Exchange coupling in the magnetic molecular cluster ${\mathrm{Mn}}_{12}\mathrm{Ac}$

The magnetic properties of the molecular cluster ${\mathrm{Mn}}_{12}\mathrm{Ac}$ are due to the four ${\mathrm{Mn}}^{3+}$ ions that have spins $S=3/2\mathrm{}$ and the eight ${\mathrm{Mn}}^{4+}$ ions with spins $S=2.\mathrm{}$ These spins are coupled by an exchange mechanism. We determine the four exchange couplings assuming a Heisenberg-type interaction between the ions. We use exact diagonalization of the spin Hamiltonian by a Lánczos algorithm and we adjust the couplings to reproduce the magnetization curve of ${\mathrm{Mn}}_{12}\mathrm{Ac}.$ We also impose the constraint of reproducing a gap of 35 K between an $S=10\mathrm{}$ ground state and a first excited state with $S=9.\mathrm{}$ We predict that there is an excited level with $S=8\mathrm{}$ at 37 K above the ground state, only slightly above the $S=9\mathrm{}$ excited state that lies at 35 K and the next excited state is an $S=9\mathrm{}$ multiplet at 67 K above the $S=10\mathrm{}$ ground state.