Theoretical Hyperfine Structure of the Molecular Hydrogen Ion at the 1 ppm Level

We revisit the $m{\alpha }^6(m/M)$ order corrections to the hyperfine splitting in the ${\mathrm{H}}_2^{+}$ ion and find a hitherto unrecognized second-order relativistic contribution associated with the vibrational motion of the nuclei. Inclusion of this correction term produces theoretical predictions which are in excellent agreement with experimental data [K. B. Jefferts, Phys. Rev. Lett. 23, 1476 (1969)], thereby concluding a nearly 50-year-long theoretical quest to explain the experimental results within their 1-ppm error. The agreement between the theory and experiment corroborates the proton structural properties as derived from the hyperfine structure of atomic hydrogen. Our work furthermore indicates that, for future improvements, a full three-body evaluation of the $m{\alpha }^6(m/M)$ correction term will be mandatory.

Figure 1

Hyperfine structure of ${\mathrm{H}}_2^{+}$ in a given vibrational quantum state $v$ and with orbital angular momentum quantum number $L=1$. Angular momentum quantum labels $F$ and $J$ are shown within parentheses.