Multiplets Spin-orbit coupling

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... 

Also shown in Figure 8.15 is the line due to removal of a 2p electron from the sulphur atom in thiophene. Spin-orbit coupling is sufficient to split the resulting core term, as it is called, into 3/2 and 1/2 states, the multiplet being inverted. 

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,... 

Transition-metal and rare-earth atoms that contain partially occupied d or f valence subshells also give rise to spectral tine structure, often with very complicated multiplet splitting [2,27,28]. The spin-unpaired valence d or f electrons can undergo spin-orbit coupling with the unpaired core electron (remaining in the orbital from which the photoelectron was removed), producing multiple non-degenerate final states manifested by broad photoelectron peaks [2,27]. 

In the presence of Coulomb correlation only, the wave function is characterized by the total spin S = SSj and the total angular momentum L = 2,1 of the 5 f electrons, and the total momentum J is given by Hund s rule (J = L S). Important spin orbit coupling will mix LS multiplets and only J remains a good quantum number. The Russell-Saunders coupling scheme is no longer valid and an intermediate coupling scheme is more appropriate. 

There is a deviation of the multiplet shape from the one found for Sm. This is explained by the change of the main quantum number and the strong spin-orbit coupling surface effects as broadening and shift of the multiplet as well as a contribution from a divalent surface can also be invoked. 

Localized d or/electrons retain their one-atom manifolds, except that states arising from different d" or f are split by crystal field and spin-orbit coupling. Multiplet splittings due to spin-orbit coupling are larger than crystal-field splittings of Af levels the converse is the case for 3d" levels. The difference in energy between d"(f) and d" (/ 1) manifolds corresponds to the amount of free atom U that is decreased due to interatomic interaction in the solid. 

The broadly successful application of these formulas to the paramagnetism of lanthanide complexes was due to the wide multiplet widths in the/block metals (large spin—orbit coupling coefficients X) and to the small effect of the ligand field on the deep-lying / orbitals. No comparably useful formula for the magnetic moments of d block complexes exists, except perhaps for the spin-only formula ... 

The spin-orbit coupling term in the Hamiltonian induces the coupling of the orbital and spin angular momenta to give a total angular momentum J = L + S. This results in a splitting of the Russell-Saunders multiplets into their components, each of which is labeled by the appropriate value of the total angular momentum quantum number J. The character of the matrix representative (MR) of the operator R(0 n) in the coupled representation is... 

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. 

In a weak m3 field the 2D multiplet is split by spin-orbit coupling into its 2Dy2(4) and 2Ds/2(6) components. The 2D3/2 state is 6(4) while the 2Ds/2 state sphts into s(2) and 6(4) components, which correlate with the intermediate field case in eq. (21). 

The first excited CFT 2 g is well separated from the ground 2T2g by the energy A0 = 10Dq. The spin-orbit coupling splits the ground term into / (ground) and / 7 multiplets separated by S0/hc = 237 cm 1, which results from the calculations in the complete d1 space spanned by ten functions (Table 70). 

Due to the increased spin-orbit coupling, the first excited CFT is split sizably and thus some multiplets span a thermally accessible region. The calculated energies of the lowest multiplets (in cm1) are ... 

Fig. 1. (a) Molecular structure of [Mn(taa)] and (b) splitting of ground multiplet under perturbations C3 twist (C3), spin-orbit coupling (LS), and E [Pg.618]

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