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

## TL;DR

This paper extends the density functional tight-binding (DFTB) method to include screened range-separated hybrid functionals, enabling accurate and efficient modeling of periodic systems with improved electronic property predictions.

## Contribution

It generalizes the theoretical foundation of DFTB to incorporate SRSH functionals with periodic boundary conditions, including techniques for handling Fock exchange in reciprocal space.

## Key findings

- Accurate prediction of polarization-induced gap renormalization in molecular crystals.
- Demonstrated convergence and efficiency for polyacenes and layered materials.
- Reduced computational cost compared to first principles methods.

## Abstract

Screened range-separated hybrid (SRSH) functionals within generalized Kohn-Sham density functional theory (GKS-DFT) have been shown to restore a general $1/(r\varepsilon)$ asymptotic decay of the electrostatic interaction in dielectric environments. Major achievements of SRSH include an improved description of optical properties of solids and correct prediction of polarization-induced fundamental gap renormalization in molecular crystals. The density functional tight-binding method (DFTB) is an approximate DFT that bridges the gap between first principles methods and empirical electronic structure schemes. While purely long-range corrected RSH are already accessible within DFTB for molecular systems, this work generalizes the theoretical foundation to also include screened range-separated hybrids, with conventional pure hybrid functionals as a special case. The presented formulation and implementation is also valid for periodic boundary conditions (PBC) beyond the $\Gamma$-point. To treat periodic Fock exchange and its integrable singularity in reciprocal space, we resort to techniques successfully employed by DFT, in particular a truncated Coulomb operator and the minimum image convention. Starting from the first principles Hartree-Fock operator, we derive suitable expressions for the DFTB method, using standard integral approximations and their efficient implementation in the DFTB+ software package. Convergence behavior is investigated and demonstrated for the polyacene series as well as two- and three-dimensional materials. Benzene and pentacene molecular and crystalline systems show the correct polarization-induced gap renormalization by SRSH-DFTB at heavily reduced computational cost compared to first principles methods.