Second-order transverse magnetic anisotropy induced by disorder in the single-molecule magnet ${\mathrm{Mn}}_{12}$

For the single-molecule magnet ${\mathrm{Mn}}_{12},$ Cornia et al. recently proposed that solvent molecules may cause the quantum tunneling that requires a lower symmetry than $S_4.$ However, magnetic quantum tunneling and electron-paramagnetic-resonance experiments suggested that the proposed theory may not correspond to the measurements. In this regard, we consider positional disorder induced by the solvent molecules and orientational disorder by the methyl groups of a ${\mathrm{Mn}}_{12}$ molecule. We calculate, within density-functional theory, the second-order transverse magnetic anisotropy parameter E and an easy-axis tilting angle $\theta$ induced by the positional disorder and the E value by the orientational disorder. We also calculate the local magnetic anisotropy and the local easy axis for each inequivalent Mn site in different environments to investigate their effects on the global E value. We find that the hydrogen bonding between a ${\mathrm{Mn}}_{12}$ molecule and the solvent molecule is crucial to obtain a substantial E value and that the E value increases upon geometry relaxations. Our calculations on relaxed geometries show that the largest calculated E value is 0.016 K and that the largest tilting angle is $0.5°.$ Our largest E value is comparable to experimental results but larger than Cornia et al.’s.