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SrTiO$_3$ is a model perovskite compound with unique properties andtechnological relevance. At 105 K it undergoes a transition from a cubic to atetragonal phase with characteristic antiferrodistortive rotations of theTiO$_6$ octahedra. Here we study systematically the effect of differentexchange correlation functionals on the structural, electronic and opticalproperties of cubic and tetragonal STO by comparing the recently implementedstrongly constrained and appropriately normed (SCAN) meta-GGA functional withthe generalized gradient approximation (PBE96 and PBEsol) and the hybridfunctional (HSE06). SCAN is found to significantly improve the description ofthe structural properties, in particular the rotational angle of the tetragonalphase, comparable to HSE06 at a computational cost similar to GGA. The additionof a Hubbard $U$-term (SCAN+$U$, $U=7.45$ eV) allows to achieve theexperimental band gap of 3.25 eV with a moderate increase in the latticeconstant, whereas within GGA+$U$ the gap is underestimated even for high $U$values. The effect of the exchange-correlation functional on the opticalproperties is progressively reduced from 1.5 eV variance in the onset of thespectrum in the independent particle picture to 0.3 eV upon inclusion ofmany-body effects within the framework of the $GW$ approximation (single-shot$G_0W_0$) and excitonic corrections by solving the Bethe-Salpeter equation(BSE). Moreover, a model BSE approach is shown to reproduce the main featuresof the optical spectrum at a lower cost compared to $G_0W_0$+BSE. Strongexcitonic effects are found in agreement with previous results and their originis analyzed based on the contributing interband transitions. Last but notleast, the effect of the tetragonal distortion on the optical spectrum isdiscussed and compared to available experimental data.

Role of the exchange-correlation functional on the structural,  electronic and optical properties of cubic and tetragonal SrTiO$_3$ including  many-body effects