Combining rules explained

In computational chemistry and molecular dynamics, the combination rules or combining rules are equations that provide the interaction energy between two dissimilar non-bonded atoms, usually for the part of the potential representing the van der Waals interaction.^[1] In the simulation of mixtures, the choice of combining rules can sometimes affect the outcome of the simulation.^[2]

Combining rules for the Lennard-Jones potential

The Lennard-Jones Potential is a mathematically simple model for the interaction between a pair of atoms or molecules.^{[3] [4]} One of the most common forms is

V _{LJ} = 4 \varepsilon \left[ \left( (\sigma)/(r) \right) ^{12} - \left( (\sigma)/(r) \right) ^{6} \right] where ε is the depth of the potential well, σ is the finite distance at which the inter-particle potential is zero, r is the distance between the particles. The potential reaches a minimum, of depth ε, when r = 2^1/6σ ≈ 1.122σ.

Lorentz-Berthelot rules

The Lorentz rule was proposed by H. A. Lorentz in 1881:^[5]

\sigma _{ij} = (\sigma _{ii} + \sigma _{jj})/(2)

The Lorentz rule is only analytically correct for hard sphere systems. Intuitively, since

\sigma _{i} , \sigma _{j} loosely reflect the radii of particle i and j respectively, their averages can be said to be the effective radii between the two particles at which point repulsive interactions become severe.

The Berthelot rule (Daniel Berthelot, 1898) is given by:^[6]

\epsilon _{ij} = \sqrt{\epsilon _{ii} \epsilon _{jj} }.

Physically, this arises from the fact that

\epsilon is related to the induced dipole interactions between two particles. Given two particles with instantaneous dipole \mu _{i,} \mu _{j} respectively, their interactions correspond to the products of \mu _{i,} \mu _{j} . An arithmetic average of \epsilon _{i} and \epsilon _{j} will not however, result in the average of the two dipole products, but the average of their logarithms would be.

These rules are the most widely used and are the default in many molecular simulation packages, but are not without failings.^{[7] [8] [9]}

Waldman-Hagler rules

The Waldman-Hagler rules are given by:^[10]

r ^{0} _{ij} = \left( ((r ^{0 )} _{i} ^{6} + (r ^{0 )} _{j} ^{6})/(2) \right) ^{1/6} and\epsilon _{ij} = 2 \sqrt{\epsilon _{i} ⋅ \epsilon _{j}} \left( ((r ^{0 )} _{i} ^{3} ⋅ (r ^{0 )} _{j} ^{3})/((r ^{0 )} _{i} ^{6} + (r ^{0 )} _{j} ^{6}) \right)

Fender-Halsey

The Fender-Halsey combining rule is given by ^[11]

\epsilon _{ij} = (2 \epsilon _{i} \epsilon _{j})/(\epsilon _{i} + \epsilon _{j})

Kong rules

The Kong rules for the Lennard-Jones potential are given by:^[12]

\epsilon _{ij} \sigma ^{6} _{ij} = \left(\epsilon _{ii} \sigma ^{6} _{ii} \epsilon _{jj} \sigma ^{6} _{jj} \right) ^{1/2} \epsilon _{ij} \sigma ^{12} _{ij} = \left[ ((\epsilon _{ii} \sigma ^{12} _{ii} ) ^{1/13} + (\epsilon _{jj} \sigma ^{12} _{jj} ) ^{1/13})/(2) \right] ^{13}

Others

Many others have been proposed, including those of Tang and Toennies^[13] Pena,^{[14] [15]} Hudson and McCoubrey^[16] and Sikora (1970).^[17]

Combining rules for other potentials

Good-Hope rule

The Good-Hope rule for Mie–Lennard‐Jones or Buckingham potentials is given by:^[18]

\sigma _{ij} = \sqrt{\sigma _{ii} \sigma _{jj} }

Hogervorst rules

The Hogervorst rules for the Exp-6 potential are:^[19]

