Quantum-Mechanical Force Balance Between Multipolar Dispersion and Pauli Repulsion in Atomic van der Waals Dimers

TL;DR

This paper derives a universal formula linking atomic van der Waals radii to multipole polarizabilities, based on a quantum-mechanical force balance between dispersion attraction and Pauli repulsion, aiding the development of accurate interatomic potentials.

Contribution

It introduces a compact, universal relation connecting atomic polarizability and vdW radius, grounded in quantum force balance, applicable across various atomic species.

Findings

The formula accurately predicts vdW radii for noble gases, alkaline-earth, and alkali atoms.

The constant A_l can be considered universal across different atoms.

The relation enables estimation of multipole polarizabilities from dipole polarizability and vdW radius.

Abstract

The structure and stability of atomic and molecular systems with van der Waals (vdW) bonding are often determined by the interplay between attractive dispersion interactions and repulsive interactions caused by electron confinement. Arising due to different mechanisms -- electron correlation for dispersion and the Pauli exclusion principle for exchange-repulsion -- these interactions do not appear to have a straightforward connection. In this paper, we use a coarse-grained approach for evaluating the exchange energy for two coupled quantum Drude oscillators and investigate the mutual compensation of the attractive and repulsive forces at the equilibrium distance within the multipole expansion of the Coulomb potential. This compensation yields a compact formula relating the vdW radius of an atom to its multipole polarizabilities, $R_{\rm vdW} = A_l^{\,}\, \alpha_l^{{2}/{7(l+1)}}$, where $l$ is the multipole rank and $A_l$ is a conversion factor. Such a relation is compelling because it connects an electronic property of an isolated atom (atomic polarizability) with an equilibrium distance in a dimer composed of two closed-shell atoms. We assess the accuracy of the revealed formula for noble-gas, alkaline-earth, and alkali atoms and show that the $A_l$ can be assumed to be universal constants. Besides a seamless definition of vdW radii, the proposed relation can also be used for the efficient determination of atomic multipole polarizabilities solely based on the corresponding dipole polarizability and the vdW radius. Finally, our work provides a basis for the construction of efficient and minimally-empirical interatomic potentials by combining multipolar interatomic exchange and dispersion forces on an equal footing.