  We examine theoretically coherent electron transport through the
single-molecule magnet Mn$_{12}$, bridged between Au(111) electrodes, using the
non-equilibrium Green's function method and the density-functional theory. We
analyze the effects of bonding type, molecular orientation, and geometry
relaxation on the electronic properties and charge and spin transport across
the single-molecule junction. We consider nine interface geometries leading to
five bonding mechanisms and two molecular orientations: (i) Au-C bonding, (ii)
Au-Au bonding, (iii) Au-S bonding, (iv) Au-H bonding, and (v) physisorption via
van der Waals forces. The two molecular orientations of Mn$_{12}$ correspond to
the magnetic easy axis of the molecule aligned perpendicular [hereafter denoted
as orientation (1)] or parallel [orientation (2)] to the direction of electron
transport. We find that the electron transport is carried by the lowest
unoccupied molecular orbital (LUMO) level in all the cases that we have
simulated. Relaxation of the junction geometries mainly shifts the relevant
occupied molecular levels toward the Fermi energy as well as slightly reduces
the broadening of the LUMO level. As a result, the current slightly decreases
at low bias voltage. Our calculations also show that placing the molecule in
the orientation (1) broadens the LUMO level much more than in the orientation
(2), due to the internal structure of the Mn$_{12}$. Consequently, junctions
with the former orientation yield a higher current than those with the latter.
Among all of the bonding types considered, the Au-C bonding gives rise to the
highest current (about one order of magnitude higher than the Au-S bonding),
for a given distance between the electrodes. The current through the junction
with other bonding types decreases in the order of Au-Au, Au-S, and Au-H.
Importantly, the spin-filtering effect in all the nine geometries stays robust
and their ratios of the majority-spin to the minority-spin transmission
coefficients are in the range of 10$^3$ to 10$^8$. The general trend in
transport among the different bonding types and molecular orientations obtained
from this study may be applied to other single-molecular magnets.
