Onset of Casimir-Polder Retardation in a Long-Range Molecular Quantum State

Two ${}^4\mathrm{He}$ atoms form a diatomic molecule with a significant vibrational wave function amplitude at interatomic separations $R>100\AA$, where the retardation switches the London $R^{-6}$ decay of the potential to the Casimir-Polder $R^{-7}$ form. It has been assumed that this effect of retardation on the long-range part of the potential is responsible for the 2 Å (4%) increase of the bond length $⟨R⟩$ of ${}^4\mathrm{He}_2$. We show that $⟨R⟩$ is, unexpectedly, insensitive to the potential at $R>20\AA$ and its increase is due to quantum electrodynamics effects computed by us from expressions valid at short $R$—beyond the validity range of Casimir-Polder theory—that seamlessly extend this theory to distances relevant for properties of long molecules.

Figure 1

The vibrational wave function of the helium dimer $\chi (R)$ (left axis), normalized such that $\chi (R{)}^2$ is the probability density of finding atoms at the distance $R$, and the retardation factor $g(R)$ showing the increasing importance of the retardation effect at large $R$ (right axis).

Figure 2

Dependence of the mean interatomic distance $⟨R⟩$ in angstroms (left axis) and of the dissociation energy $D_0$ in millikelvins (right axis) on the distance $R_d$ where the potential is damped to zero. The vertical line marks the position of the classical outer turning point.

Figure 3

Comparisons of terms constituting the $\delta V_{\mathrm{sr}\mathrm{\text{−}}\mathrm{ret}}^{\mathrm{BO}}(R)$ retardation correction.

Figure 4

Comparisons of the components of $\delta V_{\mathrm{sr}\mathrm{\text{−}}\mathrm{ret}}^{\mathrm{BO}}(R)$ with $V_{\mathrm{BO}}(R)$.