Transition Probabilities in a Dipole-Coupled System of Protons and Delocalized Electrons

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Abstract
General expressions are derived for relaxation transition probabilities in a dipole-coupled electron—proton system when the electron cannot be considered as a point-dipole. It is shown that the transition rates always maintain the same ratios to each other, regardless of the distribution of the electron wavefunction. Thus the double-resonance enhancement of the proton NMR signal is not directly affected by the delocalization of the electron. Specific expressions are given for the transition rates in four cases: the electron wavefunction concentrated at one point, equally divided between two points, uniformly distributed over a ring, and uniformly distributed over a disk.