\epsilon _{12} = (2 \epsilon _{11} \epsilon _{22})/(\epsilon _{11} + \epsilon _{22}) and\alpha _{12} = (1)/(2) (\alpha _{11} + \alpha _{22} )

Kong-Chakrabarty rules

The Kong-Chakrabarty rules for the Exp-6 potential are:^[20]

\left[ (\epsilon _{12} \alpha _{12} e ^{\alpha _{12}})/((\alpha _{12} -6 )\sigma _{12}) \right] ^{2 \sigma _{12} /\alpha _{12}} = \left[ (\epsilon _{11} \alpha _{11} e ^{\alpha _{11}})/((\alpha _{11} -6 )\sigma _{11}) \right] ^{\sigma _{11} /\alpha _{11}} \left[ (\epsilon _{22} \alpha _{22} e ^{\alpha _{22}})/((\alpha _{22} -6 )\sigma _{22}) \right] ^{\sigma _{22} /\alpha _{22}} (\sigma _{12})/(\alpha _{12}) = (1)/(2) \left( (\sigma _{11})/(\alpha _{11}) + (\sigma _{22})/(\alpha _{22}) \right)

and

(\epsilon _{12} \alpha _{12} \sigma ^{6} _{12})/((\alpha _{12} -6 )) = \left[ (\epsilon _{11} \alpha _{11} \sigma ^{6} _{11})/((\alpha _{11} -6 )) (\epsilon _{22} \alpha _{22} \sigma ^{6} _{22})/((\alpha _{22} -6 )) \right] ^{(1)/(2)}

Other rules for that have been proposed for the Exp-6 potential are the Mason-Rice rules^[21] and the Srivastava and Srivastava rules (1956).^[22]

Equations of state

Industrial equations of state have similar mixing and combining rules. These include the van der Waals one-fluid mixing rules

a _{mix} = \sum _{i} \sum _{j} y _{i} y _{j} a _{ij} b _{mix} = \sum _{i} y _{i} b _{i}

and the van der Waals combining rule, which introduces a binary interaction parameter

k _{ij} ,a _{ij} = \sqrt{a _{ii} a _{jj} }(1-k_).

There is also the Huron-Vidal mixing rule, and the more complex Wong-Sandler mixing rule, which equates excess Helmholtz free energy at infinite pressure between an equation of state and an activity coefficient model (and thus with liquid excess Gibbs free energy).

