Hyperfine tensor motion

In those calculations, the contributions from electronic orbital motion (induced by spin-orbit mixing) were estimated from crystal field theory (for the copper atom) or were neglected (for the nitrogen and hydrogen atoms). Here I discuss for the first time direct calculations of these contributions to the copper and nitrogen hyperfine tensors, as well as to the molecular -tensor. 

For the nitrogen hyperfine tensors, there is no satisfactory empirical scheme for estimating the various contributions, so that Table II compares the total observed tensor to the DSW result. The tensors are given in their principal axis system, with perpendicular to the plane of the heme and along the Cu-N bond. The small values (0.1 - 0.2 MHz) found for A O in the nonrelativistic limit are not a consequence of orbital motion (which must vanish in this limit) but are the result of inaccuracies in the decomposition of the total tensor into its components, as described above. 

Vedrine et al. (102a) have reported a different type of motion for 02 in a Ce-X zeolite which is characterized by unusual parameters gt = 2.0242, g2 = 2.0208, g3 = 2.0112, and At = 24, A2 = 66, A3 = 12 G. However, inspection of the published spectra indicates that A, and A3 are difficult to estimate. The two oxygens are apparently equivalent and the spectrum is tentatively interpreted in terms of a rotation of 02 about an axis perpendicular to the internuclear axis at 77 K, i.e., the y axis in Fig. 4. This is difficult to reconcile with the observed orthorhombic g and A tensors, which are not averaged. A rotation of 02 around the internuclear axis has also been reported by Breysse et al. (62) for 02 with equivalent nuclei adsorbed on Th02. The gxx and gyy components, which are distinct at 77 K, are averaged to gL = (gxx + gyy)/ 2 at 298 K and this is explained in terms of a rotation around the internuclear z axis. The l70 hyperfine tensor, however, is not completely averaged, ff one assumes that Ayy is zero at 77 K, as for most oxides (Appendix B), then Ayy and Axx should average to A = (Axx + Ayy)f2 = 37.5 G at 298 K. This value is substantially different from the experimental one of 65 G reported by the authors (62). 

Obviously there are limitations in the use of the 170 hyperfine tensor to derive information on the motion. When both tensors are available, the g tensor seems to give more detailed information on the type and dynamics of the motion than the hyperfine tensor (59, 66). In some cases the situation may be more complicated if the axes of the g and A tensors do not coincide, but this is difficult to measure for powder systems. 

Fig. 4 Information from nitroxide CW EPR spectra, (a) Right principal axis system of electron-Zeeman and hyperfine tensors (collinear). Left the effect of rotational dynamics on the CW EPR spectra. Fast rotation (i.e., faster than a typical rotational correlation time of Tc 10 ps) leads to the averaged spectrum. The isotropic g-value gjgo determines the center of the central line and spacing between the lines that is dominated by a- a- In the intermediate motion regime 100 ns > tc > 1 ns and the rigid limit is reached at Tj, 1 ps [19]. (b) Influence of the chemical environment on CW EPR spectra. As both, hydrophilic and polar environments lead to an increased electron spin density at the nitroxide nucleus (see gray inset), and hence the line splitting in the spectra in hydrophilic and polar surroundings is larger than in non-polar and hydrophobic environments...

Anisotropic g and Hyperfine interaction. The hyperfine tensor A for each nucleus is a real 3x3 matrix that can always be diagonalized. The components of the diagonalized hyperfine tensor consist of an isotropic part, Oo, and a purely anisotropic part, a, whose orientational average is zero. Thus, the a components are averaged out in fluid media and can only be determined in the solid state or in the case of highly restricted molecular motion. The diagonal elements... 

While in some cases considering the environment is sufficient to reproduce experimental values of the g and hyperfine tensors, there are molecules presenting fast motions in the neighborhood of the unpaired electron. Dependence of the magnetic parameters on these small geometric variations can be very significant [57, 74—76]. These motions are usually too fast with respect to the ESR time scale window so the effective contribution is a correction that can be calculated as an average over short-time dynamics calculated at a QM level [77, 78]. 

