  The electronic structure and numerous magnetic properties of MnBi magnetic
systems are investigated using local spin density approximation (LSDA) with
on-cite Coulomb correlations (LSDA+$U$) included. We show that the inclusion of
Coulomb correlations provides a much better description of equilibrium magnetic
moments on Mn atom as well as the magnetic anisotropy energy (MAE) behavior
with temperature and magneto-optical effects. We found that the inversion of
the anisotropic pairwise exchange interaction between Bi atoms is responsible
for the observed spin reorientation transition at 90 K. This interaction
appears as a result of strong spin orbit coupling on Bi atoms, large magnetic
moments on Mn atoms, significant $p-d$ hybridization between Mn and Bi atoms,
and it depends strongly on lattice constants. A better agreement with the
magneto-optical Kerr measurements at higher energies is obtained. We also
present the detailed investigation of the Fermi surface, the de Haas-van Alphen
(dHvA) effect and the X-ray magnetic circular dichroism in MnBi.
