Magnetic interactions and spin orders in Cr$_8$ and V$_8$ ring-shaped molecular magnets from non-collinear ab initio calculations

Abstract: We employ density functional theory within a non-collinear framework to investigate the magnetic properties of the octanuclear molecular rings Cr$_8$ and V$_8$. Our aim is to generalize the evaluation of the effective magnetic interactions by explicitly including non-collinear spin configurations, thereby refining our understanding of their dependence upon the underlying electronic structure and molecular geometry. By analyzing the energetics of a variety of magnetic configurations, particularly non-collinear arrangements with neighboring spins oriented along different directions, we move beyond the exchange-only Heisenberg Hamiltonian describing the low-energy sector of the excitation spectrum. This approach enables us to distinguish between in-plane and out-of-plane exchange interactions, and to incorporate biquadratic coupling terms into the effective spin Hamiltonian. We reveal significant antisymmetric exchange interactions of the Dzyaloshinskii-Moriya (DM) type whose dependence on the curvature of the annular structure is clarified by a comparison with the results obtained from linear chains of equal composition. Our work demonstrates that interactions beyond conventional exchange, particularly biquadratic anisotropic terms, in the spin Hamiltonian are essential for accurately capturing the low-energy excitations of these systems. The closest quantitative agreement with experimental results (particularly for the case of Cr$_8$) is achieved when extended Hubbard functionals are used for the evaluation of the effective magnetic couplings.

• The paper demonstrates that extended spin Hamiltonians including biquadratic and Dzyaloshinskii–Moriya interactions are essential for modeling the low-energy excitations of Cr8 and V8 rings.
• It employs non-collinear DFT with DFT+U+V to extract anisotropic exchange parameters, confirming improved experimental susceptibility agreement especially at low temperatures.
• The study highlights that curvature-induced DM effects and competition between FM and AFM couplings fundamentally govern the quantum magnetic behavior of molecular rings.

Magnetic Interactions and Spin Order in Cr$_8$ and V$_8$ Molecular Rings from Non-Collinear ab initio Calculations

Introduction and Motivation

This work presents a comprehensive non-collinear DFT investigation of the magnetic interactions in Cr$_8$ and V$_8$ octanuclear ring-shaped molecular magnets (2604.23565). These molecular systems serve as prototypical quantum magnets for exploring the intricate interplay between geometry, electronic correlation, and spin interactions at the molecular scale. The motivation stems from the recognized inadequacy of collinear spin models, particularly the isotropic Heisenberg Hamiltonian, for fully describing the low-energy excitations in such systems. The authors extend their previous work by employing a non-collinear DFT framework, allowing explicit treatment of arbitrary spin arrangements, and systematically evaluate beyond-Heisenberg couplings, such as biquadratic and antisymmetric (Dzyaloshinskii-Moriya, DM) exchange, as well as their dependence on ring curvature.

Methodology and Model Systems

The Cr$_8$ and V$_8$ molecular rings are constructed from eight transition metal $M^{3+}$ ions (M = Cr or V), each octahedrally coordinated and connected via bridging fluorine and carboxylate ligands, forming nearly planar octagons with a high degree of local symmetry.

Figure 1: Representation of the M-chain molecular structures, unit-cell and supercell, adopted in this study.

To isolate the effect of structural curvature on magnetic couplings, the authors introduce linear chain analogues with identical local bonding topology. All electronic structure calculations employ plane-wave DFT (as implemented in Quantum Espresso), using both the DFT+$U$ and extended DFT+$U$+$V$ schemes, with site- and bond-resolved Hubbard parameters extracted via linear-response theory. Non-collinear magnetic states are generated for systematic fitting of spin Hamiltonians against total energies.

Generalized Spin Hamiltonian and Extraction of Couplings

The fundamental advance in this study is the adoption of a spin Hamiltonian of the form:

$_8$0

where $_8$1 indexes cylindrical coordinates ($_8$2, $_8$3, $_8$4), $_8$5 represent anisotropic exchange, $_8$6 the corresponding biquadratic terms, and $_8$7 the DM vector.

Non-collinear DFT configurations are mapped onto combinations of anti/ferromagnetic and helical spin arrangements, enabling extraction of parameters up to fourth-nearest neighbor and analysis of in-plane versus out-of-plane anisotropy.

Figure 2: The three spin AFM non-collinear reference configurations.

Principal Results: Exchange, Biquadratic, and DM Interactions

Exchange: Directional Anisotropy and Range

The Cr$_8$8 system displays isotropic, antiferromagnetic (AFM) $_8$9 interactions (e.g., $_8$0 meV with DFT+$_8$1+$_8$2), with negligible further-neighbor couplings. By contrast, V$_8$3 exhibits ferromagnetic (FM) $_8$4 (e.g., $_8$5 meV with DFT+$_8$6+$_8$7) but sizeable antiferromagnetic $_8$8, resulting in strong competition between nearest- and next-nearest-neighbor couplings. Ring curvature ensures cylindrical, rather than Cartesian, anisotropy; exchange is modestly stronger out-of-plane ($_8$9) than in-plane.

