Combining 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. Intuitively, since ${\displaystyle \sigma _{i},\sigma _{j}}$ 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:[4]

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

Physically, this arises from the fact that ${\displaystyle \epsilon }$ is related to the induced dipole interactions between two particles. Given two particles with instantaneous dipole ${\displaystyle \mu _{i},\mu _{j}}$ respectively, their interactions correspond to the products of ${\displaystyle \mu _{i},\mu _{j}}$. An arithmetic average of ${\displaystyle \epsilon _{i}}$ and ${\displaystyle \epsilon _{j}}$ 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.[5][6][7]

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

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

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

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

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

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

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

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

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

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

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

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

and

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

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]