  We examine the recently derived quantum-mechanical relation between atomic
polarizabilities and equilibrium internuclear distances in van der Waals (vdW)
bonded diatomic systems [Phys. Rev. Lett. {\bf 121}, 183401 (2018)]. For
homonuclear dimers, this relation is described by the compact formula
$\alpha_{\rm m}^{\rm q} = \Phi R_{\rm vdW}^7$, where the constant factor in
front of the vdW radius was determined empirically. Here, we derive $\Phi =
(4\pi\epsilon_0/a_0^4) \times \alpha^{4/3}$ expressed in terms of the vacuum
electric permittivity $\epsilon_0$, the Bohr radius $a_0$, and the
fine-structure constant $\alpha$. The validity of the obtained formula is
confirmed by estimating the value of the fine-structure constant from
non-relativistic quantum-mechanical calculations of atomic polarizabilities and
equilibrium internuclear vdW distances. The presented derivation allows to
interpret the fine-structure constant as the ratio between the polarizability
densities of vacuum and matter, whereas the vdW radius becomes a geometrical
length scale of atoms endowed by the vacuum field.
