Quantum-Mechanical Relation between Atomic Dipole Polarizability and the van der Waals Radius

The atomic dipole polarizability α and the van der Waals (vdW) radius R_{vdW} are two key quantities to describe vdW interactions between atoms in molecules and materials. Until now, they have been determined independently and separately from each other. Here, we derive the quantum-mechanical relation R_{vdW}=const×α^{1/7}, which is markedly different from the common assumption R_{vdW}∝α^{1/3} based on a classical picture of hard-sphere atoms. As shown for 72 chemical elements between hydrogen and uranium, the obtained formula can be used as a unified definition of the vdW radius solely in terms of the atomic polarizability. For vdW-bonded heteronuclear dimers consisting of atoms A and B, the combination rule α=(α_{A}+α_{B})/2 provides a remarkably accurate way to calculate their equilibrium interatomic distance. The revealed scaling law allows us to reduce the empiricism and improve the accuracy of interatomic vdW potentials, at the same time suggesting the existence of a nontrivial relation between length and volume in quantum systems.