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Random-Phase Approximation and Its Applications in Computational Chemistry and Materials Science

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Random-Phase Approximation and Its Applications in Computational Chemistry and Materials Science Random-phase approximation and its applications in computational chemistry and materials science Xinguo Ren,1 Patrick Rinke,1 Christian Joas,1,2 and Matthias Scheffler1 1Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195, Berlin, Germany 2Max-Planck-Institut f¨ur Wissenschaftsgeschichte, Boltzmannstr. 22, 14195, Berlin, Germany The random-phase approximation (RPA) as an approach for computing the electronic correlation energy is reviewed. After a brief account of its basic concept and historical development, the paper is devoted to the theoretical formulations of RPA, and its applications to realistic systems. With several illustrating applications, we discuss the implications of RPA for computational chemistry and materials science. The computational cost of RPA is also addressed which is critical for its widespread use in future applications. In addition, directions of further development and current correction schemes going beyond RPA will be discussed. I. INTRODUCTION break down. The random-phase approximation actually predates Computational materials science has developed into an density-functional theory, but it took until the late 1970s indispensable discipline complementary to experimental to be formulated in the context of DFT [9] and until the material science. The fundamental aim of computational early years of this millennium to be applied as a first materials science is to derive understanding entirely from principles electronic structure method [10, 11]. We take the basic laws of physics, i.e., quantum mechanical first the renewed and widespread interest of the RPA [10–40] principles, and increasingly also to make predictions of as motivation for this review article. To illustrate the new properties or new materials for specific tasks. The unique development of a powerful physical concept, we rapid increase in available computer power together with will put the RPA into its historical context before review- new methodological developments are major factors in ing the basic theory. A summary of recent RPA results the growing impact of this field for practical applications demonstrates the strength of this approach, but also its to real materials. current limitations. In addition we will discuss some of Density functional theory (DFT) [1] has shaped the the most recent schemes of going beyond RPA, including realm of first principles materials science like no other renormalized second order perturbation theory (r2PT), method today. This success has been facilitated by the which is particularly promising in our opinion. We also computational efficiency of the local-density [2] or gener- address the issue of computational efficiency which, at alized gradient approximation [3–5] (LDA and GGA) of present, impedes the widespread use of RPA, and indi- the exchange-correlation functional that make DFT ap- cate directions for further development. plicable to polyatomic systems containing up to several thousand atoms. However, these approximations are sub- ject to several well-known deficiencies. In the quest for A. Early history finding an “optimal” electronic structure method, that combines accuracy and tractability with transferability During the 1950s, quantum many-body theory under- across different chemical environments and dimension- went a major transformation, as concepts and techniques alities (e.g. molecules/clusters, wires/tubes, surfaces, originating from quantum electrodynamics (QED)—in solids) many new approaches, improvements and refine- particular Feynman-Dyson diagrammatic perturbation ments have been proposed over the years. These have theory—were extended to the study of solids and nu- been classified by Perdew in his “Jacob’s ladder” hierar- clei. A particularly important contribution at an early chy [6]. stage of this development was the RPA, a technique in- In this context, the treatment of exchange and cor- troduced by Bohm and Pines in a series of papers pub- relation in terms of “exact-exchange plus correlation in lished in 1951–1953 [7, 41–43]. In recent years, the RPA the random-phase approximation” [7, 8] offers a promis- has gained importance well beyond its initial realm of ing avenue. This is largely due to three attractive fea- application, in computational condensed-matter physics, tures. The exact-exchange energy cancels the spurious materials science, and quantum chemistry. While the self-interaction error present in the Hartree energy ex- RPA is commonly used within its diagrammatic formu- actly. The RPA correlation energy is fully non-local lation given by Gell-Mann and Brueckner [44], it is nev- and includes long-range van der Waals (vdW) interac- ertheless instructive to briefly discuss the history of its tions automatically and seamlessly. Moreover, dynamic original formulation by Bohm and Pines. Some of the electronic screening is taken into account by summing cited references are reprinted in [45], which also recounts up a sequence of “ring” diagrams to infinite order, which the history of the RPA until the early 1960s. Historical makes EX+cRPA applicable to small-gap or metallic sys- accounts of the work of Bohm and Pines can be found in tems where finite-order many-body perturbation theories Refs. [46–48] as well as in Refs. [33, 38]. 