Metric Tensor Formulation of Strain in Density-Functional Perturbation Theory

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.