Exchange constants and second-order magnetic anisotropy in cyanide-bridged Fe2M2 single-molecule magnets

Electronic structure and intramolecular exchange constants are calculated for three cyanide-bridged single-molecule magnets, {[Tp$^{\star}$Fe$^{\mathrm{III}}$(CN)$_3$M$^{\mathrm{II}}$(DMF)$_4$]$_2$(OTf)$_2$}$\cdot$2DMF (M$^{\mathrm{II}}$=Mn, Co, Ni) (abbreviated as Fe$_2$Mn$_2$, Fe$_2$Co$_2$, and Fe$_2$Ni$_2$) that have been recently synthesized, within a generalized-gradient approximation in spin-polarized density-functional theory (DFT). Due to strong ligand fields present in the molecules, the Fe ions exhibit a low ground-state spin of $S=1/2$ in which the orbital angular momentum may not be quenched even without spin-orbit coupling. Based on the calculated electronic structures, the magnetic anisotropy for Fe$_2$Mn$_2$, Fe$_2$Co$_2$, and Fe$_2$Ni$_2$ is computed including single-electron spin-orbit coupling within a DFT formalism. The theoretical values of the induced orbital angular momentum and of the magnetic anisotropy parameters are compared to those for a single-molecule magnet Mn$_{12}$. The total magnetic anisotropy present in the three single-molecule magnets is due to competition between the magnetic anisotropy of the Fe and of the M ions.