Equation-of-Motion Coupled-Cluster Protocol for Calculating Magnetic Properties: Theory and Applications to Single-Molecule Magnets

Abstract

We present a new computational protocol for computing macroscopic magnetic properties of transition-metal complexes using equation-of-motion coupled-cluster (EOM-CC) framework. The approach follows a two-step state-interaction scheme: we first compute zero-order states using non-relativistic EOM-CC and then use these states to evaluate matrix elements of the spin-orbit and Zeeman operators. Diagonalization of the resulting Hamiltonian yields spin-orbit-and field-perturbed eigenstates.Temperature-and field-dependent magnetization and susceptibility are computed by numerical differentiation of the partition function. To compare with powder-sample experiments, these quantities are numerically averaged over field orientations. We applied this protocol to several mononuclear Fe(II) and Fe(III) single-molecule magnets (SMMs) trigonal pyramidal, linear, and trigonal bipyramidal coordination environments. The underlying electronic structure was described by the electron-attached (EOM-EA) and spin-flip (EOM-SF) variants of EOM-CC. The computed energy barriers for spin inversion, and macroscopic magnetization and susceptibility agree well with experimental data. Trends of magnetic anisotropy and spin-reversal energy barrier are explained in terms of a molecular orbital picture rigorously distilled from spinless transition density matrices between many-body states. The results illustrate excellent performances of EOM-CC for describing magnetic behavior of mononuclear transition-metal SMMs.

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

“…69 Alessio and Krylov used the EOM-SF-CCSD treatment to calculate the spin reversal energy barrier and magnetic properties of this complex. 85 Following this study, here we compute the energy barrier using the spin-orbit coupling obtained with SOMF NC-SF-TD-DFT treatment. For this complex, we use PBE0 and ωPBEh functionals to access the quartet ground state (S = 3/2) and other closely lying excited states by spin-flip excitations from a highspin reference of S = 5/2.…”

“…This differs from the EOM-SF-CCSD results, where SF 3 is about 2000 cm −1 above SF 2 . 85 We then proceed to compute the spin reversal energy barrier with 2, 3, and 5 low-lying SF states by including the SOC effects using the state-interaction procedure. 36 The inclusion of only two states is insufficient to characterize the energy barrier with SF-TD-DFT (see Table VIII), however, the energy barrier calculated with 3 states gives a value of 97 (PBE0) cm −1 and 100 (ωPBEh) cm −1 , in excellent agreement with the experimental estimate of 100 cm −1 .…”

libwfa: Wavefunction analysis tools for excited and open‐shell electronic states

“…Spin-orbit coupling splits the otherwise degenerate components of the quintet (d 6 ) state and creates a barrier U for spin reorientation. The calculations 80 yield barriers that differ by a factor of two. The computed NTOs reveal that the spin-orbit interaction involves different M s components of iron's d-orbitals, which leads to large differences in the change of angular momentum (ΔL ≈ 1 vs. ΔL ≈ 2), and, consequently, in the magnitude of the spin-orbit couplings and U.…”

“… Left : Original El‐Sayed's rules explain trends in intersystem crossing rates in terms of molecular orbitals. Right : SOC NTOs computed using EOM‐EA‐MP2 wavefunctions for Fe(II) single‐molecule magnets explain the difference in the computed spin‐reorientation barrier in terms of angular momentum change of the orbitals 80 …”

Magnetic Exchange Interactions in Binuclear and Tetranuclear Iron (III) Complexes Described by Spin-Flip DFT and Heisenberg Effective Hamiltonians

“…We follow an alternative strategy based on the spin-flip approach. Spin-flip (SF) methods [21][22][23][24][25][26][27] offer a balanced treatment of multi-configurational states of polyradicals within a single-reference formalism, making the SF-protocol suitable for applications to SMMs [28][29][30][31][32][33]. The SF approach does not rely on scrambled spin-states and does not re-quire choosing an active space.…”