comparison between gw and wave-function-based...

Comparison between GW and Wave-Function-Based Approaches:Calculating the Ionization Potential and Electron Affinity for 1DHubbard ChainsPublished as part of The Journal of Physical Chemistry A virtual special issue “Mark S. Gordon Festschrift”.

Qi Ou* and Joseph E. Subotnik

Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States

ABSTRACT: We calculate the ionization potential and electron affinity of 1DHubbard chains with a variety of different site energies from two perspectives: (i) thephysics-based GW approximation and (ii) the chemistry-based configurationinteraction (CI) approach. Results obtained from all methods are compared againstthe exact values for three classes of systems: metallic, impurity doped, and molecular(semiconducting/insulating) systems. Although all methods are reasonably accurate forweakly correlated systems, the GW method is significantly more reliable for stronglycorrelated systems with little disorder unless explicit double excitations are included inthe CI. In principle, our results should offer some intuition about the choice ofmethodologies as well as state references for different classes of physical systems.

I. INTRODUCTION

Calculations of the charged properties of materials have longbeen one of the major challenging goals in molecular systems andcondensed matter physics. For a many-electron system, theHamiltonian is given by

∑ ∑= − ∇ + +| − |≠

⎡⎣⎢

⎤⎦⎥H Vr r

r r12

( ) ( )12

1

ii i

i j i j

2

(1)

where V is a local external potential caused by nuclear-electroniccoupling. Because of the last term in eq 1, which describes theCoulomb interaction between electrons, it is exponentiallydifficult to diagonalize an N-body Hamiltonian exactly. Ofcourse, many approximate, mean-field theories have beendeveloped within the single-particle approximation for theCoulomb term, which allow one to estimate the properties oflarge systems analytically. (A good review can be found in ref 1.)For instance, one can treat electrons as essentially noninteractingparticles in theories such as the Hartree approximation or theHartree−Fock (HF) approximation2 so that the Hamiltonianbecomes straightforward to solve.When the Coulombic interactions are large and the picture of

independent electrons breaks down,3 electronic screening can nolonger be neglected. One the one hand, quantum chemistry hastraditionally sought either (i) improved exchange-correlationkernels within density-functional theory (DFT) or (ii) optimalwave function ansatzes.4−7 For example, for several recentdecades, quantum chemists were focused on local correlationtechniques to calculate ground state properties.8−13 On the otherhand, condensed matter theorists usually treat correlatedelectrons (or holes) as approximately independent quasiparticlesand correlation effects are addressed through many-bodyperturbation theory (MBPT) in terms of the self-energy of the

quasiparticles.14−16 In the framework of Green’s functions,17,18

the GW method is known to be the simplest workingapproximation for calculations of the self-energy.19−25

We believe that not yet enough work has been done bycomparing the behavior of the GW method (which is popular inthe physics community) and wave-function-based approaches(which are popular in the chemistry community) regarding thecalculation of the ionization potential and electron affinity. Thisstatement excludes recent works by (i) Bruneval et al., whocomparedGW againt DFT and HF for atoms and molecules;26,27

and (ii)McClain et al., who calculated the spectral function of theuniform electron gas via ab initio coupled-cluster theory andcompared against GW results.28 Notwithstanding refs 26−28,one difficulty in assessing different methodologies is the difficultycalculating exact benchmarks for ionization potentials andelectron affinities.With this background in mind, here we will calculate the

electron affinities and ionization potentials for a set of 1DHubbard chains29 with either eight or ten sites and differentpossible site energies using exact diagonalization of the fullHamiltonian. Thereafter, approximate results can be evaluatedfor accuracy according to both theGWmethod and various wave-function-based quantum chemistry approaches. We analyzethree different types of systems: metallic systems, doped systems,and molecular (semiconducting/insulating) systems. Althoughthe original GW method is a computationally expensive self-consistent method, we will follow the so-called G0W0 procedureproposed by Hybertsen and Louie in 198630 and perform a first-order perturbative calculation.

Received: March 31, 2016Revised: June 14, 2016

Article

pubs.acs.org/JPCA

© XXXX American Chemical Society A DOI: 10.1021/acs.jpca.6b03294J. Phys. Chem. A XXXX, XXX, XXX−XXX

pubs.acs.org/JPCA

http://dx.doi.org/10.1021/acs.jpca.6b03294

An outline of this paper is as follows. In section II, we brieflyreview our model system and the methodologies applied in thecalculation, both wave-function-based approaches and the GWmethod. In section III, we compare results given by theapproximate methods versus the exact values for three differenttypes of systems. In section IV, we discuss our results. In sectionV, we conclude.Unless otherwise specified, we use lowercase latin letters to

denote spinmolecular orbitals (MO) (a, b, c, d for virtual orbitals,i, j, k, l, m for occupied orbitals, p, q, r, s, w for arbitrary orbitals)and Greek letters (α, β, γ, δ, λ, σ, μ, ν) to denote atomic orbitals(AO). The electronic excited states obtained within the random-phase approximation (RPA) are denoted by Ψ (with uppercaseindices I and J).

