Hyperfine structure and electric quadrupole transitions in the deuterium molecular ion

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Molecular hydrogen ions are of metrological relevance due to the possibility of precise theoretical evaluation of their spectrum and of external-field-induced shifts. In homonuclear molecular ions, the electric dipole $E1$ transitions are strongly suppressed, and of primary laser spectroscopy interest is the electric quadrupole ($E2$) transition spectrum. In continuation of previous work on the ${{\mathrm{H}}_{2}}^{+}$ ion, we report here the results of the calculations of the hyperfine structure of the laser-induced electric quadrupole transitions between a large set of rovibrational states of ${{\mathrm{D}}_{2}}^{+}$; the inaccuracies of previous evaluations have been corrected. The effects of the laser polarization are studied in detail. We show that the electric quadrupole moment of the deuteron can in principle be determined with low fractional uncertainty $(\ensuremath{\simeq}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4})$ by comparing the results presented here with future data from precision spectroscopy of ${{\mathrm{D}}_{2}}^{+}$.