  The direct calculation of the elastic and piezoelectric tensors of solids can
be accomplished by treating homogeneous strain within the framework of
density-functional perturbation theory. By formulating the energy functional in
reduced coordinates, we show that the strain perturbation enters only through
metric tensors, and can be treated in a manner exactly paralleling the
treatment of other perturbations. We present an analysis of the strain
perturbation of the plane-wave pseudopotential functional, including the
internal strain terms necessary to treat the atomic-relaxation contributions.
Procedures for computationally verifying these expressions by comparison with
numerical derivatives of ground-state calculations are described and
illustrated.
