Hybrid functionals for periodic systems in the density functional tight-binding method

Over the last decades, hybrid exchange-correlation functionals have been established as a standard technique to address the delocalization problem of semi-local Kohn-Sham density functional theory (KS-DFT). The opposing delocalization in DFT and over-localization in Hartree-Fock (HF) theory gives rise to mutual error compensation when admixing exact HF-type exchange to the density functional approximation (DFA). Despite the ever-growing computational resources and algorithmic improvements, non-local HF-type exchange remains an expensive tool within first principles frameworks, particularly for periodic systems. Since its extension to purely long-range corrected hybrid functionals, the approximate density functional tight binding (DFTB) method, which is derived directly from KS-DFT, offers comparable quantum mechanical insights at a fraction of the computational cost. While the original formalism was restricted to molecules and long-range Fock exchange, this work generalizes the theoretical foundation and implementation to periodic boundary conditions beyond the Γ-point, covering the general class of range-separated hybrid functionals with Coulomb-attenuating method (CAM) type partitioning of the electron-electron interaction. Periodic boundary conditions are essential in broadening the applicability of hybrid DFTB, as most solid state systems cannot be adequately represented as cluster models. By implementing our work in the open-source software package DFTB+, a novel spectrum of methods allowing for an efficient treatment of problems beyond the reach of first principles schemes is made available to the materials science community. For the first time, we demonstrate optimally tuned screened range-separated hybrid DFTB to provide a qualitatively correct description of the polarization-induced fundamental gap renormalization in molecular crystals. Dielectric-dependent global hybrid DFTB accurately reproduces measured band gaps of simple bulk materials, that cover the entire range from narrow- to wide-gap semiconductors as well as insulators. At the same level of theory, we obtain the phonon-induced band gap renormalization of prototypical indirect semiconductors over a wide temperature range. To sample the nuclear wave function, we employ Williams-Lax theory, evaluated either by stochastic Monte-Carlo integration or a deterministic one-shot procedure, as well as classical Born-Oppenheimer molecular dynamics. Following the trend of higher-level many-body perturbation theory, such as Hedin's GW approximation, HF-type exchange admixed to the DFA systematically yields a slightly stronger electron-phonon renormalization, including zero-point corrections.