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. 2024 Jun 19;26(24):17028-17041.
doi: 10.1039/d4cp01129e.

Coupled-cluster treatment of complex open-shell systems: the case of single-molecule magnets

Maristella Alessio  1   2 , Garrette Pauley Paran  1 , Cansu Utku  1 , Andreas Grüneis  2 , Thomas-C Jagau  1
Affiliations

Coupled-cluster treatment of complex open-shell systems: the case of single-molecule magnets

Maristella Alessio et al. Phys Chem Chem Phys. .

Abstract

We investigate the reliability of two cost-effective coupled-cluster methods for computing spin-state energetics and spin-related properties of a set of open-shell transition-metal complexes. Specifically, we employ the second-order approximate coupled-cluster singles and doubles (CC2) method and projection-based embedding that combines equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) with density functional theory (DFT). The performance of CC2 and EOM-CCSD-in-DFT is assessed against EOM-CCSD. The chosen test set includes two hexaaqua transition-metal complexes containing Fe(II) and Fe(III), and a large Co(II)-based single-molecule magnet with a non-aufbau ground state. We find that CC2 describes the excited states more accurately, reproducing EOM-CCSD excitation energies within 0.05 eV. However, EOM-CCSD-in-DFT excels in describing transition orbital angular momenta and spin-orbit couplings. Moreover, for the Co(II) molecular magnet, using EOM-CCSD-in-DFT eigenstates and spin-orbit couplings, we compute spin-reversal energy barriers, as well as temperature-dependent and field-dependent magnetizations and magnetic susceptibilities that closely match experimental values within spectroscopic accuracy. These results underscore the efficiency of CC2 in computing state energies of multi-configurational, open-shell systems and highlight the utility of the more cost-efficient EOM-CCSD-in-DFT for computing spin-orbit couplings and magnetic properties of complex and large molecular magnets.

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Conflict of interest statement

There are no conflicts to declare.

Figures

Fig. 1
Fig. 1. Top: Structures of [Fe(H2O)6]2+ and [Fe(H2O)6]3+, Co(C(SiH3)3)2, and Co(C(SiMe2ONaph)3)2 complexes. For Co(C(SiMe2ONaph)3)2, the hydrogen atoms are omitted. Color code: Co – magenta, Fe – orange, Si – yellow, O – red, C – gray, H – white. Bottom: Partitioning of [Fe(H2O)6]2+, [Fe(H2O)6]3+, and Co(C(SiMe2ONaph)3)2 complexes into high-level fragment (orange and magenta for Fe and Co, respectively) and low-level fragment (blue) for the embedded EOM-CCSD calculations.
Fig. 2
Fig. 2. Simplified electron configurations of the reference and target states for [Fe(H2O)6]2+ (top), [Fe(H2O)6]3+ (middle), and Co(C(SiH3)3)2 and Co(C(SiMe2ONaph)3)2 (bottom). For [Fe(H2O)6]2+ and [Fe(H2O)6]3+, Th (and D2h) irreps are used. For Co(C(SiH3)3)2 and Co(C(SiMe2ONaph)3)2, D3d (and C2h) irreps are used. For [Fe(H2O)6]3+, λ = 0.40, λ′ = 0.34, and λ′′ and λ′′′ are arbitrarily assigned.
Fig. 3
Fig. 3. Spin–orbit splitting of the J = 9/2 ground state (states |1〉 and |2〉) of Co(C(SiH3)3)2 and Co(C(SiMe2ONaph)3)2. The energy barrier for spin-inversion is shown in red.
Fig. 4
Fig. 4. Left: Excitation energy ΔE for [Fe(H2O)6]2+ (|1〉, |2〉, |3〉 → |4〉, |5〉) computed using EOM-EA-CCSD, SCS-RI-EA-CC2, and EOM-EA-CCSD-in-DFT with cc-pVTZ basis set. Right: Absolute errors of EOM-EA-CCSD-in-DFT and SCS-RI-EA-CC2 with respect to EOM-EA-CCSD. EOM-EA-CCSD-in-DFT energies are obtained without truncation of the virtual orbital space.
Fig. 5
Fig. 5. Left: Excitation energy ΔE for [Fe(H2O)6]3+ (|1〉 → |2〉, |3〉, |4〉) computed using EOM-SF-CCSD, RI-SF-CC2, EOM-SF-CCSD-in-DFT, and SF-TD-DFT with cc-pVTZ basis set. Right: Absolute errors of EOM-SF-CCSD-in-DFT, RI-SF-CC2, and SF-TD-DFT with respect to EOM-SF-CCSD. EOM-SF-CCSD-in-DFT energies are obtained without truncation of the virtual orbital space.
Fig. 6
Fig. 6. Hole and particle NTOs of the transition density matrix between states |1〉 and |4〉 (top) and |1〉 and |5〉 (bottom) of [Fe(H2O)6]2+ computed with EOM-EA-CCSD-in-LRC-ωPBEh/cc-pVTZ. The singular values are σ = 0.97 from state |1〉 to state |4〉 and σ = 0.97 from state |1〉 to state |5〉. Red, green, and blue axes indicate x, y, and z axes. An isovalue of 0.05 was used.
Fig. 7
Fig. 7. Spin–orbit coupling constants (SOCCs) of [Fe(H2O)6]2+ (left) and [Fe(H2O)6]3+ (right) computed using EOM-CCSD and EOM-CCSD-in-DFT with the cc-pVTZ basis set. For [Fe(H2O)6]2+, the SOCC is computed between states |1〉 and |2〉, |1〉 and |3〉, and |2〉 and |3〉. For [Fe(H2O)6]3+, the SOCC is computed between state |1〉 and the triply-degenerate excited state (i.e., states |2〉, |3〉, and |4〉). EOM-CCSD-in-DFT SOCCs are obtained without truncation of the virtual orbital space.
Fig. 8
Fig. 8. Electronic states of the model system Co(C(SiH3)3)2 and the actual SMM Co(C(SiMe2ONaph)3)2 computed using EOM-EE-CCSD, CD-EE-CC2, and EOM-EE-CCSD-in-LRC-ωPBEh. The cc-pVTZ and 6-31G* basis sets were used for Co(C(SiH3)3)2 and Co(C(SiMe2ONaph)3)2, respectively.
Fig. 9
Fig. 9. Hole and particle NTOs for SOC between states |1〉 and |2〉 of Co(C(SiH3)3)2 computed with EOM-EE-CCSD-in-LRC-ωPBEh/cc-pVTZ. The singular values are σ = 0.46 and σ′ = 0.45. Red, green, and blue axes indicate x, y, and z axes. An isovalue of 0.05 was used.
Fig. 10
Fig. 10. Top: Calculated susceptibility curves of the model system Co(C(SiH3)3)2 and actual SMM Co(C(SiMe2ONaph)3)2 between 5 and 300 K under an applied field of 7 T. Bottom: Calculated magnetizations for Co(C(SiMe2ONaph)3)2 at temperatures from 2 to 15 K under magnetic fields of 1, 4, and 7 T. “av” stands for isotropic powder averaging. Experimental data for Co(C(SiMe2ONaph)3)2 were taken from ref. . The density functional is LRC-ωPBEh.
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