Differentiability of the van der Waals interaction between two atoms

In this work we improve upon previous results on the expansion of the interaction energy of two atoms. On the one hand we prove the van der Waals-London's law, assuming that only one of the ground state eigenspaces of the atoms is irreducible in an appropriate sense. On the other hand we prove strict monotonicity of the interaction energy at large distances and, under more restrictive assumptions, we provide the leading order of its first two derivatives. The first derivative is interpreted as the force in Physics. Moreover, the estimates of the first two derivatives provide a rigorous proof of the monotonicity and concavity of the interaction energy at large distances.