By utilizing the Becke 3-parameter Lee-Yang-Parr (B3LYP) hybrid exchange-correlation (XC) functional, which combines both local and non-local exchange-correlation terms, we conduct first-principles calculations to investigate the electronic, magnetic, and structural properties of small iron-oxide clusters (with \( N \le 8 \) ). Additionally, we employ the semi-local Perdew-Burke-Ernzerhof (PBE) XC-functional along with the PBE+U approximation, which accounts for on-site intra-atomic Coulomb repulsion interactions. We compare the results with those obtained using the non-local functional. Our findings show that the choice of the XC-functional strongly influences the physical properties of the iron-oxide clusters. The clusters adopt planar structures with Fe-O-Fe bonding angles close to 90 \( ^{\circ } \) , facilitating the emergence of indirect or super-exchange interaction mechanisms between the Fe atoms mediated by the connecting oxygen atoms. Overall, the stabilization of the magnetic ordering is driven by the competition between direct and indirect exchange interactions, with the latter favoring antiferromagnetism. A local density of states (LDOS) together with a Crystal Orbital Hamilton Population (COHP) analysis allows us to identify the symmetries of the participating d-orbitals on these interaction mechanisms from a local perspective, which control the magnetic properties of the iron-oxide clusters. Besides, our results highlight the importance of using XC-functionals that more accurately account for exchange and electronic correlation interactions rather than relying solely on standard semi-local functionals.
