Random-phase approximation and its applications in computational chemistry and

materials science

Xinguo Ren,1 Patrick Rinke,1 Christian Joas,1,2 and Matthias Scheffler11Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195, Berlin, Germany

2Max-Planck-Institut fur Wissenschaftsgeschichte, Boltzmannstr. 22, 14195, Berlin, Germany

The random-phase approximation (RPA) as an approach for computing the electronic correlationenergy is reviewed. After a brief account of its basic concept and historical development, the paperis devoted to the theoretical formulations of RPA, and its applications to realistic systems. Withseveral illustrating applications, we discuss the implications of RPA for computational chemistryand materials science. The computational cost of RPA is also addressed which is critical for itswidespread use in future applications. In addition, directions of further development and currentcorrection schemes going beyond RPA will be discussed.

I. INTRODUCTION

Computational materials science has developed into anindispensable discipline complementary to experimentalmaterial science. The fundamental aim of computationalmaterials science is to derive understanding entirely fromthe basic laws of physics, i.e., quantum mechanical firstprinciples, and increasingly also to make predictions ofnew properties or new materials for specific tasks. Therapid increase in available computer power together withnew methodological developments are major factors inthe growing impact of this field for practical applicationsto real materials.

Density functional theory (DFT) [1] has shaped therealm of first principles materials science like no othermethod today. This success has been facilitated by thecomputational efficiency of the local-density [2] or gener-alized gradient approximation [3–5] (LDA and GGA) ofthe exchange-correlation functional that make DFT ap-plicable to polyatomic systems containing up to severalthousand atoms. However, these approximations are sub-ject to several well-known deficiencies. In the quest forfinding an “optimal” electronic structure method, thatcombines accuracy and tractability with transferabilityacross different chemical environments and dimension-alities (e.g. molecules/clusters, wires/tubes, surfaces,solids) many new approaches, improvements and refine-ments have been proposed over the years. These havebeen classified by Perdew in his “Jacob’s ladder” hierar-chy [6].

In this context, the treatment of exchange and cor-relation in terms of “exact-exchange plus correlation inthe random-phase approximation” [7, 8] offers a promis-ing avenue. This is largely due to three attractive fea-tures. The exact-exchange energy cancels the spuriousself-interaction error present in the Hartree energy ex-actly. The RPA correlation energy is fully non-localand includes long-range van der Waals (vdW) interac-tions automatically and seamlessly. Moreover, dynamicelectronic screening is taken into account by summingup a sequence of “ring” diagrams to infinite order, whichmakes EX+cRPA applicable to small-gap or metallic sys-tems where finite-order many-body perturbation theories

break down.The random-phase approximation actually predates

density-functional theory, but it took until the late 1970sto be formulated in the context of DFT [9] and until theearly years of this millennium to be applied as a firstprinciples electronic structure method [10, 11]. We takethe renewed and widespread interest of the RPA [10–40]as motivation for this review article. To illustrate theunique development of a powerful physical concept, wewill put the RPA into its historical context before review-ing the basic theory. A summary of recent RPA resultsdemonstrates the strength of this approach, but also itscurrent limitations. In addition we will discuss some ofthe most recent schemes of going beyond RPA, includingrenormalized second order perturbation theory (r2PT),which is particularly promising in our opinion. We alsoaddress the issue of computational efficiency which, atpresent, impedes the widespread use of RPA, and indi-cate directions for further development.

A. Early history

During the 1950s, quantum many-body theory under-went a major transformation, as concepts and techniquesoriginating from quantum electrodynamics (QED)—inparticular Feynman-Dyson diagrammatic perturbationtheory—were extended to the study of solids and nu-clei. A particularly important contribution at an earlystage of this development was the RPA, a technique in-troduced by Bohm and Pines in a series of papers pub-lished in 1951–1953 [7, 41–43]. In recent years, the RPAhas gained importance well beyond its initial realm ofapplication, in computational condensed-matter physics,materials science, and quantum chemistry. While theRPA is commonly used within its diagrammatic formu-lation given by Gell-Mann and Brueckner [44], it is nev-ertheless instructive to briefly discuss the history of itsoriginal formulation by Bohm and Pines. Some of thecited references are reprinted in [45], which also recountsthe history of the RPA until the early 1960s. Historicalaccounts of the work of Bohm and Pines can be found inRefs. [46–48] as well as in Refs. [33, 38].

2

In 1933–1934, Wigner and Seitz published two paperson the band structure of metallic sodium [49, 50] in whichthey stressed the importance of the electronic correlationenergy correction to band-theory calculations of the co-hesive energy of metals. Wigner subsequently studiedthe interaction of electrons in a homogeneous electrongas (HEG) within a variational approach going beyondHartree-Fock [51]. He initially provided estimates for thecorrelation energy only in the low- and high-density lim-its, but later interpolated to intermediate densities [52].For almost two decades, Wigner’s estimate remained thestate of the art in the prototypical many-body problem ofthe HEG. According to a later statement by Herring,“themagnitude and role of correlation energy remained inad-equately understood in a considerable part of the solid-state community for many years.” [53, pp. 71-72]

Due to the long-range nature of the Coulomb inter-action and the resulting divergences, perturbative ap-proaches, so successful in other areas, had to be com-plemented with approximations that accounted for thescreening of the charge of an electron by the other elec-trons. Before the 1950s, these approximations commonlydrew on work on classical electrolytes by Debye andHuckel [54, 55] and on work on heavy atoms by Thomas[56] and Fermi [57, 58], as well as later extensions [59, 60].

As a reaction to work by Landsberg and Wohlfarth[61, 62], Bohm and Pines in 1950 reported to have beenled “independently to the concept of an effective screenedCoulomb force as a result of a systematical classical andquantum-mechanical investigation of the interaction ofcharges in an electron gas of high density” [63, p. 103].Their 1951–1953 series of papers [7, 41–43] presents thissystematical investigation. The RPA was one of severalphysically-motivated approximations in the treatment ofthe HEG which allowed them to separate collective de-grees of freedom (plasma oscillations) from single-particledegrees of freedom (which today would be called quasi-particles or charged excitations) via a suitable canonicaltransformation reminiscent of early work in QED [64–66].A similar theory was developed rather independently fornuclei by Bohr and Mottelson [67].

In their first paper, illustrating the fundamental ideaof separating single-particle and collective degrees of free-dom, Bohm and Pines introduce RPA as one of four re-quirements [41]:

“(3) We distinguish between two kinds ofresponse of the electrons to a wave. Oneof these is in phase with the wave, so thatthe phase difference between the particle re-sponse and the wave producing it is indepen-dent of the position of the particle. This isthe response which contributes to the orga-nized behavior of the system. The other re-sponse has a phase difference with the waveproducing it which depends on the position ofthe particle. Because of the general randomlocation of the particles, this second responsetends to average out to zero when we consider

a large number of electrons, and we shall ne-glect the contributions arising from this. Thisprocedure we call the random phase approxi-mation.”

In their second paper [42], Bohm and Pines develop adetailed physical picture for the electronic behavior ina HEG due to the presence of Coulomb interactions.Only in their third paper, Bohm and Pines treat the(Coulomb-)interacting HEG quantum-mechanically [7].The RPA enables Bohm and Pines to absorb the long-range Coulomb interactions into the collective behaviorof the system, leaving the single-particle degrees of free-dom interacting only via a short-range screened force.The RPA amounts to neglecting the interaction betweenthe collective and the single-particle degrees of freedom.Consequently, the momentum transfers of the Coulombpotential in Fourier space can be treated independently.The fourth paper [43] applies the new method to theelectron gas in metals, discussing both validity and con-sequences of the RPA, such as the increase in electroniceffective mass.

Within condensed-matter theory, the significance ofthe Bohm-Pines approach quickly became apparent:Renormalizing the long-range Coulomb interaction intoan effective screened interaction between new, effectivesingle-particle degrees of freedom allowed both to over-come the divergences appearing in older theories of in-teracting many-body systems and to explain the hith-erto puzzling success of the single-particle models of earlycondensed-matter theory (see, e.g., [52]). An early appli-cation of the RPA was Lindhard’s calculation of the di-electric function of the electron gas [68]. Alternative ap-proaches to and extensions of the Bohm-Pines approachwere formulated by Tomonaga [69, 70], and by Mott [71],Frohlich and Pelzer [72], and Hubbard [73, 74].

In 1956, Landau’s Fermi liquid theory [75] deliveredthe foundation for effective theories describing many-body systems in terms of quasiparticles. Brueckner [76]already in 1955 had introduced a “linked-cluster ex-pansion” for the treatment of nuclear matter (see alsoRef. [77]). In 1957, Goldstone [78], using Feynman-likediagrams (based on Ref. [79]), was able to show thatBrueckner’s theory is exact for the ground-state energyof an interacting many-fermion system. This put theanalogy between the QED vacuum and the ground stateof a many-body system on firm ground. It had been in-troduced explicitly by Miyazawa for nuclei [80] and bySalam for superconductors [81], although the essence ofthe analogy dated back to the early days of quantum fieldtheory in the 1930s.

In late 1956, Gell-Mann and Brueckner employed adiagrammatic approach for treating the problem of theinteracting electron gas. Their famous 1957 paper [44]eliminated the spurious divergences appearing in previ-ous approaches. Expressing the perturbation series forthe correlation energy of the HEG in terms of the Wigner-Seitz radius rs, they found that the divergences withinearlier calculations [e.g., 82] were mere artifacts: The log-

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arithmic divergence appearing in the perturbative expan-sion of the correlation energy is canceled by similar di-vergences in higher-order terms. Summing the diagrams(which had a ring structure) to infinite order yielded a ge-ometric series, and thus a convergent result. Gell-Mannand Brueckner derived an expression for the ground stateenergy of the interacting electron gas in the high-densitylimit. Their work, and Goldstone’s paper [78], are theearliest examples of the application of Feynman-type di-agrammatic methods in condensed-matter theory.

Many applications of the new quantum-field theoreti-cal methods followed: Gell-Mann calculated the specificheat of the high-density HEG [83]; Hubbard [84, 85] pro-vided a description of the collective modes in terms ofmany-body perturbation theory (MBPT); Sawada et al.[86, 87] demonstrated that the Gell-Mann-Brueckner ap-proach indeed contained the plasma oscillations of Bohmand Pines [7], a point around which there had been quitesome confusion initially [see, e.g., 88]. In addition, theydemonstrated that the RPA is exact in the high-densitylimit. In 1958, Nozieres and Pines formulated a many-body theory of the dielectric constant and showed theequivalence of Gell-Mann and Brueckner’s diagrammaticapproach and the RPA [89, 90].

B. RPA in modern times

Today, the concept of RPA has gone far beyond thedomain of the HEG, and has gained considerable im-portance in computational physics and quantum chem-istry. As a key example, RPA can be introduced withinthe framework of DFT [1] via the so-called adiabatic-connection fluctuation-dissipation (ACFD) theorem [9,91, 92]. Within this formulation, the unknown exactexchange-correlation (XC) energy in Kohn-Sham [2] DFTcan be formally constructed by adiabatically switching onthe Coulomb interaction between electrons, while keepingthe electron density fixed at its physical value. This is for-mulated by a coupling-strength integration under whichthe integrand is related to the linear density-responsefunction of fictitious systems with scaled Coulomb inter-action. Thus, an approximation to the response functiondirectly translates into an approximate DFT XC energyfunctional. RPA in this context is known as an orbital-dependent energy functional [93] obtained by applyingthe time-dependent Hartree approximation to the den-sity response function.

The versatility of RPA becomes apparent when con-sidering alternative formulations. For instance, the RPAcorrelation energy may be understood as the shift of thezero-point plasmon excitation energies between the non-interacting and the fully interacting system, as shown bySawada for HEG [87], and derived in detail by Furche[94] for general cases (see also Ref. [29]). In quantumchemistry, RPA can also be interpreted as an approxi-mation to coupled-cluster doubles (CCD) theory whereonly diagrams of “ring” structure are kept [13, 95]. The

equivalence of the “plasmon” and “ring-CCD formula-tion” of RPA has recently been established by Scuseriaet al. [13]. These new perspectives not only offer moreinsight into the theory, but also help to devise more ef-ficient algorithms to reduce the computational cost, e.g.,by applying the Cholesky decomposition to the “ring-CCD” equations [13].

