Self-consistent GW : an all-electron implementation with localized basis functions

Fabio Caruso,1, 2 Patrick Rinke,1, 2 Xinguo Ren,1, 2 Angel Rubio,3, 1, 2 and Matthias Scheffler1, 2

1Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany2European Theoretical Spectroscopy Facility

3Nano-Bio Spectroscopy group and ETSF Scientific Development Centre,Dpto. Fısica de Materiales, Universidad del Paıs Vasco,

CFM CSIC-UPV/EHU-MPC and DIPC, Av. Tolosa 72, E-20018 San Sebastian, Spain(Dated: March 18, 2013)

This paper describes an all-electron implementation of the self-consistent GW (sc-GW ) approach– i.e. based on the solution of the Dyson equation – in an all-electron numeric atom-centered or-bital (NAO) basis set. We cast Hedin’s equations into a matrix form that is suitable for numericalcalculations by means of i) the resolution of identity technique to handle 4-center integrals; and ii) abasis representation for the imaginary-frequency dependence of dynamical operators. In contrast toperturbative G0W0, sc-GW provides a consistent framework for ground- and excited-state propertiesand facilitates an unbiased assessment of the GW approximation. For excited-states, we benchmarksc-GW for five molecules relevant for organic photovoltaic applications: thiophene, benzothiazole,1,2,5-thiadiazole, naphthalene, and tetrathiafulvalene. At self-consistency, the quasi-particle ener-gies are found to be in good agreement with experiment and, on average, more accurate than G0W0

based on Hartree-Fock (HF) or density-functional theory with the Perdew-Burke-Ernzerhof (PBE)exchange-correlation functional. Based on the Galitskii-Migdal total energy, structural propertiesare investigated for a set of diatomic molecules. For binding energies, bond lengths, and vibrationalfrequencies sc-GW and G0W0 achieve a comparable performance, which is, however, not as goodas that of exact-exchange plus correlation in the random-phase approximation (EX+cRPA) andits advancement to renormalized second-order perturbation theory (rPT2). Finally, the improveddescription of dipole moments for a small set of diatomic molecules demonstrates the quality of thesc-GW ground state density.

Many-body perturbation theory (MBPT)1 in the GWapproach for the electron self-energy2–4 provides a natu-ral framework for an ab initio, parameter-free descriptionof photo-ionization processes and charged excitations.5

In recent years, the GW approach has become a popularmethod for the computation of band gaps and chargedexcitation energies for extended6,7 and finite systems8,9.In numerical implementations, following Hybertsen andLouie,10 it is standard practice to treat the GW self-energy as a single-shot perturbation (G0W0) acting on aKohn-Sham (KS) or Hartree-Fock (HF) reference system.Thus, excitation energies are evaluated from first-orderFeynman-Dyson perturbation theory as corrections to aset of single-particle eigenvalues.The popularity of the G0W0 approximation stems from

the substantial reduction in the complexity of Hedin’sequations at first-order perturbation theory: the KS orHF eigenstates from a self-consistent field calculation canbe used as basis functions and provide a convenient rep-resentation in which the non-interacting Green functionis diagonal. In this basis, only diagonal matrix elementsof the self-energy Σ are needed to evaluate quasi-particlecorrections at first-order. Thus, G0W0 grants a consider-able simplification of the linear algebra operations whichis decisive for applying the theory to large molecules andsolids.Although numerically more efficient than a non-

perturbative approach, G0W0 suffers from several un-desirable shortcomings such as the dependence on thestarting point,7,11–13 the violation of conservation lawsfor momentum, total energy and particle number,14–16

and – most importantly – the limited access to ground-state properties, which are kept unchanged from the pre-liminary density functional theory (DFT) or HF calcula-tions.It is known that the self-consistent GW approach (sc-

GW ) – in which both the Green function G and thescreened Coulomb interaction W are iterated to self-consistency – ameliorates most of the pathologies of per-turbative G0W0.

17 A particularly appealing feature ofthe sc-GW method consists in the possibility of treatingground- and excited-states at the same level of theory.This property arises by virtue of the non-perturbativenature of the sc-GW approach, whereby the Green func-tion is updated and encompasses many-body effects in-troduced by the self-energy. In contrast, in perturba-tive theories (which generally do not introduce updatesin the Green function) the electronic structure coincideswith that of the corresponding starting point. Therefore,density and total energy – and derived quantities suchas dipole moments, bond lengths, and binding energies –become accessible at self-consistency and reveal the qual-ity of the GW ground-state. Finally, at self-consistencyexcited- and ground-state properties are independent ofthe starting point, at least for closed shell systems,17 andprovide an unbiased assessment of the GW approach.A previous study on the homogeneous electron gas

(HEG) reported a deterioration of the sc-GW spectralproperties, as compared to G0W0.

18 This has been at-tributed to a poor description of the satellite peaksat self-consistency.18 For extended systems, the perfor-mance of sc-GW remains controversial due to the scarce

2

number of calculations for real solids.19–22 Part of thiscontroversy can be traced back to basis set problemsin early all-electron calculations23 and to the large in-fluence that pseudo-potentials may have on GW bandgaps.24 More recently, sc-GW calculations for atoms25

and molecules9,17 have shown improvements in the de-scription of the first ionization energies and for transportproperties26 of finite systems.

The price to pay in sc-GW is the demanding itera-tive procedure. The higher complexity of sc-GW arisesfor the following reasons. i) The Green function obtainedfrom the solution of the Dyson equation is in general non-diagonal. This considerably increases the computationalcost of the evaluation of the dielectric matrix. ii) Thenon-diagonal matrix elements of Σ are needed to solvethe Dyson equation. iii) Fourier transforms of dynamicalquantities are needed that introduce their own computa-tional difficulties.

In the first part of this paper, we present an all-electronimplementation of the sc-GW method in the localizedbasis-set code FHI-aims27 and propose a recipe to ef-ficiently address points i)-iii). An optimized set of lo-calized basis functions was used to represent the Greenfunction and the self-energy operator. Non-local two-particle operators, such as the screened Coulomb inter-action, were computed by means of the resolution ofthe identity technique28–30 (also known as density fittingmethod) in a general framework previously introducedby some of us.31 Finally, an auxiliary basis of Lorentzianfunctions was introduced for an efficient analytical evalu-ation of Fourier transforms between imaginary time andfrequency.

The second part of the paper, focuses on the assess-ment of ground- and excited-state properties as obtainedfrom sc-GW for molecules. The quality of the sc-GWground state was investigated by computing binding en-ergies, bond lengths, vibrational frequencies, densities,and dipole moments for a small set of hetero- and homo-atomic dimers. The full valence excitation spectra wereevaluated for a set of molecules relevant for organic pho-tovoltaic applications (thiophene, benzoithiazole, 1,2,5-thiadiazole, naphthalene, and tetrathiafulvalene). Fromthis study we conclude that sc-GW systematically im-proves the spectral properties of finite systems over theentire excitation spectrum (that is, not only for the firstionization energy) as compared to standard perturba-tive G0W0 calculations based on semi-local DFT andHF. Nonetheless, for certain starting points – exempli-fied by the PBE0 hybrid functional – G0W0 slightly out-performs sc-GW , providing ionization energies in betteragreement with experimental reference data, as also pre-viously demonstrated for benzene and the azabenzenesin Ref. 12. For structural properties the sc-GW methodyields a less satisfactory agreement with experiment. Fordimers, bond lengths and binding energies are slightlyunderestimated, and in this case there is no substantialimprovement over perturbative approaches such as G0W0

or the random-phase approximation (RPA). Finally, self-

consistency gives an accurate description of the electrondensity as manifested by the accurate dipole moments ofdiatomic molecules. These results suggest that sc-GW isa promising method for charge transfer compounds andinterfaces. However, our study also indicates the impor-tance of including higher order exchange and correlationdiagrams beyond GW to accurately describe the struc-tural properties of molecules.The paper is organized as follows: Section I gives a

brief introduction to the GW approximation recalling thebasic equations needed for the computation of the Greenfunction G and the self-energy Σ. An optimal representa-tion of Hedin’s equations in terms of localized basis func-tions and of the resolution of the indentity is presentedin Sec. II. In Sec. III we present the scheme employedin the computation of the Fourier integrals of the Greenfunction and other dynamical quantities. We report inSec. IV an assessment of sc-GW for the excitation spectraof molecules and, in Sec. V, for the ground-state prop-erties of diatomic molecules. Our conclusions and finalremarks are reported in Sec. VI.

