Scale Dependence of the Retarded van der Waals Potential

We study the ground state energy for a system of two hydrogen atoms coupled to the quantized Maxwell field in the limit $\alpha \to 0$ together with the relative distance between the atoms increasing as ${\alpha }^{-\gamma }R$, $\gamma

0$\mathrm{.}Inparticularwedetermineexplicitlythecrossoverfunctionfromthe$R^{-6}$vanderWaalspotentialtothe$R^{-7}$ retarded van der Waals potential, which takes place at scale ${\alpha }^{-2}R$.