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Zhang, Igor Ying; Rinke, Patrick; Scheffler, MatthiasWave-function inspired density functional applied to the H2/H2 + challenge

Published in:New Journal of Physics

DOI:10.1088/1367-2630/18/7/073026

Published: 01/07/2016

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Please cite the original version:Zhang, I. Y., Rinke, P., & Scheffler, M. (2016). Wave-function inspired density functional applied to the H

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Wave-function inspired density functional applied to the H2/ challenge

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New J. Phys. 18 (2016) 073026 doi:10.1088/1367-2630/18/7/073026

PAPER

Wave-function inspired density functional applied to the H2/H+2

challenge

Igor YingZhang1, PatrickRinke1,2 andMatthias Scheffler1,3

1 Fritz-Haber-Institut derMax-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany2 Department of Applied Physics, AaltoUniversity, POBox 11100, Aalto FI-00076, Finland3 Department of Chemistry andBiochemistry andMaterials Department, University of California-Santa Barbara, Santa Barbara, CA

93106-5050, USA

E-mail: zhang@fhi-berlin.mpg.de

Keywords:molecular dissociation, adiabatic-connetion path, density-functional theory, near-degeneracy correlation

AbstractWe start from the Bethe–Goldstone equation (BGE) to derive a simple orbital-dependent correlationfunctional—BGE2—which terminates the BGE expansion at the second-order, but retains the self-consistent coupling of electron-pair correlations.We demonstrate that BGE2 is size consistent andone-electron ‘self-correlation’ free. The electron-pair correlation coupling ensures the correctH2

dissociation limit and gives afinite correlation energy for any system even if it has a no energy gap.BGE2 provides a good description of bothH2 and

+H2 dissociation, which is regarded as a greatchallenge in density functional theory (DFT).We illustrate the behavior of BGE2 analytically byconsideringH2 in aminimal basis. Our analysis shows that BGE2 captures essential features of theadiabatic connection path that current state-of-the-art DFT approximations do not.

1. Introduction

Density-functional theory (DFT) is nowwidely applied in physics, chemistry,materials science and biology. Thissuccess comes from the availability of suitable approximations for the exchange–correlation (xc) functional—the only quantity that is unknown inKohn–Sham (KS)DFT.However, despite this unmatched success and theubiquitous application of common functionals, all currently available functionals suffer from certain notoriouslimitations. For example, all existing functionals fail to correctly describe the dissociations of both +H2 andH2,two very simplemolecules [1–5]. Solving the +H2 /H2dissociation problemwill not only lead to a conceptualunderstanding of why current functionals fail, but also offer potential pathways to develop better functionals,and has thus attracted increasing attention [2–19].

A viable approach in functional development is to learn fromwave-function theory in constructing nonlocalcorrelation functionals that involve unoccupiedKS orbitals and thus stand on the fifth (and currently highest)rung of Perdewʼs Jacobʼs ladder [7]. A prominent example is second-orderGörling–Levy (GL) perturbationtheory (GL2) [20] that is closely related to second-orderMøllet–Plesset (MP) perturbation theory (MP2) [21–23]inwave-function theory 4. In fact, evenMP2 can be viewed as an implicit density functional bymeans of theadiabatic connection approach [20, 24, 25]. An important feature of 2nd-order perturbation theories (PT2) suchasGL2 andMP2 is that they are one-electron ‘self-correlation’ free [1, 5], i.e. the correlation is zero for one-electron systems, and that they are size consistent (i.e. if the system is fragmented into two parts, the total energybecomes the sumof the two fragments) [26, 27]. Furthermore, PT2 is fully non-local and captures the correctR−6 decay behaviour at long distances, which is essential to provide an accurate description of weak interactions.

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14April 2016

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15 June 2016

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20 June 2016

PUBLISHED

12 July 2016

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4GL2 differs fromMP2 in the employed orbitals.WhileMP2 is formulated in terms ofHartree–Fock (HF) orbitals, GL is based onKS

orbitals. Therefore, the single-excitation contribution of second-order perturbation theory is nonzero inGL2. Inmost of currentinvestigations based onGL2 [6, 7, 30, 31, 36, 106], this single-excitation contribution is not taken into account. In this paper, we refer tosecond-order perturbationmethods that use theKS orbitals but take only double and not single excitations into account as PT2.We leave theinclusion of the single-excitation, which are important forweak interactions [51], to futurework.

© 2016 IOPPublishing Ltd andDeutsche PhysikalischeGesellschaft

Therefore, PT2 correlation is an ideal building block forfifth-level density functionals, following Perdewʼsnomenclature. This feature has been exploited in a range of promising double-hybrid functionals [28–36],whichlinearlymix generalized gradient approximations, e.g. BLYP [37, 38] or PBE [39], with both exact exchange andPT2 correlation. The admixture of semi-local exchange and correlation can be viewed as an efficient yet semi-empirical way to take higher-order perturbative contributions into account thatwould go beyond PT2 [40, 41].These double-hybrid functionals provide a satisfactory accuracy for various chemical interactions, but they failfor systemswith small KS energy gaps, e.g. heavily-stretchedH2 andmetallic systems. In such systems, two ormore determinants become degenerate in energy andmean-field theories that rely on a single referencedeterminant such asHForKS–DFTbreak down. Also, perturbation theory diverges at any order,making itessential tofind an appropriate resummation. Seidl, Perdew andKurth suggested an empirical adiabatic-connection (AC)model [20, 24, 25], namely interaction-strength interpolation [7], to implicitly resum theperturbation expansion by using only exact exchange, PT2 correlation and an explicit density functional derivedfrom the ‘point charge plus continuum’model in the strong-interaction limit where the coupling constantparameter goes to infinity [42]. Frequently, the analytic dependence on the coupling constant is approximatedby a Padé formula, whose parameters can be determined either empirically on theoretical grounds [6, 10].Recently, an explicit density functional in the strong-interaction limit was suggested. It can be constructed fromso-called co-motion functions and captures fully non-local effects in the strong-interaction limit [43].

An example of a successful resummation is the particle–hole random-phase approximation (RPA), inwhichan infinite number of ‘ring diagrams’ is summed [8, 44–49]. This has beenwidely recognized as key tomake RPAapplicable to small-gap ormetallic systems. The RPAmethod provides the correctH2 dissociation limit [2, 15].Unfortunately, it suffers from a heavy ‘self-correlation’ error for one-electron systems [1–5], and thus yields anevenworse +H2 dissociation behaviour than conventional density functionals. In addition, RPA exhibits anincorrect repulsive ‘bump’ at intermediateH2 bond distances. These deficiencies have in the past been attributedto a lack of self-consistency in RPA [2, 5], whichwas disproved by actual self-consistent calculations [15, 50].Anotherwidely accepted hypothesis attributes these deficiencies to the lack of higher order diagrams andspurred considerable beyond-RPAdevelopments in the past few years [3, 16, 18, 19, 51–56]. Other interestingdevelopments in this realm include reduced densitymatrix theory [57] or self-consistent Greenʼs functionframeworks [15, 58]. A successful beyond-RPAmethod is renormalized second-order perturbation theory(rPT2), which adds an infinite summation of the second-order exchange diagramof PT2 (termed second orderscreened exchange (SOSEX)) [4, 59–61] and renormalized single-excitation (rSE) [51, 62] diagrams on top ofRPA [16]. rPT2 does not diverge for small-gap systems and is free of one-electron ‘self-correlation’. It thusprovides the correct description of one-electron systems including +H2 and individualH atoms, but fails forH2

