Hypemetted-chain equation

Eq. (5) is useful when analyzing different approximations in the theory of inhomogeneous fluids. In particular, if all the terms involving third- and higher-order correlations in the right-hand side of Eq. (5) are neglected, and if Pi(ro))P2( o)i )Pv( o) are chosen as the densities of species for a uniform system at temperature T and the chemical potentials p,, the singlet hypemetted chain equation (HNCl) [50] results... 

Both closures have been employed for determining the distribution functions for liquids 182 the Percus-Yevick equation tends to yield better results for nonpolar systems, while the hypemetted-chain equation (with the appropriate renormalization of long-range interactions)113,183 is found to be more appropriate for polar and ionic liquids.184... 

These asymptotic results were compared with numerical results obtained by solving the hypemetted chain equation (HNC) for the pair distribution function, with Monte Carlo (MC) simulations and with a mean field approximation. The agreement between the numerical results and the asymptotic results was fairly good at distances above h — 1.5 nm. Substitution of the... 

Kovalenko A, Ten-No S, Hirata F Solution of three-dimensional reference interaction site model and hypemetted chain equations for simple point charge water by modified method of direct inversion in iterative subspace, J Comput Chem 20(9) 928-936, 1999. 

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

There are two approaches connnonly used to derive an analytical connection between g(i-) and u(r) the Percus-Yevick (PY) equation and the hypemetted chain (FfNC) equation. Both are derived from attempts to fomi fimctional Taylor expansions of different correlation fimctions. These auxiliary correlation functions include ... 

This approximation amounts to truncating the functional expansion of the excess free energy at second order in the density profile. This approach is accurate for Lennard-Jones fluids under some conditions, but has fallen out of favor because it is not capable of describing wetting transitions and coexisting liquid-vapor phases [105-107]. Incidentally, this approximation is identical to the hypemetted chain closure to the wall-OZ equation [103]. 

The free energy profile for the electron transfer reaction in a polar solvent is examined based on the extended reference interaction site method (ex-RISM) applying it to a simple model of a charge separation reaction which was previously studied by Carter and Hynes with molecular dynamics simulations. Due to the non-linear nature of the hypemetted chain (HNC) closure to solve the RISM equation, our method can shed light on the non-linearity of the free energy profiles, and we discuss these problems based on the obtained free energy profile. 

D — and D oo the smooth and generic dimension-dependence of the integrals enables one to interpolate reasonably accurate D = 3 values (rms error 1%) from the dimensional limit results. The interpolated integrals can be used either on their ovm, or in conjunction with an integral equation approximation which sums some subset of the required integrals exactly (such as the hypemetted-chain or Percus-Yevick methods) the combination methods are invariably better than either dimensional interpolation or integral equations alone. Interpolation-corrected Percus-Yevick values can be computed quite easily at arbitrary order however, errors in higher-order values are... 

In Enderby et (12) was simply assumed on the basis of an observation made by Johnson et that it is satisfied in the Bom-Green, Percus-Yevick, and hypemetted-chain approximations. [The results of our appendix are formally rather than rigorously exact, because we do not fully investigate the convergence properties of the series expansions we use, and we assume certain plausible smoothness properties of h(l 2). Thus we have given a demonstration, in the sense of Kac, rather than a proof of our equations—see footnote 2 of Mermin for a characterization of the distinction.]... 

Once potential parameters have been determined, we can start calculation downward following arrow in the figure. The first key quantity is radial distribution function g(r) which can be calculated by the use of theoretical relation such as Percus-Yevick (PY) or Hypemetted chain (HNC) integral equation. However, these equations are an approximations. Exact values can be obtained by molecular simulation. Ifg(r) is obtained accurately as functions of temperature and pressure, then all the equilibrium properties of fluids and fluid mixtures can be calculated. Moreover, information on fluid structure is contained in g(r) itself. 

A fundamental approach to liquids is provided by the integral equation methods (sometimes called distribution function methods), initiated by Kirkwood and Yvon in the 1930s. As we shall show below, one starts by writing down an exact equation for the molecular distribution function of interest, usually the pair function, and then introduces one or more approximations to solve the problem. These approximations are often motivated by considerations of mathematical simplicity, so that their validity depends on a posteriori agreement with computer simulation or experiment. The theories in question, called YBG (Yvon-Bom-Green), PY (Percus-Yevick), and the HNC (hypemetted chain) approximation, provide the distribution functions directly, and are thus applicable to a wide variety of properties. 

Two of the classic integral equation approximations for atomic liquids are the PY (Percus-Yevick) and the HNC (hypemetted chain) approximations that use the following closures... 

Computer simulations are not the only methods which can be used to calculate the dielectric constant of pure liquids. Other approaches are given by the use of integral equations, in particular, the hypemetted chain (HNC) molecular integral equation and the molecular Omstein-Zemike (OZ) theory (see Section 8.7.1 for details on such methodologies). 

A third approach is to inject particles based on a grand canonical ensemble distribution. At each predetermined molecular dynamics time step, the probability to create or destroy a particle is calculated and a random number is used to determine whether the update is accepted (the probability for both the creation and the destruction of a particle must be equal to ensure reversibility). The probability function depends on the excess chemical potential and must be calculated in a way that is consistent with the microscopic model used to describe the system. In the work of Im et al., a primitive water model is used, and the chemical potential is determined through an analytic solution to the Ornstein-Zernike equation using the hypemetted chain as a closure relation. This method is very accurate from the physical viewpoint, but it has a poorer CPU performance compared with simpler schemes based on... 

There have been other attempts to apply integral equations methods derived from Equation 3.51 or other similar expressions. One of them is the hypemetted chain (Henderson 1983), which is a generalization of the MSA theory, applicable to higher charge/potential values. It gives results comparable to the MPB ones but requires more extensive numerical evaluations. Another proposal is so-called dressed-ion theory of Kjellander and Mitchell (1994, 1997). 

Numerical results from the LOW Tables show how the saturation potential depends on Ka. Many analytical approximations have been developed for the spherical diffuse layer, but the advent of fast, efficient numerical schemes (HNC, hypemet-ted chain equation), as well as that of the mean spherical approximation (MSA) has diminished their utility (see further parts of this monograph). Moreover MSA and HNC approximations can be applied to the evaluation of most practical transport coefficients of electrolytes in a large variety of experimental situations, as it will be seen in other parts of this book. 

The function n,(l) is a member of the family of direct correlation functions (1,2,..), which is the sum of all irreducible graphs with density factors pi(l) for every field point (For a detailed discussion of correlation functions see for example Hansen and McDonald [66]). The integral equations can be obtained by differentiation of this magnitude. Functional series differentiation[67, 68] produces approximations, such as the Hypemetted Chain (HNC) and its modifications, and... 

Here, p is 1/A r. The hypemetted chain (HNC) equation (van Leeuwen et al, 1959) and the equation of Bom and Green (1949) involve similar terms, though they are in logarithmic form. The Bom-Green equation also contains the first derivative with respect to r, so it is an integrodifferential equation for g(r) in terms of V(r). 

Closure approximations to the PRISM equation are generally developed via an analogy with atomic liquids. Three of the common closures for hard spheres are the Percus-Yevick (PY), hypemetted chain (HNC), and Martynov-Sarkisov (MS) closures. It has been shown that the PY closure is the most accurate of the three, and in fact the HNC and MS closures have either no solution or unphysical solutions at low densities. The PY closure is given by. 

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