  This works extends the recent study on the dielectric permittivity of
crystals within the Hartree model [E. Cances and M. Lewin, Arch. Rational Mech.
Anal., 197 (2010) 139--177] to the time-dependent setting. In particular, we
prove the existence and uniqueness of the nonlinear Hartree dynamics (also
called the random phase approximation in the physics literature), in a suitable
functional space allowing to describe a local defect embedded in a perfect
crystal. We also give a rigorous mathematical definition of the microscopic
frequency-dependent polarization matrix, and derive the macroscopic
Maxwell-Gauss equation for insulating and semiconducting crystals, from a first
order approximation of the nonlinear Hartree model, by means of homogenization
arguments.