I. THEORETICAL FRAMEWORK

In MBPT the complexity of the many-body problemis recast into the calculation of the single-particle Greenfunction. Knowledge of the Green function grants im-mediate access to the (charged) single-particle excitationenergies of the system, to the total energy and, moregenerally, expectation values of any single-particle op-erator. Green function theory is well documented in theliterature1 and we recall here only the basic equations rel-evant for the GW approach, adhering to Hartree atomicunits h = me = e2 = 1.For a system of non-interacting electrons described

through a time-independent Hamiltonian, the Greenfunction can be written explicitly in terms of the single-particle eigenstates ψσ

n(r) and eigenvalues ǫσn:

Gσ0 (r, r

′, ω) =∑

n

ψσn(r)ψ

σ∗n (r′)

ω − (ǫσn − µ)− iη sgn(µ− ǫσn), (1)

where n and σ refer to orbital and spin quantum num-bers, respectively. µ is the electron chemical potential,and η a positive infinitesimal. In practice, ψσ

n(r) and ǫσn

are generally obtained from the self-consistent solutionof the Hartree-Fock or (generalized) KS equations.For interacting electrons, the Green function has to be

evaluated by solving the Dyson equation:

Gσ(r, r′, ω) = Gσ0 (r, r

′, ω) +

∫

dr1dr2Gσ0 (r, r1, ω)×

× [Σσ(r1, r2, ω) + ∆vσH(r1)δ(r1 − r2)−

−vσxc(r1, r2)]Gσ(r2, r

′, ω) . (2)

Here ∆vσH is the change in the Hartree potential ac-counting for density differences between Gσ

0 and Gσ, and

3

vσxc is the exchange-correlation part of the single-particleHamiltonian corresponding to the non-interacting Greenfunction G0. For example, if G0 is the Hartree-FockGreen function, then vσxc corresponds to the non-localexchange operator Σx. Alternatively, for a KS Greenfunction, vσxc is the local exchange-correlation potential.The electron self-energy Σσ encompasses all many-bodyexchange-correlation effects and therefore its practicalevaluation requires approximations. Following Hedin2,5,the self-energy can be expanded in a perturbative seriesof the screened Coulomb interaction W , with the first-order term given by the GW approximation:

Σσ(r, r′, τ) = iGσ(r, r′, τ)W (r, r′, τ) . (3)

By virtue of the time translation invariance it suffices toexpress Σ in terms of time differences (τ = t − t′). Thescreened interactionW (r, r′, ω) is in turn defined throughanother Dyson equation:

W (r, r′, ω) = v(r, r′)+∫

dr1dr2v(r, r1)χ(r1, r2, ω)W (r2, r′, ω) . (4)

Here, v is the bare Coulomb interaction 1/|r − r′| and

χ the irreducible polarizability, which in GW is approx-imated by the product of two Green functions:

χ(r, r′, τ) = −i∑

σ

Gσ(r, r′, τ)Gσ(r′, r,−τ) . (5)

The self-consistent nature of Eqs. 2-5 arises from theinterdependence of the self-energy and the Green func-tion. In the G0W0 approach, the self-energy is evaluatednon-self-consistently, and the Dyson equation is solvedapproximately in a parturbative fashion. The quasi-particle excitation energies are then obtained from thequasi-particle equation:

ǫQPn,σ = ǫσn +Re〈ψσ

n|Σσ(ǫQP

n,σ)− vσxc|ψσn〉 . (6)

In this work, Eqs. 2-5 are solved fully self-consistently.In practice, an iterative procedure requires the followingsteps:

1. Construction of an initial non-interacting Greenfunction G0 from a preliminary SCF calculationthrough Eq. 1.

2. Evaluation of the polarizability χ from Eq. 5 andFourier transformation of χ to the frequency do-main.

3. Calculation of the screened Coulomb interactionWfrom Eq. 4 and Fourier transformation of W to thetime domain.

4. Evaluation of the self-energy Σ from the Eq. 3 andFourier transformation of Σ to the frequency do-main.

5. Update of the Green function from the Dyson equa-tion (Eq. 2) and Fourier transformation of G to thetime domain.

6. Mixing of the Green function to accelerate the con-vergence of the self-consistent loop.

7. Iteration of steps 2.-5. until a convergence criterionis satisfied.

In a numerical implementation, a choice for the basisset expansion of the quantities in Eqs. 2-5 has to be made.Our choice will be discussed in the next Section.

II. SELF-CONSISTENT GW WITHIN A

LOCALIZED BASIS

Previous implementations of sc-GW were based onGaussians or Slater orbitals,18,25 full potential linear aug-mented plane-waves,20,21 real-space grids,32 and numericatom-centered orbitals (NAO).9 In the present work,the Green function G, the self-energy Σ, and all single-particle operators, are expanded in a numeric atom-centered orbital basis ϕi(r), with basis functions of theform:

ϕi(r) =ui(r)

rYlm(Ω) , (7)

where ui(r) are numerically tabulated radial functionsand Ylm(Ω) spherical harmonics. For numerical conve-nience, we work with real-valued basis functions by re-quiring – without loss of generality – that Ylm(Ω) denotesthe real part (for m = 0, . . . , l) and the imaginary part(for m = −l, . . . ,−1) of complex spherical harmonics. InFHI-aims the choice of the radial functions ui(r) is notlimited. In this work we will show results for numericallytabulated Gaussian orbital basis sets and the Tier hier-archy of FHI-aims for NAOs.27 We refer to Ref. 27 fordetails on the construction and optimization as well asthe properties of the NAO basis sets in FHI-aims.In terms of the ϕi basis functions, the Green function

can be expanded as:

Gσ(r, r′, iω) =

Nbasis∑

ijlm

ϕi(r)s−1ij G

σjl(iω)s

−1lmϕm(r′) , (8)

where sij =∫

drϕi(r)ϕj(r) is the overlap matrix tak-ing into account the non-orthonormality of the basis setand Nbasis is the total number of basis functions. In thefollowing, sums over latin indixes i, j, l,m are implicitlyassumed to run from 1 to Nbasis, whereas sums over n runover the total number of states. The coefficients Gσ

ij(iω)of the expansion are given by:

Gσij(iω) =

∫

drdr′ϕi(r)Gσ(r, r′, iω)ϕj(r

′) . (9)

4

The representation in Eq. 9 can be easily applied to thenon-interacting Green function in Eq. 1, yielding

Gσ0,ij(iω) =

∑

n

∑

lm

silcσlnc

σmnsmj

iω − (ǫσn − µ), (10)

where we introduced the expansion of the HF/KS orbitalsin the NAO basis ψσ

n(r) =∑

l cσlnϕl(r) and the Green

function was continued to the imaginary frequency axis.The matrix representation in Eq. 9 is also adopted forthe Hartree potential vσH, the self-energy Σσ, and theexchange-correlation potential vσxc.To rewrite Hedin’s equations in a matrix form suitable

for a numerical implementation, we need to introducea matrix representation for two-particle operators. Theexpansion of two-particle operators in a numerical basis,cannot be handled efficiently through Eq. 9 due to theappearance of the 4-orbital 2-electron Coulomb integralsof the form:

(ij|kl) =

∫

ϕi(r)ϕj(r)ϕk(r′)ϕl(r

′)

|r− r′|

drdr′ . (11)

The computation of the Coulomb repulsion integralsin Eq. 11 is a problem extensively discussed in theliterature33–39 and efficient techniques have been devel-oped over the years to make this calculation affordable.Numerically, the difficulty arises from the large numberof NAO pairs and from the memory requirements of stor-ing the 4-index matrix (ij|kl). In the NAO framework,the integrals in Eq. 11 are often evaluated by introducingan auxiliary basis set Pµ(r), with basis functions Pµ(r)defined to span the Hilbert space of NAO pairs

