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

TL;DR

This paper introduces a relativistic current density approach for calculating nuclear magnetic shielding and magnetizability tensors in paramagnetic molecules, simplifying computations by focusing on scalar relativistic effects within the ground-state spin density.

Contribution

It demonstrates the equivalence of existing formalisms and develops a ZORA-based method that efficiently includes relativistic effects, avoiding complex g-tensor and ZFS evaluations.

Findings

Scalar relativistic effects are crucial for HALA in light atoms.

The method accurately recovers the Curie contribution.

Neglecting spin-orbit effects on orbital contributions maintains accuracy.

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.