Combining rules

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



Diaz Peña-Pando-Renuncio[edit]

[2] [3]


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

\epsilon_{ij} = \frac{2 \epsilon_i \epsilon_j}{\epsilon_i + \epsilon_j}


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):

\sigma_{ij} = \sqrt{\sigma_{ii} \sigma_{jj}}

Hudson and McCoubrey[edit]


Hogervorst rules[edit]

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

\epsilon_{12} = \frac{2 \epsilon_{11} \epsilon_{22}}{\epsilon_{11} + \epsilon_{22}}


\alpha_{12}=\frac{1}{2} (\alpha_{11} + \alpha_{22})

Kong rules[edit]

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

 \epsilon_{ij}\sigma_{ij}^{12} =

Kong-Chakrabarty rules[edit]

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

\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}}
\frac{\sigma_{12}}{\alpha_{12}}= \frac{1}{2} \left( \frac{\sigma_{11}}{\alpha_{11}} +  \frac{\sigma_{22}}{\alpha_{22}} \right)


\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}}

Lorentz-Berthelot rules[edit]

The Lorentz rule is given by [13]

\sigma_{ij} = \frac{\sigma_{ii} + \sigma_{jj}}{2}

which is only really valid for the hard sphere model.

The Berthelot rule is given by [14]

\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}}

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]


Waldman-Hagler rules[edit]

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

r_{ij}^0 = \left( \frac{ (r_i^0)^6 + (r_j^0)^6 }{2} \right)^{1/6}


\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)


  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)

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