ϕi(r)ϕj(r) ≃∑

µ=1

CµijPµ(r) , (12)

where Cµij are the coefficients of the expansion. Due to

the high linear dependence of the NAO products, thenumber of product basis functions Naux is much smallerthan the number of NAO pairs O(N2

basis), making thenumerical evaluation of Eq. 11 affordable. This techique,known as the resolution of the identity (RI) – or alsodensity-fitting technique – was implemented in the FHI-aims code and we refer to Ref. 31 for a detailed account ofthe variational approach employed in the determinationof the RI coefficients Cµ

ij and for a review of the overallaccuracy of the RI approach for correlated calculations.In short, we used the “RI-V” variant of the RI scheme,

in which the expansion coefficients are given by:

Cµij =

∑

ν

(ij|ν)V −1νµ , (13)

where (ij|ν) ≡∫

drϕi(r)ϕj(r)Pµ(r′)/|r− r

′| and Vνµ de-notes matrix elements of the Coulomb matrix in the aux-iliary basis, i.e., Vµν =

∫

drdr′Pµ(r)Pν(r′)/|r − r

′|. Fornumerical efficiency, it is convenient to work with thegeneralized coefficients:

Mµij =

∑

ν

CνijV

1/2νµ . (14)

Following Ref. 31, one can write the RI version ofthe Dyson equation for the screened Coulomb interac-tion (Eq. 4) as:

Wµν(iω) ≡[

v−1W (iω)]

µν= [1−Π(iω)]

−1µν , (15)

where we defined [Π(iω)]µν ≡ [χ(iω)v]µν . In contrast

to the RI-based implementation of the G0W0 method31,the operator Π has to be expressed as an explicit func-tional of G. Moreover, all non-diagonal matrix elementsin the Green function have to be included. These twocriteria are satisfied by evaluating Π in terms of the Mµ

ijcoefficients, as:

[Π(iτ)]µν = −i∑

σ

∑

ijlm

MµilM

νjmG

σ

ij(iτ)Gσ

lm(−iτ). (16)

Here we defined

Gij(iτ) ≡∑

lm

s−1il Glm(iτ)s−1

mj . (17)

The quadruple sum in Eq. 16 may be reduced to dou-ble sums – with a considerable reduction of computa-tional cost – by introducing the intermediate quantityAµ

lj,σ(iτ) ≡∑

iMµilG

σ

ij(iτ). In terms of these coefficientsEq. 16 becomes:

[Π(iτ)]µν = −i∑

σ

∑

lj

Aµlj,σ(iτ)A

νjl,σ(−iτ) . (18)

The self-energy can be evaluated in terms of Eq. 15providing the following matrix representation of Eq. 3:

Σσij(iτ) =

i

2π

∑

lm

∑

µν

MµilM

νjmG

σ

lm(iτ)Wµν(iτ) . (19)

By introducing the auxiliary quantity Bµjm(iτ) =

∑

ν MνjmWµν(iτ), the self-energy can again be cast into

a double-sum form:

Σσij(iτ) =

i

2π

∑

m

∑

µ

Aµim,σ(iτ)B

µjm(iτ) . (20)

The correlation (exchange) contribution to the self-energy can be derived straightforwardly from Eq. 19 bysubstituting W with W

c

µν ≡ Wµν − δµν (Wx

µν ≡ δµν).The Hartree potential vσH is also evaluated as an explicitfunctional of G as:

vσH,ij =∑

lm

∑

µ

MµijM

µlmG

σ

lm(iτ = 0−) . (21)

Finally, the matrix representation of the Dyson equationfor the Green function completes the set of Hedin’s equa-tions:

Gσ

ij(iω) =[

Gσ

0 (iω)−1 − Σσ(iω) + vσxc −∆vσH

]−1

ij. (22)

5

FIG. 1. (Color online) Panel (a): values of ∆ – defined inEq. 24 – as a function of the number of iterations of the sc-GW loop for H2, H2O and C6H6 in their equilibrium geometryin a Tier 2 basis set. A linear mixing parameter α = 0.2 wasused for all molecules. ∆th indicates the default value of theconvergence threshold. Panel (b): values of ∆ for H2O as afunction of the number of iterations for different values of α.

Here ∆vσH is the difference between the Hartree potentialof the interacting and the non-interacting Green function.To facilitate the convergence of the sc-GW loop, the

input Green function Ginput

of the (N + 1)-th iterationis obtained from a linear mixing scheme:

Ginput

ij (iτ) = αGN

ij (iτ) + (1− α)GN−1

ij (iτ) , (23)

where GN

denotes the Green function obtained from theN -th solution of the Dyson equation, and α is the mixingparameter. As illustrated in panel (b) of Fig. 1, we findthat α = 0.2 is typically a good choice. The convergenceof the self-consistent loop is monitored looking at theaverage deviation of the Green function at each iteration∆, defined as:

∆ =1

N2basis

∑

i,j

|GN

ij (iτ = 0−)−GN−1

ij (iτ = 0−)| .

(24)

The sc-GW loop is considered converged when ∆ dropsbelow a chosen threshold ∆th. Default settings used inmost calculations are ∆th = 10−5, which suffices to con-verge both total and quasi-particle energies. The con-vergence of sc-GW is illustrated in Fig. 1, where ∆ isreported as a function of the number of iterations for H2,H2O and C6H6.Equations 15-22 constitute a matrix representation of

Hedin’s equations in the GW approach (Eqs. 2-5). Weemphasize again that in Eqs. 15-22: i) all electrons aretreated on the same quantum mechanical level, i.e. fullyself-consistently; ii) no model screening was used in thecalculation of W ; iii) all non-diagonal matrix elements ofG and Σ are correctly accounted for.The evaluation of Eqs. 16 and 19 is the most compu-

tationally demanding operation of our implementation.The scaling of the computational time as a function of the

FIG. 2. (Color online) Total time (in seconds, on a singleCPU) per iteration of the sc-GW loop, for linear hydrogenchains of different lengths. The total time required for theevaluation of the self-energy in G0W0 is included for compar-ison.

basis set size equals that of G0W0 calculations but witha larger prefactor. To illustrate this aspect, we reportin Fig. 2 the total computational time spent for a singleiteration of Eqs. 15-22 as function of the length of a lin-ear hydrogen chain in a minimal basis set (i.e., with oneNAO per atom). As compared to conventional G0W0 im-plementations, the additional computational cost arisesfrom the necessity of accounting for non-diagonal matrixelements in the calculation of G and Σ.

The only approximation introduced up to this pointis the resolution of the identity for the expansion of theproduct of NAO pairs (Eq. 12). As discussed in Ref. 31,the accuracy of the RI can be monitored systematicallyby means of two control parameters: εorth and εSVD.εorth sets the accuracy threshold for the Gram-Schmidtorthonormalization employed for the reduction of the lin-ear dependence of on-site (i.e. on the same atom) productbasis functions Pµ. In practice, by chosing smaller valuesof εorth one may increase the number of product basisfunctions used in the expansion in Eq. 12. Similarly, theparameter εSVD controls the singular value decomposi-tion (SVD) for the orthonormalization of product basisfunctions on different atoms. A more detail descriptionof the effects of these parameters can be found in Ref. 31.To show the effect of the RI scheme on the self-consistentGreen function, we report in Fig. 3 the sc-GW total en-ergy – evaluated from Eq. 30, introduced in Sec. V – ofthe water molecule as a function of εorth (left panel), andεSVD (right panel). For a wide range of values of thecontrol parameters εorth and εSVD, the changes of thetotal energy are of the order of 10−4 eV or less. In allfollowing calculations we therefore used εorth = 10−5 andεSVD = 10−5.

6

FIG. 3. sc-GW total energy of H2O as a function of the con-vergence parameters εorth (left panel) and εSVD (right panel),evaluated with a Tier 2 basis set. The number of productbasis functions corresponding to each value of εorth is alsoreported.

