On The van Vleck Paramagnetic Susceptibility of nd¹(O_h) Transition Metal Ions

In this work, we derive a mathematical expression for the van Vleck paramagnetic susceptibility of nd¹ ions embedded in octahedral crystal-field environments. The formula is derived within relativistic crystal-field theory and involves two parameters: the energy difference Δ₂ between the ground state Γ₈(t_2g) and the first excited state Γ₇(t_2g), and a dimensionless quantity ϵ that incorporates the crystal-field strength Dq, the spin-orbit coupling constant ξ_nd, and the relativistic ratio p/q. The parameter ϵ also determines the effective Bohr magneton number n_eff. Within the strong-field approximation, we show that ξ_nd can be directly estimated from Δ₂ using standard SQUID magnetometry, bypassing the need for specialized optical spectroscopy. While the excellent agreement for ReF₆ demonstrates the intrinsic link between magnetic and optical properties, we also show that for nondilute or distorted systems, such as the double perovskite Ba₂NaOsO₆, the model's deviations from optical data serve as a sensitive diagnostic tool to reveal the onset of ion-ion interactions or symmetry breaking. Our formalism thus provides a practical route for experimentalists to characterize relativistic effects and identify departures from ideal single-ion behavior using macroscopic magnetic measurements.