Following the early work on the HEG, other modelelectron systems were investigated, including the HEGsurface [96, 97], jellium slabs [22] and jellium spheres[98]. The long-range behavior of RPA for spatially well-separated closed-shell subsystems was examined by Sz-abo and Ostlund [99], as well as by Dobson [100–102].These authors showed that RPA yields the correct 1/R6

asymptotic behavior for the subsystem interaction. Inaddition, the long-range dispersion interaction of RPAis fully consistent with the monomer polarizability com-puted at the same level of theory. This is one of themain reasons for the revival of the RPA in recent years,because this long-range interaction is absent from LDA,GGA, and other popular density-functionals. Other rea-sons are the compatibility of the RPA correlation withexact exchange (which implies the exact cancellation ofthe self-interaction error present in the Hartree term) andthe applicability to metallic systems.

For the HEG it has been demonstrated that RPA is notaccurate for short-range correlation [103, 104], and hencefor a long time RPA was not considered to be valuablefor realistic systems. Perdew and coworkers investigatedthis issue [97, 105], and found that a local/semi-local cor-rection to RPA has little effect on iso-electronic energydifferences, which suggests that RPA might be accurateenough for many practical purposes. The application ofRPA to realistic systems appeared slightly later, startingwith the pioneering work of Furche [10], and Fuchs andGonze [11] for small molecules. Accurate RPA total en-ergies for closed-shell atoms were obtained by Jiang andEngel [18]. Several groups investigated molecular proper-ties, in particular in the weakly bound regime, with RPAand its variants [12, 15, 26, 28, 31, 32, 106, 107], whileothers applied RPA to periodic systems [17, 19–21, 108–110]. Harl and Kresse in particular have performed ex-tensive RPA benchmark studies for crystalline solids ofall bonding types [19, 20, 110]. At the same time, the ap-plication of RPA to surface adsorption problems has beenreported [23–25, 30, 111, 112], with considerable successin resolving the “CO adsorption puzzle”.

Most practical RPA calculations in recent years havebeen performed non-self-consistently based on a preced-ing LDA or GGA reference calculation. In these calcu-lations, the Coulomb integrals are usually not antisym-metrized in the evaluation of the RPA correlation energy,a practice sometimes called direct RPA in the quantum-chemical literature. In this paper we will denote this com-mon procedure “standard RPA” to distinguish it frommore sophisticated procedures. While a critical assess-ment of RPA is emerging and a wide variety of applica-tions are pursued, certain shortcomings of standard RPA

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have been noted. The most prominent is the systematicunderestimation of RPA binding energies [10, 20, 32], andthe failure to describe stretched radicals [16, 113, 114].Over the years several attempts have been made to im-prove upon RPA. The earliest is RPA+, where, as men-tioned above, a local/semi-local correlation correctionbased on LDA or GGA is added to the standard RPAcorrelation energy [97, 105]. Based on the observationthat in molecules the correlation hole is not sufficientlyaccurate at medium range in RPA, this has recently beenextended to a non-local correction scheme [34, 35]. Sim-ilarly, range-separated frameworks [115] have been tried,in which only the long-range part of RPA is explicitlyincluded [14, 15, 26, 116–120], whereas short/mid-rangecorrelation is treated differently. Omitting the short-range part in RPA is also numerically beneficial wherebythe slow convergence with respect to the number of basisfunctions can be circumvented. Due to this additionalappealing fact, range-separated RPA is now an activeresearch domain despite the empirical parameters thatgovern the range separation. Another route to improveRPA in the framework of ACFD is to add an fxc kernelto the response function and to find suitable approxima-tions for it [12, 33, 121, 122]. Last but not least, the CCDperspective offers a different correction in form of thesecond-order screened exchange (SOSEX) contribution[16, 95, 123], whereas the MBPT perspective inspired sin-gle excitation (SE) corrections [32]. SOSEX and SE aredistinct many-body corrections and can also be combined[124]. These corrections have a clear diagrammatic repre-sentation and alleviate the above-mentioned underbind-ing problem of standard RPA considerably [124]. An-other proposal to improve RPA by incorporating higher-order exchange effects in various ways has also been dis-cussed recently [119]. However, at this point in time, aconsensus regarding the “optimal” correction that com-bines both efficiency and accuracy has not been reached.

Although the majority of practical RPA calculationsare performed as post-processing of a preceding DFTcalculation, self-consistent RPA calculations have alsobeen performed within the optimized effective potential(OEP) framework. OEP is a procedure to find the opti-mal local multiplicative potential that minimizes orbital-dependent energy functionals. The first RPA-OEP cal-culations actually date back more than 20 years, butwere not recognized as such. Godby, Schluter and Shamsolved the Sham-Schluter equation for the GW self-energy [125, 126], which is equivalent to the RPA-OEPequation, for the self-consistent RPA KS potential of bulksilicon and other semiconductors, but did not calculateRPA ground-state energies. Similar calculations for otherbulk materials followed later by Kotani [127] and Gruninget al.[128, 129]. Hellgren and von Barth [130] and thenlater Verma and Bartlett [40] have looked at closed-shellatoms and observed that the OEP-RPA KS potentialthere reproduces the exact asymptotic behavior in thevalence region, although its behavior near the nucleusis not very accurate. Extensions to diatomic molecules

have also appeared recently [39, 40]. Our own workon the SE correction to RPA indicates that the input-orbital dependence in RPA post-processing calculationsis a significant issue. Some form of self-consistency wouldtherefore be desirable. However, due to the considerablenumerical effort associated with OEP-RPA calculations,practical RPA calculations will probably remain of thepost-processing type in the near future.

Despite RPA’s appealing features its widespread use inchemistry and materials science is impeded by its com-putational cost, which is considerable compared to con-ventional (semi)local DFT functionals. Furche’s originalimplementation based on a molecular particle-hole basisscales as O(N6) [10]. This can be reduced to O(N5) usingthe plasmon-pole formulation of RPA [94], and to O(N4)[28] when the resolution-of-identity (RI) technique is em-ployed in addition. Our own RPA implementation [131]in FHI-aims [132], which has been used in productioncalculations [24, 32, 124] before, is based on localized nu-meric atom-centered orbitals and the RI technique, andhence naturally scales as O(N4). Plane-wave based im-plementations [11, 110] also automatically have O(N4)scaling. However, in all current implementations theconvergence with respect to unoccupied states is slow.Proposals to eliminate the dependence on the unoccu-pied states [133, 134] by obtaining the response functionfrom density-functional perturbation theory [135] havenot been explored so far. In local orbital based ap-proaches the O(N4) scaling can certainly be reduced byexploiting matrix sparsity, as demonstrated recently inthe context of GW [136] or second-order Møller-Plessetperturbation theory (MP2) [137]. Also approximationsto RPA [29] or effective screening models [138] might sig-nificantly improve the scaling and the computational effi-ciency. Recently, RPA has been cast into the continuummechanics formulation of DFT [139] with considerablesuccess in terms of computational efficiency [140]. In gen-eral, there is still room for improvement, which, togetherwith the rapid increase in computer power makes us con-fident that RPA-type approaches will become a powerfultechnique in computational chemistry and materials sci-ence in the future. It would thus be desirable, if the ma-terial science community would start to build up bench-mark sets for materials science akin to the ones in quan-tum chemistry (e.g. G2 [141] or S22 [142]). These shouldinclude prototypical bulk crystals, surfaces, and surfaceadsorbates and would aid the development of RPA-basedapproaches.

II. THEORY AND CONCEPTS

RPA can be formulated within different theoreticalframeworks. One particularly convenient approach toderive RPA is the so-called “adiabatic connection (AC)”,which is a powerful mathematical technique to obtain theground-state total energy of an interacting many-particlesystem. Starting with the AC approach, the interacting

5

ground-state energy can be retrieved either by couplingto the fluctuation-dissipation theorem in the DFT con-text, or by invoking the Green-function based MBPT.RPA can be derived within both frameworks. In addi-tion, RPA is also intimately linked to the coupled-cluster(CC) theory. In this section, we will present the theoret-ical aspects of RPA from several different perspectives.

A. Adiabatic connection

The ground-state total energy of an interacting many-body Hamiltonian can formally be obtained via the ACtechnique, in which a continuous set of coupling-strength(λ) dependent Hamiltonians is introduced

H(λ) = H0 + λH1(λ), (1)

that “connect” a reference Hamiltonian H0 = H(λ = 0)

with the target many-body Hamiltonian H = H(λ = 1).

For the electronic systems considered here, H(λ) has thefollowing form:

H(λ) =

N∑

i=1

[

−1

2∇2i + vextλ (i)

]

+

N∑

i>j=1

λ

|ri − rj |, (2)

where N is the number of electrons, vextλ is a λ-dependentexternal potential with vextλ=1(r) = vext(r) being the phys-ical external potential of the fully-interacting system.Note that in general vextλ can be non-local in space forλ 6= 1. Hartree atomic units ~ = e = me = 1 are usedhere and in the following. The reference Hamiltonian H0,given by Eq. (2) for λ = 0, is of the mean-field (MF) type,i.e., a simple summation over single-particle Hamiltoni-ans:

H0 =

N∑

i=1

[

−1

2∇2i + vextλ=0(i)

]

=N∑

i=1

[

−1

2∇2i + vext(ri) + vMF(i)

]

. (3)

In Eq. (3), vMF is a certain (yet-to-be-specified) mean-field potential arising from the electron-electron interac-tion. It can be the Hartree-Fock (HF) potential vHF orthe Hartree plus exchange-correlation potential vHxc inDFT. Given Eq. (2) and (3), the perturbative Hamilto-

nian H1(λ) in Eq. (1) becomes

H1(λ) =N∑

i>j=1

1

|ri − rj |+

1

λ

N∑

i=1

[

vextλ (ri) − vextλ=0(i)]

,

=N∑

i>j=1

1

|ri − rj |+

1

λ

N∑

i=1

[

vextλ (ri) − vext(ri) − vMF(i)]

.

(4)

In the AC construction of the total energy, we introducethe ground-state wave function |Ψλ〉 for the λ-dependentsystem such that

H(λ)|Ψλ〉 = E(λ)|Ψλ〉 . (5)

Adopting the normalization condition 〈Ψλ|Ψλ〉 = 1, theinteracting ground-state total energy can then be ob-tained with the aid of the Hellmann-Feynman theorem,

E(λ = 1) =E0 +

∫ 1

0

dλ×

〈Ψλ|

(

H1(λ) + λdH1(λ)

dλ

)

|Ψλ〉 , (6)

where

E0 = E(0) = 〈Ψ0|H0|Ψ0〉 (7)

is the zeroth-order energy. We note that the choice ofthe adiabatic-connection path in Eq. (6) is not unique.In DFT, the path is chosen such that the electron densityis kept fixed at its physical value along the way. This im-plies a non-trivial (not explicitly known) λ-dependence

of H1(λ). In MBPT, one often chooses a linear connec-

tion path — H1(λ) = H1 (and hence dH1(λ)/dλ = 0).In this case, a Taylor expansion of |Ψλ〉 in terms of λ inEq. (6) leads to standard Rayleigh-Schrodinger pertur-bation theory (RSPT) [143].

B. RPA derived from ACFD

Here we briefly introduce the concept of RPA in thecontext of DFT, which serves as the foundation for mostpractical RPA calculations in recent years. In Kohn-Sham (KS) DFT, the ground-state total energy for aninteracting N -electron systems is an (implicit) functionalof the electron density n(r) and can be conveniently splitinto four terms:

E[n(r)] = Ts[ψi(r)] +Eext[n(r)] +EH[n(r)] +Exc[ψi(r)] .(8)

Ts is the kinetic energy of the KS independent-particlesystem, Eext the energy due to external potentials,EH the classic Hartree energy, and Exc the exchange-correlation energy. In the KS framework, the electrondensity is obtained from the single-particle KS orbitalsψi(r) via n(r) =

∑occi |ψocc

i (r)|2. Among the four termsin Eq. (8), only Eext[n(r)] and EH[n(r)] are explicit func-tionals of n(r). Ts is treated exactly in KS-DFT in termsof the single-particle orbitals ψi(r) which themselves arefunctionals of n(r).

