Lennard-Jones Potential
The Lennard-Jones (LJ) potential is a mathematically simple model that approximates the interaction between a pair of neutral atoms or molecules. A common form of this interatomic potential was first proposed in 1924 by John Lennard-Jones. The most common expressions of the LJ potential are:
where ε is the depth of the potential well, σ is the finite distance at which the inter-particle potential is zero, r is the distance between the particles, and rm is the distance at which the potential reaches its minimum. At rm, the potential function has the value −ε. The distances are related as rm = 21/6σ ≈ 1.122σ. These parameters can be fitted to reproduce experimental data or accurate quantum chemistry calculations. Due to its computational simplicity, the LJ potential is used extensively in computer simulations even though more accurate potentials exist.
The first term, which is the repulsive term, describes Pauli repulsion at short ranges due to overlapping electron orbitals, and the second term, which is the attractive long-range term, describes attraction at long ranges. Differentiating the LJ potential with respect to r gives an expression for the net inter-molecular force between 2 molecules. This inter-molecular force may be attractive or repulsive, depending on the value of r. When r is very small, the molecules repel each other. Whereas the functional form of the attractive term has a clear physical justification, the repulsive term has no theoretical justification. Technically, LJ potential is a relatively good approximation. Due to its simplicity, it is often used to describe the properties of gases and to model dispersion and overlap interactions in molecular models. It is especially accurate for noble gas atoms and is a good approximation at long and short distances for neutral atoms and molecules. The lowest-energy arrangement of an infinite number of atoms described by a LJ 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. Real materials include body-centered cubic structures also.
Reference
Jones, J. E. (1924). On the Determination of Molecular Fields. II. From the Equation of State of a Gas. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 106(738), 463–477. doi:10.1098/rspa.1924.0082.