Lennard-Jones 12-6 FunctionsΒΆ
The Lennard-Jone 12-6 function is the most commonly used for vdw interactions due to its simplistic form for computations. It is expressed in various equivalent forms and they can be inter-changed via algebraic operations.
\[V \left(r \right) = \epsilon \left\{ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right\}\]
\[V \left(r \right) = \epsilon \left\{ \left(\frac{R}{r}\right)^{12} - 2 \left(\frac{R}{r}\right)^{6} \right\}\]
\[V \left(r \right) = \frac{A}{r^{12}} - \frac{B}{r^{6}}\]
where \(\epsilon\) is the well depth, which measures the strength of the interaction; \(\sigma\) is the steric parameter, which is the distance when the LJ function changes sign. R is the distance at which V is at the minimum. They are related to each other as follows:
\[R = 2^{ \frac{1}{6}} \sigma\]
Normally, the third form is used in the programming since it is easier to compute, with A and B having the following relationships:
\[A = 4 \epsilon \sigma^{12}\]
\[B = 4 \epsilon \sigma^{6}\]
Diagram below shows the vdw interactions between two methyl carbon atoms, with the parameters obtained from the CHARMM FF.
