Hypemetted chains

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 ... 

P. Attard, G. N. Patey. Hypemetted-chain closure with bridge diagrams. Asymmetric hard sphere mixtures. J Chem Phys 92 4970-4982, 1990. 

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... 

Lantto, L.J. (1980) Fermi hypemetted-chain calculations ofthe electron-gas correlations, Phys. Rev., B22, 1380-1393. 

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]. 

From the various possible closures, the mean spherical approximation (MSA) [189] has found particularly wide attention in phase equilibrium calculations of ionic fluids. The Percus-Yevick (PY) closure is unsatisfactory for long-range potentials [173, 187, 190]. The hypemetted chain approximation (HNC), widely used in electrolyte thermodynamics [168, 173], leads to an increasing instability of the numerical algorithm as the phase boundary is approached [191]. There seems to be no decisive relation between the location of this numerical instability and phase transition lines [192-194]. Attempts were made to extrapolate phase transition lines from results far away, where the HNC is soluble [81, 194]. 

An extension of the HNC approach is the reference hypemetted chain (RHNC) approximation [52-54]. In this approach, studying a system preliminarily requires that we determine the properties of a reference fluid (RF). Usually, it has to be assumed that the properties of the actual system are closed to those of the RF. For example, the interaction potential of the system under study can be written as the sum of the RF potential and of a small perturbation, namely, u r) = wRF(r) + Au(r). Correspondingly, one has y(r) = yKF(r) + Ay(r), and the direct correlation function is given by... 

Having at our disposal accurate structural and thermodynamic quantities for HS fluid, the latter has been naturally considered as a RF. Although real molecules are not hard spheres, mapping their properties onto those of an equivalent HS fluid is a desirable goal and a standard procedure in the liquid-state theory, which is known as the modified hypemetted chain (MHNC) approximation. According to Rosenfeld and Ashcroft [27], it is possible to postulate that the bridge function of the actual system of density p reads... 

The main question is whether the hydrated ions behave as hard spheres while this seems plausible for ions much larger than the water molecules, it is probably not entirely applicable to small ions, whose hydration shells continuously change. Marcelj a calculated recently the double layer interaction8 using the anisotropic hypemetted chain method and potentials of mean force between pairs of ions in water, provided by Monte Carlo simulations. This... 

Bom-Green-Yvon 1.3.69 Carnahan-Starling I.3.69ff hypemetted chain 1.3.69 Percus-Yevick 1.3.69... 

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. 

Bacquet, R. and P. Rossky. (1984). Ionic Atmosphere of Rodlike Polyelectrolytes. A Hypemetted Chain Study. J. Phys. Chem. 88 2660. 

Figure 9.13 Energy per He atom as a fiinction of ID density from the diffusion Monte Carlo (DMC) calculation (circles and foil curve) of Boninsegni and Moroni [58]. Crosses are hypemetted chain results of Krotscheck and Miller [57]. Analogous DMC results were obtained by Gordillo, Boronat, Casulleras [59].

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... 

FIG. 17 Distribution functions P(r) for the systems with Manning parameter 4.2 and 10.5 from Figure 16. The solid lines are the results from the molecular dynamics simulations while the dashed lines are the predictions from hypemetted chain theory. (See Ref. 36.)... 

Indeed, f is found first to increase with t, but from a certain value on it decreases and would finally even become negative. Note that this would reverse the drift direction in electrophoresis measurements. While nonlinear and linearized PB theory coincide with the data and with each other for small Manning parameter, they completely fail to predict the back-bending, which already sets in at comparatively small values of t Hypemetted chain... 

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