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: Hückel [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- Fröhlich 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. It had been in- response of the electrons to a wave. One troduced explicitly by Miyazawa for nuclei [80] and by of these is in phase with the wave, so that Salam for superconductors [81], although the essence of the phase difference between the particle re- the analogy dated back to the early days of quantum field sponse and the wave producing it is indepen- theory in the 1930s. dent of the position of the particle. This is In late 1956, Gell-Mann and Brueckner employed a the response which contributes to the orga- diagrammatic approach for treating the problem of the nized behavior of the system. The other re- interacting electron gas. Their famous 1957 paper [44] sponse has a phase difference with the wave eliminated the spurious divergences appearing in previ- producing it which depends on the position of ous approaches. Expressing the perturbation series for the particle. Because of the general random the correlation energy of the HEG in terms of the Wigner- location of the particles, this second response Seitz radius rs, they found that the divergences within tends to average out to zero when we consider earlier calculations [e.g., 82] were mere artifacts: The log- 3 arithmic divergence appearing in the perturbative expan- equivalence of the “plasmon” and “ring-CCD formula- sion of the correlation energy is canceled by similar di- tion” of RPA has recently been established by Scuseria vergences in higher-order terms. Summing the diagrams et al. [13]. These new perspectives not only offer more (which had a ring structure) to infinite order yielded a ge- insight into the theory, but also help to devise more ef- ometric series, and thus a convergent result. Gell-Mann ficient algorithms to reduce the computational cost, e.g., and Brueckner derived an expression for the ground state by applying the Cholesky decomposition to the “ring- energy of the interacting electron gas in the high-density CCD” equations [13]. limit. Their work, and Goldstone’s paper [78], are the Following the early work on the HEG, other model earliest examples of the application of Feynman-type di- electron systems were investigated, including the HEG agrammatic methods in condensed-matter theory. surface [96, 97], jellium slabs [22] and jellium spheres Many applications of the new quantum-field theoreti- [98]. The long-range behavior of RPA for spatially well- cal methods followed: Gell-Mann calculated the specific separated closed-shell subsystems was examined by Sz- heat of the high-density HEG [83]; Hubbard [84, 85] pro- abo and Ostlund [99], as well as by Dobson [100–102]. vided a description of the collective modes in terms of These authors showed that RPA yields the correct 1/R6 many-body perturbation theory (MBPT); Sawada et al. asymptotic behavior for the subsystem interaction. In [86, 87] demonstrated that the Gell-Mann-Brueckner ap- addition, the long-range dispersion interaction of RPA proach indeed contained the plasma oscillations of Bohm is fully consistent with the monomer polarizability com- and Pines [7], a point around which there had been quite puted at the same level of theory. This is one of the some confusion initially [see, e.g., 88]. In addition, they main reasons for the revival of the RPA in recent years, demonstrated that the RPA is exact in the high-density because this long-range interaction is absent from LDA, limit. In 1958, Nozières and Pines formulated a many- GGA, and other popular density-functionals. Other rea- body theory of the dielectric constant and showed the sons are the compatibility of the RPA correlation with equivalence of Gell-Mann and Brueckner’s diagrammatic exact exchange (which implies the exact cancellation of approach and the RPA [89, 90]. the self-interaction error present in the Hartree term) and the applicability to metallic systems. For the HEG it has been demonstrated that RPA is not B. RPA in modern times accurate for short-range correlation [103, 104], and hence for a long time RPA was not considered to be valuable Today, the concept of RPA has gone far beyond the for realistic systems. Perdew and coworkers investigated domain of the HEG, and has gained considerable im- this issue [97, 105], and found that a local/semi-local cor- portance in computational physics and quantum chem- rection to RPA has little effect on iso-electronic energy istry. As a key example, RPA can be introduced within differences, which suggests that RPA might be accurate the framework of DFT [1] via the so-called adiabatic- enough for many practical purposes. The application of connection fluctuation-dissipation (ACFD) theorem [9, RPA to realistic systems appeared slightly later, starting 91, 92]. Within this formulation, the unknown exact with the pioneering work of Furche [10], and Fuchs and exchange-correlation (XC) energy in Kohn-Sham [2] DFT Gonze [11] for small molecules. Accurate RPA total en- can be formally constructed by adiabatically switching on ergies for closed-shell atoms were obtained by Jiang and the Coulomb interaction between electrons, while keeping Engel [18]. Several groups investigated molecular proper- the electron density fixed at its physical value. This is for- ties, in particular in the weakly bound regime, with RPA mulated by a coupling-strength integration under which and its variants [12, 15, 26, 28, 31, 32, 106, 107], while the integrand is related to the linear density-response others applied RPA to periodic systems [17, 19–21, 108– function of fictitious systems with scaled Coulomb inter- 110]. Harl and Kresse in particular have performed ex- action. Thus, an approximation to the response function tensive RPA benchmark studies for crystalline solids of directly translates into an approximate DFT XC energy all bonding types [19, 20, 110]. At the same time, the ap- functional. RPA in this context is known as an orbital- plication of RPA to surface adsorption problems has been dependent energy functional [93] obtained by applying reported [23–25, 30, 111, 112], with considerable success the time-dependent Hartree approximation to the den- in resolving the “CO adsorption puzzle”. sity response function. Most practical RPA calculations in recent years have The versatility of RPA becomes apparent when con- been performed non-self-consistently based on a preced- sidering alternative formulations. For instance, the RPA ing LDA or GGA reference calculation. In these calcu- correlation energy may be understood as the shift of the lations, the Coulomb integrals are usually not antisym- zero-point plasmon excitation energies between the non- metrized in the evaluation of the RPA correlation energy, interacting and the fully interacting system, as shown by a practice sometimes called direct RPA in the quantum- Sawada for HEG [87], and derived in detail by Furche chemical literature. In this paper we will denote this com- [94] for general cases (see also Ref. [29]). In quantum mon procedure “standard RPA” to distinguish it from chemistry, RPA can also be interpreted as an approxi- more sophisticated procedures. While a critical assess- mation to coupled-cluster doubles (CCD) theory where ment of RPA is emerging and a wide variety of applica- only diagrams of “ring” structure are kept [13, 95]. The tions are pursued, certain shortcomings of standard RPA 4 have been noted. The most prominent is the systematic have also appeared recently [39, 40]. Our own work underestimation of RPA binding energies [10, 20, 32], and on the SE correction to RPA indicates that the input- the failure to describe stretched radicals [16, 113, 114]. orbital dependence in RPA post-processing calculations Over the years several attempts have been made to im- is a significant issue. Some form of self-consistency would prove upon RPA. The earliest is RPA+, where, as men- therefore be desirable. However, due to the considerable tioned above, a local/semi-local correlation correction numerical effort associated with OEP-RPA calculations, based on LDA or GGA is added to the standard RPA practical RPA calculations will probably remain of the correlation energy [97, 105]. Based on the observation post-processing type in the near future. that in molecules the correlation hole is not sufficiently Despite RPA’s appealing features its widespread use in accurate at medium range in RPA, this has recently been chemistry and materials science is impeded by its com- extended to a non-local correction scheme [34, 35]. Sim- putational cost, which is considerable compared to con- ilarly, range-separated frameworks [115] have been tried, ventional (semi)local DFT functionals. Furche’s original in which only the long-range part of RPA is explicitly implementation based on a molecular particle-hole basis included [14, 15, 26, 116–120], whereas short/mid-range scales as O(N 6) [10]. This can be reduced to O(N 5) using correlation is treated differently. Omitting the short- the plasmon-pole formulation of RPA [94], and to O(N 4) range part in RPA is also numerically beneficial whereby [28] when the resolution-of-identity (RI) technique is em- the slow convergence with respect to the number of basis ployed in addition. Our own RPA implementation [131] functions can be circumvented. Due to this additional in FHI-aims [132], which has been used in production appealing fact, range-separated RPA is now an active calculations [24, 32, 124] before, is based on localized nu- research domain despite the empirical parameters that meric atom-centered orbitals and the RI technique, and govern the range separation. Another route to improve hence naturally scales as O(N 4). Plane-wave based im- 4 RPA in the framework of ACFD is to add an fxc kernel plementations [11, 110] also automatically have O(N ) to the response function and to find suitable approxima- scaling. However, in all current implementations the tions for it [12, 33, 121, 122]. Last but not least, the CCD convergence with respect to unoccupied states is slow. perspective offers a different correction in form of the Proposals to eliminate the dependence on the unoccu- second-order screened exchange (SOSEX) contribution pied states [133, 134] by obtaining the response function [16, 95, 123], whereas the MBPT perspective inspired sin- from density-functional perturbation theory [135] have gle excitation (SE) corrections [32]. SOSEX and SE are not been explored so far. In local orbital based ap- distinct many-body corrections and can also be combined proaches the O(N 4) scaling can certainly be reduced by [124]. These corrections have a clear diagrammatic repre- exploiting matrix sparsity, as demonstrated recently in sentation and alleviate the above-mentioned underbind- the context of GW [136] or second-order Møller-Plesset ing problem of standard RPA considerably [124]. An- perturbation theory (MP2) [137]. Also approximations other proposal to improve RPA by incorporating higher- to RPA [29] or effective screening models [138] might sig- order exchange effects in various ways has also been dis- nificantly improve the scaling and the computational effi- cussed recently [119]. However, at this point in time, a ciency. Recently, RPA has been cast into the continuum consensus regarding the “optimal” correction that com- mechanics formulation of DFT [139] with considerable bines both efficiency and accuracy has not been reached. success in terms of computational efficiency [140]. In general, there is still room for improvement, which, together Although the majority of practical RPA calculations with the rapid increase in computer power makes us con- are performed as post-processing of a preceding DFT fident that RPA-type approaches will become a powerful calculation, self-consistent RPA calculations have also technique in computational chemistry and materials sci- been performed within the optimized effective potential ence in the future. It would thus be desirable, if the ma- (OEP) framework. OEP is a procedure to find the opti- terial science community would start to build up bench- mal local multiplicative potential that minimizes orbital- mark sets for materials science akin to the ones in quan- dependent energy functionals. The first RPA-OEP cal- tum chemistry (e.g. G2 [141] or S22 [142]). These should culations actually date back more than 20 years, but include prototypical bulk crystals, surfaces, and surface were not recognized as such. Godby, Schlüter and Sham adsorbates and would aid the development of RPA-based solved the Sham-Schlüter equation for the GW self- approaches. energy [125, 126], which is equivalent to the RPA-OEP equation, for the self-consistent RPA KS potential of bulk silicon and other semiconductors, but did not calculate RPA ground-state energies. Similar calculations for other II. THEORY AND CONCEPTS bulk materials followed later by Kotani [127] and Grüning et al.[128, 129]. Hellgren and von Barth [130] and then RPA can be formulated within different theoretical later Verma and Bartlett [40] have looked at closed-shell frameworks. One particularly convenient approach to atoms and observed that the OEP-RPA KS potential derive RPA is the so-called “adiabatic connection (AC)”, there reproduces the exact asymptotic behavior in the which is a powerful mathematical technique to obtain the valence region, although its behavior near the nucleus ground-state total energy of an interacting many-particle is not very accurate. Extensions to diatomic molecules system. Starting with the AC approach, the interacting 5 ground-state energy can be retrieved either by coupling In the AC construction of the total energy, we introduce to the fluctuation-dissipation theorem in the DFT con- the ground-state wave function |Ψλi for the λ-dependent text, or by invoking the Green-function based MBPT. system such that RPA can be derived within both frameworks. In addition, RPA is also intimately linked to the coupled-cluster H(λ)|Ψλi = E(λ)|Ψλi . (5) (CC) theory. In this section, we will present the theoretical aspects of RPA from several different perspectives. Adopting the normalization condition hΨλ|Ψλi = 1, the interacting ground-state total energy can then be obtained with the aid of the Hellmann-Feynman theorem,

