The changes in density, wave functions, and self-consistent potentials of solids, in response to small atomic displacements or infinitesimal homogeneous electric fields, are considered in the framework of the density-functional theory. A variational principle for second-order derivatives of the energy provides a basis for efficient algorithmic approaches to these linear responses, such as the state-by-state conjugate-gradient algorithm presented here in detail. The phase of incommensurate perturbations of periodic systems, that are, like phonons, characterized by some wave vector, can be factorized: the incommensurate problem is mapped on an equivalent one presenting the periodicity of the unperturbed ground state. The singularity of the potential change associated with an homogeneous field is treated by the long-wave method. The efficient implementation of these theoretical ideas using plane waves, separable pseudopotentials, and a nonlinear exchange-correlation core correction is described in detail, as well as other technical issues.

DOI:https://doi.org/10.1103/PhysRevB.55.10337