  Magnetoelectric responses are a fundamental characteristic of materials that
break time-reversal and inversion symmetries (notably multiferroics) and,
remarkably, of "topological insulators" in which those symmetries are unbroken.
Previous work has shown how to compute spin and lattice contributions to the
magnetoelectric tensor. Here we solve the problem of orbital contributions by
computing the frozen-lattice electronic polarization induced by a magnetic
field. One part of this response (the "Chern-Simons term") can appear even in
time-reversal-symmetric materials and has been previously shown to be quantized
in topological insulators. In general materials there are additional orbital
contributions to all parts of the magnetoelectric tensor; these vanish in
topological insulators by symmetry and also vanish in several simplified models
without time-reversal and inversion those magnetoelectric couplings were
studied before. We give two derivations of the response formula, one based on a
uniform magnetic field and one based on extrapolation of a long-wavelength
magnetic field, and discuss some of the consequences of this formula.
