Lennard Jones potentials (Definition&formula) caculation and it's applications(LJ potentials)

Definition:

 The Lennard-Jones potential (also referred to as the L-J potential, 6-12 potential or, less commonly, 12-6 potential) is a mathematically simple model that describes the interaction between a pair of neutral atoms or molecules. A form of the potential was first proposed in 1924 by John Lennard-Jones.

where ε = Depth of the potential well

 σ = (finite) Distance at which the inter-particle potential is zero

 r = Distance between the particles.

These parameters can be fitted to reproduce experimental data or accurate quantum chemistry calculations. The r⁻¹² term describes Pauli repulsion at short ranges due to overlapping electron orbitals and the r⁻⁶ term describes attraction at long ranges (van der Waals force, or dispersion force).

Applications:

The Lennard-Jones 12-6 (LJ) potential is an important model for exploring the  behaviour of simple fluids, and has been used to study vapour-liquid and liquidliquid equilibria, melting, behaviour of fluids confined within small pores, small atomic clusters, a variety of surface and transport properties, and so on. It has also been widely used as a reference fluid in perturbation treatments for more complex fluids. In many ways, the LJ model is oversimplified ; for example, the form of the repulsive part of the potential is incorrect, being insufficiently repulsive at short distances, the C6 dispersion coefficient is too high while higher dispersion coefficients are neglected, and its application to dense systems neglects three-body forces completely. Nevertheless, it captures much of the essential physics of simple fluids.

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 is 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 short range part of the repulsion term can be seen.

LJ potential for Argon dimer is as shown in the diagram.