Differentiability of the van der Waals interaction between two atoms

TL;DR

This paper rigorously analyzes the differentiability and monotonicity of the van der Waals interaction energy between two atoms, proving fundamental physical laws and providing detailed derivative estimates under specific mathematical conditions.

Contribution

It establishes the van der Waals-London law under new assumptions and proves strict monotonicity and derivative estimates of the interaction energy at large distances.

Findings

Proof of van der Waals-London law under new conditions

Demonstration of strict monotonicity of interaction energy

Derivation of leading order of first two derivatives of energy

Abstract

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.