First-order spin-orbit splitting

Only spatially degenerate states exhibit a first-order zero-field splitting. This condition restricts the phenomenon to atoms, diatomics, and highly symmetric polyatomic molecules. For a comparison with experiment, computed matrix elements of one or the other microscopic spin-orbit Hamiltonian have to be equated with those of a phenomenological operator. One has to be aware of the fact, however, that experimentally determined parameters are effective ones and may contain second-order contributions. Second-order SOC may be large, particularly in heavy element compounds. As discussed in the next section, it is not always distinguishable from first-order effects. 

A phenomenological spin-orbit Hamiltonian, formulated in terms of tensor operators, was presented already in the subsection on tensor operators. Few experimentalists utilize an effective Hamiltonian of this form (see Eq. [159]). Instead, shift operators are used to represent space and spin angular 

For the sign of the spin-orbit parameter Aso, the following conventions apply  

Typically, states with less than half-filled shells (particle states) are regular, whereas states with more than half-filled shells (hole states) are inverted. 

To determine the first-order splitting pattern of an atomic state in terms of the phenomenological spin-orbit parameter Aso [Eq. 203], we utilize 

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When A > 0 the first excited level is +( ). Now the above discussed first-order spin-orbital splitting of 2 -term occurs for the excited levels ( ), +(3) that are parallel in the strong crystal field limit. 

The operator [157] is a phenomenological spin-orbit operator. In addition to being useful for symmetry considerations, Eq. [157] can be utilized for setting up a connection between theoretically and experimentally determined fine-structure splittings via the so-called spin-orbit parameter Aso (see the later section on first-order spin-orbit splitting). In terms of its tensor components, the phenomenological spin-orbit Hamiltonian reads... 

To determine the first-order spin-orbit splitting pattern of an orbitally degenerate electronic state, we shall make use of the energy expression obtained from the phenomenological operator, which in this case reduces to Aso A S because only the z component of the spin-orbit operator is involved. 

Table 14 First-Order Spin-Orbit Splitting of a 3II State...

The ground state of boron exhibits first-order spin-orbit splitting, the energy of the P3/2 level being 16 cm" above the ground-state level P /2 [6]. Determine the first-order spin-orbit correction A so( Pi/2). 

HOMO-LUMO gap stays less than 7% with respect to the SR values. In Figure 15.15, scalar relativistic and first-order spin-orbit split valence molecular orbitals levels are shown. 

Figure 15.15 Scalar relativistic (left-hand side) and first-order spin-orbit split (right-hand side) valence molecular orbital levels for Pu + C28 (T )...

Spin-Spin Coupling Although we focus on spin-orbit coupling (SOC), we need to consider the tensorial structure of electronic spin-spin coupling Second-order SOC mimics perfectly first-order spin-spin coupling and vice versa, so that they cannot be told apart (see the later section on second-order spin-orbit splitting). 

We therefore turn to the second triplet possibility, EJin d ). First-order spin-orbit coupling splits this orbitally degenerate state as sketched below. We include the D h symmetry labels and also values of the cylindrical quantum number which measures the total parallel angular momentum [22]. 

In any future reexamination of the spectrum of the ReFg anion it may be anticipated that a moderately strong band should be found between ca. 5-10 kK., since the spin-orbit splitting of the zT g ground level again amounts to 3/2 f in the first order. 

We consider first the MCD caused by the spin-orbit splitting of the excited state. If an excited state has both spin and orbital degeneracy then it will be split to first order in the spin-orbit coupling. We denote that splitting by... 

Is there any first-order fine-structure splitting in the electronic ground state, due to either electronic spin-orbit or spin-spin coupling ... 

Exploiting the fact that in first-order perturbation theory all sublevels exhibit identical J 2 and S1 eigenvalues, respectively, yields a spin-orbit splitting of... 

Fig. 12. First-order splittings for Fei Vc > Vls- The spin-orbit splittings are drawn to an arbitrary scale. The numbers indicate the total degeneracy, (a) d2 in octahedral interstice, (b) d8 in octahedral interstice, (c) d8 in tetrahedral interstice, (d) d2 in tetrahedral interstice...

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