dissociation if breaking spin symmetry is not allowed [3, 16, 52 and also below]. Further improvements havebeen stipulated in the context of the couple-cluster (CC) theory [18, 52–56] or the Bethe–Salpeter equation[3, 19, 22]. Thesemethods, although proposed fromdifferent perspectives, can all be interpreted as attempts toexplicitly introducemore ‘selective summations to infinite order’ in the density functional perturbationframework. Even though thesemethods improve over standard RPA schemeswith varying degrees of success forthe +H2 /H2 dissociation problem, no improvement to date removes the ‘bump’while simultaneously yieldingthe correct dissociation limit for +H2 andH2, indicating the difficulty of understanding this problem in anyperturbative framework. In this context, Aggelen et al recently proposed the particle–particle randomphaseapproximation (pp–RPA). It describes pairingmatrix fluctuations by the particle–particle (pp)-propagator, andcan also be viewed as aCCdoubles (CCD) restricted to ladder diagrams [55, 56]. The pp-RPA captures thedissociation limit inH2 and eliminates the ‘self-correlation’ error in +H2 , but results in amore repulsive ‘bump’than standardRPA [55].

In this paper, we lay the ground for an efficient orbital-dependent correlation functional based on the Bethe–Goldstone equation (BGE) [63], which is derived from the correlation of two particles [22]. As the BGE is thesimplest approximationwhich provides the exact solution for one- and two-electron systems, it is a goodstarting point to understand the aforementioned +H2 /H2dissociation challenge. In section 2, we formulate theBGE in the context ofDFT through the adiabatic connection approach. In contrast to the normal resummationstrategy in density functional perturbation theory, we propose a new correlation functional by terminating theBGE expansion at the second order (BGE2). As shown in section 3, this BGE2 approximation gives a gooddescription of both +H2 andH2 dissociations, without requiring any higher order connectedGoldstone diagramsthat are commonly believed to be necessary.We further analyse BGE2 analytically in theminimal basis H2

model.We show that BGE2 is size-extensive and free of one-electron ‘self-correlation’.

2

New J. Phys. 18 (2016) 073026 I YZhang et al

2. BGE in density functional theory (DFT)

In theAC approach ofDFT [20, 24, 25], the non-interacting KS system is connected to the physical systemby anadiabatic path. The density n along the path is fixed to the exact ground-state density. TheHamiltonian for afamily of partially interactingN-electron systems in this path is controlled by a coupling-constant parameterλ,(atomic units are used hereafter unless stated otherwise):

l l l= + - = + Dl l lH H V v H . 1s seeˆ ˆ ( ˆ ˆ ) ˆ ( )

Here, Hsˆ is theHamiltonian of the non-interacting KS system

⎡⎣⎢

⎤⎦⎥å= - + rH v

1

2, 2s

i

N

i s i2ˆ ( ) ( )

where rvs ( ) is amultiplicative one-electron potential

= + + +r r r r rv v v v v 3s ext H x c( ) ( ) ( ) ( ) ( ) ( )comprising the external potential (vext ) arising from theCoulomb interaction between the electrons and thenuclei, theHartree potential (vH ), and the exchange (vx) and correlation (vc) potential. The operator lv is alsomultiplicative and constrained to satisfy =v 00 and = +v v v1 H xcˆ ˆ ˆ . Thus, =l=H Hs0

ˆ ˆ , while H1ˆ is the

Hamiltonian of the fully interacting system. From the perspective ofmany-body perturbation theory,lD = -l lV vee

ˆ ˆ is a perturbation of the non-interactionKSHamiltonian, which does not change the ground-state density n. By using coordinate scaling [20, 25, 64], it was shown that

⎡⎣⎢

⎤⎦⎥ål l

dd

a l add

= + + = =la a

=

-r rr

rr

v v vE n

nv

E n

n, , , , 4i i

i

N

ii

i1H x

c 1c

cˆ ( ) ( ) [ ]( )

( ) [ ]( )

( )!

where a a a a=an x y z n x y z, , , ,3( ) ( ), and arv ,ic ( ) is the correlation potential of the scaled correlation energywith respect to the normal density n.

In the AC framework [20, 24, 25], the xc functional can be interpreted as the coupling-constant integration

ò òll

l=¶¶

=l

lE nE n

Vd d , 5xc0

1xc

0

1

xc[ ] [ ] ( )

where lE nxc [ ] is the xc functional for a given coupling-constant λ [65]

l l= áY + Y ñ - áF F ñ -l l lE n T V T E n . 6n n n nxc ee H[ ] ∣ ˆ ˆ ∣ ∣ ˆ ∣ [ ] ( )lVxc is the corresponding xc potential for a given coupling constantλ.We can further define the exchange lE nx [ ]

and correlation lE nc [ ] components separately

l l= áF F ñ - =

=áY Y ñ - áF F ñ

l

l ll

ll

E n V E n E n

E n H H . 7

n n

n n n n

x ee H x

c

[ ] ( ∣ ˆ ∣ [ ]) [ ][ ] ∣ ˆ ∣ ∣ ˆ ∣ ( )

Here Yln is the ground-state wave-function on the ACpathwith the coupling constantλ, which gives the same

ground-state density n as the physical system (l = 1). Y = Fn n0 is thus the ground-state wave-function of the

non-interacting KS system. E nH [ ] is theHartree energy

ò=-

r rr r

r rE n

n n1

2d d 8H 1 2

1 2

1 2

[ ] ( ) ( )∣ ∣

( )

which is an explicit functional of the density. Immediately, we have =lV E nx x [ ], as the exchange densityfunctional lE nx [ ]defined in thismanner is linear in the coupling constantλ. And the correspondingHartreepotential vH is written as

åf f f f f fá ñ = á ñv 9a ai

occ

a i a iH∣ ∣ ∣ ( )

with the definition of the two-electron four-center integral as

* *òf y f f

f y f fá ñ =

-r r

r r r r

r rd d . 10i j k l

i j k l1 2

1 2 1 2

1 2

∣( ) ( ) ( ) ( )

∣ ∣( )

In contrast, it is in general not possible to obtain the exact lE nxc [ ] for any l ¹ 0, since the electron–electronrepulsion operatorVee

ˆ appears explicitly in theHamiltonian, and the ground-state wave-function Yln cannot be

obtained exactly. This is also true for two-electron systems, although the ground-state wave-function is now justa simple electron-pair function

3

New J. Phys. 18 (2016) 073026 I YZhang et al

Y = Yl . 11n ab ( )

As one of themotivations in this paper is to construct a functional which can provide an accurate description forbothH2 and

+H2 dissociations, we start from the BGE of lH [22], which is derived from the correlation of twoparticles, and is thus the exact solution for one- and two-electron systems. The BGE explicitly solves theSchrödinger equation for each electron pair ab interacting through a perturbation lH1

ˆ ( )

l ll l

+ Y = Y

- Y = Y

H H E

E H H . 12

s ab ab ab

ab s ab ab

1

1

[ ˆ ˆ ( )][ ˆ ] ˆ ( ) ( )