A. Adiabatic connection 1 E(λ =1)=E0 + dλ× The ground-state total energy of an interacting many- Z0 body Hamiltonian can formally be obtained via the AC dHˆ1(λ) hΨλ| Hˆ1(λ)+ λ |Ψλi , (6) technique, in which a continuous set of coupling-strength dλ ! (λ) dependent Hamiltonians is introduced where Hˆ (λ)= Hˆ0 + λHˆ1(λ), (1) (0) E0 = E = hΨ0|H0|Ψ0i (7) that “connect” a reference Hamiltonian Hˆ0 = Hˆ (λ = 0) with the target many-body Hamiltonian Hˆ = Hˆ (λ = 1). is the zeroth-order energy. We note that the choice of the adiabatic-connection path in Eq. (6) is not unique. For the electronic systems considered here, Hˆ (λ) has the In DFT, the path is chosen such that the electron density following form: is kept fixed at its physical value along the way. This im- N N plies a non-trivial (not explicitly known) λ-dependence ˆ 1 2 ext λ of Hˆ (λ). In MBPT, one often chooses a linear connec- H(λ)= − ∇i + vλ (i) + , (2) 1 2 |ri − rj | i=1 i>j=1 tion path — Hˆ (λ) = Hˆ (and hence dHˆ (λ)/dλ = 0). X X 1 1 1 In this case, a Taylor expansion of |Ψλi in terms of λ in ext where N is the number of electrons, vλ is a λ-dependent Eq. (6) leads to standard Rayleigh-Schrödinger pertur- ext r ext r external potential with vλ=1( )= v ( ) being the phys- bation theory (RSPT) [143]. ical external potential of the fully-interacting system. ext Note that in general vλ can be non-local in space for λ 6= 1. Hartree atomic units ~ = e = me = 1 are used B. RPA derived from ACFD here and in the following. The reference Hamiltonian H0, given by Eq. (2) for λ = 0, is of the mean-field (MF) type, Here we briefly introduce the concept of RPA in the i.e., a simple summation over single-particle Hamiltoni- context of DFT, which serves as the foundation for most ans: practical RPA calculations in recent years. In Kohn- Sham (KS) DFT, the ground-state total energy for an N 1 interacting N-electron systems is an (implicit) functional Hˆ = − ∇2 + vext (i) 0 2 i λ=0 of the electron density n(r) and can be conveniently split i=1 X into four terms: N 1 2 ext MF = − ∇ + v (ri)+ v (i) . (3) r r r r r 2 i E[n( )] = Ts[ψi( )]+ Eext[n( )]+ EH[n( )]+ Exc[ψi( )] . i=1 X (8) T is the kinetic energy of the KS independent-particle In Eq. (3), vMF is a certain (yet-to-be-specified) mean- s system, Eext the energy due to external potentials, field potential arising from the electron-electron interac- E the classic Hartree energy, and E the exchange- tion. It can be the Hartree-Fock (HF) potential vHF or H xc Hxc correlation energy. In the KS framework, the electron the Hartree plus exchange-correlation potential v in density is obtained from the single-particle KS orbitals DFT. Given Eq. (2) and (3), the perturbative Hamilto- r r occ occ r 2 ψi( ) via n( ) = |ψi ( )| . Among the four terms nian Hˆ (λ) in Eq. (1) becomes i 1 in Eq. (8), only Eext[n(r)] and EH[n(r)] are explicit func- P tionals of n(r). Ts is treated exactly in KS-DFT in terms N N r ˆ 1 1 ext r ext of the single-particle orbitals ψi( ) which themselves are H1(λ)= + vλ ( i) − vλ=0(i) , r |ri − rj | λ functionals of n( ). i>j=1 i=1 X X All the many-body complexity is contained in the un- N N known XC energy term, which is approximated as an 1 1 ext r ext r MF = + vλ ( i) − v ( i) − v (i) . explicit functional of n(r) (and its local gradients) in con- |ri − rj | λ i>j=1 i=1 X X ventional functionals, (LDA, GGAs, meta-GGAs), and as (4) a functional of the ψi(r)’s in more advanced functionals 6

(hybrid density functionals, RPA, etc.). Different exist- Here ing approximations to Exc can be classified into a hier- r r′ λ ′ hΨλ|δnˆ( )δnˆ( )|Ψλi ′ archical scheme known as “Jacob’s ladder” [6] in DFT. n (r, r )= − δ(r − r ) , (16) xc n(r) However, what if one would like to improve the accuracy of Exc in a more systematic way? For this purpose it is is the formal expression for the so-called XC hole, with illuminating to start with the formally exact way of con- δnˆ(r) =n ˆ(r) − n(r) being the fluctuation of the density r r structing Exc using the AC technique discussed above. operatorn ˆ( ) around its expectation value n( ). Equa- As alluded to before, in KS-DFT the AC path is chosen tion (16) shows that the XC hole is related to the density- such that the electron density is kept fixed. Equation (6) density correlation function. In physical terms, it de- for the exact ground-state total-energy E = E(λ = 1) scribes how the presence of an electron at point r depletes then reduces to the density of all other electrons at another point r′. 1 N In the second step, the density-density correlations 1 1 (fluctuations) in Eq. (16) are linked to the response E = E0+ dλhΨλ| |Ψλi 2 |ri − rj| properties (dissipation) of the system through the zero- Z0 i=6 j=1 X temperature fluctuation-dissipation theorem [144] 1 N d ext r ∞ + dλhΨλ| vλ ( i)|Ψλi ′ 1 λ ′ dλ hΨλ|δnˆ(r)δnˆ(r )|Ψλi = − dωImχ (r, r ,ω) , (17) Z0 i=1 π X Z0 1 1 = E + dλ drdr′× where χλ(r, r′,ω) is the linear density response func- 0 2 Z0 ZZ tion of the (λ-scaled) system. Using Eqs. (15-17) and nˆ(r) [ˆn(r′) − δ(r − r′)] v(r, r′) = 1/|r − r′|, we arrive at the renowned ACFD hΨ | |Ψ i λ |r − r′| λ expression for the XC energy in DFT r r ext r ext r 1 1 + d n( ) vλ=1( ) − vλ=0( ) , (9) E = dλ drdr′v(r, r′) × xc 2 Z Z0 ZZ where ∞ 1 λ r r′ r r′ r N − dωImχ ( , ,ω) − δ( − )n( ) π 0 nˆ(r)= δ(r − r ) (10) Z i 1 1 i=1 r r′ r r′ X = dλ d d v( , ) × r 2π 0 is the electron-density operator, and n( ) = Z ∞ ZZ r 1 λ ′ ′ hΨλ|nˆ( )|Ψλi for any 0 ≤ λ ≤ 1. − dωχ (r, r ,iω) − δ(r − r )n(r) . π For the KS reference state |Ψ0i (given by the Slater Z0 determinant of the occupied single-particle KS orbitals (18) {ψ (r)}) we obtain i The reason that the above frequency integration can be N 1 performed along the imaginary axis originates from the E = hΨ | − ∇2 + vext (r ) |Ψ i λ r r′ 0 0 2 λ=0 i 0 analytical structure of χ ( , ,ω) and the fact that it be- i=1 X comes real on the imaginary axis. The ACFD expression in Eq. (18) transforms the problem of computing the XC = T [ψ (r)] + drn(r)vext (r) , (11) s i λ=0 energy to one of computing the response functions of a Z and thus series of fictitious systems along the AC path, which in practice have to be approximated as well. r r r ext r In this context the random-phase approximation is a E =Ts [ψi( )] + d n( )vλ=1( )+ Z particularly simple approximation of the response func- 1 1 nˆ(r) [ˆn(r′) − δ(r − r′)] tion: dλ drdr′hΨ | |Ψ i . λ r r′ λ 2 0 | − | χλ (r, r′,iω)= χ0(r, r′,iω)+ Z ZZ (12) RPA dr dr χ0(r, r ,iω)λv(r − r )χλ (r , r′,ω). Equating (8) and (12), and noticing 1 2 1 1 2 RPA 2 Z r r′ (19) 1 ′ n( )n( ) EH[n(r)] = drdr (13) 2 |r − r′| χ0(r, r ,iω) is the independent-particle response function Z 1 r r r ext r of the KS reference system at λ = 0 and is known ex- Eext[n( )] = d n( )vλ ( ) , (14) =1 plicitly in terms of the single-particle KS orbitals ψi(r), Z orbital energies ǫ and occupation factors f one obtains the formally exact expression for the XC en- i i ∗ ∗ ′ ′ ergy (fi − fj)ψ (r)ψj (r)ψ (r )ψi(r ) χ0(r, r′,iω)= i j . λ ′ ǫ − ǫ − iω 1 n (r, r )n(r) ij i j E = dλ drdr′ xc . (15) X xc 2 |r − r′| (20) Z ZZ 7