FIG. 4. (Color online) Comparison of the analytic structure ofthe real (in blue, above) and imaginary part (in orange, below)of fn(iω) for different values of bn, and a matrix element ofthe Green function (black).

III. DISCRETIZATION OF THE FOURIER

INTEGRALS

In this implementation, we solve Eqs. 15-22 in imagi-nary time and frequency, taking advantage of the reducednumber of frequency points required to describe Gσ(iω)and other dynamical quantities, as compared to real fre-quency implementations. In a mixed time-frequency for-malism convolutions on the frequency axis can be ex-pressed, by virtue of the convolution theorem, as prod-ucts on the time axis after a Fourier transform. Due tothe slow decay of Gσ(iω) at large frequencies, the com-putation of Fourier transforms may require extended anddense frequency grids. We obviate this problem by intro-ducing a basis for the frequency/time dependence of alldynamical quantities. This permits an analytic evalua-tion of the Fourier integrals – as one can choose basisfunctions with a Fourier transform known analytically –and substantially reduces the number of frequency pointsneeded to converge the calculation.

Following the approach introduced in Ref. 40, we ex-pand the Green function in a set of Lorentzian functions

20 40 60 80

Frequency points

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

MA

E

60 poles

10 poles

20 poles

30 poles

40 poles

50 poles

FIG. 5. (Color online) Mean absolute error (MAE) introducedby Fourier transforming the Green function of N2 for differentnumbers of poles and frequency points.

of the form fn(iω) = 1/(bn+ iω), with Fourier transformf(iτ) = 1/(2π)e−bnτ . The parameters bn are fixed at thebeginning of the calculation and are distributed logarith-mically in the energy range covered by the Kohn-Sham orHartree-Fock eigenvalues ǫσn. Although in principle otherfunctions could be used, the functions fn(iω) constitute anatural choice for the expansion of the Green function, asthe frequency dependence of fn(iω) closely resembles theanalytic structure of G and captures the 1/iω behaviourat large frequency. This is illustrated in Fig. 4, where thereal and imaginary parts of fn(iω) – with different valuesof bn – are compared to a Green function matrix elementfor the Ne atom. The Green function can be expandedin the basis of Lorentzian functions as:

Gσij(iω) =

Npoles∑

n=1

[

αnijf

Ren (iω) + βn

ijfImn (iω)

]

, (25)

where Npoles denotes the number of functions fn, and

fRe (Im)n (iω) the real (imaginary) part of fn(iω). Thereal and imaginary part of the Green function have beentreated separately to maintain a real-valued linear-leastsquare problem, leading in turn to real-valued coefficientsαn and βn. Since the Fourier transform of the fn(iω) isknown, the coefficients αn and βn also determine the ex-pansion of the Green function in imaginary time. Expan-sions similar to Eq. 25 were employed also for the Fouriertransform of χ, W and Σ.The imaginary time and frequency axes are then dis-

cretized on exponentially spaced grids composed of Nω

points in the range 0, ωmax, and by 2Nτ + 1 points inthe range −τmax, τmax. The grid points ωn and inte-gration weights w(ωn) are defined as:

ωk = ω0

[

e(k−1)h − 1]

, w(ωk) = hω0e(k−1)h (26)

and similarly for τk and w(τk). The constant h is ob-tained by imposing the maximum frequency ωmax fromthe constraint ωmax = ω0

[

eNωh − 1]

and the parame-

7

ter ω0 sets the initial spacing of the grid. Typical val-ues adopted in our calculations are ωmax = 5000 Ha,τmax = 1000 Ha−1 and ω0 = τ0 = 0.001.

The error introduced by the Fourier transform canbe quantified for functions known analytically on boththe (imaginary) frequency and time axes such as, for in-stance, the non-interacting Green function given in Eq. 1.In Fig. 5, we report the mean absolute error (MAE) inthe Fourier transform of the non-interacting Green func-tion Gσ

0 (iω) of the nitrogen dimer N2, averaged over allmatrix elements. The MAE drops exponentially whenincreasing the number of functions fn, and few tens offrequency points suffice to converge the Fourier integralswith an accuracy of the order of 10−8. In our calculationswe used Nω = Nτ = 60 as default parameters.

IV. SPECTRAL PROPERTIES OF MOLECULES

We turn now to the spectral properties in sc-GW . Atself-consistency, the excitation spectrum is given by thespectral function:

A(ω) = −1/π

∫

dr limr′→r

Im G(r, r′, ω)

= −1/π Tr [Im G(ω)] , (27)

where the Green function has to be evaluated on thereal frequency axis. To evaluate Eq. 27, we first obtainthe real frequency self-energy by means of the analyticcontinuation based on a two-pole fitting scheme41. In thisapproach, the matrix elements of the self-energy in theimaginary frequency domain (i.e., the Fourier transformof Eq. 20) are fit by polynomials of the form:

Σ(iω) ≃

2∑

n=1

aniω + bn

. (28)

Here the matrix element indices were suppressed for no-tational simplicity and the coefficients an and bn are de-termined by means of a non-linear least-square fit, solvedwith a Levenberg-Marquardt algorithm. By replacing iωby ω in Eq. 28 the self-energy can then be evaluatedon the real frequency axis. Once the real-frequency self-energy is obtained, the Dyson equation is solved directlyin real frequency on a fine, equally spaced grid. The re-sulting Green function is used to determine the sc-GWspectral function A(ω).Previous works41 have indicated that the two-pole

model presented in Eq. 28 reliably reproduce quasi-particle energies with an average relative error of the0.2% for solids. The parameter η in the denominatorof Eq. 1, necessary to avoid the numerical divergence ofG0 is set to η = 10−4. This parameter contributes negli-gibly to the broadening of the spectral function and hasno effect on the position of the quasi-particle peaks.As an example, we report the sc-GW spectral func-

tions of H2O, NH3 and N2 in Fig. 6 calculated using ba-sis sets of increasing size. The sc-GW spectral function

shows sharp δ-function-like peaks at the quasi-particleenergies. The absence of broadening in the quasi-particlepeaks in Fig. 6 may be associated with a infinite lifetimeof the corresponding quasi-particle states, as expectedfor states close to the Fermi energy. As discussed inSec. IVA, higher energy excitations may decay throughthe formation of electron-hole pairs, leading to a finitelifetime and thus to a more pronounced broadening of thequasi-particle peaks. In panels a), b) and c) in Fig. 6 wereport the spectral function corresponding to the high-est occupied quasi-particle states evaluated with a Tier1, Tier 2 and Tier 3 basis; panels d), e) and f) showthe peaks corresponding to the lowest unoccupied quasi-particle states. The G0W0@HF and sc-GW ionizationenergies are reported in panels g), h) and i) of Fig. 6 asa function of the basis set size. The G0W0 ionization en-ergy is calculated from the linearized quasi-particle equa-tion (Eq. 6), whereas in sc-GW it is extracted from thehighest (valence) peak of the spectra shown in panels a),b) and c).

For the quasi-particle energies corresponding to occu-pied states, the largest change is observed going fromTier 1 to Tier 2. For N2 for example, we observe a changein the HOMO of approximately 0.2 eV going from Tier1 (which consists of 14 NAO basis functions per atom)to Tier 2 (39 NAO per atom). A further increase ofthe basis set from Tier 2 to Tier 3 (55 NAO per atom)leads to changes of the order of 5 meV in the HOMO –as illustrated in the right panels of Fig. 6. Lower lyingquasi-particle peaks show a similar convergence behav-ior as the HOMO. H2O, and NH3 exhibit a qualitativelysimilar behavior. Surprisingly, for all systems consideredhere, sc-GW data converge faster with the basis set sizethan perturbative G0W0 calculations. In the following,we will focus on closed shell molecules, which in manyinstances do not have a stable anionic state. Therefore,we will focus on the spectral function of occupied statesonly.

