Koopmans spectral functionals: an open-source periodic-boundary implementation

Koopmans' spectral functionals aim to describe simultaneously ground state properties and charged excitations of atoms, molecules, nanostructures and periodic crystals. This is achieved augmenting standard density functionals with simple but physically motivated orbital-density-dependent corrections. These corrections act on a set of localized orbitals that, in periodic systems, resembles maximally localized Wannier function. At variance with a direct supercell implementation, we discuss here i) the complex but efficient formalism required for a periodic-boundary code using explicit Brillouin zone sampling, and ii) the calculation of the screened Koopmans' corrections with density-functional perturbation theory. In addition to delivering improved scaling with system size, the present development makes the calculation of band structures with Koopmans functionals straightforward. The implementation in the Quantum ESPRESSO distribution and the application to prototypical insulating and semiconducting systems are presented and discussed.