Hyper-netted chain: Difference between revisions

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, $E(r)$ , 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]

@@ Line 1: / Line 1: @@
-The HNC equation has a clear physical basis in the Kirkwood superposition approximation \cite{MP_1983_49_1495}.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
+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 \cite{P_1959_25_0792}, Morita and Hiroike, 1960 \cite{PTP_1958_020_0920,PTP_1959_021_0361,PTP_1960_023_0829,PTP_1960_023_1003,PTP_1960_024_0317,PTP_1961_025_0537},
+groups, namely: van Leeuwen, Groeneveld and de Boer, 1959 (Ref. 2). Morita and Hiroike, 1960 (Ref.s 3-8),
-Rushbrooke, 1960 \cite{P_1960_26_0259}, Verlet, 1960 \cite{NC_1960_18_0077_nolotengo}, and Meeron, 1960 \cite{JMP_1960_01_00192}. 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>
-The HNC closure can be written as (5.7)
+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>
 or
-:<math>c(r)= h(r) - \beta \Phi(r) - \ln {\rm g}(r)</math>
+:<math>c\left(r\right)= h(r) - \beta \Phi(r) - \ln {\rm g}(r)</math>
-or (Eq. 12 \cite{MP_1983_49_1495})
+or (Eq. 12 Ref. 1)
 :<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 \cite{MP_1988_65_0599}.
+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==
+#[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/10.1016/0031-8914(59)90004-7  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)]
+#[http://dx.doi.org/10.1143/PTP.20.920  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)]
+#[http://dx.doi.org/10.1143/PTP.21.361 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)]
+#[http://dx.doi.org/10.1143/PTP.23.829 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)]
+#[http://dx.doi.org/10.1143/PTP.23.1003 Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. I" Progress of Theoretical Physics '''23''' pp. 1003-1027 (1960)]
+#[http://dx.doi.org/10.1143/PTP.24.317 Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. II: Multicomponent Systems" Progress of Theoretical Physics '''24''' pp. 317-330 (1960)]
+#[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)]
+#L. Verlet "On the Theory of Classical Fluids.", Il Nuovo Cimento '''18''' pp. 77- (1960)
+#[http://dx.doi.org/10.1063/1.1703652 Emmanuel Meeron "Nodal Expansions. III. Exact Integral Equations for Particle Correlation Functions", Journal of Mathematical Physics '''1''' pp.  192-201 (1960)]
+#[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)]
+[[Category: Integral equations]]

Hyper-netted chain: Difference between revisions

Latest revision as of 22:54, 1 April 2011

References[edit]

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