To investigate the performance of the GW approxima-tion at self-consistency, we have performed sc-GW calcu-lations for a set of molecules relevant for organic photo-voltaic applications. We report in Fig. 7 the comparisonbetween experimental42–46 and theoretical ionization en-ergies evaluated from sc-GW and G0W0 based on theHF, PBE, and PBE0 starting points, for thiophene, ben-zothiazole, 1, 2, 5-thiadiazole, naphthalene, and tetrathi-afulvalene. For an unbiased assessment, it would be de-sirable to benchmark sc-GW against higher level theo-ries, since in experiment the distinction between verticaland adiabatic ionization energies is difficult and vibra-tional effects are always present. For naphthalene thecoupled cluster singles doubles with perturbative triples(CCSD(T)) method, that is currently considered as thegold standard in quantum chemistry, gives a vertical ion-ization potential of 8.241 eV47, which sc-GW underes-timates (-7.48 eV). For this molecule, the difference be-tween the vertical and the adiabatic ionization potentialis only 0.1 eV in CCSD(T). For thiophene, CCSD(T)

8

FIG. 6. (Color online) Spectral function calculated from Eq. 27 using a sc-GW Green function for H2O, NH3 and N2 with Tier1 (black), Tier 2 (orange) and Tier 3 (blue, dashed) NAO basis sets. Panels a), b) and c) show the peaks corresponding to thefirst valence states, peaks relatives to conduction states are reported in panels d), e) and f). Finally, the panels g), h), and i)report the convergence of the HOMO level as a function of the number of numerical orbitals used in the basis set.

calculations of the adiabatic ionization energy obtain8.888 eV48, in good agreement with experiment, whereassc-GW yields 8.45 eV. Zero-point vibration effects aresmall and cancel with core-correlation and relativistic ef-fects. However, the authors of this study indicate thatthe geometry of the cation differs considerably from thatof the molecule, but did not give values for the verticalionization energy. It therefore remains an open ques-tion, by how much vertical and adiabatic ionization po-tentials differ for thiophene. For benzothiazole, 1, 2, 5-thiadiazole, and tetrathiafulvalene we were not able tofind CCSD(T) calculations for the vertical ionization po-tential.

Despite the tendency to underestimate the first ion-ization energy, for these systems sc-GW ionization ener-gies are in good agreement with experiment, and givea good overall description of the excitation spectrum:full self-consistency leads to an average error of 0.4 eV(with a maximum error of 1.2 eV) between the experi-mental and theoretical ionization energies, whereas HF-and PBE-based G0W0 differs on average by 0.7 eV (witha maximum error of 1.5 eV for G0W0@HF, and 1.6 eVfor G0W0@PBE). Interestingly, G0W0@PBE0 ionizationenergies are close to the sc-GW ones. Moreover, the

G0W0@PBE0 spectrum is in slightly better agreementwith experiments with an average deviation of 0.3 eV(with a maximum error of 1.2 eV) – as recently also re-ported for benzene and the azabenzenes in Ref. 12.

For small molecules, the improvements of the spectralproperties at self-consistency can partially be traced backto corrections of the over- or under-screening inG0W0. InPBE based G0W0 calculations, the small HOMO-LUMOgap induces an overestimation of the screening in theCoulomb interaction. This is the origin of a systematicerror in the G0W0@PBE quasi-particle energies. Simi-lar considerations are easily generalized to the HF start-ing point, where HOMO-LUMO gaps are generally toolarge due to the missing correlation energy. PBE0, on theother hand, often gives a good compromise. Therefore,the over- and under-screening is reduced in G0W0@PBE0and the resulting excitation spectrum is similar to sc-GW , where – because of the self-consistent calculation ofW – this problem is mitigated. Based on these results,PBE0 emerges as an optimal starting point for the per-turbative calculation of the spectral properties. It wasargued that the screened Coulomb interaction obtainedfrom sc-GW may also be underscreened due to the lackof electron-hole interactions – typically accounted for by

9

FIG. 7. (Color online) Comparison between theoretical andexperimental vertical ionization energies of thiophene, ben-zothiazole, 1, 2, 5−thiadiazole, naphthalene, and tetrathiaful-valene. Experimental photoemission data are from Refs. 42–46. The molecular geometries were optimized with PBE in aTier 2 basis set and are reported on the right. G0W0 ioniza-tion energies are obtained with a Tier 4 basis set, the sc-GW

ones with a Tier 2 basis.

vertex corrections49. This would in principle lead to asystematic overestimation of the quasi-particle energies,as reported in Ref. 50 for semiconductors. Figure 7, onthe other hand, indicates a slight underestimation of thesc-GW quasi-particle energies, confirming the expecta-tion that effects of the electron-hole attraction on the

screened Coulomb interaction are small in molecules withlarge HOMO-LUMO gaps.As alluded to in the introduction, previous sc-GW

studies have reported conflicting conclusions on the accu-racy of the spectral properties.9,18–20,25 Consequently, noconsensus has so far been reached in this respect. sc-GWcalculations for the homogeneous electron gas (HEG) in-dicated a deterioration of the spectra as compared toperturbative G0W0 based on the local-density approx-imation (LDA).18 For the HEG, Holm and von Barthobserved a transfer of spectral weight from the plas-mon satellite to the quasi-particle peak in self-consistentcalculations.18 This results in a weaker plasmon peak anda broader valence band, that worsens the agreement withphoto-emission experiments for metallic sodium.The first self-consistent calculation for real systems

– performed for potassium and silicon in the pseudo-potential approximation – confirmed the picture outlinedby Holm and von Barth, indicating a deterioration of theband width and band gap at self-consistency.19 In a laterwork, Ku and Eguiluz attributed the origin of this fail-ure to the pseudo-potential approximation, emphasizingthe importance of accounting for core-valence coupling inthe determination of the screening20. However, severalgroups.23,51,52 have questioned the convergence of thesecalculations with respect to the number of bands. Nev-ertheless, these earlier studies gave the impression thatfull self-consistency deteriorates the spectral propertiescompared to perturbative G0W0, and that it is not rec-ommended to perform sc-GW calculations. However, inour opinion, the scarce numerical evidence for realisticsystems is not enough to corroborate this notion.It was argued that the deterioration of spectra in sc-

GW might arise due to the iterative construction of thepolarizability χ as the product of two Green functions53.This would systematically weaken the incoherent part ofthe Green function, and reduce the intensity of the plas-mon satellites. For molecules however, this mechanismdoes not apply since quasi-particle peaks carry integerspectral weight, and no plasmon satellites are observed.For extended systems, this mechanism might effectivelydeteriorate the sc-GW spectral function, as for the homo-geneous electron gas. Nonetheless, more investigationsare needed to provide a general and systematic assess-ment of sc-GW for real solids.

A. Lifetimes of quasi-particle peaks

To facilitate the comparison between G0W0@PBE0and sc-GW , we report in Fig. 8 the full sc-GW andG0W0@PBE0 spectral function for thiophene and 1,2,5-thiadiazole. Figure 8 illustrates that even if the peakpositions in sc-GW and G0W0@PBE0 are very similar,there are qualitative differences. In G0W0@PBE0 thelower lying peaks are considerably more broadened thanthe corresponding sc-GW peaks.

We observe that quasi-particle peaks corresponding

10

FIG. 8. (Color online) Comparison between sc-GW and G0W0@PBE0 spectral functions for thiophene and 1, 2, 5−thiadiazoleevaluated with a Tier 2 basis set. Vertical dashed lines indicates experimental photoemission data from Refs. 42 and 43,respectively. G0W0@PBE0 spectra are obtained with a Tier 4 basis set, the sc-GW spectra with a Tier 2 basis.

to high-energy excitations are accompanied by a finitebroadening. The broadening in turn, being inverselyproportional to the lifetime of the corresponding quasi-particle state, yields important information on the dy-namics and damping of excitations. A careful quanti-tative evaluation of lifetimes would require to properlyaccount for effects beyond the GW approximation (suchas the coupling between electrons and vibrations and thesatisfaction of selection rules in the decay process) andgoes beyond the purpose of the present work. There-fore, the following discussion will be limited to explainwhy even the quasi-particle excitations of molecules havefinite lifetimes and we will briefly characterize their start-ing point dependence.

