Current Density Formulation of Nuclear Magnetic Shielding and Magnetizability Tensors in Paramagnetic Molecules in the Presence of Relativistic Effects

Abstract

This work presents the computation of nuclear magnetic shielding and magnetizability tensors for paramagnetic molecules, using a magnetically induced current density framework to account for orbital and spin contributions. We demonstrate that the methodology proposed by Soncini[1] is physically equivalent to the formalisms of Pennanen and Vaara[2] and Franzke et al.[3], provided that scalar and spin-orbit relativistic effects are included within the ground-state spin density. In our model, these corrections are implemented through a Zeroth-Order Regular Approximation (ZORA) formulation of the current density. 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. This methodology offers a straightforward computational route that bypasses the complex evaluation of g-tensors and Zero-Field Splitting (ZFS) Hamiltonians, requiring only a ground-state spin density incorporating relativistic effects. Notably, scalar relativistic effects are shown to be essential for capturing the Heavy-Atom Light-Atom (HALA) effect in 1H and 13C shieldings. To maintain efficiency, relativistic effects on the orbital contribution are neglected as they are negligible for light atoms. This approach represents an optimal compromise for paramagnetic complexes involving transition metals up to the second row, where the HALA effect is primarily driven by scalar relativistic corrections within the ground-state spin density. Neglecting spin-orbit terms in the orbital contribution significantly streamlines the calculation without loss of accuracy, providing the pNMR community with a robust tool for characterizing open-shell systems.

Figures (3)

• Figure 1: Isotropic contribution to the $^1$H nuclear magnetic shielding density for metallocenes at 298.15 K. Top: orbital shielding densities, defined from equation (\ref{['orbshield']}) using the CTOCD method (with the CSGT option of the shift function); middle: the spin contribution, defined from equation (\ref{['spinshield']}) using a scalar relativistic spin density throughout the first term of the ZORA spin current; and bottom: the total shielding density (sum of the two contributions). Isosurface values are set at $\pm 10$ ppm/$a_0^3$.
• Figure 2: Total ZORA Spin Current density maps induced by a magnetic field $B_x\mathbf{e}_x$ pointing toward the reader, for the maximum polarized state $N_\alpha>N_\beta$, in O2 ($2S+1=3$).
• Figure 3: Total ZORA Spin Current density map of NO2 ($2S+1=2$) induced by a magnetic field $B_x\mathbf{e}_x$ pointing toward the reader for the maximum polarized state $N_\alpha>N_\beta$.