II. THEORYII.A. Exact Diagonalization versus Orbital Energies. For

a system with N electrons, the exact ionization potential Eion andaffinity energy Eaff can be obtained from the total energydifference

= −−E E EN Nion 0

10 (2)

= − +E E EN Naff 0 0

1(3)

Alternatively, the easiest scheme to estimate ionization energies(Eion) and electron affinities (Eaff) is to evaluate the highestoccupied molecular orbital (HOMO) energy and the lowestunoccupied molecular orbital (LUMO) energy,31 i.e.,

ε≈ −Eion homo (4)

ε≈ −Eaff lumo (5)

Here, the orbital energies are taken from a HF calculation (or, ofcourse, a DFT calculation). For a Hartree calculation, theionization energies (electron affinities) are approximated by theHOMO (LUMO) energy plus the orbital exchange potential.

∑ε≈ − + | =E pi ip p( ) HOMO or LUMOpi

H

(6)

where (pq|rs) = ∫ ′ϕ ϕ ϕ ϕ′ ′

− ′

* *

d dr rr r r r

r r

( ) ( ) ( ) ( )p q r s refers to the Coulomb

interaction.The zero of energy is a slow electron infinitely faraway. As defined in eq 2−6, for a system that does not want tolose or accept another electron, the ionization potential ispositive and the electron affinity is negative.II.B. IP/EA-CISD. Beyond the Hartree/HF orbital energies,

the simplest quantum chemistry approximation for Eion issometimes referred to as IP-CISD (Ionization Potential-Configuration Interaction with Singles and Doubles).32 IP-CISD describes ionized states as 1h and 2h1p excitations from aclosed-shell reference (usually HF) and the amplitudes of thetarget states are found by diagonalizing the bare Hamiltonian H.Here h denotes a hole and p denotes a particle. In some cases, IP-CISD can provide an accurate ionization potential for closed-shell systems.32 Mathematically for IP-CISD, one constructs avariational wave function ansatz |Ψ⟩ = ∑iti|Φi⟩ + ∑ijatij

a|Φija⟩. By

including one set of orbital excitations on top of the hole, onehopes to recover a reasonable amount of electronic correlationfollowing ionization. To solve for a stationary state withcoefficients ti and tij

a, one diagonalizes a four-by-four-blockHamiltonian

=⎛⎝⎜⎜

⎞⎠⎟⎟H

H H

H H

SS SD

DS DD (7)

where Hi,kSS = ⟨Φi|H|Φk⟩, Hi,klb

SD = ⟨Φi|H|Φklb ⟩, Haji,k

DS = ⟨Φija|H|Φk⟩,

and Haji,klbDD = ⟨Φij

a|H|Φklb ⟩. The same logic above can be applied

when the electron affinity (Eaff) is computed, which is the amountof energy released when an electron is added to an unoccupiedorbital. The simplest Hamiltonian for evalulating the electronaffinity (titled EA-CISD) contains the followingmatrix elements:Ha,c

SS = ⟨Φa|H|Φc⟩, Ha,jdcSD = ⟨Φa|H|Φj

cd⟩, Habi,cDS = ⟨Φi

ab|H|Φc⟩, andHabi,jdc

DD = ⟨Φiab|H|Φj

cd⟩.It should be emphasized that neither IP-CISD nor EA-CISD is

very computationally cheap. These methods require thediagonalization of a matrix of size Nocc

2(Nvirt + 1)2 orNvirt

2(Nocc + 1)2. Nevertheless, they are the most inexpensive,straightforward wave function approaches.Finally, we mention that, in principle, one can compute IP/

EA-CISD energies using either a Hartree or HF reference. Belowwe will perform such calculations with both references for theHubbard model and, after extracting the electron affinities andionization potentials, we will compare results.

