# Combining rules

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. One of the most common forms is

${\displaystyle V_{LJ}=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]}$

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 = 21/6σ ≈ 1.122σ.

### Lorentz-Berthelot rules

The Lorentz rule was proposed by H. A. Lorentz in 1881:[3]

${\displaystyle \sigma _{ij}={\frac {\sigma _{ii}+\sigma _{jj}}{2}}}$

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

The Berthelot rule (Daniel Berthelot, 1898) is given by:[4]

${\displaystyle \epsilon _{ij}={\sqrt {\epsilon _{ii}\epsilon _{jj}}}}$

These rules are the most widely used and are the default in many molecular simulation packages, but are not without failings.[5][6][7]

### Waldman-Hagler rules

The Waldman-Hagler rules are given by:[8]

${\displaystyle r_{ij}^{0}=\left({\frac {(r_{i}^{0})^{6}+(r_{j}^{0})^{6}}{2}}\right)^{1/6}}$

and

${\displaystyle \epsilon _{ij}=2{\sqrt {\epsilon _{i}\cdot \epsilon _{j}}}\left({\frac {(r_{i}^{0})^{3}\cdot (r_{j}^{0})^{3}}{(r_{i}^{0})^{6}+(r_{j}^{0})^{6}}}\right)}$

### Fender-Halsey

The Fender-Halsey combining rule is given by [9]

${\displaystyle \epsilon _{ij}={\frac {2\epsilon _{i}\epsilon _{j}}{\epsilon _{i}+\epsilon _{j}}}}$

### Kong rules

The Kong rules for the Lennard-Jones potential are given by:[10]

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

### Others

Many others have been proposed, including those of Tang and Toennies[11] Pena,[12][13] Hudson and McCoubrey[14] and Sikora(1970).[15]

## Combining rules for other potentials

### Good-Hope rule

The Good-Hope rule for MieLennard‐Jones or Buckingham potentials is given by:[16]

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

### Hogervorst rules

The Hogervorst rules for the Exp-6 potential are:[17]

${\displaystyle \epsilon _{12}={\frac {2\epsilon _{11}\epsilon _{22}}{\epsilon _{11}+\epsilon _{22}}}}$

and

${\displaystyle \alpha _{12}={\frac {1}{2}}(\alpha _{11}+\alpha _{22})}$

### Kong-Chakrabarty rules

The Kong-Chakrabarty rules for the Exp-6 potential are:[18]

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

and

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

Other rules for that have been proposed for the Exp-6 potential are the Mason-Rice rules[19] and the Srivastava and Srivastava rules (1956).[20]

