Theory of NMR chemical shift in an electronic state with arbitrary degeneracy

TL;DR

This paper develops a comprehensive theory for NMR shielding tensors in electronic states with arbitrary degeneracy, valid under strong spin-orbit coupling, and demonstrates its importance through ab initio calculations on lanthanide-doped crystals.

Contribution

The paper introduces a generalized theory of NMR shielding tensors for degenerate electronic states, extending previous models to strong spin-orbit coupling regimes.

Findings

Generalized Zeeman and hyperfine tensors expressed for all degeneracy ranks.

Ab initio calculations show neglected contributions can exceed 50% of paramagnetic shift.

Theory applicable to lanthanide-doped fluorite crystals with 4-fold degeneracy.

Abstract

We present a theory of nuclear magnetic resonance (NMR) shielding tensors for electronic states with arbitrary degeneracy. The shieldings are here expressed in terms of generalized Zeeman ($g^{(k)}$) and hyperfine ($A^{(k)}$) tensors, of all ranks $k$ allowed by the size of degeneracy. Contrary to recent proposals [T. O. Pennanen and J. Vaara, Phys. Rev. Lett. 100, 133002 (2008)], our theory is valid in the strong spin-orbit coupling limit. Ab initio calculations for the 4-fold degenerate $\Gamma_8$ ground state of lanthanide-doped fluorite crystals CaF$_2$:Ln (Ln = Pr$^{2+}$, Nd$^{3+}$, Sm$^{3+}$, and Dy$^{3+}$) show that previously neglected contributions can account for more than 50% of the paramagnetic shift.