  We have developed and implemented a formalism for computing the structural
response of a periodic insulating system to a homogeneous static electric field
within density-functional perturbation theory (DFPT). We consider the
thermodynamic potentials E(R,eta,e) and F(R,eta,e) whose minimization with
respect to the internal structural parameters R and unit cell strain eta yields
the equilibrium structure at fixed electric field e and polarization P,
respectively. First-order expansion of E(R,eta,e) in e leads to a useful
approximation in which R(P) and eta(P) can be obtained by simply minimizing the
zero-field internal energy with respect to structural coordinates subject to
the constraint of a fixed spontaneous polarization P. To facilitate this
minimization, we formulate a modified DFPT scheme such that the computed
derivatives of the polarization are consistent with the discretized form of the
Berry-phase expression. We then describe the application of this approach to
several problems associated with bulk and short-period superlattice structures
of ferroelectric materials such as BaTiO3 and PbTiO3. These include the effects
of compositionally broken inversion symmetry, the equilibrium structure for
high values of polarization, field-induced structural phase transitions, and
the lattice contributions to the linear and the non-linear dielectric
constants.
