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The '''combining rules''' are geometric expressions designed to provide the interaction energy between two dissimilar non-bonded atoms (here labelled <math>i</math> and <math>j</math>). Most of the rules are designed to be used with a specific [[Idealised models| interaction potential]] in mind. (''See also'' [[Mixing rules]]).
The '''combining rules''' (also known as ''mixing rules'') are geometric expressions designed to provide the interaction energy between two dissimilar non-bonded atoms (here labelled <math>i</math> and <math>j</math>). Most of the rules are designed to be used with a specific [[Idealised models| interaction potential]] in mind.
==Admur and Mason==
For the [[second virial coefficient]] of a mixture
<ref>[http://dx.doi.org/10.1063/1.1724353 I. Amdur and E. A. Mason "Properties of Gases at Very High Temperatures",  Physics of Fluids '''1''' pp. 370-383 (1958)]</ref>
:<math>B_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}</math>
==Böhm-Ahlrichs==
==Böhm-Ahlrichs==
<ref>[http://dx.doi.org/10.1063/1.444057 Hans‐Joachim Böhm and Reinhart Ahlrichs "A study of short‐range repulsions", Journal of Chemical Physics '''77''' pp. 2028- (1982)]</ref>
<ref>[http://dx.doi.org/10.1063/1.444057 Hans‐Joachim Böhm and Reinhart Ahlrichs "A study of short‐range repulsions", Journal of Chemical Physics '''77''' pp. 2028- (1982)]</ref>
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==Hudson and McCoubrey==
==Hudson and McCoubrey==
<ref>[http://dx.doi.org/10.1039/TF9605600761 G. H. Hudson and J. C. McCoubrey "Intermolecular forces between unlike molecules. A more complete form of the combining rules", Transactions of the Faraday Society '''56''' pp.  761-766 (1960)]</ref>
<ref>[http://dx.doi.org/10.1039/TF9605600761 G. H. Hudson and J. C. McCoubrey "Intermolecular forces between unlike molecules. A more complete form of the combining rules", Transactions of the Faraday Society '''56''' pp.  761-766 (1960)]</ref>
==Hogervorst rules==
The Hogervorst rules for the [[Exp-6 potential]] <ref>[http://dx.doi.org/10.1016/0031-8914(71)90138-8    W. Hogervorst "Transport and equilibrium properties of simple gases and forces between like and unlike atoms", Physica '''51''' pp. 77-89 (1971)]</ref>:
:<math>\epsilon_{12} = \frac{2 \epsilon_{11} \epsilon_{22}}{\epsilon_{11} + \epsilon_{22}}</math>
and
:<math>\alpha_{12}=\frac{1}{2} (\alpha_{11} + \alpha_{22})</math>
==Kong rules==
==Kong rules==
The Kong rules for the [[Lennard-Jones model]] are given by (Table I in
<ref>[http://dx.doi.org/10.1063/1.1680358 Chang Lyoul Kong "Combining rules for intermolecular potential parameters. II. Rules for the Lennard-Jones (12–6) potential and the Morse potential", Journal of Chemical Physics '''59''' pp. 2464-2467 (1973)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1680358 Chang Lyoul Kong "Combining rules for intermolecular potential parameters. II. Rules for the Lennard-Jones (12–6) potential and the Morse potential", Journal of Chemical Physics '''59''' pp. 2464-2467 (1973)]</ref>):
 
:<math>\epsilon_{ij}\sigma_{ij}^{6}=\left(\epsilon_{ii}\sigma_{ii}^{6}\epsilon_{jj}\sigma_{jj}^{6}\right)^{1/2}</math>
 
