Spin-polarized density functional theory (DFT) calculations are conducted using the projector augmented wave (PAW) method , as incorporated in the Vienna Ab initio Simulation Package (VASP) . The valence states explicitly include the 3p, 3d, and 4s orbitals of Ti, along with the 2s and 2p orbitals of O. Electronic exchange and correlation are handled within the framework of the semi-local generalized gradient approximation (GGA) functional, specifically the Perdew-Burke-Ernzerhof (PBE) formulation , supplemented by the van der Waals (vdW) correction according to the Tkatchenko-Scheffler (TS) method . To obtain a more precise representation of the electronic structure of TiO2, an on-site Coulomb interaction correction (DFT+U) is applied to the Ti 3d electrons, following the Dudarev approach , with an effective Hubbard parameter (Ueff) set to 3.5 eV , which has been identified as an optimal value for accurately describing the electronic characteristics and energetics of oxygen vacancies. A kinetic energy cutoff of 500 eV is employed for the plane-wave basis set.