Second-Order Magnetic Properties in Paramagnetic Molecules From a Current Density Formulation Including Scalar Relativistic Effects

This work presents the theoretical background for the computation of nuclear magnetic shielding, nuclear hyperfine and magnetizability tensors of paramagnetic molecules, using a magnetically induced current density framework to account for both orbital and spin contributions. The resulting magnetizability tensor is fully consistent with the general Van Vleck formulation, recovering the temperature-dependent Curie contribution through the explicit integration of the magnetically induced spin current density. The methodology proposed herein provides a straightforward computational route that bypasses the complex evaluation of g-tensors and Zero-Field Splitting (ZFS) Hamiltonians. While the theoretical framework is general, we present applications rooted on physically motivated approximations where scalar relativistic effects are incorporated through corrections based on the Zeroth-Order Regular Approximation (ZORA) Hamiltonian within the ground-state spin density. This approach combines a relativistic self-consistent field (SCF) calculation for the ground-state spin density with a non-relativistic, origin-independent current density calculation for the orbital contribution. This hybrid strategy is shown to capture the Heavy-Atom Light-Atom (HALA) effect in 1H and 13C shieldings, particularly in paramagnetic molecular systems containing transition metals up to the 3d series. By restricting the relativistic treatment to the spin density, where scalar relativistic effects are dominant, and neglecting such effects on the orbital contribution of light atoms, this method offers a good compromise between computational efficiency and accuracy for the characterization of large open-shell molecular systems.