Editing Reference hyper-netted chain

(Note: the reference-HNC (RHNC) is somtimes referred to as the modified-HNC (MHNC) \cite{PRA_1979_20_001208}).
 Rosenfeld and Ashcroft (1979) \cite{PRA_1979_20_001208}
proposed the `anzatz of universality':
{\it ``...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 $\eta$
(related to the hard-sphere diameter) by requiring thermodynamic consistency.\\
Fred Lado  \cite{PRA_1973_08_002548} and Rosenfeld and Ashcroft (1979) \cite{PRA_1979_20_001208}
noticed that the OZ equation can always be written in the form
\begin{equation}
\gamma_{12} =  \rho \int_V (h_{13} - \gamma_{13}) h_{23} ~{\rm d}(3)
\end{equation}
In view of this a hybrid solution between the HNC
approximation and the Malijevsky-Labik formula for hard spheres \cite{MP_1987_60_0663,MP_1989_67_0431}
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 \cite{PLA_1982_89_0196,PRA_1983_28_002374}.\\
The RHNC closure is given by (Eq. 17 \cite{PRA_1983_28_002374})
\begin{equation}
c(r) = h(r) - \ln [g(r) e^{\beta \phi(r)}] + B_0(r)
\end{equation} 
along with the constraint (Eq. 18 \cite{PRA_1983_28_002374})
\begin{equation}
\rho \int [g(r) - g_0(r)] \delta B_0(r) {\rm d}{\bf r}_3 = 0 
\end{equation}
Incorporating a reference potential $\phi_0(r)= \phi_0(r;\sigma,\epsilon)$
this equation becomes  (Eqs. 19a and 19b \cite{PRA_1983_28_002374})
\begin{equation}
\rho \int [g(r) - g_0(r)] \sigma \frac{\partial B_0(r)}{\partial \sigma} {\rm d}{\bf r}_3 = 0 
\end{equation}
and 
\begin{equation}
\rho \int [g(r) - g_0(r)] \epsilon \frac{\partial B_0(r)}{\partial \epsilon} {\rm d}{\bf r}_3 = 0 
\end{equation}
These are the conditions that will determine the optimum values of
$\sigma$ and $\epsilon$ that minimise the free energy. 
The RHNC satisfies the 'Hiroike' termodynamic relation test \cite{PRA_1983_28_002374,JPSJ_1957_12_00326}, i.e.
\begin{equation}
\left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p
\end{equation}
For highly asymmetric mixtures see \cite{JCP_2005_123_174508}.

==References==
@@ Line 1: / Line 1: @@
-The '''reference hyper-netted chain''' (RHNC) is sometimes referred to as the modified-HNC (MHNC) (Ref. 1)
+(Note: the reference-HNC (RHNC) is somtimes referred to as the modified-HNC (MHNC) \cite{PRA_1979_20_001208}).
-Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':
+ Rosenfeld and Ashcroft (1979) \cite{PRA_1979_20_001208}
- "...the bridge functions.. ..constitute the same family of curves, irrespective of the assumed pair potential"
+proposed the `anzatz of universality':
-The basis of the method is to solve the modified [[HNC]] equation
+{\it ``...the bridge functions.. ..constitute the same family of curves, irrespective of the assumed pair potential".}
-(with inclusion of the one-parameter [[bridge function]]s
+The basis of the method is to solve the modified HNC equation
-appropriate to [[hard sphere model | hard spheres]]), and determine the only free parameter <math>\eta</math>
+(with inclusion of the one-parameter bridge functions
-(related to the hard-sphere diameter) by requiring [[thermodynamic consistency]].
