Pivotal Role of Intersite Hubbard Interactions in Fe-doped $\alpha$-MnO$_2$

We present a first-principles investigation of the structural, electronic, and magnetic properties of the pristine and Fe-doped $\alpha$-MnO$_2$ using density-functional theory with extended Hubbard functionals. The onsite $U$ and intersite $V$ Hubbard parameters are determined from first principles and self-consistently using density-functional perturbation theory in the basis of L\"owdin-orthogonalized atomic orbitals. For the pristine $\alpha$-MnO$_2$ we find that the so-called C2-AFM spin configuration is the most energetically favorable, in agreement with the experimentally observed antiferromagnetic ground state. For the Fe-doped $\alpha$-MnO$_2$ two types of doping are considered: Fe insertion in the $2 \times 2$ tunnels and partial substitution of Fe for Mn. We find that the interstitial doping preserves the C2-AFM spin configuration of the host lattice only when both onsite $U$ and intersite $V$ Hubbard corrections are included, while for the substitutional doping the onsite Hubbard $U$ correction alone is able to preserve the C2-AFM spin configuration of the host lattice. The oxidation state of Fe is found to be $+2$ and $+4$ in the case of the interstitial and substitutional doping, respectively, while the oxidation state of Mn is $+4$ in both cases. This work paves the way for accurate studies of other MnO$_2$ polymorphs and complex transition-metal compounds when the localization of $3d$ electrons occurs in the presence of strong covalent interactions with ligands.