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The structural, electronic, optical, elastic, and ferroelectric characteristics of cubic strontium titanate ($SrTiO_3$) with the space group $(Pm\bar3m)$, have been computationally assessed through density functional theory (DFT). In this analysis, the Projector Augmented Wave (PAW) pseudo-potential was employed in conjunction with the Perdew Burke Ernzerhof (PBE) exchange-correlation functional within the framework of the Generalized Gradient Approximation (GGA). Two distinct scenarios were considered: one with and one without the incorporation of the Hubbard U potential [U(Ti$_{3d}$ & O$_{2p}$) = 5eV]. These calculations were carried out using the Quantum Espresso computational tool. One of the key aspects under investigation was the band gap of cubic $SrTiO_3$ (STO), which was determined by analyzing the electronic band structure at the Gamma point. The results revealed a band gap value of 2.2135eV when utilizing the standard DFT method and 3.7625eV when applying the DFT+U approach. Notably, the DFT+U method demonstrated a closer alignment with experimental data. In the context of DFT+U, the PDOS shifts arise from the on-site electron interaction correction. It was observed that the Ti$_{3d}$ and O$_{2p}$ orbitals exhibited a robust hybridization, signifying a covalent bond, substantiated by the overlapping charge density evident in the 2D contour map, Conversely, an ionic bond prevailed between strontium (Sr) and oxygen (O) atoms. Both DFT and DFT+U methods are employed to compute complex dielectric function components, and optical parameters such as reflectivity, refractive index, extinction coefficient, absorption coefficient, and the electron energy-loss spectrum. It was elucidated that the inclusion of the Hubbard U potential played a pivotal role in density functional theory calculations, resulting in a narrower and more experimentally congruent optical spectrum, a significant contrast to standard DFT calculations, which lack this corrective measure and consequently generate broader and less precise electronic bands and optical spectra, particularly in cases where strong electron-electron interactions prevail. Thermal properties like entropy and heat capacity are typically less sensitive to on-site electron correlations, which Hubbard U in DFT+U addresses. Therefore, both standard DFT and DFT+U yield similar results for these properties as electron correlations have a relatively minor impact in this context and have good agreement with the Debye model. Elastic properties, including the Bulk Modulus, Young's Modulus, Shear Modulus, and Poisson's Ratio, were assessed via both cases, yielding closely aligned values, indicative of their consistency and agreement.
