  We study quantum-classical escape-rate transitions for uniaxial and biaxial
models with finite spins S=10 (such as Mn_12Ac and Fe_8) and S=100 by a direct
numerical approach. At second-order transitions the level making a dominant
contribution into thermally assisted tunneling changes gradually with
temperature whereas at first-order transitions a group of levels is skipped.
For finite spins, the quasiclassical boundaries between first- and second-order
transitions are shifted, favoring a second-order transition: For Fe_8 in zero
field the transition should be first order according to a theory with S \to
\infty, but we show that there are no skipped levels at the transition.
Applying a field along the hard axis in Fe_8 makes transition the strongest
first order. For the same model with S=100 we confirmed the existence of a
region where a second-order transition is followed by a first-order transition
[X. Martines Hidalgo and E. M. Chudnovsky, J. Phys.: Condensed Matter (in
press)].
