Hypernetted-chain

D. Beglov and B. Roux. Numerical solutions of the hypernetted chain equation for a solute of arbitrary geometry in three dimensions. J. Chem. Phys., 103 360-364, 1995. 

The general equation can be further reduced to the case of infinite dilution limit, a binary mixmre, ionic solutions, and so on. These equations are supplemented by closure relations such as the Percus-Yevick (PY) and hypernetted chain (HNC) approximations. 

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. 

One should perhaps mention some other closures that are discussed in the literature. One possibility is to combine the PY approximation for the hard core part of the potential and then use the HNC approximation to compute the corrections due to the attractive forces. Such an approach is called the reference hypernetted chain or RHNC approximation [48,49]. Recently, some new closures for a mixture of hard spheres have been proposed. These include one by Rogers and Young [50] (RY) and the Martynov-Sarkisov [51] (MS) closure as modified by Ballone, Pastore, Galli and Gazzillo [52] (BPGG). The RY and MS/BPGG closure relations take the forms... 

When all terms in Eq. (54) are neglected, Eqs. (32)-(34) form a closed set, which can be solved. This kind of approximation has been called by Zhou and Stell the hydrostatic hypernetted chain approximation (HHNCl) [91] a procedure for the numerical solution of the HHNCl equations can be found in Ref. 45. 

We apply the singlet theory for the density profile by using Eqs. (101) and (103) to describe the behavior of associating fluids close to a crystalline surface [120-122], First, we solve the multidensity OZ equation with the Percus-Yevick closure for the bulk partial correlation functions, and next calculate the total correlation function via Eq. (68) and the direct correlation function from Eq. (69). The bulk total direct correlation function is used next as an input to the singlet Percus-Yevick or singlet hypernetted chain equation, (6) or (7), to obtain the density profiles. The same approach can be used to study adsorption on crystalline surfaces as well as in pores with walls of crystalline symmetry. 

To solve the replica OZ equations, they must be completed by closure relations. Several closures have been tested against computer simulations for various models of fluids adsorbed in disordered porous media. In particular, common Percus-Yevick (PY) and hypernetted chain approximations have been applied [20]. Eq. (21) for the matrix correlations can be solved using any approximation. However, it has been shown by Given and Stell [17-19] that the PY closure for the fluid-fluid correlations simplifies the ROZ equation, the blocking effects of the matrix structure are neglected in this... 

Friedman (1971). This approach is described as the hypernetted chain procedure and in it ion-ion pair potentials are expressed as the sum of four terms. These are ... 

Similarly, Wei et al. [18] made analogous predictions of C < 0 using hypernetted-chain theory and incorporating molecular polarizability. However, the discussion was again restricted to the plausibility of negative C under -control. 

Figure 13. Comparison of prediction of the wall-PRISM theory with the hypernetted chain...

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... 

One such equation derived from statistical mechanics is the hypernetted-chain (HNC) approximation,... 

An analysis of clusters expansion to higher order (as compared to PY equation) leads to the hypernetted-chain (HNC) approximation [44—46]. In other words, directly solving the OZ relation in conjunction with Eq. (28) is possible, under a drastic assumption on B(r). The total correlation function is given simply by... 

Other closure relations are the hypernetted chain approximation (HNQ, defined by [25,32]... 

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... 

One more relation is required to achieve closure, i.e., to determine the two types of correlation functions. The most commonly used relations are the Percus-Yevick (PY) and the hypernetted chain (HNC) approximations [47-49]. From graph or diagram expansion of the total correlation function in powers of the density n(r) and resummation, an exact relation between the total and direct correlation functions is obtained, namely... 

A closure similar to the RHNC is the modified hypernetted chain (MHNC), introduced by Rosenfeld and Ashcroft [67]. This approximation is based on the empirical observation that the bridge functions for a wide variety of pair potentials belong to the same family of curves. This means that the bridge functions calculated for a suitable reference fluid can be used to a good approximation for another fluid. The reference fluid is usually... 

An approximate functional for Ap, is derived in Matubayasi et al. [15,16] and Takahashi et al. [19]. The functional is constructed by adopting the Percus-Yevick-type approximation in the unfavorable region of the solute-solvent interaction and the hypernetted-chain-type approximation in the favorable region. Ap, is then given... 

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