  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.
