Relativistic shift

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... 

The total relativistic shift of frequencies (e.g. the largest s-electron shift) is included. 

The largest relativistic shift in the ground state is included. 

Many possibilities to search for systematic errors. For example, we can exclude any line or atom to avoid possible effects of unknown line blending, calibration errors, etc. The opposite signs and different values of the relativistic shifts for different lines give us a very efficient method to control the systematic effects. 

Fig. 4. Relativistic shifts for s d"-s d" excitation energies of transition-metal atoms. Differences between excitation energies from Hartree-Fock and relativistic Hartree-Fock calculations are plotted. (Reproducedfrom Ref 174 by permission of the authors and the American Institute of Physics.)...

Darwin) corrections at the Hartree-Fock level. Their results for the d" s d" s separation are shown on Fig. 4. While the common wisdom is roughly verified, the relativistic shifts for the later members of the 3d series are probably larger than many would have expected. They certainly cannot be neglected if very high accuracy is required. (A similar statement applies to molecules (see e.g. Cuj below).) Of course Fig. 4 assumes that the relativistic correction can be separated from correlation effects. Martin and Hay reasoned that, since the relativistic correction scales roughly as the valence s-orbital population, correlation effects which change this population would have the most important influence. By using the Cl coefficient for the most important (s -> p ) contribution they estimated that, for Ni, correlation could reduce the relativistic correction of 0.35 eV by about 0.07 eV (probably an upper limit). 

Solids may be structurally disordered or crystalline. Perfect crystals with completely periodic structures do not exist in Nature. However, most of the discussion here will be based on such idealiized models, and the electronic structure is described in terms of band structures, dispersion relations between formal one-electron energies, s, and wavevectors, k e(k). First, in Section 2, we illustrate how the relativistic shifts (mass-velocity and Darwin) of parts of the band structure with respect to each other may affect the physical properties, including the crystal structure. The second subject (Section 3) treated in this chapter concerns the simultaneous influence of the crystal symmetry and the SO-coupling on e(fe), spin splitting effects, i.e. effects which are without atomic counterparts. 

For solids with heavy atoms, relativistic shifts may affect the bonding properties, and also optical properties may be influenced. The relativistic shifts of the 5d bands relative to the s-p bands in gold change the main inter band edge more than 1 eV. Already Pyykko and Desclaux mentioned [1] that the fact that gold is yellow is a result of relativistic effects. These are indirect [2] (see also the introduction. Sect. 1), and the picture was confirmed by relativistic band structure calculations [3,4]. Also the optical properties of semiconductors are influenced by relativistic shifts which affect the gap between occupied and empty states, see for example Ref. [5]. Two additional examples may be mentioned where relativistic shifts in the energy band structure drastically influence the physical properties. First,... 

Figure 1. Self-consistent scalar-relativistic band structme for CsAu. The Cs-5s and -5p states were included as band states (not shown). In a band calculation without relativistic shifts included there is no gap at R, and in that model CsAu would be a (semi-) metal.

A more recent example of how relativistic shifts of the bands can influence the crystal structme of a solid was presented by Sbhnel et al. [12] who performed ab initio calculations for gold halides. By comparing relativistic to non-relativistic calculations it was found that the fact that Au compounds assume chain-like structures and not (like Cu- and Ag halides) cubic (or hexagonal) structures is indeed a result of relativistic shifts, mainly of Au-6s and -6p states. 

There are cases where the inclusion of the relativistic shifts of the bands is essential, but where inclusion of spin-orbit coupling is not needed. In... 

There are three main effects of relativity on the electronic (band) structure (i) scalar-relativistic shift of bands, frequently connected with a considerable change of the band width in comparison with the related non-relativistic calculation (ii) spin-orbit (s-o) splitting of degenerate band states, most notably in the vicinity of high-symmetry points in fe-space (iii) in combination with spin polarization that breaks the time-inversion symmetry, s-o coupling may reduce the crystal symmetry. 

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