Relativistic spin-orbit splitting

Relativistic mean field (RMF) models have been applied successfully to describe properties of rinite nuclei. In general ground state energies, spin-orbit splittings, etc. can be described well in terms of a few parameters ref. [18]. Recently it has lead to the suggestion that the bulk SE is strongly correlated with the neutron skin [19, 20] (see below). In essence the method is based upon the use of energy-density functional (EDF) theory. 

Z, as determined from spin-orbit splittings in atomic spectra (5). The Jtf is small relativistic correction, important mainly for heavy atoms (Z > 50). The values for the elements with the p configuration (N, P, As, Sb) were obtained by interpolation. (Taken from Barnes and Smith (5), appeared also in Jameson and Gutowsky (35)]. 

The only term depending upon % is the last on the left hand side and this is just the spin-orbit couphng. If this term is dropped x-independent solutions, corresponding to zero spin-orbit couphng, are obtained. However, the large relativistic shifts, the mass velocity and Darwin terms are retained. In Fig. 3, for example, the relativistic levels remain in place but each of the spin-orbit split pairs is replaced by the average energy level... 

Recently we developed a new approach which improves the sensitivity to a variation of a by more than an order of magnitude [1,2]. The relative value of any relativistic corrections to atomic transition frequencies is proportional to a2. These corrections can exceed the fine structure interval between the excited levels by an order of magnitude (for example, an s-wave electron does not have the spin-orbit splitting but it has the maximal relativistic correction to energy). The relativistic corrections vary very strongly from atom to atom and can have opposite signs in different transitions (for example, in s-p and d-p transitions). Thus, any variation of a could be revealed by comparing different transitions in different atoms in cosmic and laboratory spectra. 

Fig. 6. 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 DF atomic energies are from [23] and the HF values are from [17].

The photoionisation spectrum of the OSO4 molecule is one example were we definitely see the relativistic influence. Fig. 5 compares the experimental results with our DVM-DFS method and extensive HF-CI calculations. The splitting of the 3tj levels (originating from the spin-orbit splitting of the 5d-wavefunctions of Os) are quite well-reproduced. 

The valence structure of xenon is similar to that of argon in that the valence p manifolds each have one state with a spectroscopic factor near unity and the inner-valence s manifold is severely split. The additional feature of xenon is the possibility of testing relativistic calculations of the orbitals. The spin—orbit splitting of the Sp /2 and 5pi/2 manifolds can be experimentally resolved. 

Thus, the main relativistic effects are (1) the radical contraction and energetic stabilization of the s and p orbitals which in turn induce the radial expansion and energetic destabilization of the outer d and f orbitals, and (2) the well-known spin-orbit splitting. These effects will be pronounced upon going from As to Sb to Bi. Associated with effect (1), it is interesting to note that the Bi atom has a tendency to form compounds in which Bi is trivalent with the 6s 6p valence configuration. For this tendency of the 6s electron pair to remain formally unoxidized in bismuth compounds (i.e. core-like nature of the 6s electrons), the term inert pair effect or nonhybridization effect has been often used for a reasonable explanation. In this context, the relatively inert 4s pair of the As atom (compared with the 5s pair of Sb) may be ascribed to the stabilization due to the d-block contraction , rather than effect (1) . On the other hand, effect (2) plays an important role in the electronic and spectroscopic properties of atoms and molecules especially in the open-shell states. It not only splits the electronic states but also mixes the states which would not mix in the absence of spin-orbit interaction. As an example, it was calculated that even the ground state ( 2 " ) of Bij is 25% contaminated by Hg. In the Pauli Hamiltonian approximation there is one more relativistic effect called the Dawin term. This will tend to counteract partially the mass-velocity effect. 

As discussed in section 2.8, relativistic effects on the valence electronic structure of atoms are dominated by spin-orbit splitting of (nl) states into (nlj) subshells, and stabilization of s- and p-states relative to d- and f-states. Here we examine consequences of relativistic interactions for cubo-octahedral metal-cluster complexes of the type [M6X8Xfi] where M=Mo, Nb, W and X=halogen, which have a well defined solution chemistry, and are building blocks for many interesting crystal structures. 

It was not until the 1970s that the full relevance of relativistic effects in heavy-element chemistry was discovered. However, for the sixth row (W---Bi), relativistic effects are comparable to usual shell-structure effects and therefore provide an explanation for many unusual properties of gold chemistry155-159. The main effects on atomic orbitals are (i) the relativistic radial contraction and energetic stabilization of the s and p shells, (ii) the spin-orbit splitting and (iii) the relativistic radial expansion and energetic destabilization of the outer d and f shells. 

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