All the many-body complexity is contained in the un-known XC energy term, which is approximated as anexplicit functional of n(r) (and its local gradients) in con-ventional functionals, (LDA, GGAs, meta-GGAs), and asa functional of the ψi(r)’s in more advanced functionals

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(hybrid density functionals, RPA, etc.). Different exist-ing approximations to Exc can be classified into a hier-archical scheme known as “Jacob’s ladder” [6] in DFT.However, what if one would like to improve the accuracyof Exc in a more systematic way? For this purpose it isilluminating to start with the formally exact way of con-structing Exc using the AC technique discussed above.As alluded to before, in KS-DFT the AC path is chosensuch that the electron density is kept fixed. Equation (6)for the exact ground-state total-energy E = E(λ = 1)then reduces to

E = E0+

∫ 1

0

dλ〈Ψλ|1

2

N∑

i6=j=1

1

|ri − rj ||Ψλ〉

+

∫ 1

0

dλ〈Ψλ|

N∑

i=1

d

dλvextλ (ri)|Ψλ〉

= E0+1

2

∫ 1

0

dλ

∫∫

drdr′×

〈Ψλ|n(r) [n(r′) − δ(r− r

′)]

|r− r′|

|Ψλ〉

+

∫

drn(r)[

vextλ=1(r) − vextλ=0(r)]

, (9)

where

n(r) =

N∑

i=1

δ(r− ri) (10)

is the electron-density operator, and n(r) =〈Ψλ|n(r)|Ψλ〉 for any 0 ≤ λ ≤ 1.

For the KS reference state |Ψ0〉 (given by the Slaterdeterminant of the occupied single-particle KS orbitals{ψi(r)}) we obtain

E0 = 〈Ψ0|N∑

i=1

[

−1

2∇2 + vextλ=0(ri)

]

|Ψ0〉

= Ts [ψi(r)] +

∫

drn(r)vextλ=0(r) , (11)

and thus

E =Ts [ψi(r)] +

∫

drn(r)vextλ=1(r)+

1

2

∫ 1

0

dλ

∫∫

drdr′〈Ψλ|n(r) [n(r′) − δ(r− r

′)]

|r− r′|

|Ψλ〉 .

(12)

Equating (8) and (12), and noticing

EH[n(r)] =1

2

∫

drdr′n(r)n(r′)

|r− r′|

(13)

Eext[n(r)] =

∫

drn(r)vextλ=1(r) , (14)

one obtains the formally exact expression for the XC en-ergy

Exc =1

2

∫

dλ

∫∫

drdr′nλxc(r, r

′)n(r)

|r− r′|

. (15)

Here

nλxc(r, r′) =

〈Ψλ|δn(r)δn(r′)|Ψλ〉

n(r)− δ(r− r

′) , (16)

is the formal expression for the so-called XC hole, withδn(r) = n(r) − n(r) being the fluctuation of the densityoperator n(r) around its expectation value n(r). Equa-tion (16) shows that the XC hole is related to the density-density correlation function. In physical terms, it de-scribes how the presence of an electron at point r depletesthe density of all other electrons at another point r

′.In the second step, the density-density correlations

(fluctuations) in Eq. (16) are linked to the responseproperties (dissipation) of the system through the zero-temperature fluctuation-dissipation theorem [144]

〈Ψλ|δn(r)δn(r′)|Ψλ〉 = −1

π

∫ ∞

0

dωImχλ(r, r′, ω) , (17)

where χλ(r, r′, ω) is the linear density response func-tion of the (λ-scaled) system. Using Eqs. (15-17) andv(r, r′) = 1/|r − r

′|, we arrive at the renowned ACFDexpression for the XC energy in DFT

Exc =1

2

∫ 1

0

dλ

∫∫

drdr′v(r, r′) ×

[

−1

π

∫ ∞

0

dωImχλ(r, r′, ω) − δ(r− r′)n(r)

]

=1

2π

∫ 1

0

dλ

∫∫

drdr′v(r, r′) ×

[

−1

π

∫ ∞

0

dωχλ(r, r′, iω) − δ(r− r′)n(r)

]

.

(18)

The reason that the above frequency integration can beperformed along the imaginary axis originates from theanalytical structure of χλ(r, r′, ω) and the fact that it be-comes real on the imaginary axis. The ACFD expressionin Eq. (18) transforms the problem of computing the XCenergy to one of computing the response functions of aseries of fictitious systems along the AC path, which inpractice have to be approximated as well.

In this context the random-phase approximation is aparticularly simple approximation of the response func-tion:

χλRPA(r, r′, iω) = χ0(r, r′, iω) +∫

dr1dr2χ0(r, r1, iω)λv(r1 − r2)χλRPA(r2, r

′, ω).

(19)

χ0(r, r1, iω) is the independent-particle response functionof the KS reference system at λ = 0 and is known ex-plicitly in terms of the single-particle KS orbitals ψi(r),orbital energies ǫi and occupation factors fi

χ0(r, r′, iω) =∑

ij

(fi − fj)ψ∗i (r)ψj(r)ψ∗

j (r′)ψi(r′)

ǫi − ǫj − iω.

(20)

7

From equations (18) and (19), the XC energy in RPA canbe separated into an exact exchange (EX) and the RPAcorrelation term,

ERPAxc = EEX

x + ERPAc , (21)

where

EEXx =

1

2

∫∫

drdr′v(r, r′) ×

[

−1

π

∫ ∞

0

dωχ0(r, r′, iω) − δ(r− r′)n(r)

]

= −

occ∑

ij

∫∫

drdr′ψ∗i (r)ψj(r)v(r, r′)ψ∗

j (r′)ψi(r′)

(22)

and

ERPAc = −

1

2π

∫∫

drdr′v(r, r′) ×

∫ ∞

0

dω

[∫ 1

0

dλχλRPA(r, r′, iω) − χ0(r, r′, iω)

]

=1

2π

∫ ∞

0

dωTr[

ln(1 − χ0(iω)v) + χ0(iω)v]

.

(23)

For brevity the following convention

Tr [AB] =

∫∫

drdr′A(r, r′)B(r′, r) (24)

has been used in Eq. (23).

C. RPA derived from MBPT

An alternative to ACFD is to compute the interactingground-state energy by performing an order-by-order ex-pansion of Eq. (6). To this end, it is common to choosea linear AC path, i.e., in Eq. (4) vextλ = vext + (1−λ)vMF

such that

H1(λ) = H1 =

N∑

i>j=1

1

|ri − rj |−

N∑

i=1

vMFi . (25)

Now equation (6) reduces to

E = E0 +

∫ 1

0

dλ〈Ψλ|H1|Ψλ〉 . (26)

A Taylor expansion of |Ψλ〉 and a subsequent λ inte-gration lead to an order-by-order expansion of the in-teracting ground-state total energy, e.g., the first-ordercorrection to E0 is given by

E(1) =

∫ 1

0

dλ〈Ψ0|H1|Ψ0〉

= 〈Ψ0|H1|Ψ0〉

= EH + EEXx − EMF, (27)

EMP2c =

+

FIG. 1: Goldstone diagrams for the MP2 correlation energy.The two graphs describe respectively the second-order direct

process, and the second-order exchange process. The upgoingsolid line represents a particle associated with an unoccupiedorbital energy ǫa, the downgoing solid line represents a holeassociated with an occupied orbital energy ǫi, and the dashedline denotes the bare Coulomb interaction.

where EH and EEXx are the classic Hartree and exact ex-

change energy defined in Eq. (13) and (22), respectively.EMF = 〈Ψ0|v

HF|Ψ0〉 is the “double-counting” term dueto the MF potential vMF, which is already included inH0. The sum of E0 and the first-order term E(1) yieldsthe Hartree-Fock energy, and all higher-order contribu-tions constitute the so-called correlation energy.

The higher-order terms can be evaluated using the di-agrammatic technique developed by Goldstone [78]. Forinstance, the second-order energy in RSPT is given by

E(2) =∑

n>0

|〈Φ0|H1|Φn〉|2

E0 − E(0)n

=

occ∑

i

unocc∑

a

|〈Φ0|H1|Φai 〉|

2

E0 − E(0)i,a

+

occ∑

ij

unocc∑

ab

|〈Φ0|H1|Φabij 〉|

2

E0 − E(0)ij,ab

(28)

where |Φ0〉 = |Ψ0〉 is the ground state of the reference

Hamiltonian H0, and |Φn〉 for n > 0 correspond to its

excited states with energy E(0)n = 〈Φn|H0|Φn〉. |Φn〉

can be classified into singly-excited configurations |Φi,a〉,doubly-excited configurations |Φij,ab〉, etc.. The summa-tion in Eq. 28 terminates at the level of double excita-tions. This is because H only contains one- and two-particle operators, and hence does not couple the groundstate |Φ0〉 to triple and higher-order excitations. We willexamine the single-excitation contribution in Eq. (28)in detail in Section IV C. Here it suffices to say thatthis term is zero for the HF reference and therefore isnot included in MP2. The double-excitation contribu-tion can be further split into two terms, correspondingto the second-order direct and exchange energy in MP2,whose representation in terms of Goldstone diagrams isdepicted in Fig. 1. The rules to evaluate Goldstone di-agrams can be found in the classic book by Szabo andOstlund [143].

The Goldstone approach is convenient for the lowestfew orders, but becomes cumbersome or impossible forarbitrarily high orders, the evaluation of which is essen-tial when an order-by-order perturbation breaks downand a “selective summation to infinite order” has to beinvoked. In this case, it is much more convenient to ex-press the total energy in terms of the Green function

8

and the self-energy, as done, e.g., by Luttinger and Ward[145]. Using the Green function language, the ground-state total-energy can be expressed as [145, 146],

E = E0 +1

2

∫ 1

0

dλ

λ

(

1

2π

∫ ∞

−∞

dωTr[

G0(iω)Σ(iω, λ)]

)

(29)

= E0 +1

2

∫ 1

0

dλ

λ

(

1

2π

∫ ∞

−∞

dωTr [G(iω, λ)Σ∗(iω, λ)]

)

(30)

where G0 and G(λ) are single-particle Green functionscorresponding to the non-interacting Hamiltonian H0

and the scaled interacting Hamiltonian H(λ), respec-tively. Σ∗(λ) and Σ(λ) are the proper (irreducible) andimproper (reducible) self-energies of the interacting sys-tem with interaction strength λ. [Proper self-energy dia-grams are those which cannot be split into two by cuttinga single Green function line.] Note that in Eq. (29) and(30), the trace convention of Eq. (24) is implied.

The above quantities satisfy the following relationship

G(iω, λ) = G0(iω) +G0(iω)Σ(iω, λ)G0(iω)

= G0(iω) +G0(iω)Σ∗(iω, λ)G(iω, λ). (31)

From Eq. (31) the equivalence of Eq. (29) and Eq. (30)is obvious. In Eq. (29), a perturbation expansion ofthe λ-dependent self-energy Σλ(iω) naturally translatesinto a perturbation theory of the ground-state energy.In particular, the linear term of Σλ(iω) yields the first-order correction to the ground-state energy, i.e., E(1)

in Eq. (27). All higher-order (n ≥ 2) contributions ofΣλ(iω), here denoted Σc, define the so-called correla-tion energy. In general the correlation energy cannot betreated exactly. A popular approximation to Σc is theGW approach, which corresponds to a selective summa-tion of self-energy diagrams with ring structure to infi-nite order, as illustrated in Fig. (2a). Multiplying G0 tothe GW self-energy ΣGWc (iω) as done in Eq. (29) andperforming the λ integration, one obtains the RPA cor-relation energy

ERPAc =

1

2

∫ 1

0

dλ

λ

(

1

2π

∫ ∞

−∞

dωTr[

G0(iω)ΣGWc (iω, λ)]

)

.

(32)This illustrates the close connection between RPA andthe GW approach. A diagrammatic representation ofERPA

c is the shown in Fig. (2b). We emphasize thatthe diagrams in Fig. (2a) and (2b) are Feynman dia-grams, i.e., the arrowed lines should really be interpretedas propagators, or Green functions. A similar represen-tation of ERPA

c can be drawn in terms of Goldstone di-agrams [143], as shown in Fig (2b). However, cautionshould be applied, because the rules for evaluating thesediagrams are different (see e.g., Ref. 143, 146), and theprefactors in Fig. (2b) are not present in the correspond-ing Goldstone diagrams. The leading term in RPA cor-responds to the second-order direct term in MP2.

(a)

ΣGWc (λ) = λ2 + λ3 + · · ·

(b)

ERPAc = − 1

4− 1

6− · · ·

(c)

ERPAc = + + ...

