  High energy beams incident on a fixed target may scatter against atomic
electrons. To a first approximation, one can treat these electrons as at rest.
For precision experiments, however, it is important to be able to estimate the
size of, and when necessary calculate, sub-leading corrections. We discuss
atomic binding corrections to relativistic lepton-electron scattering. We
analyze hydrogen in detail, before generalizing our analysis to multi-electron
atoms. Using the virial theorem, and many-body sum rules, we find that the
corrections can be reduced to measured binding energies, and the expectation
value of a single one-body operator. We comment on the phenomenological impact
for neutrino flux normalization and an extraction of hadronic vacuum
polarization from elastic muon electron scattering at MUonE.
