  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 work,
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 derived formula for noble
gases, alkaline earth and alkali atoms and show that the $A_l$ can be assumed
to be universal constants for each $l$ after considering the uncertainties in
the measured values of $R_{\rm vdW}$. Our results enable a practical definition
of atomic vdW radii from calculated or measured polarizabilities, an accurate
estimation of multipole polarizabilities from the dipole one, and the
construction of efficient and minimally empirical interatomic potential models.