Notes and References

1. Halgren. Thomas A.. 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. September 1992. 114. 20. 7827–7843. 10.1021/ja00046a032.
2. Desgranges. Caroline. Delhommelle. Jerome. Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. III. Impact of combining rules on mixtures properties. The Journal of Chemical Physics. 14 March 2014. 140. 10. 104109. 10.1063/1.4867498. 24628154 . 2014JChPh.140j4109D .
3. Fischer . Johann . Wendland . Martin . October 2023 . On the history of key empirical intermolecular potentials . Fluid Phase Equilibria . en . 573 . 113876 . 10.1016/j.fluid.2023.113876. 2023FlPEq.57313876F . free .
4. Lenhard . Johannes . Stephan . Simon . Hasse . Hans . June 2024 . On the History of the Lennard-Jones Potential . Annalen der Physik . en . 536 . 6 . 10.1002/andp.202400115 . 0003-3804.
5. Lorentz. H. A.. Ueber die Anwendung des Satzes vom Virial in der kinetischen Theorie der Gase. Annalen der Physik. 1881. 248. 1. 127–136. 10.1002/andp.18812480110. 1881AnP...248..127L .
6. 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)
7. DELHOMMELLE. JÉRÔME. MILLIÉ. PHILIPPE. Inadequacy of the Lorentz-Berthelot combining rules for accurate predictions of equilibrium properties by molecular simulation. Molecular Physics. 20 April 2001. 99. 8. 619–625. 10.1080/00268970010020041. 2001MolPh..99..619D . 94931352 .
8. Boda. Dezső. Henderson. Douglas. The effects of deviations from Lorentz–Berthelot rules on the properties of a simple mixture. Molecular Physics. 20 October 2008. 106. 20. 2367–2370. 10.1080/00268970802471137. 2008MolPh.106.2367B . 94289505 .
9. 10.1063/1.1610435 . 119 . Modeling alkane+perfluoroalkane interactions using all-atom potentials: Failure of the usual combining rules . 2003 . The Journal of Chemical Physics . 9145–9162 . Song . W. . Rossky . P. J. . Maroncelli . M.. 17 . 2003JChPh.119.9145S .
10. Waldman. Marvin. Hagler. A.T.. New combining rules for rare gas van der waals parameters. Journal of Computational Chemistry. September 1993. 14. 9. 1077–1084. 10.1002/jcc.540140909. 16732612 .
11. 10.1063/1.1701284 . 36 . Second Virial Coefficients of Argon, Krypton, and Argon-Krypton Mixtures at Low Temperatures . 1962 . The Journal of Chemical Physics . 1881–1888 . Fender . B. E. F. . Halsey . G. D.. 7 . 1962JChPh..36.1881F .
12. Kong. Chang Lyoul. Combining rules for intermolecular potential parameters. II. Rules for the Lennard-Jones (12–6) potential and the Morse potential. The Journal of Chemical Physics. 1973. 59. 5. 2464–2467. 10.1063/1.1680358. 1973JChPh..59.2464K .
13. Tang. K. T.. Toennies. J. Peter. New combining rules for well parameters and shapes of the van der Waals potential of mixed rare gas systems. Zeitschrift für Physik D . March 1986. 1. 1. 91–101. 10.1007/BF01384663. 1986ZPhyD...1...91T . 122224768 .
14. Diaz Peña. M.. Combination rules for intermolecular potential parameters. I. Rules based on approximations for the long-range dispersion energy. The Journal of Chemical Physics. 1982. 76. 1. 325–332. 10.1063/1.442726. 1982JChPh..76..325D .
15. Diaz Peña. M.. 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.. The Journal of Chemical Physics. 1982. 76. 1. 333–339. 10.1063/1.442727. 1982JChPh..76..333D .
16. Hudson. G. H.. McCoubrey. J. C.. Intermolecular forces between unlike molecules. A more complete form of the combining rules. Transactions of the Faraday Society. 1960. 56. 761. 10.1039/TF9605600761.
17. Sikora. P T. Combining rules for spherically symmetric intermolecular potentials. Journal of Physics B . November 1970. 3. 11. 1475–1482. 10.1088/0022-3700/3/11/008. 1970JPhB....3.1475S .
18. Good. Robert J.. New Combining Rule for Intermolecular Distances in Intermolecular Potential Functions. The Journal of Chemical Physics. 1970. 53. 2. 540–543. 10.1063/1.1674022. 1970JChPh..53..540G . free.
19. Hogervorst. W.. Transport and equilibrium properties of simple gases and forces between like and unlike atoms. Physica. January 1971. 51. 1. 77–89. 10.1016/0031-8914(71)90138-8. 1971Phy....51...77H .
20. Kong. Chang Lyoul. Chakrabarty. Manoj R.. Combining rules for intermolecular potential parameters. III. Application to the exp 6 potential. The Journal of Physical Chemistry. October 1973. 77. 22. 2668–2670. 10.1021/j100640a019.
21. The Intermolecular Potentials of Helium and Hydrogen. 10.1063/1.1740100. 1954JChPh..22..522M . 22. 1954. The Journal of Chemical Physics. 522. Mason . Edward A. . Rice . William E. . 3 .
22. Srivastava. B. N.. Srivastava. K. P.. Combination Rules for Potential Parameters of Unlike Molecules on Exp-Six Model. The Journal of Chemical Physics. 1956. 24. 6. 1275. 10.1063/1.1742786. 1956JChPh..24.1275S .