FIG. 2: Feynman diagrams for the GW self-energy (a), Feyn-man diagrams for the RPA correlation energy (b), and Gold-stone diagrams for the RPA correlation energy (c). Solid linesin (a) and (b) (with thick arrows) represent fermion propa-gators G, and those in (c) (with thin arrows) denote parti-cle (upgoing line) or hole states (down-going line) withoutfrequency dependence. Dashed lines correspond to the bareCoulomb interaction v in all graphs.

We note that starting from Eq. (29) this procedurenaturally gives the perturbative RPA correlation energybased on any convenient non-interacting reference Hamil-tonian H0, such as Hartree-Fock or local/semi-local KS-DFT theory. If one instead starts with Eq. (30) andapplies the GW approximation therein, G(λ, iω) andΣ∗(λ, iω) become the self-consistent GW Green functionand self-energy. As a result the improper self-energy di-agrams in Eq. (29), which are neglected in the perturba-tive GW approach (known as G0W 0 in the literature),are introduced and the total energy differs from that ofthe RPA. An in-depth discussion of self-consistent GWand its implications can be found in [147–150].

D. Link to coupled cluster theory

In recent years, RPA has also attracted considerableattention in the quantum chemistry community. One keyreason for this is its intimate relationship with coupledcluster (CC) theory, which has been very successful foraccurately describing both covalent and non-covalent in-teractions in molecular systems. To understand this re-lationship, we will give a very brief account of the CCtheory here. More details can for instance be found ina review paper by Bartlett and Musia l [151]. The essen-tial concept of CC builds on the exponential ansatz forthe many-body wave function Ψ for correlated electronic

9

systems

|Ψ〉 = eT |Φ〉. (33)

|Φ〉 is a non-interacting reference state, usually chosen to

be the HF Slater determinant, and T is a summation ofexcitation operators of different order,

T = T1 + T2 + T3 + · · · + Tn + · · · , (34)

with T1, T2, T3, · · · being the single, double, and tripleexcitation operators, etc. These operators can be mostconveniently expressed using the language of second-quantization, namely,

T1 =∑

i,a

tai c†aci,

T2 =1

4

∑

ij,ab

tabij c†ac

†b cj ci, (35)

· · ·

Tn =1

(n!)2

∑

ijk··· ,abc···

tabc···ijk··· c†ac

†b c

†c · · · ck cj ci, (36)

where c† and c are single-particle creation and annihila-tion operators and tai , tabij , . . . are the so-called CC singles,doubles, . . . amplitudes yet to be determined. As before,i, j, . . . refer to occupied single-particle states, whereasa, b, · · · refer to unoccupied (virtual) ones. Acting with

Tn on the non-interacting reference state |Φ0〉 generatesn-order excited configuration denoted |Φabc···ijk···〉:

Tn|Φ0〉 =∑

i>j>k··· ,a>b>c···

tabc···ijk··· |Φabc···ijk···〉. (37)

The next question is how to determine the expansioncoefficients tabc···ijk···? The CC many-body wave function

in Eq. (33) has to satisfy the many-body Schrodingerequation,

HeT |Φ〉 = EeT |Φ〉 , (38)

or

e−T HeT |Φ〉 = E|Φ〉 . (39)

By projecting Eq. (39) onto the excited configurations|Φabc···ijk···〉, which have zero overlap with the non-interacting

ground-state configuration |Φ〉, one obtains a set of cou-pled non-linear equations for the CC amplitudes tabc···ijk··· ,

〈Φabc···ijk··· |e−T HeT |Φ〉 = 0 . (40)

These can be determined by solving Eqs. (40) self-consistently.

In analogy to the Goldstone diagrams, equation (40)can be represented pictorially using diagrams, as illus-trated by Cızek [152] in 1966. In practice, the expan-

sion of the T operator has to be truncated. One popular

choice is the CC doubles (CCD) approximation, or T2approximation [152], that retains only the double exci-tation term in Eq. (34). The graphical representation ofCCD contains a rich variety of diagrams including ringdiagrams, ladder diagrams, the mixture of the two, etc.If one restricts the choice to the pure ring diagrams, aspracticed in early work on the HEG [95, 153], the CCDequation is reduced to the following simplified form [13],

B +AT + TA+ TBT = 0. (41)

A, B, T are all matrices of rank Nocc ·Nvir with Nocc andNvir being the number of occupied and unoccupied single-particle states, respectively. Specifically we have Aia,jb =(ǫi− ǫa)δijδab−〈ib|aj〉, Bia,jb = 〈ij|ab〉, and Tia,jb = tabij ,where the Dirac notation for the two-electron Coulombrepulsion integrals

〈pq|rs〉 =

∫∫

drdr′ψ∗p(r)ψr(r)ψ∗

q (r′)ψs(r′)

|r− r′|

(42)

has been adopted.Equation (41) is mathematically known as the Riccati

equation [154]. Solving this equation yields the ring-CCDamplitudes T rCCD, with which the RPA correlation en-ergy can written as

ERPAc =

1

2Tr(

BT rCCD)

=1

2

∑

ij,ab

Bia,jbTrCCDjb,ia . (43)

The CCD formulation of RPA as given by Eq. (41) and(43) was shown by Scuseria et al. [13] to be analyti-cally equivalent to the plasmonic formulation of RPA.The latter has recently been discussed in detail by Furche[38, 94], and and hence will not be presented in this re-view. Technically, the solution of the Riccati equation(41) is not unique, due to the non-linear nature of theequation. One therefore has to make a judicious choicefor the ring-CCD amplitudes in practical RPA calcula-tions [114].

III. ALGORITHMS AND IMPLEMENTATIONS

A. RPA implementations and scaling

In this section we will briefly review different imple-mentations of the RPA approach, since scaling and effi-ciency are particularly important for a computationallyexpensive approach like the RPA. Also, for historical rea-sons, the theoretical formulation of RPA is often linkedclosely to a certain implementation. Similar to conven-tional DFT functionals, implementations of RPA can bebased on local orbitals (LO), or on plane waves, or on(linearized) augmented plane waves (LAPW). LO im-plementations have been reported for the developmentversion of Gaussian [14, 16, 116], the development ver-sion of Molpro [15, 121], FHI-aims [131] and Turbomole[10, 28]. Plane-wave based implementations can be found

10

in ABINIT [11], VASP [19, 110], and Quantum-Espresso[21, 133]. An early implementation by Miyake et al. [108]was based on LAPW.

Furche’s original implementation uses a molecularparticle-hole basis and scales as O(N6) [10], where N isthe number of atoms in the system (unit cell). This canbe reduced to O(N5) using the plasmon-pole formula-tion of RPA [94], and to O(N4) [28] when the resolution-of-identity (RI) technique is employed in addition. Ourown RPA implementation [131] in FHI-aims [132] is de-scribed in Appendix A. It is based on localized numericatom-centered orbitals and the RI technique, and hencenaturally scales as O(N4).

The key in the RI-RPA implementation is to expandthe occupied -virtual orbital pair products φ∗i (r)φj(r) ap-pearing in Eq. (20) in terms of a set of auxiliary ba-sis functions (ABFs) {Pµ(r)}. In this way, one can re-duce the rank of the matrix representation of χ0 fromNocc ∗Nvir to Naux with Naux ≪ Nocc ∗Nvir. Here Naux,Nocc, and Nvir denote the number of ABFs, and the num-bers of occupied and unoccupied (virtual) single-particleorbitals respectively. With both χ0 and the coulomb ker-nel v represented in terms of the ABFs, the RPA correla-tion energy expression in Eq. (23) can be re-interpretedas a matrix equation of rank Naux, which is numericallyvery cheap to evaluate. The dominating step then be-comes the build of the matrix form of χ0 which scalesas O(N4). We refer the readers to Appendix A andRef. [131] for further details.

Plane-wave based implementations [11, 110] automat-ically have O(N4) scaling. In a sense the plane-wavebased RPA implementation is very similar in spirit to thelocal-orbital-based RI-RPA implementation. In the for-mer case the plane waves themselves serve as the above-mentioned ABFs.

B. Speed-up of RPA with iterative methods

The RPA correlation energy in (23) can also be rewrit-ten as follows,

ERPAc = −

1

2π

∫ ∞

0

dω

Naux∑

µ

[

ln(

εDµ (iω))

+ 1 − εDµ (iω)]

,

(44)where εDµ (iω) is the µth eigenvalue of the dielectric func-

tion ε(iω) = 1 − χ0(iω)v represented in the ABS. Alleigenvalues are larger than or equal to 1. From (44) itis clear that eigenvalues which are close to 1 have a van-ishing contribution to the correlation energy. For a setof different materials, Wilson et al. [134] observed thatonly a small fraction of the eigenvalues differs signifi-cantly from 1, which suggests that the full spectrum ofε(iω) is not required for accurate RPA correlation en-ergies. This opens up the possibility of computing theRPA correlation energy by obtaining the “most signifi-cant” eigenvalues of ε(iω) (or equivalently χ0(iω)v) from

an iterative diagonalization procedure, instead of con-structing and diagonalizing the full ε(iω) or χ0(iω)v ma-trices. In practice this can be conveniently done by re-sorting to the linear response technique of density func-tional perturbation theory (DFPT) [135] and has beenproposed and implemented in the two independent worksof Galli and coworkers [21, 134], and of Nguyen andde Gironcoli [133] within a pseudopotential plane-waveframework. In these (plane-wave based) implementationsthe computational cost is reduced from N2

pw-χNoccNvir to

N2pw-ψN

2occNeig, where Npw-χ and Npw-ψ are the numbers

of plane waves to expand the response function χ0 andthe single-particle orbitals ψ respectively, and Neig is thenumber of dominant eigenvalues. In this way, althoughthe formal scaling is still O(N4), one achieves a large re-duction of the prefactor, said to be 100-1000 [133]. Thisprocedure is in principle applicable to RI-RPA implemen-tation in local-basis sets, too, but has, to the best of ourknowledge, not been reported so far.

IV. COMPUTIONAL SCHEMES BEYOND RPA

In this Section we will give a brief account of the ma-jor activities for improving the standard RPA, aiming atbetter accuracy.

A. Semi-local and non-local corrections to RPA

It is generally accepted that long-range interactions arewell described within RPA, whereas short-range correla-tions are not adequate [104]. This deficiency manifestsitself most clearly in the pair-correlation function of theHEG, which spuriously becomes negative when the sepa-ration between two electrons gets small [103, 104]. Basedon this observation, Perdew and coworkers [97, 105] pro-posed a semi-local correction to RPA, termed as RPA+

ERPA+c = ERPA

c + EGGAc − EGGA-RPA

c , (45)

where EGGAc is the GGA correlation energy, and

EGGA-RPAc represents the random-phase approximation

within GGA. Thus the difference between EGGAc and

EGGA-RPAc gives a semi-local correction to RPA for in-

homogeneous systems. As mentioned before in the intro-duction, the RPA+ scheme, although conceptually ap-pealing, and good for total energies [18], does not sig-nificantly improve the description of energy differences,in particular the atomization energies of small molecules[10]. This failure has been attributed to the inaccuracyof RPA in describing the multi-center non-locality of thecorrelation hole, which cannot be corrected by semi-localcorrections of the RPA+ type [34, 35]. A fully non-localcorrection (nlc) to RPA has recently been proposed byRuzsinszky, Perdew, and Csonka [35]. It takes the fol-

11

lowing form

Enlcc =

∫

drn(r)[

ǫGGA(r) − ǫGGA-RPA(r)]

[1 − αF (f(r)]] ,

(46)where ǫGGA(r) and ǫGGA-RPA(r) are the GGA energydensity per electron and its approximate value withinRPA, respectively. α is an empirical parameter yet tobe determined, and F is a certain functional of f(r), thedimensionless ratio measuring the difference between theGGA exchange energy density and the exact-exchangeenergy density at a given point r,

f(r) =ǫGGAx (r) − ǫexactx (r)

ǫGGAx (r)

. (47)

One may note that by setting α = 0 in Eq. (46) the usualRPA+ correction term is recovered. A simple choiceof the functional form F (f) = f turns out to be goodenough for fitting atomization energies, but the correctdissociation limit of H2 given by standard RPA is de-stroyed. To overcome this problem, Ruzsinszky et al.chose a more complex form of F ,

F (f) = f [1 − 7.2f2][1 + 14.4f2]exp(−7.2f2), (48)

which ensures the correct dissociation limit, while yield-ing significantly improved atomization energies for α = 9.Up to now the correction scheme of Eq. (46) has notbeen widely benchmarked except for a small test set of10 molecules where the atomization energy has been im-proved by a factor of two [35].

