DfPT

Density functional perturbation theory (DFPT) is a linear-response framework built on density functional theory (DFT) that computes the response of a periodic electronic system to small external perturbations. Instead of using finite differences of total energy or forces, DFPT derives and solves perturbation equations for the Kohn–Sham orbitals and the electron density, yielding response functions directly in reciprocal space.

DFPT is used to evaluate a range of perturbations, including atomic displacements (phonons), uniform electric fields, and strain. From these perturbations, it provides the dynamical matrix to obtain phonon frequencies and eigenvectors at any wave vector q, the dielectric tensor, Born effective charges, and piezoelectric coefficients. It can also be extended to compute electron–phonon coupling constants and related properties such as superconducting parameters through perturbations of the electron–nuclei potential and charge density.

In insulating and semiconducting materials, DFPT incorporates non-analytic corrections to describe long-range Coulomb interactions, enabling correct LO–TO phonon splitting at the zone center. In metals, the method requires careful treatment of partial occupancies and smearing. While primarily a harmonic theory, DFPT can be used to explore anharmonic effects only indirectly; other approaches are needed for explicit anharmonicity.

Implementation and availability: DFPT is implemented in several plane-wave and basis-set codes, including Quantum ESPRESSO and ABINIT, among others. It generally offers efficient phonon calculations with respect to finite-difference methods, but still requires dense sampling of phonon wave vectors and can be computationally demanding for large systems. The accuracy of DFPT depends on the underlying exchange–correlation functional and the chosen pseudopotentials or projector augmented-wave datasets.