The physical origin of the lifetime is simple. The holecreated in a lower valence (or core) state by the photoe-mission process can in principle decay to the HOMO, orto states energetically close to the Fermi energy. The en-ergy released in this process has to be converted into aninternal excitation of the system, since isolated moleculescannot dissipate energy. If the energy released is largerthan the HOMO-LUMO gap, a particle-hole pair can becreated. This opens up a scattering or decay channel forthe hole, which therefore acquires a lifetime. The energythreshold for electron-hole formation is then given by∆ ≡ EGS

HOMO − EGSgap, with E

GSHOMO and EGS

gap the HOMOlevel and the HOMO-LUMO gap of the starting point,respectively. In other words, only quasi-particle stateswith an energy below ∆ may decay, and acquire a finitebroadening. This argument is general and does not onlyapply to GW . What is particular to G0W0 is that therelevant gap for this process is determined by the start-ing point; in this case the DFT functional for the groundstate.

To illustrate this effect on the broadening of the quasi-particle peaks, we report in Fig. 9 the spectrum of ben-zene evaluated from G0W0 based on different startingpoints and at self-consistency. Values of ∆ for the dif-ferent exchange-correlation functionals are reported asvertical dashed lines (in green). PBE has the smallestHOMO-LUMO gap (∼ 5.2 eV) and we observe a notice-able peak broadening (i.e., short lifetime) at ∼14 eV inthe G0W0@PBE0 spectrum. A systematic increase of thebroadening is then observed the further a state lies be-low ∆. Adding exact exchange to the DFT functionaland increasing its admixture opens the HOMO-LUMOgap. The onset of the finite lifetime subsequently movesto lower energies.In sc-GW , the broadening of the quasi-particle peaks

is consistent with the HOMO-LUMO gap at the GWlevel, and the ambiguity of the starting point depen-dence is lifted. Consequently, for benzene only peaks be-low −19 eV acquire a small finite broadening. Based onthe sc-GW results, the large broadening observed in theG0W0@PBE0 spectrum can be attributed to the smallHOMO-LUMO gap of the underlying PBE0 calculation.The inclusion of a fraction of exact exchange partiallyameliorates this problem, but not fully. This indicatesthat the calculation of lifetimes presents an additionalsituation in which – due to the severe dependence on thestarting point – resorting to full self-consistency is essen-tial.

B. Independence of the starting point

Previous work showed that partially self-consistentapproaches such as eigenvalue self-consistent GW 10 or

11

FIG. 9. (Color online) Spectral function of benzene evaluatedfrom sc-GW and G0W0 based on PBE, PBE0 with differ-ent mixture of exact exchange (EX), and HF. Vertical dashedlines (in green) indicate the energy threshold ∆ for the for-mation of an electron-hole pair, which depends explicitly onthe HOMO and LUMO levels of the underlying DFT/HF cal-culation. The quasiparticle peaks acquire a finite broadeningonly for states below ∆. For peaks above ∆, the residualbroadening stems from the parameter η = 10−4 discussed inthe text.

quasi-particle self-consistency,53,54 reduce the startingpoint dependence but they do not eliminate it.12,55 Onlyfull self-consistency successfully removes any dependenceon the starting point, as we discussed in Ref. 17. This is amajor advantage of the sc-GW scheme, as it allows a sys-tematic assessment of the GW approximation unbiasedby spurious dependence on the input Green function.To illustrate the independence of the starting point

of the sc-GW Green function, we report in Fig. 10 thespectral function of the carbon monoxide molecule as afunction of the number of iterations of the Dyson equa-tion initialized with HF, PBE, and PBE0. After just afew iterations of the sc-GW loop, the quasi-particle peaksin the spectral function are located at the same energiesdemonstrating the independence of the starting point insc-GW .

V. GROUND-STATE PROPERTIES FROM THE

GW APPROXIMATION

A. Galitskii-Migdal total energy

In MBPT, the total energy Etot may be regarded asan explicit functional of the Green function, i.e. Etot =Etot[G]. The functional dependence of Etot[G] on G isnot uniquely defined and different total energy function-als have been proposed over the years. Some example are

FIG. 10. (Color online) Spectral function of the CO dimerevaluated for the first five iterations of the Dyson equationfrom the HF, PBE, and hybrid PBE0 starting points. Thefirst iteration corresponds to the G0W0 approximation.

the Luttinger-Ward56 and the Klein functional57 – whichare stationary at the self-consistent Green function58 –and the Galitskii-Migdal formula59. The total energyobtained from different functionals may differ in princi-ple, if evaluated with a given Green function. However,if the Green function is obtained self-consistently solv-ing the Dyson equation, all functionals yield the same,unique total energy. Since we are interested in sc-GWtotal energies, all total energy functionals are equivalentand we use the Galitskii-Migdal formula because of itssimplicity:

EGM = −i

∫

∑

σ

dω

2πTr [ω + h0]G

σ(ω)+Eion . (29)

Here h0 is the single-particle term of the many-bodyHamiltonian, i.e. the sum of the external potential dueto the nuclei and the kinetic energy operator, and Eion

accounts for the repulsive nuclear energy. As discussedin Ref. 32, Eq. 29 can be computed directly in imagi-nary frequency taking advantage of the reduced size ofthe integration grid needed to describe G and Σ on theimaginary axis. In the present work, we cast Eq. 29 intoa more suitable form for numerical implementations60:

EGM = −i∑

ij,σ

Gσ

ij(τ = 0−)[2tji + 2vextji + vHji +Σxji,σ]

− i∑

ij,σ

∫

dω

2πG

σ

ij(ω)Σcji,σ(ω)e

iωη + Eion , (30)

where t denotes the kinetic-energy operator, vH andvext the Hartree and external potential, and Σx and Σc

the exchange and correlation parts of the self-energy,

12

FIG. 11. (Color online) BSSE corrected binding energy of COevaluated with Tier 1, 2, 3 and 4. Each curve is aligned attheir respective Tier 4 value. All data are in eV.

respectively. We used G ≡ s−1Gs−1, with sij =∫

drϕi(r)ϕj(r). A derivation of Eq. 30 is reported inAppendix A. We emphasize that Eq. 30 is exact if eval-uated with the exact self-energy and Green function. InEq. 30 contributions arising from time-independent oper-ators can be easily evaluated by simple matrix products.The correlation energy on the other hand requires a fre-quency integration which can be evaluated directly onthe imaginary frequency axis without resorting to theanalytic continuation32. We evaluated Eq. 30 in theGW approximation. In sc-GW , both Σ and G are self-consistent solution of the Dyson equation, whereas forG0W0 the self-energy is evaluated only once and G isthe non-interacting Green function of the DFT/HF cal-culation. The latter procedure corresponds to a first-order perturbative correction of the DFT/HF total en-ergy, with a perturbing potential given by [Σ(ω)− vxc].

