# 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, , in the exact convolution equation for . 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 hyper-netted chain omits the bridge function, i.e. , thus
the cavity correlation function becomes

The hyper-netted chain closure relation can be written as

or

or (Eq. 12 Ref. 1)

where is the intermolecular pair potential. 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 hyper-netted chain equation for ionic systems (see Ref. 12).

## References[edit]

- G. A. Martynov; G. N. Sarkisov "Exact equations and the theory of liquids. V" Molecular Physics
**49**pp.1495-1504 (1983) - 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) - 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) - 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) - 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) - Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics
**23**pp. 1003-1027 (1960) - Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. II: Multicomponent Systems" Progress of Theoretical Physics
**24**pp. 317-330 (1960) - 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) - G. S. Rushbrooke "On the hyper-chain approximation in the theory of classical fluids" Physica
**26**pp. 259-265 (1960) - L. Verlet "On the Theory of Classical Fluids.", Il Nuovo Cimento
**18**pp. 77- (1960) - Emmanuel Meeron "Nodal Expansions. III. Exact Integral Equations for Particle Correlation Functions", Journal of Mathematical Physics
**1**pp. 192-201 (1960) - M. Kinoshita; M. Harada "Numerical solution of the HNC equation for ionic systems", Molecular Physics
**65**pp. 599-618 (1988)