Spin-orbit splitting Hartree-Fock

The existence of an anomaly in the spectrum of Ba+ had been known for many years. It was commented on by Saunders et al. [213], who remarked the perturbation. .. appears to be of a novel type but could advance no explanation for it. The realisation that this is due to a near-critical potential barrier effect came much later [212]. Hartree-Fock calculations were used to establish that the n/ wavefunctions of Ba+ become resonantly localised in the inner well of a double-well potential around n = 5. As a consequence, all the low n orbitals become distinctly bimodal or hybrid in character (as can be seen in fig. 5.12) with peaks in both the inner and the outer potential wells. This results in (i) an unusual course in the quantum defects for the nf series they are not constant, but depend strongly on the energy, despite the absence of any local perturbation (ii) a pronounced intensity anomaly in the 5d — nf excitation series, which was first noted by Roig and Tondello [214] and (iii) an anomaly in the course of the spin-orbit splittings for the first members of the series, both of which can be explained on the basis of the bimodal character of the... 

Fig. 5.18. Spin-orbit splittings for the nf series of Ba+, showing that good agreement is obtained between g Hartree and experimental values. For comparison, Dirac-Fock (labelled DHF) and multiconfigurational Dirac-Fock (labelled MCDF) curves computed from the same code are also shown (after J.-P. Con-nerade and K. Dietz [230]).

Table 14.4 Total electronic and selected orbital energies obtained within the Hartree-Fock model tor the neutral Rn atom from Ret. [16], The relative errors (in %) with respect to the tour-component Dirac-Coulomb (DC) reterence values are presented in parentheses. The SO (spin-orbit splitting) entry is the energy difference between orbitals =/+l/2 and y=/-l/2.

This pattern is indeed followed quite faithfully when the principal quantum number n is 2, 3, and 4. For n = 5 there is a marked reduction in the increase from p to p compared to the previous series, but the pattern still holds. For = 6, however, we observe a decrease of ionization potential from p to p and an increase from p to p. These trends are depicted in figure 1.1, where the experimental values are shown with predictions Ifom nonrelativistic Hartree-Fock calculations. It is clear that the nonrelativistic calculations do not produce the correct trend. The explanation lies in the spin-orbit splitting of the p shell. 

Fig. 5 Relativistic stabilization of the ns and npi/2 orbitals and the spin-orbit splitting of the np orbitals for the noble gases Xe, Rn and element 118. The Dirac-Fock atomic energies are from [21] and the Hartree-Fock (nonrelativistic) values are from [8]...

These were calculated by the method of Barnes and Smith from spin-orbit splittings in atomic spectra without relativistic correction (open circles). Values calculated from relativistic Hartree-Fock-Slater atomic wave functions are included for comparison (filled circles). 

Table 3.3 Spin-orbit splitting of orbital levels of the uranium atom obtained with different relativistic Hamiltonians and Hartree-Fock calculations averaging over the [Rn]5P6d iP ground state configuration...

As with ligand-field absorption spectra, the Nephelauxetic effect [41, 42] will also impact L-edge XAS data. For L-edge spectra, the effect of a reduction in interelectron repulsion is also to reduce the ffee-ion state splitting and make the spectra somewhat more orbital like. This is demonstrated by the sequence of simulated low-spin Fe(III) L-edge X-ray absorption spectra shown in Fig. 13, where the e - e repulsion is reduced systematically from [i = 100% to 60%. This starts from 80% of the Hartree-Fock calculated values of the Slater Condon Shortley parameters for e - e repulsion. 

If spin-orbit effects are considered in ECP calculations, additional complications for the choice of the valence basis sets arise, especially when the radial shape of the / -f-1/2- and / — 1/2-spinors differs significantly. A noticeable influence of spin-orbit interaction on the radial shape may even be present in medium-heavy elements as 53I, as it is seen from Fig. 21. In many computational schemes the orbitals used in correlated calculations are generated in scalar-relativistic calculations, spin-orbit terms being included at the Cl step [244] or even after the Cl step [245,246]. It therefore appears reasonable to determine also the basis set contraction coefficients in scalar-relativistic calculations. Table 9 probes the performance of such basis sets for the fine structure splitting of the 531 P ground state in Kramers-restricted Hartree-Fock [247] and subsequent MRCI calculations [248-250], which allow the largest flexibility of... 

State (Thole et al. 1985) for all the rare-earth ions and for all ionization states known to be relevant to the solid state. The electrostatic and exchange parameters were all scaled down to 80% of their Hartree-Fock (HF) values. The spin-orbit parameter was adjusted by a factor to correct the energy splitting between 3d /2 and 3d3/2 peaks. The resulting line intensities were broadened by a life-time broadening function for comparison with the observed 3d-4f spectra from metallic rare-earth samples. We omit here details of the experimental and theoretical procedures which can be foimd in the paper by Thole et al. (1985). 

The configuration sharing on the basis 4s4p2+4s24d+4s25s has been evaluated by means of the Slater-Condon parametric method. The spin-orbit constants derived from experiment are compared with ab initio (Hartree-Fock) and with semiempirical predictions [4]. The ground term splitting has been also calculated by means of the Dirac-Fock method [7]. 

Sulphur Heterocyclic Radicals.—Most studies in this field have used theoretical methods to derive the spin-density distribution and hyperfine splitting constants. In a few cases the unrestricted Hartree-Fock (UHF) formalism has been applied (PPP approximation). (The inclusion of d-orbitals slightly improved the spin-density results in the p-model. )... 

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