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 a 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 a. 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.