Editing Hyper-netted chain
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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 | 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 [[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 relations | closure relation]] 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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:<math> c\left( r \right)= g(r) - \omega(r) </math> | :<math> c\left( r \right)= g(r) - \omega(r) </math> | ||
where <math>\Phi(r)</math> is the [[intermolecular pair potential]]. | 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 | 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)] |