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.