Exchange coupling and contribution of induced orbital angular momentum of low-spin Fe 3 + ions to magnetic anisotropy in cyanide-bridged Fe 2 M 2 molecular magnets: Spin-polarized density-functional calculations

Electronic structure and intramolecular exchange constants are calculated for three cyanide-bridged molecular magnets, ${[T{p}^{\ensuremath{\star}}{\mathrm{Fe}}^{III}{(\mathrm{C}\mathrm{N})}_{3}{M}^{II}{(\mathrm{DMF})}_{4}]}_{2}{(\mathrm{O}Tf)}_{2}∙2\mathrm{DMF}$ $({M}^{II}=\mathrm{Mn},\mathrm{Co},\mathrm{Ni})$ (abbreviated as ${\mathrm{Fe}}_{2}{\mathrm{Mn}}_{2}$, ${\mathrm{Fe}}_{2}{\mathrm{Co}}_{2}$, and ${\mathrm{Fe}}_{2}{\mathrm{Ni}}_{2}$) that have been recently synthesized, within a generalized-gradient approximation in spin-polarized density-functional theory (DFT). Here $T{p}^{\ensuremath{\star}}={[{\mathrm{C}}_{3}{(\mathrm{C}{\mathrm{H}}_{3})}_{2}\mathrm{H}{\mathrm{N}}_{2}]}_{3}\mathrm{B}\mathrm{H}$, $\mathrm{O}Tf={\mathrm{O}}_{3}{\mathrm{SCF}}_{3}$, and $\mathrm{DMF}=\mathrm{HCON}{(\mathrm{C}{\mathrm{H}}_{3})}_{2}$. Due to strong ligand fields present in the ${[T{p}^{\ensuremath{\star}}{\mathrm{Fe}}^{III}{(\mathrm{C}\mathrm{N})}_{3}]}^{\ensuremath{-}}$ units, the ${\mathrm{Fe}}^{3+}$ ions exhibit a low ground-state spin of $S=1∕2$. Our calculations show that the metal ions in the ${\mathrm{Fe}}_{2}{\mathrm{Mn}}_{2}\phantom{\rule{0.3em}{0ex}}$ molecule interact antiferromagnetically via cyanide ligands, while those in the ${\mathrm{Fe}}_{2}{\mathrm{Co}}_{2}$ and ${\mathrm{Fe}}_{2}{\mathrm{Ni}}_{2}$ molecule interact ferromagnetically. The calculations also suggest that the smallest gaps between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) for ${\mathrm{Fe}}_{2}{\mathrm{Mn}}_{2}$, ${\mathrm{Fe}}_{2}{\mathrm{Co}}_{2}$, and ${\mathrm{Fe}}_{2}{\mathrm{Ni}}_{2}$ are 0.12, 0.03, and $0.33\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. Based on the calculated electronic structures, the second-order magnetic anisotropy is computed including single-electron spin-orbit coupling within a DFT formalism. In comparison to a prototype single-molecule magnet ${\mathrm{Mn}}_{12}$, the three cyanide-bridged molecular magnets are found to bear substantial transverse magnetic anisotropy that becomes 15%--36% of molecular longitudinal anisotropy. Spin-orbit coupling arising from the low-spin ${\mathrm{Fe}}^{3+}$ and high-spin ${\mathrm{Co}}^{2+}$ ions induces significant orbital angular momentum that contributes to the total magnetic anisotropy of the three cyanide-bridged molecular magnets. The induced orbital angular momentum is 4--8 times those calculated for ${\mathrm{Mn}}_{12}$. The total magnetic anisotropy present in the three molecular magnets is due to competition between the magnetic anisotropy of the ${\mathrm{Fe}}^{3+}$ and of the ${M}^{2+}$ ions. In the ${\mathrm{Fe}}_{2}{\mathrm{Mn}}_{2}$ and ${\mathrm{Fe}}_{2}{\mathrm{Ni}}_{2}\phantom{\rule{0.3em}{0ex}}$ molecules, the anisotropy is primarily due to the ${\mathrm{Fe}}^{3+}$ ions, while in the ${\mathrm{Fe}}_{2}{\mathrm{Co}}_{2}\phantom{\rule{0.3em}{0ex}}$ molecule, the single-ion anisotropy of the ${\mathrm{Co}}^{2+}$ ions counters the ${\mathrm{Fe}}^{3+}$ contributions. These results are supported by previously reported magnetic measurements.