Here, we consider the electron–electron interactionVeeˆ of electron pair ab explicitly, while leaving the

interactionwith the otherN-2 electrons on themeanfield level vabMFˆ . For two electronswe trivially have =v 0ab

MFˆ .However, formore than two electronswewould have tomake this approximation explicitly. The resultingperturbation operator is

l l= - +lH V v v 13ab1 eeMFˆ ( ) ˆ ˆ ˆ ( )

with the definition of vabMFˆ as

å å f f f fáF F ñ = á ñ= ¹

v1

2, 14ab ab ab

i a b j a b

occ

i j i jMF

, ,

∣ ˆ ∣ ∣∣ ( )

where fi{ } are theKS orbitals and f f f f f f f f f f f fá ñ = á ñ - á ñi j i j i j i j i j j i∣∣ ∣ ∣ . TheKS orbitals can be used togenerate an antisymmetric non-interacting KS electron-pair function

f ff f

F =1, 21

2

1 1

2 2. 15ab

a b

a b

( )( ) ( )( ) ( )

( )

Here the numbers 1 and 2 are a short-hand notation for the tuple of space and spin variables of the first andsecond electron, respectively. The corresponding non-interactingGreenʼs function ¢ ¢G E1, 2; 1 , 2 ; ab0 ( ) for thiselectron pair ab is

⎛⎝⎜⎜

⎞⎠⎟⎟ * å å å¢ ¢ =

F- -

+F- -

+F- -

Y ¢ ¢= <

G EE E E

1, 2; 1 , 2 ;1, 2 1, 2 1, 2

1 , 2 , 16abab

ab a b i a b r

unoccir

ab i r r s

unoccrs

ab r sab0

,

( ) ( ) ( ) ( ) ( ) ( )

where i{ }are theKS eigenvalues

f f f f f f= á ñ = á ñ + á ñH T v . 17i i s i i i i s i∣ ˆ ∣ ∣ ˆ ∣ ∣ ∣ ( )

As Hsˆ (equation (2)) is a one-electron operator, it is also possible to reorganize the eigenvalues in terms of each

electron pair ab

= + = áF F ñH 18ab a b ab s ab∣ ˆ ∣ ( )

which could be considered as the zero-order approximation of the electron pair energy Eab.Nowwe introduce thefirst approximation to the BGE.Weneglect the single excitation contribution, i.e., the

first sum in equation (16)

⎛⎝⎜

⎞⎠⎟ * å¢ ¢ »

F- -

+F- -

Y ¢ ¢<

G EE E

1, 2; 1 , 2 ;1, 2 1, 2

1 , 2 . 19abab

ab a b r s

unoccrs

ab r sab0 ( ) ( ) ( ) ( ) ( )

Aswill be discussed in section 4.1, this approximation, together with the other two approximations wewillmakelater, is essential for achieving an efficient correlation functional which, however, keeps the exact solution for theH2 dissociation limit in theminimal basis.

With this approximation, the electron-pair function Yab can bewritten as

ò

å

l l

l ll l

Y = ¢ ¢ ¢ ¢ ¢ ¢ F ¢ ¢

= FáF Y ñ

- -+

F- -

áF Y ñ

d d G E H

H

E EH

1, 2 1 2 1, 2; 1 , 2 ; 1 , 2 1 , 2

1, 21, 2

. 20

ab ab ab

abab ab

ab a b rs

unoccrs

ab r srs ab

0 1

11

( ) ( ) ˆ ( )( ) ( )

( ) ∣ ˆ ( )∣ ( ) ∣ ˆ ( )∣ ( )

It is convenient to introduce intermediate normalization áF Y ñ = 1ab ab∣ . Together with the expression of theexpectation value of the perturbation energy

l l- - = áF Y ñE H 21ab a b ab ab1∣ ˆ ( )∣ ( )

4

New J. Phys. 18 (2016) 073026 I YZhang et al

the BGE electron pair function Y 1, 2ab( ) becomes

å l lY = F +F- -

áF Y ñ< E

H1, 2 1, 21, 2

. 22ab abr s

unoccrs

ab r srs ab1( ) ( ) ( ) ∣ ˆ ( )∣ ( )

Since the BGE electron pair function Yab appears on both sides of this equation, both Yab andEab (equation (23))contain an infinite sequence ofGoldstone diagrams [22, 23], as one can easily see by inserting equation (22) intoequation (21)

ål l

l l l l- - = áF F ñ +

áF F ñáF Y ñ- -<

E HH H

E. 23ab a b ab ab

r s

unoccab rs rs ab

ab r s1

1 1∣ ˆ ( )∣ ∣ ˆ ( )∣ ∣ ˆ ( )∣ ( )

Wenow consider the different terms step by step. First, we expand the leab1st ( ) termon the right-hand side

l l l

l l

= áF F ñ

= áF + - F ñl

e H

V v v . 24

ab ab ab

ab ab ab

1st1

eeMF

( ) ∣ ˆ ( )∣∣ ˆ ˆ ˆ ∣ ( )

It is thefirst-order correction to the non-interaction electron pair energy ab defined in equation (18). Utilizingthe definitions of lv (equation (4)) and vab

MFˆ (equation (14)) for the fully-interacting system (l = 1), we have

å f f

= áF + - - F ñ

= - -

= +=

e V v v v

v v v

e e

1

1

2

1 1 , 25

ab ab ab ab

i a bi i

a b

1stee

MFH xc

,xHF

H xc

1st 1st

( ) ∣ ˆ ˆ ˆ ˆ ∣

( ˆ ˆ ) ˆ

( ) ( ) ( )

where 1a1st ( ) is the corresponding first-order correction of the non-interaction electron energy a for the fully

interacting system. And vxHFˆ is theHartree–Fock like exact exchange operator defined as

åf f f f f fá ñ = - á ñv . 26a ai

occ

a i i axHF∣ ˆ ∣ ∣ ( )

Together with the the non-interaction electron pair energy ab, this leads to the electron-pair total energy atthefirst-ordermany-body perturbation level, which contains only the exact exchange.Next we define the BGEelectron-pair correlation energy leab

BGE ( ) as

l l= - - -e E e . 27ab ab a b abBGE 1st( ) ( ) ( )

For two electrons, leabBGE ( ) is the total correlation energy lE nc

BGE [ ]( ). Formore than two electronswe have tosumup the the correlation energies of all electron pairs:

ål l=<

E n e . 28a b

occ

abcBGE BGE[ ]( ) ( ) ( )

Finally, the BGE total energy for the fully interacting system (l = 1) becomes

å å= + +

= +<

E e n

E E n

1 1

1 , 29

a

occ

a aa b

occ

abtotBGE 1st BGE

totEX

cBGE

( ) [ ]( )

[ ]( ) ( )

where EtotEX is the exact-exchange total energy in theKS–DFT framework. BGE is thus exact for one- and two-

electron systems, but approximate formore electrons, because interaction terms between three ormoreelectrons aremissing.

