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

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.