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

Abstract

Low‐energy spectra of single‐molecule magnets (SMMs) are often described by Heisenberg Hamiltonians. 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 12 binuclear complexes with J‐values ranging from −6 to −132 cm−1, our benchmark calculations using the short‐range hybrid ωPBEh functional and 6‐31G(d,p) basis set agree well with the experimentally derived values (mean absolute error of 4.7 cm−1). For the tetranuclear SMMs, the computed J constants are within 6 cm−1 from the experimentally derived values. We explore the range of applicability of the Heisenberg model by analyzing bonding patterns in these Fe(III) complexes using natural orbitals (NO), their occupations, and the number of effectively unpaired electrons. 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.

Spin–orbit couplings within spin-conserving and spin-flipping time-dependent density functional theory: Implementation and benchmark calculations

“…This limitation was overcome by using non-collinear formulation. [52][53][54] Previous benchmark studies for organic diradicals 54 , binuclear Cu(II) 62 and Fe(III) 58 SMMs illustrated robust performance of the non-collinear SF-TD-DFT, especially when combined with the functionals from the PBE family. Here, we use the best performing functionals to assess the performance of the SF-TD-DFT for computing SOCs in selected diradicals and in a Fe(III) SMM.…”

“…Benchmark calculations on other classes of molecules have also shown superior performance of NC-SF-TD-DFT with PBE0 and ωPBEh. 58,62 Following previous studies, 35,36,41 we computed SOCCs between the lowest triplet ( 3 B 2 ) and singlet ( 1 A 1 ) states of these diradicals (CH 2 , NH + 2 , SiH 2 , and PH + 2 ). We used NC-SF-TD-DFT and considered B3LYP, PBE0, ωPBEh, ωB97X-D, and ωB97M-V. Table V presents the SOMF SOCCs in these diradicals using NC-SF-TD-DFT, com-paring them with SOMF SOCCs obtained with EOM-SF-CCSD.…”

“…The SF approach extends Kohn-Sham TD-DFT to treat certain types of strong correlation, such as bond-breaking, conical intersections, and systems with two or more unpaired electrons. 55,[57][58][59] This work describes the implementation of SOCs using TD-DFT and SF-TD-DFT within the Q-Chem electronic structure package 60,61 and presents benchmark results for molecules featuring different types of electronic structure: e.g., closed-shell organic molecules, diradicals, and a molecular magnet. We compare the results obtained with different DFT and wave-function-based methods and assess the effect of specific density functionals and basis sets on the SOC.…”

Validation of the Green’s Function Approximation for the Calculation of Magnetic Exchange Couplings

“…This expression can be straightforwardly generalized to the multi-center case. , We refer the reader to the pioneering work of Noodleman and the widely available literature regarding methods based on energy differences to extract magnetic exchange couplings from DFT calculations. ,, Another approach has been proposed that deserves to be highlighted. Using the spin-flip time-dependent DFT, Mayhall et al derived a method for mapping single spin-flip energies into the spectrum of an effective Hamiltonian that can be used to extract the exchange couplings. , …”

“…Using the spin-flip timedependent DFT, Mayhall et al derived a method for mapping single spin-flip energies into the spectrum of an effective Hamiltonian that can be used to extract the exchange couplings. 33,34 A second family of methods, very popular in the solid-state physics community, is based on local spin rotations, where the mapping from the DFT energy and the model spin Hamiltonian is based on the energy derivative with respect to differential local spin rotations. The underlying idea is that if both low-energy spectra match, then for a system of two coupled centers, one can write…”