Hyper-netted chain: Difference between revisions
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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, <math>E(r)</math>, in the exact convolution equation for <math>g(r)</math>. The 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, <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 HNC omits the Bridge function, i.e. <math> B(r) =0 </math>, thus | ||
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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 | 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== | ==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)] |
Revision as of 16:30, 26 February 2007
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 HNC omits the Bridge function, i.e. , thus the cavity correlation function becomes
The HNC closure can be written as
or
or (Eq. 12 Ref. 1)
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
- 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)