+appropriate to hard spheres), and determine the only free parameter $\eta$
-Fred Lado  (Ref. 2) and Rosenfeld and Ashcroft (1979) (Ref. 3)
+(related to the hard-sphere diameter) by requiring thermodynamic consistency.\\
-noticed that the [[Ornstein-Zernike relation]]  can always be written in the form
+Fred Lado  \cite{PRA_1973_08_002548} and Rosenfeld and Ashcroft (1979) \cite{PRA_1979_20_001208}
-:<math>\gamma_{12} =  \rho \int_V (h_{13} - \gamma_{13}) h_{23} ~{\rm d}(3)</math>
+noticed that the OZ equation can always be written in the form
-In view of this a hybrid solution between the [[HNC| hyper-netted chain]]
+\begin{equation}
-approximation and the Malijevsky-Labik formula for hard spheres (Ref. 4 and 5)
+\gamma_{12} =  \rho \int_V (h_{13} - \gamma_{13}) h_{23} ~{\rm d}(3)
-was proposed. The philosophy of this method is that the  bridge functional of the
+\end{equation}
-liquid is fairly similar to that of the hard sphere fluid.
+In view of this a hybrid solution between the HNC
-(See also Ref.s 6 and 7)
+approximation and the Malijevsky-Labik formula for hard spheres \cite{MP_1987_60_0663,MP_1989_67_0431}
-The RHNC closure is given by (Eq. 17 Ref. 7)
+was proposed.\\
-:<math> c\left(r\right) = h(r) - \ln [g(r) e^{\beta \Phi(r)}] + B_0(r) </math>
+The philosophy of this method is that the  bridge functional of the
-along with the constraint (Eq. 18 Ref. 7)
+liquid is fairly similar to that of the
-:<math>\rho \int [g(r) - g_0(r)] \delta B_0(r) dr_3 = 0 </math>
+hard sphere fluid.
-where <math>\Phi(r)</math> is the [[intermolecular pair potential]].
+See also \cite{PLA_1982_89_0196,PRA_1983_28_002374}.\\
-Incorporating a reference potential <math>\Phi_0(r)= \Phi_0(r;\sigma,\epsilon)</math>
+The RHNC closure is given by (Eq. 17 \cite{PRA_1983_28_002374})
-this equation becomes  (Eqs. 19a and 19b in Ref. 7)
+\begin{equation}
-:<math>\rho \int [g(r) - g_0(r)] \sigma \frac{\partial B_0(r)}{\partial \sigma}dr_3 = 0 </math>
+c(r) = h(r) - \ln [g(r) e^{\beta \phi(r)}] + B_0(r)
+\end{equation}
+along with the constraint (Eq. 18 \cite{PRA_1983_28_002374})
+\begin{equation}
+\rho \int [g(r) - g_0(r)] \delta B_0(r) {\rm d}{\bf r}_3 = 0
+\end{equation}
+Incorporating a reference potential $\phi_0(r)= \phi_0(r;\sigma,\epsilon)$
+this equation becomes  (Eqs. 19a and 19b \cite{PRA_1983_28_002374})
+\begin{equation}
+\rho \int [g(r) - g_0(r)] \sigma \frac{\partial B_0(r)}{\partial \sigma} {\rm d}{\bf r}_3 = 0
+\end{equation}
 and
-:<math>\rho \int [g(r) - g_0(r)] \epsilon \frac{\partial B_0(r)}{\partial \epsilon} dr_3 = 0 </math>
+\begin{equation}
-These are the conditions that will determine the optimum values of <math>\sigma</math> and <math>\epsilon</math>
+\rho \int [g(r) - g_0(r)] \epsilon \frac{\partial B_0(r)}{\partial \epsilon} {\rm d}{\bf r}_3 = 0
-that minimise the free energy.
+\end{equation}
-The RHNC satisfies the 'Hiroike' termodynamic relation test Ref. 7 and 9, i.e.
+These are the conditions that will determine the optimum values of
-:<math>\left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p</math>
+$\sigma$ and $\epsilon$ that minimise the free energy.
-For highly asymmetric mixtures see Ref. 9.
+The RHNC satisfies the 'Hiroike' termodynamic relation test \cite{PRA_1983_28_002374,JPSJ_1957_12_00326}, i.e.
+\begin{equation}
+\left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p
+\end{equation}
+For highly asymmetric mixtures see \cite{JCP_2005_123_174508}.
 ==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]]