Isotropic g tensor

Lso is the commutator with the electron spin Zeeman Hamiltonian (assuming isotropic g tensor, Hso = gS- Bo), Lrs = Lzfs (the sub-script RS stands for coupling of the rotational and spin parts of the composite lattice) is the commutator with the ZFS Hamiltonian and Lr = —ir, where is a stationary Markov operator describing the conditional probability distribution, P(QolQ, t), of the orientational degrees of freedom through ... 

The g tensors and hyperfine constants of the paramagnetic centers observed in irradiated NH4Y zeolites after various activation treatments are given in Table VIII. Despite the abundance of experimental results, many of the structures proposed for these centers should be regarded as suggestive rather than definitive, as previously noted by Kasai and Bishop (264). Neither the axially symmetric g tensor of the V center associated with two aluminum atoms nor the isotropic g tensor of the V center associated with one aluminum atom reported by Vedrine et al. (266) is consistent with the symmetry of the respective models proposed above. 

This is supported by comparison with the isoelectronic ion SO3, which has an isotropic g tensor at 2.0036 in K2CH2(S03)2 (376), C103, which exhibits an axial g tensor (gL = 2.008, g = 2.007) in NH4C104 (377), and POr (gL = 2.001, gn = 1.999) in Na2HP03-5H20 (378). However, on surfaces there are only two reports of 25-electron radicals adsorbed on surfaces. SO3 adsorbed at the surface of MgO gives an isotropic signal with... 

A further term that can contribute to E(1)yAa is the ZFS (59,60). As implied by its name, ZFS splits the components of a state in the absence of a magnetic field. For states that are only spin degenerate, ZFS occurs when the spin S>l/2. Like the g-tensor, ZFS causes the axis of spin quantization to deviate from the direction of the magnetic field. The consequences with respect to spin integration and orientational averaging are similar to those caused by the use of a non-isotropic g-tensor. ZFS is made up of two terms, one second-order in spin-orbit coupling and the other from spin-spin coupling (59). The calculation of ZFS within DFT has been the subject of several recent publications (61-65). 

The additional requirements introduced by the non-isotropic g-tensor or ZFS complicate the MCD formalism somewhat but do not introduce significant additional computational cost nor do they prevent the definition of a C term parameter consistent with Eq. (33) (provided that kT En)yAaB). 

Eq. (4), which relates the observed contact shifts of the proton resonances to their isotropic contact coupling constants, and hence to the spin densities on the ring carbon atoms, is valid only for systems with isotropic g-tensors. To obtain an estimate of the errors which might arise from its application to low spin ferric heme compounds, we shall briefly consider a more general form of the equation, which was given by Jesson (46) for tetragonal systems with more than one populated electronic state. 

As shown in Figure 2.2, the isotropic g tensor shift (Agiso) is almost linearly dependent on the NO bond length, whereas it does not display any regular trend with respect to out-of-plane motion. 

Figure 2,2 Computed HCC and isotropic g tensor shift along a Car-Parrinello molecular dynamic trajectory of PROXYL in the gas phase.

While the effective g value is expressed in terms of three principal values directed along three axes or directions in a single crystal, only the principal values of g can be extracted from the powder spectrum rather than the principal directions of the tensor with respect to the molecular axes. (Therefore it is more correct to label the observed g values as gi, g2, g3 rather than g gyy, in a powder sample.) In the simplest case, an isotropic g tensor can be observed, such that all three principal axes of the paramagnetic center are identical (x = y = z and therefore gi= gi = g-i). In this case, only a single EPR line would be observed (in the absence of any hyperfine interaction). With the exception of certain point defects in oxides and the presence of signals from conduction electrons, such high symmetry cases are rarely encountered in studies of oxides and surfaces. 

The equations for other solution averaging conditions are given in references (18) and (19). In the special case of a spin-only state with an isotropic g-tensor (i.e. gy = gi and g n = gsi = 2) equation (9) reduces... 

Table 16-1. Energies, electric dipolar moments, net atomic populations, vibrational polarizabilities and mean vibrational molecular polarization, magnetizability and contributions thereto, isotropic g tensor and nuclear and electronic paramagnetic and diamagnetic contributions thereto, principal moments of inertia and rotational parameters calculated for H2 C N2 in seven structural isomers... 

A further feature is that low concentrations of Cr(V) species (>0.1 pM) can be detected selectively in aqueous solutions at room temperature, in the presence of large excesses of Cr species in other oxidation states. An important contribution to the structural characterization of Cr(V) complexes came with the development of empirical parameters that related the values of the isotropic g tensor (giso) to the ligand donor groups and the coordination number (18). In the same paper, it was shown that there were linear correlations between these EPR parameters and those of analogous isoelectronic V(IV) complexes. This correlation also assisted in the characterization of new Cr(V) complexes, since many V(IV) complexes are sufficiently stable to be crystallized, and their structures have been determined by X-ray diffraction (XRD). This, together with the determination of a number of structures of Cr(V) species by XRD (12,... 

Here D denotes the axial fine structure parameter, whereas E describes the orthorhombic fine structure parameter [5]. The influence of an axial (i.e. D O and E=0) and an orthorhombic fine structure splitting (i.e. D O and E O) on a powder spectrum is shown in Fig. 3 for an isotropic g-tensor and for S equal to (e.g. Cr +). 

In the general case, the g-factor is a second-rank tensor. For the simplified case of an isotropic g-tensor, (g = g = isotrop), the Zeeman term becomes BBo/h) pSx +... 

Equation (5.14) amounts to a description of the anisotropic coupling A°° + A originating from the interaction of an electron spin magnetic moment i = , UsgS with the proton magnetic moment The isotropic part of gT which is usually called the pseudo contact interaction is only zero for isotropic g tensors. Since in (5.14) the g tensor is known from EPR data, the remaining parameters which describe A are a, and the direction cosines, nj. 

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