With the definition of leabBGE ( ), equation (23) becomes

å

å

å å

ll l l

l l

l l l ll l

l l l l l l

l l l l

=áF F ñáF Y ñ

+ - D

=áF F ñáF F ñ

+ - D

+áF F ñáF F ñáF Y ñ

+ - D + - D

=

<

<

< <

eH H

e e

H H

e e

H H H

e e e e

, 30

abr s

unoccab rs rs ab

ab ab abrs

r s

unoccab rs rs ab

ab ab abrs

r s

unocc

p q

unoccab rs rs pq pq ab

ab ab abrs

ab ab abpq

BGE2

1 1BGE 1st

1 1BGE 1st

1 1 1

BGE 1st BGE 1st

( ) ∣ ˆ ( )∣ ∣ ˆ ( )∣( ) ( )

∣ ˆ ( )∣ ∣ ˆ ( )∣( ) ( )

∣ ˆ ( )∣ ∣ ˆ ( )∣ ∣ ˆ ( )∣( ( ) ( ) )( ( ) ( ) )

( )

where D = + - -abrs

r s a b. The second term on the third line of equation (30) emerges whenwe replaceYab by equation (22). This expansion reveals that the BGE correlation energy contains an infinite summation of asequence ofGoldstone pair diagrams [22, 23]. Therefore, this sequence ofGoldstone diagrams contains only twohole lines, representing the electron pair ab, and the infinite summation goes through all the ladder diagrams

5

New J. Phys. 18 (2016) 073026 I YZhang et al

over two particle lines (seefigure 1). In other words, the intermediate pairs always propagate as electrons [22].Conversely, equation (30) has to be solved self-consistently, as the electron-pair energyEab depends on itself.

As alluded to before, the BGE accounts for the correlation of two electrons and thus provides the exactsolution for one- and two-electron systems.However, equation (30) is not exact, becausewe omitted the singleexcitations at the very beginning.Nonetheless, our approximation should still be able to capture the subtle(near)-degeneracy static correlation effects atH2 dissociation and eliminate the one-electron ‘self-correlation’error as does the configuration-interactionmethodwith double excitations (CID) or theCCDmethod. Asmentioned above, the electron-pair correlationwith all ladder diagrams can also be introduced frompairingmatrix fluctuation in the pp-RPA [56]. Nesbet [66] has demonstrated that BGE is equivalent to the so-calledindependent electron-pair approximation (IEPA) in quantum chemistry [23], which can be considered as anintermediate approximation betweenCID andMP2.However, we prefer to keep the BGE acronym(equation (30)) as it ismore compact and easily linked to an expansion ofGoldstone diagrams [23, 67, 68]whichis helpful for further discussions (figure 1).

3. The second-order BGE approximation

3.1.Derivation of BGE2The BGE electron-pair correlation energy eab

BGE in equation (30) contains an infinite sequence of particle–particle ladder diagrams (pp-ladder resummation), as shown infigure 1, because the BGEwave function Yab alsoappears on the right-hand side of the equation. In addition, equation (30) should be solved iteratively as eab

BGE

appears on both sides of the equation (eab-coupling effect). These twomechanisms cooperate to deliver anaccurate description of exchange and correlation in one- and two-electron systems. It has been argued that anexplicit (or implicit) resummation of a selected series of diagrams (e.g., the pp-ladder resummation shown infigure 1) is necessary to remove the divergence at degeneracies of any finite-order perturbation theory[7, 8, 16, 19, 47]. However, in this workwewill show that the same effect can be achieved by the eab-couplingeffect atfinite orders of perturbation theory. This allows us to terminate the BGE expansion at the second order,as long aswe retain the eab-coupling effect. This second-order BGE (BGE2) is the second approximationwemake:

å

å

å

ll l

l l

ll l

l f f f fl l

=áF F ñ

+ - D

=áF F ñ+ - D

=á ñ

+ - D

<

<

<

eH

e e

V

e e

e e. 31

abr s

unoccab rs

ab ab abrs

r s

unoccab rs

ab ab abrs

r s

unocca b r s

ab ab abrs

BGE22

12

BGE2 1st

2ee

2

BGE2 1st

2 2

BGE2 1st

( ) ∣ ∣ ˆ ( )∣ ∣( ) ( )

∣ ∣ ˆ ∣ ∣( ) ( )

∣ ∣∣ ∣( ) ( )

( )

Herewe have utilized the fact that both vabMFˆ and lv are one-electron operators, which do not contribute to the

expectation value between the ground state and a double excitation.Wewill show in section 4.1 that BGE2 only dissociatesH2 in aminimal basis correctly, if we remove eab

1st fromthe denominator. So in the followingwe drop eab

1st. This is ourfinal approximation:

ål

l f f f fl

»á ñ

- D<

ee

. 32abr s

unocca b r s

ab abrs

BGE22 2

BGE2( )

∣ ∣∣ ∣( )

( )

leabBGE2 ( ) now appears as a simple sum-over-state formula that is similar to standard PT2 and thus exhibits the

same computational scaling.Wewill again need to sumall electron pairs to obtain the full BGE2 correlationenergy lEc

BGE2 ( ) for systemswithmore than two electrons

Figure 1.TheGoldstone diagrams in the BGE are an infinite sequence of particle–particle ladder diagrams (pp-ladder) [22]. Thesquiggly lines refer to the bare Coulomb interaction.

6

New J. Phys. 18 (2016) 073026 I YZhang et al

ål l=<

E e . 33a b

occ

abcBGE2 BGE2( ) ( ) ( )

Adistinct advantage of equation (32) is that the dependence on the coupling constantλ is now simple andwell-defined. Following equation (5)we can easily obtain the BGE2 electron-pair correlation potential lvab

BGE2 ( ) as

⎛⎝⎜

⎞⎠⎟

åll f f f f

l

l ll

=á ñ

- D

´ +- D

<

ve

v

e

2

11

234

abr s

unocca b r s

ab abrs

ab

ab abrs

BGE22

BGE2

BGE2

BGE2

( )∣ ∣∣ ∣

( )

( )( )

( )

and then for amany electron system

ål l=<

V v . 35a b

occ

abcBGE2 BGE2( ) ( ) ( )

The BGE2 correlation potential also has to be solved iteratively, which prevents us frommaking furtheranalyticalmanipulations. However, forH2 in aminimal basis, the BGE2 correlation energy and potential can besolved analytically, whichwill give usmore insight into BGE2. This will be discussed later in section 4.1.

At this point, wewill recap the approximationsmade in the derivation of the BGE2 correlation functional:

(1) From the outset we chose a pair theory. The full BGE (equations (11) and (12)) explicitly treats interactions inone electron pair and is exact for one- and two-electron systems. Formore than two electrons, the interactionfromother electrons can be taken into account in ameanfield fashion (equation (14)). Then the correlationenergy sums up the correlations of all electron pairs (equations (28) and (33)).

(2)The BGE can be solved by means of Greenʼs functions (equation (20)). Here we omit the single excitationcontribution in the construction of the non-interactingGreenʼs functionG0 (equation (19)).We argue thatthis approximation is justified and captures the subtle xc effects in one- and two-electron systems as doesCID orCCD.

(3)The next approximation is to terminate the BGE expansion at the second order (equation (31)). This impliesthat we remove the infinite summation of particle–particle ladder diagrams (figure 1). The resulting BGE2approximation still retains the eab-coupling effect.Wewill show later that the eab-coupling at second-order inperturbation theory is sufficient to capture correlations that emerge from (near)-degeneracies and thathigher-order connectedGoldstone diagrams are then not needed.However, if we are far fromdegeneracies(e.g., H2 in themiddle of dissociation)BGE2 alone is not sufficient and higher order diagramswould berequired.

