Valence spin-orbit coupling

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]. 

The Wlc total atomization energy at 0 K of aniline, 1468.7 kcal/mol, is in satisfying agreement with the value obtained from heats of formation in the NIST WebBook 39), 1467.7 0.7 kcal/mol. (Most of the uncertainty derives from the heat of vaporization of graphite.) The various contributions to this result are (in kcal/mol) SCF limit 1144.4, valence CCSD correlation energy limit 359.0, connected triple excitations 31.7, inner shell correlation 7.6, scalar relativistic effects -1.2, atomic spin-orbit coupling -0.5 kcal/mol. Extrapolations account for 0.6, 12.1, and 2.5 kcal/mol, respectively, out of the three first contributions. 

For the heavier elements of the Periodic Table, say the third transition series and the actinoids, the approximation that spin—orbit coupling is so small it can be treated as a perturbation on free-ion terms fails. Spin-orbit coupling rises rapidly with nuclear charge while interelectronic repulsion terms decrease with the diffuseness of the valence electron density of larger atoms. 

Matrix elements for the valence functions were taken with the effective core potential the coulomb and exchange terms were handled exactly, numerically, without any parameterization and a Phillips-Kleinman projection operator term was also used. Spin-orbit coupling effects amongst the valence orbitals were treated semi-empirically using the operator... 

Another series of composite computational methods, Weizmann-n (Wn), with n = 1-4, have been recently proposed by Martin and co-workers W1 and W2 in 1999 and W3 and W4 in 2004. These models are particularly accurate for thermochemical calculations and they aim at approximating the CBS limit at the CCSD(T) level of theory. In all Wn methods, the core-valence correlations, spin-orbit couplings, and relativistic effects are explicitly included. Note that in G2, for instance, the single-points are performed with the frozen core (FC) approximation, which was discussed in the previous section. In other words, there is no core-valence effect in the G2 theory. Meanwhile, in G3, the corevalence correlation is calculated at the MP2 level with a valence basis set. In the Wn methods, the core-valence correlation is done at the more advanced CCSD(T) level with a specially designed core-valence basis set. 

The inclusion of relativistic corrections in the GGA thus does not resolve the problems of the GGA with the 5d transition metals, suggesting that the nonlocal contributions to Exc n beyond the first density gradient are important in these systems. In addition, the spin-orbit coupling of the valence electrons, neglected in this work, could be partially responsible for this discrepancy [25,30]. 

Figure 2 Comparison of AVBbuik, AVBC]asttT, and the AVB K1(iei, where AVBI11D(iei was obtained using eq. 1 with an optimal AU value of 3.1 eV. Here AVB U1 kand AVBduster have been area normalized. Insert Illustration of the spin-orbital coupling effects in the X-ray absorption L2)3 edge spectra and 5d valence band.

Table 3.12. Crystal-field splitting (Acf) and spin-orbit coupling (Aso) energies for ZnO single crystal bulk samples (b) and ZnO thin films (f) depending on the assumed valence-band orderinga...

FIGURE 6 Energy splitting at the top of the valence bands of GaN under the influence of crystal-field and spin-orbit coupling. The figure is not drawn to scale. 

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