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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'') for binary [[mixtures]]  are variously given by
==Böhm-Ahlrichs==
====Admur and Mason====
<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>
For the [[second virial coefficient]] of a mixture
==Diaz Peña-Pando-Renuncio==
<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>
<ref>[http://dx.doi.org/10.1063/1.442726 M. Diaz Peña, C. Pando, and J. A. R. Renuncio "Combination rules for intermolecular potential parameters. I. Rules based on approximations for the long-range dispersion energy", Journal of Chemical Physics  '''76''' pp. 325- (1982)]</ref>
:<math>B_{ij} = \frac{\left(B_{ii}^{1/3}+B_{jj}^{1/3}\right)^3}{8}</math>
<ref>[http://dx.doi.org/10.1063/1.442727 M. Diaz Peña, C. Pando, and J. A. R. Renuncio "Combination rules for intermolecular potential parameters. II. Rules based on approximations for the long-range dispersion energy and an atomic distortion model for the repulsive interactions", Journal of Chemical Physics '''76''' pp. 333- (1982)]</ref>
====Berthelot rule====
==Fender-Halsey==
:<math>\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}}</math>
The Fender-Halsey combining rule for the [[Lennard-Jones model]] is given by <ref>[http://dx.doi.org/10.1063/1.1701284 B. E. F. Fender and G. D. Halsey, Jr. "Second Virial Coefficients of Argon, Krypton, and Argon-Krypton Mixtures at Low Temperatures", Journal of Chemical Physics '''36''' pp. 1881-1888 (1962)]</ref>
====Hudson and McCoubrey====
:<math>\epsilon_{ij} = \frac{2 \epsilon_i \epsilon_j}{\epsilon_i + \epsilon_j}</math>
==Gilbert-Smith==
The Gilbert-Smith rules for the [[Born-Huggins-Meyer potential]]<ref>[http://dx.doi.org/10.1063/1.1670463 T. L. Gilbert "Soft‐Sphere Model for Closed‐Shell Atoms and Ions", Journal of Chemical Physics '''49''' pp. 2640- (1968)]</ref><ref>[http://dx.doi.org/10.1063/1.431848 T. L. Gilbert, O. C. Simpson, and M. A. Williamson "Relation between charge and force parameters of closed‐shell atoms and ions", Journal of Chemical Physics '''63''' pp. 4061- (1975)]</ref><ref>[http://dx.doi.org/10.1103/PhysRevA.5.1708 Felix T. Smith "Atomic Distortion and the Combining Rule for Repulsive Potentials", Physical Review A '''5''' pp. 1708-1713 (1972)]</ref>.
==Good-Hope rule==
The Good-Hope rule for [[Mie potential |Mie]]–[[Lennard-Jones model |Lennard‐Jones]] or [[Buckingham potential]]s <ref>[http://dx.doi.org/10.1063/1.1674022  Robert J. Good and Christopher J. Hope "New Combining Rule for Intermolecular Distances in Intermolecular Potential Functions", Journal of Chemical Physics '''53''' pp. 540- (1970)]</ref> is given by (Eq. 2):
 
:<math>\sigma_{ij} = \sqrt{\sigma_{ii} \sigma_{jj}}</math>
==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>
====Kong rules====
==Hogervorst rules==
<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>
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>:
====Lorentz rule====
 
:<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==
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>):
 
:<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>
 
and
 
:<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==
The Lorentz rule is given by <ref>[http://dx.doi.org/10.1002/andp.18812480110 H. A. Lorentz "Ueber die Anwendung des Satzes vom Virial in der kinetischen Theorie der Gase", Annalen der Physik '''12''' pp. 127-136 (1881)]</ref>
:<math>\sigma_{ij} = \frac{\sigma_{ii} + \sigma_{jj}}{2}</math>
:<math>\sigma_{ij} = \frac{\sigma_{ii} + \sigma_{jj}}{2}</math>
See also [[Lennard-Jones model]]
 
====Tang and Toennies====
which is only really valid for the [[hard sphere model]].
 
The Berthelot rule is given by <ref>[http://visualiseur.bnf.fr/Document/CadresPage?O=NUMM-3082&I=1703 Daniel Berthelot "Sur le mélange des gaz", Comptes rendus hebdomadaires des séances de l’Académie des Sciences, '''126''' pp. 1703-1855 (1898)]</ref>
 
:<math>\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}}</math>
 
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/10.1063/1.1610435 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== 
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>.
==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==
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>.
==Tang and Toennies==
<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==
<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>
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:
 
:<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/>
'''Related reading'''
'''Related reading'''
*[http://dx.doi.org/10.1063/1.442726 M. Diaz Peña, C. Pando, and J. A. R. Renuncio "Combination rules for intermolecular potential parameters. I. Rules based on approximations for the long-range dispersion energy", Journal of Chemical Physics  '''76''' pp. 325- (1982)]
*[http://dx.doi.org/10.1021/ja00046a032 Thomas A. Halgren "The representation of van der Waals (vdW) interactions in molecular mechanics force fields: potential form, combination rules, and vdW parameters", Journal of the American Chemical Society '''114''' pp. 7827-7843 (1992)]
*[http://dx.doi.org/10.1063/1.442727 M. Diaz Peña, C. Pando, and J. A. R. Renuncio "Combination rules for intermolecular potential parameters. II. Rules based on approximations for the long-range dispersion energy and an atomic distortion model for the repulsive interactions", Journal of Chemical Physics '''76''' pp. 333- (1982)]
*[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)]
*[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)]
[[category: mixtures]]
[[category: mixtures]]

Latest revision as of 11:48, 16 January 2015

The combining rules are geometric expressions designed to provide the interaction energy between two dissimilar non-bonded atoms (here labelled and ). Most of the rules are designed to be used with a specific interaction potential in mind. (See also Mixing rules).

