# softmatter:Lennard-Jones Potential

The Lennard-Jones Potential.
The Lennard-Jones potential (LJ) is used to model the excluded volume interactions and van der Waals interaction attraction of neutral atoms. The commonly used 6-12 form of the potential is as follows:

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

Where ε is the well depth, σ is the characteristic diameter (typically the diameter of the smallest particle), and r is the radial separation of the two atoms.

## van der Waals Attraction

In theory, the van der Waals interaction for atoms with similar ionization frequencies and where the dispersion (London) interactions are dominant is proportional to $- (\frac{\alpha_{01}\alpha_{02}}{r^{6}})$ where α01 and α02 are the polarizabilities of atom 1 and atom 2 respectively. Again, this assumes that dispersion (London) forces are dominant and that there are no permanent dipoles (Keesom forces) or induced dipoles (Debye forces). In the LJ construction, the term $\left(\frac{1}{r}\right)^{6}$ is used to describe this attractive van der Waals interaction [1].

## Excluded Volume Interaction

As the seperation distance between atoms decreases, the electron clouds will eventually overlap, resulting in a very strong repulsion that rapidly increases as interatomic spacing is further decreased. In the LJ construction, the term $\left(\frac{1}{r}\right)^{12}$describes this repulsive interaction. The 12th power is used for two main reasons: it is very steep, rapidly becoming dominant as r is small and it is also a multiple of the 6th power allowing for efficient computation [1].

## LJ in Simulation

### Typical implementation

Typically, the LJ potential is truncated at a distance of 2.5σ. Computationally, this drastically reduces the number of force calculations needed at each time step because the potential interaction is only calculated on neighbors within a distance of 2.5σ away from a particle, rather than over the entire box. At distance of 2.5σ, the potential has acquired a value of less than 1/60th of the well depth. In dynamical simulation methods (such as Molecular Dynamics and Brownian Dynamics), it is not favorable to have a potential that has a discontinous jump at the tail. As such, the potential is typically shifted by the value at 2.5σ such that the potential has a value of zero at the tail. Mathematically this is represented by [2]:

$U_{LJ}(r) = \begin{cases} 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right] - 4\varepsilon \left[ \left(\frac{\sigma}{r_c}\right)^{12} - \left(\frac{\sigma}{r_c}\right)^{6} \right] &r

where rc is the cutoff. While less common, a cutoff value of rc = 2(2(1/6)) ~ 2.245 is also used, where 2(1/6) corresponds to minimum of the potential well.

### Truncation and long range corrections

When one uses a truncated pair potential, it would be assumed that the contributions of the long tails would be negligible, however, in practice this is not true. By assuming that at large r values the density is equal to the average density, ρ, we can integrate and arrive at an expression that gives a corrective term for the overall potential energy[2]:

$PE^{tail} = \frac{8}{3}\pi\rho\epsilon\sigma^{3} \left[\frac{1}{3} \left(\frac{\sigma}{r_c}\right)^9 -\left(\frac{\sigma}{r_c}\right)^3\right]$