  We report the importance of anisotropic Coulomb interactions in DFT+U
calculations of the electronic and magnetic properties of Mn$_3$O$_4$. The
effects of anisotropic interactions in Mn$^{2+}$ and Mn$^{3+}$ are separately
examined by defining two different sets of Hubbard parameters: $U^{2+}$ and
$J^{2+}$ for Mn$^{2+}$ and $U^{3+}$ and $J^{3+}$ for Mn$^{3+}$. The anisotropic
interactions in Mn$^{3+}$ have a significant impact on the physical properties
of Mn$_3$O$_4$ including local magnetic moments, canted angle, spontaneous
magnetic moment, and superexchange coupling, but those in Mn$^{2+}$ do not make
any noticeable difference. Weak ferromagnetic interchain superexchange,
observed in experiments, is predicted only if a sizable anisotropic interaction
is considered in Mn$^{3+}$. By analyzing the eigenoccupations of the on-site Mn
density matrix, we found that the spin channel involving Mn$^{3+}$
$d_{x^2-y^2}$ orbitals, which governs the 90$^\circ$ correlation superexchange,
is directly controlled by the anisotropic interactions. These findings
demostrate that the exchange correction $J$ for the intraorbital Coulomb
potential is of critical importance for first-principles description of reduced
Mn oxides containing Mn$^{3+}$ or Mn$^{4+}$.
