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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(r) =0 } , thus the cavity correlation function becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln y (r) = h(r) -c(r) \equiv \gamma (r)}

The HNC closure can be written as (5.7)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \left[ \gamma (r) \right] = e^{[-\beta \Phi (r) + \gamma (r)]} - \gamma (r) -1}

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(r)= h(r) - \beta \Phi(r) - \ln {\rm g}(r)}

or (Eq. 12 \cite{MP_1983_49_1495})

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c\left( r \right)= 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}.