The interaction of standard models particles with the axionic Dark Matter field may generate oscillating nuclear electric dipole moments (EDMs), oscillating nuclear Schiff moments and oscillating nuclear magnetic quadrupole moments (MQMs) with a frequency corresponding to the axions Compton frequency. Within an atom or a molecule an oscillating EDM, Schiff moment or MQM can drive transitions between atomic or molecular states. The excitation events can be detected, for example, via subsequent fluorescence or photoionization. Here we calculate the rates of such transitions. If the nucleus has octupole deformation or quadrupole deformation then the transition rate due to Schiff moment and MQM can be up to $10^{-16}$ transition per molecule per year. In addition, an MQM-induced transition may be of M2-type, which is useful for the elimination of background noise since M2-type transitions are suppressed for photons.
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