Determining electron–nucleus distances and Fermi contact couplings from ENDOR spectra†
The hyperfine coupling between an electron spin and a nuclear spin depends on the Fermi contact coupling aiso and, through dipolar coupling, the distance r between the electron and the nucleus. It is measured with electron-nuclear double resonance (ENDOR) spectroscopy and provides insight into the electronic and spatial structure of paramagnetic centers. The analysis and interpretation of ENDOR spectra is commonly done by ordinary least-squares fitting. As this is an ill-posed, inverse mathematical problem, this is challenging, in particular for spectra that show features from several nuclei or where the hyperfine coupling parameters are distributed. We introduce a novel Tikhonov-type regularization approach that analyzes an experimental ENDOR spectrum in terms of a complete non-parametric distribution over r and aiso. The approach uses a penalty function similar to the cross entropy between the fitted distribution and a Bayesian prior distribution that is derived from density functional theory calculations. Additionally, we show that smoothness regularization, commonly used for a similar purpose in double electron–electron resonance (DEER) spectroscopy, is not suited for ENDOR. We demonstrate that the novel approach is able to identify and quantitate ligand protons with electron–nucleus distances between 4 and 9 Å in a series of vanadyl porphyrin compounds.