  The methods of density-functional perturbation theory may be used to
calculate various physical response properties of insulating crystals including
elastic, dielectric, Born charge, and piezoelectric tensors. These and other
important tensors may be defined as second derivatives of the total energy with
respect to atomic-displacement, electric-field, or strain perturbations, or as
mixed derivatives with respect to two of these perturbations. The resulting
tensor quantities tend to be coupled in complex ways in polar crystals, giving
rise to a variety of variant definitions. For example, it is generally
necessary to distinguish between elastic tensors defined under different
electrostatic boundary conditions, and between dielectric tensors defined under
different elastic boundary conditions. Here, we describe an approach for
computing all of these various response tensors in a unified and systematic
fashion. Applications are presented for two materials, wurtzite ZnO and
rhombohedral BaTiO3, at zero temperature.
