  Most experimental data on the single-molecule magnet Mn$_{12}$ acetate have
been successfully explained by the assumption that the Mn$_{12}$ acetate has an
effective ground-state spin of S=10. However, the effect of the low-lying
excited manifolds caused by interactions between Mn spins has not been well
understood. To investigate the features of the low-lying excited manifolds, the
intramolecular exchange interactions are calculated using density-functional
theory (DFT). With the calculated exchange parameters, the energy gap between
the S=10 ground-state and the first excited-state manifold is calculated by
diagonalization of the Heisenberg Hamiltonian. The upper limit on the energy
gap is about 40.5 K which is likely to be overestimated due to incomplete
treatment of the Coulomb potential within DFT. It is found that there are
several S=9 low-energy excited-state manifolds above the S=10 ground-state
manifold. The magnetic anisotropy barriers for the low-lying spin excitations
are calculated using DFT. Based on the calculations, it is found that the
anisotropy barriers for the low-lying excited manifolds are approximately the
same as that for the ground-state manifold, which is applicable to other
single-molecule magnets such as Mn$_4$.
