Koopmans Spectral Functionals in Periodic Boundary Conditions

Abstract

Koopmans spectral functionals aim to describe simultaneously ground-state properties and charged excitations of atoms, molecules, nanostructures, and periodic crystals. This is achieved by 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, resemble maximally localized Wannier functions. At variance with the original, direct supercell implementation (Phys. Rev. X 2018, 8, 021051), 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 open-source Quantum ESPRESSO distribution and the application to prototypical insulating and semiconducting systems are presented and discussed.

koopmans: an open-source package for accurately and efficiently predicting spectral properties with Koopmans functionals

“…This was shown for a large set of molecules relevant for photovoltaic applications, 39 with Koopmans functionals yielding ultraviolet photoemission spectra that agree quantitatively with experiment. One can see similar accuracy in the prediction of band gaps and band structures of periodic systems; 41,43,49 in a study of prototypical semiconductors and insulators, Koopmans functionals were found to yield band gaps with a mean absolute error of 0.22 eV, compared to 0.18 eV when using self-consistent GW with vertex corrections. 41 Importantly, alignment between the valence band edge and the vacuum level was also very good: across six semiconductors the mean absolute error was 0.19 eV, compared to 0.39 eV for G 0 W 0 and 0.49 eV for self-consistent GW with vertex corrections.…”

“…This perturbation is the Hartree-plus-exchange-correlation potential generated when adding/removing an infinitesimal fraction of an electron to/from orbital i. One determines ∆ i n self-consistently via DFPT, 49 and then the screening parameters are given by…”

“…This choice of basis allows us to take advantage of the translational symmetry 43 of periodic systems: the DFPT expression for the screening coefficients (Equation 10) can be decomposed into a set of independent problems (monochromatic perturbations), one for each q point sampling the Brillouin zone of the primitive cell. 49 The now q-dependent charge density variation ∆n 0i q (r)…”

“…induced by the perturbing potential v 0i pert,q is obtained self-consistently via DFPT (Equations 15-17 of Ref. 49), and then the screening coefficients are obtained by summing over q:…”

“…Thus, as the supercell size (or equivalently the number of q-points in the primitive cell) increases, the kcw.x DFPT approach becomes more and more computationally efficient. 49 That said, kcw.x makes two approximations that kcp.x does not: the DFPT approach expands the total energy only to second order when computing screening parameters (see Section 1.3.2), and it does not optimize the variational orbitals. These are instead defined via Wannier functions, which often closely resemble the minimizing orbitals of the Koopmans energy functional.…”