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HfO₂ shows different polymorphs, including monoclinic and orthorhombic ones, that exhibit singular properties. Moreover, the character of HfO₂ is also influenced by the Zr atoms as a doping agent. Here, an extensive study of the monoclinic <i>P2₁/c</i> and the orthorhombic <i>Pca2₁</i> polymorphs of HfO₂, Hf_0.75Zr_0.25O₂, and Hf_0.5Zr_0.5O₂ is reported. For all six systems, density functional theory (DFT) methods based on generalized gradient approximations (GGAs) were first used; then the GGA + U method was settled and calibrated to describe the electrical and optical properties of polymorphs and the responses to the oxygen vacancies. Zr had different effects in relation to the polymorph; moreover, the amount of Zr led to important differences in the optical properties of the <i>Pca2₁</i> polymorph. Finally, oxygen vacancies were investigated, showing an important modulation of the properties of Hf_xZr_yO₂ nanostructures. The combined GGA and GGA + U methods adopted in this work generate a reasonable prediction of the physicochemical properties of o- and m-Hf_xZr_yO₂, identifying the effects of doping phenomena.