Principal values hyperfine coupling tensors

The authors also reconsidered radical 20, which, in the current work, was formed by x-irradiation and studied at 77 K. The experimentally determined hyperfine coupling tensors for the two methylene protons have the same principal values and agreed with the tensors found in the earlier work.16 These tensors are presumed to be the average of the two individual proton tensors. By assuming that C5 is sp2 hybridized the authors were able to calculate, from the observed average tensors, that Ami = 1.06 mT and Amai = 3.45 mT for the methylene protons. Using the individual tensors for the methylene protons and a 3.6 mT isotropic coupling for the C4(H) proton, a satisfactory simulation of the... 

The spectra of polycrystalline samples are broad and represent the envelope of all the anisotropic couplings together with the g-anisotropy. Although in favourable cases it is possible to extract the principal values of the g- and hyperfine coupling tensors from such... 

Here it is interesting to continue with the discussion of using the ENDOR data to discern radical geometry. Results for the cytosine reduction product observed in irradiated single crystals of l-MethylCytosine 5-FluoroUracil are shown in Table 18-2 [34]. First one notes the three principal values of the hyperfine coupling tensor. For an ordinary rr-electron radical with unit spin density on the central... 

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. 

Matrix Principal values (G) and direction cosines of the hyperfine coupling tensors Ref. 

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 (Table I) is toown to be associated with the direction of the >C-H... 

An alternative method to determine A and Q coupling tensors simultaneously by fitting directly to the experimental ENDOR frequencies [28] provides a means to estimate the uncertainties of the principal values and directions, similarly as for the hyperfine coupling tensor in Table 2.1. 

In the example shown in Eig. 2.11 [29] the paramagnetic species is the carbon dioxide radical anion, CO2, in an X-irradiated lithium formate crystal. ENDOR lines from several Li nuclei (/ = 3/2) at different lattice positions occur. The principal values and directions of the hyperfine coupling tensors provide a means to obtain the geometric arrangement of the ions about CO2. The analysis is based on the point dipolar approximation, in which the maximum value, bmax, in MHz of the dipolar coupling is given by... 

Principal values of hyperfine coupling tensors for H (and other I = V2 nuclei) can often be obtained by analysis of the ENDOR powder spectra. Visual analysis is sometimes sufficient, in other cases computer simulation is required to refine the analysis. 

Any orientation in a molecular frame of the g tensor and of hyperfine coupling tensors A of several nuclei of any spin. For nuclei of spin I = Vi the satellite lines resulting from the forbidden Am/ = 1 transitions are taken into account. The spectral simulations resort to the second order treatment of the spin Hamiltonian by Iwasaki [19]. The optimized principal values of g and A tensors can be obtained from the least- squares fitting of spectra. 

Powder ENDOR lines are usually broadened by the anisotropy of the hyperfine couplings. The parameters of well resolved spectra can be extracted by a visual analysis analogous to that applied in ESR. The principle is indicated in Fig. 3.25 for an 5 = V2 species with anisotropic H hyperfine structure, where the hyperfine coupling tensor of axial symmetry is analysed under the assumption that 0 < A < Aj. < 2 vh- The lines for electronic quantum numbers ms = V2 and -Vi, centered at the nuclear frequency vh 14.4 MHz at X-band, are separated by distances equal to the principal values of the hyperfine coupling tensor as indicated in the figure. Absorption-like peaks separated by A in the 1st derivative spectrum occur due to the step-wise increase of the amplitude in the absorption spectrum, like in powder ESR spectra (Section 3.4.1). The difference in amplitude commonly observed between the ms = /2 branches is caused by the hyperfine enhancement effect on the ENDOR intensities first explained by Whiffen [45a]. The effect of hyperfine enhancement is apparent in Figs. 3.25 and 3.26. 

Fig. 3.26 Schematic powder ENDOR spectra of an S = Vi species with axially symmetric g and H hyperfine structure. ENDOR spectra with the magnetic field locked at g and gi, respectively, are single-crystal like due to angular selection. The lines for electronic quantum numbers ms = Vi and -V2 are separated by distances equal to A and Aj, the principal values of the hyperfine coupling tensor as indicated in the figure...

The first three terms are usually the ones of relevance for the ESR analysis, where D and A are the zero-field (or fine structure) and hyperfine coupling tensors. They are represented by 3-3 symmetric matrices and specified by three principal values and three principal directions as for the -tensor. The remaining nuclear Zeeman and quadrupole (/ > Vi) terms do not affect the ESR spectra, unless they are of comparable magnitude to the hyperfine coupling, but must be taken into account in the analysis of ENDOR and ESEEM spectra. The spin Hamiltonian formalism introduced by M.H.L. Pryce and A. Abragam [79] is used explicitly or implicitly in the ESR literature as a convenient way to summarise resonance parameters. 

