Reference hyper-netted chain

(Note: the reference-HNC (RHNC) is somtimes referred to as the modified-HNC (MHNC) Ref.1).

Rosenfeld and Ashcroft (1979) (Ref. 1) 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

$\gamma _{12}=\rho \int _{V}(h_{13}-\gamma _{13})h_{23}~{\rm {d}}(3)$

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})

$c\left(r\right)=h(r)-\ln[g(r)e^{\beta \phi (r)}]+B_{0}(r)$

along with the constraint (Eq. 18 \cite{PRA_1983_28_002374})

$\rho \int [g(r)-g_{0}(r)]\delta B_{0}(r)dr_{3}=0$

Incorporating a reference potential $\phi_0(r)= \phi_0(r;\sigma,\epsilon)$ this equation becomes (Eqs. 19a and 19b \cite{PRA_1983_28_002374})

$\rho \int [g(r)-g_{0}(r)]\sigma {\frac {\partial B_{0}(r)}{\partial \sigma }}dr_{3}=0$

and

$\rho \int [g(r)-g_{0}(r)]\epsilon {\frac {\partial B_{0}(r)}{\partial \epsilon }}dr_{3}=0$

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.

$\left.{\frac {\partial U}{\partial V}}\right\vert _{T}=T\left.{\frac {\partial p}{\partial T}}\right\vert _{V}-p$

For highly asymmetric mixtures see \cite{JCP_2005_123_174508}.

References

1. [PRA_1979_20_001208]