B. Screened second-order exchange (SOSEX)

The SOSEX correction [16, 95, 123] is an importantroute to go beyond standard RPA. This concept can bemost conveniently understood within the context of thering-CCD formulation of RPA as discussed in Sec. II D.If in Eq. (43), the anti-symmetrized Coulomb integrals

Bia,jb = 〈ij|ab〉 − 〈ij|ba〉 are inserted instead of the un-symmetrized Coulomb integrals, the RPA+SOSEX cor-relation energy expression is obtained

ERPA+SOSEXc =

1

2

∑

ij,ab

T rCCDia,jb Bia,jb . (49)

This approach, which was first used by Freeman [95],and recently examined by Gruneis et al. for solids [123]and Paier [16] for molecular properties, has received in-creasing attention in the RPA community. In contrast toRPA+, this scheme has the attractive feature that it im-proves both total energies and energy differences simul-taneously. Although originally conceived in the CC con-text, SOSEX has a clear representation in terms of Gold-stone diagrams, as shown in Fig. 3 (see also Ref. [123]),which can be compared to the Goldstone diagrams forRPA in Fig. (2c). From Fig. 3, it is clear that the lead-

ESOSEXc = + + ...

FIG. 3: Goldstone diagrams for SOSEX contribution. Therules to evaluate Goldstone diagrams can be found inRef. [143].

ing term in SOSEX corresponds to the second-order ex-change term of MP2. In analogy, the leading term inRPA corresponds to the second-order direct term of MP2.Physically the second-order exchange diagram describesa (virtual) process in which two particle-hole pairs arecreated spontaneously at a given time. The two particles(or equivalently the two holes) then exchange their po-sitions, and two (already exchanged) particle-hole pairsannihilate themselves simultaneously at a later time. InSOSEX, similar to RPA, a sequence of higher-order dia-grams are summed up to infinity. In these higher-orderdiagrams, after the initial creation and exchange process,one particle-hole pair is scattered into new positions re-peatedly following the same process as in RPA, until itannihilates simultaneously with the other pair at the endof the process.

SOSEX is one-electron self-correlation free and ame-liorates the short-range over-correlation problem of RPAto a large extent, leading to significantly better total en-ergies [95, 123]. More importantly, the RPA underesti-mation of atomization energies is substantially reduced.However, the dissociation of covalent diatomic molecules,which is well described in RPA, worsens considerably asdemonstrated in Ref. [16] and to be shown in Fig. 6. Itwas argued that the self-correlation error present in RPAmimics static correlation, which becomes dominant in thedissociation limit of covalent molecules [114].

C. Single excitation correction and its combinationwith SOSEX

In most practical calculations, RPA and SOSEX corre-lation energies are evaluated using input orbitals from apreceding KS or generalized KS (gKS) [155] calculation.In this way both RPA and SOSEX can be interpreted asinfinite-order summations of selected types of diagramswithin the MBPT framework introduced in Sec. II C, asis evident from Figs (2c) and (3). This viewpoint is help-ful for identifying contributions missed in RPA throughthe aid of diagrammatic techniques. An an example, thesecond-order energy in RSPT in Eq. (28) have contribu-tions from single excitations (SE) and double excitations.The latter gives rise to the familiar MP2 correlation en-ergy, which is included in the RPA+SOSEX scheme as

12

the leading term. The remaining SE term is given by

ESEc =

occ∑

i

unocc∑

a

|〈Φ0|H1|Φai 〉|

2

E0 − E(0)i,a

=∑

ia

|〈ψi|vHF − vMF|ψa〉|

2

ǫi − ǫa(50)

=∑

ia

|〈ψi|f |ψa〉|2

ǫi − ǫa(51)

where vHF is the self-consistent HF single-particle poten-tial, vMF is the mean-field potential associated with the

reference Hamiltonian, and f = −∇2/2 + vext + vHF isthe single-particle HF Hamiltonian (also known as theFock operator in the quantum chemistry literature). Adetailed derivation of Eq. (50) using second-quantizationcan be found in the supplemental material of Ref. [32].The equivalence of Eqs. (51) and (50) can be readily

confirmed by observing the relation between f and the

single-particle reference Hamiltonian hMF: f = hMF +

vHF − vMF, and the fact 〈ψi|hMF|ψa〉 = 0. Obviously

for a HF reference where vMF = vHF, Eq. (51) becomeszero, a fact known as Brillouin theorem [143]. Therefore,as mentioned in Section II C, this term is not present inMP2 theory which is based on the HF reference. Wenote that a similar SE term also appears in 2nd-orderGorling-Levy perturbation theory (GL2) [156, 157], abinitio DFT [158], as well as in CC theory [151]. How-ever, the SE terms in different theoretical frameworksdiffer quantitatively. For instance, in GL2 vMF shouldbe the exact-exchange OEP potential instead of the ref-erence mean-field potential.

In Ref. [32] we have shown that adding the SE term ofEq. (51) to RPA significantly improves the accuracy ofvdW-bonded molecules, which the standard RPA schemegenerally underbinds. This improvement carries over toatomization energies of covalent molecules and insulat-ing solids as shown in Ref. [124]. It was also observedin Ref. [32] that a similar improvement can be achievedby replacing the non-self-consistent HF part of the RPAtotal energy by its self-consistent counterpart. It ap-pears that, by iterating the exchange-only part towardsself-consistency, the SE effect can be accounted for effec-tively. This procedure is termed “hybrid-RPA”, and hasbeen shown to be promising even for surface adsorptionproblems [30].

The SE energy in Eq. (51) is a second-order term inRSPT, which suffers from the same divergence problemas MP2 for systems with zero (direct) gap. To overcomethis problem, in Ref. [32] we have proposed to sum overa sequence of higher-order diagrams involving only singleexcitations. This procedure can be illustrated in terms ofGoldstone diagrams as shown in Fig. 4. This summationfollows the spirit of RPA and we denote it renormalizedsingle excitations (rSE) [32]. The SE contribution to the2nd-order correlation energy in Eq. (51), represented bythe first diagram in Fig 4, constitutes the leading term

ErSEc =

+ ia + ... a

bi+

∆via

i

∆vai

ja

∆via

∆vij

∆v

∆via

∆

∆vbiaj

vab

2nd−order 3rd−order

FIG. 4: Goldstone diagrams for renormalized single excitationcontributions. Dashed line ending with a cross denotes thematrix element ∆vpq = 〈ψp|v

HF − vMF|ψq〉.

in the rSE series. A preliminary version of rSE, whichneglects the “off-diagonal” terms of the higher-order SEdiagrams (by setting i = j = · · · and a = b = · · · ),was benchmarked for atomization energies and reactionbarrier heights in Ref. [124]. Recently we were able toalso include the “off-diagonal” terms, leading to a refinedversion of rSE. This rSE “upgrade” does not affect theenergetics of strongly bound molecules, as those bench-marked in Ref. [124]. However, the interaction energiesof weakly bound molecules improve considerably. A moredetailed description of the computational procedure andextended benchmarks for rSE will be reported in a forth-coming paper [159]. However, we note that all the rSEresults reported in Section (Sec. V) correspond to theupgraded rSE.

Diagrammatically, RPA, SOSEX and rSE are threedistinct infinite series of many-body terms, in which thethree leading terms correspond to the three terms in 2nd-order RSPT. Thus it is quite natural to include all threeof them, and the resultant RPA+SOSEX+rSE schemecan be viewed a renormalization of the normal 2nd-orderRSPT. Therefore we will refer to RPA+SOSEX+rSE as“renormalized second-order perturbation theory” or r2PTin the following.

D. Other “beyond-RPA” activities

There have been several other attempts to go beyondRPA. Here we will only briefly discuss the essential con-cepts behind these approaches without going into details.The interested reader is referred to the corresponding ref-erences. Following the ACFD formalism, as reviewed inSec. II B, one possible route is to improve the interact-ing density response function. This can be convenientlydone by adding the exchange correlation kernel (fxc) oftime-dependent DFT [160, 161], that is omitted in RPA.Fuchs et al. [162], as well as Heßelmann and Gorling [122]have added the exact-exchange kernel to RPA, a schemetermed by these authors as RPA+X or EXX-RPA, forstudying the H2 dissociation problem. RPA+X or EXX-RPA displays a similar dissociation behavior for H2 asRPA: accurate at infinite separation, but slightly repul-sive at intermediate bond lengths. This scheme howevergives rise to a noticeable improvement on the total en-ergy [121]. Furche and Voorhis examined the influenceof several different local and non-local kernels on the at-

13

omization energies of small molecules and the bindingenergy curves of rare-gas dimers [12]. They found thatsemilocal fxc kernels lead to a diverging pair density atsmall inter-particle distances, and it is necessary to go tonon-local fxc kernels to cure this. More work along theselines has to be done before conclusions can be drawn andaccurate kernels become available.

Quantum chemistry offers another route to go beyondstandard RPA by including higher-order exchange ef-fects (often termed as “RPAx”). There the two-electronCoulomb integrals usually appear in an antisymmetrizedform, whereby exchange-type contributions, which areneglected in standard RPA, are included automatically[99, 163]. The RPA correlation energy can be expressedas a contraction between the ring-CCD amplitudes andthe Coulomb integrals (see Eq. (43)), or alternatively be-tween the coupling-strength-averaged density matrix andthe Coulomb integrals (see Ref. [117]). Different flavors ofRPA can therefore be constructed depending on whetherone antisymmetrizes the averaged density matrix and/or

the Coulomb integrals (see Angyan et al. [119]). Accord-ing to our definitions in this article, these schemes are cat-egorized as different ways to go beyond standard RPA,while in the quantum chemistry community they mightbe referred to simply as RPA. The SOSEX correction,discussed in Sec. IV B, can also be rewritten in terms ofa coupling-strength-averaged density matrix [117]. An-other interesting scheme was proposed by Heßelmann[36], in which RPA is corrected to be exact at the thirdorder of perturbation theory. These corrections showpromising potential for the small molecules consideredin Ref. [36]. However, more benchmarks are needed fora better assessment.

Both standard RPA and RPA with the exchange-typecorrections discussed above have been tested in a range-separation framework by several authors [14, 15, 26, 116–120]. As mentioned briefly in the introduction, the con-cept of range separation is similar to the RPA+ proce-dure, in which only the long-range behavior of RPA is re-tained. However, instead of additional corrections RPAat the short-range is completely removed and replacedby semi-local or hybrid functionals. The price to payare empirical parameters that control the range separa-tion. The gain is better accuracy in describing molecularbinding energies [14, 15], and increased computationalefficiency. The latter is due to a reduction in the num-ber of required basis functions to converge the long-rangeRPA part, which is no longer affected by the cusp condi-tion. More details on range-separated RPA can be foundin the original references [14, 15, 26, 116–120]. Com-pared to the diagrammatic approaches discussed before,the range-separation framework offers an alternative andcomputationally more efficient way to handle short-rangecorrelations, albeit at the price of introducing some em-piricism into the theory.

V. APPLICATIONS

A. Molecules

RPA based approaches have been extensively bench-marked for molecular systems, ranging from the disso-ciation behavior of diatomic molecules [10, 11, 14, 15,120, 122, 138], atomization energies of small covalentmolecules [10, 16, 34, 35, 124], interaction energies ofweakly bonded molecular complex [26, 31, 117–119], andchemical reaction barrier heights [124]. The behaviorof RPA for breaking covalent bonds was examined inits early-day’s applications [10, 11], and today is stilla topic of immense interest [15, 120, 122, 138]. Atom-ization energies of covalent molecules are somewhat dis-appointing, because standard RPA, as well as its localcorrection (RPA+), is not better than semi-local DFTfunctionals [10]. This issue was subsequently referred toas “the RPA atomization energy puzzle” [34]. A solu-tion can be found in the beyond RPA schemes SOSEX[16, 123] and SE [32, 124, 159]. Another major applica-tion area of RPA are weakly bonded molecules. Due tothe seamless inclusion of the ubiquitous vdW interaction,RPA clearly improves over conventional DFT function-als, including hybrids. This feature is very importantfor systems where middle-ranged non-local electron cor-relations play a significant role, posing great challengesto empirical or semi-empirical pairwise-based correctionschemes. Finally, for activation energies it turned outthat standard RPA performs remarkably well [38, 124]and the beyond RPA correction schemes that have beendeveloped so far do not improve the accuracy of the stan-dard RPA [124].

