Nonrelativistic exchange

The derivation of its lowest order contribution, i.e. the exchange energy, is discussed in some detail in Appendix B, illustrating in particular the UV-renormalisation required. The final result, that has been obtained by a number of authors [93,94,95,96,18, 19], can be expressed as the nonrelativistic exchange energy density multiplied by a relativistic correction factor. 

In the leading nonrelativistic approximation the denominator of the photon propagator cancels the exchanged momentum squared in the numerator, and we immediately obtain the Hamiltonian for the interaction of two magnetic moments, reproducing the above result of classical electrodynamics. 

To account for the interchannel coupling, or, which is the same, electron correlation in calculations of photoionization parameters, various many-body theories exist. In this paper, following Refs. [20,29,30,33], the focus is on results obtained in the framework of both the nonrelativistic random phase approximation with exchange (RPAE) [55] and its relativistic analogy the relativistic random phase approximation (RRPA) [56]. RPAE makes use of a nonrelativistic HF approximation as the zero-order approximation. RRPA is based upon the relativistic Dirac HF approximation as the zero-order basis, so that relativistic effects are included not as perturbations but explicitly. Both RPAE and RRPA implicitly sum up certain electron-electron perturbations, including the interelectron interaction between electrons from... 

Schwarz values of a turn out to lie between Slater s value of one and the Gaspar-Kohn-Sham (GKS) value of 2/3, as would those of a method constructed to give the total nonrelativistic atomic electronic energy. All values are used in the following spin density-functional expression for the exchange and correlation (XC) energy,... 

In the atomic context the need for relativistic corrections to Exc[n] is obvious and has led to the development of the relativistic LDA (RLDA) [5,6,24]. On the basis of RLDA calculations for metallic Au and Pt, MacDonald et al. [25,26] have concluded that in solids relativistic contributions to Exc[n] can produce small but significant modifications of measurable quantities, as eg. the Fermi surface area. On the other hand, it has been shown [7] that the RLDA suffers from several shortcomings, eg. from a drastic overestimation of transverse exchange contributions, thus making the RLDA a less reliable tool than its nonrelativistic counterpart. As relativistic corrections are clearly misrepresented by the RLDA, it seems worthwhile to reinvestigate the role of relativistic arc-effects in solids on the basis of a more accurate form for Exc[n. ... 

We start with a discussion of the role of relativistic xc-contributions for the band structure of Au and Pt on the basis of the RGGA, thus repeating the analysis of MacDonald et al. [25,26] with a more appropriate form for Exc[n. As relativistic effects are most important near the atomic nucleus, first the core levels are analyzed (on the basis of the Kohn-Sham eigenvalues, as usual). Table 2 shows the core levels of Au (relative to the Fermi level) for several xc-functionals, i.e. the nonrelativistic LDA, the RLDA, the nonrelativistic GGA, the RGGA for both exchange and correlation as well as a combination of the... 

As the understanding of chemical bonding was advanced through such concepts as covalent and ionic bond, lone electron pairs etc., the theory of intermolecular forces also attempted to break down the interaction energy into a few simple and physically sensible concepts. To describe the nonrelativistic intermolecular interactions it is sufficient to express them in terms of the aforementioned four fundamental components electrostatic, induction, dispersion and exchange energies. 

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. 

The relative success of the nonrelativistic LDA is to some extent due to a fortunate partial cancellation of errors between the exchange and the correla-... 

Quite generally, it must be stated that some additional effort is required to develop the RDFT towards the same level of sophistication that has been achieved in the nonrelativistic regime. In particular, all exchange-correlation functionals, which are available so far, are functionals of the density alone. An appropriate extension of the nonrelativistic spin density functional formalism on the basis of either the time reversal invariance or the assembly of current density contributions (which are e.g. accessible within the gradient expansion) is one of the tasks still to be undertaken. 

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