Relativistic exchange-correlation functional

Relativistic density functional theory (RDFT), including relativistic exchange-correlation functionals [4, Chapter 4]. 

It is impossible to develop a current-dependent relativistic exchange-correlation functional, which is computationally tractable and reduces to spin-density functional theory in the weakly relativistic limit. One reason is that there is no local approximation to such a functional since j vanishes for any homogeneous system. This means that the relativistic electron gas cannot serve as a starting point. More insight is gained from a Gordon decomposition of the current density j (see e.g. Refs. [7, 24]), which shows that j consists of an orbital part and the curl of a magnetisation density fh. 

Even with this approximation, we cannot assume that the relativistic exchange-correlation functional is the same as the nonrelativistic functional, though we might reasonably suppose that an expansion of the relativistic functional will yield the nonrelativistic functional with correction terms. This proves to be the case for the exchange energy of a relativistic uniform electron gas, for which we can write the exchange energy per electron as... 

In subsection 3.1, we will present GGA and LDA calculations for Au clusters with 6first principles method outlined in section 2, which employs the same scalar-relativistic pseudo-potential for LDA and GGA (see Fig 1). These calculations show the crucial relevance of the level of density functional theory (DFT), namely the quality of the exchange-correlation functional, to predict the correct structures of Au clusters. Another, even more critical, example is presented in subsection 3.2, where we show that both approaches, LDA and GGA, predict the cage-like tetrahedral structure of Au2o as having lower energy than amorphous-like isomers, whereas for other Au clusters, namely Auig, Au ... 

Atomic units are used in all equations and all considerations concern non-relativistic quantum mechanics in Born-Oppenheimer approximation. Square brackets, as in E[p] for instance, are used to indicate that the relevant quantity is a functional i.e. the correspondence between a function in real space p = p(r) and a real number (energy in this example). Abbreviations or acronyms denoting different approximate exchange-correlation functionals reflect their common usage in the literature. They are collected in Appendix. Unless specified, the equations are given for the spin-compensated case. 

In addition to the ab initio approach to relativistic electronic structure of molecules, four-component Kohn-Sham programs, which approximate the electron-electron interaction by approximate exchange-correlation functionals from density functional theory, have also been developed (Liu et al. 1997 Sepp et al. 1986). However, we concentrate on the ab initio methods and refer the reader to Chapter 4, which treats relativistic density functional theory (RDFT). 

EXc[n+,n ] is the exchange-correlation functional. Now, the (non-relativistic) ground state energy is ... 

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 ... 

As mentioned above, the nuclei are assumed to be fixed and are thus nothing more than sources of an external electrostatic potential in which the electrons move. If there is no magnetic field external to the molecule under consideration, and if external electric fields are time-independent, we arrive at the so-called electrostatic limit of relativistic density functional theory. Note that most molecular systems fall within this regime. In this case, one can prove the relativistic Hohen-berg-Kohn theorem using the charge density, p(r) = J f), only. This leads to a definition of an exchange-correlation functional -Exc[p( )]... 

The combination of the Dirac-Kohn-Sham scheme with non-relativis-tic exchange-correlation functionals is sometimes termed the Dirac-Slater approach, since the first implementations for atoms [13] and molecules [14] used the Xa exchange functional. Because of the four-component (Dirac) structure, such methods are sometimes called fully relativistic although the electron interaction is treated without any relativistic corrections, and almost no results of relativistic density functional theory in its narrower sense [7] are included. For valence properties at least, the four-component structure of the effective one-particle equations is much more important than relativistic corrections to the functional itself. This is not really a surprise given the success of the Dirac-Coulomb operator in wave function based relativistic ab initio theory. Therefore a major part of the applications of relativistic density functional theory is done performed non-rela-tivistic functionals. 

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