Coupling tensor direct

The hyperfine constant a in Eq. (1) was also taken to be a scalar quantity for the hydrogen atom however, it is in general a tensor because of the various directional interactions in a paramagnetic species. The hyperfine term in the spin Hamiltonian is more correctly written as S-a-I, where a is the hyperfine coupling tensor. 

For free atoms this value can be calculated from the atomic wave-functions so that, to a first approximation, the p-electron density at the magnetic nucleus under study can be calculated from the ratio of the experimental and the atomic coupling constants. Furthermore, the direction of the largest component of the anisotropic coupling tensor coincides with the direction of the p-orbital. This is thus an important factor in the identification of the radical species. 

In the analysis of the ENDOR data for Radical I all three hyperfine coupling tensors fit the expected directions of the crystal structure very closely. These tensors were produced by fitting >90 accurately measured data points to theoretical equations with a total rms error of ca. 0.25 MHz. For Radical II the tensors are just as accurate, but the expected directions are off by ca. 10°. It can be seen that the Amid direction for both the >N6-H and C8-H couplings are both 8.3° from the computed ring perpendicular. This suggests that there is some slight deviation from planarity for this radical. 

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. 

In the absence of magnetic field, the three triplet levels are separated in energy by the zero field splitting (roughly 10 -I0" cm" in short biradicals) and are quantized with respect to the molecular axes dictated by the principal directions of the electron spin-spin dipolar coupling tensor primarily responsible for the zero-field splitting. (Spin-orbit coupling affects the... 

Anisotropy of the Spin-Spin Coupling Tensor. - In NMR experiments performed in anisotropic liquid crystal (LC) phases or in the solid state, the anisotric part of an indirect nuclear spin-spin coupling tensor J appears combined with the direct dipolar coupling D. The NMR spin Hamiltonian appropriate for spin 1/2 nuclei in molecules partially oriented in uniaxial LC solvents can be written in the high field approximation as... 

The orientational distribution fimction P (cos 0) enters the shape of the wideline spectrum 5(f2) in a slightly hidden way. The angular dependence of the resonance frequency is given by (3.1.23) via the orientation of the magnetic field in the principal axes system XYZ of the coupling tensor (cf. Fig. 3.1.2), while the orientational distribution function specifies the distribution of the preferential direction n in a molecule-fixed coordinate frame (Fig. 3.2.2(a)). Figure 3.2.3 shows the relationship between the different coordinate frames and the definition of the relative orientation angles. 

The orientational distribution function P cos of axially symmetric coupling tensors (t] — 0) in the laboratory frame can be read directly from the NMR spectrum S S2). Its expansion coefficients x/ can be transformed to those, of the orientational distribution function P(cos0) of the molecules in the sample fixed frame by using the known orientation angles 0 and 0 of the principal-axes frame in the molecule-fixed coordinate frame, and the orientation angle of ihe sample frame in the laboratory frame. By convention, the preferential axis n of the sample is parallel to the Zs-axis of the sample frame. 

Fig. 3.2.3 Relationships between coordinate systems for the description of molecular order. The orientation of the laboratory coordinate system in the principal axes system of the coupling tensor determines the angular dependence of the resonance frequency. The orientations of the preferential sample direction in a molecule-fixed coordinate frame determines the orientational distribution function.

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