Relativistic Solid State Theory

In general, relativistic effects are less important in the electronic structure theory of solids than in the theory of atoms. The reason is that the physical and chemical properties of solids are mainly related to the valence states, the outermost electrons in the constituent atoms, whereas the relativistic effects are largest near the atomic nuclei. The valence electrons have only small probability amplitudes near the nuclear site, and relativis- 

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. 

Often the electronic spin mainly plays the indirect role through the Pauli principle, but in some cases it is directly the source of physical processes. Magnetism is an obvious example, and the combination of spin- and orbital moments determines the magnetic properties of materials. The theory must therefore be able to treat spin-polarization and SO-coupling simultaneously. This is therefore the third subject dealt with here. Magnetoelastic and magnetooptic effects are related to this and are discussed in Section 5. 

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, 

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. 

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DFT-Based Pseudopotentials. - The model potentials and shape-consistent pseudopotentials as introduced in the previous two sections can be characterized by a Hartree-Fock/Dirac-Hartree-Fock modelling of core-valence interactions and relativistic effects. Now, Hartree-Fock has never been popular in solid-state theory - the method of choice always was density-functional theory (DFT). With the advent of gradient-corrected exchange-correlation functionals, DFT has found a wide application also in molecular physics and quantum chemistry. The question seems natural, therefore Why not base pseudopotentials on DFT rather than HF theory ... 

The Kohn-Sham-Dirac equation (27) introduced in the last section is the basis of most relativistic electronic structure calculations in solid state theory. There are certain aspects which make the numerical solution of this four-component equation more involved than its non-relativistic coimterpart The Hamiltonian of the Kohn-Sham-Dirac equation is, unlike its Schrodinger equivalent and unlike the field-theoretical Hamiltonian (7) with the properly chosen normal order, not bounded below. In the limit of free, non-interacting particles the solutions of the Kohn-Sham-Dirac equation are plane waves with energies e(k) = cVk -I- c, where positive energies correspond to electrons and states with negative energy can be interpreted as positrons. For numerical procedures, which preferably use variational techniques to find electronic solutions, this property of the Dirac operator causes a severe problem, which can be circumvented by certain techniques like the application of a squared Dirac operator or a projection onto the properly chosen electronic states according to their above definition after Eq. (19). 

Eschrig, H., Richter, M., Opahle, I. Relativistic solid state calculations. In Schwerdtfeger, P. (ed.) Relativistic Electronic Structure Theory, Part I, pp. 723-776. Elsevier, Amsterdam (2002)... 

Based on an empirical correlation between adsorption enthalpies of single atoms on Au surfaces with their sublimation enthalpy (see Fig. 38), for Cn a value of A/is = 39( ( kJ moP (= A.% kcal moP ) results [136]. This value is significantly lower compared to a theoretical prediction based on solid-state theory using relativistic Dirac-Kohn-Sham calculations, which predicted that Cn is a semiconductor with a cohesive energy of about 110 kJ moP [138]. 

Applications have been stressed in many recent reviews (see, e.g.. Refs. 8, 10-13), so we shall be brief here, commenting shortly on selected classes of chemically important applications (see also Lanthanides and Actinides and Relativistic Effects of the Superheavy Elements). For applications in solid state theory we refer to an article in this encyclopedia and to recently published literature, ... 

Ebert, H., and Battocletti, M., 1996, Spin and orbital polarized relativistic multiple scattering theory - with applications to Fe, Co, Ni and Fe cCoi- c , Solid State Commun. 98 785. 

The LDA approach originated from solid-state physics where, the Hartree-Fock approximation being less useful, it is mandatory to take electron correlation from the beginning into account, and this is almost always done in the framework of density functional theory. As the non-relativistic density functional had to be changed to take relativity into account [41], the... 

Earlier it was mentioned that the relativistic theory of electronic states in solids in many respects is identical to that of atoms. Since this is well described elsewhere, this section will only deal with some features of specific implementations of the theory in actual calculation methods used for solids, and the importance of relativistic effects — apart from those already discussed — will be illustrated by examples. Although Section 3 did refer to results of LMTO calculations, we did not describe how these included relativity. This section will deal with these items in the form of an overview, and the basic band structure calculations described relate to the density-functional theory [62,63]. Since magnetism is one of the most important solid state physics fields we shall discuss the simultaneous inclusion of spin-polarization and relativistic effects, in particular the spin-orbit coupling. In that context it appears that several of the materials where such effects are particularly large and interesting are those where electron... 

The collection presented here is far from being complete. Extended bibliographies including more than 10.000 references on relativistic theory in chemistry and physics have been published by Pekka Pyykko [32-34]. We took much advantage of his careful and patient work when preparing this chapter. Specialized on solid state effects are recent reviews on magnetooptical Kerr spectra [35] and on density functional theory applied to 4f and 5f elements and metallic compounds [24]. 

The field of relativistic electronic structure theory is generally not part of theoretical chemistry education, and is therefore not covered in most quantum chemistry textbooks. This is due to the fact that only in the last two decades have we learned about the importance of relativistic effects in the chemistry of heavy and super-heavy elements. Developments in computer hardware together with sophisticated computer algorithms make it now possible to perform four-component relativistic calculations for larger molecules. Two-component and scalar all-electron relativistic schemes are also becoming part of standard ab-initio and density functional program packages for molecules and the solid state. The second volume of this two-part book series is therefore devoted to applications in this area of quantum chemistry and physics of atoms, molecules and the solid state. Part 1 was devoted to fundamental aspects of relativistic electronic structure theory. Both books are in honour of Pekka Pyykko on his 60 birthday - one of the pioneers in the area of relativistic quantum chemistry. 

C. R. Jacob, M. Jansen, V. KeUb, A. V. Mudring, A. J. Sadlej, T. Saue, T. Sbhnel, F. E. Wagner. The quadrupole moment of the 3/2+ nuclear ground state of Au from electric field gradient relativistic coupled cluster and density-functional theory of small molecules and the solid state. /. Chem. Phys., 122 (2005) 124317. 

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