# Magnetic exchange interactions in binuclear and tetranuclear iron (III) complexes described by spin-flip DFT and Heisenberg effective Hamiltonians

S. Kotaru, M. Alessio, S. Kaehler, and A.I. Krylov
J. Comp. Chem. , in press (2022)

Low-energy spectra of single-molecule magnets (SMMs) are often described by the Heisenberg Hamiltonian. Within this formalism, exchange interactions between magnetic centers determine the ground-state multiplicity and energy separation between the ground and excited states. In this contribution, we extract exchange coupling constants (J) for a set of iron (III) binuclear and tetranuclear complexes from all-electron calculations using non-collinear spin-flip time-dependent density functional theory (NC-SF-TDDFT). For the series of binuclear complexes with $J$-values ranging from -6 to -132 cm-1 our benchmark calculations using the short-range hybrid LRC-$\omega$PBEh functional and 6-31G(d,p) basis set agree well (mean absolute error of 4.7 cm-1) with the experimentally derived values. For the tetranuclear SMMs, the computed $J$ constants are within 6 cm-1 from the values extracted from the experiment. We explore the range of applicability of the Heisenberg model by analyzing the radical character in the binuclear iron (III) complexes using natural orbitals (NO) and their occupations. On the basis of the number of effectively unpaired electrons and the NO occupancies, we attribute larger errors observed in strongly anti-ferromagnetic species to an increased ionic character. The results illustrate the efficiency of the spin-flip protocol for computing the exchange couplings and the utility of the NO analysis in assessing the validity of effective spin Hamiltonians.