Hyper-netted chain

From SklogWiki
Jump to: navigation, search

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 hyper-netted chain 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 hyper-netted chain closure relation 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)

where \Phi(r) 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]

  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)