Reference hyper-netted chain

(Note: the reference-HNC (RHNC) is sometimes 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 (Ref. 2) and Rosenfeld and Ashcroft (1979) (Ref. 3) noticed that the OZ equation can always be written in the form

${\gamma }_{12}=\rho {\int }_V(h_{13}-{\gamma }_{13})h_{23}\mathrm{d}(3)$

In view of this a hybrid solution between the HNC 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)

$c(r)=h(r)-\ln [g(r)e^{\beta \varphi (r)}]+B_0(r)$

along with the constraint (Eq. 18 Ref. 7)

$\rho \int [g(r)-g_0(r)]\delta B_0(r)d{r}_3=0$

Incorporating a reference potential ${\varphi }_0(r)={\varphi }_0(r;\sigma ,ϵ)$ this equation becomes (Eqs. 19a and 19b in Ref. 7)

$\rho \int [g(r)-g_0(r)]\sigma \frac{\partial B_0(r)}{\partial \sigma }d{r}_3=0$

and

$\rho \int [g(r)-g_0(r)]ϵ\frac{\partial B_0(r)}{\partial ϵ}d{r}_3=0$

These are the conditions that will determine the optimum values of $\sigma$ and $ϵ$ that minimise the free energy. The RHNC satisfies the 'Hiroike' termodynamic relation test Ref. 7 and 8, i.e.

${\frac{\partial U}{\partial V}|}_T=T{\frac{\partial p}{\partial T}|}_V-p$

For highly asymmetric mixtures see Ref. 9.

References

1. [PRA_1979_20_001208]
2. [PRA_1973_08_002548]
3. [PRA_1979_20_001208]
4. [MP_1987_60_0663]
5. [MP_1989_67_0431]
6. [PLA_1982_89_0196]
7. [PRA_1983_28_002374]
8. [JPSJ_1957_12_00326]
9. [JCP_2005_123_174508]