In the following, we will discuss the performance ofRPA and its variants using representative examples toillustrate the aforementioned points.

1. Dissociation of diatomic molecules

The dissociation of diatomic molecules is an importanttest ground for electronic structure methods. The perfor-mance of RPA on prototypical molecules has been exam-ined in a number of studies [10, 11, 14, 15, 113, 120, 122,138]. Here we present a brief summary of the behaviorof RPA-based approaches based on data generated usingour in-house code FHI-aims [132]. The numerical detailsand benchmark studies of our RPA implementation havebeen presented in Ref. [131]. In Fig. 5 the binding en-ergy curves obtained with PBE, MP2, and RPA-basedmethods are plotted for four molecular dimers, includingtwo covalent molecules (H2 and N2), one purely vdW-bonded molecule (Ar2), and one with mixed character(Be2). Dunning’s Gaussian cc-pV6Z basis [164, 165] wasused for H2 and N2, aug-cc-pV6Z for Ar2, and cc-pV5Zfor Be2. Currently, no larger basis seems to be avail-able for Be2, but this will not affect the discussion here.Basis-set superposition errors (BSSE) are corrected us-

14

ing the Boys-Bernardi counterpoise procedure [166]. Alsoplotted in Fig. 5 are accurate theoretical reference datafor H2, Ar2, and Be2 coming respectively from the fullCI approach [167], the Tang-Toennies model [168], andthe extended germinal model [169]. To visualize the cor-responding asymptotic behavior more clearly, the largebond distance regime of all curves is shown in Fig. 6.

RPA and RPA+ dissociate the covalent molecules cor-rectly to their atomic limit at large separations, albeitfrom above after going through a positive “bump” at in-termediate bond distances. The fact that spin-restrictedRPA calculations yield the correct H2 dissociation limit isquite remarkable, given the fact that most spin-restrictedsingle-reference methods, including local and semi-localDFT, Hartree-Fock, as well as the coupled cluster meth-ods, yield an dissociation limit that is often too high inenergy, as illustrated in Fig. 5 for PBE. MP2 fails moredrastically, yielding diverging results in the dissociationlimit for H2 and N2. The RPA+ binding curves followthe RPA ones closely, with only minor differences. TherSE corrections are also quite small in this case, shift-ing the RPA curves towards larger binding energies, withthe consequence that the binding energy dips slightly be-low zero in the dissociation limit (see the N2 examplein Fig. 6). This shift however leads to better molecu-lar binding energies around the equilibrium where RPAsystematically underbinds. The SOSEX correction, onthe other hand, leads to dramatic changes. Although“bump” free, SOSEX yields dissociation limits that aremuch too large, even larger than PBE. This effect carriesover to r2PT at large bond distances where rSE does notreduce the SOSEX overestimation.

For the purely dispersion-bonded dimer Ar2, all RPA-based approaches, as well as MP2, yield the correctC6/R

6 asymptotic behavior, whereas the semi-local PBEfunctional gives a too fast exponential decay. Quantita-tively, the C6 dispersion coefficient is underestimated by∼ 9% within RPA (based on a PBE reference) [170], andSOSEX or rSE will not change this. In contrast, MP2overestimates the C6 value by ∼ 18% [170]. Around theequilibrium point, RPA and RPA+ underbind Ar2 sig-nificantly. The rSE correction improves the results con-siderably, bringing the binding energy curve into closeagreement with the Tang-Toennies reference curve. TheSOSEX correction, on the other hand, does very little inthis case. As a consequence, r2PT resembles RPA+rSEclosely in striking contrast to the covalent molecules.

Be2 represents a most complex situation, in which bothstatic correlation and long-range vdW interactions playan important role. In the intermediate regime, the RPAand RPA+ binding energy curves display a positive bumpwhich is much more pronounced than for purely covalentmolecules. At very large bonding distance, the curvescross the energy-zero line and eventually approach theatomic limit from below. The rSE correction moves thebinding energy curve significantly further down, givingbinding energies in good agreement with the referencevalues, whereas in the intermediate region a small pos-

itive bump remains. The SOSEX correction exhibits acomplex behavior. While it reducing the bump it con-comitantly weakens binding around the equilibrium dis-tance. Combining the corrections from SOSEX and rSE,the r2PT does well at intermediate and large bondingdistances, but the binding energy at equilibrium is stillnoticeably too small. Regarding MP2, it is impressive toobserve that this approach yields a binding energy curvethat is in almost perfect agreement with the reference inthe asymptotic region, although a substantial underbind-ing can be seen around the equilibrium region.

Summarizing this part, RPA with and without correc-tions shows potential, but at this point, none of the RPA-based approaches discussed above can produce quantita-tively accurate binding energy curves for all bonding sit-uations. It is possible, but we consider it unlikely, thatiterating RPA to self-consistency will change this result.Apart from applications to neutral molecules, RPA andRPA+SOSEX studies have been carried out for the dis-sociation of charged molecules. RPA fails drastically inthis case [113], giving too low a total energy in the disso-ciation limit. Adding SOSEX to RPA fixes this problem,although the correction now overshoots (with the excep-tion of H+

2 ) (see Ref. [16, 38, 113, 114] for more details).

2. Atomization energies: the G2-I set

One important molecular property for thermochem-istry is the atomization energy, given by Emol −

∑

iEati

where Emol is the ground-state energy of a molecule andEati that of the i-th isolated atom. According to this def-

inition, the negative of the atomization energy gives theenergy to break the molecule into its individual atoms.Here we examine the accuracy of RPA-based approachesfor atomization energies of small molecules. The RPA re-sults for a set of 10 small organic molecules were reportedin Furche’s seminal work [10] where the underestimationof RPA for atomization energies was first observed. Thisbenchmark set is included in their recent review [38].A widely accepted representative set for small organicmolecules is the G2-I set [141], that contains 55 cova-lent molecules and will be used as an illustrative examplehere. The RPA-type results for the G2-I set have recentlybeen reported in the work by Paier et al. [16, 124].

In Fig 7 we present in a bar graph the mean absolutepercentage error (MAPE) for the G2-I atomization ener-gies obtained by four RPA-based approaches in additionto GGA-PBE, the hybrid density-functional PBE0, andMP2. The actual values for the the mean error (ME),mean absolute error (MAE), MAPE, and maximum ab-solute percentage error (MaxAPE) are listed in table IIin Appendix B. The calculations were performed us-ing FHI-aims [131, 132] with Dunning’s cc-pV6Z basis[164, 165]. Reference data are taken from Ref. [171] andcorrected for zero-point energies. Fig 7 and Table II il-lustrate that among the three traditional approaches, thehybrid functional PBE0 performs best, with a ME close

15

1 2 3 4 5 6

-4

-2

0

2

4

Bin

ding

ene

rgy

(eV

)1 2 3 4 5 6

-12

-8

-4

0

4

8

12

2 3 4 5 6 7-300

-200

-100

0

100

200

3 4 5 6 7 8

Bond length (Å)

-12

-8

-4

0

4

8B

indi

ng e

nerg

y (m

eV)

PBEMP2RPARPA+RPA+rSERPA+SOSEXr2PTAccurate

H2

N2

Be2Ar

2

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

2 3 4 5 6 7 8-2

0

2

4

Bin

ding

ene

rgy

(eV

)

2 3 4 5 6 7-10

-5

0

5

10

15

20

5 6 7 8 9 10-4

-3

-2

-1

0

1

5 6 7 8 9

Bond length (Å)

-6

-4

-2

0

Bin

ding

ene

rgy

(meV

)

PBEMP2RPARPA+RPA+rSERPA+SOSEXr2PTAccurate

H2 N

2

Be2Ar

2

FIG. 6: Asymptotic region of the curves in Fig. 5.

to the “chemical accuracy” (1 kcal/mol = 43.4 meV).MP2 comes second, and PBE yields the largest error andshows a general trend towards overbinding. ConcerningRPA-based approaches, standard RPA leads to ME andMAE that are even larger than the corresponding PBEvalues, with a clear trend of underbinding. RPA+ doesnot improve the atomization energies [10]. All this isin line with previous observations [10, 16, 124, 131]. Asshown in the previous section, the underbinding of RPAfor small molecules can be alleviated by adding SOSEX

or rSE correction. And the combination of the two (i.e.r2PT) further brings the MAE down to 3.3 kcal/mol,comparable to the corresponding PBE0 value. Whetherthis mechanism of this improvement can be interpretedin terms of the “multi-center non-locality of the correla-tion hole” as invoked by Ruzsinszky et al. [34, 35] or not,is not yet clear at the moment.

16

0

1

2

3

4

5

6

7

8M

AP

E (

%)

PB

EP

BE

0M

P2

RP

AR

PA

+R

PA

+SO

SE

XR

PA

+rS

Er2

PT

FIG. 7: Mean absolute percentage error (MAPE) for theG2-I atomization energies obtained with four RPA-based ap-proaches in addition to PBE, PBE0 and MP2.

3. vdW interactions: S22 set

As discussed above, one prominent feature of RPA isthat it captures vdW interactions that are of paramountimportance for non-covalently bonded systems. Bench-marking RPA-based methods for vdW bonded systemsis an active research field [12, 14, 15, 19, 21, 26, 31, 32,107, 117, 118]. Here we choose the S22 test set [142] asthe illustrating example to demonstrate the performanceof RPA for non-covalent interactions. This test set con-tains 22 weakly bound molecular complex of different sizeand bonding type (7 of hydrogen bonding, 8 of dispersionbonding, and 7 of mixed nature). Since its inception ithas been widely adopted as the benchmark or trainingdataset for computational schemes that aim at dealingwith non-covalent interactions [172–178] including RPA-based approaches [26, 31, 32, 118]. The consensus emerg-ing from these studies is as expected: RPA improves thebinding energies considerably over semilocal functionals.

Quantitatively the MAEs given by standard RPA re-ported for S22 by different groups show an unexpectedspread. Specifically Zhu et al. [26] report an MAE of2.79 kcal/mol, Ren et al. 39 meV or 0.90 kcal/mol [32],and Eshuis and Furche 0.41 kcal/mol [31]. The authors ofthe latter study investigated this issue in detail [179] andconcluded that the discrepancy is due to basis set incom-pleteness and BSSE. Using Dunning’s correlation consis-tent basis sets plus diffuse functions and extrapolatingto the complete basis set (CBS) limit Eshuis and Furcheobtained a MAE of 0.79 kcal/mol with 0.02 kcal/mol un-certainty [179]. These authors confirmed our observationthat standard RPA generally underbinds weakly boundmolecules. The basis set we have used, NAO tier 4 plusdiffuse functions from aug-cc-pV5Z (denoted as “tier 4+ a5Z-d”) [32, 131], yields RPA results very close to theCBS limit. The results reported in Ref. [32] are, in our

0

10

20

30

40

50

60

MA

PE

(%

)

PB

E

PB

E0

MP

2

RP

AR

PA

+R

PA

+SO

SE

XR

PA

+rS

Er2

PT

FIG. 8: The MAPEs for the S22 test set obtained with RPA-based approaches in addition to PBE, PBE0, and MP2. The“tier 4 + a5Z-d” basis set was used in the calculations.

opinion, the most reliable RPA results (based on the PBEreference) for S22 so far.

