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. 2021 Jun 10;125(22):4727-4740.
doi: 10.1021/acs.jpca.1c00397. Epub 2021 May 28.

Resolution of Low-Energy States in Spin-Exchange Transition-Metal Clusters: Case Study of Singlet States in [Fe(III)4S4] Cubanes

Giovanni Li Manni  1 , Werner Dobrautz  1 , Nikolay A Bogdanov  1 , Kai Guther  1 , Ali Alavi  1   2
Affiliations

Resolution of Low-Energy States in Spin-Exchange Transition-Metal Clusters: Case Study of Singlet States in [Fe(III)4S4] Cubanes

Giovanni Li Manni et al. J Phys Chem A. .

Abstract

Polynuclear transition-metal (PNTM) clusters owe their catalytic activity to numerous energetically low-lying spin states and stable oxidation states. The characterization of their electronic structure represents one of the greatest challenges of modern chemistry. We propose a theoretical framework that enables the resolution of targeted electronic states with ease and apply it to two [Fe(III)4S4] cubanes. Through direct access to their many-body wave functions, we identify important correlation mechanisms and their interplay with the geometrical distortions observed in these clusters, which are core properties in understanding their catalytic activity. The simulated magnetic coupling constants predicted by our strategy allow us to make qualitative connections between spin interactions and geometrical distortions, demonstrating its predictive power. Moreover, despite its simplicity, the strategy provides magnetic coupling constants in good agreement with the available experimental ones. The complexes are intrinsically frustrated anti-ferromagnets, and the obtained spin structures together with the geometrical distortions represent two possible ways to release spin frustration (spin-driven Jahn-Teller distortion). Our paradigm provides a simple, yet rigorous, route to uncover the electronic structure of PNTM clusters and may be applied to a wide variety of such clusters.

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Conflict of interest statement

The authors declare no competing financial interest.

Figures

Figure 1
Figure 1
Genealogical branching diagrams describe the spin coupling of a given unpaired electron with all the previous ones in a cumulative manner. The node weights are given by the van Vleck–Sherman formula (eq 6) but can also be computed as the sum of the node weights connected from the left, indicating the number of possible paths. The orange and green paths identify two configurations out of the g(12,0) = 132 possible for a spin-exchange system containing 12 unpaired electrons coupled to a singlet. The green configuration is derived from the orange one by a double spin flip, involving orbital 5 through 8. The blue path is one of the g(20,0) = 16796 configurations to couple 20 unpaired electrons to form a singlet spin state.
Figure 2
Figure 2
Schematic representation (a) and actual structure (b) of compound (1) used in our investigations. The magnetic centers form a distorted tetrahedron with two longer Fe–Fe bond distances (AB and CD, orange lines in (a,b), 2.846 Å) and four shorter Fe–Fe bond distances (AC, AD, BC, and BD, green lines in (a,b), 2.752 Å). In (b), white, gray, yellow, and blue spheres represent H, C, S, and Fe atoms, respectively. (c) Three possible orderings of the localized orbitals of the four magnetic centers. L and S refer to long and short bonds, respectively. ABCD is the one utilized in this work. For a perfect tetrahedron, the three orderings would be equivalent. For compound (2), a similar structure is considered except that system (2) features two short bonds (2.741 Å) and four long bonds (2.788 Å).
Figure 3
Figure 3
Genealogical branching diagram of a system containing four magnetic centers (FeA, FeB, FeC, and FeD), total spin Stotal = 0 (singlet), and five unpaired electrons on each site with parallel spins (Slocal = 5/2). The dashed red vertical lines separate four domains, each describing the spin coupling of the electrons residing on one of the four magnetic centers with all the previous ones. Gray nodes and arcs refer to non-Hund configurations which play a marginal role in the electronic wave function when the atom-separated ordering is utilized. The vanishing node weights for the gray nodes lead to the important reduction of the total number of spin-flip configurations, from a total of g(20,0) = 16796 (see Figure 1) to only 252 CSFs. The 252 CSFs can further be classified as: 1 for (Γ(5) ⊗ Γ(5)) and (Γ(0) ⊗ Γ(0)), 25 for (Γ(4) ⊗ Γ(4)) and (Γ(1) ⊗ Γ(1)), and 100 for (Γ(3) ⊗ Γ(3)) and (Γ(2) ⊗ Γ(2)), as discussed in section 2.3.
Figure 4
Figure 4
Hamiltonian matrices of exclusively exchange-coupled open-shell CSFs (including non-Hund spin-flip excitations) of a (12e,12o) active space, for an N4 model system in the same geometry as the iron atoms in Figure 2. The active space consists of the 12 2p orbitals of the nitrogens and their electrons. The Hamiltonian matrix of this simple model mimics well the one corresponding to the [Fe(III)4S4(SCH3)4] compounds, with the exception that each site features a local spin S = 3/2, and the intermediate pair states may only have spin SAB ranging from 0 to 3. On the left, the orbitals are ordered as 2pAx, 2pBx, 2pCx, 2pDx, 2pAy,..., while on the right, the localized orbitals are ordered in the atom-separated manner described in the text. There is a striking effect on the sparsity and quasi-block-diagonal form of the CI matrix by MO localization and ordering in conjunction with a spin-adapted basis. Red and blue squares represent negative and positive Hamiltonian matrix elements, respectively. The sign coherence (same sign) of the Hamiltonian matrix elements that follows the atom-separated ordering is another aspect that is worth mentioning that might have important implications in understanding the sign problem in fermionic many-body wave functions. Nondrawn squares (white) are zero entries of the CI Hamiltonian matrix. On the right, the small 20 by 20 sub-block in the top-left corner (green background) corresponds to the CSFs depicted in Figure 3, while the remaining sub-blocks (bottom right) correspond to non-Hund spin-flip excitations.
Figure 5
Figure 5
Spin-adapted FCIQMC dynamics for the six lowest singlet spin states of compound (1) within the CAS(20e,20o) and a walker population of 1 × 106 walkers. The colors used for the trajectories correspond to the ones utilized in Figure 3 to identify the leading components of the six singlet states.
Figure 6
Figure 6
Dominant electronic configurations of the ground-state wave functions of fully oxidized [Fe(III)4S4(SCH3)4] complexes (1) and (2).
Figure 7
Figure 7
Singly occupied 3d orbitals of the Fe(III) magnetic centers of compound (1) (top two rows) used for the CAS(20e,20o) calculations and doubly occupied 3p orbitals of the bridging sulfur atom (bottom row) added in the enlarged CAS(44e,32o).
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