Editing Hyper-netted chain
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The | 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 (HNC) 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 | 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 | ||
the | 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 | The HNC closure 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 HNC 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). | |||
The | |||
==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)] | ||
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#[http://dx.doi.org/10.1143/PTP.25.537 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)] | #[http://dx.doi.org/10.1143/PTP.25.537 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)] | ||
#[http://dx.doi.org/10.1016/0031-8914(60)90020-3 G. S. Rushbrooke "On the hyper-chain approximation in the theory of classical fluids" Physica '''26''' pp. 259-265 (1960)] | #[http://dx.doi.org/10.1016/0031-8914(60)90020-3 G. S. Rushbrooke "On the hyper-chain approximation in the theory of classical fluids" Physica '''26''' pp. 259-265 (1960)] | ||
# | #[http://dx.doi.org/ NC_1960_18_0077_nolotengo] | ||
#[http://dx.doi.org/ | #[http://dx.doi.org/ JMP_1960_01_00192] | ||
#[http://dx.doi.org/10.1080/00268978800101271 M. Kinoshita; M. Harada "Numerical solution of the HNC equation for ionic systems", Molecular Physics '''65''' pp. 599-618 (1988)] | #[http://dx.doi.org/10.1080/00268978800101271 M. Kinoshita; M. Harada "Numerical solution of the HNC equation for ionic systems", Molecular Physics '''65''' pp. 599-618 (1988)] | ||