Higher-order corrections to spin-spin scalar interactions in HD+ and H2+

The largest hyperfine interaction coefficients in the hydrogen molecular ion HD⁺, i.e., the electron-proton and electron-deuteron spin-spin scalar interactions, are calculated with estimated uncertainties slightly below 1 ppm. The (Z^{α ) 2}E_F relativistic correction for which a detailed derivation is presented, QED corrections up to the order α³ln²(α ) along with an estimate of higher-order terms, and nuclear structure corrections are taken into account. Improved results are also given for the electron-proton interaction coefficient in H₂⁺, in excellent agreement with rf spectroscopy experiments. In HD⁺, a 4 σ difference is found in the hyperfine splitting of the (v ,L )=(0 ,3 )→(9 ,3 ) two-photon transition that was recently measured with high precision. The origin of this discrepancy is unknown.