B. Structural parameters of diatomic molecules

Total energy differences are more important than abso-lute total energies, as they give information on structuralproperties and on the strength of chemical bonds. Here,we restrict the discussion to the ground-state propertiesof dimers at their equilibrium geometry. The reader is re-ferred to Ref. 61 for an assessment of the sc-GW methodin the dissociation limit.In the following, we provide an assessment of the sc-

GW method for bond lengths, binding energies, and vi-brational frequencies based on the calculations of the po-tential energy curve of LiH, LiF, HF, CO, H2, and N2. Toillustrate the convergence with the basis set, we reportin Fig. 11 the binding energy of the carbon monoxidedimer evaluated with sc-GW , RPA@PBE, G0W0@PBE,and PBE-based renormalized second-order perturbationtheory62,63 (rPT2) using increasingly larger NAO basissets (Tier 1-4).The mean absolute errors of theoretical bond lengths,

binding energies, and vibrational frequencies as com-pared to experiment are reported in Fig. 12. The

FIG. 12. (Color online) Mean absolute error (MAE) of bondlengths (upper panel), binding energies (central panel), andvibrational frequencies (lower panel) of LiH, LiF, HF, CO,H2, and N2 evaluated at different levels of theory. The esti-mated zero-point motion correction has been subtracted fromthe experimental binding energies (reported from Ref. 64).Calculations were done with a Tier 3 basis set (the largestbasis set available) for H2, LiH, and HF, whereas a Tier 4basis was used for N2 and CO. Numerical values are reportedin Tables III, II, and IV in the Appendix.

corresponding numerical values are reported in Ap-pendix B. Since our calculations are performed in theBorn-Oppenheimer approximation with clamped nuclei,we compared our calculations with zero-point motioncorrected experimental binding energies from Ref. 64.PBE, HF, and several perturbative approaches basedon MBPT, namely G0W0, EX+cRPA, and rPT2 are in-cluded for comparison.Self-consistency provides better bond lengths than per-

turbative G0W0 calculations. However, the accuracyachieved by sc-GW for the bond lengths is still compara-ble to perturbative RPA and not as good as rPT2@PBE,which includes higher order exchange and correlation di-agrams.The binding energies obtained fromG0W0 based on HF

and PBE, reported in Table II, are systematically over-estimated. Self-consistent GW over-corrects this trendand yields binding energies that slightly underestimateexperiment. RPA@HF and sc-GW give a very similardescription of the binding energy, the deviation betweenthe two methods being approximately 10−20 meV. This

13

similarity is expected for two reasons: First, in diatomicmolecules screening is small, thus the sc-GW Green func-tion resembles the Hartree-Fock one (since in absenceof polarization W reduces to the bare Coulomb inter-action). Second, the RPA total energy is a variationalfunctional of the Green function, and therefore RPA to-tal energies are close to sc-GW ones, if the input Greenfunction is close enough to the sc-GW Green function16.Larger discrepancies between perturbative RPA and sc-GW are to be expected for the structural properties ofsystems for which the sc-GW density is substantiallydifferent as compared to HF or semi-local DFT. Exam-ple of these material are molecular interfaces and chargetransfer compounds, where the ground-state density (andthe charge transfer) depends strongly on the level align-ment between the individual components of the system.This will be addressed in future works. Also, for bind-ing energies and bond lengths, sc-GW is outperformedby rPT2@PBE, which illustrates the importance of in-cluding exchange and correlation diagrams beyond theGW approximation for a systematic improvement of theground-state properties of finite systems.For vibrational frequencies, the dependence on the

starting point is larger than for binding energies or bondlengths. In this case, the best agreement with experi-ment is achieved with the PBE functional, whereas forHF the errors are substantially larger. Similarly, PBE-basedG0W0 and EX+cRPA deviate less from experimentthan HF-based schemes. For instance, the mean absoluteerror of EX+cRPA@HF is approximately a factor of twolarger that EX+cRPA@PBE, and the same is observedfor G0W0. Consequently, sc-GW gives smaller MAEscompared to HF-based schemes, but does not improveover PBE-based perturbative schemes.

C. Density and dipole moments at self-consistency

In perturbative approaches, such asG0W0, EX+cRPA,and rPT2, the electron density of a system is definedby the eigenstates of the unperturbed reference Hamil-tonian – although in principle perturbative correctionsto the eigenstates of the unpertubed Hamiltonian couldbe calculated. This constitutes a major drawback, as itis in part responsible for the well-known starting pointdependence of perturbative schemes. Self-consistency, onthe other hand, permits us to incorporate exact exchangeand dynamical correlation effects into the electron den-sity. To illustrate this aspect in sc-GW , we discuss in thefollowing the effects of self-consistency on the density andthe dipole moment of diatomic molecules.Figure 13 shows the density difference between sc-GW

and HF for the hydrogen fluoride dimer. The densitydifference illustrates the effects of GW correlation on theelectron density. sc-GW enhances the angular distribu-tion of the electron density exhibiting more pronouncedangular correlation. Moreover, density is shifted fromthe two lobes perpendicular to the molecular axis to the

FIG. 13. (Color online) Difference between the sc-GW andHartree-Fock densities for the hydrogen fluoride dimer at itsexperimental equilibrium geometry (d = 0.917 A). At self-consistency, electron density is shifted from negative (dark)to positive regions (light). Units are A−3 and the calculationwere performed using a Tier 3 basis set.

TABLE I. Comparison between experimental65 and theoreti-cal dipole moments evaluated from sc-GW , PBE, and HF attheir corresponding equilibrium bond lengths. All values arein Debye.

LiH HF LiF CO MAE

Exp. 5.88 1.82 6.28 0.11 -

sc-GW 5.90 1.85 6.48 0.07 0.07

PBE 5.63 1.77 6.12 0.20 0.14

PBE0 5.77 1.81 6.20 0.09 0.06

HF 6.04 1.89 6.46 -0.13 0.17

bond region, leading to a reduction of the dipole mo-ment as compared to HF, which is in better agreementwith experiment (see Table I).The dipole moment provides a systematic way to quan-

tify the quality of the electron density of a system, as itis directly comparable with experimental data. We re-port in Table I the dipole moment of LiH, LiF, HF andCO evaluated from sc-GW , PBE, and HF. sc-GW dipolemoments are in good agreement with experiments andreduce the deviation from experiment by approximatelya factor of two compared to HF and PBE, that tend tounder and overestimate, respectively. The quality of thedipole moment for the small set of molecules presentedhere, indicates that sc-GW is a promising method for thedescription of charge-trasfer compounds, such as molec-ular interfaces and hetero-structures.

VI. CONCLUSIONS

We have presented an all-electron implementation ofthe fully self-consistent GW method based on a nu-meric atom-centered orbital basis in the FHI-aims code27.Self-consistent GW is based on the iterative solution of

14

Hedin’s equations with the GW self-energy and polar-izability, and therefore it is significantly different frompartially self-consistent approaches based on perturba-tion theory, such as the quasi-particle self-consistent GWscheme53,54 and self-consistency in the eigenvalues10,66.

In our implementation, the two-particle operators aretreated efficiently by means of the resolution of identitytechnique31. The introduction of an auxiliary basis forthe representation of the frequency and time dependenceof dynamic operators, facilitates an accurate evaluationof Fourier integrals that require just few tens of gridpoints. These ingredients allow us to reformulate Hedin’sequations in a matrix form, that can be solved with stan-dard linear algebra packages.

We presented an assessment of the spectral proper-ties of five molecules of interest for organic photo-voltaicapplications: thiophene, benzothiazole, 1,2,5-thiadiazole,naphthalene, and tetrathiafulvalene. For these systems,the quasi-particle energies extracted from the sc-GWspectral function are found to be in good agreement withexperimental photo-emission data for all valence states.The sc-GW excitation spectrum systematically improvesover perturbative G0W0 based on semi-local DFT andHartree-Fock. This is interpreted as a consequence ofthe mitigation of over- and under-screening errors char-acteristic of G0W0@PBE and G0W0@HF, respectively.G0W0 based on PBE0, on the other hand, provides re-sults in slightly better agreement with experimental datathan sc-GW . Thus, the PBE0 functional appears to beclose to an optimal starting point for perturbative calcu-lations.

Self-consistent GW total energies based on theGalitskii-Migdal formula permit an assessment ofground-state and structural properties of molecules. Fora small set of diatomic molecules we evaluated bind-ing energies, bond lengths and vibrational frequencies.The bond lengths improve at self-consistency, but stillhave an accuracy comparable to other perturbative meth-ods such as exact-exchange with correlation from therandom-phase approximation. Binding energies are typi-cally underestimated compared to experimental referencedata and do not substantially improve over perturbativeG0W0 calculations. Our results indicate and quantify theimportance of including vertex corrections – or alterna-tively, higher order correlation and exchange diagrams –in order to achieve an accurate description of the struc-tural properties of molecules.

Finally, the dipole moments of a set of hetero-atomicdimers were studied to investigate the accuracy of the sc-GW density. Compared to Hartree-Fock and PBE, thesc-GW dipole moments are found in better agreementwith experiment. These results indicate that sc-GW is apromising approach for the description of charge-transfercompounds and hetero-junctions, where the relative or-dering of the frontier orbitals influences the charge trans-fer at the interface.

