Exchange-correlation relativistic functionals

From what has been said already with respect to the variational collapse and the minimax principle, it is clear from the beginning that the standard derivation of the Hohenberg-Kohn theorems [386], which are the fundamental theorems of nonrelativistic DFT and establish a variational principle, must be modified compared to nonrelativistic theory [383-385]. Also, we already know that the electron density is only the zeroth component of the 4-current, and we anticipate that the relativistic, i.e., the fundamental, version of DFT should rest on the 4-current and that different variants may be derived afterwards. The main issue of nonrelativistic DFT for practical applications is the choice of the exchange-correlation energy functional [387], an issue of equal importance in relativistic DFT [388,389] but beyond the scope of this book. 

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

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 summary, the RLDA addresses relativistic corrections to Ec n on the same limited level of sophistication as the NRLDA does for the nonrelativistic correlation energy functional. Even more than in the case of exchange, nonlocal corrections seem to be required for a really satisfactory description of (relativistic) correlation effects in atoms. 

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). 

Most frequently, however, a purely density-dependent version of RDFT is used. In this context we have examined the role of relativistic corrections to the exchange-correlation (xc) energy functional. In view of the limited accuracy of the relativistic local density approximation (RLDA) (Das etal. 1980 Engel etal. 1995a Ramana et... 

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

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