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We present a first-principles investigation of the structural, electronic,and magnetic properties of pyrolusite ($\beta$-MnO$_2$) using conventional andextended Hubbard-corrected density-functional theory (DFT+$U$ and DFT+$U$+$V$).The onsite $U$ and intersite $V$ Hubbard parameters are computed usinglinear-response theory in the framework of density-functional perturbationtheory. We show that while the inclusion of the onsite $U$ is crucial todescribe the localized nature of the Mn($3d$) states, the intersite $V$ is keyto capture accurately the strong hybridization between neighboring Mn($3d$) andO($2p$) states. In this framework, we stabilize the simplified collinearantiferromagnetic (AFM) ordering (suggested by the Goodenough-Kanamori rule)that is commonly used as an approximation to the experimentally-observednoncollinear screw-type spiral magnetic ordering. A detailed investigation ofthe ferromagnetic and of other three collinear AFM spin configurations is alsopresented. The findings from Hubbard-corrected DFT are discussed using twokinds of Hubbard manifolds - nonorthogonalized and orthogonalized atomicorbitals - showing that special attention must be given to the choice of theHubbard projectors, with orthogonalized manifolds providing more accurateresults than nonorthogonalized ones within DFT+$U$+$V$. This work paves the wayfor future studies of complex transition-metal compounds containing stronglylocalized electrons in the presence of pronounced covalent interactions.

Importance of intersite Hubbard interactions in $\beta$-MnO$_2$: A first-principles DFT+$U$+$V$ study