In Fig 8 the relative errors (in percentage) of five RPA-based schemes are plotted for the molecules of the S22set. Results for PBE, PBE0, and MP2 are also pre-sented for comparison. For MP2 and RPA-based meth-ods, the relative errors (in percentage) for the 22 indi-vidual molecules are further demonstrated in Fig 9. Thereference data were obtained using CCSD(T) and prop-erly extrapolated to the CBS limit by Takatani et al.[180]. MP2 and RPA results are taken from Ref. [32].The RPA+rSE, RPA+SOSEX, and r2PT results are pre-sented for the first time. Further details for these calcu-lations and an in-depth discussion will be presented ina forthcoming paper [159]. Figure 8 shows that PBEand PBE0 fail drastically in this case, because thesetwo functionals do not capture vdW interactions by con-struction, whereas all other methods show significant im-provement. Figure 9 further reveals that MP2 describesthe hydrogen-bonded systems very accurately, but vastlyoverestimates the strength of dispersion interactions, par-ticularly for the π-π stacking systems. Compared toMP2, RPA provides a more balanced description of allbonding types, but shows a general trend to underbind.It has been shown that this underbinding is significantlyreduced by adding SE corrections [32]. The renormal-ized SE correction presented here gives rise to a moresystematic correction to RPA, as can be seen from Fig 9.SOSEX shows a similar correction pattern as rSE for hy-drogen bonding and mixed interactions, but has littleeffect on the dispersion interaction. The r2PT scheme,that combines SOSEX and rSE corrections, overshootsfor hydrogen bonding, but on average improves the de-scription of the other two bonding types. Figure 9 alsoreveals that π-π stacking configurations, as exemplifiedby the benzene dimer in the slip parallel geometry (#

17

1 2 3 4 5 6 7 8 9 10 11 12 13 1415 1617 18 19 20 21 22

S22 molecules

-60

-40

-20

0

20

40

60

80R

elat

ive

erro

r (%

)MP2RPARPA+rSERPA+SOSEXr2PT

Hydrogen Dispersion Mixed

FIG. 9: The relative errors (in %) for the individual S22molecules obtained with RPA-based approaches in additionto MP2. Connection lines are just guide to eyes

11), represent the most challenging case for RPA-basedmethods. The relative error of RPA for this case is thelargest. rSE provides little improvement, whereas SO-SEX worsens slightly. More work is needed to understandthe origin of this failure.

The detailed errors (ME, MAE, MAPE, and MaxAPE)for S22 are presented in table III in Appendix B. Amongthe approaches we have investigated, RPA+rSE givesthe smallest MAE for S22, whiles r2PT gives the small-est MAPE. Due to space restrictions it is not possi-ble to include the multitude of computational schemesthat have emerged in recent years for dealing with non-covalent interactions [172–178] in Tab. III. Compared tothese approaches, the RPA-based approaches presentedhere are completely parameter-free and systematic in thesense that they have a clear diagrammatic representa-tion. Thus RPA-based approaches are expected to havea more general applicability, and may well serve as thereference for benchmarking other approaches for systemswhere CCSD(T) calculations are not feasible.

4. Reaction barrier heights

One stringent test for an electronic structure methodis its ability to predict chemical reaction barrier heights,i.e., the energy difference between the reactants and theirtransition state. This is a central quantity that dic-tates chemical kinetics. Semi-local density approxima-tions typically underestimate barrier heights [16, 181].RPA has already been benchmarked for barrier heightsin two independent studies [38, 124]. Both studiesused the test sets of 38 hydrogen-transfer barrier heights(HTBH38) and 38 non-hydrogen-transfer barrier heights(NHTBH38) designed by Zhao et al. [182, 183] (togethercoined as BH76 in Ref. [184]). HTBH38 contains the for-ward and inverse barrier heights of 19 hydrogen transfer

0

0.1

0.2

0.3

0.4

0.5

MA

E (

eV)

PB

E

PB

E0

MP

2

RP

A

RP

A+

RP

A+S

OS

EX

RP

A+r

SE

r2P

T

FIG. 10: The MAEs for the HTBH38 (full bars) andNHTBH38 (hatched bars) test sets obtained with RPA-basedapproaches in addition to PBE, PBE0, and MP2. The cc-pV6Z basis set was used in the calculations.

reactions, whereas NHTBH38 contains 19 reactions in-volving heavy atom transfers, nucleophilic substitutions,association, and unimolecular processes. The referencedata were obtained using the “Weizmann-1” theory [185]– a procedure to extrapolate the CCSD(T) results – orby other “best theoretical estimates” [183]. Paier et al.[124] presented results for standard RPA and “beyondRPA” approaches based on the PBE reference, wherea two-point cc-pVTZ→cc-pVQZ basis-set extrapolationstrategy is used. In the work of Eshuis et al. [38], stan-dard RPA results based on both PBE and TPSS [186]references were presented, where the Def2-QZVP basis[187] was used. The RPA@PBE results for BH76 re-ported by both groups are very close, with an ME/MAEof -1.35/2.30 kcal/mol from the former and -1.65/3.10kcal/mol from the latter.

The performance of RPA-based approaches, as well asPBE, PBE0, and MP2 for HTBH38/NHTBH38 test setsis demonstrated by the MAE bar graph in Fig. 10. Thecalculations were done using FHI-aims with the cc-pV6Zbasis set. The ME, MAE, and the maximal absoluteerror (MaxAE) are further presented in table IV in Ap-pendix B. In this case we do not present the relativeerrors, which turn out to be very sensitive to the compu-tational parameters due to some small barrier heights inthe test set, and hence cannot be used as a reliable mea-sure of the performance of the approaches examined here.Compared to the results reported in Ref. [124], besidesa different basis set (cc-pV6Z instead of cc-pVTZ→cc-pVQZ extrapolation), we also used the refined rSE cor-rection (see discussion in Section IV C) for the RPA+rSEand r2PT results in Tab. IV, which gives slightly betterresults in this case.

On average, PBE underestimates the reaction bar-rier heights substantially, a feature that is well-knownfor GGA functionals. The hybrid PBE0 functional re-

18

duces both the ME and MAE by more than a factor oftwo. However, the remaining error is still sizable. Stan-dard RPA does magnificently and shows a significant im-provement over PBE0. The performance of RPA+ isagain very similar to standard RPA. As already notedin Ref. [124], both the rSE and the SOSEX correctiondeteriorate the performance of RPA. This is somewhatdisappointing, and highlights the challenge for designingsimple, generally more accurate corrections to RPA. For-tunately, the errors of rSE and SOSEX are now in theopposite direction, and largely canceled out when com-bining the two schemes. Indeed, the AEs and MAEs ofr2PT are not far from their RPA counterparts, althoughthe individual errors are more scattered in r2PT as man-ifested by the larger maximal absolute error.

B. Crystalline solids

Crystalline solids are an important domain for RPAbased approaches, in particular because the quantumchemical hierarchy of benchmark approaches cannot eas-ily be transfered to periodic systems. Over the years RPAcalculations have been performed for a variety of systemssuch as Si [108, 109, 133], Na [108], h-BN [17], NaCl [109],rare-gas solids [19], graphite [20, 27], and benzene crys-tals [21]. The most systematic benchmark study of RPAfor crystalline solids was conducted by Harl, Schmika,and Kresse [20, 110]. These authors reported “technicallyconverged” calculations using their VASP code and theprojector augmented plane wave method for atomizationenergies, lattice constants, and bulk moduli of 24 repre-sentative 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-als (Na, Al, Cu, Rh, Pd, Ag). The error analysis of theirRPA and RPA+ results, based on a PBE reference, aswell as the LDA and PBE results are presented in Tab. I.As is clear from Tab. I, the RPA lattice constants andbulk moduli are better than in LDA and PBE. The at-omization energies, however, are systematically underes-timated in RPA, and the MAE in this case is even largerthan that of PBE. This behavior is very similar to that forthe atomization energies in the G2 set discussed above.Harl et al. also observed that the error of RPA does notgrow when going to heavier atoms, or open-shell systemsin contrast to LDA or PBE [110]. The RPA+ results arein general slightly worse than those of standard RPA.

The performance of RPA has not been extensivelybenchmarked for vdW-bound solids. However, from thestudies on h-BN [17], rare-gas crystals [19], benzene crys-tal [21], and graphite [20], standard RPA does an excel-lent job regarding the equilibrium lattice constant andcohesive energy, whereas semi-local DFT fails miserably,yielding typically too weak binding and too large lat-tice constant. LDA typically gives a finite binding (of-ten overbinding) for weakly bonded solids, but its per-formance varies significantly from system to system, and

TABLE I: ME, MAE, MAPE (%), and MaxAPE (%) for theatomization energies (in eV/atom), lattice constants (in A),and bulk moduli (in Gpa) of 24 crystalline solids. Results aretaken from Ref. [110]. The experimental atomization ener-gies Ref. [188] are corrected for temperature effect (based onthermochemical correction data) [189] and zero-point vibra-tional energy. The experimental lattice constants have beencorrected for anharmonic expansion effect.

Atomization energies

ME (eV) MAE (eV) MAPE (%) MaxAPE (%)

LDA −0.74 0.74 18.0 32.7PBE 0.15 0.17 4.5 15.4RPA 0.30 0.30 7.3 13.5RPA+ 0.35 0.35 8.7 15.0

Lattice constants

ME (A) MAE (A) MAPE (%) MaxAPE (%)

LDA −0.045 0.045 1.0 3.7PBE 0.070 0.072 1.4 2.7RPA 0.016 0.019 0.4 0.9RPA+ 0.029 0.030 0.6 1.1

Bulk Moduli

ME (GPa) MAE (GPa) MAPE (%) MaxAPE (%)

LDA 9 11 9.6 31.0PBE −11 11 10.7 23.7RPA −1 4 3.5 10.0RPA+ −3 5 3.8 11.4

cannot be trusted in general, as the functional by con-struction does not contain the necessary physics to re-liably describe this phenomenon. In a recent work [27],it was shown that RPA also reproduces the correct 1/d3

asymptotics between graphite layers as analytically pre-dicted by Dobson et al. [190]. This type of behaviorcannot be described by LDA, GGAs, and hybrid func-tionals.

For solids attempts have also been made to go beyondthe standard RPA. For a smaller test set of 11 insulators,Paier et al. [124] showed that adding SOSEX correctionsto RPA the MAE of atomization energies is reduced from0.35 eV/atom to 0.14 eV/atom. By replacing the non-self-consistent HF energy by its self-consistent counter-part, which mimicks the effect of adding single excita-tion corrections [32], reduces the MAE further to 0.09eV/atom [124]. Thus the trend in periodic insulators isagain in line with what has been observed for molecularatomization energies. The effects of SOSEX and rSE cor-rections for metals, and for other properties such as thelattice constants and bulk moduli have not been reportedyet.

19

C. Adsorption at surfaces

The interaction of atoms and molecules with surfacesplays a significant role in many phenomena in surfacescience and for industrial applications. In practical cal-culations, the super cells needed to model the surfacesare large and a good electronic structure approach hasto give a balanced description for both the solid and theadsorbate, as well as the interface between the two. Mostapproaches today perform well for either the solid or theisolated adsorbate (e.g. atoms, molecules, or clusters),but not for the combined system, or are computationallytoo expensive to be applied to large super cells. This isan area where we believe RPA will prove to be advanta-geous.

The systems to which RPA has been applied include Xeand 3,4,9,10-perylene-tetracarboxylic acid dianhydride(PTCDA) adsorbed on Ag(111) [23]; CO on Cu(111)[20, 24] and other noble/transition metal surfaces [25];benzene on Ni(111) [25], Si(001) [112], and the graphitesurface[111]; and graphene on Ni(111) [191, 192], andCu(111), Co(0001) surfaces [192]. In all these applica-tions, RPA has been very successful.

To illustrate how RPA works for an adsorbate system,here we briefly describe the RPA study of CO@Cu(111)following Ref. [24]. The work was motivated by the so-called “CO adsorption puzzle” – LDA and several GGAspredict the wrong adsorption site for CO adsorbed onseveral noble/transition metal surfaces at low coverage[193]. For instance, for the (111) surface of Cu and PtDFT within local/semi-local approximations erroneouslyfavor the highly-coordinated hollow site, whereas exper-iments clearly show that the singly-coordinated on-topsite is the energetically most stable site [194, 195]. Thisposed a severe challenge to the first-principles modelingof molecular adsorption problems and the question arose,at what level of approximation can the correct physics berecovered. In our study, the Cu surface was modeled us-ing systematically increased Cu clusters cut out of theCu(111) surface. Following a procedure proposed by Huet al. [196], the RPA adsorption energy was obtainedby first converging its difference to the PBE values withrespect to cluster size, and then adding the convergeddifference to the periodic PBE results. The RPA adsorp-tion energies for both the on-top and fcc (face centeredcubic) hollow sites are presented in Fig. 11, together withthe results from LDA, AM05 [197], PBE, and the hybridPBE0 functional. Fig. 11 reveals what happens in theCO adsorption puzzle when climbing the so-called Ja-cob’s ladder in DFT [6] — going from the first two rungs(LDA and GGAs) to the fourth (hybrid functionals), andfinally to the 5th (RPA and other functionals that explic-itly depend on unoccupied KS states). Along the way themagnitude of the adsorption energies on both sites arereduced, but the effect is more pronounced for the fcchollow site. The correct energy ordering is already re-stored in PBE0, but the energy between the two sites istoo small. RPA not only gives the correct adsorption site,

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Ead

s(eV

)

on-top sitehollow site

AM05 PBE PBE0 RPA@PBE0RPA@PBELDA

FIG. 11: Adsorption energies for CO adsorbed on the on-top and fcc hollow sites of the Cu(111) surface as obtainedusing LDA, AM05, PBE, PBE0, and RPA. RPA results arepresented for both PBE and PBE0 references, and they differvery little.

but also produces a reasonable adsorption energy differ-ence of 0.22 eV, consistent with experiments. This resultwas later confirmed by the periodic RPA calculations ofHarl and Kresse in Ref. [20], with only small numeri-cal differences arising from the different implementationsand different convergence strategy.

