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 $Phi = (4piepsilon_0/a_0^4) times alpha^{4/3}$ 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.
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