From equations (18) and (19), the XC energy in RPA can be separated into an exact exchange (EX) and the RPA correlation term, MP2 + Ec = RPA EX RPA Exc = Ex + Ec , (21) where FIG. 1: Goldstone diagrams for the MP2 correlation energy. EX 1 r r′ r r′ The two graphs describe respectively the second-order direct Ex = d d v( , ) × 2 exchange ZZ process, and the second-order process. The upgoing ∞ solid line represents a particle associated with an unoccupied 1 0 r r′ r r′ r − dωχ ( , ,iω) − δ( − )n( ) orbital energy ǫa, the downgoing solid line represents a hole π 0 associated with an occupied orbital energy ǫi, and the dashed occZ r r′ ∗ r r r r′ ∗ r′ r′ line denotes the bare Coulomb interaction. = − d d ψi ( )ψj ( )v( , )ψj ( )ψi( ) ij ZZ X H EX (22) where E and Ex are the classic Hartree and exact exchange energy deﬁned in Eq. (13) and (22), respectively. MF HF and E = hΨ0|v |Ψ0i is the “double-counting” term due MF 1 to the MF potential v , which is already included in ERPA = − drdr′v(r, r′) × 0 (1) c 2π H . The sum of E0 and the ﬁrst-order term E yields ∞ ZZ 1 the Hartree-Fock energy, and all higher-order contribu- λ r r′ r r′ tions constitute the so-called correlation energy. dω dλχRPA( , ,iω) − χ0( , ,iω) Z0 Z0 The higher-order terms can be evaluated using the di- 1 ∞ agrammatic technique developed by Goldstone [78]. For = dωTr ln(1 − χ0(iω)v)+ χ0(iω)v . 2π instance, the second-order energy in RSPT is given by Z0 (23) ˆ 2 (2) |hΦ0|H1|Φni| E = (0) For brevity the following convention n>0 E0 − En X occ unocc a 2 occ unocc ab 2 |hΦ |Hˆ |Φ i| |hΦ0|Hˆ1|Φ i| Tr [AB]= drdr′A(r, r′)B(r′, r) (24) 0 1 i ij = (0) + (0) ZZ i a E0 − Ei,a ij ab E0 − Eij,ab X X X X has been used in Eq. (23). (28)

where |Φ0i = |Ψ0i is the ground state of the reference C. RPA derived from MBPT Hamiltonian Hˆ0, and |Φni for n > 0 correspond to its (0) excited states with energy En = hΦn|Hˆ0|Φni. |Φni An alternative to ACFD is to compute the interacting can be classified into singly-excited configurations |Φi,ai, ground-state energy by performing an order-by-order ex- doubly-excited configurations |Φij,abi, etc.. The summa- pansion of Eq. (6). To this end, it is common to choose tion in Eq. 28 terminates at the level of double excita- ext ext MF ˆ a linear AC path, i.e., in Eq. (4) vλ = v + (1 − λ)v tions. This is because H only contains one- and two- such that particle operators, and hence does not couple the ground N N state |Φ0i to triple and higher-order excitations. We will ˆ ˆ 1 MF examine the single-excitation contribution in Eq. (28) H1(λ)= H1 = − vi . (25) |ri − rj | in detail in Section IVC. Here it suffices to say that i>j=1 i=1 X X this term is zero for the HF reference and therefore is Now equation (6) reduces to not included in MP2. The double-excitation contribu-

1 tion can be further split into two terms, corresponding to the second-order direct and exchange energy in MP2, E = E0 + dλhΨλ|Hˆ1|Ψλi . (26) whose representation in terms of Goldstone diagrams is Z0 depicted in Fig. 1. The rules to evaluate Goldstone di- A Taylor expansion of |Ψ i and a subsequent λ inte- λ agrams can be found in the classic book by Szabo and gration lead to an order-by-order expansion of the in- Ostlund [143]. teracting ground-state total energy, e.g., the first-order The Goldstone approach is convenient for the lowest correction to E is given by 0 few orders, but becomes cumbersome or impossible for 1 arbitrarily high orders, the evaluation of which is essen- (1) ˆ E = dλhΨ0|H1|Ψ0i tial when an order-by-order perturbation breaks down Z0 and a “selective summation to infinite order” has to be ˆ = hΨ0|H1|Ψ0i invoked. In this case, it is much more convenient to ex- H EX MF = E + Ex − E , (27) press the total energy in terms of the Green function 8 and the self-energy, as done, e.g., by Luttinger and Ward (a) [145]. Using the Green function language, the ground- state total-energy can be expressed as [145, 146], GW 2 3 Σc (λ) = λ + λ + ··· 1 1 dλ 1 ∞ E = E + dωTr G0(iω)Σ(iω,λ) 0 2 λ 2π Z0 Z−∞ (29) (b) 1 1 dλ 1 ∞ = E + dωTr [G(iω,λ)Σ∗(iω,λ)] 0 2 λ 2π Z0 Z−∞ RPA 1 1 (30) Ec = − 4 − 6 − ··· where G0 and G(λ) are single-particle Green functions (c) corresponding to the non-interacting Hamiltonian H0 and the scaled interacting Hamiltonian H(λ), respec- ∗ tively. Σ (λ) and Σ(λ) are the proper (irreducible) and RPA + + ... Ec = improper (reducible) self-energies of the interacting system with interaction strength λ. [Proper self-energy diagrams are those which cannot be split into two by cutting a single Green function line.] Note that in Eq. (29) and FIG. 2: Feynman diagrams for the GW self-energy (a), Feyn- (30), the trace convention of Eq. (24) is implied. man diagrams for the RPA correlation energy (b), and Gold- The above quantities satisfy the following relationship stone diagrams for the RPA correlation energy (c). Solid lines 0 0 0 in (a) and (b) (with thick arrows) represent fermion propa- G(iω,λ) = G (iω)+ G (iω)Σ(iω,λ)G (iω) gators G, and those in (c) (with thin arrows) denote parti- = G0(iω)+ G0(iω)Σ∗(iω,λ)G(iω,λ). (31) cle (upgoing line) or hole states (down-going line) without frequency dependence. Dashed lines correspond to the bare From Eq. (31) the equivalence of Eq. (29) and Eq. (30) Coulomb interaction v in all graphs. is obvious. In Eq. (29), a perturbation expansion of the λ-dependent self-energy Σλ(iω) naturally translates into a perturbation theory of the ground-state energy. We note that starting from Eq. (29) this procedure In particular, the linear term of Σλ(iω) yields the first- (1) naturally gives the perturbative RPA correlation energy order correction to the ground-state energy, i.e., E based on any convenient non-interacting reference Hamil- in Eq. (27). All higher-order (n ≥ 2) contributions of λ tonian H0, such as Hartree-Fock or local/semi-local KS- Σ (iω), here denoted Σc, define the so-called correla- DFT theory. If one instead starts with Eq. (30) and tion energy. In general the correlation energy cannot be applies the GW approximation therein, G(λ,iω) and treated exactly. A popular approximation to Σc is the Σ∗(λ,iω) become the self-consistent GW Green function GW approach, which corresponds to a selective summa- and self-energy. As a result the improper self-energy di- tion of self-energy diagrams with ring structure to infi- agrams in Eq. (29), which are neglected in the perturba- nite order, as illustrated in Fig. (2a). Multiplying G0 to 0 0 GW tive GW approach (known as G W in the literature), the GW self-energy Σc (iω) as done in Eq. (29) and are introduced and the total energy differs from that of performing the λ integration, one obtains the RPA cor- the RPA. An in-depth discussion of self-consistent GW relation energy and its implications can be found in [147–150]. 1 1 dλ 1 ∞ ERPA = dωTr G0(iω)ΣGW (iω,λ) . c 2 λ 2π c Z0 Z−∞ (32) D. Link to coupled cluster theory This illustrates the close connection between RPA and the GW approach. A diagrammatic representation of RPA Ec is the shown in Fig. (2b). We emphasize that In recent years, RPA has also attracted considerable the diagrams in Fig. (2a) and (2b) are Feynman dia- attention in the quantum chemistry community. One key grams, i.e., the arrowed lines should really be interpreted reason for this is its intimate relationship with coupled as propagators, or Green functions. A similar represen- cluster (CC) theory, which has been very successful for RPA tation of Ec can be drawn in terms of Goldstone di- accurately describing both covalent and non-covalent in- agrams [143], as shown in Fig (2b). However, caution teractions in molecular systems. To understand this re- should be applied, because the rules for evaluating these lationship, we will give a very brief account of the CC diagrams are different (see e.g., Ref. 143, 146), and the theory here. More details can for instance be found in prefactors in Fig. (2b) are not present in the correspond- a review paper by Bartlett and Musial[151]. The essen- ing Goldstone diagrams. The leading term in RPA cor- tial concept of CC builds on the exponential ansatz for responds to the second-order direct term in MP2. the many-body wave function Ψ for correlated electronic 9 systems choice is the CC doubles (CCD) approximation, or T2 approximation [152], that retains only the double exci- ˆ |Ψi = eT |Φi. (33) tation term in Eq. (34). The graphical representation of CCD contains a rich variety of diagrams including ring |Φi is a non-interacting reference state, usually chosen to diagrams, ladder diagrams, the mixture of the two, etc. be the HF Slater determinant, and Tˆ is a summation of If one restricts the choice to the pure ring diagrams, as excitation operators of different order, practiced in early work on the HEG [95, 153], the CCD equation is reduced to the following simplified form [13], Tˆ = Tˆ1 + Tˆ2 + Tˆ3 + ··· + Tˆn + ··· , (34) B + AT + T A + TBT = 0. (41) with Tˆ1, Tˆ2, Tˆ3, ··· being the single, double, and triple excitation operators, etc. These operators can be most A, B, T are all matrices of rank Nocc ·Nvir with Nocc and conveniently expressed using the language of second- Nvir being the number of occupied and unoccupied single- quantization, namely, particle states, respectively. Specifically we have Aia,jb = ab (ǫi − ǫa)δij δab −hib|aji, Bia,jb = hij|abi, and Tia,jb = tij , ˆ a † T1 = ti câcî, where the Dirac notation for the two-electron Coulomb i,a repulsion integrals X 1 ab † † ∗ ∗ ′ ′ Tˆ = t cˆ cˆ cˆ cˆ , (35) ψ (r)ψr(r)ψ (r )ψs(r ) 2 4 ij a b j i hpq|rsi = drdr′ p q (42) ij,ab r r′ X | − | ··· ZZ 1 has been adopted. Tˆ = tabc···cˆ† cˆ†cˆ† ··· cˆ cˆ cˆ , (36) n (n!)2 ijk··· a b c k j i Equation (41) is mathematically known as the Riccati ijk··· ,abc··· X equation [154]. Solving this equation yields the ring-CCD amplitudes T rCCD, with which the RPA correlation en- wherec ˆ† andc ˆ are single-particle creation and annihila- a ab ergy can written as tion operators and ti , tij , ... are the so-called CC singles, doubles, ... amplitudes yet to be determined. As before, 1 1 ERPA = Tr BT rCCD = B T rCCD . (43) i,j,... refer to occupied single-particle states, whereas c 2 2 ia,jb jb,ia virtual ij,ab a,b, ··· refer to unoccupied ( ) ones. Acting with X Tˆn on the non-interacting reference state |Φ0i generates abc··· The CCD formulation of RPA as given by Eq. (41) and n-order excited configuration denoted |Φijk··· i: (43) was shown by Scuseria et al. [13] to be analytically equivalent to the plasmonic formulation of RPA. ˆ abc··· abc··· Tn|Φ0i = tijk··· |Φijk··· i. (37) The latter has recently been discussed in detail by Furche i>j>k··· ,a>b>c··· X [38, 94], and and hence will not be presented in this review. Technically, the solution of the Riccati equation The next question is how to determine the expansion (41) is not unique, due to the non-linear nature of the coefficients tabc···? The CC many-body wave function ijk··· equation. One therefore has to make a judicious choice in Eq. (33) has to satisfy the many-body Schrödinger for the ring-CCD amplitudes in practical RPA calcula- equation, tions [114]. ˆ ˆ Heˆ T |Φi = EeT |Φi , (38) III. ALGORITHMS AND IMPLEMENTATIONS or

