Girish Mahajan (Editor)

Combining rules

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In computational chemistry and molecular dynamics, the combination rules or combining rules are equations that provide the interaction energy between two dissimilar non-bonded atoms, usually for the part of the potential representing the van der Waals interaction. In the simulation of mixtures, the choice of combining rules can sometimes affect the outcome of the simulation.

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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

V L J = 4 ε [ ( σ r ) 12 ( σ r ) 6 ]

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 = 21/6σ ≈ 1.122σ.

Lorentz-Berthelot rules

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

σ i j = σ i i + σ j j 2

The Lorentz rule is only analytically correct for hard sphere systems.

The Berthelot rule (Daniel Berthelot, 1898) is given by:

ϵ i j = ϵ i i ϵ j j

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:

r i j 0 = ( ( r i 0 ) 6 + ( r j 0 ) 6 2 ) 1 / 6

and

ϵ i j = 2 ϵ i ϵ j ( ( r i 0 ) 3 ( r j 0 ) 3 ( r i 0 ) 6 + ( r j 0 ) 6 )

Fender-Halsey

The Fender-Halsey combining rule is given by

ϵ i j = 2 ϵ i ϵ j ϵ i + ϵ j

Kong rules

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

ϵ i j σ i j 6 = ( ϵ i i σ i i 6 ϵ j j σ j j 6 ) 1 / 2 ϵ i j σ i j 12 = [ ( ϵ i i σ i i 12 ) 1 / 13 + ( ϵ j j σ j j 12 ) 1 / 13 2 ] 13

Others

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

Good-Hope rule

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

σ i j = σ i i σ j j

Hogervorst rules

The Hogervorst rules for the Exp-6 potential are:

ϵ 12 = 2 ϵ 11 ϵ 22 ϵ 11 + ϵ 22

and

α 12 = 1 2 ( α 11 + α 22 )

Kong-Chakrabarty rules

The Kong-Chakrabarty rules for the Exp-6 potential are:

[ ϵ 12 α 12 e α 12 ( α 12 6 ) σ 12 ] 2 σ 12 / α 12 = [ ϵ 11 α 11 e α 11 ( α 11 6 ) σ 11 ] σ 11 / α 11 [ ϵ 22 α 22 e α 22 ( α 22 6 ) σ 22 ] σ 22 / α 22 σ 12 α 12 = 1 2 ( σ 11 α 11 + σ 22 α 22 )

and

ϵ 12 α 12 σ 12 6 ( α 12 6 ) = [ ϵ 11 α 11 σ 11 6 ( α 11 6 ) ϵ 22 α 22 σ 22 6 ( α 22 6 ) ] 1 2

Other rules for that have been proposed for the Exp-6 potential are the Mason-Rice rules and the Srivastava and Srivastava rules (1956).

References

Combining rules Wikipedia