The Lennard-Jones potential

  The Lennard-Jones 12-6 potential is given by the expression  

$$\phi_{\rm LJ} (r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]$$ (5)

for the interaction potential between a pair of atoms. The total potential of a system containing many atoms is then given by 2.2.

This potential has an attractive tail at large r, it reaches a minimum around 1.122$\,\sigma$, and it is strongly repulsive at shorter distance, passing through 0 at $r=\sigma$ and increasing steeply as r is decreased further.

The term $\sim 1/r^{12}$, dominating at short distance, models the repulsion between atoms when they are brought very close to each other. Its physical origin is related to the Pauli principle: when the electronic clouds surrounding the atoms starts to overlap, the energy of the system increases abruptly. The exponent 12 was chosen exclusively on a practical basis: equation (2.3) is particularly easy to compute. In fact, on physical grounds an exponential behavior would be more appropriate.

The term $\sim 1/r^6$, dominating at large distance, constitute the attractive part. This is the term which gives cohesion to the system. A 1/r⁶ attraction is originated by van der Waals dispersion forces, originated by dipole-dipole interactions in turn due to fluctuating dipoles. These are rather weak interactions, which however dominate the bonding character of closed-shell systems, that is, rare gases such as Ar or Kr. Therefore, these are the materials that a LJ potential could mimic fairly well. The parameters $\varepsilon$ and $\sigma$ are chosen to fit the physical properties of the material.

On the other hand, a LJ potential is not at all adequate to model situations with open shells, where strong localized bonds may form (as in covalent systems), or where there is a delocalized ``electron sea'' where the ions sit (as in metals). In these systems the two-body interactions scheme itself fails very badly. Potentials for these systems will be discussed in chapter 4.

However, regardless of how well it is able to model actual materials, the LJ 12-6 potential constitutes nowadays an extremely important model system. There is a vast body of papers who investigated the behavior of atoms interacting via LJ on a variety of different geometries (solids, liquids, surfaces, clusters, two-dimensional systems, etc). One could say that LJ is the standard potential to use for all the investigations where the focus is on fundamental issues, rather than studying the properties of a specific material. The simulation work done on LJ systems helped us (and still does) to understand basic points in many areas of condensed matter physics, and for this reason the importance of LJ cannot be underestimated.

When using the LJ potentials in simulation, it is customary to work in a system of units where $\sigma = 1$ and $\varepsilon = 1$.The example codes accompanying these notes follow this convention.