Editing Reference hyper-netted chain

The '''reference hyper-netted chain''' (RHNC) is sometimes referred to as the modified-HNC (MHNC) (Ref. 1)
Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':
 "...the bridge functions.. ..constitute the same family of curves, irrespective of the assumed pair potential"
The basis of the method is to solve the modified HNC equation
(with inclusion of the one-parameter bridge functions
appropriate to hard spheres), and determine the only free parameter <math>\eta</math>
(related to the hard-sphere diameter) by requiring thermodynamic consistency.
Fred Lado  (Ref. 2) and Rosenfeld and Ashcroft (1979) (Ref. 3)
noticed that the OZ equation can always be written in the form
:<math>\gamma_{12} =  \rho \int_V (h_{13} - \gamma_{13}) h_{23} ~{\rm d}(3)</math>
In view of this a hybrid solution between the [[HNC| hyper-netted chain]]
approximation and the Malijevsky-Labik formula for hard spheres (Ref. 4 and 5)
was proposed. The philosophy of this method is that the  bridge functional of the
liquid is fairly similar to that of the hard sphere fluid.
(See also Ref.s 6 and 7)
The RHNC closure is given by (Eq. 17 Ref. 7)
:<math> c\left(r\right) = h(r) - \ln [g(r) e^{\beta \phi(r)}] + B_0(r) </math>
along with the constraint (Eq. 18 Ref. 7)
:<math>\rho \int [g(r) - g_0(r)] \delta B_0(r) dr_3 = 0 </math>
Incorporating a reference potential <math>\phi_0(r)= \phi_0(r;\sigma,\epsilon)</math>
this equation becomes  (Eqs. 19a and 19b in Ref. 7)
:<math>\rho \int [g(r) - g_0(r)] \sigma \frac{\partial B_0(r)}{\partial \sigma}dr_3 = 0 </math>
and 
:<math>\rho \int [g(r) - g_0(r)] \epsilon \frac{\partial B_0(r)}{\partial \epsilon} dr_3 = 0 </math>
These are the conditions that will determine the optimum values of <math>\sigma</math> and <math>\epsilon</math>
that minimise the free energy. 
The RHNC satisfies the 'Hiroike' termodynamic relation test Ref. 7 and 9, i.e.
:<math>\left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p</math>
For highly asymmetric mixtures see Ref. 9.

==References==
#[http://dx.doi.org/10.1103/PhysRevA.20.1208 Yaakov Rosenfeld and N. W. Ashcroft "Theory of simple classical fluids: Universality in the short-range structure", Physical Review A '''20''' pp. 1208 - 1235 (1979)]
#[http://dx.doi.org/10.1103/PhysRevA.8.2548  F. Lado "Perturbation Correction for the Free Energy and Structure of Simple Fluids", Physical Review A    '''8''' 2548 - 2552 (1973)]
#[http://dx.doi.org/10.1080/00268978700100441 Anatol Malijevský and Stanislav Labík "The bridge function for hard spheres", Molecular Physics, '''60''' pp. 663-669 (1987)]
#[http://dx.doi.org/10.1080/00268978900101181 Stanislav Labík and Anatol Malijevský "Bridge function for hard spheres in high density and overlap regions", Molecular Physics, '''67''' pp. 431-438 (1989)]
#[http://dx.doi.org/10.1016/0375-9601(82)90207-9  F. Lado  "A local thermodynamic criterion for the reference-hypernetted chain equation", Physics Letters A '''89''' pp. 196-198 (1982)]
#[http://dx.doi.org/10.1103/PhysRevA.28.2374  F. Lado, S. M. Foiles and N. W. Ashcroft, "Solutions of the reference-hypernetted-chain equation with minimized free energy", Physical Review A '''28''' 2374 - 2379 (1983)]
#[http://dx.doi.org/10.1143/JPSJ.12.326 Kazuo Hiroike "Radial Distribution Function of Fluids I", Journal of the Physical Society of Japan '''12''' pp. 326-334 (1957)]
#[http://dx.doi.org/10.1143/JPSJ.12.864 Kazuo Hiroike "Radial Distribution Function of Fluids II", Journal of the Physical Society of Japan '''12''' pp. pp. 864-873 (1957)]
#[http://dx.doi.org/10.1063/1.2102891     S. Amokrane, A. Ayadim, and J. G. Malherbe "Structure of highly asymmetric hard-sphere mixtures: An efficient closure of the Ornstein-Zernike equations", Journal of Chemical Physics, '''123''' 174508 (2005)]



[[Category: Integral equations]]