Past exam of the quantum condensed matter course of the University of Cambridge / 2009 / Section B / B1 / a

The attractive part of the Lennard-Jones potential represents the London dispersion force between fluctuating atomic electric dipoles. The dipole-dipole interaction is proportional to $r^{-3}$; for nonpolar ground-state atoms its first-order expectation vanishes. In second-order perturbation theory, virtual excitations give a negative energy shift proportional to the square of that interaction, hence $-A/r^6$ with $A>0$. This applies at separations large compared with atomic size but small enough that electromagnetic retardation is negligible.

The repulsive term models the steep energy increase when closed electron shells overlap, involving the Pauli exclusion principle and increased electronic energy. The $r^{-12}$ power is an empirical convenient approximation, not a universal power derived from the exclusion principle; its being the square of $r^{-6}$ makes computation convenient. At still larger separations retardation changes the attractive asymptotics, outside this model.