  To understand the effect of molecular environment on the electronic and
magnetic properties of the single-molecule magnet (SMM) Mn$_{12}$, we explore
two possible means for adding extra electrons to molecule. We explore both
substitution of Mn ions by Fe ions and the inclusion of neighboring electronic
donors. For both possibilities we calculate, within density-functional theory,
the electronic structure, the total ground-state spin and ordering, the
magnetic anisotropy barrier, the transverse magnetic anisotropy parameter $E$
which is responsible for some measured tunneling, and the tilting angle of the
easy axis from the z axis. Our calculations show that the total spin increases
with increasing number of extra electrons except for the case of Mn$_{8}$Fe$_4$
where the resulting ground state has a low spin. The calculated energy gaps
between the unoccupied and the occupied orbitals exhibit no clear trend as a
function of number of extra electrons. The calculated magnetic anisotropy
barrier decreases with increasing number of extra electrons and can be directly
chased to a quenching of Jahn-Teller distortions at the sites where the
additional electrons are localized. The values of $E$ and the easy-axis tilting
angles for the geometries with one- and two-extra electrons are significantly
larger than those induced by solvent disorder.
