Electronic g-tensor

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The Mu spin Hamiltonian, with the exception of the nuclear terms, was first determined by Patterson et al. (1978). They found that a small muon hyperfine interaction axially symmetric about a (111) crystalline axis (see Table I for parameters) could explain both the field and orientation dependence of the precessional frequencies. Later /xSR measurements confirmed that the electron g-tensor is almost isotropic and close to that of a free electron (Blazey et al., 1986 Patterson, 1988). One of the difficulties in interpreting the early /xSR spectra on Mu had been that even in high field there can be up to eight frequencies, corresponding to the two possible values of Ms for each of the four inequivalent (111) axes. It is only when the external field is applied along a high symmetry direction that some of the centers are equivalent, thus reducing the number of frequencies. 

As mentioned at the beginning of this section the size of the pseudocontact shifts in the NMR spectra could in principle be calculated for all the low spin ferric heme compounds if detailed data on the electronic g-tensors were available (Jesson (47)). Unfortunately the EPR data on the azides can not be used directly, because these complexes are not in a pure low spin state under the conditions of the NMR experiments (see section VI C). For the compounds in Figs. 10 through 20 no. successful single-crystal EPR studies were as yet reported. However only g-values determined in frozen solutions are presently available (Blumberg and Peisach (70) Salmeen and Palmer (95a)), e.g. for dicyanoferri-porphin at 1.4 °Kgi = 3.64, g 2.29, and gs 1.0 were found. 

S is an effective electronic spin (S = 1/2), / is a function which depends on the averaging conditions in the solution, g is the electronic g-tensor for the jth state, and A the contact coupling tensor which describes the interactions of the ith proton with the electronic spin in the jth state. The summations are over the populated states, and Ej are the state energies (Fig. 24). Under the averaging conditions in solutions of hemoproteins, with rr <=> 10 8 sec, and gn — gx >0.01 at 220 Me / is... 

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... 

Fig. 20. Spin-orbit mixing mechanism for orbital g-shifts in substituted phenoxyl radicals. The electronic g-tensor is perturbed by spin-orbit effects which can be viewed as orbital rotation elements. The perturbation of theg t term arises from mixing perpendicularly oriented valence orbitals on the same atom under the G orbital operator and summing these individual contributions over all atoms to produce the resultant molecular g-shift.

Section 4.1.1 reviews second harmonic generation (SHG) for para-nitroaniline (PNA), Section 4.1.2 the polarizability and second hyperpolarizability of nitrogen and benzene, Section 4.1.3 the second hyperpolarizability of Cgo, Section 4.2 the excited state polarizability of pyrimidine and r-tetrazine. Section 4.3 three-photon absorption, and finally, in Section 4.5 the electronic g-tensor and the hyperfine coupling tensor are reviewed as examples of open shell DFT response properties. 

The electronic g-tensor is a fundamental parameter in descriptions of the electronic Zeeman effect and one of the key elements in characterization of EPR spectra. It couples the external magnetic field, B, with the total spin angular momentum, S, of the molecule and is conventionally evaluated as the second derivative of the molecular energy ... 

Computation of the spin-orbit contribution to the electronic g-tensor shift can in principle be carried out using linear density functional response theory, however, one needs to introduce an efficient approximation of the two-electron spin-orbit operator, which formally can not be described in density functional theory. One way to solve this problem is to introduce the atomic mean-field (AMEI) approximation of the spin-orbit operator, which is well known for its accurate description of the spin-orbit interaction in molecules containing heavy atoms. Another two-electron operator appears in the first order diamagnetic two-electron contribution to the g-tensor shift, but in most molecules the contribution of this operator is negligible and can be safely omitted from actual calculations. These approximations have effectively resolved the DET dilemma of dealing with two-electron operators and have so allowed to take a practical approach to evaluate electronic g-tensors in DET. Conventionally, DET calculations of this kind are based on the unrestricted... 

Table 12. Electronic g-tensors of transition metals compounds evaluated with various exchange-correlation functionals.Reproduced from [150]...

Electronic g-tensor shifts are given in the principal axis system. Values are in ppt. 

The development of DFT computations of electronic g-tensors has mainly focused on improving the accuracy and applicability for isolated systems, while only little attention has been devoted to account for environmental effects. Most studies of solvent or matrix effects on electronic g-tensors have adopted the supermolecular approach, in which the solvent molecules are explicitly introduced into the model used in the calculations. Recently, we developed an electronic g-tensor formalism in which solvent effects are accounted for by the polarizable continuum model [154]. We applied this approach to investigate solvent effects on electronic g-tensors of di-r-butyl nitric oxide (N-I) and diphenyl nitric oxide (N-II). Calculations were... 

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... 

Today, the role relativistic effects play for NMR and EPR parameters has been appreciated to very different extents for different properties and by different communities of experimentalists and theoreticians. For example, it has been known early on in the EPR community that the electronic g-tensors of EPR spectroscopy are basically dominated by spin-orbit coupling and are thus intrinsically relativistic [2]. On the other hand, in spite of much early work on relativistic theories of NMR chemical shifts, and much associated recent cori5)utational developments and applications [3,4,5,6,7], most users of NMR spectroscopy still seem largely unaware of the important role of relativistic effects. This holds in particular for the role of spin-orbit effects, in what is often simply called heavy-atom effects on NMR chemical shifts. This can be seen easily when inspecting most NMR textbooks and much of the research literature. 

In this article, we will concentrate on NMR chemical shifts, for which the inportance of spin-free (scalar) relativistic (SFR) and spin-orbit (SO) contributions needs to be better appreciated. Reviews of relativistic calculations of spin-spin coupling constants are available [6,7]. Articles on conten jorary quantum chemical calculations of electronic g-tensors have been published elsewhere [5,8]. Other EPR parameters like zero-field splittings and hyperfine coupling constants are also strongly affected by relativity and are covered. 

Ab initio and Density Functional Calculations of Electronic g-Tensors for Organic Radicals M. Kaupp in EPR Spectroscopy of Free Radicals in Solids. Trends in Methods and Applications (Hrsg. A. Lund, M. Shiotani) Kluwer, Dordrecht, 2003, in press. 

In Equation (2), D and E are the axial and rhombic zero-field splitting (ZFS) parameters, respectively, and g is the electronic g tensor. The magnetic hyperfine interactions of the electronic system with the Fe nucleus are described by S-a-I, and —is the nuclear Zeeman term. The quadrupole interaction involves the traceless EFG tensor. The EFG tensor has principal components Vyy, and The asymmetry parameter t] = Vxx- VyyyiV can be confined to 0 < 7 < 1 if the convention V zl > I Vyy > V xl is adopted. A quadrupole doublet... 

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