Defects in materials significantly influence their properties and enhance functionality. Hybrid functionals like HSE06, though effective for describing defects, face challenges in geometry optimization for large supercells. The r2SCAN+rVV10+U+Ud method provides a computationally efficient alternative. By selecting appropriate U and Ud values for the d orbitals of host and defect atoms, this method accurately describes defects in materials. Our study on small polaron defects in layered transition-metal oxides demonstrates this. Using literature values for U and Ud, we investigated birnessite (KnMnO2, n = 0.03) and KnNiO2, n = 0.03. With one K atom intercalated in a supercell, both materials show a localized eg polaronic state on the transition metal ion reduced by the K atom, when the geometry is calculated using published U values. The expected Jahn-Teller distortion is not observed when U=Ud=0. In layered cobalt oxide with additional potassium ions (KnCoO2, n = 1.03), a single extra K atom in the supercell leads to four localized electrons in the band gap, using standard U values, and even for U=Ud=0. Monolayer MoS2 exhibits intriguing properties and potential technological applications when subjected to strain. A recent experimental study reported that the bandgap of monolayer MoS2 on a mildly curved graphite surface decreases by 400 meV/% strain under biaxial strain with a Poisson’s ratio of 0.44. We conducted density functional theory (DFT) calculations on a free-standing MoS2 monolayer using the generalized gradient approximation (GGA) PBE, the hybrid functional HSE06, and many-body perturbation theory with the GW approximation using PBE wavefunctions (G0W0@PBE). Our findings indicate that under biaxial strain with the experimental Poisson’s ratio, the bandgap decreases at rates of 63 meV/% strain (PBE), 73 meV/% strain (HSE06), and 43 meV/% strain (G0W0@PBE), which are significantly lower than the experimental rate. Additionally, PBE predicts a reduction rate of 90 meV/% strain for a Poisson’s ratio of 0.25. Spin-orbit correction (SOC) has minimal impact on the bandgap or its strain dependence. We also observed a semiconductor-to-metal transition at 10% tensile biaxial strain and a shift from a direct to an indirect bandgap, aligning with previous theoretical studies. / Physics