Combining rules

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).

Fender-Halsey

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

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

Gilbert-Smith

The Gilbert-Smith rules for the Born-Huggins-Meyer potential.

Good-Hope rule

The Good-Hope rule for Mie–Lennard‐Jones or Buckingham potentials is given by (Eq. 2):

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

Hogervorst rules

The Hogervorst rules for the Exp-6 potential :

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

and

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

Kong rules

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

$\epsilon _{ij}\sigma _{ij}^{6}=\left(\epsilon _{ii}\sigma _{ii}^{6}\epsilon _{jj}\sigma _{jj}^{6}\right)^{1/2}$

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

Kong-Chakrabarty rules

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

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

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

and

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

Lorentz-Berthelot rules

The Lorentz rule is given by

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

which is only really valid for the hard sphere model.

The Berthelot rule is given by

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

These rules are simple and widely used, but are not without their failings .

Mason-Rice rule

The Mason-Rice rule for the Exp-6 potential .

Sikora rules

The Sikora rules for the Lennard-Jones model .

Waldman-Hagler rules

The Waldman-Hagler rules are given by:

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

and

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

References

Related reading

• 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)