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 Γ8 ground state of lanthanide-doped fluorite crystals CaF2: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.Introduction. Paramagnetic nuclear magnetic resonance (pNMR) spectroscopy is a fundamental tool for probing static and dynamic local magnetic properties of materials [1] and metallo-proteins [2]. pNMR plays a central role in the elucidation of the quantum dynamics of single-molecule magnets and antiferromagnetic spin rings [3]. Moreover, it is an increasingly central technique for probing strong spin-orbit coupled electronic states in a dissipative environment, as shown by a recent study of quantum tunneling processes [4], silent in ac-susceptibility experiments, in lanthanide-based (Dy 3+ and Tb 3+ ) molecular nanomagnets.Despite the central role of pNMR in the development of new magnetic materials, only very recently ab initio approaches have been developed for the calculation of fundamental pNMR observables, such as nuclear shielding tensors [5]. In particular, by generalizing the work of Moon and Patchkovskii [6] on Kramers doublets, the paper by Pennanen and Vaara [7] stands out as the first comprehensive formulation of NMR shielding tensors in terms of molecular response properties, like the secondrank (first-rank in spin) EPR g and hyperfine A tensors, routinely computed via ab initio methods [5,[7][8][9][10]. The work by Pennanen and Vaara [7] aims at proposing an NMR-theory for arbitrary electronic ground state degeneracy. However, this task is only accomplished in [7] for the weak spin-orbit coupling limit, and to date no theory fully accounts for truly general (spin and orbital) electronic degeneracy, as that arising in strong spin-orbit coupled lanthanide nanomagnets.In this Letter we propose a general theory for the ab initio calculation of pNMR chemical shifts as function of molecular tensors of all ranks consistent with the size of degeneracy, which applies to arbitrarily strong spinorbit coupling (SOC) limit, and any degree of electronic degeneracy.Theory. In the linear response regime, the electroninduced magnetic field B I ind experienced by a nucleus I in a molecule immersed in a magnetic field B, is proportional to the field itself, so that B I ind = −B · σ I . This leads to an electron-mediated interaction energy between external field and the magnetic dipole moment µ I of nucleus I which is bilinear in the field and the nuclear