Integral equations hypernetted chain

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. 

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... 

Much more flexible is to describe the solute-solvent system within the supercell technique employing a set of 3D plane waves. The convolution in the OZ equation can be then approximated by means of the 3D fast Fourier transform (3D-FFT) procedure. For the first time, this approach was elaborated by Beglov and Roux [16] for numerical solution of the 3D-OZ integral equation with the hypernetted chain (HNC) closure for the 3D distribution of a simple LJ solvent around non-polar solutes... 

During the last 20 years, a quantitative description of conductance and self-diffusion up to 1-M solutions has been achieved by the use of modern gij functions coming from integral equation techniques such as the hypernetted chain (HNC) equation or mean spherical approximation... 

To complete our theoretical treatment of liquids, we need a procedure to calculate g(r). It so happens that there is no exact equation for g(r), but there are several accurate approximate equations. Four equations that have had some success for fluids are the Kirkwood equation, the Born-Green-Yvon equation, the Percus-Yevick equation and the Hypernetted chain equation (McQuarrie, 1976). All four of these equations are integral equations for g(r) in terms of u(r). 

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... 

In the previous section we have described how to implement TPTl for a mixture of Lennard-Jones chains with a FENE bonding potential. Before considering binary mixtures, however, we shall restrict our attention to the particular case of a one component system of polymers. In order to describe the thermodynamic properties of such a system, we will consider two TPTl implementations, which we denote TPTl-MSA and TPTl-RHNC. In TPTl-MSA, we employ the fiilly analytic equation of state described in the previous section. In TPTl-RHNC, the Lennard-Jones reference system is described by means of the Reference Hypernetted Chain theory (RHNC). This is an integral equation theory which can only be solved numerically. 

In order to get these values, Ramanathan and Friedman and later on Ramanathan, Krishnan and Friedman used the hypernetted chain theory (HNC). This integral equation theory is supposed to be a somewhat better theoretical approach than PB at the primitive model level, but this is not really important in this context. What matters is that the following values were obtained for A+ (the hydration overlap energy when a cation... 

From the preceding sections, it seems evident that a real description of ion specificities in solutions can only be done if the geometry and the properties of water molecules are explicitly taken into account. Such models are called non-primitive or Born-Oppenheimer models. In the 1970s and 1980s, they were developed in two different directions. In particular, integral equation theories, such as the hypernetted chain (HNC) approach, were extended to include angle-dependent interaction potentials. The site-site Ornstein-Zernike equation with a HNC-like closure and the molecular Ornstein-Zernike equation are examples. For more information, see Ref. 17. 

Indeed, nature is much more comphcated. Even systems that show the same Hofmeister series may have different interactions that nnderpin the behaviour. As ion behaviour depends on the environment, it is also difficult to interpret widely different experiments by the same type of interactions. That the same series is found in two experiments is not a proof that the same interactions govern both experiments. This is a frequent mistake. How subtle these interactions can be is illustrated in Chap. 11, where the potentials of mean force from molecular dynamics simulations are taken and inserted into Poisson-Boltzmann calculations. Figure 4 of this chapter demonstrates that the double layer pressure increases in the non-Hofrneister sequence NaCl < Nal < NaBr at large separations (> 1.0 nm). Frequent changes in this sequence occur as the surfaces approach closer to each other, including NaCl < NaBr < Nal and NaCl > NaBr > Nal. Even if the underlying approximations do not fully reflect the real world, this result is far too complicated to be interpreted by simple rules. Note that integral equations such as the hypernetted chain theory are a more powerful alternative to the Poisson-Boltzmann equation, see Chap. 10. 

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