ˆ ˆ e−T Heˆ T |Φi = E|Φi . (39) A. RPA implementations and scaling

By projecting Eq. (39) onto the excited configurations In this section we will briefly review different imple- abc··· |Φijk··· i, which have zero overlap with the non-interacting mentations of the RPA approach, since scaling and effi- ground-state configuration |Φi, one obtains a set of cou- ciency are particularly important for a computationally abc··· pled non-linear equations for the CC amplitudes tijk··· , expensive approach like the RPA. Also, for historical reasons, the theoretical formulation of RPA is often linked abc··· −Tˆ ˆ Tˆ hΦijk··· |e He |Φi = 0 . (40) closely to a certain implementation. Similar to conventional DFT functionals, implementations of RPA can be These can be determined by solving Eqs. (40) self- based on local orbitals (LO), or on plane waves, or on consistently. (linearized) augmented plane waves (LAPW). LO im- In analogy to the Goldstone diagrams, equation (40) plementations have been reported for the development can be represented pictorially using diagrams, as illus- version of Gaussian [14, 16, 116], the development ver- trated by C´ıˇzekˇ [152] in 1966. In practice, the expansion of Molpro [15, 121], FHI-aims [131] and Turbomole sion of the Tˆ operator has to be truncated. One popular [10, 28]. Plane-wave based implementations can be found 10 in ABINIT [11], VASP [19, 110], and Quantum-Espresso an iterative diagonalization procedure, instead of con- [21, 133]. An early implementation by Miyake et al. [108] structing and diagonalizing the full ε(iω) or χ0(iω)v ma- was based on LAPW. trices. In practice this can be conveniently done by re- Furche’s original implementation uses a molecular sorting to the linear response technique of density func- particle-hole basis and scales as O(N 6) [10], where N is tional perturbation theory (DFPT) [135] and has been the number of atoms in the system (unit cell). This can proposed and implemented in the two independent works be reduced to O(N 5) using the plasmon-pole formula- of Galli and coworkers [21, 134], and of Nguyen and tion of RPA [94], and to O(N 4) [28] when the resolution- de Gironcoli [133] within a pseudopotential plane-wave of-identity (RI) technique is employed in addition. Our framework. In these (plane-wave based) implementations 2 own RPA implementation [131] in FHI-aims [132] is de- the computational cost is reduced from Npw-χNoccNvir to 2 2 scribed in Appendix A. It is based on localized numeric Npw-ψNoccNeig, where Npw-χ and Npw-ψ are the numbers atom-centered orbitals and the RI technique, and hence of plane waves to expand the response function χ0 and 4 naturally scales as O(N ). the single-particle orbitals ψ respectively, and Neig is the The key in the RI-RPA implementation is to expand number of dominant eigenvalues. In this way, although ∗ r r 4 the occupied-virtual orbital pair products φi ( )φj ( ) ap- the formal scaling is still O(N ), one achieves a large re- pearing in Eq. (20) in terms of a set of auxiliary ba- duction of the prefactor, said to be 100-1000 [133]. This r sis functions (ABFs) {Pµ( )}. In this way, one can re- procedure is in principle applicable to RI-RPA implemen- duce the rank of the matrix representation of χ0 from tation in local-basis sets, too, but has, to the best of our Nocc ∗ Nvir to Naux with Naux ≪ Nocc ∗ Nvir. Here Naux, knowledge, not been reported so far. Nocc, and Nvir denote the number of ABFs, and the numbers of occupied and unoccupied (virtual) single-particle orbitals respectively. With both χ0 and the coulomb kernel v represented in terms of the ABFs, the RPA correla- IV. COMPUTIONAL SCHEMES BEYOND RPA tion energy expression in Eq. (23) can be re-interpreted as a matrix equation of rank Naux, which is numerically In this Section we will give a brief account of the ma- very cheap to evaluate. The dominating step then be- jor activities for improving the standard RPA, aiming at comes the build of the matrix form of χ0 which scales better accuracy. as O(N 4). We refer the readers to Appendix A and Ref. [131] for further details. Plane-wave based implementations [11, 110] automatically have O(N 4) scaling. In a sense the plane-wave A. Semi-local and non-local corrections to RPA based RPA implementation is very similar in spirit to the local-orbital-based RI-RPA implementation. In the for- It is generally accepted that long-range interactions are mer case the plane waves themselves serve as the above- well described within RPA, whereas short-range correla- mentioned ABFs. tions are not adequate [104]. This deficiency manifests itself most clearly in the pair-correlation function of the HEG, which spuriously becomes negative when the sepa- B. Speed-up of RPA with iterative methods ration between two electrons gets small [103, 104]. Based on this observation, Perdew and coworkers [97, 105] pro- The RPA correlation energy in (23) can also be rewrit- posed a semi-local correction to RPA, termed as RPA+ ten as follows, RPA+ RPA GGA GGA-RPA Ec = Ec + Ec − Ec , (45) Naux 1 ∞ ERPA = − dω ln εD(iω) + 1 − εD(iω) , where EGGA is the GGA correlation energy, and c 2π µ µ c 0 µ GGA-RPA Z Ec represents the random-phase approximation X GGA (44) within GGA. Thus the difference between Ec and D GGA-RPA where εµ (iω) is the µth eigenvalue of the dielectric func- Ec gives a semi-local correction to RPA for in- tion ε(iω) = 1 − χ0(iω)v represented in the ABS. All homogeneous systems. As mentioned before in the intro- eigenvalues are larger than or equal to 1. From (44) it duction, the RPA+ scheme, although conceptually ap- is clear that eigenvalues which are close to 1 have a van- pealing, and good for total energies [18], does not sig- ishing contribution to the correlation energy. For a set nificantly improve the description of energy differences, of different materials, Wilson et al. [134] observed that in particular the atomization energies of small molecules only a small fraction of the eigenvalues differs signifi- [10]. This failure has been attributed to the inaccuracy cantly from 1, which suggests that the full spectrum of of RPA in describing the multi-center non-locality of the ε(iω) is not required for accurate RPA correlation en- correlation hole, which cannot be corrected by semi-local ergies. This opens up the possibility of computing the corrections of the RPA+ type [34, 35]. A fully non-local RPA correlation energy by obtaining the “most signifi- correction (nlc) to RPA has recently been proposed by cant” eigenvalues of ε(iω) (or equivalently χ0(iω)v) from Ruzsinszky, Perdew, and Csonka [35]. It takes the fol- 11 lowing form

Enlc = drn(r) ǫGGA(r) − ǫGGA-RPA(r) [1 − αF (f(r)]] , SOSEX c Ec = + + ... Z (46) where ǫGGA(r) and ǫGGA-RPA(r) are the GGA energy density per electron and its approximate value within FIG. 3: Goldstone diagrams for SOSEX contribution. The RPA, respectively. α is an empirical parameter yet to rules to evaluate Goldstone diagrams can be found in be determined, and F is a certain functional of f(r), the Ref. [143]. dimensionless ratio measuring the difference between the GGA exchange energy density and the exact-exchange energy density at a given point r, ing term in SOSEX corresponds to the second-order ex- ǫGGA(r) − ǫexact(r) change term of MP2. In analogy, the leading term in f(r)= x x . (47) ǫGGA(r) RPA corresponds to the second-order direct term of MP2. x Physically the second-order exchange diagram describes One may note that by setting α = 0 in Eq. (46) the usual a (virtual) process in which two particle-hole pairs are RPA+ correction term is recovered. A simple choice created spontaneously at a given time. The two particles of the functional form F (f) = f turns out to be good (or equivalently the two holes) then exchange their po- enough for fitting atomization energies, but the correct sitions, and two (already exchanged) particle-hole pairs dissociation limit of H2 given by standard RPA is de- annihilate themselves simultaneously at a later time. In stroyed. To overcome this problem, Ruzsinszky et al. SOSEX, similar to RPA, a sequence of higher-order dia- chose a more complex form of F , grams are summed up to infinity. In these higher-order diagrams, after the initial creation and exchange process, F (f)= f[1 − 7.2f 2][1 + 14.4f 2]exp(−7.2f 2), (48) one particle-hole pair is scattered into new positions re- peatedly following the same process as in RPA, until it which ensures the correct dissociation limit, while yield- annihilates simultaneously with the other pair at the end ing significantly improved atomization energies for α = 9. of the process. Up to now the correction scheme of Eq. (46) has not SOSEX is one-electron self-correlation free and ame- been widely benchmarked except for a small test set of liorates the short-range over-correlation problem of RPA 10 molecules where the atomization energy has been im- to a large extent, leading to significantly better total en- proved by a factor of two [35]. ergies [95, 123]. More importantly, the RPA underestimation of atomization energies is substantially reduced. B. Screened second-order exchange (SOSEX) However, the dissociation of covalent diatomic molecules, which is well described in RPA, worsens considerably as demonstrated in Ref. [16] and to be shown in Fig. 6. It The SOSEX correction [16, 95, 123] is an important was argued that the self-correlation error present in RPA route to go beyond standard RPA. This concept can be mimics static correlation, which becomes dominant in the most conveniently understood within the context of the dissociation limit of covalent molecules [114]. ring-CCD formulation of RPA as discussed in Sec. IID. If in Eq. (43), the anti-symmetrized Coulomb integrals B˜ia,jb = hij|abi−hij|bai are inserted instead of the un- symmetrized Coulomb integrals, the RPA+SOSEX correlation energy expression is obtained C. Single excitation correction and its combination with SOSEX 1 ERPA+SOSEX = T rCCDB˜ . (49) c 2 ia,jb ia,jb ij,ab In most practical calculations, RPA and SOSEX corre- X lation energies are evaluated using input orbitals from a This approach, which was first used by Freeman [95], preceding KS or generalized KS (gKS) [155] calculation. and recently examined by Grüneis et al. for solids [123] In this way both RPA and SOSEX can be interpreted as and Paier [16] for molecular properties, has received in- infinite-order summations of selected types of diagrams creasing attention in the RPA community. In contrast to within the MBPT framework introduced in Sec. IIC, as RPA+, this scheme has the attractive feature that it im- is evident from Figs (2c) and (3). This viewpoint is help- proves both total energies and energy differences simul- ful for identifying contributions missed in RPA through taneously. Although originally conceived in the CC con- the aid of diagrammatic techniques. An an example, the text, SOSEX has a clear representation in terms of Gold- second-order energy in RSPT in Eq. (28) have contribu- stone diagrams, as shown in Fig. 3 (see also Ref. [123]), tions from single excitations (SE) and double excitations. which can be compared to the Goldstone diagrams for The latter gives rise to the familiar MP2 correlation en- RPA in Fig. (2c). From Fig. 3, it is clear that the lead- ergy, which is included in the RPA+SOSEX scheme as 12

