Nuclear scattering events with large momentum transfer in atomic, molecular, or solid-state systems may result in electronic excitations. In the context of atomic scattering by dark matter (DM), this is known as the Migdal effect, but the same effect has also been studied in molecules in the chemistry and neutron scattering literature. Here we present two distinct Migdal-like effects from DM scattering in molecules, which we collectively refer to as the molecular Migdal effect: a centerof-mass recoil, equivalent to the standard Migdal treatment, and a non-adiabatic coupling resulting from corrections to the Born-Oppenheimer approximation. The molecular bonds break spherical symmetry, leading to large daily modulation in the Migdal rate from anisotropies in the matrix elements. Our treatment reduces to the standard Migdal effect in atomic systems but does not rely on the impulse approximation or any semiclassical treatments of nuclear motion, and as such may be extended to models where DM scatters through a long-range force. We demonstrate all of these features in a few simple toy models of diatomic molecules, namely H+ 2 , N2, and CO, and find total molecular Migdal rates competitive with those in semiconductors for the same target mass. We discuss how our results may be extended to more realistic targets comprised of larger molecules which could be deployed at the kilogram scale.

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The Migdal effect occurs when a particle collides with an atom, and the lag between the movement of the nucleus and that of the electron cloud results in ionization. It was first predicted in 1939, but to date it has yet to be experimentally observed. In order to measure the effect, one must remove many of the standard assumptions that make calculations easier in order to obtain a result that is realistically measurable. The original calculation was performed assuming a free atom, but most realistic experiments require the use of semiconductors in which atoms are constrained to a lattice. Modern calculations take these lattice effects into account, but they still use the soft-limit assumption that the momentum of the electron cloud will be significantly less than that of the nucleus. This assumption breaks down in key regions of the parameter space where calibration experiments would be most sensitive to a signal. Thus, in order to make an accurate prediction for the signature of the Migdal effect, we determine whether removing the soft-limit assumption significantly changes the result of the calculation.