Lennard-Jones potential

Neutral atoms and molecules are subject to two distinct forces in the limit of large distance and short distance: an attractive force at long ranges (van der Waals force, or dispersion force) and a repulsive force at short ranges (the result of overlapping electron orbitals, referred to as Pauli repulsion from Pauli exclusion principle). The Lennard-Jones potential (also referred to as the L-J potential, 6-12 potential or, less commonly, 12-6 potential) is a simple mathematical model that represents this behavior. It was proposed in 1931 by John Lennard-Jones of Bristol University.

Lennard-Jones potential for argon dimer

The L-J potential is of the form

$V(r)=4\epsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]$

where $\epsilon$ is the depth of the potential well and $\sigma$ is the (finite) distance at which the potential is zero. The function is also often written as

$V(r)=\epsilon \left[\left({\frac {r_{min}}{r}}\right)^{12}-2\left({\frac {r_{min}}{r}}\right)^{6}\right]$

where r_min = 2^1/6σ is the distance at the minimum of the potential.

These parameters can be fitted to reproduce experimental data or deduced from results of accurate quantum chemistry calculations. The $\left({\frac {1}{r}}\right)^{12}$ term describes repulsion and the $\left({\frac {1}{r}}\right)^{6}$ term describes attraction. The force function is the negative of the gradient of the above potential:

$\mathbf {F} (r)=-\nabla V(r)=-{\frac {d}{dr}}V(r){\hat {\mathbf {r} }}=4\epsilon \left(12\,{\frac {{\sigma }^{12}}{{r}^{13}}}-6\,{\frac {{\sigma }^{6}}{{r}^{7}}}\right){\hat {\mathbf {r} }}$

The L-J potential is approximate. The form of the repulsion term has no theoretical justification; the repulsion force should depend exponentially on the distance, but the repulsion term of the L-J formula is more convenient due to the ease and efficiency of computing r¹² as the square of r⁶. The attractive long-range potential, however, is derived from dispersion interactions. The L-J potential is a relatively good approximation and due to its simplicity often used to describe the properties of gases, and to model dispersion and overlap interactions in molecular models. It is particularly accurate for noble gas atoms and is a good approximation at long and short distances for neutral atoms and molecules. On the graph, Lennard-Jones potential for argon dimer is shown. Small deviation from the accurate empirical potential due to incorrect long range part of the repulsion term can be seen.

The lowest energy arrangement of an infinite number of atoms described by a Lennard-Jones potential is a hexagonal close-packing. On raising temperature, the lowest free energy arrangement becomes cubic close packing and then liquid. Under pressure the lowest energy structure switches between cubic and hexagonal close packing.

Other more recent methods, such as the Stockmayer equation and the so-called multi equation, describe the interaction of molecules more accurately. Quantum chemistry methods, Møller-Plesset perturbation theory, coupled cluster method or full configuration interaction can give extremely accurate results, but require large computational cost.

See also

• Embedded atom model
• Morse potential
• Force field (chemistry)