the leading term. The remaining SE term is given by ∆ via ∆ ∆ via via occ unocc i ∆ ∆v a ˆ a 2 vij ab SE |hΦ0|H1|Φi i| rSE a i + a + i + ... Ec Ec = (0) = j b E0 − E ∆v ∆ ∆v i a i,a ai vaj bi X X 2nd−order 3rd−order |hψ |vˆHF − vˆMF|ψ i|2 = i a (50) ǫ − ǫ ia i a FIG. 4: Goldstone diagrams for renormalized single excitation X contributions. Dashed line ending with a cross denotes the |hψ |fˆ|ψ i|2 HF MF = i a (51) matrix element ∆vpq = hψp|vˆ − vˆ |ψqi. ǫ − ǫ ia i a X wherev ˆHF is the self-consistent HF single-particle poten- in the rSE series. A preliminary version of rSE, which tial,v ˆMF is the mean-field potential associated with the neglects the “off-diagonal” terms of the higher-order SE reference Hamiltonian, and fˆ = −∇2/2 +v êxt +v ˆHF is diagrams (by setting i = j = ··· and a = b = ··· ), the single-particle HF Hamiltonian (also known as the was benchmarked for atomization energies and reaction Fock operator in the quantum chemistry literature). A barrier heights in Ref. [124]. Recently we were able to detailed derivation of Eq. (50) using second-quantization also include the “off-diagonal” terms, leading to a refined can be found in the supplemental material of Ref. [32]. version of rSE. This rSE “upgrade” does not affect the The equivalence of Eqs. (51) and (50) can be readily energetics of strongly bound molecules, as those bench- confirmed by observing the relation between fˆ and the marked in Ref. [124]. However, the interaction energies single-particle reference Hamiltonian hˆMF: fˆ = hˆMF + of weakly bound molecules improve considerably. A more detailed description of the computational procedure and vˆHF − vˆMF, and the fact hψ |hˆMF|ψ i = 0. Obviously i a extended benchmarks for rSE will be reported in a forth- for a HF reference wherev ˆMF =v ˆHF, Eq. (51) becomes coming paper [159]. However, we note that all the rSE zero, a fact known as Brillouin theorem [143]. Therefore, results reported in Section (Sec. V) correspond to the as mentioned in Section IIC, this term is not present in upgraded rSE. MP2 theory which is based on the HF reference. We Diagrammatically, RPA, SOSEX and rSE are three note that a similar SE term also appears in 2nd-order distinct infinite series of many-body terms, in which the Görling-Levy perturbation theory (GL2) [156, 157], ab three leading terms correspond to the three terms in 2nd- initio DFT [158], as well as in CC theory [151]. How- order RSPT. Thus it is quite natural to include all three ever, the SE terms in different theoretical frameworks of them, and the resultant RPA+SOSEX+rSE scheme differ quantitatively. For instance, in GL2 vMF should can be viewed a renormalization of the normal 2nd-order be the exact-exchange OEP potential instead of the ref- RSPT. Therefore we will refer to RPA+SOSEX+rSE as erence mean-field potential. “renormalized second-order perturbation theory” or r2PT In Ref. [32] we have shown that adding the SE term of in the following. Eq. (51) to RPA significantly improves the accuracy of vdW-bonded molecules, which the standard RPA scheme generally underbinds. This improvement carries over to atomization energies of covalent molecules and insulat- D. Other “beyond-RPA” activities ing solids as shown in Ref. [124]. It was also observed in Ref. [32] that a similar improvement can be achieved There have been several other attempts to go beyond by replacing the non-self-consistent HF part of the RPA RPA. Here we will only briefly discuss the essential con- total energy by its self-consistent counterpart. It ap- cepts behind these approaches without going into details. pears that, by iterating the exchange-only part towards The interested reader is referred to the corresponding ref- self-consistency, the SE effect can be accounted for effec- erences. Following the ACFD formalism, as reviewed in tively. This procedure is termed “hybrid-RPA”, and has Sec. IIB, one possible route is to improve the interact- been shown to be promising even for surface adsorption ing density response function. This can be conveniently problems [30]. done by adding the exchange correlation kernel (fxc) of The SE energy in Eq. (51) is a second-order term in time-dependent DFT [160, 161], that is omitted in RPA. RSPT, which suffers from the same divergence problem Fuchs et al. [162], as well as Heßelmann and Görling [122] as MP2 for systems with zero (direct) gap. To overcome have added the exact-exchange kernel to RPA, a scheme this problem, in Ref. [32] we have proposed to sum over termed by these authors as RPA+X or EXX-RPA, for a sequence of higher-order diagrams involving only single studying the H2 dissociation problem. RPA+X or EXX- excitations. This procedure can be illustrated in terms of RPA displays a similar dissociation behavior for H2 as Goldstone diagrams as shown in Fig. 4. This summation RPA: accurate at infinite separation, but slightly repul- follows the spirit of RPA and we denote it renormalized sive at intermediate bond lengths. This scheme however single excitations (rSE) [32]. The SE contribution to the gives rise to a noticeable improvement on the total en- 2nd-order correlation energy in Eq. (51), represented by ergy [121]. Furche and Voorhis examined the influence the first diagram in Fig 4, constitutes the leading term of several different local and non-local kernels on the at- 13 omization energies of small molecules and the binding V. APPLICATIONS energy curves of rare-gas dimers [12]. They found that semilocal fxc kernels lead to a diverging pair density at A. Molecules small inter-particle distances, and it is necessary to go to non-local fxc kernels to cure this. More work along these RPA based approaches have been extensively bench- lines has to be done before conclusions can be drawn and marked for molecular systems, ranging from the disso- accurate kernels become available. ciation behavior of diatomic molecules [10, 11, 14, 15, 120, 122, 138], atomization energies of small covalent Quantum chemistry offers another route to go beyond molecules [10, 16, 34, 35, 124], interaction energies of standard RPA by including higher-order exchange ef- weakly bonded molecular complex [26, 31, 117–119], and fects (often termed as “RPAx”). There the two-electron chemical reaction barrier heights [124]. The behavior Coulomb integrals usually appear in an antisymmetrized of RPA for breaking covalent bonds was examined in form, whereby exchange-type contributions, which are its early-day’s applications [10, 11], and today is still neglected in standard RPA, are included automatically a topic of immense interest [15, 120, 122, 138]. Atom- [99, 163]. The RPA correlation energy can be expressed ization energies of covalent molecules are somewhat dis- as a contraction between the ring-CCD amplitudes and appointing, because standard RPA, as well as its local the Coulomb integrals (see Eq. (43)), or alternatively be- correction (RPA+), is not better than semi-local DFT tween the coupling-strength-averaged density matrix and functionals [10]. This issue was subsequently referred to the Coulomb integrals (see Ref. [117]). Different flavors of as “the RPA atomization energy puzzle” [34]. A solu- RPA can therefore be constructed depending on whether tion can be found in the beyond RPA schemes SOSEX one antisymmetrizes the averaged density matrix and/or [16, 123] and SE [32, 124, 159]. Another major application area of RPA are weakly bonded molecules. Due to the Coulomb integrals (see Angyán´ et al. [119]). Accord- the seamless inclusion of the ubiquitous vdW interaction, ing to our definitions in this article, these schemes are cat- RPA clearly improves over conventional DFT function- egorized as different ways to go beyond standard RPA, als, including hybrids. This feature is very important while in the quantum chemistry community they might for systems where middle-ranged non-local electron cor- be referred to simply as RPA. The SOSEX correction, relations play a significant role, posing great challenges discussed in Sec. IVB, can also be rewritten in terms of to empirical or semi-empirical pairwise-based correction a coupling-strength-averaged density matrix [117]. An- schemes. Finally, for activation energies it turned out other interesting scheme was proposed by Heßelmann that standard RPA performs remarkably well [38, 124] [36], in which RPA is corrected to be exact at the third and the beyond RPA correction schemes that have been order of perturbation theory. These corrections show developed so far do not improve the accuracy of the stan- promising potential for the small molecules considered dard RPA [124]. in Ref. [36]. However, more benchmarks are needed for In the following, we will discuss the performance of a better assessment. RPA and its variants using representative examples to illustrate the aforementioned points. Both standard RPA and RPA with the exchange-type corrections discussed above have been tested in a range- separation framework by several authors [14, 15, 26, 116– 1. Dissociation of diatomic molecules 120]. As mentioned briefly in the introduction, the concept of range separation is similar to the RPA+ proce- The dissociation of diatomic molecules is an important dure, in which only the long-range behavior of RPA is re- test ground for electronic structure methods. The perfor- tained. However, instead of additional corrections RPA mance of RPA on prototypical molecules has been exam- at the short-range is completely removed and replaced ined in a number of studies [10, 11, 14, 15, 113, 120, 122, by semi-local or hybrid functionals. The price to pay 138]. Here we present a brief summary of the behavior are empirical parameters that control the range separa- of RPA-based approaches based on data generated using tion. The gain is better accuracy in describing molecular our in-house code FHI-aims [132]. The numerical details binding energies [14, 15], and increased computational and benchmark studies of our RPA implementation have efficiency. The latter is due to a reduction in the num- been presented in Ref. [131]. In Fig. 5 the binding en- ber of required basis functions to converge the long-range ergy curves obtained with PBE, MP2, and RPA-based RPA part, which is no longer affected by the cusp condi- methods are plotted for four molecular dimers, including tion. More details on range-separated RPA can be found two covalent molecules (H2 and N2), one purely vdW- in the original references [14, 15, 26, 116–120]. Com- bonded molecule (Ar2), and one with mixed character pared to the diagrammatic approaches discussed before, (Be2). Dunning’s Gaussian cc-pV6Z basis [164, 165] was the range-separation framework offers an alternative and used for H2 and N2, aug-cc-pV6Z for Ar2, and cc-pV5Z computationally more efficient way to handle short-range for Be2. Currently, no larger basis seems to be avail- correlations, albeit at the price of introducing some em- able for Be2, but this will not affect the discussion here. piricism into the theory. Basis-set superposition errors (BSSE) are corrected us- 14 ing the Boys-Bernardi counterpoise procedure [166]. Also itive bump remains. The SOSEX correction exhibits a plotted in Fig. 5 are accurate theoretical reference data complex behavior. While it reducing the bump it con- for H2, Ar2, and Be2 coming respectively from the full comitantly weakens binding around the equilibrium dis- CI approach [167], the Tang-Toennies model [168], and tance. Combining the corrections from SOSEX and rSE, the extended germinal model [169]. To visualize the cor- the r2PT does well at intermediate and large bonding responding asymptotic behavior more clearly, the large distances, but the binding energy at equilibrium is still bond distance regime of all curves is shown in Fig. 6. noticeably too small. Regarding MP2, it is impressive to RPA and RPA+ dissociate the covalent molecules cor- observe that this approach yields a binding energy curve rectly to their atomic limit at large separations, albeit that is in almost perfect agreement with the reference in from above after going through a positive “bump” at in- the asymptotic region, although a substantial underbind- termediate bond distances. The fact that spin-restricted ing can be seen around the equilibrium region. RPA calculations yield the correct H2 dissociation limit is Summarizing this part, RPA with and without correc- quite remarkable, given the fact that most spin-restricted tions shows potential, but at this point, none of the RPA- single-reference methods, including local and semi-local based approaches discussed above can produce quantita- DFT, Hartree-Fock, as well as the coupled cluster meth- tively accurate binding energy curves for all bonding sit- ods, yield an dissociation limit that is often too high in uations. It is possible, but we consider it unlikely, that energy, as illustrated in Fig. 5 for PBE. MP2 fails more iterating RPA to self-consistency will change this result. drastically, yielding diverging results in the dissociation Apart from applications to neutral molecules, RPA and limit for H2 and N2. The RPA+ binding curves follow RPA+SOSEX studies have been carried out for the dis- the RPA ones closely, with only minor differences. The sociation of charged molecules. RPA fails drastically in rSE corrections are also quite small in this case, shift- this case [113], giving too low a total energy in the disso- ing the RPA curves towards larger binding energies, with ciation limit. Adding SOSEX to RPA fixes this problem, the consequence that the binding energy dips slightly be- although the correction now overshoots (with the excep- + low zero in the dissociation limit (see the N2 example tion of H2 ) (see Ref. [16, 38, 113, 114] for more details). in Fig. 6). This shift however leads to better molecular binding energies around the equilibrium where RPA systematically underbinds. The SOSEX correction, on 2. Atomization energies: the G2-I set the other hand, leads to dramatic changes. Although “bump” free, SOSEX yields dissociation limits that are One important molecular property for thermochem- much too large, even larger than PBE. This effect carries mol at istry is the atomization energy, given by E − i Ei over to r2PT at large bond distances where rSE does not where Emol is the ground-state energy of a molecule and reduce the SOSEX overestimation. at P Ei that of the i-th isolated atom. According to this def- For the purely dispersion-bonded dimer Ar2, all RPA- inition, the negative of the atomization energy gives the based approaches, as well as MP2, yield the correct energy to break the molecule into its individual atoms. 6 C6/R asymptotic behavior, whereas the semi-local PBE Here we examine the accuracy of RPA-based approaches functional gives a too fast exponential decay. Quantita- for atomization energies of small molecules. The RPA re- tively, the C6 dispersion coefficient is underestimated by sults for a set of 10 small organic molecules were reported ∼ 9% within RPA (based on a PBE reference) [170], and in Furche’s seminal work [10] where the underestimation SOSEX or rSE will not change this. In contrast, MP2 of RPA for atomization energies was first observed. This overestimates the C6 value by ∼ 18% [170]. Around the benchmark set is included in their recent review [38]. equilibrium point, RPA and RPA+ underbind Ar2 sig- A widely accepted representative set for small organic nificantly. The rSE correction improves the results con- molecules is the G2-I set [141], that contains 55 cova- siderably, bringing the binding energy curve into close lent molecules and will be used as an illustrative example agreement with the Tang-Toennies reference curve. The here. The RPA-type results for the G2-I set have recently SOSEX correction, on the other hand, does very little in been reported in the work by Paier et al. [16, 124]. this case. As a consequence, r2PT resembles RPA+rSE In Fig 7 we present in a bar graph the mean absolute closely in striking contrast to the covalent molecules. percentage error (MAPE) for the G2-I atomization ener- Be2 represents a most complex situation, in which both gies obtained by four RPA-based approaches in addition static correlation and long-range vdW interactions play to GGA-PBE, the hybrid density-functional PBE0, and an important role. In the intermediate regime, the RPA MP2. The actual values for the the mean error (ME), and RPA+ binding energy curves display a positive bump mean absolute error (MAE), MAPE, and maximum ab- which is much more pronounced than for purely covalent solute percentage error (MaxAPE) are listed in table II molecules. At very large bonding distance, the curves in Appendix B. The calculations were performed us- cross the energy-zero line and eventually approach the ing FHI-aims [131, 132] with Dunning’s cc-pV6Z basis atomic limit from below. The rSE correction moves the [164, 165]. Reference data are taken from Ref. [171] and binding energy curve significantly further down, giving corrected for zero-point energies. Fig 7 and Table II il- binding energies in good agreement with the reference lustrate that among the three traditional approaches, the values, whereas in the intermediate region a small pos- hybrid functional PBE0 performs best, with a ME close 15

