Theory of Field-Dependent NMR Shifts in Paramagnetic Molecules

NMR chemical shifts depend on the applied magnetic flux density, and this becomes more and more important as stronger and stronger magnetic fields are becoming available. Herein we develop a theory of the field dependence of NMR shifts of paramagnetic molecules in solution. Our derivation leads to two distinct approaches: a finite-field approach that describes the shift up to infinite order in the applied field B₀ but requires numerical integration for the orientational average, and a second-order approach that is valid up to second order in B₀. In this latter approach, the orientational average can be performed analytically, and the field dependence cleanly separates into two additive terms: the well-known "indirect" field dependence due to incomplete averaging in solution and the "direct" field dependence due to the nonlinear response to the external field. In analogy to the diamagnetic case, the direct field dependence involves a fourth-order tensor τ whose elements are fourth derivatives of the electronic Helmholtz free energy. Generalizing the Van den Heuvel-Soncini equation, we provide analytical sum-over-states equations for these higher-order derivatives. Using the NiSAL-HDPT complex as an example, we demonstrate the applicability of the second-order approach at room temperature and the highest commercially available field strength and show that it agrees well with the field dependence measured experimentally.