Many-body random phase approximation

In the 1950s, many basic nuclear properties and phenomena were qualitatively understood in terms of single-particle and/or collective degrees of freedom. A hot topic was the study of collective excitations of nuclei such as giant dipole resonance or shape vibrations, and the state-of-the-art method was the nuclear shell model plus random phase approximation (RPA). With improved experimental precision and theoretical ambitions in the 1960s, the nuclear many-body problem was born. The importance of the ground-state correlations for the transition amplitudes to excited states was recognized. 

Many-body calculations which go beyond the Hartree-Fock model can be performed in two ways, i.e. using either a variational or a perturbational procedure. There are a number of variational methods which account for correlation effects superposition-of-configurations (or configuration interaction (Cl)), random phase approximation with exchange, method of incomplete separation of variables, multi-configuration Hartree-Fock (MCHF) approach, etc. However, to date only Cl and MCHF methods and some simple versions of perturbation theory are practically exploited for theoretical studies of many-electron atoms and ions. 

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... 

A comparison between experimental and theoretical values for the J (2p) parameter in neon is shown in Fig. 2.14. (The corresponding comparison between experimental and theoretical values for the partial cross section experimental data are given by the solid curve surrounded by a hatched area which takes into account the error bars. Theoretical results from advanced photoionization theories (many-body perturbation theory, R-matrix theory, and random-phase approximation) are represented by the other lines, and they are in close agreement with the experimental data (for details see [Sch86]). The theoretical / (2p) data of Fig. 2.13 are also close to the experimental values, except in the threshold region. 

Next the results from the relativistic random-phase approximation (RRPA) and the many-body perturbation theory (MBPT), also shown in Table 5.1, will be discussed. Because both calculations include basically the same electron-electron interactions, rather good agreement exists, and it is sufficient to concentrate only on the RRPA model. 

Actually, for the jellium model dfco) s 0. This result has been obtained in the random phase approximation (RPA) in Ref. [190]. It is easily established as a rigorous many-body result for the jellium model [191]. To define dj (co) we... 

A. more sophisticated approximation to calculate the Auger rate involves the inclusion of many-body effects in the medium excitations. This can be achieved by using the many-body response function of the interacting-electron system v(r, r, ru) [19]. In the random-phase approximation (RPA), X is obtained in a self-consistent way ... 

The random phase approximation (RPA) was first introduced into many-body theory by Pines and Bohm.This approximation was shown to be equivalent to the TDHF for the linear opticcd response of many-electron systems by Lindhard. ° (See, for example, Chapter 8.5 in ref 83. The electronic modes are identical to the transition densities of the RPA eigenvalue equation.) The textbook of D. J. Thouless contains a good overview of Hailree—Fock and TDHF theory. 

Thouless, D. J., The Quantum Mechanics of Many-Body Systems, Academic Press, New York, 1961. The transition from a restricted to an unrestricted wave function is part of the Hartree-Fock stability problem and is closely related to the random phase approximation. The original source of information on this problem is this book. 

The photoabsorption spectrum a(co) of a cluster measures the cross-section for electronic excitations induced by an external electromagnetic field oscillating at frequency co. Experimental measurements of a(co) of free clusters in a beam have been reported, most notably for size-selected alkali-metal clusters [4]. Data for size-selected silver aggregates are also available, both for free clusters and for clusters in a frozen argon matrix [94]. The experimental results for the very small species (dimers and trimers) display the variety of excitations that are characteristic of molecular spectra. Beyond these sizes, the spectra are dominated by collective modes, precursors of plasma excitations in the metal. This distinction provides a clear indication of which theoretical method is best suited to analyze the experimental data for the very small systems, standard chemical approaches are required (Cl, coupled clusters), whereas for larger aggregates the many-body perturbation methods developed by the solid-state community provide a computationally more appealing alternative. We briefly sketch two of these approaches, which can be adapted to a DFT framework (1) the random phase approximation (RPA) of Bohm and Pines [95] and the closely related time-dependent density functional theory (TD-DFT) [96], and (2) the GW method of Hedin and Lundqvist [97]. 

The random phase approximation (RPA) expression for 2 in 15.1 has been shown to be adequate for GeTe [93]. In fact, local field effects and other many body effects in the GW approximation or from the solution of the Bethe—Salpeter equation have also... 

Another nonrelativistic many-body theory which has a relativistic extension is the random phase approximation. This theory is discussed in detail by W. Johnson in this volume, so we will not consider it further, other than to remark that- one possible attraction of this technique is that oscillator strengths it produces are gauge invariant. 

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. 

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