Salient Signature of van der Waals Interactions

van der Waals interactions govern the physics of a plethora of molecular structures. It is well known that the leading term in the distance-based London expansion of the van der Waals energy for atomic and molecular dimers decays as $1\mathrm{/}{R}^6$, where $R$ is the dimer distance. Using perturbation theory, we find the leading term in the distance-based expansion of the intracule pair density at the interatomic distance. Our results unveil a universal $1\mathrm{/}{R}^3$ decay, which is less prone to numerical errors than the $1\mathrm{/}{R}^6$ dependency, and it is confirmed numerically in H${}_2$ and He${}_2$ molecules. This $\mathit{\text{signature}}$ of van der Waals interactions can be directly used in the construction of approximate pair density and energy functionals including vdW corrections.