Precision spectroscopy of the hydrogen molecular ions HD$^+$

Abstract: Expectation values of the Breit operators and the $Q$ terms are calculated for HD$^+$ with the vibrational number $v=0-4$ and the total angular momentum $L=0-4$. Relativistic and radiative corrections to some ro-vibrational transition frequencies are determined. Numerical uncertainty in $R_{\infty}\alpha^2$ order correction is reduced to sub kHz or smaller. Our work provides an independent verification of Korobov's calculations [Phys. Rev. A {\bf74}, 052506 (2006); {\bf77}, 022509 (2008)].