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):

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

Kong rules

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

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

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

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)