4 12 H N 2 8 2 2 4 0 0 -2 PBE -4 MP2 -8 -4 Binding energy (eV) RPA RPA+ -12 1 2 3RPA+rSE 4 5 6 1 2 3 4 5 6 RPA+SOSEX 200 8 Be Ar2 r2PT 2 4 Accurate 100 0 0 -4 -100 -8 -200 -12 Binding energy (meV) -300 3 4 5 6 7 8 2 3 4 5 6 7 Bond length (Å)

FIG. 5: Dissociation curves for H2, N2, Ar2, and Be2 using PBE, MP2, and RPA-based methods. All RPA-based methods use PBE orbitals as input. “Accurate” reference curves are obtained with the full CI method for H2 [167], the Tang-Toennies potential model for Ar2 [168], and the extended germinal model for Be2 [169].

4 20 H N 2 15 2 2 10 5 0 0 -5 Binding energy (eV) -2 -10 2 3 4 5 6 7 8 2 3 4 5 6 7 1 0 Be Ar 2 2 PBE 0 MP2 -2 RPA -1 RPA+ RPA+rSE -2 -4 RPA+SOSEX r2PT -3 Accurate Binding energy (meV) -6 -4 5 6 7 8 9 5 6 7 8 9 10 Bond length (Å)

FIG. 6: Asymptotic region of the curves in Fig. 5. to the “chemical accuracy” (1 kcal/mol = 43.4 meV). or rSE correction. And the combination of the two (i.e. MP2 comes second, and PBE yields the largest error and r2PT) further brings the MAE down to 3.3 kcal/mol, shows a general trend towards overbinding. Concerning comparable to the corresponding PBE0 value. Whether RPA-based approaches, standard RPA leads to ME and this mechanism of this improvement can be interpreted MAE that are even larger than the corresponding PBE in terms of the “multi-center non-locality of the correla- values, with a clear trend of underbinding. RPA+ does tion hole” as invoked by Ruzsinszky et al. [34, 35] or not, not improve the atomization energies [10]. All this is is not yet clear at the moment. in line with previous observations [10, 16, 124, 131]. As shown in the previous section, the underbinding of RPA for small molecules can be alleviated by adding SOSEX 16

8 60 7 50 6 40 5 4 30 3 MAPE (%)

MAPE (%) 20 2 10 1 0 0

RPA r2PT PBE MP2 RPA+ r2PT PBE0 PBE PBE0 MP2 RPA RPA+ RPA+rSE RPA+SOSEX RPA+SOSEX RPA+rSE

FIG. 7: Mean absolute percentage error (MAPE) for the FIG. 8: The MAPEs for the S22 test set obtained with RPA- G2-I atomization energies obtained with four RPA-based ap- based approaches in addition to PBE, PBE0, and MP2. The proaches in addition to PBE, PBE0 and MP2. “tier 4 + a5Z-d” basis set was used in the calculations.

3. vdW interactions: S22 set opinion, the most reliable RPA results (based on the PBE reference) for S22 so far. As discussed above, one prominent feature of RPA is In Fig 8 the relative errors (in percentage) of five RPA- that it captures vdW interactions that are of paramount based schemes are plotted for the molecules of the S22 importance for non-covalently bonded systems. Bench- set. Results for PBE, PBE0, and MP2 are also pre- marking RPA-based methods for vdW bonded systems sented for comparison. For MP2 and RPA-based meth- is an active research field [12, 14, 15, 19, 21, 26, 31, 32, ods, the relative errors (in percentage) for the 22 indi- 107, 117, 118]. Here we choose the S22 test set [142] as vidual molecules are further demonstrated in Fig 9. The the illustrating example to demonstrate the performance reference data were obtained using CCSD(T) and prop- of RPA for non-covalent interactions. This test set con- erly extrapolated to the CBS limit by Takatani et al. tains 22 weakly bound molecular complex of different size [180]. MP2 and RPA results are taken from Ref. [32]. and bonding type (7 of hydrogen bonding, 8 of dispersion The RPA+rSE, RPA+SOSEX, and r2PT results are pre- bonding, and 7 of mixed nature). Since its inception it sented for the first time. Further details for these calcu- has been widely adopted as the benchmark or training lations and an in-depth discussion will be presented in dataset for computational schemes that aim at dealing a forthcoming paper [159]. Figure 8 shows that PBE with non-covalent interactions [172–178] including RPA- and PBE0 fail drastically in this case, because these based approaches [26, 31, 32, 118]. The consensus emerg- two functionals do not capture vdW interactions by con- ing from these studies is as expected: RPA improves the struction, whereas all other methods show significant im- binding energies considerably over semilocal functionals. provement. Figure 9 further reveals that MP2 describes Quantitatively the MAEs given by standard RPA re- the hydrogen-bonded systems very accurately, but vastly ported for S22 by different groups show an unexpected overestimates the strength of dispersion interactions, par- spread. Specifically Zhu et al. [26] report an MAE of ticularly for the π-π stacking systems. Compared to 2.79 kcal/mol, Ren et al. 39 meV or 0.90 kcal/mol [32], MP2, RPA provides a more balanced description of all and Eshuis and Furche 0.41 kcal/mol [31]. The authors of bonding types, but shows a general trend to underbind. the latter study investigated this issue in detail [179] and It has been shown that this underbinding is significantly concluded that the discrepancy is due to basis set incom- reduced by adding SE corrections [32]. The renormal- pleteness and BSSE. Using Dunning’s correlation consis- ized SE correction presented here gives rise to a more tent basis sets plus diffuse functions and extrapolating systematic correction to RPA, as can be seen from Fig 9. to the complete basis set (CBS) limit Eshuis and Furche SOSEX shows a similar correction pattern as rSE for hy- obtained a MAE of 0.79 kcal/mol with 0.02 kcal/mol un- drogen bonding and mixed interactions, but has little certainty [179]. These authors confirmed our observation effect on the dispersion interaction. The r2PT scheme, that standard RPA generally underbinds weakly bound that combines SOSEX and rSE corrections, overshoots molecules. The basis set we have used, NAO tier 4 plus for hydrogen bonding, but on average improves the de- diffuse functions from aug-cc-pV5Z (denoted as “tier 4 scription of the other two bonding types. Figure 9 also + a5Z-d”) [32, 131], yields RPA results very close to the reveals that π-π stacking configurations, as exemplified CBS limit. The results reported in Ref. [32] are, in our by the benzene dimer in the slip parallel geometry (# 17

80 0.5 MP2 RPA 60 RPA+rSE RPA+SOSEX 0.4 40 r2PT 0.3 20

0 0.2 MAE (eV) -20 0.1 Relative error (%) -40 Hydrogen Dispersion Mixed 0 -60 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 PBE MP2 RPA r2PT PBE0 RPA+ RPA+rSE S22 molecules RPA+SOSEX

FIG. 9: The relative errors (in %) for the individual S22 FIG. 10: The MAEs for the HTBH38 (full bars) and molecules obtained with RPA-based approaches in addition NHTBH38 (hatched bars) test sets obtained with RPA-based to MP2. Connection lines are just guide to eyes approaches in addition to PBE, PBE0, and MP2. The cc- pV6Z basis set was used in the calculations.

