Hyper-netted chain

The HNC equation has a clear physical basis in the Kirkwood superposition approximation (Ref. 1). 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 (Ref. 2). Morita and Hiroike, 1960 (Ref.s 3-8), Rushbrooke, 1960 (Ref. 9), Verlet, 1960 (Ref. 10), and Meeron, 1960 (Ref. 11). 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

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

or

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

or (Eq. 12 Ref. 1)

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

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 Ref. 12).

References

1. [MP_1983_49_1495]
2. [P_1959_25_0792]
3. [PTP_1958_020_0920]
4. [PTP_1959_021_0361]
5. [PTP_1960_023_0829]
6. [PTP_1960_023_1003]
7. [PTP_1960_024_0317]
8. [PTP_1961_025_0537]
9. [P_1960_26_0259]
10. [NC_1960_18_0077_nolotengo]
11. [JMP_1960_01_00192]
12. [MP_1988_65_0599]