Explicit Relativistic Exchange-Correlation Functionals

The derivation of explicit energy functionals in nonrelativistic DFT follows a variety of avenues. The present day standard is the LDA, 

An alternative approach under the heading of weighted density approximation (WDA) attempts to model the density dependence of the pair correlation function of inhomogeneous systems. 

Compared with the nonrelativistic case, the derivation of explicit relativistic functionals is not as fully developed. Concerning the RLDA both the x-only limit and the correlation contribution in the so-called random phase approximation (RPA) are available. We discuss the RLDA in Section 4.1. Relativistic gradient corrections for E , on the other hand, have not been evaluated at all, although the basic technique for their derivation can be extended to the relativistic regime. In view of the absence of explicit results we only illustrate this method for the case of in Appendix D. An extension of the WDA scheme to relativistic systems (RWDA) [92, 36] is summarised in Section 4.2. However, no information on the RWDA beyond the longitudinal x-only limit is available. Moreover, it should be emphasised at the very outset that on the present level of sophistication neither the RLDA nor the RWDA contain radiative corrections. The issue of vacuum corrections in xc[ ] is discussed in detail in Appendix B and will not be addressed in this section. 

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As a fully nonlocal alternative to these explicit density functionals orbital-dependent (implicit) density functionals have been suggested. In addition to the exact exchange [51,52] some approximate correlation functionals are available, both empirical [166] and first-principles forms [57,59,60], As is already clear from Section 3.4 this concept can also be used in the relativistic situation. The status of relativistic implicit functionals [54] will be reviewed in Section 4.1. In particular, the various ingredients of the exact exchange will be analyzed. Subsequently the results obtained with the exact exchange will then serve as reference data for the analysis of the RLDA and RGGA. 

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