Prediction of spin orientations in terms of HOMO-LUMO interactions using spin-orbit coupling as perturbation

The preferred spin orientation of a magnetic ion can be predicted on the basis of density functional theory (DFT) calculations including electron correlation and spin-orbit coupling (SOC). However, most chemists and physicists are unaware of how the observed and/or calculated spin orientations are related to the local electronic structures of the magnetic ions. The objective of this article is to provide a conceptual framework of thinking about and predicting the preferred spin orientation of a magnetic ion by examining the relationship between the spin orientation and the local electronic structure of the ion. In general, a magnetic ion (i.e., an ion possessing unpaired spins) in a solid or a molecule is surrounded with main-group ligand atoms to form a polyhedron, and the d-states of the polyhedron are split because the antibonding interactions of the metal d-orbitals with the p orbitals of the surrounding ligands depend on the symmetries of the orbitals involved. The magnetic ion of the polyhedron has a certain preferred spin direction because its split d-states interact among themselves under SOC and because the energy lowering associated with the SOC-induced interactions depends on spin orientation. The preferred spin direction can be readily predicted on the basis of perturbation theory, in which the SOC is taken as perturbation and the split d-states as unperturbed states, by inspecting the magnetic quantum numbers of its d-orbitals present in the HOMO and LUMO of the polyhedron. Experimentally, the determination of the preferred spin orientations of magnetic ions requires a sophisticated level of experiments. Theoretically, it requires an elaborate level of DFT electronic structure calculations. We show that the outcomes of such experimental measurements and theoretical calculations can be predicted by a simple perturbation theory analysis.