(4)The final approximation (equation (32)) removes the first-order perturbation term eab1st from the

denominator of equation (31). On the one hand, this omission removes the difficulty of having to considerthe unknown density adaptive operator lv explicitly along the ACpath (see equations (4) and (24)). On theother hand, wewill show in section 4.1 that togetherwith the other two approximations, this approximationis necessary to deliver an accurate description ofH2 dissociation in aminimal basis.

3.2. Analysis of the eab-coupling effect in BGE2Comparing to the standard PT2 expression, the only difference in the BGE2 correlation expression(equations (32) and (33)) is that the BGE2 electron-pair correlation leab

BGE2 ( ) itself appears in the denominator.leab

BGE2 ( ) should acquire a finite value to prevent the numerical divergence for small-gap systemswhereD 0ab

rs . To distinguish BGE2 fromPT2,modified particle and hole lines (double line) are introduced infigure 2 to represent the eab-coupling effect in the BGE2methodwhich corrects the double-excitation energiesD ab

rs and should be solved iteratively.Wewill demonstrate the accuracy of the eab-coupling effect later bothnumerically (section 3.3) and analytically (section 4.1). In this section, wewill provide amany-bodyperturbation theory perspective of the eab-coupling effect.

In quantum chemistry, the configuration interaction equationwith singles and doubles (CISD) is usuallysolvedwith iterative techniques [69], to avoid a direct diagonalization of the largeHamiltonianmatrices inconfiguration space. Pople et al [70] demonstrated that such iterative algorithms aremore than just a technicaltrick. From amany-body perturbation theory viewpoint, each iteration introduces higher order terms. Forexample, after the second iteration, the second- and third-order terms emerge in theCISD energy expression butwith a scaledweight [70, 71]. This would also be true for the eab-coupling effect in the complete BGE expansion(equation (30)). But canwewrite down a perturbative expansion for the second-order BGE expression(equation (32) andfigure 2)? In otherwords, does the eab-coupling effect introduce higher order perturbationterms during the iterative procedure? In this section, wewill answer these questions step-by-step.

7

New J. Phys. 18 (2016) 073026 I YZhang et al

Todemonstrate the behavior of BGE2 for (near)-degeneracies, we analyse its limit as D abrs goes to zero. To

do sowe need to introduce a level-shift (L) into the expression of the BGE2 correlation energy (equation (32))

å

å

ll f f f fl

l f f f f

=á ñ

+ - D +

=-á ñD +

-

<

<

-

ee L L

Lx L1 , 36

abr s

unocca b r s

ab abrs

r s

unocca b r s

abrs

BGE22 2

BGE2

2 21

( )∣ ∣∣ ∣

( ) ( )∣ ∣∣ ∣( )

( ( )) ( )

where

l

=+

D +x L

e L

L. 37ab

abrs

BGE2

( ) ( ) ( )

To be able to expand equation (36) into a geometric series we require

- < <x L1 1. 38( ) ( )This leads to the following constraint for L

⎜ ⎟⎛⎝

⎞⎠ l> - D +L emax 0,

1

2. 39ab

rsabBGE2( ( )) ( )

By definitionwe have e 0abBGE2 and D 0ab

rs . In addition, lD > eabrs

abBGE2∣ ( )∣holds for insulators,most

semi-conductors and even formost of the double excitations in small-gap systems (excluding cases where abrefers to the highest occupiedmolecular orbital (HOMO)).We can then always choose L= 0.

However, we cannot take L= 0when the energy gap of double excitations from theHOMO to the LUMO( D ab

rs ) tends to zero and lD eabrs

abBGE2∣ ( )∣. Then only a positive level shift l> - D +L eab

rsab

1

2BGE2( ( ))will

guarantee a convergent geometric expansion.Wewill discuss the positive level shift later and first analyse the L= 0 case. Expanding equation (36) into a

geometric series yields

ååll f f f f

= -á ñD +=

¥

<

eL

x L , 40abn r s

unocca b r s

abrs

nBGE2

0

2 2

( )∣ ∣∣ ∣( )

( ) ( )

which for L= 0 becomes

ååll f f f f

l= -á ñD=

¥

<+

e e . 41abn r s

unocca b r s

abrs n ab

nBGE2

0

2 2

1BGE2( )

∣ ∣∣ ∣( )

( ) ( )

This expression allows us to analyze the BGE2 correlation energy from amany-body perspective by iterating theright-hand side.Here, we examine the simplest two terms. Thefirst term in the BGE2 expansion (n= 0) isnothing but the standard PT2 correlation energy

åll f f f f

= -á ñ

D<

e . 42abr s

unocca b r s

abrs

BGE2,1st2 2

( )∣ ∣∣ ∣

( )

TheGoldstone diagrams of the PT2 correlation are shown infigure (2). The second termbecomes

åll f f f f

=á ñ

D´

<

e S, 43abr s

unocca b r s

abrs

BGE2,2nd4 2

( )∣ ∣∣ ∣

( )

Figure 2.The diagramatic representation of PT2 andBGE2.Double lines in the BGE2 diagram represent a correction to the doubleexcitation energies due to the eab-coupling effect, which should be solved iteratively. The squiggly lines refer to the bare Coulombinteraction.

8

New J. Phys. 18 (2016) 073026 I YZhang et al

where S is the normalization of the first-order perturbative wave-function Fab1st of the electron-pair ab

å

å

f f f f

f f f f

F ñ=á ñ

DF ñ

=á ñ

D= áF F ñ

<

<

S 44

abr s

a b r s

abrs rs

r s

unocca b r s

abrs ab ab

1st

2

21st 1st

∣∣∣

∣

∣ ∣∣ ∣( )

∣ ( )

eabBGE2,2nd is a fourth-order perturbation in terms of the coupling constantλ. This expansion includes only even

powers of the perturbation. On the other hand, eabBGE2,2nd can be interpreted as 32 quadruple-excitation

Goldstone diagramswhich, however, are both disconnected. The eab-coupling effect in BGE2 does therefore notproduce higher-order connectedGoldstone diagrams. Using the so-calledHugenholtz diagram rule [72], these32 quadruple-excitations can be represented by twoHugenholtz diagrams, which are shown in figure 3.

Recently, a self-consistent Greenʼs function schemewas proposed at 2nd order as well [58]. This self-consistent second-order self-energymethod also exhibits promising performance for systemswith strongcorrelation. It would be very interesting to compare the diagrams of theGreenʼs function theorywith BGE2 inthe future.