ACKNOWLEDGMENTS

We would like to thank Christoph Friedrich for fruit-ful discussions. AR acknowledges financial support fromthe European Research Council Advanced Grant DY-Namo (ERC-2010-AdG -Proposal No. 267374) Span-ish Grants (FIS2011-65702- C02-01 and PIB2010US-00652), ACI-Promociona (ACI2009-1036), Grupos Con-solidados UPV/EHU del Gobierno Vasco (IT-319-07)and European Commission projects CRONOS (280879-2 CRONOS CP-FP7) and THEMA(FP7-NMP-2008-SMALL-2, 228539).

Appendix A: Rewriting the Galitskii-Migdal formula

In Hartree atomic units (h = e = m = 1), the elec-tronic contribution of the Galitskii-Migdal total energyis1:

EGM = −i∑

σ

∫

dr dt limr′→r

t′→t+

[

i∂

∂t+ h0

]

Gσ(rt, r′t′) .

(A1)Here, h0 is the single-particle term of the many-bodyHamiltonian, i.e., the sum of the kinetic energy operatorand the external potential due to the nuclei. Introduc-ing the equation of motion for the interacting Green’sfunction (see e.g. Ref. 67)

[

i∂

∂t+

∇2r

2− vH(r)− vext(r)

]

Gσ(rt, r′t′)−

−

∫

dr′′dt′′Σσ(rt, r′′t′′)Gσ(r′′t′′, r′t′) =

= δ(r− r′)δ(t− t′) , (A2)

Eq. A1 can be simplified by eliminating the partialderivative with respect to time, obtaining:

EGM = −i∑

σ

∫

dr dr′dt dt′

limr′→r

t′→t+

[(

−∇2r+ 2vext(r) + vH(r)

)

δ(r− r′)δ(t− t′)

+ Σσ(rt, r′t′)]Gσ(r′t′, r′t′). (A3)

Making use of the matrix representation of the Greenfunction Gσ(r, r′, τ) =

∑

ij ϕi(r)Gσ

ij(τ)ϕj(r′) – with

Gσ≡ s−1Gσs−1 – the first three terms in Eq. A3 can

be rewritten as:

−i∑

σ

∫

dr limr′→r

[

−∇2r+ 2vext(r) + vH(r)

]

×

×∑

ij

ϕi(r)Gσ

ij(τ = 0−)ϕj(r′) =

−i∑

σ

∑

ij

Gσ

ij(τ = 0−)[

2tji + 2vextji + vHji]

. (A4)

15

For time independent Hamiltonians, the Green func-tion depends only on time differences τ ≡ t − t′. Inthe last step of Eq. A4, we defined the matrix rep-resentation of the kinetic energy operator as tij =∫

drϕi(r)[

−∇

2r

2

]

ϕj(r), and use a similar representation

for vextji and vHji. Finally, the last term in Eq. A3can be rearranged by using the Fourier transformof the Green function and the self-energy G(t, t′) =∫ +∞

−∞

dω2π e

−iω(t−t′)G(ω), and substituting the matrix rep-resentation of G:

− i∑

σ

∫

dr dr′′dt dt′′ limr′→r

t′→t+

Σσ(rt, r′′t′′)Gσ(r′′t′′, r′t′) =

− i∑

σ

∑

ij

∫

dω

2πΣσ

ji(ω)Gσij(ω)e

iωη . (A5)

Summing Eqs. A4 and A5 and separating the self-energyin its exchange and correlation components Σij(ω) =

Σxij+Σc

ij(ω), one finally arrives at the expression reportedin Eq. 30. We refer to Ref. 68, for a discussion on theevaluation of the Galitskii-Migdal formula on the imagi-nary frequency axis.

Appendix B: Binding energies, bond lengths, and

vibrational frequencies of diatomic molecules

In Tables II and III we report counterpoise corrected69

binding energies and bond lengths for H2, LiH, LiF, HF,N2, and CO. The corresponding vibrational frequency arereported in Table IV. The sc-GW results are comparedwith experimental values64,65 and several perturbativeapproaches based on MBPT, namely G0W0, EX+cRPA,and rPT2@PBE62. HF and PBE are included for com-parison.

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TABLE II. sc-GW , and perturbativeG0W0 binding energies (evaluated from the Galitskii-Migdal formula) of diatomic moleculescompared to (zero point motion corrected) experimental reference data taken from Ref. 64. We report perturbative RPA, HF,PBE, and renormalized second-order perturbation theory (rPT2) for comparison. Calculations were done with a Tier 3 basisset (the largest basis set available) for H2, LiH, and HF, whereas a Tier 4 basis was used for N2 and CO. The mean absoluteerrors (MAE) are reported in panel (b) of Fig. 12. All values are in eV.

H2 LiH HF LiF N2 CO MAE

Exp -4.75 -2.52 -6.12 -6.02 -9.91 -11.24

sc-GW -4.41 -2.16 -5.55 -5.50 -8.42 -10.19 0.72

G0W0@HF -5.05 -2.72 -6.45 -6.60 -10.61 -11.88 0.46

G0W0@PBE -5.44 -2.94 -6.46 -6.37 -11.82 -12.16 0.77

G0W0@PBE0 -5.32 -2.90 -6.49 -6.67 -11.50 -12.34 0.78

RPA@HF -4.41 -2.17 -5.54 -5.52 -8.51 -10.19 0.70

RPA@PBE -4.68 -2.32 -5.60 -5.43 -9.54 -10.48 0.42

rPT2@PBE -4.71 -2.49 -5.93 -5.90 -9.42 -11.06 0.72

HF -3.64 -1.49 -4.22 -3.95 -5.10 -7.62 2.42

PBE -4.54 -2.32 -6.17 -6.03 -10.58 -11.67 0.26

TABLE III. sc-GW and perturbative G0W0 bond lengths of diatomic molecules compared to experimental reference data takenfrom Ref. 65. RPA, HF, PBE, and renormalized second-order perturbation theory (rPT2) are included for comparison. Meanabsolute errors (MAE) are reported in panel (a) of Fig. 12. All values are in A.

H2 LiH HF LiF N2 CO MAE

Exp. 0.741 1.595 0.917 1.564 1.098 1.128

sc-GW 0.735 1.579 0.919 1.586 1.085 1.118 0.011

G0W0@HF 0.733 1.560 0.919 1.579 1.093 1.119 0.012

G0W0@PBE 0.746 1.582 0.938 1.593 1.116 1.143 0.017

G0W0@PBE0 0.741 1.564 0.932 1.590 1.100 1.136 0.014

RPA@HF 0.734 1.587 0.914 1.576 1.087 1.117 0.009

RPA@PBE 0.745 1.597 0.927 1.589 1.107 1.137 0.010

rPT2@PBE 0.739 1.597 0.914 1.578 1.091 1.125 0.005

HF 0.734 1.606 0.898 1.560 1.066 1.102 0.017

PBE 0.751 1.605 0.930 1.575 1.104 1.136 0.010

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17

TABLE IV. sc-GW , and perturbative G0W0 vibrational frequencies of diatomic molecules compared to experimental refer-ence data taken from Ref. 65. RPA, HF, PBE, and renormalized second-order perturbation theory (rPT2) are included forcomparison. The mean absolute errors (MAE) are reported in panel (c) of Fig. 12. All values are in cm−1.

H2 LiH HF LiF N2 CO MAE

Exp. 4401 1405 4138 911 2359 2170

sc-GW 4533 1743 4266 971 2543 2322 166

G0W0@HF 4585 1827 4341 1010 2490 2647 252

G0W0@PBE 4341 1743 4130 971 2346 2322 105

G0W0@PBE0 4425 1813 4273 922 2386 2222 109

RPA@HF 4533 1685 5512 952 2544 2321 360

RPA@PBE 4357 1691 4757 933 2354 2115 172

HF 4567 1473 4569 949 2736 2448 226

PBE 4320 1364 3991 899 2328 2128 59

rPT2@PBE 4460 1605 4620 922 2507 2251 163

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