Relativistic Effects in Ligand Field Theory (II): Optical and Magnetic Properties of d1 Atoms in Cubic and Tetragonal Symmetries

Abstract

The relativistic motion of an electron in the nd atomic subshell subjected to an electrostatic ligand field and a magnetic field is studied. The ligand and magnetic fields are treated as perturbations on equal footing, employing the atomic Dirac spinors nd_3/2 and nd_5/2 as basic vectors according to relativistic ligand field theory. The parametrized ligand field allows for the description of $C_{4v}^{\prime }$, $D_{4h}^{\prime }$, $O_h^{\prime }$, and $T_d^{\prime }$ symmetries as particular cases. The wave functions of the ground state manifold Γ_i(²D_3/2) are employed to calculate the Zeeman coefficients and consequently the Curie constant according to Van Vleck’s theory of paramagnetism. The theoretical framework is illustrated in Rb₂TaCl₆, ReF₆, Ba₂NaOsO₆, VO(R-acac)₂, and Cs[MoOCl₄] d¹-systems. It is indicated how correlating the optical transition energies with the magnetic susceptibility enables experimental determination of the theory’s parameters: the relativistic ratio p/q, the spin–orbit coupling constant ξ_nd, and the ligand field strengths Dq, Ds, and Dt. Then, from the deduced parameters, the X-band signals of the electron paramagnetic resonance spectrum are predicted. A strong dependency of the magnetic properties upon the relativistic ratio p/q is observed. Energies of the well-separated (ca. 330 cm^–1) optical transitions Γ₆(²D_3/2) ← Γ₇(²D_3/2) and Γ₇(²D_5/2) ← Γ₇(²D_3/2) for Cs[MoOCl₄] are predicted. Our results suggest that the reported spin–orbit coupling constant ξ_5d for ReF₆ might have been overestimated by more than 1000 cm^–1, offering valuable insights into the accuracy of previous experimental and theoretical studies.