Precision studies of the hydrogen molecular ion and its isotopologues

We present systematic calculations of the leading order relativistic corrections including the hyperfine splitting (HFS) for a wide range of rotational and vibrational states of the HT + molecular ion. We also calculate the DC and AC Stark effects for the molecular hydrogen ions H_2^+ and H D + in the non-relativistic approximation. The influence of the DC Stark polarizability effect on the hyperfine substates of a ro-vibrational state is carefully analyzed. Our results enable the detailed evaluation of certain systematic shifts of the transition frequencies for the purpose of ultra-high-precision optical, microwave, or radio-frequency spectroscopy of the hydrogen molecular ions in a trap.