Hyper-netted chain: Difference between revisions

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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 [[Cluster diagrams | elementary clusters]], <math>E(r)</math>, in the exact convolution equation for <math>g(r)</math>. The  hyper-netted chain  approximation was developed almost simultaneously by various
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 [[Cluster diagrams | elementary clusters]], <math>E(r)</math>, in the exact convolution equation for <math>g(r)</math>. 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),
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. <math> B(r) =0 </math>, thus
Rushbrooke, 1960 (Ref. 9), Verlet, 1960 (Ref. 10), and Meeron, 1960 (Ref. 11). The hyper-netted chain omits the [[bridge function]], i.e. <math> B(r) =0 </math>, thus
the  [[cavity correlation function]] becomes
the  [[cavity correlation function]] becomes
:<math>\ln y (r) =  h(r)  -c(r) \equiv \gamma (r)</math>
:<math>\ln y (r) =  h(r)  -c(r) \equiv \gamma (r)</math>
The  hyper-netted chain closure can be written as
The  hyper-netted chain [[Closure relations | closure relation]] can be written as
:<math>f \left[ \gamma (r) \right] = e^{[-\beta \Phi (r) + \gamma (r)]} - \gamma (r) -1</math>
:<math>f \left[ \gamma (r) \right] = e^{[-\beta \Phi (r) + \gamma (r)]} - \gamma (r) -1</math>
or
or
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or (Eq. 12 Ref. 1)
or (Eq. 12 Ref. 1)
:<math> c\left( r \right)= g(r) - \omega(r) </math>
:<math> c\left( r \right)= g(r) - \omega(r) </math>
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).
where <math>\Phi(r)</math> 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==
==References==
#[http://dx.doi.org/10.1080/00268978300102111 G. A. Martynov; G. N. Sarkisov "Exact equations and the theory of liquids. V" Molecular Physics '''49''' pp.1495-1504 (1983)]
#[http://dx.doi.org/10.1080/00268978300102111 G. A. Martynov; G. N. Sarkisov "Exact equations and the theory of liquids. V" Molecular Physics '''49''' pp.1495-1504 (1983)]

Latest revision as of 22:54, 1 April 2011

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]

  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)
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