Relativistic random phase approximation , with

An important advantage of hrst method is the possibility of using different expressions for the atomic potential, and the calculations can be done not only for a purely Coulomb interaction, but in the multiconhguration interaction approximation, the Hartree-Fock-Dirac approximation, and the relativistic random phase approximation with exchange effects. The most exact relativistic calculations were done in [12] for the polarizability of the ground state of a helium-like atom. 

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... 

The results of a spin-polarization measurement of xenon photoelectrons with 5p5 2P3/2 and 5p5 2P1/2 final ionic states are shown in Fig. 5.21 together with the results of theoretical predictions. Firstly, there is good agreement between the experimental data (points with error bars) and the theoretical results (solid and dashed curves, obtained in the relativistic and non-relativistic random-phase approximations, respectively). This implies that relativistic effects are small and electron-electron interactions are well accounted for. (In this context note that the fine-structure splitting in the final ionic states has also to be considered in... 

In the paragraphs below we review some of the recent progress on relativi tlc many-body calculations which provide partial answers to the first of these questions and we also describe work on the Brelt Interaction and QED corrections which addresses the second question. We begin in Section IT with a review of applications of the DF approximation to treat inner-shell problems, where correlation corrections are insignificant, but where the Breit Interaction and QED corrections are important. Next, we discuss, in Section III, the multiconfiguration Dirac-Fock (MCDF) approximation which is a many-body technique appropriate for treating correlation effects in outer shells. Finally, in Section IV, we turn to applications of the relativistic random-phase approximation (RRPA) to treat correlation effects, especially in systems involving continuum states. 

During this same period theoretical techniques which can account for electron correlations have been developed and refined. Many calculations have been carried out using many-body perturbation theory (MBPT), R-matrix theory,6 the random phase approximation with exchange (RPAE), and other related techniques. This article will focus on nonrelativistic calculations since relativistic calculations such as the RRPA will be covered in the article by W, Johnson in this volume. 

Saue and Jensen used linear response theory within the random phase approximation (RPA) at the Dirac level to obtain static and dynamic dipole polarizabilities for Cu2, Ag2 and Au2 [167]. The isotropic static dipole polarizability shows a similar anomaly compared with atomic gold, that is, Saue and Jensen obtained (nonrelativ-istic values in parentheses) 14.2 for Cu2 (15.1 A ), 17.3 A for Ag2 (20.5 A ), and 12.1 A for Au2 (20.2 A ). They also pointed out that relativistic and nonrelativistic dispersion curves do not resemble one another for Auz [167]. We briefly mention that Au2 is metastable at 5 eV with respect to 2 Au with a barrier to dissociation of 0.3 eV [168, 169]. 

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. 

Relativistic correlation contributions in the LDA have so far only been considered on the basis of a partial resummation of those terms in the perturbation expansion in which are the most relevant in the high density limit. This contribution is either called the ring approximation (in accordance with its diagrammatic form) or, most often, the random phase approximation (RPA - which we shall use here) to ef The detailed discussion of is... 

Apart from primary structural and energetic data, which can be extracted directly from four-component calculations, molecular properties, which connect measured and calculated quantities, are sought and obtained from response theory. In a pilot study, Visscher et al. (1997) used the four-component random-phase approximation for the calculation of frequency-dependent dipole polarizabilities for water, tin tetrahydride and the mercury atom. They demonstrated that for the mercury atom the frequency-dependent polarizability (in contrast with the static polarizability) cannot be well described by methods which treat relativistic effects as a perturbation. Thus, the varia-tionally stable one-component Douglas-Kroll-Hess method (Hess 1986) works better than perturbation theory, but differences to the four-component approach appear close to spin-forbidden transitions, where spin-orbit coupling, which the four-component approach implicitly takes care of, becomes important. Obviously, the random-phase approximation suffers from the lack of higher-order electron correlation. 

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