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

TL;DR

This paper establishes a quantum-mechanical relation between atomic dipole polarizability and the van der Waals radius, providing a unified way to determine vdW radii and improve interatomic potential calculations.

Contribution

It derives a new quantum-mechanical formula linking vdW radius and polarizability, differing from classical assumptions, and validates it across 72 elements.

Findings

The relation $R_{vdW} \\propto \\alpha^{1/7}$ fits data for 72 elements.

The average polarizability provides accurate interatomic distances in heteronuclear dimers.

The scaling law enhances the accuracy of vdW potential models.

Abstract

The atomic dipole polarizability, $\alpha$, and the van der Waals (vdW) radius, $R_{\rm vdW}$, 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 $R_{\rm vdW} = const. \times\alpha^{1/7}$ which is markedly different from the common assumption $R_{\rm vdW} \propto \alpha^{1/3}$ 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 $\alpha = (\alpha_A + \alpha_B)/2$ provides a remarkably accurate way to calculate their equilibrium interatomic distance. The revealed scaling law allows to reduce the empiricism and improve the accuracy of interatomic vdW potentials, at the same time suggesting the existence of a non-trivial relation between length and volume in quantum systems.