  The systematic shifts of the transition frequencies in the molecular hydrogen
ions are of relevance to ultra-high-resolution radio-frequency, microwave and
optical spectroscopy of these systems, performed in ion traps. We develop the
ab-initio description of the interaction of the electric quadrupole moment of
this class of molecules with the static electric field gradients present in ion
traps. In good approximation, it is described in terms of an effective
perturbation hamiltonian. An approximate treatment is then performed in the
Born-Oppenheimer approximation. We give an expression of the electric
quadrupole coupling parameter valid for all hydrogen molecular ion species and
evaluate it for a large number of states of H2+, HD+, and D2+. The systematic
shifts can be evaluated as simple expectation values of the perturbation
hamiltonian. Results on radio-frequency (M1), one-photon electric dipole (E1)
and two-photon E1 transitions between hyperfine states in HD+ are reported. For
two-photon E1 transitions between rotationless states the shifts vanish. For a
subset of rovibrational one-photon transitions the quadrupole shifts range from
0.2 to 10 Hz for an electric field gradient of 0.1 GV/m2. We point out an
experimental procedure for determining the quadrupole shift which will allow
reducing its contribution to the uncertainty of unperturbed rovibrational
transition frequencies to the 1.10^(-15) relative level and, for selected
transitions, even below it. The combined contributions of black-body radiation,
Zeeman, Stark and quadrupole effects are considered for a large set of
transitions and it is estimated that the transition frequency uncertainty of
selected transitions can be reduced below the 1.10^(-15) level.
