Simulation of nitrogen nuclear spin magnetization of liquid solved nitroxides

Nitroxide radicals are widely used in electron paramagnetic resonance (EPR) applications. Nitroxides are stable organic radicals containing the N–O˙ group with hyperfine coupled unpaired electron and nitrogen nuclear spins. In the past, much attention was devoted to studying nitroxide EPR spectra and electron spin magnetization evolution under various experimental conditions. However, the dynamics of nitrogen nuclear spin has not been investigated in detail so far. In this work, we performed quantitative prediction and simulation of nitrogen nuclear spin magnetization evolution in several magnetic resonance experiments. Our research was focused on fast rotating nitroxide radicals in liquid solutions. We used a general approach allowing us to compute electron and nitrogen nuclear spin magnetization from the same time-dependent spin density matrix obtained by solving the Liouville/von Neumann equation. We investigated the nitrogen nuclear spin dynamics subjected to various radiofrequency magnetic fields. Furthermore, we predicted a large dynamic nuclear polarization of nitrogen upon nitroxide irradiation with microwaves and analyzed its effect on the nitroxide EPR saturation factor.


S1. Simulated frequency scan EPR spectra for the electron spin T2 relaxation time estimation
This section contains simulated frequency scan CW-EPR spectra, which were used to illustrate our computational method and to estimate electron spin T2 relaxation times according to the formula (30) of the main text.
See the next page … Figure S1. Simulated evolution of the transverse electron spin magnetization of liquid solved nitroxides in CW-EPR experiments with varied MW frequency. The values of 0 , 0 , ̇ and Δ can be deduced from the data in the plots. 1 = 2 × 10 −5 T and Φ 0 = 2 were used in all simulations.

S2. Excitation and relaxation of nitroxide longitudinal electron spin magnetization
The left column of this table contains the simulation of the longitudinal nitroxide electron spin magnetization Mz(t) upon application of tp = 5 µs (2.5 µs for 0 ≥ 2 T) long MW pulse followed by an observation period when the irradiation is switched off. The right column shows the logarithm of the function ( ) = ( − ( )) ( − ( )) ⁄ for the time ≥ .
See the next page … Figure S2. Simulated recovery of the longitudinal electron spin magnetization upon the application of a linearly polarized MW field with B1 = 0.0002 T tuned to the central nitroxide line frequency 0 , which is determined for each B0 value and indicated in the plots. The computations were performed with the same relaxation superoperators R as the corresponding spectra in the Figure S1.

S3. Simulation of experimental data
This section demonstrates the simulation of the real experimental data using the computational method based on the direct solution of the LvN equation developed in this paper.
a) TEMPO in water in B0 = 0.337 T The experimental data were obtained by measuring X-band CW-EPR spectra of water solved TEMPO with a concentration of 1mM at 290 K. For the simulation of these experimental data, we used the simulation parameters given in the With these parameters we simulated the evolution of the transverse electron spin magnetization in the frequency scan CW-EPR experiment shown in Fig.3A (left). The frequency sweep time and range are indicated at the figure bottom and top axes, respectively. The curves mx(δν) and my(δν) were transformed into the field sweep spectra mx(δB) and my(δB) by flipping the x-axis and rescaling MHz to Gauss. Further, the derivative spectra were computed and an additional Gaussian line broadening of 0.08 mT (equal for all three lines) was introduced by convoluting the derivative spectra with a Gaussian curve. The normalized result of this computation compared to the experimental spectrum is shown in Fig. 3A (right).