Nowwe consider the case of lD eabrs

abBGE2∣ ( )∣, where the (near)-degeneracy effects are dominant. As

mentioned above, to guarantee a convergent geometric expansion, a positive level-shift l> - D +L eab

rsab

1

2BGE2( ( )) is required. By inserting the binomial series +e Lab

nBGE2( ) into equation (36)weobtain the corresponding geometric series

⎜ ⎟⎛⎝

⎞⎠ åå ål

l f f f fl= -

á ñD +

¥-

<+

en

mL

Le . 45ab

n m

nn m

r s

unocca b r s

abrs n ab

mBGE22 2

1BGE2( )

∣ ∣∣ ∣( )

( ) ( )

Wenote that the L=0 case in equation (41) is a special case of equation (45). For positive level shifts, it can beeasily proven that the first two expansion terms of equation (45) are the same as in equations (42) and (43), onlythat the level-shift L appears in the denominator.We plot the perturbative expansion of the BGE2 correlation infigure 4. For the L=0 case, the BGE2 correlation can be interpreted based on the standard perturbativeexpansion.However, if lD eab

rsabBGE2∣ ( )∣, a positive level shift is required to guarantee awell-defined

perturbative expansion of the BGE2 correlation. Infigure 4, we show thefirst- and second-order geometricexpansion of the direct termof the BGE2 correlation. The exchange part can be expanded in the sameway. Thicklines are introduced to represent a positive level-shift L to the double excitation energies D ab

rs . For both cases,higher order excitations are not involved, but only rescale weights of the existing contributions. The S is themodified Swith a positive level-shift L:

Figure 3.The BGE2 diagrams at second order are rescaled PT2-like diagrams represented by two disconnectedHugenholtz diagramswith quadrupole excitations, which can be expanded into 32 disconnectedGoldstone diagrams [22, 72].

Figure 4.The perturbative expansion of the direct term in the BGE2 correlation. Double lines in the BGE2diagram represent acorrection to the double excitation energies due to the eab-coupling effect, which should be solved iteratively. Thin lines for the case ofL=0 are the particles and holes for the standardmany-body perturbation theory (top). However, thick lines in the perturbationexpansion for the >L 0 case suggest a constant level-shift L to the double excitation energies (bottom). The squiggly lines refer to thebare Coulomb interaction.

9

New J. Phys. 18 (2016) 073026 I YZhang et al

åf f f f

=á ñD +< L

S . 46r s

unocca b r s

abrs

2

2

∣ ∣∣ ∣( )

( )

The divergence ofMP andGLperturbation theories has beenwidely discussed in quantum chemistry [73–77]. Small-gap systemswith strong (near)-degeneracy effects are, of course, one kind of failure, as theperturbation energy diverges at anyfinite order. For non-degenerate systems, where D ab

rs is not exactly zero, theMP (orGL) perturbation expansion does not diverge. However, Leininger et al found that the perturbationexpansion also does not converge toward the exact solution even for very simple systems such asNe, F−, andCl−

[74]. The individual terms in the perturbative expansion of these systems do not diverge, but exhibit anoscillatory divergence with increasing order [73, 74]. From amathematical point of view theMP expansion fails,if the single reference, e.g., HF orKS, is far from the exact ground state [75–77]. However, there is no simplediagnostic tool to determinewhen amulti-reference problembreaks theMP expansion.

In this workwe propose to use the condition lD eabrs

abBGE2∣ ( )∣as a simple criterion to judge the divergence

of a single-reference perturbationmethod. If theHOMO-LUMOgap of a given single reference is smaller thanthe absolute value of the corresponding BGE2 electron-pair correlation energy, it is not advisable to use aperturbativemethod based on this single reference, because the BGE2 correlation cannot be expanded directlywithout a proper level-shift L (equation (36)). The value of the level shift L then quantifies themulti-referencenature of each electron pair in the investigated systems.

3.3.H2 and+H2 dissociation

OurBGE2 xc functional encompasses the exact exchange and the BGE2 correlation term (equations (32) and(33))

l= + =E E E 1 . 47xcBGE2

xEX

cBGE2 ( ) ( )

It has been implemented in the FritzHaber Institute ab initiomolecular simulations (FHI-aims) code package[78, 79]. Due to its simple sum-over-state formula, BGE2 has the same computational scaling as standard PT2 interms of both time andmemory. Although the eab-coupling requires an iterative solution, convergence is fast inour experience, and an accuracy of -10 8 Hartree can be achievedwithin a few iterations.

Infigure 5we plot theH2 and+H2 dissociation curves for variousmethods (BGE2, PBE, PBE0, PT2, RPA, and

rPT2). All results are obtainedwith input KS orbitals from a PBE0 calculation [80–82]. All calculations,including theCISD reference, are carried out with FHI-aims using theNAO-VCC-5Z basis set [79, 83]. Infigure 6we show the same curves for RPA, rPT2 andBGE2 for different starting points (PBE, PBE0 andHF).

In the followingwewill analyse the performance of the different approaches shown infigure 5 class by class.In the dissociation of +H2 only non-local exact exchange is required and correlation is absent, while dissociatedH2 contains strong (near)-degeneracy static correlation, which current DFTmethods typically underestimate.As illustrated infigure 5, the PBE0 functional fails in both cases, yielding a heavy one-electron ‘self-correlation’error (around 66mHartree in the +H2 dissociation limit) and a significant underestimation of the (near)-degeneracy static correlation limit (around 119mHartree in theH2 dissociation limit).

Figure 5.H2 (A) and+H2 (B) dissociation curves without breaking spin symmetry. All calculations, including the configuration

interactionmethodwith singles and doubles (CISD), have been carried out with FHI-aims [78] and theNAO-VCC-5Z basis set [79].For one and two electron systems, CISDprovides the exact curves, which are thus denoted asCI. TheHF, PBE and PBE0 results areobtained self-consistently. TheHForbitals are employed to evaluate theCISD results, and the PT2, RPA, rPT2, andBGE2 calculationsare on top of a PBE0 reference. In the smaller panel forH2 dissociation, the total energies of two isolatedHydrogen atoms are plottedfor eachmethod. And for +H2 dissociation, the smaller panel shows the total energies of one isolatedHydrogen atom.

10

New J. Phys. 18 (2016) 073026 I YZhang et al

Fifth-level correlation functionals, e.g., PT2, RPA and rPT2, are fully non-local. Figure 5 reveals that PT2 isexact for one-electron systems such as +H2 dissociation, in agreement with previous investigations [1–5], butcompletely fails for heavily stretchedH2, where static correlation becomes dominant. RPA exhibits the oppositebehavior. It is accurate in theH2 dissociation limit, but its severe one-electron ‘self-interaction’ or delocalizationerror [84, 85] affects +H2 dissociation adversely. Adding the SOSEX term toRPA removes the one-electron self-interaction error again such that rPT2 dissociates +H2 correctly, but simultaneously the performance forH2

deteriorates [15]. Henderson et alhave ascribed this behavior to a removal of static correlation by the SOSEXterm [4].

Figure 5 reveals that the BGE2 functional is free of one-electron ‘self-correlation’ and thus dissociates +H2

correctly. It also delivers the correctH2 dissociation limit and reduces the incorrect repulsive ‘bump’ that RPAexhibits at intermediate bond lengths.We attribute this consistent improvement to three factors. (1)The one-electron ‘self-correlation’ error is removed at the PT2 level. (2) (Near)-degeneracy static correlation (for twoelectrons) is incorporated by the eab-couplingmechanism at the second-order perturbation level without havingto invoke higher-order Goldstone diagrams. (3)Due to the systematic nature of the approximations wemade,we can trace the repulsive ‘bump’ back to the 2nd order approximation, whenwewent from the full BGE toBGE2. In other words higher-order pp-ladder diagramswill alleviate the ‘bump’ [19].