The work of Rohlfing and Bredow on Xe and PTCDAadsorbed at Ag(111) surface represents the first RPAstudy regarding surface adsorption problems, where theauthors explicitly demonstrated that RPA yields the ex-pected −C3/(d−d0)3 behavior for large molecule-surfaceseparations d. Schimka et al. extended the RPA bench-mark studies of the CO adsorption problem to more sur-faces [25]. They found that RPA is the only approach sofar that gives both good adsorption energies as well assurface energies. GGAs and hybrid functionals at mostyield either good surface energies, or adsorption ener-gies, but never both. Goltl and Hafner investigated theadsorption of small alkanes in Na-exchanged chabaziteusing RPA and several other approaches. They foundthat the “hybrid RPA” scheme, as proposed in Ref. [32]and further examined in Ref. [124], provides the mostaccurate description of the system compared to the al-ternatives e.g. DFT-D [198] and vdW-DF [199]. Morerecently RPA was applied to the adsorption of benzeneon the Si(001) surface by Kim et al [112], graphene on theNi(111) surface by Mittendorfer et al [191] and by Olsenet al. [192], and additionally graphene on Cu(111) andCo(0001) surfaces by the latter authors. In all these stud-ies, RPA is able to capture the delicate balance betweencovalent and dispersive interactions, and yields quanti-tatively reliable results. We expect RPA to become in-creasingly more important in surface science with risingcomputer power and more efficient implementations.

20

VI. DISCUSSION AND OUTLOOK

RPA is an important concept in physics and has a morethan 50 year old history. Owing to its rapid developmentin recent years, RPA has been established as a powerfulfirst-principles electronic-structure method with signifi-cant implications for quantum chemistry, computationalphysics, and materials science in the foreseeable future.The rise of the RPA method in electronic structure the-ory, and its recent generalization to r2PT, were borne outby realizing that traditional DFT functionals (local andsemi-local approximations) are encountering noticeableaccuracy and reliability limits and that hybrid densityfunctionals are not sufficient to overcome them. Withthe rapid development of computer hardwares and algo-rithms, it is not too ambitious to expect RPA-based ap-proaches to become (or at least to inspire) main-streamelectronic-structure methods in computational materialsscience and engineering in the coming decades. At thispoint it would be highly desirable if the community wouldstart to build up benchmark sets for materials scienceakin to the ones in quantum chemistry (e.g. G2 [141] orS22 [142]). These should include prototypical bulk crys-tals, surfaces, and surface adsorbates and would aid thedevelopment of RPA-based approaches.

At this point, we would like to indicate several direc-tions for future developments of RPA-based methods.

i. Improved accuracy : Although RPA does not sufferfrom the well-documented pathologies of LDA andGGAs, its quantitative accuracy is not always whatis desired, in particular for atomization energies.To improve on this and to make RPA worth thecomputational effort, further corrections to RPAare necessary. To be useful in practice, these shouldnot increase the computational cost significantly.The r2PT approach as presented in Sec. IV andbenchmarked in Sec. V is one example of this kind.More generally the aim is to develop RPA-basedcomputational schemes that are close in accuracyto CCSD(T), but come at a significantly reducednumerical cost. More work can and should be donealong this direction.

ii. Reduction of the computational cost : The majorfactor that currently prevents the widespread useof RPA in materials science is its high numeri-cal cost compared to traditional DFT methods.The state-of-the-art implementations still have anO(N4) scaling, as discussed in Sec. III. To enlargethe domain of RPA applications, a reduction of thisscaling behavior will be highly desirable. Ideas canbe borrowed from O(N) methods [137] developedin quantum-chemistry (in particular in the contextof MP2) or compression techniques applied in theGW context [136].

iii. RPA forces : For a ground-state method, one cru-cial component that is still missing in RPA are

atomic forces. Relaxations of atomic geometriesthat are common place in DFT and that make DFTsuch a powerful method are currently not possi-ble with RPA or have at least not been demon-strated yet. An efficient realization of RPA forceswould therefore extend its field of application tomany more interesting and important material sci-ence problems.

iv. Self-consistency : Practical RPA calculations arepredominantly done in a post-processing manner,in which single-particle orbitals from KS or gen-eralized KS calculations are taken as input for aone-shot RPA calculation. This introduces unde-sired uncertainties, although the starting-point de-pendence is often not very pronounced, if one re-stricts the input to KS orbitals. A self-consistentRPA approach can be defined within KS-DFTvia the optimized effective potential method [93],and has already been applied in a few instances[39, 40, 127, 128, 130]. However, in its current re-alizations self-consistent RPA is numerically verychallenging, and a more practical, robust and nu-merically more efficient procedure will be of greatinterest.

With all these developments, we expect RPA and itsgeneralizations will play increasingly important roles incomputational materials science in the near future.

Appendix A: RI-RPA implementation in FHI-aims

In this section we will briefly describe how RPAis implemented in the FHI-aims code [132] using theresolution-of-identity (RI) technique. More details canbe found in Ref. [131]. For a different formulation of RI-RPA see Ref. [28, 38]. We start with the expression forthe RPA correlation energy in Eq. (20), which can beformally expanded in a Taylor series,

ERPAc = −

1

π

∫ ∞

0

dω∞∑

n=2

1

2nTr[

(

χ0(iω)v)2]

. (A1)

Applying RI to RPA in this context means to representboth χ0(iω) and v in an appropriate auxiliary basis set.Eq. (A1) can then be cast into a series of matrix op-erations. To achieve this we perform the following RIexpansion

ψ∗i (r)ψj(r) ≈

Naux∑

µ=1

CµijPµ(r) , (A2)

where Pµ(r) are auxiliary basis functions, Cµij are the ex-pansion coefficients, and Naux is the size of the auxiliarybasis set. Here C serves as the transformation matrixthat reduces the rank of all matrices from Nocc ∗ Nvir

to Naux, with Nocc, Nvir and Naux being the number

21

of occupied single-particle orbitals, unoccupied (virtual)single-particle orbitals, and auxiliary basis functions, re-spectively. The determination of the C coefficients isnot unique, but depends on the underlying metric. Inquantum chemistry the “Coulomb metric” is the stan-dard choice where the C coefficients are determined byminimizing the Coulomb repulsion between the residualsof the expansion in Eq. (A2) (for details see Ref. [131]and references therein). In practice, sufficiently accu-rate auxiliary basis set can be constructed such thatNaux << Nocc ∗ Nvir, thus reducing the computationaleffort considerably. A practically accurate and efficientway of constructing auxiliary basis set {Pµ(r)} and theirassociated {Cµij} for atom-centered basis functions of gen-

eral shape has been presented in Ref. [131].Combining Eq. (20) with (A2) yields

χ0(r, r′, iω) =∑

µν

∑

ij

(fi − fj)CµijC

νji

ǫi − ǫj − iωPµ(r)Pν(r′)

=∑

µν

χ0µν(iω)Pµ(r)Pν(r′), (A3)

where

χ0µν(iω) =

∑

ij

(fi − fj)CµijC

νji

ǫi − ǫj − iω. (A4)

Introducing the Coulomb matrix

Vµν =

∫∫

drr′Pµ(r)v(r, r′)Pν(r′) , (A5)

we obtain the first term in (A1)

E(2)c = −

1

4π

∫ ∞

0

dω

∫

· · ·

∫

drdr1dr2dr′×

χ0(r, r1)v(r1, r2)χ0(r2, r′)v(r′, r)

= −1

4π

∫ ∞

0

dω∑

µν,αβ

χ0µν(iω)Vναχ

0αβ(iω)Vβ,µ

= −1

4π

∫ ∞

0

dωTr[

(

χ0(iω)V)2]

. (A6)

This term corresponds to the 2nd-order direct correla-tion energy also found in MP2. Similar equations holdfor the higher order terms in (A1). This suggests thatthe trace operation in Eq. (23) can be re-interpreted as asummation over auxiliary basis function indices, namely,Tr [AB] =

∑

µν AµνBνµ, provided that χ0(r, r′, iω) and

v(r, r′) are represented in terms of a suitable set of auxil-iary basis functions. Equations (23), (A4), and (A5) con-stitute a practical scheme for RI-RPA. In this implemen-tation, the most expensive step is Eq. (A4) for the con-struction of the independent response function, scaling

as N2auxNoccNvir. We note that the same is true for stan-

dard plane-wave implementations [110] whereNaux corre-sponds to the number of plane waves used to expand theresponse function Npw-χ. In that sense RI-based local-basis function implementations are very similar to plane-wave-based or LAPW-based implementations, where theplane waves themselves or the mixed product basis serveas the auxiliary basis set.

Appendix B: Error statistics for G2-I, S22, andNHBH38/NHTBH38 test sets

Tables II, III, and IV present a more detailed erroranalysis for the G2-I, S22, and NHBH38/NHTBH38 testsets. Given are the mean error (ME), the mean ab-solute error (MAE), the mean absolute percentage er-ror (MAPE), the maximum absolute percentage error(MaxAPE), and the maximum absolute error (MaxAE).

TABLE II: ME (in eV), MAE (in eV), MAPE (%),MaxAPE(%) for atomization energies of the G2-I set obtainedwith four RPA-based approaches in addition to PBE, PBE0and MP2. A negative ME indicates overbinding (on average)and a positive ME underbinding. The cc-pV6Z basis set wasused.

ME MAE MAPE MaxAPE

PBE -0.28 0.36 5.9 38.5PBE0 0.07 0.13 2.6 20.9MP2 -0.08 0.28 4.7 24.6RPA 0.46 0.46 6.1 24.2RPA+ 0.48 0.48 6.3 24.9RPA+SOSEX 0.22 0.25 4.2 36.7RPA+rSE 0.30 0.31 4.0 22.5r2PT 0.07 0.14 2.6 24.9

TABLE III: ME (in meV), MAE (in meV), MAPE (%), andMaxAPE(%) for the S22 test set [142] obtained with five RPA-based approaches in addition to PBE, PBE0, and MP2 ob-tained with FHI-aims. The basis set “tier 4 + a5Z-d” [131]was used in all calculations.

ME MAE MAPE MaxAPEPBE 116.2 116.2 57.8 170.3PBE0 105.7 106.5 55.2 169.1MP2 -26.5 37.1 18.7 85.1RPA 37.8 37.8 16.1 28.7RPA+ 51.2 51.2 21.9 39.4RPA+rSE 14.1 14.8 7.7 24.8RPA+SOSEX 15.4 18.0 10.5 34.6r2PT -8.4 21.0 7.1 30.7

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, asobtained using FHI-aims. The cc-pV6Z basis set was used inall calculations. Negative ME indicates an underestimationof the barrier height on average.

HTBH38 NHTBH38

ME MAE MaxAE ME MAE MaxAE

PBE -0.399 0.402 0.863 -0.365 0.369 1.320PBE0 -0.178 0.190 0.314 -0.134 0.155 0.609MP2 0.131 0.169 0.860 0.215 0.226 1.182RPA 0.000 0.066 0.267 -0.065 0.081 0.170RPA+ 0.005 0.069 0.294 -0.068 0.084 0.168RPA+rSE -0.170 0.187 0.809 -0.251 0.252 0.552RPA+SOSEX 0.243 0.244 0.885 0.185 0.188 0.781r2PT 0.072 0.084 0.453 -0.001 0.129 0.432

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