Many-body perturbation theory at the level of the GW and the BSE (43) was used to compute the band structure and the dielectric response of bulk anatase TiO2. The GW and BSE calculations were performed on top of eigenvalues and eigenfunctions obtained from DFT. We used the planewave pseudopotential implementation of DFT as provided by the package Quantum ESPRESSO. For the GW and BSE calculations, we used the BerkeleyGW package (41). The DFT calculations were performed using the generalized gradient approximation as in the Perdew-Burke-Ernzerhof (PBE) scheme for the exchange-correlation functional. The Ti norm-conserving pseudopotential was generated in the Rappe-Rabe-Kaxiras-Joannopoulos scheme, including semicore 3s and 3p states. While standard structural and electronic quantities are already converged in DFT with an energy cutoff of 90 Ry, the energy cutoff used here was raised to 160 Ry to properly include the high number of bands necessary to reach convergence for the many-body evaluated properties. Bulk anatase TiO2 was modeled on a body-centered tetragonal lattice containing two Ti atoms and four O atoms (primitive cell) with lattice parameters (optimized at the PBE level) a = b = 3.79 Å and c = 9.66 Å. The experimental lattice constants at RT are a = b = 3.78 Å and c = 9.51 Å. Scaling these parameters to zero temperature via a linear extrapolation of the temperature dependence of the lattice constant at high temperature, appearing in (44), yields a = b = 3.78 Å and c = 9.49 Å.

The ground-state electronic density is properly described with a coarse 4 × 4 × 4 k-point grid for sampling of the Brillouin zone. The GW quasiparticle corrections to the DFT eigenvalues were performed at the one-shot level of theory (G0W0). For the computation of the polarizability and inverse dielectric matrices in BerkeleyGW, we used a total of 2474 conduction bands (CBs) and G-vectors with kinetic energies up to 46 Ry, whereas the self-energy operator was computed using 2472 unoccupied bands and a G-vector cutoff energy of 46 and 160 Ry for the screened and bare Coulomb matrices, respectively. The coarse 4 × 4 × 4 k-point grid sampling is sufficient for the description of the quasiparticle corrections, while a high number of bands are mandatory to get a proper description of screening effects and many-body corrections. The electronic band structure was finally obtained by interpolating GW corrections on top of a more refined DFT calculation with a 16 × 16 × 16 grid. The fully converged BSE results shown in the main text were obtained with BerkeleyGW. We used a shifted grid with up to 16 × 16 × 16 k-points (4096 irreducible k-points). The six lowest CBs and six topmost valence bands (VBs) were included to solve the excitonic Hamiltonian. Spin-polarized calculations were performed to highlight possible dark excitons due to triplet excitations, but no measurable differences with respect to the spin-restricted results were obtained. More details are provided in (7).

To estimate the role of the electron-acoustic phonon coupling in the electronic and optical properties of anatase TiO2, we performed frozen phonon DFT + GW + BSE calculations by applying a strain of 0.2% along the [010] crystallographic direction. The results of these calculations are shown in Fig. 3 (A to C).

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