Relativistic Weighted Density Approximation

In the nonrelativistic LDA one finds partial, but by no means satisfactory cancellation of self-interaction effects between Eh and The WDA 

In order to apply the WDA in the relativistic regime, a fully covariant extension of the concept of the pair correlation function would be desirable. To our knowledge this is, however, not available. Nevertheless, if one restricts the discussion to the (instantaneous) longitudinal limit, one can express xc[ ] Eq. (3.21), via a relativistic pair correlation function defined in analogy to the nonrelativistic case as 

In fact, (4.13) is also satisfied by the x-only limit of g, i.e. its lowest order contribution in e. In the relativistic case only this limit of the pair correlation function of the RHEG, g ikplr — Ikp), specified in Eq. (B.68), is known (within the no-pair approximation [19,102]), so that we restrict the subsequent discussion to the x-only limit. 

For the transition from the x-only RLDA to the x-only RWDA one replaces the constant kp inside — r% kp) by a local screening momentum 

The main advantage of this approximation is that it is exact for two-electron systems (if the correct kf(r) = 0 is utilised in (4.15) before performing the functional differentiation (3.17) required for its application) and also correctly accounts for the self-interaction energies of individual closed shells if a shellpartitioning scheme is used [71]. Furthermore, the RWDA reproduces the asymptotic r proportionality of the exact x-only potential (although with the incorrect prefactor of 1/2 [103]). 

---

We also mention that recently a density functional approach to excited states of relativistic systems has been formulated [45], using ensembles of unequally weighted states. This formalism is restricted to the electrostatic limit and the no-sea approximation (see Section 3.2). Moreover, it remains unclear how the spontaneous emission of photons, which is possible in QED in contrast to the standard nonrelativistic many-body theory, is handled for the excited states involved. 

---