Interaction of Radical Pairs Through-Bond and Through-Space: Scope and Limitations of the Point−Dipole Approximation in Electron Paramagnetic Resonance Spectroscopy

The validity of the popular point−dipole approximation for interpretation of the zero-field splitting (ZFS) parameter (D-value) in EPR spectroscopy is studied. This approximation is of central importance for the determination of distances by analysis of EPR data. In this work, a detailed experimental (EPR spectroscopy and X-ray crystallography) and theoretical study for a model system (2,2′,5,5′-tetra(tert-butyl)-4,4′-bis(ethoxy-carbonyl)-3,3′-bipyrrolyl-1,1′-dioxyl) was performed to understand the scope and limitations of the point−dipole model in EPR spectroscopy. For this diradical, the radical−radical distance derived with the point−dipole approximation deviates significantly (by ∼40%) from the results derived from the X-ray analysis. Explicit quantum chemical calculation of the D-value on the basis of B3LYP density functional calculations leads to excellent quantitative agreement with the measured D-value. The quantitative accuracy of the employed methodology was confirmed for two additional systems that have previously been experimentally characterized. We therefore analyzed the contributions to the D-value of the target system in detail. This analysis leads to insight into the reasons for the failure of the point−dipole approximation. The analysis was then extended to an in silico study of five classes of model systems. Linkers of varying length and bond saturation were introduced between the radical-carrying groups. This allows for the analysis of the distance dependence of the D-parameter as well as the through-bond and through-space spin−spin interaction. From these results we established the limitations of the point−dipole approximation. The results of this analysis demonstrate that even very modest amounts of spin delocalization can cause significant deviations from pure point−dipole behavior and consequently cause the EPR derived distances to deviate from the N−O midpoint distance by up to several angströms. If unsaturated linkers are used, the distance dependence of D does not follow the inverse cubic behavior predicted by the point−dipole model. However, for commonly used nonaromatic nitroxide rings connected by a saturated linker, the point−dipole approximation works well. Among the various point−dipole variants tested in this work for delocalized spins, the most successful one is based on distributed point−dipoles with spin populations derived from quantum chemical calculations. The distance dependence of the isotropic Heisenberg exchange parameter J has also been studied theoretically. The decay was found to be monoexponential with a decay constant of ∼1 Å−1. Thus at linker lengths between 6−8 carbon atoms between a nitroxide radical pair, a switch from the strong to the weak exchange limit is predicted.