## References

1. ^ Halgren, Thomas A. (September 1992). "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 (20): 7827–7843. doi:10.1021/ja00046a032.
2. ^ Desgranges, Caroline; Delhommelle, Jerome (14 March 2014). "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. 140 (10): 104109. Bibcode:2014JChPh.140j4109D. doi:10.1063/1.4867498.
3. ^ Lorentz, H. A. (1881). "Ueber die Anwendung des Satzes vom Virial in der kinetischen Theorie der Gase". Annalen der Physik. 248 (1): 127–136. Bibcode:1881AnP...248..127L. doi:10.1002/andp.18812480110.
4. ^ Daniel Berthelot "Sur le mélange des gaz", Comptes rendus hebdomadaires des séances de l’Académie des Sciences, 126 pp. 1703-1855 (1898)
5. ^ DELHOMMELLE, JÉRÔME; MILLIÉ, PHILIPPE (20 April 2001). "Inadequacy of the Lorentz-Berthelot combining rules for accurate predictions of equilibrium properties by molecular simulation". Molecular Physics. 99 (8): 619–625. Bibcode:2001MolPh..99..619D. doi:10.1080/00268970010020041.
6. ^ Boda, Dezső; Henderson, Douglas (20 October 2008). "The effects of deviations from Lorentz–Berthelot rules on the properties of a simple mixture". Molecular Physics. 106 (20): 2367–2370. Bibcode:2008MolPh.106.2367B. doi:10.1080/00268970802471137.
7. ^ Song, W.; Rossky, P. J.; Maroncelli, M. (2003). "Modeling alkane+perfluoroalkane interactions using all-atom potentials: Failure of the usual combining rules". The Journal of Chemical Physics. 119: 9145–9162. Bibcode:2003JChPh.119.9145S. doi:10.1063/1.1610435.
8. ^ Waldman, Marvin; Hagler, A.T. (September 1993). "New combining rules for rare gas van der waals parameters". Journal of Computational Chemistry. 14 (9): 1077–1084. doi:10.1002/jcc.540140909.
9. ^ Fender, B. E. F.; Halsey, G. D. (1962). "Second Virial Coefficients of Argon, Krypton, and Argon‐Krypton Mixtures at Low Temperatures". The Journal of Chemical Physics. 36: 1881–1888. Bibcode:1962JChPh..36.1881F. doi:10.1063/1.1701284.
10. ^ Kong, Chang Lyoul (1973). "Combining rules for intermolecular potential parameters. II. Rules for the Lennard-Jones (12–6) potential and the Morse potential". The Journal of Chemical Physics. 59 (5): 2464. Bibcode:1973JChPh..59.2464K. doi:10.1063/1.1680358.
11. ^ Tang, K. T.; Toennies, J. Peter (March 1986). "New combining rules for well parameters and shapes of the van der Waals potential of mixed rare gas systems". Zeitschrift für Physik D. 1 (1): 91–101. Bibcode:1986ZPhyD...1...91T. doi:10.1007/BF01384663.
12. ^ Diaz Peña, M. (1982). "Combination rules for intermolecular potential parameters. I. Rules based on approximations for the long-range dispersion energy". The Journal of Chemical Physics. 76 (1): 325. Bibcode:1982JChPh..76..325D. doi:10.1063/1.442726.
13. ^ Diaz Peña, M. (1982). "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. 76 (1): 333. Bibcode:1982JChPh..76..333D. doi:10.1063/1.442727.
14. ^ Hudson, G. H.; McCoubrey, J. C. (1960). "Intermolecular forces between unlike molecules. A more complete form of the combining rules". Transactions of the Faraday Society. 56: 761. doi:10.1039/TF9605600761.
15. ^ Sikora, P T (November 1970). "Combining rules for spherically symmetric intermolecular potentials". Journal of Physics B. 3 (11): 1475–1482. Bibcode:1970JPhB....3.1475S. doi:10.1088/0022-3700/3/11/008.
16. ^ Good, Robert J. (1970). "New Combining Rule for Intermolecular Distances in Intermolecular Potential Functions". The Journal of Chemical Physics. 53 (2): 540. Bibcode:1970JChPh..53..540G. doi:10.1063/1.1674022.
17. ^ Hogervorst, W. (January 1971). "Transport and equilibrium properties of simple gases and forces between like and unlike atoms". Physica. 51 (1): 77–89. Bibcode:1971Phy....51...77H. doi:10.1016/0031-8914(71)90138-8.
18. ^ Kong, Chang Lyoul; Chakrabarty, Manoj R. (October 1973). "Combining rules for intermolecular potential parameters. III. Application to the exp 6 potential". The Journal of Physical Chemistry. 77 (22): 2668–2670. doi:10.1021/j100640a019.
19. ^ "The Intermolecular Potentials of Helium and Hydrogen". The Journal of Chemical Physics. 22: 522. 1954. Bibcode:1954JChPh..22..522M. doi:10.1063/1.1740100.
20. ^ Srivastava, B. N.; Srivastava, K. P. (1956). "Combination Rules for Potential Parameters of Unlike Molecules on Exp-Six Model". The Journal of Chemical Physics. 24 (6): 1275. Bibcode:1956JChPh..24.1275S. doi:10.1063/1.1742786.