# Lennard-Jones Potential

### Arthur La Porta

The Lennard-Jones potential is a common way of representing the force
between molecules in a fluid. We need a force which is weakly attractive
at long distances, but which reverts to a hard-sphere repulsion if the molecules
get too close.
The potential is defined by

*U*(*r*) = 4 *ε* [(*σ*/*r*)^{12} - (*σ*/*r*)^{6}]
where *σ* determines the distance at which the two particles are at
equilibrium, and *ε* is the strength of the interaction.
The first term is responsible for the repulsion at short distance and the second
term is responsible for the attraction at long distance.
A plot of the Lennard-Jones potential is shown below.

The force is defined by F(r) = -∂U/∂r, so the force associated with
the Lennard-Jones potential is

*R*(*r*) = 4*ε* [12*σ*^{12}/*r*^{13} - 6*σ*^{6}/*r*^{7}]
A sketch of the force associated with the Lennard-Jones potential is shown below.

By setting the force expression to zero, it is possible to show that the equilibrium
distance for the Lennard-Jones potenial is 2^{1/6}*σ* and that the
potential at this distance is -*ε*.

One practical problem with the Lennard-Jones potential is that it has infinite
range. It is impossible for two particles go far enough apart to escape their
attractive force. The force gets infinitely weak at large distance, so this
is not a physical problem. But from a computational point of view, it means that the
force must be calculated for every pair of particles in the system. This will
limit the speek of the computation for large systems.
For this reason, the Lennard-Jones force is often cut off
at a finite distance, often about 3*σ*. The effect on the dynamics
is small, but the speed-up of the computation can be dramatic.

There are cases where only the hard-sphere repulsion of the Lennard-Jones
potential is necessary (for instance, in our simulation of the crystal).
In such cases, we can either cut off the force
at the equilibrium distance so there is no attractive force,
or simply omit the attractive term from
the potential.