Thermal averaging of Hamiltonians in NMR revisited

In this Note, we reexamine an earlier alternative proposal for averaging NMR spectroscopic parameters in the fast-exchange regime [D. H. Jones, N. D. Kurur, and D. Pl Weitekamp, Bull. Magn. Reson. 14, 214-219 (1992)], which argued that neglecting the contributions of nuclear spin energy in thermal averaging can lead to significant errors in the interpretation of NMR kinetic data while also violating the basic tenets of statistical thermodynamics. While other groups were quick to point out that this alternative proposal gave predictions that were inconsistent with prior experimental observations, subsequent theoretical and experimental work [L. J. Mueller and D. P. Weitekamp, Science 283, 61-65 (1999)] demonstrated that taking into account nuclear spin energies in exchange dynamics can indeed lead to a detectable contribution to the traditional averaging result. While the intuition that including nuclear spin energies in exchange dynamics could lead to an observable effect was ultimately found to be correct, the reason behind the failure of the earlier alternative averaging proposal was never fully articulated. In this work, it is shown that while both the traditional and alternative averaging proposals involve a thermal average over the system-environment interaction, the alternative averaging proposal also includes an additional thermal average over the Hamiltonian for the environment, which is the source of its error.