  Natural optical activity is the paradigmatic example of an effect originating
in the weak spatial inhomogeneity of the electromagnetic field on the atomic
scale. In molecules, such effects are well described by the multipole theory of
electromagnetism, where the coupling to light is treated semiclassically beyond
the electric-dipole approximation. That theory has two shortcomings: it is
limited to bounded systems, and its building blocks - the multipole transition
moments - are origin dependent. In this work, we recast the multipole theory in
a translationally-invariant form that remains valid for crystals. Working in
the independent-particle approximation, we introduce "intrinsic" multipole
transition moments that are origin independent and transform covariantly under
gauge transformations of the Bloch eigenstates. Electric-dipole transitions are
given by the interband Berry connection, while magnetic-dipole and
electric-quadrupole transitions are described by matrix generalizations of the
intrinsic magnetic moment and quantum metric. In addition to multipole-like
terms, the response of crystals at first order in the wavevector of light
contains band-dispersion terms that have no counterpart in molecular theories.
The full response is broken down into magnetoelectric and quadrupolar parts,
which can be isolated in the static limit where electric and magnetic fields
become decoupled. The rotatory-strength sum rule for crystals is found to be
equivalent to the topological constraint for a vanishing chiral magnetic effect
in equilibrium, and the formalism is validated by numerical tight-binding
calculations.
