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# 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 [4]

$\epsilon_{ij} = \frac{2 \epsilon_i \epsilon_j}{\epsilon_i + \epsilon_j}$

## Gilbert-Smith

The Gilbert-Smith rules for the Born-Huggins-Meyer potential[5][6][7].

## Good-Hope rule

The Good-Hope rule for MieLennard‐Jones or Buckingham potentials [8] is given by (Eq. 2):

$\sigma_{ij} = \sqrt{\sigma_{ii} \sigma_{jj}}$

## Hogervorst rules

The Hogervorst rules for the Exp-6 potential [10]:

$\epsilon_{12} = \frac{2 \epsilon_{11} \epsilon_{22}}{\epsilon_{11} + \epsilon_{22}}$

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 [11]):

$\epsilon_{ij}\sigma_{ij}^{6}=\left(\epsilon_{ii}\sigma_{ii}^{6}\epsilon_{jj}\sigma_{jj}^{6}\right)^{1/2}$
$\epsilon_{ij}\sigma_{ij}^{12} = \left[ \frac{ (\epsilon_{ii}\sigma_{ii}^{12})^{1/13} + (\epsilon_{jj}\sigma_{jj}^{12})^{1/13} }{2} \right]^{13}$

## Kong-Chakrabarty rules

The Kong-Chakrabarty rules for the Exp-6 potential [12] 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 [13]

$\sigma_{ij} = \frac{\sigma_{ii} + \sigma_{jj}}{2}$

which is only really valid for the hard sphere model.

The Berthelot rule is given by [14]

$\epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}}$

These rules are simple and widely used, but are not without their failings [15] [16] [17] [18].

## Mason-Rice rules

The Mason-Rice rules for the Exp-6 potential [19].

## Srivastava and Srivastava rules

The Srivastava and Srivastava rules for the Exp-6 potential [20].

## Sikora rules

The Sikora rules for the Lennard-Jones model [21].

## Waldman-Hagler rules

The Waldman-Hagler rules [23] are given by:

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

and

$\epsilon_{ij} = 2 \sqrt{\epsilon_i \cdot \epsilon_j} \left( \frac{ (r_i^0)^3 \cdot (r_j^0)^3 }{ (r_i^0)^6 + (r_j^0)^6 } \right)$