11), represent the most challenging case for RPA-based methods. The relative error of RPA for this case is the reactions, whereas NHTBH38 contains 19 reactions in- largest. rSE provides little improvement, whereas SO- volving heavy atom transfers, nucleophilic substitutions, SEX worsens slightly. More work is needed to understand association, and unimolecular processes. The reference the origin of this failure. data were obtained using the “Weizmann-1” theory [185] The detailed errors (ME, MAE, MAPE, and MaxAPE) – a procedure to extrapolate the CCSD(T) results – or for S22 are presented in table III in Appendix B. Among by other “best theoretical estimates” [183]. Paier et al. the approaches we have investigated, RPA+rSE gives [124] presented results for standard RPA and “beyond the smallest MAE for S22, whiles r2PT gives the small- RPA” approaches based on the PBE reference, where est MAPE. Due to space restrictions it is not possi- a two-point cc-pVTZ→cc-pVQZ basis-set extrapolation ble to include the multitude of computational schemes strategy is used. In the work of Eshuis et al. [38], stan- that have emerged in recent years for dealing with non- dard RPA results based on both PBE and TPSS [186] covalent interactions [172–178] in Tab. III. Compared to references were presented, where the Def2-QZVP basis these approaches, the RPA-based approaches presented [187] was used. The RPA@PBE results for BH76 re- here are completely parameter-free and systematic in the ported by both groups are very close, with an ME/MAE sense that they have a clear diagrammatic representa- of -1.35/2.30 kcal/mol from the former and -1.65/3.10 tion. Thus RPA-based approaches are expected to have kcal/mol from the latter. a more general applicability, and may well serve as the The performance of RPA-based approaches, as well as reference for benchmarking other approaches for systems PBE, PBE0, and MP2 for HTBH38/NHTBH38 test sets where CCSD(T) calculations are not feasible. is demonstrated by the MAE bar graph in Fig. 10. The calculations were done using FHI-aims with the cc-pV6Z basis set. The ME, MAE, and the maximal absolute 4. Reaction barrier heights error (MaxAE) are further presented in table IV in Ap- pendix B. In this case we do not present the relative One stringent test for an electronic structure method errors, which turn out to be very sensitive to the compu- is its ability to predict chemical reaction barrier heights, tational parameters due to some small barrier heights in i.e., the energy difference between the reactants and their the test set, and hence cannot be used as a reliable mea- transition state. This is a central quantity that dic- sure of the performance of the approaches examined here. tates chemical kinetics. Semi-local density approxima- Compared to the results reported in Ref. [124], besides tions typically underestimate barrier heights [16, 181]. a different basis set (cc-pV6Z instead of cc-pVTZ→cc- RPA has already been benchmarked for barrier heights pVQZ extrapolation), we also used the refined rSE cor- in two independent studies [38, 124]. Both studies rection (see discussion in Section IVC) for the RPA+rSE used the test sets of 38 hydrogen-transfer barrier heights and r2PT results in Tab. IV, which gives slightly better (HTBH38) and 38 non-hydrogen-transfer barrier heights results in this case. (NHTBH38) designed by Zhao et al. [182, 183] (together On average, PBE underestimates the reaction bar- coined as BH76 in Ref. [184]). HTBH38 contains the for- rier heights substantially, a feature that is well-known ward and inverse barrier heights of 19 hydrogen transfer for GGA functionals. The hybrid PBE0 functional re- 18 duces both the ME and MAE by more than a factor of two. However, the remaining error is still sizable. Stan- TABLE I: ME, MAE, MAPE (%), and MaxAPE (%) for the atomization energies (in eV/atom), lattice constants (in A),˚ dard RPA does magnificently and shows a significant im- and bulk moduli (in Gpa) of 24 crystalline solids. Results are provement over PBE0. The performance of RPA+ is taken from Ref. [110]. The experimental atomization ener- again very similar to standard RPA. As already noted gies Ref. [188] are corrected for temperature effect (based on in Ref. [124], both the rSE and the SOSEX correction thermochemical correction data) [189] and zero-point vibra- deteriorate the performance of RPA. This is somewhat tional energy. The experimental lattice constants have been disappointing, and highlights the challenge for designing corrected for anharmonic expansion effect. simple, generally more accurate corrections to RPA. For- Atomization energies tunately, the errors of rSE and SOSEX are now in the opposite direction, and largely canceled out when com- ME (eV) MAE (eV) MAPE (%) MaxAPE (%) bining the two schemes. Indeed, the AEs and MAEs of LDA −0.74 0.74 18.0 32.7 r2PT are not far from their RPA counterparts, although PBE 0.15 0.17 4.5 15.4 the individual errors are more scattered in r2PT as man- RPA 0.30 0.30 7.3 13.5 ifested by the larger maximal absolute error. RPA+ 0.35 0.35 8.7 15.0 Lattice constants ME (A)˚ MAE(A)˚ MAPE (%) MaxAPE (%) B. Crystalline solids LDA −0.045 0.045 1.0 3.7 Crystalline solids are an important domain for RPA PBE 0.070 0.072 1.4 2.7 RPA 0.016 0.019 0.4 0.9 based approaches, in particular because the quantum RPA+ 0.029 0.030 0.6 1.1 chemical hierarchy of benchmark approaches cannot eas- ily be transfered to periodic systems. Over the years RPA Bulk Moduli calculations have been performed for a variety of systems ME (GPa) MAE (GPa) MAPE (%) MaxAPE (%) such as Si [108, 109, 133], Na [108], h-BN [17], NaCl [109], rare-gas solids [19], graphite [20, 27], and benzene crys- LDA 9 11 9.6 31.0 PBE −11 11 10.7 23.7 tals [21]. The most systematic benchmark study of RPA RPA −1 4 3.5 10.0 for crystalline solids was conducted by Harl, Schmika, RPA+ −3 5 3.8 11.4 and Kresse [20, 110]. These authors reported “technically converged” calculations using their VASP code and the projector augmented plane wave method for atomization energies, lattice constants, and bulk moduli of 24 representative crystals, including ionic compounds (MgO, LiF, NaF, LiCl, NaCl), semiconductors (C, Si, Ge, SiC, AlN, AlP, AlAs, GaN, GaP, GaAs, InP, InAs, InSb), and met- cannot be trusted in general, as the functional by con- als (Na, Al, Cu, Rh, Pd, Ag). The error analysis of their struction does not contain the necessary physics to re- RPA and RPA+ results, based on a PBE reference, as liably describe this phenomenon. In a recent work [27], well as the LDA and PBE results are presented in Tab. I. it was shown that RPA also reproduces the correct 1/d3 As is clear from Tab. I, the RPA lattice constants and asymptotics between graphite layers as analytically pre- bulk moduli are better than in LDA and PBE. The at- dicted by Dobson et al. [190]. This type of behavior omization energies, however, are systematically underes- cannot be described by LDA, GGAs, and hybrid func- timated in RPA, and the MAE in this case is even larger tionals. than that of PBE. This behavior is very similar to that for the atomization energies in the G2 set discussed above. Harl et al. also observed that the error of RPA does not For solids attempts have also been made to go beyond grow when going to heavier atoms, or open-shell systems the standard RPA. For a smaller test set of 11 insulators, in contrast to LDA or PBE [110]. The RPA+ results are Paier et al. [124] showed that adding SOSEX corrections in general slightly worse than those of standard RPA. to RPA the MAE of atomization energies is reduced from The performance of RPA has not been extensively 0.35 eV/atom to 0.14 eV/atom. By replacing the non- benchmarked for vdW-bound solids. However, from the self-consistent HF energy by its self-consistent counter- studies on h-BN [17], rare-gas crystals [19], benzene crys- part, which mimicks the effect of adding single excita- tal [21], and graphite [20], standard RPA does an excel- tion corrections [32], reduces the MAE further to 0.09 lent job regarding the equilibrium lattice constant and eV/atom [124]. Thus the trend in periodic insulators is cohesive energy, whereas semi-local DFT fails miserably, again in line with what has been observed for molecular yielding typically too weak binding and too large lat- atomization energies. The effects of SOSEX and rSE cor- tice constant. LDA typically gives a finite binding (of- rections for metals, and for other properties such as the ten overbinding) for weakly bonded solids, but its per- lattice constants and bulk moduli have not been reported formance varies significantly from system to system, and yet. 19

C. Adsorption at surfaces 1.6 on-top site 1.4 hollow site The interaction of atoms and molecules with surfaces plays a significant role in many phenomena in surface 1.2 science and for industrial applications. In practical cal- 1.0 culations, the super cells needed to model the surfaces (eV) 0.8 ads are large and a good electronic structure approach has E to give a balanced description for both the solid and the 0.6 adsorbate, as well as the interface between the two. Most 0.4 approaches today perform well for either the solid or the isolated adsorbate (e.g. atoms, molecules, or clusters), 0.2 but not for the combined system, or are computationally 0.0 PBE0 RPA@PBE too expensive to be applied to large super cells. This is LDA AM05 PBE RPA@PBE0 an area where we believe RPA will prove to be advanta- FIG. 11: Adsorption energies for CO adsorbed on the on- geous. top and fcc hollow sites of the Cu(111) surface as obtained The systems to which RPA has been applied include Xe using LDA, AM05, PBE, PBE0, and RPA. RPA results are and 3,4,9,10-perylene-tetracarboxylic acid dianhydride presented for both PBE and PBE0 references, and they differ (PTCDA) adsorbed on Ag(111) [23]; CO on Cu(111) very little. [20, 24] and other noble/transition metal surfaces [25]; benzene on Ni(111) [25], Si(001) [112], and the graphite surface[111]; and graphene on Ni(111) [191, 192], and Cu(111), Co(0001) surfaces [192]. In all these applications, RPA has been very successful. To illustrate how RPA works for an adsorbate system, but also produces a reasonable adsorption energy differ- here we briefly describe the RPA study of CO@Cu(111) ence of 0.22 eV, consistent with experiments. This result following Ref. [24]. The work was motivated by the so- was later confirmed by the periodic RPA calculations of called “CO adsorption puzzle” – LDA and several GGAs Harl and Kresse in Ref. [20], with only small numeri- predict the wrong adsorption site for CO adsorbed on cal differences arising from the different implementations several noble/transition metal surfaces at low coverage and different convergence strategy. [193]. For instance, for the (111) surface of Cu and Pt DFT within local/semi-local approximations erroneously The work of Rohlfing and Bredow on Xe and PTCDA favor the highly-coordinated hollow site, whereas exper- adsorbed at Ag(111) surface represents the first RPA iments clearly show that the singly-coordinated on-top study regarding surface adsorption problems, where the site is the energetically most stable site [194, 195]. This authors explicitly demonstrated that RPA yields the ex- 3 posed a severe challenge to the first-principles modeling pected −C3/(d−d0) behavior for large molecule-surface of molecular adsorption problems and the question arose, separations d. Schimka et al. extended the RPA bench- at what level of approximation can the correct physics be mark studies of the CO adsorption problem to more sur- recovered. In our study, the Cu surface was modeled us- faces [25]. They found that RPA is the only approach so ing systematically increased Cu clusters cut out of the far that gives both good adsorption energies as well as Cu(111) surface. Following a procedure proposed by Hu surface energies. GGAs and hybrid functionals at most et al. [196], the RPA adsorption energy was obtained yield either good surface energies, or adsorption ener- by first converging its difference to the PBE values with gies, but never both. Göltl and Hafner investigated the respect to cluster size, and then adding the converged adsorption of small alkanes in Na-exchanged chabazite difference to the periodic PBE results. The RPA adsorp- using RPA and several other approaches. They found tion energies for both the on-top and fcc (face centered that the “hybrid RPA” scheme, as proposed in Ref. [32] cubic) hollow sites are presented in Fig. 11, together with and further examined in Ref. [124], provides the most the results from LDA, AM05 [197], PBE, and the hybrid accurate description of the system compared to the al- PBE0 functional. Fig. 11 reveals what happens in the ternatives e.g. DFT-D [198] and vdW-DF [199]. More CO adsorption puzzle when climbing the so-called Ja- recently RPA was applied to the adsorption of benzene cob’s ladder in DFT [6] — going from the first two rungs on the Si(001) surface by Kim et al [112], graphene on the (LDA and GGAs) to the fourth (hybrid functionals), and Ni(111) surface by Mittendorfer et al [191] and by Olsen finally to the 5th (RPA and other functionals that explic- et al. [192], and additionally graphene on Cu(111) and itly depend on unoccupied KS states). Along the way the Co(0001) surfaces by the latter authors. In all these stud- magnitude of the adsorption energies on both sites are ies, RPA is able to capture the delicate balance between reduced, but the effect is more pronounced for the fcc covalent and dispersive interactions, and yields quanti- hollow site. The correct energy ordering is already re- tatively reliable results. We expect RPA to become in- stored in PBE0, but the energy between the two sites is creasingly more important in surface science with rising too small. RPA not only gives the correct adsorption site, computer power and more efficient implementations. 20

