Effect of spin-orbit coupling

The spin-allowed d-d bands which dominate the visible spectrum of complexes of many transition metal ions are rather broad, with half-widths of ca. 3000 cm This means that if there is some interaction within the system which has been overlooked, its presence will not be apparent from the electronic spectrum unless it causes splittings in the energy levels of perhaps 1000 cm In fact, two such effects have been omitted—spin-orbit coupling and the Jahn-Teller effect. 

Data are adapted from Handbook of Atomic Data by S. Fraga, J. Karawowski and K. M. S. Saxena, Elsevier, Amsterdam, 1976. 

A more recent compilation is given by J. Bendix, M. Brorson and C. E. Schaffer, Inorg. Chem. (1993) 32, 2838 but the definitions that they adopt, although with merit, are not those generally encountered in the literature. 

This is why our discussion of crystal and ligand theories has been exemplified by complexes formed by elements of the first transition series. Had we included spin-orbit coupling within the crystal field model (and in a more complete treatment this would have been done), within the crystal field model it would have appeared with its free-ion value. Not surprisingly, in ligand field theory it becomes a parameter which, characteristically, is found to have a value somewhat lower than that found for the free ion. 

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Owing to the effects of spin-orbit coupling all the triplet terms, except S, are split into three components. For example, in the case of a term, with L = I and S = l,J can take the values 2, 1, 0 (Equation 7.15). 

A corresponding formula (Eq. 5.8), due to Van Vleck, has been derived for free atoms in which the effects of spin-orbit coupling can be ignored. 

In this section the effect of spin-orbit coupling on radiative and radiationless intercombinational transitions (transitions occurring between states of different multiplicity) will be investigated. We will be particularly concerned with the use of internal and external heavy atoms to induce spin-orbit coupling. The effect of heavy atoms on intercombinational processes occurring in aromatic hydrocarbons, carbonyl compounds, and heterocyclic compounds will be discussed. 

The labelling of terms as S,L,J,Mj) is preferable when one takes into account the effect of spin-orbit coupling, since / and Mj remain good quantum numbers even after this perturbation is accounted for. In detail, the effect of spin-orbit coupling over a many-electron atomic term is evaluated by writing the spin-orbit operator in terms of the total angular and spin momentum, L and 5 ... 

In Equation 1.5, A is the spin-orbit coupling within a given Russell-Saunders multiplet, which is related to the spin-orbit coupling constant of the ion, C, by the relation A = f/2.S, the + sign applying for n <7 and the - sign for n > 7 [ 15]. The effect of spin-orbit coupling is to split the terms in multiplets with same L,... 

The effects of spin-orbit coupling on geometric phase may be illustrated by imagining the vibronic coupling between the two Kramers doublets arising from a 2E state, spin-orbit coupled to one of symmetry 2A. The formulation given below follows Stone [24]. The four 2E components are denoted by e, a), e a), e+ 3), c p), and those of 2A by coa), cop). The spin-orbit coupling operator has nonzero matrix elements... 

The calculations gave a bond distance of 2.43 A and a bond energy of about 35 kcal/mol, including the effects of spin-orbit coupling. An experimental value of 50 5 kcal/mol was reported in 1974.88... 

Table 4. CASPT2 excitation energies for the Pt atom including the effect of spin-orbit coupling...

If we consider the effects of spin-orbit coupling, we see that in many ways it is analogous to the influence exerted by a magnetic field. Similarly to before, three terms in Eq. (5) depend on spin-orbit coupling, the excitation energy toj (and thus /), the transition dipoles, and the populations of the components of the ground state. 

The model describes the simultaneous perturbation of the sixfold degenerate 2T2 basis by the effects of spin-orbit coupling and an axially symmetric ligand field distortion,... 

The correctness of the sequence of ionic states based on the CNDO ASCF calculations has been confirmed by studying the effect of spin—orbit coupling on the UPS of the rhenium analog HRe(CO)5 (87, 161, 174). To consider the effects of spin-orbit coupling it is necessary to employ the double group C . 

Example 8.2-1 Examine the effect of spin-orbit coupling on the states that result from an intermediate field of O symmetry on the Russell-Saunders term 4F. Correlate these states with those produced by the effect of a weak crystal field of the same symmetry on the components produced by spin-orbit coupling on the 4F multiplet. 

The solution is summarized in Figure 8.1. The4F state has L 3, and so (from Tabic 7.1) it is split by an intermediate field into three states which belong to the IRs A2 Ti T2 =r2 r4 r5. To examine the effect of spin—orbit coupling on these intermediate-field states, we use the fact that if -tp fi"fi, where fi forms a basis for T and x1 forms a basis for F, then ip = fix forms a basis for the direct product (DP) representation F F. Here S=3/2, and the representation 1 / is Tg (Table 8.2). Take the DPs of Tg with 1 2, T4, and P to obtain... 

Investigate the effect of spin-orbit coupling on the crystal-field levels of a Ce3+ ion substituting for Ca2 + in CaF2 with a nearest-neighbor O2 ion (see Problem 7.2). Is any further splitting of these levels to be expected if the site symmetry at Ce3+ is lowered to Cs by a further crystal-field perturbation that is weaker than //s.i. ... 

Several simple models exist5 that approximately describe the temperature dependence of x for transition metal cations that do not represent spin-only centers. As one example that is applicable to coordination complexes at low temperatures, the Kotani theory6 incorporates the effects of spin-orbit coupling into the Van Vleck equation and describes y(T) as a function of the spin-orbit coupling energy C,. 

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