Comparison of Magnetization Tunneling in the Giant-Spin and Multi-Spin Descriptions of Single-Molecule Magnets

We perform a mapping of the spectrum obtained for a triangular Mn$_{3}$ single-molecule magnet (SMM) with idealized $C_{3}$ symmetry via exact diagonalization of a multi-spin (MS) Hamiltonian onto that of a giant-spin (GS) model which assumes strong ferromagnetic coupling and a spin $S$~=~6 ground state. Magnetic hysteresis measurements on this Mn$_{3}$ SMM reveal clear evidence that the steps in magnetization due to magnetization tunneling obey the expected quantum mechanical selection rules [J. Henderson \textit{et al.}, Phys. Rev. Lett. \textbf{103}, 017202 (2009)]. High-frequency EPR and magnetization data are first fit to the MS model. The tunnel splittings obtained via the two models are then compared in order to find a relationship between the sixth order transverse anisotropy term $B_6^6 $ in GS model and the exchange constant $J$ coupling the Mn$^{III}$ ions in the MS model. We also find that the fourth order transverse term $B_4^3 $\textbf{ }in the GS model is related to the orientation of JahnTeller axes of Mn$^{III}$ ions, as well as $J$