Energies of multiplets

The spin-orbit coupling in the atomic system manifests itself in a further splitting of atomic terms into multiplets (Table 8.5). 

The orbital angular momentum /, adds to the spin angular momentum , for each electron and yields the total angular momentum of an electron 

The quantisation of the total angular momentum for a given electron is given as 

The corresponding spin-orbitals are labelled according to Table 8.6. 

The angular momenta for individual electrons can be summed up to yield the total angular momentum of an atom 

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Table 19-2. Matrices for the energies of multiplets of a tetrahedral d complex"...

Ziegler T, Rauk A and Baerends E J 1977 On the calculation of multiplet energies by the Hartree-Fock-Slater method Theor. Chim. Acta 43 261-71... 

Ziegler, T., Rauk, A., Baerends, E. J., 1977, On the Calculation of Multiplet Energies by the Hartree-Fock-Slater Method , Theor. Chim. Acta, 43, 261. 

When the network chains contain ionic groups, there will be additional forces that affect their swelling properties. Translational entropy of counterions, Coulomb interactions, and ion pair multiplets are forces that lead to interesting phenomena in ion-containing gels. These phenomena were studied in detail by Khokhlov and collaborators [74-77]. The free energy of the networks used by this group is... 

Table 6.1 Energies of the low-lying J-multiplets predicted within the CASSCF/RASSI approach for the free Ln3+ ions. ANO-RCC of single-zeta quality were employed. The active space of the CASSCF method included only n electrons spanning the 4f shell. 

Table 6.2 Energies of the lowest spin-free states originating from the SH multiplet and the energies of the low-lying Kramers doublets of the DyZn3 complex.

As was mentioned in Section 6.2.3, phenomenological CF models have been extensively used in the past to rationalize the electronic structure of lanthanide complexes [4, 40-50], and they are still in use nowadays [38, 39, 78, 79]. The ab initio approach described above is, in principle, able to provide accurate energies and wave functions of multiplets in lanthanide complexes. Nevertheless, even if ab initio calculations appear to be successful, CF description is still of great... 

Dysprosium activated minerals have liuninescence in the visible part of the spectrum. The spectra of Dy + in minerals are mainly characterized by narrow fines near 480 and 575 nm, accompanied by the weaker ones near 660 and 752 nm corresponding to transitions from level F j2 to the levels of multiplets Hj and (Tarashchan 1978). Consequently, the spectra are not changed with delay time and excitation energy and all luminescence fines of Dy " are characterized by decay time. The best excitation at 350 nm is connected with 4/-4/ ffi5/2- G7 2 transition. Such liuninescence is detected in... 

It has been empirically known that the energies of the lowest excited state of tetrahedrally coordinated metals decrease in the order Cr + < Mn + < Fe ". As in the case of 3cf elements, this tendency has been considered to originate from the difference in covalency, which reduced two-electron repulsion between the electrons occupying 3d orbitals. Recently this question was treated using first-principles electronic-structure calculation (Ishii et al. 2002). The same tendencies were found as for the 3d ions. Distance dependent multiplet-energy diagrams for these elements have been obtained (Fig. 5.34), which enable us to envisage the typical shapes of the possible emissions. As in... 

Figure 6 shows the process in a schematic way. Baer describes the two electron excitations of direct and inverse photoemission as two-step processes in which first Ep is reached and then the emission of an electron (direct) or of a photon (inverse photoemission) to vacuum occurs from Ep. A and A+ are the energies associated with the two first steps. They are counted from Ep, and they are to be considered as the minimum energies necessary to create the f and f final state. If the localized level response is a final state multiplet, therefore, Uh = A+ -I- A is given by the sum of the smallest measured energies of the multiplet. 

We see from the above discussion that staying within the DFT it is not possible to describe the multiplet structure of the d-shell (incidentally all the success stories reported so far are limited to the p-shells [82,113] whereas in the d-shells only average energies of several multiplet states would be reproduced [114]). In this context it seems to be necessary to analyze the attempts to achieve it which are available in the literature (see [87,115]). These attempts, however, have a nature absolutely different from the DFT itself so they will be described in an appropriate place below. 

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 ... 

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