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}.