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Improved values of hyperfine coefficients related to the electronicspin-orbit and electron-nucleus spin-spin tensor interactions in the HD$^+$molecular ion are obtained through numerical calculation of relativisticcorrections at the $m\alpha^6$ order and radiative corrections at the$m\alpha^7\ln(\alpha)$ order. The theoretical accuracy is improved by more thanone order of magnitude. Some deviations with recent high-precisionro-vibrational spectroscopy experiments are observed, in contrast with the goodagreement obtained in H$_2^+$.

Higher-order corrections to the spin-orbit and spin-spin tensor interactions in HD$^+$