  It has been shown within density-functional theory that in Mn$_{12}$-acetate
there are effects due to disorder by solvent molecules and a coupling between
vibrational and electronic degrees of freedom. We calculate the in-plane
principal axes of the second-order anisotropy caused by the second effect and
compare them with those of the fourth-order anisotropy due to the first effect.
We find that the two types of the principal axes are not commensurate with each
other, which results in a complete quenching of the tunnel-splitting
oscillation as a function of an applied transverse field.
