Macroscopic polarization from electronic wave functions

The dipole moment of any finite and neutral system, having a square-integrable wave function, is a well-defined quantity. The same quantity is ill-defined for an extended system, whose wave function invariably obeys periodic (Born–von Kármán) boundary conditions. Despite this fact, macroscopic polarization is a theoretically accessible quantity, for either uncorrelated or correlated many-electron systems: in both cases, polarization is a rather “exotic” observable. For an uncorrelated—either Hartree–Fock or Kohn–Sham—crystalline solid, polarization has been expressed and computed as a Berry phase of the Bloch orbitals (since 1993). The case of a correlated and/or disordered system received a definitive solution only very recently (1998): This latest development allows us to present here the whole theory from a novel, and very general, viewpoint. The modern theory of polarization is even relevant to the foundations of density functional theory in extended systems.