:<math> \epsilon_{ij}\sigma_{ij}^{12} =
\left[
\frac{
  (\epsilon_{ii}\sigma_{ii}^{12})^{1/13}
  +
  (\epsilon_{jj}\sigma_{jj}^{12})^{1/13}
  }{2}
\right]^{13}
</math>
==Kong-Chakrabarty  rules==
The Kong-Chakrabarty rules for the [[Exp-6 potential]] <ref>[http://dx.doi.org/10.1021/j100640a019 Chang Lyoul Kong , Manoj R. Chakrabarty "Combining rules for intermolecular potential parameters. III. Application to the exp 6 potential", Journal of Physical Chemistry '''77''' pp. 2668-2670 (1973)]</ref> are given by (Eqs. 2-4):
 
:<math>\left[ \frac{\epsilon_{12}\alpha_{12} e^{\alpha_{12}}}{(\alpha_{12}-6)\sigma_{12}} \right]^{2\sigma_{12}/\alpha_{12}}=
\left[ \frac{\epsilon_{11}\alpha_{11} e^{\alpha_{11}}}{(\alpha_{11}-6)\sigma_{11}} \right]^{\sigma_{11}/\alpha_{11}}
\left[ \frac{\epsilon_{22}\alpha_{22} e^{\alpha_{22}}}{(\alpha_{22}-6)\sigma_{22}} \right]^{\sigma_{22}/\alpha_{22}}
</math>


:<math>\frac{\sigma_{12}}{\alpha_{12}}= \frac{1}{2} \left( \frac{\sigma_{11}}{\alpha_{11}} +  \frac{\sigma_{22}}{\alpha_{22}} \right)</math>
<math>\epsilon_{ij}\sigma_{ij}^{6}=\left(\epsilon_{ii}\sigma_{ii}^{6}\epsilon_{jj}\sigma_{jj}^{6}\right)^{\frac{1}{2}}</math>


and
<math> \epsilon_{ij}\sigma_{ij}^{12} = \frac{\epsilon_{ii}\sigma_{ii}^{12}}{2^{13}}\left[ 1+\left( \frac{\epsilon_{jj}\sigma_{jj}^{12}}{\epsilon_{ii}\sigma_{ii}^{12}} \right)^{\frac{1}{13}}\right]^{13} </math>
 
:<math>\frac{\epsilon_{12}\alpha_{12}\sigma_{12}^6}{(\alpha_{12}-6)} = \left[\frac{\epsilon_{11}\alpha_{11}\sigma_{11}^6}{(\alpha_{11}-6)} \frac{\epsilon_{22}\alpha_{22}\sigma_{22}^6}{(\alpha_{22}-6)}   \right]^{\frac{1}{2}}</math>


==Lorentz-Berthelot rules==
==Lorentz-Berthelot rules==
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These rules are simple and widely used, but are not without their failings <ref>[http://dx.doi.org/10.1080/00268970010020041 Jérôme Delhommelle; Philippe Millié "Inadequacy of the Lorentz-Berthelot combining rules for accurate predictions of equilibrium properties by molecular simulation", Molecular Physics '''99''' pp. 619-625  (2001)]</ref>
These rules are simple and widely used, but are not without their failings <ref>[http://dx.doi.org/10.1080/00268970010020041 Jérôme Delhommelle; Philippe Millié "Inadequacy of the Lorentz-Berthelot combining rules for accurate predictions of equilibrium properties by molecular simulation", Molecular Physics '''99''' pp. 619-625  (2001)]</ref>
<ref>[http://dx.doi.org/10.1080/00268970802471137 Dezso Boda and Douglas Henderson "The effects of deviations from Lorentz-Berthelot rules on the properties of a simple mixture", Molecular Physics '''106''' pp. 2367-2370 (2008)]</ref>
<ref>[http://dx.doi.org/10.1080/00268970802471137 Dezso Boda and Douglas Henderson "The effects of deviations from Lorentz-Berthelot rules on the properties of a simple mixture", Molecular Physics '''106''' pp. 2367-2370 (2008)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1610435 W. Song, P. J. Rossky, and M. Maroncelli "Modeling alkane+perfluoroalkane interactions using all-atom potentials: Failure of the usual combining rules", Journal of Chemical Physics '''119''' pp. 9145- (2003)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1610435 W. Song, P. J. Rossky, and M. Maroncelli "Modeling alkane+perfluoroalkane interactions using all-atom potentials: Failure of the usual combining rules", Journal of Chemical Physics '''119''' pp. 9145- (2003)]</ref>.
<ref>[http://dx.doi.org/10.1063/1.4867498  Caroline Desgranges and Jerome Delhommelle "Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. III. Impact of combining rules on mixtures properties", Journal of Chemical Physics '''140''' 104109 (2014)]</ref>.


