Vibrationally induced fourth-order magnetic anisotropy and tunnel splittings in Mn₁₂

From density-functional-theory (DFT) based methods we calculate the vibrational spectrum of the Mn_{12}O_{12}(COOH)_{16}(H_2 O)_4 molecular magnet. Calculated infrared intensities are in accord with experimental studies. There have been no ab initio attempts at determining which interactions account for the fourth-order anisotropy. We show that vibrationally induced distortions of the molecule contribute to the fourth-order anisotropy Hamiltonian and that the magnitude and sign of the effect (-6.2 K) is in good agreement with all experiments. Vibrationally induced tunnel splittings in isotopically pure and natural samples are predicted.