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The largest hyperfine interaction coefficients in the hydrogen molecular ionHD$^+$, i.e. the electron-proton and electron-deuteron spin-spin scalarinteractions, are calculated with estimated uncertainties slightly below 1~ppm.The $(Z\alpha)^2 E_F$ relativistic correction, for which a detailed derivationis presented, QED corrections up to the order $\alpha^3 \ln^2 (\alpha)$ alongwith an estimate of higher-order terms, and nuclear structure corrections aretaken into account. Improved results are also given for the electron-protoninteraction coefficient in H$_2^+$, in excellent agreement with RF spectroscopyexperiments. In HD$^+$, a 4$\sigma$ difference is found in the hyperfinesplitting of the $(v,L)=(0,3) \to (9,3)$ two-photon transition that wasrecently measured with high precision. The origin of this discrepancy isunknown.

Higher-order corrections to spin-spin scalar interactions in HD$^+$ and H$_2^+$