Multiplet energies

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. 

Fig.5.25, Distance-dependent multiplet-energy diagram for Mn +, Cr + and (Ishii et al.

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... 

Fig. 8. Energy of the 3 Pr. multiplet of I. aCl3 Pr3 as a function of the host lattice equatorial and apical La—Cl distances (from Gregorian et al., 1989). The circles correspond to the multiplet energies of Pr3+ in RC13 (R = La, Pr, Nd, Gd) and the respective host-lattice bond distances at ambient pressure. Triangles denote the high-pressure...

First-principles calculations of multiplet 4.1.2. Multiplet energies with introduc- ... 

By solving this, the multiplet energies E and the coefficients of the many-electron wave functions are obtained. 

Multiplet energies with introduction of reduction factors... 

Fig. 4. Calculated multiplet energy levels from 0 to 50000 cm-1 obtained by first-principles calculations (left), first-principles calculations considering scaling factors (center) and semiempirical calculations (right) for each trivalent...

The first-principles results corrected with the scaling factor are shown in figs. 4 and 5 together with the first-principles results without corrections and the semiempirical results. As shown in these figures the calculated multiplet energies are significantly improved by introduction of the scaling factor. 

The theoretical spectra considering the lattice relaxation effects are shown in fig. 12. The separation between peaks decreases upon expansion of the bond lengths. For example, the separation between peaks A and B is 1.03 eV, 0.87 eV, 0.75 eV for 0%, +2%, +4% relaxed models, respectively. Comparing with the experimental spectrum, the multiplet energy splitting is reproduced best of all for the +2% relaxed model. Contrary to the peak positions, the peak intensities are not significantly influenced by the bonds lengthening. 

Fig. 13. Theoretical multiplet energy levels and their configuration compositions for isolated Pr3+ ion.

Fig. 16. Configuration analysis of 4f5d multiplet energy levels for Pr3+ LiYF4.

The theoretical spectra for the three relaxed models are shown in fig. 24. As the bond lengths increase, the separations of the peaks decrease. The theoretical multiplet energy levels for these three models are compared in fig. 25. As shown in the previous section, the separation between peak a and peaks 02, <23 originates from the spin-oibit splittings of the Pr 4f levels which are distinguished by red and blue lines. The origin of the separation between peaks <24,... 

Furthermore, the intensities of peaks 4/4. an are almost unchanged while there is a reversal of the relative intensities of the two peaks denoted as a<, as the bond lengths increase. As shown in this example, the Pr-F bond lengths influence not only the multiplet energy levels but also the peak intensities. 

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