b) TEMPOL in water in B0 = 9.2 T
Here we simulated the field scan CW EPR spectrum of low concentrated water solved TEMPOL at B0 = 9.2 T and room temperature published by [c1]*. The simulation parameters are listed in the In our simulation we used the same values for g-tensor and hyperfine tensor A as were used in [].
The relaxation superoperator R was determined for the magnetic field B0 = 9.2 T with the same parameters K, T, Δτ, σ ϕ , I, τ c η and T, which are specified in the caption of the Fig. S1. Note that these parameters (Δτ = 0.1 ps and σ ϕ = 2.3 ∘ ) yield a rotational correlation time τ c = 20 ps, is the same as used in [c1]. With these parameters the frequency scan absorption and dispersion EPR spectra were simulated (see Fig. S3.B, left). Similar as in the previous case, these spectra were transformed into the field sweep spectra. To account for the admixture of the dispersion component in the measured data, the function S = mxcos(α) + mysin(α), where α is the phase correction angle, was computed. In order to introduced additional broadening, the spectrum S was convoluted with a Lorenz curve with a width of 0.08 mT, which is the same value as used in [c1]. Further the derivative of the spectrum was computed, normalized by its maximal value and compared to the experimental data (see Fig S3.B, right).

S4. Comparison of the simulated spectra by direct solution of the LvN equation to the spectra obtained by EasySpin
In this section we compared our simulated nitroxide CW-EPR spectra, which we obtained by the direct solution of the LvN equation, to the corresponding spectra obtained by EasySpin package. Four CW-EPR spectra for B0 = 1.2 T, B0 = 3.35T, B0 = 9.4 T and B0 = 14 T, which are shown in Fig S1, were also simulated with the EasySpin package ('garlic') using the same g-tensor, hyperfine interaction tensor A and rotational correlation time as we used for the simulation of the Fig. S1 spectra. Additionally, we introduced 0.02 mT Lorentzian line broadening for all spectra in EasySpin simulation, which did not account for the effect of the B1 field value on the spectral shape. To remind you, our spectra in Fig. S1 were simulated with B1 = 0.02 mT. Figure S4. Red curves with DiSoLiNE legend are the same as in Fig S1. Blue curves are the spectra obtained by the EasySpin package with the same g-tensor, hyperfine interaction tensor and rotational correlation time.

S5. Nitroxide nitrogen NMR lines simulated with and without electron spin interaction with B1 field
This chapter illustrates the frequency scan CW NMR spectra, which were simulated with (left) and without (right) electron spin interaction with the field B1(t), that is, with the Hamiltonians ̂( ) = − 1 ( )(̂+̂) and ̂( ) = − 1 ( )̂ , respectively. The computations were performed with the same relaxation superoperators R as the corresponding spectra in the Figure S1 and S2.

S6. Determination of the nitrogen nuclear spin T1 relaxation time
The plots on the left side illustrate the time dependence of the simulated longitudinal nitrogen nuclear spin magnetization, , during 25 μs long MW irradiation time (tp = 25 μs) followed by a 25 μs long period when the irradiation is switched off and the nitroxide spin system returns to the thermal equilibrium. The plots on the right side show the corresponding logarithm of the function ( ) = ( − ( )) ( − ( )) ⁄ for the time ≥ , which allows us to determine nuclear T1 relaxation time as the point where the function Ln(Y(t)) is equal to -1. In this way, we obtained T1 = 6.7 μs for B0 = 1.2 T and T1 = 7.0 μs for B0 = 14 T. Figure S6. (left) Time dependence of the longitudinal nitroxide nitrogen nuclear spin magnetization upon the MW, which is applied the central nitroxide EPR line for 25 μs and then switched off. The simulation of these curves was performed with the same parameters as the corresponding blue curves shown in the main text. (right) Auxiliary function Ln(Y(t)), which is plotted to determine nitrogen nuclear spin relaxation time T1.

S7. Nitroxide electron spin saturation factor
Simulated longitudinal nitroxide electron spin magnetization, which was used to determine the saturation factors shown in the TABLE I. In this simulation, the nitroxide central EPR line is permanently irradiated with the microwave. Figure S7. Time dependence of the longitudinal electron spin magnetization under the action of the MW irradiation with B1 = 0.0002 T. The upper two plots were simulated with the same relaxation superoperators R as the corresponding spectra in the Figure S1, S2, S5, and S6. The lower two plots were simulated with the relaxation superoperators R, determined by a faster rotational diffusion, specified in the caption of FIG. 8 of the main text.