VI. DISCUSSION AND OUTLOOK atomic forces. Relaxations of atomic geometries that are common place in DFT and that make DFT RPA is an important concept in physics and has a more such a powerful method are currently not possi- than 50 year old history. Owing to its rapid development ble with RPA or have at least not been demon- in recent years, RPA has been established as a powerful strated yet. An efficient realization of RPA forces first-principles electronic-structure method with signifi- would therefore extend its field of application to cant implications for quantum chemistry, computational many more interesting and important material sci- physics, and materials science in the foreseeable future. ence problems. The rise of the RPA method in electronic structure the- iv. Self-consistency: Practical RPA calculations are ory, and its recent generalization to r2PT, were borne out predominantly done in a post-processing manner, by realizing that traditional DFT functionals (local and in which single-particle orbitals from KS or gen- semi-local approximations) are encountering noticeable eralized KS calculations are taken as input for a accuracy and reliability limits and that hybrid density one-shot RPA calculation. This introduces unde- functionals are not sufficient to overcome them. With sired uncertainties, although the starting-point de- the rapid development of computer hardwares and algo- pendence is often not very pronounced, if one re- rithms, it is not too ambitious to expect RPA-based ap- stricts the input to KS orbitals. A self-consistent proaches to become (or at least to inspire) main-stream RPA approach can be defined within KS-DFT electronic-structure methods in computational materials via the optimized effective potential method [93], science and engineering in the coming decades. At this and has already been applied in a few instances point it would be highly desirable if the community would [39, 40, 127, 128, 130]. However, in its current re- start to build up benchmark sets for materials science alizations self-consistent RPA is numerically very akin to the ones in quantum chemistry (e.g. G2 [141] or challenging, and a more practical, robust and nu- S22 [142]). These should include prototypical bulk crys- merically more efficient procedure will be of great tals, surfaces, and surface adsorbates and would aid the interest. development of RPA-based approaches. At this point, we would like to indicate several direc- With all these developments, we expect RPA and its tions for future developments of RPA-based methods. generalizations will play increasingly important roles in computational materials science in the near future. i. Improved accuracy: Although RPA does not suffer from the well-documented pathologies of LDA and GGAs, its quantitative accuracy is not always what Appendix A: RI-RPA implementation in FHI-aims is desired, in particular for atomization energies. To improve on this and to make RPA worth the computational effort, further corrections to RPA In this section we will briefly describe how RPA are necessary. To be useful in practice, these should is implemented in the FHI-aims code [132] using the not increase the computational cost significantly. resolution-of-identity (RI) technique. More details can The r2PT approach as presented in Sec. IV and be found in Ref. [131]. For a different formulation of RI- benchmarked in Sec. V is one example of this kind. RPA see Ref. [28, 38]. We start with the expression for More generally the aim is to develop RPA-based the RPA correlation energy in Eq. (20), which can be computational schemes that are close in accuracy formally expanded in a Taylor series, to CCSD(T), but come at a significantly reduced ∞ ∞ 1 1 2 numerical cost. More work can and should be done ERPA = − dω Tr χ0(iω)v . (A1) c π 2n along this direction. Z0 n=2 X h i ii. Reduction of the computational cost: The major Applying RI to RPA in this context means to represent factor that currently prevents the widespread use both χ0(iω) and v in an appropriate auxiliary basis set. of RPA in materials science is its high numeri- Eq. (A1) can then be cast into a series of matrix op- cal cost compared to traditional DFT methods. erations. To achieve this we perform the following RI The state-of-the-art implementations still have an expansion O(N 4) scaling, as discussed in Sec. III. To enlarge the domain of RPA applications, a reduction of this Naux ∗ r r µ r scaling behavior will be highly desirable. Ideas can ψi ( )ψj ( ) ≈ Cij Pµ( ) , (A2) µ=1 be borrowed from O(N) methods [137] developed X in quantum-chemistry (in particular in the context r µ of MP2) or compression techniques applied in the where Pµ( ) are auxiliary basis functions, Cij are the ex- GW context [136]. pansion coefficients, and Naux is the size of the auxiliary basis set. Here C serves as the transformation matrix iii. RPA forces: For a ground-state method, one cru- that reduces the rank of all matrices from Nocc ∗ Nvir cial component that is still missing in RPA are to Naux, with Nocc, Nvir and Naux being the number 21

2 of occupied single-particle orbitals, unoccupied (virtual) as NauxNoccNvir. We note that the same is true for stan- single-particle orbitals, and auxiliary basis functions, re- dard plane-wave implementations [110] where Naux corre- spectively. The determination of the C coefficients is sponds to the number of plane waves used to expand the not unique, but depends on the underlying metric. In response function Npw-χ. In that sense RI-based local- quantum chemistry the “Coulomb metric” is the stan- basis function implementations are very similar to plane- dard choice where the C coefficients are determined by wave-based or LAPW-based implementations, where the minimizing the Coulomb repulsion between the residuals plane waves themselves or the mixed product basis serve of the expansion in Eq. (A2) (for details see Ref. [131] as the auxiliary basis set. and references therein). In practice, sufficiently accurate auxiliary basis set can be constructed such that Naux << Nocc ∗ Nvir, thus reducing the computational Appendix B: Error statistics for G2-I, S22, and effort considerably. A practically accurate and efficient NHBH38/NHTBH38 test sets way of constructing auxiliary basis set {Pµ(r)} and their associated {Cµ } for atom-centered basis functions of gen- ij Tables II, III, and IV present a more detailed error eral shape has been presented in Ref. [131]. analysis for the G2-I, S22, and NHBH38/NHTBH38 test Combining Eq. (20) with (A2) yields sets. Given are the mean error (ME), the mean ab- µ ν solute error (MAE), the mean absolute percentage er- (fi − fj)C C χ0(r, r′,iω)= ij ji P (r)P (r′) ror (MAPE), the maximum absolute percentage error ǫ − ǫ − iω µ ν µν ij i j (MaxAPE), and the maximum absolute error (MaxAE). X X 0 r r′ = χµν (iω)Pµ( )Pν ( ), (A3) µν X TABLE II: ME (in eV), MAE (in eV), MAPE (%), where MaxAPE(%) for atomization energies of the G2-I set obtained with four RPA-based approaches in addition to PBE, PBE0 (f − f )Cµ Cν 0 i j ij ji and MP2. A negative ME indicates overbinding (on average) χµν (iω)= . (A4) ǫ − ǫ − iω and a positive ME underbinding. The cc-pV6Z basis set was ij i j X used. Introducing the Coulomb matrix ME MAE MAPE MaxAPE PBE -0.28 0.36 5.9 38.5 ′ ′ ′ Vµν = drr Pµ(r)v(r, r )Pν (r ) , (A5) PBE0 0.07 0.13 2.6 20.9 ZZ MP2 -0.08 0.28 4.7 24.6 we obtain the first term in (A1) RPA 0.46 0.46 6.1 24.2 RPA+ 0.48 0.48 6.3 24.9 ∞ RPA+SOSEX 0.22 0.25 4.2 36.7 (2) 1 r r r r′ Ec = − dω ··· d d 1d 2d × RPA+rSE 0.30 0.31 4.0 22.5 4π 0 r2PT 0.07 0.14 2.6 24.9 0 Z Z 0 Z ′ ′ χ (r, r1)v(r1, r2)χ (r2, r )v(r , r) 1 ∞ = − dω χ0 (iω)V χ0 (iω)V 4π µν να αβ β,µ Z0 µν,αβ ∞ X TABLE III: ME (in meV), MAE (in meV), MAPE (%), and 1 2 = − dωTr χ0(iω)V . (A6) MaxAPE(%) for the S22 test set [142] obtained with five RPA- 4π based approaches in addition to PBE, PBE0, and MP2 ob- Z0 h i tained with FHI-aims. The basis set “tier 4 + a5Z-d” [131] This term corresponds to the 2nd-order direct correla- was used in all calculations. tion energy also found in MP2. Similar equations hold ME MAE MAPE MaxAPE for the higher order terms in (A1). This suggests that PBE 116.2 116.2 57.8 170.3 the trace operation in Eq. (23) can be re-interpreted as a PBE0 105.7 106.5 55.2 169.1 summation over auxiliary basis function indices, namely, MP2 -26.5 37.1 18.7 85.1 0 r r′ RPA 37.8 37.8 16.1 28.7 Tr [AB] = µν Aµν Bνµ, provided that χ ( , ,iω) and v(r, r′) are represented in terms of a suitable set of auxil- RPA+ 51.2 51.2 21.9 39.4 RPA+rSE 14.1 14.8 7.7 24.8 iary basis functions.P Equations (23), (A4), and (A5) con- RPA+SOSEX 15.4 18.0 10.5 34.6 stitute a practical scheme for RI-RPA. In this implemen- r2PT -8.4 21.0 7.1 30.7 tation, the most expensive step is Eq. (A4) for the construction of the independent response function, scaling 22

TABLE IV: ME, MAE, and MaxAE (in eV) for the HTBH38 [182] and NHTBH38 [183] test sets obtained with four RPA- based approaches in addition to PBE, PBE0, and MP2, as obtained using FHI-aims. The cc-pV6Z basis set was used in all calculations. Negative ME indicates an underestimation of the barrier height on average. HTBH38 NHTBH38 ME MAE MaxAE ME MAE MaxAE PBE -0.399 0.402 0.863 -0.365 0.369 1.320 PBE0 -0.178 0.190 0.314 -0.134 0.155 0.609 MP2 0.1310.169 0.860 0.2150.226 1.182 RPA 0.000 0.066 0.267 -0.065 0.081 0.170 RPA+ 0.005 0.069 0.294 -0.068 0.084 0.168 RPA+rSE -0.170 0.187 0.809 -0.251 0.252 0.552 RPA+SOSEX 0.243 0.244 0.885 0.185 0.188 0.781 r2PT 0.072 0.084 0.453 -0.001 0.129 0.432

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