Combining rules

Combining rules for the Lennard-Jones potential

The Lennard-Jones Potential is a mathematically simple model for the interaction between a pair of atoms or molecules. One of the most common forms is

${\displaystyle V_{LJ}=4\epsilon \left[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right]}$

where ε is the depth of the potential well, σ is the finite distance at which the inter-particle potential is zero, r is the distance between the particles. The potential reaches a minimum, of depth ε, when r = 2^1/6σ ≈ 1.122σ.

Lorentz-Berthelot rules

The Lorentz rule was proposed by H. A. Lorentz in 1881:

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

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:

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

Physically, this arises from the fact that ${\displaystyle ϵ}$ 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 ${\displaystyle {ϵ}_i}$ and ${\displaystyle {ϵ}_j}$ 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.

Waldman-Hagler rules

The Waldman-Hagler rules are given by:

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

and

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

Fender-Halsey

The Fender-Halsey combining rule is given by

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

Kong rules

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

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

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

Others

Many others have been proposed, including those of Tang and Toennies Pena, Hudson and McCoubrey and Sikora (1970).