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

${ϵ}_{ij}=\frac{2{ϵ}_i{ϵ}_j}{{ϵ}_i+{ϵ}_j}$

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 :

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

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 ):

${ϵ}_{ij}\underset{ij}{\overset{6}{\sigma }}={({ϵ}_{ii}\underset{ii}{\overset{6}{\sigma }}{ϵ}_{jj}\underset{jj}{\overset{6}{\sigma }})}^{1/2}$

${ϵ}_{ij}\underset{ij}{\overset{12}{\sigma }}={[\frac{({ϵ}_{ii}\underset{ii}{\overset{12}{\sigma }}{)}^{1/13}+({ϵ}_{jj}\underset{jj}{\overset{12}{\sigma }}{)}^{1/13}}{2}]}^{13}$

Kong-Chakrabarty rules

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

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

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

and

$\frac{{ϵ}_{12}{\alpha }_{12}\underset{12}{\overset{6}{\sigma }}}{({\alpha }_{12}-6)}={[\frac{{ϵ}_{11}{\alpha }_{11}\underset{11}{\overset{6}{\sigma }}}{({\alpha }_{11}-6)}\frac{{ϵ}_{22}{\alpha }_{22}\underset{22}{\overset{6}{\sigma }}}{({\alpha }_{22}-6)}]}^{\frac{1}{2}}$

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

${ϵ}_{ij}=\sqrt{{ϵ}_{ii}{ϵ}_{jj}}$

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

Mason-Rice rules

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

Srivastava and Srivastava rules

The Srivastava and Srivastava rules 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:

$r_{ij}^0={(\frac{(r_i^0{)}^6+(r_j^0{)}^6}{2})}^{1/6}$

and

${ϵ}_{ij}=2\sqrt{{ϵ}_i\cdot {ϵ}_j}(\frac{(r_i^0{)}^3\cdot (r_j^0{)}^3}{(r_i^0{)}^6+(r_j^0{)}^6})$

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)