Simplify to the case with isotropic g to discuss the possibility to obtain the relative signs of the principal values of the hyperfine coupling tensor. Consider the angular dependences of the two H hfc lines in the xy-plane of rotation for the cases a) Axx > 0, Ayy > 0, b) Axx > 0, Ayy < 0. For simplicity assume that x and y are principal axes for the tensor. 

Fig. 4.7 Anisotropic ESR spectra of monovalent copper complex in frozen toluene solution, (a) 2 mm band ESR (140 GHz), (b) 3 cm band ESR (9.6 GHz) experimental spectnim, (b ) simulated spectrum with g = (2.0078, 2.0059, 1.9991), A(P ) = (20, 23, 20) G, A(P) = (8, 11, 8) G, and A(Cu) = (20, 0, 17) G for the principal values of the g-tensor and hyperfine coupling tensors of two inequivalent phosphorous and one Cu atom. The spectra are adapted from [R.R. Rakhimov et al. Chem. Phys. Letters 255, 156 (1996)] with permission from Elsevier...

The principal values and even the orientation of the principal axes of the Na hyperfine coupling tensor with respect to axes of the g tensor could be determined from Mims and Davies pulsed ENDOR spectra, refer to Section 2.3.3 in Chapter 2. The values Axx( Na) = Ayy( Na) = 6.3 and Azz( Na) = 10.9 MHz were obtained by simulation taking angular selection into account. The so-called hyperfine enhancement of ENDOR intensities due to the interaction between the radio frequency field and the electron spin could lead to pronounced differences in the ENDOR intensities between signals from different rris electron spin states in experiments at conventional MW frequencies such as in X-band, but also at the W-band. The Na (I = 3/2) nuclear quadrupole tensor is almost coaxial to the A tensor, 2zz = 0.48 MHz, Qyy = -0.07 MHz, and Qxx = -0.41 MHz. Simulation of orientation-selective ENDOR spectra as described in [26, 33] serves to refine the principal values of the hyperfine coupling tensors estimated from experiment. In... 

Ay are the principal values of the g and hyperfine coupling tensors respectively. 

Table 2 Principal values of reduced hyperfine coupling tensors for Mu-substituted radicals in single-crystal naphthalene (A g = /Also + B i). Also given are the isotropic solution couplings and the tensors obtained for the CH2 protons of analogous hydrogen radicals by ENDOR spectroscopy.

The ESR spectrum of C6H6 " trapped in CFCI3 at 15 K is shown in Figure la and agrees with that reported previously [18]. The principal values of the hyperfine coupling were obtained from previous ESR and ENDOR measurements [17, 18]. The best agreement with experiment was obtained with the axes oriented as in Table 4. In the latter study, the simulated ENDOR spectra were insensitive to the orientation of the tensor axes, however, and the assignment was made on the basis of molecular orbital calculations [9]. The tensor data are reproduced here for convenience (see Table 4). 

Radical IV can be considered as a unique phosphorus radical species. Reduction of the parent macrocycle with sodium naphtalenide in THF at room temperature gave a purple solution. The FPR spectrum displayed a signal in a 1 2 1 pattern, with flp(2P)=0.38 mT. DFT calculations on radical IV models indicated a P-P distance of 2.763 A (P - P is3.256 A in the crystal structure of the parent compound and the average value of a single P-P bond is 2.2 A). According to the authors, the small coupling constant arises from the facts that the principal values of the hyperfine tensor are of opposite sign and that the a P P one electron bond results from overlap of two 3p orbitals [88]. 

The polyerystalline spectrum of N02 on MgO is somewhat complex, but it yields an unambiguous g and hyperfine tensor which can be checked by comparison with data for NO2 in single crystals. For N02 on MgO, principal values of the hyperfine tensor are m = 53.0, 21 = 49.0, and a31 = 67.0 G (29). It should be noted here that neither the signs of the coupling constants nor their directions relative to the molecular coordinates... 

Figure 11 The magnetic field dependence of H-ESEEM spectra obtained for Fe(II)NO-TauD samples treated with aKG and taurine deuterated at both Cl andC2. Data were obtained using the ratio method described for Figure 9. The field positions displayed are (a) 172.8 mT (b) IQO.OmT (c) 290.0mT and (d) 345.0 mT. Simulated H-ESEEM spectra (dashed lines) for the Ci,C2-deuterated taurine are plotted along with the data. For the stronger coupled Ci deuteron, Hamiltonian parameters identical to those of Figure 10 were used. Hamiltonian parameters used for a second deuteron on C2 were principal deuteriiun hyperfine values, —0.13, —0.13, 0.26 MHz Euler angles for hyperfine tensor, 0, 63°, 0 e q Q, 0.20 MHz q, 0 and Euler angles relating nqi to hyperfine, 0, 23°, 0...

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