Infigure 6we illustrate the starting-point dependence of RPA, rPT2 andBGE2 by evaluating all threeapproaches for a PBE, a PBE0 and aHF reference. The starting-point dependence in RPA is quite pronounced,which is akin to themuch investigated starting-point dependence in the correspondingGWapproach forexcitation spectra [86–89]. Judging by figure 6, the starting-point dependence of rPT2 andBGE2 appears to be aspronounced as for RPA. Although it remains to be seen in the future, if this statement can be generalized.

4. Promising properties of the BGE2 correlation functional

Inwave-function theory, there is a systematic way to improve the theoretical approach for the electroniccorrelation energy [23, 66, 90, 91]. However, it is challenging to designmethods that fulfil a certain number ofexact conditions and constraints [91]. One requirement is that the solution for one- and two-electron systemssuch asH2/

+H2 is exact. Another requirement is size consistency [23, 26, 27], i.e., the ground-state total energyitself is an extensive quality, which should be asymptotically proportional to the system size [23]. These twocriteria are widely used to judge the universal applicability of a given theoreticalmethod from small isolatedmolecules to extensive solids.Wewill analyse analytically, howwell BGE2 fares forH2 in aminimal basis.

4.1.H2 inminimal basisThe BGE2 approximation is not exact for two-electron systems.However, in section 3.3, we reported asignificant improvement of BGE2 over PT2 andRPA forH2 dissociation. In this section, we analyze the BGE2correlation functional (equation (32)) and its correlation potential (equation (34)) forH2 in aminimal basis.Wedemonstrate that the BGE2 approximation captures essential features of the adiabatic connection path thatcurrent state-of-the-art approximations do not.

Figure 6.TheH2 dissociations calculated byRPA (A), rPT2 (B) andBGE2 (C). The nomenclature adopted here: F@SC is the advancedfunctionals (F), i.e. RPA, rPT2, and BGE2, respectively, evaluatedwith the orbitals of different self-consistent (SC) schemes, i.e. HF,PBE0, and PBE.

11

New J. Phys. 18 (2016) 073026 I YZhang et al

Theminimal basis ofH2 consists of one bonding (y+) and one anti-bonding (y-)KS orbital determined bythe ¥D h symmetry:

yf f

=- -

+ r

r rR R

S2 2, 48s s

R R

1 1 1 2

1, 2

( )( ) ( )

( )

whereR1 andR2 are the two atomic positions, f -r Rs1 1 2( ) is the normalized s1 atomic orbital located at theeach hydrogen atom, and SR R,1 2

is the overlap integral of the two atomic orbitals. In thisminimal basisrepresentation, the BGE2 correlation energy (equation (32)) and its potential with respect to the couplingconstantλ (equation (34)) can be derived analytically

ll

ll

l

=- +

=-+

EB B A

VA

B A

4

22

4, 49

cBGE2

2 2

cBGE2

2 2

( )

( ) ( )

where y y y y= á ñ+ + - -A 2∣ ∣ ∣ and = D ++--B . Nowwe see that several important features of the ACpath are

captured by the BGE2 approximation:(i)When l 0, we have

ll

=¶

= - =l

¢

=

VV A

BE0 2 2 50c

BGE2 cBGE2

0

cPT2( ) ( ) ( )

which shows that the initial slope of the BGE2 correlation potential is twice the energy in PT2. Comparing to theexact condition, i.e. ¢ =V E0 2c c

GL2( ) , the single-excitation contribution is ignored during the first approximationin this work (equations (16) and (19)).

(ii) In theH2 dissociation limit ( - ¥R R1 2∣ ∣ and = D ++--B 0)wefind for the initial slope

= - ¥¢V 2 A

BcBGE2 . The BGE2 correlation potential itself tends to a constant value that is independent ofλ

thanks to the eab-coupling effect:

l f f f f -á ñ- ¥V . 51R R s s s scBGE2

1 1 1 11 2( )∣ ∣ ( )∣ ∣

The coupling-constant integration (equation (5)) is trivial to carry out and the correlation energy EcBGE2 has the

same value asVcBGE2. This is the exact correlation energy in theminimal basis [23]. In conjunctionwith the exact

exchange energy, it completely cancels out the undesired error originating from theHartree approximation, andthus guarantees the correct dissociation limit.

If we do notmake the fourth approximation, i.e. to remove the first-order perturbation term eab1st in the

denominator of equation (31), the parameterBnow equals lD +++--

++e1st ( ). By using the definitions of lv andeab

1st (equations (4) and (24))we have l++e1st ( ) in theH2 dissociation limit

l l f f f f» - á ñ++ - ¥e . 52R R s s s s1st

1 1 1 11 2( )∣ ∣ ( )∣ ∣

Here, for simplicity, we neglect the exchange rv ix ( ) and the scaled correlation potential arv ,ic ( ) (equation (4)),since they are small compared to theHartree energyEH. The resulting correlation energy

f f f f= - á ñE s s s sc2

5 1 1 1 1∣ recovers only 90%of the exact value in theminimal basis (equation (51)). Thismotivates a posterioriwhywemade the second approximation, i.e. omitted the single excitation contribution. Inpractice this single excitation contribution is non-zero in aDFT framework. However, ourminimal basisconsideration shows that if wewere to include it, H2would no longer dissociate correctly, unless wewould alsoinclude higher-order particle–particle ladder diagrams (the third approximation), whichwould significantlyincrease the computational cost. The BGE2 correlation functional proposed in this work is thus the simplestapproximation that provides the exactH2 dissociation in theminimal basis. Any further improvement overBGE2 has to simultaneously deal with all approximations in a proper and balancedway.

(iii)The second-order derivative of the BGE2 correlation energy l l = ¶ ¶V Vc c ( ) is

ll

l=

+¢¢

-V

A B

A B

24

1 4. 53c

BGE22 3

2 2 5 2( )

( )( )

AsA andB are positive, l¢¢V 0cBGE2 ( ) , indicating that the first derivative ofVc

BGE2 ismonotonically increasingwithin l0 1

l dl l+¢ ¢V V . 54cBGE2

cBGE2( ∣ ∣) ( ) ( )

Considering that l¢V 0cBGE2 ( ) for both l = 0 and l = 1, lVc

BGE2 ( ) captures the convex shape of the exactAC correlation path [6, 10, 92].

(v)Theλ-dependent ACpath is closely connected to the behavior under uniformdensity scaling [93], forwhich the low-density limit is related to the strong correlation limit l ¥. The exact correlation functional

12

New J. Phys. 18 (2016) 073026 I YZhang et al

should reach afinite value in the strong correlation limit [6, 92, 93], which is satisfied by the BGE2modelaccording to equation (49) ( -V Ac when l ¥).

Figure 7 presents the correlation potentials of RPA andBGE2 forH2 dissociation in theminimal basis. At theequilibrium geometry (R=0.8Å)where the energy gap is large, the BGE2model exhibits a quasi-linearbehavior. For stretched geometries (R=3.0 and 6.0Å), the eab-coupling then bends the correlation potential,whereas RPAoverestimates the initial slopes. The BGE2 correlation potentials are similar to those ofconfiguration interaction calculations with a quadrupole-ζ basis set [92].

