Exchange energy functional relativistic

A relativistic extension of the OPM on the longitudinal no-pair level has been put forward by Talman and collaborators [35] (and recently been applied to atoms [36]). Further extension to a covariant exchange energy functional is straightforward on the basis of the RKS propagator G s,... 

Table 1 Energies (in KeV) of single positive ions evaluated with (AH) a full relativistic kinetic energy functional without exchange [15] the c -order semi-relativistic functional (Eq. 46) without (1) and with (2) the relativistic exchange correction ((f-term), all using near-nuclear corrections, compared to Dirac-Fock (DF) values.

Before the progress with the relativistic gradient expansion of the kinetic energy took place, and due to a growing interest of applying the Kohn-Sham scheme of density functional theory [19] in the relativistic framework, an explicit functional for the exchange energy of a relativistic electron gas was found [20,21] ... 

The expressions (27-28) give the first correction given by the fully relativistic functionals, respectively for the kinetic and exchange energy densities (Eqs. 8 and 17) when performing a weak relativistic limit ppic small) ... 

Note that replacing c 2 by zero in this procedure, we find all the TFD expresions, and if we make C 2 = 0 we supress the relativistic exchange corrections. When we include these in our calculations we have always chosen the value of 3jr/2, the one which matches the weak relativistic limit of the fully relativistic energy functional. 

Nevertheless, when we include the near nuclear corrections (where the fully relativistic kinetic energy is used), the truncation of the energy functional only in the outer region up to order both in the kinetic and exchange energies turns out to be an adequate approximation. 

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. 

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 ... 

The presently available explicit approximations for the relativistic xc-energy functional are presented in Section 4. Both implicit functionals (as the exact exchange) and explicit density functionals (as the RLDA and RGGA) are discussed (on the basis of the information on the RHEG in Appendix C and that on the relativistic gradient expansion in Appendix E). Section 4 also contains a number of illustrative results obtained with the various functionals. However, no attempt is made to review the wealth of RDFT applications in quantum chemistry (see e.g.[74-88]) and condensed matter theory (see e.g.[89-l(X)]) as well as the substantial literature on nonrelativistic xc-functionals (see e.g.[l]). In this respect the reader is referred to the original literature. The review is concluded by a brief summary in Section 5. 

---