Böhm-Ahlrichs[edit]

[1]

Diaz Peña-Pando-Renuncio[edit]

[2] [3]

Fender-Halsey[edit]

The Fender-Halsey combining rule for the Lennard-Jones model is given by [4]

Gilbert-Smith[edit]

The Gilbert-Smith rules for the Born-Huggins-Meyer potential[5][6][7].

Good-Hope rule[edit]

The Good-Hope rule for MieLennard‐Jones or Buckingham potentials [8] is given by (Eq. 2):

Hudson and McCoubrey[edit]

[9]

Hogervorst rules[edit]

The Hogervorst rules for the Exp-6 potential [10]:

and

Kong rules[edit]

The Kong rules for the Lennard-Jones model are given by (Table I in [11]):

Kong-Chakrabarty rules[edit]

The Kong-Chakrabarty rules for the Exp-6 potential [12] are given by (Eqs. 2-4):

and

Lorentz-Berthelot rules[edit]

The Lorentz rule is given by [13]

which is only really valid for the hard sphere model.

The Berthelot rule is given by [14]

These rules are simple and widely used, but are not without their failings [15] [16] [17] [18].

Mason-Rice rules[edit]

The Mason-Rice rules for the Exp-6 potential [19].

Srivastava and Srivastava rules[edit]

The Srivastava and Srivastava rules for the Exp-6 potential [20].

Sikora rules[edit]

The Sikora rules for the Lennard-Jones model [21].

Tang and Toennies[edit]

[22]

Waldman-Hagler rules[edit]

The Waldman-Hagler rules [23] are given by:

and

References[edit]

  1. Hans‐Joachim Böhm and Reinhart Ahlrichs "A study of short‐range repulsions", Journal of Chemical Physics 77 pp. 2028- (1982)
  2. M. Diaz Peña, C. Pando, and J. A. R. Renuncio "Combination rules for intermolecular potential parameters. I. Rules based on approximations for the long-range dispersion energy", Journal of Chemical Physics 76 pp. 325- (1982)
  3. M. Diaz Peña, C. Pando, and J. A. R. Renuncio "Combination rules for intermolecular potential parameters. II. Rules based on approximations for the long-range dispersion energy and an atomic distortion model for the repulsive interactions", Journal of Chemical Physics 76 pp. 333- (1982)
  4. B. E. F. Fender and G. D. Halsey, Jr. "Second Virial Coefficients of Argon, Krypton, and Argon-Krypton Mixtures at Low Temperatures", Journal of Chemical Physics 36 pp. 1881-1888 (1962)
  5. T. L. Gilbert "Soft‐Sphere Model for Closed‐Shell Atoms and Ions", Journal of Chemical Physics 49 pp. 2640- (1968)
  6. T. L. Gilbert, O. C. Simpson, and M. A. Williamson "Relation between charge and force parameters of closed‐shell atoms and ions", Journal of Chemical Physics 63 pp. 4061- (1975)
  7. Felix T. Smith "Atomic Distortion and the Combining Rule for Repulsive Potentials", Physical Review A 5 pp. 1708-1713 (1972)
  8. Robert J. Good and Christopher J. Hope "New Combining Rule for Intermolecular Distances in Intermolecular Potential Functions", Journal of Chemical Physics 53 pp. 540- (1970)
  9. 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)
  10. W. Hogervorst "Transport and equilibrium properties of simple gases and forces between like and unlike atoms", Physica 51 pp. 77-89 (1971)
  11. 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)
  12. 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)
  13. H. A. Lorentz "Ueber die Anwendung des Satzes vom Virial in der kinetischen Theorie der Gase", Annalen der Physik 12 pp. 127-136 (1881)
  14. Daniel Berthelot "Sur le mélange des gaz", Comptes rendus hebdomadaires des séances de l’Académie des Sciences, 126 pp. 1703-1855 (1898)
  15. 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)
  16. 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)
  17. 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)
  18. 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)
  19. Edward A. Mason and William E. Rice "The Intermolecular Potentials of Helium and Hydrogen", Journal of Chemical Physics 22 pp. 522- (1954)
  20. 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)
  21. P. T. Sikora "Combining rules for spherically symmetric intermolecular potentials", Journal of Physics B: Atomic and Molecular Physics 3 pp. 1475- (1970)
  22. 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)
  23. 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)

Related reading