Electronic and Structural Factors Controlling the Spin Orientations of Magnetic Ions

Abstract

Magnetic ions M in discrete molecules and extended solids form ML n complexes with their first-coordinate ligand atoms L. The spin moment of M in a complex ML n prefers a certain direction in coordinate space because of spin–orbit coupling (SOC). In this minireview, we examine the structural and electronic factors governing the preferred spin orientations. Elaborate experimental measurements and/or sophisticated computational efforts are required to find the preferred spin orientations of magnetic ions, largely because the energy scale of SOC is very small. The latter is also the very reason why one can readily predict the preferred spin orientation of M by analyzing the SOC-induced highest occupied molecular orbital (HOMO)–lowest unoccupied molecular orbital (LUMO) interactions of the ML n complexes in terms of qualitative perturbation theory. The strength of this HOMO–LUMO interaction depends on the spin orientation, which is governed by the selection rules based on the minimum |ΔL z | value (i.e., the minimum difference in the magnetic quantum numbers) between the HOMO and LUMO. With the local z axis of ML n chosen as its n-fold rotational axis, the preferred spin orientation is parallel to the z axis (∥z) when |ΔL z | = 0 but perpendicular to the z axis (⊥z) when |ΔL z | = 1.

Physical Properties of Molecules and Condensed Materials Governed by Onsite Repulsion, Spin-Orbit Coupling and Polarizability of Their Constituent Atoms

“…However, this d-state split pattern cannot explain the uniaxial magnetism of linear FeL 2 , because the resulting configuration (1a) 2 (1e) 2 (2e) 2 for the Fe 2+ (d 6 ) ion does not have an unevenly occupied degenerate level (Figure 6d). Thus, we arrive at an impasse of the one-electron picture, as the d-state split pattern needed to explain the uniaxial magnetism of a high-spin d 6 ion (d 7 ion) does not explain that of the high-spin d 7 ion (d 6 ion) [6,7]. In terms of the CFT, the d-state split pattern of any transition metal cation at a linear coordination site is given by 1e < 2e < 1a.…”

“…In short, only the ground state (i.e., the L = 2 state) correctly predicted by the Aufbau principle when combined with the d-state split pattern of 1a < 1e < 2e. As discussed above, the correct ground state for a linear ML 2 system containing a high-spin d 7 ion is not correctly predicted by the Aufbau principle. It is important to be aware that, though useful as a conceptual framework for discussion, the Aufbau principle can lead to such failures.…”

“…If the Aufbau principle were to be combined with the d-state split pattern of 1a < 1e < 2e, the following relative stabilities can be deduced from Figure 9: In short, only the ground state (i.e., the L = 2 state) correctly predicted by the Aufbau principle when combined with the d-state split pattern of 1a < 1e < 2e. As discussed above, the correct ground state for a linear ML2 system containing a high-spin d 7 ion is not correctly predicted by the Aufbau principle. It is important to be aware that, though useful as a conceptual framework for discussion, L = 2 < L = 1 < L = 3, and L = 2 < L = 0 (A) < L = 0 (B).…”

“…Let us consider a linear molecule FeL2 containing a high-spin Fe 2+ (d 6 , S = 2) ion, where the ligand L is bulky, for example, L = C[Si(CH3)3]3. With the local z-axis of FeL2 taken along the axis of its rotational symmetry, i.e., along the L-Fe-L direction (Figure 5a), the five d-states, xy, x 2 -y 2 , xz, yz and 3z 2 -r 2 , of FeL2 are grouped into the 1a, 1e and 2e sets [6,7] The crystal-field theory (CFT) predicts that the d-states of a transition-metal cation at a linear coordination site are split as 1e < 2e < 1a (Figure 5b), so that the ground-state electron configuration of FeL2 containing a Fe 2+ (d 6 , S = 2) ion is given by (1e) 3 (2e) 2 (1a) 1 (Figure 6a).…”

“…Let us consider a linear molecule FeL 2 containing a high-spin Fe 2+ (d 6 , S = 2) ion, where the ligand L is bulky, for example, L = C[Si(CH 3 ) 3 ] 3 . With the local z-axis of FeL 2 taken along the axis of its rotational symmetry, i.e., along the L-Fe-L direction (Figure 5a), the five d-states, xy, x 2 -y 2 , xz, yz and 3z 2 -r 2 , of FeL 2 are grouped into the 1a, 1e and 2e sets [6,7],…”