# 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"
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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 Ornstein-Zernike relation 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 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)

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

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

$\rho \int [g(r) - g_0(r)] \delta B_0(r) dr_3 = 0$

where $\Phi(r)$ is the intermolecular pair potential. Incorporating a reference potential $\Phi_0(r)= \Phi_0(r;\sigma,\epsilon)$ this equation becomes (Eqs. 19a and 19b in Ref. 7)

$\rho \int [g(r) - g_0(r)] \sigma \frac{\partial B_0(r)}{\partial \sigma}dr_3 = 0$

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 Ref. 7 and 9, 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 Ref. 9.