The g-tensor

The g-tensor (magnetogyric ratio tensor) appears as the crossing contribution to the energy 

The electronic g-tensor is a fundamental quantity in electron spin resonance (ESR) spectroscopy. Its experimental reading is based on the assumption that the following equation is fulfilled 

The g-tensor and the G-tensor can be obtained as the second derivatives of the energy splitting and its square, respectively [38] 

Methods of evaluating the molecular g-tensor at the ab initio level represent a contemporary problem and are still developing [38-45]. Three variants of the molecular orbital calculations are used the restricted Hartree-Fock, the unrestricted Hartree-Fock and the generalised Hartree-Fock methods. 

The differential g-tensor is given as the sum of two contributions the diamagnetic and paramagnetic terms. Since the diamagnetic term is much lower than the paramagnetic one, it is often neglected. 

In the original Schonland procedure [9] the. g-factor was measured by ESR as a function of the angle of rotation of the crystal with respect to the magnetic field in three different mutually orthogonal planes xy, yz and zx. Data are frequently collected over a full period of 180 but are often summarised in diagrams of the type in Fig. 3.9 to more clearly display the match of the g-factor along the X , Y , and Z axes. 

It is customary to use the same symbol for the measured g-factor and the tensor g. For the rotation in the xy-plane one has  

The procedure can give rise to an ambiguity in the determination of the g-tensor when the rotation is done in opposite senses between the different planes. An ambiguity of this type occurs for instance when different crystals are used for the measurements in the different planes. It is then difficult to experimentally determine the sense of rotation, a crystal may for instance have been mounted upside 

The ambiguity gives rise to just two different tensors and can be resolved by measurements in a fourth skew plane [9]. As a rule only one of the two tensors reproduces the experimentally determined -factor variation in the skew plane. A simpler procedure is applicable when the principal values can be read from the experimental powder spectrum. Only one of the two sets of principal values obtained from the analysis usually matches the experimental data. 

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The axial resonance is assigned to ruthenium A with its D4h local symmetry (compare gj = 2.51, gy = 1.64 in traw.s-RuCl4(PEt3)J) while the rhombic signal is assigned to ruthenium B , where the local symmetry is D2h and three different components of the g-tensor are expected. 

In the framework of this standard model, the g tensor of the iS = 2 state is given by the simple expression... 

According to Eqs. (8) and (9), the g tensor of [4Fe-4S] clusters depends on four ferrous gi, 2, gi, gi) and two ferric gs,gs) local g tensors. This holds true even if the mixed-valence pair is fully delocalized Ca = Cb = i Owing to the low-lying excitation energies of the... 

Despite its weakness, the anisotropy of the g tensor of iron-sulfur centers can be used to determine the orientation of these centers or that of the accommodating polypeptide in relation to a more complex system such as a membrane-bound complex. For this purpose, the EPR study has to be carried out on either partially or fully oriented systems (oriented membranes or monocrystals, respectively). Lastly, the sensitivity of the EPR spectra of iron-sulfur centers to structural changes can be utilized to monitor the conformational changes induced in the protein by different factors, such as the pH and the ionic strength of the solvent or the binding of substrates and inhibitors. We return to the latter point in Section IV. 

Since Mossbauer spectroscopy is sensitive to all terms in the SH, it is also sensitive to the ZFS and the g-tensor. The theory of both interactions can be approached along the same lines as explained in some detail in Appendix 1 (Part III, 3 of CD-ROM) [89, 90]. This becomes somewhat elaborate for the ZFS while the g-tensor is more readily approached. Both quantities have been previously treated in some detail and a protracted discussion would be inappropriate here [9, 79, 89-93]. In general, the accuracy with which both quantities can be calculated from DFT is rather moderate and a combination of ligand field theory and DFT or some... 

Schreckenbach, G., Ziegler, T., 1997b, Calculation of the g-Tensor of Electron Paramagnetic Resonance Spectroscopy Using Gauge-Including Atomic Orbitals and Density Functional Theory , J. Phys. Chem. A, 101, 3388. 

Density functional theory (DFT) calculations to interpret the powder ENDOR and HYSCORE spectra Establish the use of the g-tensor parameters to detect the presence of dimers... 

High-field EPR (HFEPR) spectroscopy greatly improves the resolution of the EPR signals for spectral features such as the g-tensor. Deviations of the g-value from free electron g=2.0023 are due to spin-orbital interactions, which are one of the most important structural characteristics (Kevan and Bowman 1990). Using a higher frequency results in enhanced spectral resolution in accordance with the resonance equation ... 

The 327-670 GHz EPR spectra of canthaxanthin radical cation were resolved into two principal components of the g-tensor (Konovalova et al. 1999). Spectral simulations indicated this to be the result of g-anisotropy where gn=2.0032 and gi=2.0023. This type of g-tensor is consistent with the theory for polyacene rc-radical cations (Stone 1964), which states that the difference gxx gyy decreases with increasing chain length. When gxx-gyy approaches zero, the g-tensor becomes cylindrically symmetrical with gxx=gyy=g and gzz=gn. The cylindrical symmetry for the all-trans carotenoids is not surprising because these molecules are long straight chain polyenes. This also demonstrates that the symmetrical unresolved EPR line at 9 GHz is due to a carotenoid Jt-radical cation with electron density distributed throughout the whole chain of double bonds as predicted by RHF-INDO/SP molecular orbital calculations. The lack of temperature... 

Determination of g-tensor components from resolved 327-670 GHz EPR spectra allows differentiation between carotenoid radical cations and other C-H jt-radicals which possess different symmetry. The principal components of the g-tensor for Car"1 differ from those of other photosynthetic RC primary donor radical cations, which are practically identical within experimental error (Table 9.2) (Robinson et al. 1985, Kispert et al. 1987, Burghaus et al. 1991, Klette et al. 1993, Bratt et al. 1997) and exhibit large differences between gxx and gyy values. 

When an organic radical is located near a high-spin metal ion, the g-tensor of the radical depends on the exchange interaction between the radical and the metal ion. 

The g-tensor principal values of radical cations were shown to be sensitive to the presence or absence of dimer- and multimer-stacked structures (Petrenko et al. 2005). If face-to-face dimer structures occur (see Scheme 9.7), then a large change occurs in the gyy component compared to the monomer structure. DFT calculations confirm this behavior and permitted an interpretation of the EPR measurements of the principal g-tensor components of the chlorophyll dimers with stacked structures like the P 00 special dimer pair cation radical and the P700 special dimer pair triplet radical in photosystem I. Thus dimers that occur for radical cations can be deduced by monitoring the gyy component. 

SCHEME 9.7 The geometries of Dp2+, Dp3+, and Dp4+ used in the g-tensor calculations (Dp is the p-dimethylenebenzene molecule). Face-to-face configurations PD1, PD2, and PD3 are shown for clarity. (From Petrenko, A., Chem. Phys. Lett., 406, 327, 2005. With permission.)... 

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