2 In 1933–1934, Wigner and Seitz published two papers a large number of electrons, and we shall ne- on the band structure of metallic sodium [49, 50] in which glect the contributions arising from this. This they stressed the importance of the electronic correlation procedure we call the random phase approxi- energy correction to band-theory calculations of the co- mation.” hesive energy of metals. Wigner subsequently studied In their second paper [42], Bohm and Pines develop a the interaction of electrons in a homogeneous electron detailed physical picture for the electronic behavior in gas (HEG) within a variational approach going beyond a HEG due to the presence of Coulomb interactions. Hartree-Fock [51]. He initially provided estimates for the Only in their third paper, Bohm and Pines treat the correlation energy only in the low- and high-density lim- (Coulomb-)interacting HEG quantum-mechanically [7]. its, but later interpolated to intermediate densities [52]. The RPA enables Bohm and Pines to absorb the long- For almost two decades, Wigner’s estimate remained the range Coulomb interactions into the collective behavior state of the art in the prototypical many-body problem of of the system, leaving the single-particle degrees of free- the HEG. According to a later statement by Herring,“the dom interacting only via a short-range screened force. magnitude and role of correlation energy remained inad- The RPA amounts to neglecting the interaction between equately understood in a considerable part of the solid- the collective and the single-particle degrees of freedom. state community for many years.” [53, pp. 71-72] Consequently, the momentum transfers of the Coulomb Due to the long-range nature of the Coulomb inter- potential in Fourier space can be treated independently. action and the resulting divergences, perturbative ap- The fourth paper [43] applies the new method to the proaches, so successful in other areas, had to be com- electron gas in metals, discussing both validity and con- plemented with approximations that accounted for the sequences of the RPA, such as the increase in electronic screening of the charge of an electron by the other elec- effective mass. trons. Before the 1950s, these approximations commonly Within condensed-matter theory, the significance of drew on work on classical electrolytes by Debye and the Bohm-Pines approach quickly became apparent: H¨uckel [54, 55] and on work on heavy atoms by Thomas Renormalizing the long-range Coulomb interaction into [56] and Fermi [57, 58], as well as later extensions [59, 60]. an effective screened interaction between new, effective As a reaction to work by Landsberg and Wohlfarth single-particle degrees of freedom allowed both to over- [61, 62], Bohm and Pines in 1950 reported to have been come the divergences appearing in older theories of in- led “independently to the concept of an effective screened teracting many-body systems and to explain the hith- Coulomb force as a result of a systematical classical and erto puzzling success of the single-particle models of early quantum-mechanical investigation of the interaction of condensed-matter theory (see, e.g., [52]). An early appli- charges in an electron gas of high density” [63, p. 103]. cation of the RPA was Lindhard’s calculation of the di- Their 1951–1953 series of papers [7, 41–43] presents this electric function of the electron gas [68]. Alternative ap- systematical investigation. The RPA was one of several proaches to and extensions of the Bohm-Pines approach physically-motivated approximations in the treatment of were formulated by Tomonaga [69, 70], and by Mott [71], the HEG which allowed them to separate collective de- Fr¨ohlich and Pelzer [72], and Hubbard [73, 74]. grees of freedom (plasma oscillations) from single-particle In 1956, Landau’s Fermi liquid theory [75] delivered degrees of freedom (which today would be called quasi- the foundation for effective theories describing many- particles or charged excitations) via a suitable canonical body systems in terms of quasiparticles. Brueckner [76] transformation reminiscent of early work in QED [64–66]. already in 1955 had introduced a “linked-cluster ex- A similar theory was developed rather independently for pansion” for the treatment of nuclear matter (see also nuclei by Bohr and Mottelson [67]. Ref. [77]). In 1957, Goldstone [78], using Feynman-like In their first paper, illustrating the fundamental idea diagrams (based on Ref. [79]), was able to show that of separating single-particle and collective degrees of free- Brueckner’s theory is exact for the ground-state energy dom, Bohm and Pines introduce RPA as one of four re- of an interacting many-fermion system. This put the quirements [41]: analogy between the QED vacuum and the ground state “(3) We distinguish between two kinds of of a many-body system on firm ground.
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