Prediction of Spin Orientations in Terms of HOMO–LUMO Interactions Using Spin–Orbit Coupling as Perturbation

For most chemists and physicists, electron spin is merely a means needed to satisfy the Pauli principle in electronic structure description. However, the absolute orientations of spins in coordinate space can be crucial in understanding the magnetic properties of materials with unpaired electrons. At low temperature, the spins of a magnetic solid may undergo long-range magnetic ordering, which allows one to determine the directions and magnitudes of spin moments by neutron diffraction refinements. 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. This is true even for most crystallographers who determine the directions and magnitudes of spin moments because, for them, they are merely the parameters needed for the diffraction refinements. 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 M (i.e., an ion possessing unpaired spins) in a solid or a molecule is surrounded with main-group ligand atoms L to form an MLn polyhedron, where n is typically 4-6, and the d states of MLn are split because the antibonding interactions of the metal d orbitals with the p orbitals of the surrounding ligands L depend on the symmetries of the orbitals involved.1 The magnetic ion M of MLn has a certain preferred spin direction because its split d states interact among themselves under SOC.2,3 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 MLn polyhedron. This is quite analogous to how chemists predict whether a chemical reaction is symmetry-allowed or symmetry-forbidden in terms of the HOMO-LUMO interactions by simply inspecting the symmetries of the frontier orbitals.4,5 Experimentally, the determination of the preferred spin orientations of magnetic ions requires a sophisticated level of experiments, for example, neutron diffraction measurements for magnetic solids with an ordered spin state at a very low temperature. Theoretically, it requires an elaborate level of electronic structure calculations, namely, DFT calculations including electron correlation and SOC. We show that the outcomes of such intricate experimental measurements and theoretical calculations can be predicted by a simple perturbation theory analysis.