The anisotropy of the g and hyperfine tensor leads to a dependence of the spectral line shape of nitroxides on the reorientation rate in soft matter or liquid solution. In the simplest case, nitroxide motion can be considered as isotropic Brownian rotational diffusion and can then be characterized by a single rotational correlation time Zr. To understand Zr, one can consider the reorientation of the molecular z axis caused by stochastic molecular motion. With the angle 0 between the orientation of this axis at zero time and the orientation at time t, the correlation fimction (cos0) exhibits exponential decay with time constant (the brackets () denote the average over a large ensemble of nitroxide molecules). Starting from the rigid limit, exemplified by a solid sample at very low... 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

It is important to note that the proportional relationship between Amax, Amid, and Amin for these couplings is the same for 100% spin density, and for the present case with approximately 50% spin density. When this is so it indicates that there is no rocking motion at the radical site. This is good evidence therefore that the radical site is essentially planar. The best evidence for radical planarity comes from the analysis of the direction cosines associated with each principal values of the hyperfine coupling tensor. The direction of Amin (Table 18-2) is known to be associated with the direction of the >C-H bond, while the direction associated with the Amid indicates the direction of the n-clcctron orbital. These directions are easily calculated from the crystal structure, and are included in Table 18-2. One sees that the direction associated with Amid deviates only 2.0° from the computed perpendicular to the ring plane, while the direction of Amin, deviates only 2.8° from the computed direction of the C6-H bond. The errors listed on these values are at the 95% confidence level. This is very clear evidence that the radical shown here is planar in the solid-state. Any torsional motion of the C6-H would lead to asymmetries of the hyperfine coupling tensor, and would not produce the observed agreement between the direction cosines and the known directions obtained from the crystal structure. 

The molecular motion of redox couples within polymer-coated electrodes has recently been investigated by making use of both nitroxide spin probes and various cationic spin probes [94-97]. Spin probes, such as the nitroxide probe TEMPO (see Sect. 2.1.2) and its derivatives, have well-defined electrochemistry and their ESR spectra in viscous media exhibit effects due to incomplete rotational averaging of the g and hyperfine coupling constant tensors. Analysis of the spectra [98] allows deductions to be made concerning the molecular rotation. Such analysis has been performed for spin probes incorporated into various polymer films. 

Rotational motion exerts fluctuating magnetic fields on the electron spin and also averages the various tensor components (see Fig. 2). Nitro-xide EPR spectra are profoundly influenced by the rate of this motion relative to the range of electron spin precession frequencies within each hyperfine line. Thus, one can use EPR spectroscopy to determine tr accurately. There are distinct temporal regimes in which spin label spectra require different types of data collection methods and data analysis to achieve this. These regimes are discussed below. 

For nitroxides in dilute liquid solution, the generally anisotropic spin Hamilton operator is simplified tremendously and, if imresolved proton hyperfine couplings are treated as line broadening, only the electron-Zeeman interaction and the hyperfine coupling to the magnetic nucleus (7 = 1) remain [20]. The g- and hyperfine (4-) tensors are averaged to isotropic values due to fast motion of the spin probe and the resonance condition for the irradiated microwave becomes... 

Lunsford [3b] and Hoffman and Nelson [23] first reported the ESR spectra for adsorbed NO molecules. Then, Kasai [4b] revealed that ESR spectra of NO probe molecules are very sensitive to the interaction with metal ions and Lewis acid sites in zeolites. The earlier ESR studies of the NO/zeolite system have been summarized in several review papers [3a, 4a, 8]. A number of ESR studies have been also carried out for NO adsorbed on metal oxides such as MgO and ZnO as reviewed by Che and Giamello [5]. Modern ESR techniques such as pulsed ESR [25-27], ENDOR (Electron Nuclear Double Resonance) [26], and multi-frequency (X-, Q-, and W-band) ESR [28] are especially useful for an unambiguous identification of the ESR magnetic parameters (g, hyperfine A, and quadrupole tensors, etc.) and, consequently, for a detailed characterization of structural changes and motional dynamics involved. Some recent advancements in ESR studies on NO adsorbed on zeolites are presented in this section. 

A the total (isotropic + dipolar) hyperfine coupling, and Q the quadrupole tensor. The rapid tumbling of the molecules due to Brownian motion averages the anisotropic parts of the Y and the A tensor, leaving behind their isotropic parts, g and A, respectively. The quadrupolar term averages to zero since it has no isotropic part, thereby yielding... 

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