Hypernetted chain method

A number of approximate integral equations for the radial distribution function g(r) of fluids have been proposed in recent years. Two particularly useful approximations are the Percus-Yevick (PY)1,2 and the Convolution Hypernetted Chain (CHNC)3-4 equations. In this paper an efficient numerical method of solving these equations is described and the results obtained bv applying the method to the PY equation are discussed. A later paper will describe the behavior of the... 

Another possible approach solving the equilibrium distribution for an electric double layer is offered by integral equation theories [22]. They are based on approximate relationships between different distribution functions. The two most common theories are Percus-Yevick [23] and Hypernetted Chain approximation (HNQ [24], where the former is a good method for short range interactions and the latter is best for long-range interactions. They were both developed around 1960, but are still used. The correlation between two particles can be divided into two parts, one is the direct influence of particle j on particle i and the other originates from the fact that all other particles correlate with particle j and then influence particle i in precisely... 

In the layer with decreased relative permittivity surrounding the ions, the free energy of the solvent is lower than in the absence of the electric field of the ions. The approach of the ions towards one another requires the mutual inter-penetration of the solvate spheres, i.e., the release of a certain amount of solvent from ihc solvate sphere of the ions. This process needs work, and this work appears as a repulsive force between the ions, (This effect lends stability to electrolyte solutions, for in the absence of such repulsive forces, attraction between the charges would favour the precipitation of the solid salts.) By taking into account such repulsive forces, it was possible to interpret the positive deviation of the average activity coefficients of the ions from the Debye-Hiickel limiting law (hypernetted chain equations, HNC, calculation by the Monte Carlo method [Ra 68, Ra 70],... 

From a theoretical point of view the hypernetted chain (HNC) and Percus-Yevick (PY) equations are better approximations. Although both can be solved only by numerical methods, they offer the opportunity to study any model potential if appropriate computer facilities are available. 

Baxter (1968b) showed that the Ornstein-Zernike equation could, for some simple potentials, be written as two one-dimensional integral equations coupled by a function q(r). In the PY approximation for hard spheres, for instance, the q(r) functions are easily solved, and the direct-correlation function c(r) and the other thermodynamic properties can be obtained analytically. The pair-correlation function g(r) is derived from q(r) through numerical solution of the integral equation which governs g(r) for which a method proposed by Perram (1975) is especially useful. Baxter s method can also be used in the numerical solution of more complicated integral equations such as the hypernetted-chain (HNC) approximation in real space, avoiding the need to take Fourier transforms. An equivalent set of relations to Baxter s equations was derived earlier by Wertheim (1964). 

Theory nowadays overcomes the limitation of concentration range by integral equation and simulation methods. Mean spherical approximation (MSA) and hypernetted chain approximation (HNC) are the most important features yielding modern analytical transport equations over extended concentration ranges. Nevertheless, the IcCM expressions maintain their importance. They are reliable expressions for the determination of limiting values of the transport properties at infinite dilution of the electrolyte as a convenient basis for the provision of... 

There are a number of many-body systems which exhibit quantum effects on a macroscopic scale. These include liquid and crystal states of both He-3 and He-4, the electron gas, and neutron matter which probably constitutes the interior of pulsors. In addition, "nuclear matter" - a hypothetical extensive system of nucleons has been studied for the insight one may gain into the nature of finite nuclei. The theoretical studies of these systems have by now a long history, but are by no means concluded. In the last few years, significant advances have been made. This has come in part from the maturity of and gradual unification of many-body theory, in part from the development and application of powerful new expansion procedures, especially varieties of hypernetted-chain equations (3 ) and finally to the growing power of computer simulation methods for quantum systems. This article is intended as a review of some recent development in computational methods for extensive quantum systems, and of the relation between results so obtained to the evolution of other theoretical work. 

The results summarised here are for pure water at the temperature 25°C and the density 1.000 g cm , and are obtained by solving numerictdly the Ornstein-Zemike (OZ) equation for the pair correlation functions, using a closure that supplements the hypernet-ted chain (HNC) approximation with a bridge function. The bridge function is determined from computer simulations as described below. The numerical method for solving the OZ equation is described by Ichiye and Haymet and by Duh and Haymet. ... 

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