Relativistic Solid State Calculations

This chapter deals exclusively with the ground state problem. Even this task has to be simplified further for a treatment of most of the realistic cases one has to rely on the adiabatic approximation in which the electronic subsystem is treated in the frame of nuclear positions at rest. For many aspects, relativistic corrections are more important than non-adiabatic corrections. Generally, the former inerease with increasing nuclear charge while the latter decrease with increasing nuclear mass. The adiabatic electronic states (groxond state and quasi-particles) constitute what is commonly called the electronic structure of a solid. 

In this chapter, natural xmits (atomic units, a.u.) will be used throughout by putting 

Four-component Dirac spinor quantities are given in the standard representation in which the Dirac matrices are (in terms of 2 x 2 blocks) 

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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)... 

Chemistry of the Heaviest Elements ) sufficiently accurate, predict —A// (F1) = 68 kJ mol somewhat less volatile than Cn [146]. It was argued, that on transition metal surfaces FI should adsorb stronger than Cn, due to chemical forces since the 7pi/2(Fl) atomic orbital is less stabilized than the 7i(Cn) atomic orbital. On the other hand, relativistic solid-state calculation yield cohesive energies (equivalent to sublimation enthalpies) which are higher for Cn (1.1 eV [138]) compared to FI (0.5 eV [147]). This prediction is at variance with the calculation from [146] under the assumption that sublimation enthalpies correlate with the adsorption enthalpies on Au surfaces see Fig. 38. 

A list of recent solid-state calculations is given in Refs. [43-45]. We mention only a few of the most recent results discussing relativistic effects. Christensen and Kolar revealed very large relativistic effects in electronic band structure calculations for CsAu... 

Solid-state calculations were performed on the Rf metal [191]. The structural and electronic properties were evaluated by the first principles DFT in scalar relativistic formalism with and without SO coupling and compared with its 5d homolog Hf. It is found that Rf should crystalhze in the hexagonal close packed stmcture as Hf. However, under pressure, it should have a different sequence of phase transitions than Hf hep bcc instead of hep co bee. An explanation is offered for this difference in terms of the competition between the band structure and the Ewald energy contributions. 

The non-relativistic version of DVME method was developed in 1998 and was applied to the analysis of multiplet spectra of ruby [6-8]. This method was later applied to the analysis of a variety of TM-doped solid-state-laser materials [9-11]. The relativistic version of DVME method was developed in 2000. However, at that time, it was still difficult to calculate multiplet spectra of RE ions due to the limited performance of available computers. On the other hand, the relativistic... 

It is worth noticing that wide varieties of topics from such scientific fields as atomic physics, relativistic effects, photoelectron, X-ray and Auger processes, metal physics, chemical bonding affects, solid state chemistry and the others weis presented in the first workshop. This was perfectly astoimding because most calculating methods of the molecular orbitals have been applied only to specialized fields in which a researcher, who has proposed and formulated the calculating method, has been involved. 

Within a quasiatomic interpretation, these results can be understood if two types of 4/ orbital are allowed to coexist in the solid. These are denoted as 4/ (the usual, localised type), and a new kind of orbital, denoted 4/, which can coexist with the ground state, as originally suggested by Band and Fomichev [623] for atoms of the 4/ sequence. An interprets tion along such lines has been pursued by performing ab initio relativistic Dirac-Fock calculations for atoms with modified boundary conditions at... 

Fig. 22. The magnetic loop splitting energy Eml of the ground state, normalized to a(Za)AE ° , which is the leading term in a non-relativistic expansion. The dashed line refers to a relativistic point nucleus calculation whereas the solid line indicates the corresponding value for extended nuclei. The difference between both charge distribution models is notable. Note the logarithmic scale of the ordinate where 10° indicates 1.

Figure 3. Relativistic Dirac-Hartree-Fock (DHF) and experimental (Exp.) term energies of LSI levels arising from the ns np configuration of the group 4 elements (J=0 solid lines, J=1 dotted lines, J=2 dashed lines). The experimental result for Eka-Pb actually corresponds to the result of a high level relativistic coupled-cluster calculation [79]. The corresponding results for the nonrelativistic P, and S states (dot-dashed lines) were obtained from Hartree-Fock (HF) calculations.

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,... 

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 enormous progress in the calculation of solid state properties in the past decades has been pushed by the development of a nrunber of distinct band structure schemes like KKR-, ASW-, LMTO-, LCAO-, PP- and (L)APW-methods which differ essentially in their representation of basis functions. For all of the mentioned methods there exist by now full potential codes which also incorporate relativistic effects in one way or another. 

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). 

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. 

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