  The purpose of this paper is to provide a mathematical analysis of the
Adler-Wiser formula relating the macroscopic relative permittivity tensor to
the microscopic structure of the crystal at the atomic level. The technical
level of the presentation is kept at its minimum to emphasize the mathematical
structure of the results. We also briefly review some models describing the
electronic structure of finite systems, focusing on density operator based
formulations, as well as the Hartree model for perfect crystals or crystals
with a defect.
