Hyper-netted chain

The HNC equation has a clear physical basis in the Kirkwood superposition approximation \cite{MP_1983_49_1495}.The hyper-netted chain approximation is obtained by omitting the elementary clusters, $E(r)$, in the exact convolution equation for $g(r)$. The hyper-netted chain (HNC) approximation was developed almost simultaneously by various groups, namely: van Leeuwen, Groeneveld and de Boer, 1959 \cite{P_1959_25_0792}, Morita and Hiroike, 1960 \cite{PTP_1958_020_0920,PTP_1959_021_0361,PTP_1960_023_0829,PTP_1960_023_1003,PTP_1960_024_0317,PTP_1961_025_0537}, Rushbrooke, 1960 \cite{P_1960_26_0259}, Verlet, 1960 \cite{NC_1960_18_0077_nolotengo}, and Meeron, 1960 \cite{JMP_1960_01_00192}. The HNC omits the Bridge function, i.e. $B(r)=0$, thus the cavity correlation function becomes

$\ln y(r)=h(r)-c(r)\equiv \gamma (r)$

The HNC closure can be written as (5.7)

$f[\gamma (r)]=e^{[-\beta \Phi (r)+\gamma (r)]}-\gamma (r)-1$

or

$c(r)=h(r)-\beta \Phi (r)-\ln {\rm {g}}(r)$

or (Eq. 12 \cite{MP_1983_49_1495})

$c(r)=g(r)-\omega (r)$

The HNC approximation is well suited for long-range potentials, and in particular, Coulombic systems. For details of the numerical solution of the HNC for ionic systems see \cite{MP_1988_65_0599}.