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

Screened range-separated hybrid (SRSH) functionals within generalized Kohn-Sham density functional theory (GKS-DFT) have been shown to restore a general $1/\left(r\varepsilon \right)$ 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 $\mathrm{\Gamma }$ 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 $\mathrm{dftb}+$ 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.

Figure 1

Total wall-clock time of a RSH-DFTB $\mathrm{\Gamma }$-point calculation with the LCY-PBE xc functional, in comparison with conventional PBE-DFTB, as performed on a single CPU core (AMD Ryzen 7 PRO 5850U). For LCY-PBE-DFTB, the total time spent on an average self-consistent cycle, the construction of the RSH contribution to the total Hamiltonian and atomic force evaluation is indicated as well. GaAs supercells are employed as model system. LCY-PBE-DFTB exhibits slightly subcubic scaling, which might indicate that the asymptotic limit is not yet reached.

Figure 2

Parallel performance of $\mathrm{dftb}+$, when performing range-separated calculations beyond the $\mathrm{\Gamma }$ point (${\varepsilon }_{\mathrm{screen}}$ is set at ${10}^{-7}\mathrm{a}.\mathrm{u}.$). The primitive GaAs unit cell, sampled by a $9\times 9\times 9$ Monkhorst-Pack $k$-point set, served as model system to obtain the (I/O time removed) wall-clock time of the entire $\mathrm{dftb}+$run (with one MPI process corresponding to one processor core). Up to about 100 cores the parallel efficiency is excellent, however, a slight change in slope due to incipient internode communication between 20 and 30 cores is observed. The parallel efficiency then decreases to about $50\%\text{–}60\%$ when further increasing the core count. Considering the extremely small test system of only two atoms, this saturation is expected. The employed HPC provides nodes of two Intel Xeon E5-2690v4 CPUs (2.6 GHz, 14 cores each), resulting in 28 cores per node, whereas internode communication is based on Intel's Omni-Path network architecture.

Figure 3

Band-gap convergence behavior of the periodic implementations ($\mathrm{\Gamma }$ point, TCI, MIC) for the polyacene series, in comparison with the nonperiodic algorithm. In the limit of dense $k$ points or large clusters and supercells, respectively, all implementations agree ($E_{\mathrm{gap}}=2.9\mathrm{eV}$).

Figure 4

Band structures of the primitive polyacene unit cell, aligned at their respective valence band maximum (VBM). Fully converged RSH-DFTB calculations, with $1\times 1\times 15$ $k$-point sampling in the self-consistent run that produced the ground-state density, performed in the TCI and MIC schemes, are compared to a band structure that exhibits artifacts due to an insufficient truncation of the Coulomb interaction for small BvK supercells. The first Brillouin zone was sampled between $B^{\prime }=(0.0,0.0,-0.5)$ and $B=(0.0,0.0,+0.5)$.

Figure 5

Absolute deviation in the total energy and band gap with respect to their fully converged reference. For periodic RSH-DFTB calculations, the total energy of a system is empirically found to converge more rapidly than the Kohn-Sham band gap. For GaAs bulk it was found that setting the TCI cutoff as half of the minimum lattice vector norm (${\mathrm{TCI}}^{\prime }$), the total energy and band gap converge more quickly when compared to the scheme described in Sec. 3b. The choice of coupling the TCI cutoff to the $k$-point sampling can therefore significantly influence the convergence behavior of a system.

Figure 6

Fundamental gaps of gaseous and crystalline benzene and pentacene, as obtained by conventional PBE-DFTB and optimally tuned screened range-separated hybrid functionals (OT-SRSH-DFTB), compared with $G_0{W}_0$ calculations. To maintain comparability with Ref. [77], crystalline gaps were aligned to the middle of their respective gas-phase gap. $G_0{W}_0$ values were taken from Refs. [83, 88] (gas phase) and Ref. [77] (bulk).