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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''' are geometric expressions designed to provide the interaction energy between two dissimilar non-bonded atoms (here labelled UNIQa5d53e6b5f34ffec-math-0000006C-QINU and UNIQa5d53e6b5f34ffec-math-0000006D-QINU). Most of the rules are designed to be used with a specific [[Idealised models| interaction potential]] in mind. (''See also'' [[Mixing rules]]). |
| ==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>
| | UNIQa5d53e6b5f34ffec-ref-0000006E-QINU |
| ==Diaz Peña-Pando-Renuncio== | | ==Diaz Peña-Pando-Renuncio== |
| <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>
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| <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>
| | UNIQa5d53e6b5f34ffec-ref-00000070-QINU |
| ==Fender-Halsey== | | ==Fender-Halsey== |
| 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> | | The Fender-Halsey combining rule for the [[Lennard-Jones model]] is given by UNIQa5d53e6b5f34ffec-ref-00000071-QINU |
| :<math>\epsilon_{ij} = \frac{2 \epsilon_i \epsilon_j}{\epsilon_i + \epsilon_j}</math> | | :UNIQa5d53e6b5f34ffec-math-00000072-QINU |
| ==Gilbert-Smith== | | ==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>. | | The Gilbert-Smith rules for the [[Born-Huggins-Meyer potential]]UNIQa5d53e6b5f34ffec-ref-00000073-QINUUNIQa5d53e6b5f34ffec-ref-00000074-QINUUNIQa5d53e6b5f34ffec-ref-00000075-QINU. |
| ==Good-Hope rule== | | ==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): | | The Good-Hope rule for [[Mie potential |Mie]]–[[Lennard-Jones model |Lennard‐Jones]] or [[Buckingham potential]]s UNIQa5d53e6b5f34ffec-ref-00000076-QINU is given by (Eq. 2): |
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| :<math>\sigma_{ij} = \sqrt{\sigma_{ii} \sigma_{jj}}</math> | | :UNIQa5d53e6b5f34ffec-math-00000077-QINU |
| ==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>
| | UNIQa5d53e6b5f34ffec-ref-00000078-QINU |
| ==Hogervorst rules== | | ==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>: | | The Hogervorst rules for the [[Exp-6 potential]] UNIQa5d53e6b5f34ffec-ref-00000079-QINU: |
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| :<math>\epsilon_{12} = \frac{2 \epsilon_{11} \epsilon_{22}}{\epsilon_{11} + \epsilon_{22}}</math> | | :UNIQa5d53e6b5f34ffec-math-0000007A-QINU |
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| :<math>\alpha_{12}=\frac{1}{2} (\alpha_{11} + \alpha_{22})</math> | | :UNIQa5d53e6b5f34ffec-math-0000007B-QINU |
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| ==Kong rules== | | ==Kong rules== |
| The Kong rules for the [[Lennard-Jones model]] are given by (Table I in | | 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>):
| | UNIQa5d53e6b5f34ffec-ref-0000007C-QINU): |
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| :<math>\epsilon_{ij}\sigma_{ij}^{6}=\left(\epsilon_{ii}\sigma_{ii}^{6}\epsilon_{jj}\sigma_{jj}^{6}\right)^{1/2}</math> | | :UNIQa5d53e6b5f34ffec-math-0000007D-QINU |
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| :<math> \epsilon_{ij}\sigma_{ij}^{12} = | | :UNIQa5d53e6b5f34ffec-math-0000007E-QINU |
| \left[
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| \frac{
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| (\epsilon_{ii}\sigma_{ii}^{12})^{1/13}
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| +
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| (\epsilon_{jj}\sigma_{jj}^{12})^{1/13}
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| }{2}
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| \right]^{13}
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| </math>
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| ==Kong-Chakrabarty rules== | | ==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): | | The Kong-Chakrabarty rules for the [[Exp-6 potential]] UNIQa5d53e6b5f34ffec-ref-0000007F-QINU are given by (Eqs. 2-4): |
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| :<math>\left[ \frac{\epsilon_{12}\alpha_{12} e^{\alpha_{12}}}{(\alpha_{12}-6)\sigma_{12}} \right]^{2\sigma_{12}/\alpha_{12}}= | | :UNIQa5d53e6b5f34ffec-math-00000080-QINU |
| \left[ \frac{\epsilon_{11}\alpha_{11} e^{\alpha_{11}}}{(\alpha_{11}-6)\sigma_{11}} \right]^{\sigma_{11}/\alpha_{11}}
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| \left[ \frac{\epsilon_{22}\alpha_{22} e^{\alpha_{22}}}{(\alpha_{22}-6)\sigma_{22}} \right]^{\sigma_{22}/\alpha_{22}}
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| </math>
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| :<math>\frac{\sigma_{12}}{\alpha_{12}}= \frac{1}{2} \left( \frac{\sigma_{11}}{\alpha_{11}} + \frac{\sigma_{22}}{\alpha_{22}} \right)</math> | | :UNIQa5d53e6b5f34ffec-math-00000081-QINU |
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| :<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> | | :UNIQa5d53e6b5f34ffec-math-00000082-QINU |
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| ==Lorentz-Berthelot rules== | | ==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> | | The Lorentz rule is given by UNIQa5d53e6b5f34ffec-ref-00000083-QINU |
| :<math>\sigma_{ij} = \frac{\sigma_{ii} + \sigma_{jj}}{2}</math> | | :UNIQa5d53e6b5f34ffec-math-00000084-QINU |
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| which is only really valid for the [[hard sphere model]]. | | which is only really valid for the [[hard sphere model]]. |
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| 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> | | The Berthelot rule is given by UNIQa5d53e6b5f34ffec-ref-00000085-QINU |
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| :<math>\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}}</math> | | :UNIQa5d53e6b5f34ffec-math-00000086-QINU |
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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 UNIQa5d53e6b5f34ffec-ref-00000087-QINU |
| <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>
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| <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>
| | UNIQa5d53e6b5f34ffec-ref-00000089-QINU |
| <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>.
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| ==Mason-Rice rules== | | ==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>. | | The Mason-Rice rules for the [[Exp-6 potential]] UNIQa5d53e6b5f34ffec-ref-0000008B-QINU. |
| ==Srivastava and Srivastava rules== | | ==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>. | | The Srivastava and Srivastava rules for the [[Exp-6 potential]] UNIQa5d53e6b5f34ffec-ref-0000008C-QINU. |
| ==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]] UNIQa5d53e6b5f34ffec-ref-0000008D-QINU. |
| ==Tang and Toennies== | | ==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>
| | UNIQa5d53e6b5f34ffec-ref-0000008E-QINU |
| ==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: | | The Waldman-Hagler rules UNIQa5d53e6b5f34ffec-ref-0000008F-QINU are given by: |
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| :<math>r_{ij}^0 = \left( \frac{ (r_i^0)^6 + (r_j^0)^6 }{2} \right)^{1/6}</math> | | :UNIQa5d53e6b5f34ffec-math-00000090-QINU |
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| :<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> | | :UNIQa5d53e6b5f34ffec-math-00000091-QINU |
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| ==References== | | ==References== |
| <references/>
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| '''Related reading''' | | '''Related reading''' |
| *[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/10.1021/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)] |
| [[category: mixtures]] | | [[category: mixtures]] |