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

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 $Γ_8$ 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.