On The van Vleck Paramagnetic Susceptibility of nd1(Oh) 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(t2g) and the first excited state Γ7(t2g), and a dimensionless quantity ϵ that incorporates the crystal-field strength Dq, the spin-orbit coupling constant ξnd, and the relativistic ratio p/q. The parameter ϵ also determines the effective Bohr magneton number neff. Within the strong-field approximation, we show that ξnd 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.