Self-consistent $\mathrm{DFT}+U$ method for real-space time-dependent density functional theory calculations

We implemented various DFT+$U$ schemes, including the Agapito, Curtarolo, and Buongiorno Nardelli functional (ACBN0) self-consistent density-functional version of the $\mathrm{DFT}+U$ method [Phys. Rev. X 5, 011006 (2015)] within the massively parallel real-space time-dependent density functional theory (TDDFT) code octopus. We further extended the method to the case of the calculation of response functions with real-time TDDFT+$U$ and to the description of noncollinear spin systems. The implementation is tested by investigating the ground-state and optical properties of various transition-metal oxides, bulk topological insulators, and molecules. Our results are found to be in good agreement with previously published results for both the electronic band structure and structural properties. The self-consistent calculated values of $U$ and $J$ are also in good agreement with the values commonly used in the literature. We found that the time-dependent extension of the self-consistent DFT+$U$ method yields improved optical properties when compared to the empirical TDDFT+$U$ scheme. This work thus opens a different theoretical framework to address the nonequilibrium properties of correlated systems.

Figure 1

(a) A localized orbital, represented on a subgrid (violet circle), overlaps with the border of the simulation box. Red points indicate grid points that belong to the subgrid of the localized orbital but not to the regular real-space grid. (b) The modified subgrid of the orbital, which is compatible with the periodicity of the solid and with the real-space grid. A phase shift is added to the points which are originally outside of the simulation box.

Figure 2

Top: Convergence of the band gap of bulk NiO versus the truncation radius of the $d$ orbitals of nickel. We used $U_{\mathrm{eff}}=U-J$ with $U=5.0$ eV and $J=0.95$ eV, as chosen in Ref. [28]. The back dot corresponds to the value obtained in Ref. [28] for an atomic sphere radius of 2.1 bohrs. Bottom: Number of grid points covered by the $d$ orbitals of Ni. Two dashed vertical lines indicate, respectively, the cutoff radius $R_c$ of the nonlocal part of the pseudopotential used and the smallest dimension of the simulation box $R_{\mathrm{box}}$.

Figure 3

Band structure of wurtzite zinc oxide, calculated from our implementation (solid black lines) compared to the result of the original ACBN0 paper (dashed red lines) [16].

Figure 4

Same as Fig. 3, but for ${\mathrm{TiO}}_2$ in its rutile phase.

Figure 5

Left: Band structure of bulk topological insulation ${\mathrm{Sb}}_2{\mathrm{Te}}_3$, calculated using the LDA functional (dashed red lines) and using the ACBN0 functional (solid green lines). Right: Details of the band-structure around the $\mathrm{\Gamma }$ point.

Figure 6

Left: Same as the left panel of Fig. 5, but now including full spin-orbit coupling (see the text for details of dealing with spinors in the ACBN0 functional). Right: The $p$-orbital character of the band structure around the $\mathrm{\Gamma }$ point.

Figure 7

Top: Binding energy versus atomic distance for the ${\mathrm{O}}_2$ molecule, obtained from PBE (blue curve), PBE+$U$ with $U=4\mathrm{eV}$ (red curve), and using the ACBN0 functional (orange curve). Bottom: Corresponding absolute value of the atomic force. The vertical line indicates the experimental equilibrium distance of 1.208 Å [40].

Figure 8

Calculated optical absorption spectra of bulk NiO obtained using the ACBN0 functional (solid red line) and the usual DFT+$U$ scheme with $U_{\mathrm{eff}}=5\mathrm{eV}$ (dashed blue line). The experimental data are taken from Ref. [41]. A Gaussian broadening of $\eta =0.5\mathrm{eV}$ is added to reproduce the experimental broadening.

Figure 9

Calculated optical absorption spectra of bulk MnO obtained using the ACBN0 functional (red line). The experimental data are taken from Messick et al. [42] and Ksendzov et al. [43, 44]. A Gaussian broadening of $\eta =0.4\mathrm{eV}$ is added to reproduce the experimental broadening.