Designing novel cathode materials for Li-ion batteries necessitates accurate first-principles predictions of their properties. Density-functional theory (DFT) employing standard (semi-)local functionals encounters challenges due to pronounced self-interaction errors within the partially filled d shells of transition-metal (TM) elements. Here, we demonstrate the efficacy of DFT with extended Hubbard functionals in accurately predicting the "digital" change in oxidation states of TM ions for mixed-valence phases at intermediate Li concentrations in phospho-olivine and spinel cathode materials. The resulting voltages exhibit remarkable agreement with experimental data [1,2].Our approach involves the use of onsite U and intersite V Hubbard parameters computed from density-functional perturbation theory with Lowdin-orthogonalized atomic orbitals, thus circumventing empirical factors [3,4]. Notably, the incorporation of intersite Hubbard V interactions proves indispensable for the precise prediction of thermodynamic quantities, particularly when electronic localization occurs amid inter-atomic orbital hybridization.Extending our investigation to vibrational spectroscopy, we calculate Raman spectra for these materials and find very good agreement with available experimental data [5]. Leveraging the robust and accurate computational framework based on extended Hubbard functionals, we embark on a high-throughput screening of Li-containing compounds, aiming to unveil novel and promising candidates for cathodes [6]. Our findings underscore the pivotal role of advanced DFT methodologies in advancing the discovery and design of high-performance battery materials.[1] I. Timrov, F. Aquilante, M. Cococcioni, N. Marzari, PRX Energy 1, 033003 (2022).[2] I. Timrov, M. Kotiuga, N. Marzari, Phys. Chem. Chem. Phys. 25, 9061 (2023).[3] I. Timrov, N. Marzari, M. Cococcioni, Phys. Rev. B 98, 045141 (2021).[4] I. Timrov, N. Marzari, M. Cococcioni, Phys. Rev. B 103, 085127 (2018).[5] L. Bastonero, N. Marzari, I. Timrov, in preparation.[6] C. Malica, L. Bastonero, M. Bercx, N. Marzari, I. Timrov, in preparation.