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 <i>n</i>d 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 <i>n</i>d_3/2 and <i>n</i>d_5/2 as basic vectors according to relativistic ligand field theory. The parametrized ligand field allows for the description of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mo>'</mml:mo></mml:mrow></mml:msubsup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mo>'</mml:mo></mml:mrow></mml:msubsup></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>'</mml:mo></mml:mrow></mml:msubsup></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mo>'</mml:mo></mml:mrow></mml:msubsup></mml:math> symmetries as particular cases. The wave functions of the ground state manifold Γ_<i>i</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 <i>p</i>/<i>q</i>, the spin-orbit coupling constant ξ_<i>n</i>d, and the ligand field strengths <i>Dq</i>, <i>Ds,</i> and <i>Dt</i>. 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 <i>p</i>/<i>q</i> 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.