  Inelastic neutron scattering has been used to determine the microscopic
Hamiltonian describing two high-spin variants of the high-anisotropy Mn$_6$
nanomagnet. The energy spectrum of both systems is characterized by the
presence of several excited total-spin multiplets partially overlapping the
S=12 ground multiplet. This implies that the relaxation processes of these
molecules are different from those occurring in prototype giant-spin
nanomagnets. In particular, we show that both the height of the energy barrier
and resonant tunnelling processes are greatly influenced by low-lying excited
total-spin multiplets.