==Mason-Rice rules==
==Mason-Rice rule==
The Mason-Rice rules for the [[Exp-6 potential]] <ref>[http://dx.doi.org/10.1063/1.1740100 Edward A. Mason and William E. Rice "The Intermolecular Potentials of Helium and Hydrogen", Journal of Chemical Physics '''22''' pp. 522- (1954)]</ref>.
The Mason-Rice rule for the [[Exp-6 potential]] <ref>[http://dx.doi.org/10.1063/1.1740100 Edward A. Mason and William E. Rice "The Intermolecular Potentials of Helium and Hydrogen", Journal of Chemical Physics '''22''' pp. 522- (1954)]</ref>.
==Srivastava and Srivastava rules==
The Srivastava and Srivastava rules for the [[Exp-6 potential]] <ref>[http://dx.doi.org/10.1063/1.1742786  B. N. Srivastava and K. P. Srivastava "Combination Rules for Potential Parameters of Unlike Molecules on Exp‐Six Model", Journal of Chemical Physics '''24''' pp. 1275-1276 (1956)]</ref>.
==Sikora rules==
==Sikora rules==
The Sikora rules for the [[Lennard-Jones model]] <ref>[http://dx.doi.org/10.1088/0022-3700/3/11/008 P. T. Sikora "Combining rules for spherically symmetric intermolecular potentials", Journal of Physics B: Atomic and Molecular Physics '''3''' pp. 1475- (1970)]</ref>.
The Sikora rules for the [[Lennard-Jones model]] <ref>[http://dx.doi.org/10.1088/0022-3700/3/11/008 P. T. Sikora "Combining rules for spherically symmetric intermolecular potentials", Journal of Physics B: Atomic and Molecular Physics '''3''' pp. 1475- (1970)]</ref>.
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<ref>[http://dx.doi.org/10.1007/BF01384663 K. T. Tang and J. Peter Toennies "New combining rules for well parameters and shapes of the van der Waals potential of mixed rare gas systems", Zeitschrift für Physik D Atoms, Molecules and Clusters '''1''' pp. 91-101 (1986)]</ref>
<ref>[http://dx.doi.org/10.1007/BF01384663 K. T. Tang and J. Peter Toennies "New combining rules for well parameters and shapes of the van der Waals potential of mixed rare gas systems", Zeitschrift für Physik D Atoms, Molecules and Clusters '''1''' pp. 91-101 (1986)]</ref>
==Waldman-Hagler rules==
==Waldman-Hagler rules==
The Waldman-Hagler rules <ref>[http://dx.doi.org/10.1002/jcc.540140909 M. Waldman and A. T. Hagler "New combining rules for rare-gas Van der-Waals parameters", Journal of Computational Chemistry '''14''' pp.  1077-1084 (1993)]</ref> are given by:
<ref>[http://dx.doi.org/10.1002/jcc.540140909 M. Waldman and A. T. Hagler "New combining rules for rare-gas Van der-Waals parameters", Journal of Computational Chemistry '''14''' pp.  1077-1084 (1993)]</ref>
 
:<math>r_{ij}^0 = \left( \frac{ (r_i^0)^6 + (r_j^0)^6 }{2} \right)^{1/6}</math>
 
and
 
:<math>\epsilon_{ij} = 2 \sqrt{\epsilon_i \cdot \epsilon_j} \left( \frac{ (r_i^0)^3 \cdot  (r_j^0)^3 }{ (r_i^0)^6  + (r_j^0)^6 }  \right)</math>
 
==References==
==References==
<references/>
<references/>
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