Triplet terms

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). 

The splitting of triplet terms of helium is unusual in two respects. First, multiplets may be inverted and, second, the splittings of the multiplet components do not obey the splitting rule of Equation (7.20). For this reason we shall discuss fine stmcture due to spin-orbit coupling in the context of the alkaline earth atomic spectra where multiplets are usually normal and... 

These transitions correspond to the electronic promotion —> with the promoted electron maintaining it.s- spin unaltered. The orbital multiplicity of the configuration i.s 6 and so corresponds to two orbital triplet terms Ti, and Tjg- If, on the other hand, the promoted electron changes its spin, the orbital multiplicity is again 6 but the two T terms arc now spin triplets, T g and A weak band attributable to the spin-forbidden Mi, transition is indeed... 

Figure 3-9. Four spin-triplet terms arise for...

Figure 3-16. Octahedral field spin-triplet terms arising for d ...

Figure 3-17. A partial correlation diagram for d showing weak- (left) and strong-field (right) limits for spin-triplet terms.

Some of the corresponding combinations of singlet terms and of intercombinations between singlet and triplet terms of Pb III have been found by the writer. A more comprehensive account of the spectra of mercurylike atoms will be given at a later date. 

For certainty, assume that, as in the earlier example of Section 2, the JT term is the orbital triplet T2a of Oh. The matrix JT Hamiltonian of the orbital triplet term linearly coupled to a threefold degenerate t2g mode can be written as ... 

The other two components of the vibronic triplet term, transforming as y and z, can be obtained by applying symmetry operations of the group Oh to the x component in equation (31). Thus, as in the example of Section 4, the junction rule gives the same result as the symmetry projection techniques. The corresponding eigenvalues are... 

Now we turn our attention to the d2 octahedral complexes. For this configuration, there are two triplet terms, 3F (ground term) and 3P. In an octahedral crystal field, there are three configurations t g < t e < e2, in increasing... 

Note that the energies of the triplet terms can be obtained independently using diagonal-sum rule with the microstates Ms = +1... 

It is interesting to note that the triplet terms have only one parameter 3 other than the common additive constant °, similar to the d electron configurations where the terms of highest spin-multiplicity of a dN configuration have only one parameter B. 

According to the restricted accuracy of the CASPT2 calculations [66], it is not excluded that the substates of a different triplet term, namely of 82, can become the lowest ones under inclusion of spin-orbit coupling. 82 would lead to Aj, A2, and 8 substate symmetries in Cjv- In this case, transitions between the ground state A and the sublevels of symmetries A and 81 are formally dipole allowed. With this alternative assignment one would also expect to have two allowed transitions and one forbidden transition. 

Figure 7.5 A correlation diagram showing the effect of octahedral ligand field strength on the energies of the triplet terms of a d metal ion...

The third lucky case is the so-called T g gg problem in a cubic system. The JT instability of the orbital triplet term (the T term) is due to its linear coupling to twofold degenerate vibrations, Qe and Qe, shown in Fig. 3b and c [1,2]. As it is similar to the tetragonal E g b g case, for this cooperative JT effect we do not go into its details. Just note that there are three minimum points in this case and there are three matrix positions on the main diagonal allowing the respective simultaneous coordinate shift to these three vertices. 

A further approximation to the GVB wavefunction is the deletion of the triplet term in Eq. (205) for each geminal. This results in the perfect pairing... 

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