Determining electron-nucleus distances and Fermi contact couplings from ENDOR spectra

Abstract

The hyperfine coupling between an electron spin and a nuclear spin depends on the Fermi contact coupling a_iso 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_iso. 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.

Fig. 1

(a) Simulated powder Mims ENDOR spectra (colored) of protons with typical axial hyperfine tensors, with τ = 150ns, together with simulated spectra without τ suppression (gray). The shape and the width of the spectrum depend on a_iso and T. Spectra are centered around the nuclear Larmor frequency of the proton $v_I\left({ }^1\text{H}\right)$, were simulated at 344 mT, and are scaled to maximum 1. (b) Hyperfine parameters used for the spectra in (a), in addition to $\eta$ = 0. The top and bottom horizontal axes show the dipolar coupling constant T and the associated effective electron–nucleus distance r. The vertical axes show a_iso and the associated contact electron spin density $\rho (\mathit{R})$, expressed as a fraction in relation to the contact spin density in a hydrogen atom, ρ₀, corresponding to a_iso,0 =1420MHz.

Fig. 2

(a) Lewis structures of the four model compounds examined in this study, plus VO-TPP axially coordinated by tetrahydrofuran (THF). The different types of protons are indicated in color. (b) Hyperfine coupling parameters determined from DFT calculations for protons from all structures.

Fig. 3

Demonstration of linear dependence within the basis set of Mims ENDOR spectra. (a) An anisotropic spectrum (a_iso = 0.62MHz, T = 0.64MHz), simulated at B₀ = 350mT, is fit accurately by a linear combination of basis spectra with a small dipolar contribution and large variation in contact coupling (a_iso = −0.1–1.6MHz, T = 0.079MHz). The resulting superposition (red) reproduces the anisotropic spectrum almost perfectly. The combinations or r and a_iso are shown in (b). Color saturation of the dots for the basis spectra indicates relative weights in the linear combination.

Fig. 4

(a) The solution for the ordinary least square fitting approach for VO-TPP, eqn (9). (b) The same data analyzed with smoothness regularization as shown in eqn (11). (c) Regularization with penalty function based on the $\tilde{P}$ shown in Fig. 5 and using eqn (12). For (b) and (c) the regularization parameters were selected via the AIC method. The top panels show the experimental spectrum in red and the fit to it in blue, as well as the residuals with a magnification factor of 10. The bottom panels show the density distribution over distance r and contact coupling a_iso that was fit to the spectrum, with contour levels at 0.02, 0.05, 0.1, 0.2, 0.5, 0.75, and 0.9 of the distribution maximum. The DFT results for the protons in VO-TPP (pyrrole, ortho, meta, and para) are shown as gray dots. The dotted lines in the lower panel of (b) indicate locations of constant $\left|a_{\text{iso}}+2T\right|\approx 1.4$ MHz and of constant $\left|a_{\text{iso}}-T\right|\approx 0.15$ MHz.

Fig. 5

The probability distribution $\tilde{P}$, overlaid on the results from the DFT calculations (gray). Contour levels are at 0.05, 0.1, 0.2, 0.5, 0.75, and 0.9 of the distribution maximum.

Fig. 6

Analysis of the four model vanadyl porphyrin compounds, (a) VO-TPP, (b) VO-DPP, (c) VO-OEP, and (d) VO-EP. The top panels show the experimental spectrum in red and the fit to it in blue, as well as the residuals with a magnification factor of 10. The bottom panels show the distance r vs. contact coupling a_iso distribution that was fit to the spectrum, with contour levels at 0.02, 0.05, 0.1, 0.2, 0.5, 0.75, and 0.9 of the distribution maximum, as well as its projections along a_iso and r. Results from the DFT simulations for protons of the same type as in the individual compounds are shown as gray dots.

Fig. 7

Ligand proton quantitation. (a) Integration areas that were used to calculate the relative abundances of the proton types. For better illustration, the contour plot of P_fit for VO-EP is shown as well. (b) Relative proton abundances for each compound. The upper bar for each compound shows the expected stoichiometric ratios, obtained from the Lewis structure, and the lower bar shows the ones obtained from integration of the fitted hyperfine parameter distribution.

Fig. 8

Analysis of the ENDOR spectrum of VO-EP using a three-dimensional distribution over r, a_iso, and the rhombicity parameter $\eta$. Shown in (a) is the spectrum (red) and the fit (blue) to it, including the residuals. The projection of the fitted three-dimensional distribution P(r,a_iso,η) along $\eta$ is displayed in b). Panels c) and d) show the integrals of P(r,a_iso,η) along a_iso and r, respectively. Contour levels are drawn at 0.02, 0.05, 0.1, 0.2, 0.5, 0.75, and 0.9 of the maximum of the distribution. The DFT results for protons in VO-EP are shown as gray circles. The regularization parameter α_P was determined via the AIC method, and 7440 basis functions were used.