Anisotropic Spin-Orbit Coupling

In their early study of the ruby spectrum Sugano and Tanabe [44] attributed the zfs to the trigonal anisotropy of the spin-orbit coupling. In a simplified form the anisotropic s.o.c. hamiltonian may be written as  

This hamiltonian has cylindrical symmetry and may be used to introduce trigonal or tetragonal anisotropy, depending on whether the principal z axis is oriented along a C3 or C4 symmetry axis. The second-quantized form of the intra-r29 part of this operator is given in Eq. 39. 

Now this operator can directly be applied to the spherical eigenvectors of Table 2. For the Ms = + 3/2 and Ms = + 1/2 components of 4A2g one obtains  

Hence the E level (Ms = 1/2) is connected to the 2P term via Ch and Ci elements in a 2 1 ratio, while the E level (Ms = 3/2) is connected via the Ci part of the hamiltonian. This anisotropy gives rise to the following second-order contribution to the zfs  

A more general expression involving states of the (t2g)2(eg)1 configuration has recently been presented by Dubicki [45], It should be noted that Eq. 41 applies both to tetragonal and trigonal zfs. This is a consequence of the pseudo-spherical symmetry of the t2g-shell and will be further elaborated in the subsequent section. 

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The first term is characterized by a scalar, 7, and it is the dominant term. Be aware of a convention disagreement in the definition of this term instead of -27, some authors write -7, or 7, or 27, and a mistake in sign definition will turn the whole scheme of spin levels upside down (see below). The second and third term are induced by anisotropic spin-orbit coupling, and their weight is predicted to be of order Ag/ge and (Ag/ge)2, respectively (Moriya 1960), when Ag is the (anisotropic) deviation from the free electron -value. The D in the second term has nothing to do with the familiar axial zero-field splitting parameter D, but it is a vector parameter, and the x means take the cross product (or vector product) an alternative way of writing is the determinant form... 

Matrix asymmetry is linked to the existence of anisotropic spin-orbit coupling,123 and the presence of sufficiently low-symmetry electric fields (static and dynamic). 

Table 19 Sublevels of the 4A2 ground state and of the 2E spin-flip octahedral excited state split by spin-orbit coupling from NEVPT2 calculations and their AILFT equivalents adopting a model of anisotropic spin-orbit coupling (A) and their shifts by reduction of the values of B and C from those of a NEVPT2 treatment to effective values b deduced from high-resolute d-d absorption and emission spectra of Cr(acaca)3c (B)...

Anisotropic Spin-Orbit Coupling in paramagnetic resonance. 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

Nitrosyl complexes of both S = and S = are common. The S = nitrosyl complexes or iron and copper have slightly anisotropic g tensors (Ag/g < 2) with the anisotropy provided primarily by contributions of orbital angular momentum from the metal d orbitals. As an example, we consider the model proposed by Kon and Katakoa (1969) for ferroprotoheme-NO complexes, which is also applicable to the ferrous-nitrosyl complexes of heme proteins. In the complexes studied by these workers, the axial ligands are a nitrogenous base and NO. They proposed that the unpaired electron resides primarily in the metal d z orbital. The spin-orbit coupling would then mix contributions from the d, and dy orbitals. 

General relationships between AOM and crystal field parameters are shown in Table 23. Using the AOM one can easily compute the electronic energy levels, inclusive of spin-orbit coupling, without any symmetry assumption or perturbation procedure, and it is also easy to account for the different chemical natures of the ligands and for differences in bond distances. It is also possible to handle anisotropic n interactions, which can be expected to occur with pyridine or pyridine iV-oxide ligands.366,367 General review articles on the AOM and its applications have already appeared.364,368-371... 

We should note that if g = ge, the contact shift is isotropic (independent of orientation). If g is different from ge and anisotropic (see Section 1.4), then the contact shift is also anisotropic. The anisotropy of the shift is due to the fact that (1) the energy spreading of the Zeeman levels is different for each orientation (see Fig. 1.16), and therefore the value of (Sz) will be orientation dependent and (2) the values of (5, A/s Sz S, Ms) of Eq. (1.31) are orientation dependent as the result of efficient spin-orbit coupling. On the contrary, the contact coupling constant A is a constant whose value does not depend on the molecular orientation. 

The crystal field [57], which contains both electrostatic and hopping contributions [58], acts on the orbits of the inner d and f electrons. That is, the electron orbits reflect the anisotropic crystalline environment, and adding spin-orbit coupling translates this anisotropic electron motion into magnetic anisotropy. 

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