function cap(ra, rp) for pair interactions plus an indirect term that reflects these interactions mediated by all other particles y ... 

From the view point of statistical mechanical theory there are little more than plausible arguments about the universality class of primitive models. The MSA is analytical and thus it yields classical exponents. Therefore, applications to calculate critical exponents or to prove the apparent mean-field criticality of ionic systems [230] are not meaningful [300]. By construction, other integral equation theories also are also not suitable for determining critical exponents [301], The ultimate goal of an RG analysis of the RPM or UPM has not yet been reached. 

Mode coupling theory of binary mixtures where the constituents are of rather different sizes is a challenging task, as we have already discussed while addressing the mass depenence of diffusion. In addition to the problem with proper formulation of mode coupling terms, there is an additional difficulty of the nonavailability of the equilibrium two-particle correlation functions The existing integral equation theories become unstable when the size ratio exceeds a certain (low) value, like 1.5 or so [195],... 

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