Tensors magnetic coupling

Throughout this paper the principal values of magnetic coupling tensors are denoted by lower indices x, y, z if the corresponding principal axes coincide with the g tensor axes. In all other cases, indices 1, 2, 3 are used... 

Figure 1 The principal sources of structural data are the NOEs, which give information on the spatial proximity d of protons coupling constants, which give information on dihedral angles < i and residual dipolar couplings, which give information on the relative orientation 0 of a bond vector with respect to the molecule (to the magnetic anisotropy tensor or an alignment tensor). Protons are shown as spheres. The dashed line indicates a coordinate system rigidly attached to the molecule.

The leading term in T nuc is usually the magnetic hyperfine coupling IAS which connects the electron spin S and the nuclear spin 1. It is parameterized by the hyperfine coupling tensor A. The /-dependent nuclear Zeeman interaction and the electric quadrupole interaction are included as 2nd and 3rd terms. Their detailed description for Fe is provided in Sects. 4.3 and 4.4. The total spin Hamiltonian for electronic and nuclear spin variables is then ... 

To compare measured magnetic coupling constants with values obtained from theoretical calculations, the signs of the magnetic parameters should be known. In the following, different approaches for the determination of absolute and relative signs of the principal values of hf and quadrupole tensors will be discussed. 

The isotropic shift. The isotropic shift is the sum of two contributions the contact and the dipolar contributions. The former is due to the presence of unpaired electron density on the resonating nucleus. The latter arises from the anisotropy of the magnetic susceptibility tensor, modulated by the distance between the unpaired electron and the resonating nucleus, and is also dependent on the orientation of the metal nucleus vector with respect to the principal axes of the magnetic susceptibility tensor. Some problems arise when the spin delocalization is taken into account in calculating the dipolar coupling, but we will not address this problem except when strictly necessary. 

In addition to the isomer shift and the quadrupole splitting, it is possible to obtain the hyperfine coupling tensor from a Mossbauer experiment if a magnetic field is applied. This additional parameter describes the interactions between impaired electrons and the nuclear magnetic moment. Three terms contribute to the hyperfine coupling (i) the isotropic Fermi contact, (ii) the spin—dipole... 

From the lJ residual dipolar coupling the molecular magnetic anisotropy tensor is obtained, which differs from the metal contribution by an extent which depends on the magnetic anisotropy of the diamagnetic part. For example, in cytochrome b the diamagnetic, the paramagnetic and the total susceptibility anisotropy values are A Xax = —0.8, 2.8, 2.20 x 10 32 m3, respectively, and Axrh = 0.1, —1.1, — 1.34 x 10-32 m3, respectively [60]. The corresponding tensors sum up as expected. 

Some the best-known work of Grein and his coworkers involves the development of methods for the calculation of hyperfine coupling constants.141 More recently the focus has shifted to calculating magnetic g-tensors from highly correlated wavefunctions. Grein s current interests include the study of stereoelectronic effects (such as the anomeric and reverse anomeric effects in acetal-like systems) in organic chemistry, a topic to which he has made important contributions.142... 

The most comprehensive information obtained from a Mossbauer spectrum is contained in Bint that depends on the magnetic hyperfine tensor A and, through (S), on the ZFS, the electronic g tensor (and exchange couplings when we consider polynuclear systems). For samples containing randomly oriented molecules, such as poly crystalline powders or molecules in frozen solution, the Mossbauer spectrum depends on the orientation of the molecule relative to the direction of the applied field,4 6 which is fixed in the laboratory and is generally either parallel or perpendicular to the direction of Mossbauer radiation. As a consequence, the spectrum is a powder average from which we have to extract the various tensor quantities of... 

The hyperfine coupling tensor (A) describes the interaction between the electronic spin density and the nuclear magnetic momentum, and can be split into two terms. The first term, usually referred to as Fermi contact interaction, is an isotropic contribution also known as hyperfine coupling constant (HCC), and is related to the spin density at the corresponding nucleus n by [25]... 

The presence of magnetic moments /lia, b, of nuclei A,B,... in a molecule are responsible for the two observables of the NMR experiment that are most frequently utilized in chemical applications. They are physically observed in form of quantized energy differences AE that can be measured very precisely. These two observables are the nuclear shielding tensor cr for nucleus A and the so-called indirect reduced coupling tensor KAB for a pair of nuclei A,B. Both crA and Kab are second-rank tensors that are defined via the phenomenological Hamiltonians... 

Table 1 Nonrelativistic one-electron magnetic terms in the Hamiltonian. Their derivatives with respect to /m and/or or [ip enter the expressions for the nuclear shielding and spin-spin coupling tensors via the perturbation operators ) ancj g is the spin-operator for an electron, va a distance vector with respect to nucleus A etc. ...

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. 

Where (3 is the Bohr magneton, H0 is the applied magnetic field, g is the g-tensor, S is the electron spin, I is the nuclear spin, gn and (3n.are the nuclear splitting factor and the nuclear magneton. The hyperfine coupling tensor A consists of an isotropic contact interaction... 

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