We also include the common [1/1]-PadéACmodel [6, 10]

lll

=+

= -+

Va

bE

a

b

a b

b1,

ln 1. 55c

PadecPade

2( ) ( ) ( )

The twoparameters are determined byfixing the initial slope to E2 cPT2 and the correlation energy to Ec

BGE2. Asillustrated infigure 7, the [1/1]-Padé formula describes the ACpath at the equilibrium geometrywell, butexhibits a tendency to underestimate the curvature and overestimate the correlation potential at stretchedgeometries. A better agreement can be expected by using amore sophisticated [2/2]-Padé formula [6, 10].

4.2. Size consistency for the ground-state energy calculationRecently, the application of advanced correlationmethods, e.g.,MP2 [94–96], RPA [60, 62, 96, 97], and coupled-cluster theories [98–100], inmaterials science has attracted increased attention. In this paper, we adopt theK-dependence criterion [27, 101–104] to demonstrate the size consistency of the BGE2 correlation functionaland its applicability to complex extendedmaterials. The advantage and usage of theK-dependence criterion hasbeen discussed comprehensively in [27]. In short, the number (K) of k-points in the Brillouin zone is a directmeasure of system size in periodic boundary conditions. Then a size-consistentmethodmust have anasymptoticK1 dependence.

Before we turn to BGE2, wefirst discuss Brillouin–Wigner second-order perturbation theory (BW2)[102, 105]. The size consistency of BW2has been disproved in [27]. This helps us to better understand the sizeconsistency of BGE2which shares a very similar sum-over-state formula as BW2.

In periodic boundary conditions, the BW2 correlation is given by

åå å

å å å

f f f f

f f f f

=á ñ

+ - D

»á ñ

- D

< <

< <

EE E

E. 56

a b r s k k k

ak bk rk sk

ak bkrk sk

a b r s k k k

ak bk rk sk

ak bkrk sk

cBW2

2

cBW2

xEX

2

cBW2

b r s

a b r s

a b

r s

b r s

a b r s

a b

r s

∣ ∣∣ ∣

∣ ∣∣ ∣( )

Here, fikiis a canonicalHF orKS spin-orbital in the ith bandwithwave vector ki. Due tomomentum

conservation, the summation only goes over threewave vectors (k k k, ,b r s), giving rise to a factorK3. It can be

proven that f f f fá ñak bk rk sk2

a b r s∣ ∣∣ ∣ exhibits an asymptotic -K 2 dependence. Ex

EX and D ak bkrk sk

a b

r s scale asK1 and

K0, respectively [27]. If the denominator scales the same as ExEX (thefirst line in equation (56)), the overall scaling

Figure 7.Correlation potentials ofH2 in aminimal basis at the equilibrium geometry (R=0.8 Å, upper panel), an intermediate bonddistance (R=3.0 Å, middel panel), and a large bond distance (R=6.0 Å, lower panel) for RPA, the [1/1]-Padémodel, and BGE2.The parameters in the [1/1]-Padémodel are determined by fixing the initial slope to E2 c

PT2 and the correlation energy to EcBGE2.

13

New J. Phys. 18 (2016) 073026 I YZhang et al

of EcBW2 becomesK0, whichwould not be size consistent. Removing Ex

EX from the denominator (second line inequation (56)) changes the scaling of Ec

BW2 to K1 2, which is still not size consistent. These non-physical sizedependencesmake the second-order energy per unit cell, E Kc

BW2 , go to zero as ¥K . It has been arguedthat the presence of an extensive quantity, +E Ec

BW2xEX, in the denominator is responsible for the lack of size

consistency [27, 105].The BGE2 correlation energy (equations (32) and (33)) in periodic boundary conditions takes the form

åå

åå

ll f f f f

l

l l

=á ñ

- D

=

<

<

ee

E e . 57

abr s k k

ak bk rk sk

ab ak bkrk sk

a b kab

BGE22 2

BGE2

cBGE2 BGE2

r s

a b r s

a b

r s

b

( )∣ ∣∣ ∣

( )

( ) ( ) ( )

Compared to the BW2 correlation energy, the only difference in BGE2 is the appearance of the correlationcoupling for each electron pair ab, i.e. the eab-coupling. Following a similar approach as for BW2, it is easy toprove that the electron-pair correlation term eab

BGE2 scales asK0 [27]. Due tomomentum conservation thesummation in the BGE2 correlation energy Ec

BGE2 only runs over onewave vector kb (see equation (57)).Therefore, Ec

BGE2 scales asK1 and thus is size consistent.

4.3.Orbital invarianceIt should be stated that an electron-pair approximation such as BGE2 does not have a derivedwave function. Itthus breaks another important feature: the orbital invariance [23], i.e. the total energies cannot be determineduniquelywith respect to rotations among occupied and/or unoccupied orbitals. However, the simple sum-over-state formula of BGE2 (equations (32) and (57)) clearly suggests that the eab-coupling decays very quickly tostandard PT2when the energy difference becomes larger. Therefore, we expect that this orbital invariancedeficiency does not affect real applications.We leave a detailed examination of this issue to the future work.

5. Conclusion

In this work, we present a new insight into theH2/+H2 challenge inDFT.We establish the BGE in the context of

DFT through the AC approach. BGE is the simplest approximation to provide the exact solution for one- andtwo-electron systems.We propose a simple orbital-dependent correlation functional, BGE2, by terminating theBGE expansion at the second order, but reversing the eab-coupling effect in BGE. BGE2 has a similar sum-over-state formula as the standard PT2, thus sharing the same computational scaling as PT2 in terms of both time andmemory.We demonstrate that the eab-coupling iteration procedure at the second-order expansion does notinvoke higher-order connectedGoldstone diagrams, but partially captures themulticenter character of eachelectron pair, especially in heavily stretchedH2. A remarkable improvement of BGE2 over PT2 andRPA in theH2/

+H2 challenge can be observed, which suggests that the one-electron ‘self-correlation’ and the two-electron(near)-degeneracy static correlation can be included simultaneously well at the second-order perturbation levelin conjunctionwith a proper treatment of themulti-reference contributions of each electron pair. In addition ofthe size consistency, the advantage of the BGE2 correlation functional has been further demonstrated usingH2

inminimal basis.However, for systemswith large energy gaps ormore electrons, BGE2 reduces to standard PT2 since the

effect of the eab-coupling is nearly damped out. Further development on top of BGE2 could proceed as follows:(1) From the semi-empirical double hybrid perspective, the BGE2 correlation formula could be a promisingsubstitute of normal PT2,which opens an opportunity to extend the double-hybrid scheme into the realmoftransitionmetals while keeping the accuracy achieved formain group elements [30]. (2) From theACmodelingperspective, we can improve the BGE2model by satisfyingmore physical constraints. For example, in the strongcorrelation limit l ¥, the BGE2model depends asymptotically on l-1 rather than the correct l-1 2 [7, 92].Promisingly, the plain sum-over-state PT2-like formulamakes it easy to fulfill the correct asymptotic behaviorby introducing an additional term that depends on l-3 2 in the denominator of the BGE2 formula(equation (31)). (3) From themany-body perturbation theory perspective, it would be appealing to renormalizethe BGE2 scheme in rPT2 [16], whichwould be an alternative to construct advanced orbital-dependentfunctionals systematically.

Acknowledgments

IYZ thanks Professor XinXu for helpful discussion.Work at Aaltowas supported by the Academy of Finlandthrough its Centres of Excellence Programme (2012-2014 and 2015-2017) under project numbers 251748 and284621.

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