Hyper-netted chain

The hyper-netted chain (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 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 hyper-netted chain 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. G. A. Martynov; G. N. Sarkisov "Exact equations and the theory of liquids. V" Molecular Physics 49 pp.1495-1504 (1983)
2. J. M. J. van Leeuwen, J. Groeneveld and J. de Boer "New method for the calculation of the pair correlation function. I" Physica 25 pp. 792-808 (1959)
3. Tohru Morita "Theory of Classical Fluids: Hyper-Netted Chain Approximation, I: Formulation for a One-Component System", Progress of Theoretical Physics 20 pp. 920 -938 (1958)
4. Tohru Morita "Theory of Classical Fluids: Hyper-Netted Chain Approximation. II: Formulation for Multi-Component Systems" Progress of Theoretical Physics 21 pp. 361-382 (1959)
5. Tohru Morita "Theory of Classical Fluids: Hyper-Netted Chain Approximation. III: A New Integral Equation for the Pair Distribution Function" Progress of Theoretical Physics 23 pp. 829-845 (1960)
6. Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics 23 pp. 1003-1027 (1960)
7. Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. II: Multicomponent Systems" Progress of Theoretical Physics 24 pp. 317-330 (1960)
8. Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. III: General Treatment of Classical Systems" Progress of Theoretical Physics 25 pp. 537-578 (1961)
9. G. S. Rushbrooke "On the hyper-chain approximation in the theory of classical fluids" Physica 26 pp. 259-265 (1960)
10. L. Verlet "On the Theory of Classical Fluids.", Il Nuovo Cimento 18 pp. 77- (1960)
11. Emmanuel Meeron "Nodal Expansions. III. Exact Integral Equations for Particle Correlation Functions", Journal of Mathematical Physics 1 pp. 192-201 (1960)
12. M. Kinoshita; M. Harada "Numerical solution of the HNC equation for ionic systems", Molecular Physics 65 pp. 599-618 (1988)