Non-coincident matrix axes

In general, the g- and nuclear hyperline coupling matrices, g and A can be written in diagonal form with three principal values, i.e., gx, gy, g. and A,x, Aiy, Aiz. In textbooks on ESR6a 30,33 35 it is usually assumed that the same set of principal axes diagonalizes all the relevant matrices. While this is sometimes true, there are many instances where the principal axes are non-coincident.36 

Kneubuhl37,38 has given a detailed group theoretical analysis of symmetry restrictions on the orientations of g- and hyperfine matrix principal axes. His results are summarized in Table 4.9. 

For a nucleus sharing all the molecular symmetry elements (e.g., the metal nucleus in a mononuclear complex), the hyperfine matrix is subject to the same 

Although symmetry considerations often permit g- and hyperfine matrix principal axes to be non-coincident, there are relatively few cases of such noncoincidence reported in the literature. Most of the examples discussed by Pilbrow and Lowrey in their 1980 review36 cite cases of transition metal ions doped into a host lattice at sites of low symmetry. This is not to say that matrix axis non-coincidence is rare but that the effects have only rarely been recognized. 

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It is relatively easy to understand the significance of the non-coincident matrix axes in these cases. For the Co2C2 cluster, the C2v molecular symmetry permits a specific prediction of the possible matrix axis orientations. The g-matrix principal axes must be coincident with the molecular symmetry axes. The two cobalt nuclei are located in a reflection plane (which we label xz) so that symmetry requires the y-axis to be a principal axis for all three matrices. The other two axes may be rotated, relative to the molecular x- and z-axes, by /J. (Since the two nuclei are symmetrically equivalent, the rotations must be equal and opposite.)... 

The low-spin manganese(n) complex [Mn(dppe)2-(CO)(CNBu)]2+ gave us a textbook example of a well-behaved ESR spectrum characterized by coincident g- and hyperfine-matrix principal axes. The nearly identical complex [Mn(dppm)2(CO)(CN)]+, (dppm = Ph2PCH2PPh2) (ref. 25) provides us with a good example of non-coincident principal axes. The frozen solution spectrum (Figure 4.8) shows that the parallel features are not evenly spaced. 

In these cases, the g-matrix is nearly isotropic, but the principal axes of the two 59Co hyperfine matrices are non-coincident. The largest hyperfine matrix component (ay = 66.0 G in the case of the Co-Co-Fe-S cluster) results in 15 features, evenly spaced (apart from small second-order shifts). Another series of features, less widely spaced, shows some variation in spacing and, in a few cases, resolution into components. This behavior can be understood as follows Suppose that the hyperfine matrix y-axes are coincident and consider molecular orientations with the magnetic field in the vz-plane. To first order, the resonant field then is ... 

Because of the non-coincidence of the g- and -matrix principal axes, the various parallel features correspond to different orientations of the magnetic field in the g-matrix principal axis system. These orientations are given in Table 4.16. 

The next thing to notice is that the widely spaced features are very approximately equally spaced. This suggests that the g- and -matrix principal axes are non-coincident. You might think that a simple application of the above equations would suffice for a complete analysis. It is not quite so simple, and a nonlinear least-squares program is required.1,6 Table 7.4 shows the fitted parameters. 

When treating CF parameters in any of the two formalisms, non-specialists often overlook that the coefficients of the expansion of the CF potential (i.e. the values of CF parameters) depend on the choice of the coordinate system, so that conventions for assigning the correct reference framework are required. The conventional choice in which parameters are expressed requires the z-direction to be the principal symmetry axis, while the y-axis is chosen to coincide with a twofold symmetry axis (if present). Finally, the x-axis is perpendicular to both y- and z-axes, in such a way that the three axes form a right-handed coordinate system [31]. For symmetry in which no binary axis perpendicular to principal symmetry axis exists (e.g. C3h, Ctt), y is usually chosen so as to set one of the B kq (in Wybourne s approach) or Aq with q < 0 (in Stevens approach) to zero, thereby reducing the number of terms providing a non-zero imaginary contribution to the matrix elements of the ligand field Hamiltonian. Finally, for even lower symmetry (orthorhombic or monoclinic), the correct choice is such that the ratio of the Stevens parameter is restrained to X = /A (0, 1) and equivalently k =... 

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