Biquadratic Couplings

Fitting spin-rotation energetics reveals negligible biquadratic terms for Cr$_8$0, but substantial BQ couplings for V$_8$1 (e.g., $_8$2 meV, $_8$3 meV in DFT+$_8$4). These favor in-plane spin alignment and introduce additional anisotropy, nontrivial in magnitude compared to the leading bilinear terms.

Figure 3: Non-collinear DFT total energy profiles obtained rotating a single spin in Cr$_8$5 from the AFM ground state configuration in each direction.

Figure 4: Non-collinear DFT total energy profiles obtained rotating a single spin in V$_8$6 from the FM ground state configuration in each direction.

Dzyaloshinskii–Moriya (DM) Interactions

Despite the absence of explicit spin-orbit coupling (SOC) in the calculations, significant DM-like antisymmetric exchange arises, particularly for the $_8$7-component in both rings ($_8$8 meV for Cr$_8$9 ring, $_8$0 meV for V$_8$1 ring with DFT+$_8$2+$_8$3). These DM terms are attributed in Cr$_8$4 primarily to geometric effects (curvature-induced basis transformation), as chain analogues show vanishingly small values. For V$_8$5, however, DM couplings are nonzero even in the chain, indicating a possible intrinsic electronic origin unrelated to geometrical curvature.

Figure 5: Magnetic spin configurations used to evaluate $_8$6 in Cr$_8$7 and V$_8$8.

Figure 6: Helical and non-helical spin configurations used to to evaluate the $_8$9 antisymmetric exchange couplings in chain model.

Quantitative Agreement with Experiment and Hamiltonian Validation

Exact diagonalization of the fitted spin models enables direct calculation of the magnetic susceptibilities, which are compared to experimental data for Cr$_8$0. The comprehensive Hamiltonian—including extended exchange, biquadratic, and DM terms—matches experimental $_8$1 curves over a broad temperature range, particularly improving low-$_8$2 agreement (below 10 K), as shown in the right panel of Figure 7. This endorses the necessity of including geometrically-induced DM and higher-order interactions even in nominally "simple" molecular magnets. DFT+$_8$3+$_8$4 reproduces experimental trends more accurately than standard DFT+$_8$5, confirming the importance of inter-site correlation treatments.

Figure 7: Magnetic susceptibility of Cr$_8$6 for different model Hamiltonian truncations alongside experimental data; DM and biquadratic terms are needed for quantitative agreement at low $_8$7.

Figure 8: Magnetic susceptibility of V$_8$8 for the DFT+$_8$9+$M^{3+}$0 Hamiltonian, highlighting the transition from FM to AFM-like low-$M^{3+}$1 behavior upon including longer-range interactions.

Strikingly, for V$M^{3+}$2, the inclusion of antiferromagnetic $M^{3+}$3 couplings completely inverts the low-$M^{3+}$4 susceptibility behavior from FM to AFM-like, despite all ab initio predicted interactions being either FM or weakly AFM. This transition, and the associated radical restructuring of the low-energy spectrum, is demonstrated and matches the behavior predicted for quantum spin rings experiencing frustration between FM and AFM interactions.

Theoretical and Practical Implications

This work establishes several key insights for the molecular nanomagnetism domain:

• Essential role of non-Heisenberg terms: Both strong biquadratic exchange and DM components must be included to accurately model low-energy excitations, particularly as ring curvature and open 3d-shell occupancies increase.
• Curvature-driven DM interactions: Even in the absence of heavy elements or explicit SOC, ring geometry induces substantial DM-like exchange which meaningfully alters magnetic response.
• Electronic origins for nonzero DM in V$M^{3+}$5: The anomalous DM terms in the V$M^{3+}$6 chain point to a mechanism—potentially linked to the less-than-half-filled $M^{3+}$7 subshell of V$M^{3+}$8—requiring further many-body and current-density functional studies.
• Inter-site correlation criticality: DFT+$M^{3+}$9+$U$0 functionals markedly improve the quantitative match with experiment, highlighting the necessity of accurate correlated electron treatments for reliable predictions.

From a practical standpoint, these findings impact the rational design of quantum magnetic materials for information storage, spintronics, and quantum coherence applications, where the correct hierarchy of competing interactions determines coherence times, excitation spectra, and operational temperature windows.

Future Directions

Immediate research avenues include explicit assessment of SOC, current-induced DM and biquadratic mechanisms, and the role of correlated spin-currents and non-collinear exchange-correlation fields beyond traditional functional constructions. Time-dependent DFT with Hubbard corrections may enable direct magnon spectrum prediction. The impact of chemical substitution, mechanical deformation, and symmetry breaking on these competing couplings merits investigation for targeted functionalization of molecular quantum devices.

Conclusion

A rigorous non-collinear ab initio treatment of Cr$U$1 and V$U$2 molecular rings demonstrates that extended spin Hamiltonians—including biquadratic and DM-type couplings—are indispensable for quantitative description of spectral and magnetic properties. The research clarifies the interplay between geometry, electronic filling, and magnetism and sets a methodological benchmark for both fundamental exploration and applied quantum materials design.