Quadratic Spin-Orbit Mechanism of the Electronic g-Tensor

Abstract

Understanding how the electronic g-tensor is linked to the electronic structure is desirable for the correct interpretation of electron paramagnetic resonance spectra. For heavy-element compounds with large spin-orbit (SO) effects, this is still not completely clear. We report our investigation of quadratic SO contributions to the g-shift in heavy transition metal complexes. We implemented third-order perturbation theory in order to analyze the contributions arising from frontier molecular spin orbitals (MSOs). We show that the dominant quadratic SO term─spin-Zeeman (SO²/SZ)─generally makes a negative contribution to the g-shift, irrespective of the particular electronic configuration or molecular symmetry. We further analyze how the SO²/SZ contribution adds to or subtracts from the linear orbital-Zeeman (SO/OZ) contribution to the individual principal components of the g-tensor. Our study suggests that the SO²/SZ mechanism decreases the anisotropy of the g-tensor in early transition metal complexes and increases it in late transition metal complexes. Finally, we apply MSO analysis to the investigation of g-tensor trends in a set of closely related Ir and Rh pincer complexes and evaluate the influence of different chemical factors (the nuclear charge of the central atom and the terminal ligand) on the magnitudes of the g-shifts. We expect our conclusions to aid the understanding of spectra in magnetic resonance investigations of heavy transition metal compounds.

Figure 1

Structure of the M(II) d⁷ complexes (M = Ir and Rh) investigated and their orientation in the Cartesian coordinate system. The overall charge is zero except for complexes 5 and 7, which have neutral ligands. Complexes 1, 2, 3, 6, and 7 as well as Rh3 and Rh6 have previously been synthesized and characterized.⁻

Figure 2

Orientation of the principal axes of the g-tensor in complex 6 from the 4c/PBE0/TZ/vac calculation and the contributions to the corresponding components of the g-shift from the SO/OZ and SO²/SZ terms calculated by using the PT/PBE/DZ/vac approach (see Section 5, Methods).

Figure 3

(a) Simplified MSO diagram calculated for compound 6 (Ir–Cl) with KS/PBE/DZ/vac and (b) portion of the diagram with highlighted occupied ↔ vacant couplings and their contributions to individual components of the g-shift. The component of the angular momentum operator involved in the integral (see Section 2, Theoretical Background) is given beside each contribution. All couplings that contribute at least 10% of Δg_u^term (where the term = SO/OZ or SO²/SZ) are included. Note that while the SO/OZ mechanism acts between MSOs with the same spin, the SO²/SZ mechanism facilitates couplings between α and β MSOs.

Figure 4

(a) Trend of the g-shift components along a series of Ir(II) compounds calculated with 4c/PBE0/TZ/vac (full line) or PT/PBE/DZ/vac (dashed line, eq 9). The ligand series is ordered according to the largest component Δg_y in the 4c calculation. For comparison of calculations with available experimental values and the effect of the solvent, see Tables S4 and S5, respectively. (b) Orientation of the principal axes of the g-tensor with respect to the molecular geometry of complex 1 (Ir–NH₂) from 4c/PBE0/TZ/vac. The orientation does not change qualitatively throughout the series.

Figure 5

Decomposition of the g-shift components obtained with PT (PT/PBE/DZ/vac) into contributions from the SO/OZ, SO²/SZ, and SO²/OZ mechanisms along a series of Ir(II) compounds. The full value (PT) is equal to the sum of the individual contributions as defined in eq 9. For numerical data, see Table S6.

Figure 6

Energy diagram showing selected frontier MSOs calculated for complexes 7, 6, 3, and 1 at the nonrelativistic (KS/PBE/DZ/vac) level. Note the energy offset due to the neutral character of the NH₃ ligand and the different overall charge of the complex. The values of the g-shift components indicate the contributions of the labeled MSO couplings arising from the SO/OZ term (β ↔ β couplings) or the SO²/SZ term (α ↔ β couplings). All indicated couplings arise from the y component of the operators.

Figure 7

Dependence of the Δg arising from the largest coupling between a pair of MSOs (PT/PBE/DZ/vac) through the (a) SO/OZ term (β-NB ↔ SUMO) or (b) SO²/SZ term (α-NB ↔ SUMO) on the energy separation of the two MSOs, 1/ΔE or 1/ΔE² (see the Theoretical Background section for the relevant equations).