II.C. ΔSCF and ΔCISD. Another way to evaluate theionization potential and electron affinity is to compute E0

N−1,E0N, and E0

N+1 directly, and here we will calculate such quantities intwo ways: (i) a simple self-consistent field (SCF) calculation,denoted ΔSCF; (ii) and a more sophisticated configurationinteraction singles and doubles (CISD) calculation, denotedΔCISD.Mathematically for CISD, one constructs a wave function

according to the ansatz

∑ ∑|Ψ⟩ = |Ψ ⟩ + |Φ ⟩ + |Φ ⟩t t tai

ia

ia

ijabijab

ijab

HF HF(8)

where |ΨHF⟩ is the HF ground state with the coefficient tHF, and|Φi

a⟩ and |Φijab⟩ are single and double excitations with coefficients

tia and tij

ab, respectively. One diagonalizes a CISD Hamiltonian ofthe form

=† †

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟H

E H

H H

H H H

0

0

( ) ( )

HF0D

SS SD

0D SD DD(9)

where EHF is the HF ground state energy, Hiajb0D = ⟨ΨHF|H|Φij

ab⟩,Hia j b

S S = ⟨Φ ia |H |Φ j

b⟩ , Hia , j b k cSD = ⟨Φ i

a |H |Φ k cj b ⟩ , and

Hiajb,kcldDD = ⟨Φij

ab|H|Φklcd⟩. Note that the manifold of single

excitations does not couple to the ground state according toBrillouin’s theorem.

II.D.GWMethod. II.D.1. Green’s Function Review.The one-body time-ordered Green’s function Gt can be written as

′ ′ ≡ ⟨Ψ | ̂ ̂ ̂ ′ ′ |Ψ ⟩†G t t T a t a tr r r r( , ; , ) [ ( ) ( )]t N N0 0 (10)

where |Ψ0N⟩ is the ground state for an interacting N-body system,

a ̂(rt) is the electron annihilation operator which annihilates anelectron at (r, t), and T̂ is the Fermionic time-ordering operator.Equation 10 is the starting point for MBPT.For our purposes (to get the ionization potentials and electron

affinities), GW calculations require only the noninteractingGreen’s function where |Ψ0

N⟩ is a Slater determinant. In thefrequency domain, one can then write down the expression forthe noninteracting time-ordered Green’s function as (in the MObasis)

The Journal of Physical Chemistry A Article

DOI: 10.1021/acs.jpca.6b03294J. Phys. Chem. A XXXX, XXX, XXX−XXX

B


∑ ∑ωϕ ϕω ε η

ϕ ϕω ε η

ω ω

′ =′

− −+

′− +

≡ ′ − ′

G

G G

r rr r r r

r r r r

( , , )( ) ( )

i

( ) ( )

i(11)

( , , ) ( , , ) (12)

t

i

i i

i a

a a

a

R Aocc virt

Equation 11 defines the “G” in the GW method. In eq 12, wehave defined Gocc

R and GvirtA as per the usual convention that

electrons move forward in time and holes move backward intime. Thus, it is straightforward to show that iℏGocc

R (r,r′,t−t′)(iℏGvirt

A (r,r′,t−t′)) is the probability amplitude for thepropagation of an additional electron (hole) from (r′, t′) to(r, t) for a noninteracting system.II.D.2. Screened Coulomb Interaction. To express the “W” in

the GW approximation, we need a few definitions. Classically, ifan external potential u(r,t) is applied to a solid, the screening ofthe solid is defined in terms of a dielectric constant. In many-body theory, the inverse of the dielectric response is defined as

δδ

ϵ ′ ′ ≡′ ′

−

=

t tV tu t

r rr

r( , ; , )

( , )( , )

u

1

0 (13)

where V(r,t) is the combination of the external potential and theCoulomb potential created by the induced charge (with densityρ)

∫ ρ≡ + ′ ′ ′V t u t v tr r r r r r( , ) ( , ) d ( , ) ( , )(14)

The inverse of the dielectric response can be rewritten as

∫δ δ δρδ

ϵ ′ ′ = + ′ ′′ ′

−′ ′

=

t t vt

u tr r r r r

rr

( , ; , ) d ( , )( , )

( , )ttu

1rr

0 (15)

In linear response theory, the response function (or the so-calledRPA polarizability) χ is defined to describe the change of theinduced charge density if the external potential undergoes a smallchange

χ δρδ

′ ′ ≡′ ′ =

t tt

u tr r

rr

( , ; , )( , )

( , )u 0 (16)

Using the chain rule for functional derivatives, we can express χ interms of the noninteracting response function as

∫

∫

χ δρδ

δδ

′ ′ = ″ ″″ ″

″ ″′ ′

≡ ″ ″ ″ ″ ϵ ″ ″ ′ ′

=

−

t t tt

V tV tu t

t P t t t t

r r rr

rrr

r r r r r

( , ; , ) d d( , )

( , )( , )( , )

(17)

d d ( , ; , ) ( , ; , ) (18)

u 0

1

where we define the noninteracting response function as

δρδ

′ ′ ≡′ ′ =

P t tt

V tr r

rr

( , ; , )( , )

( , )u 0 (19)

In the GW approximation, P(r,r′,t−t′) can be expressed as themultiplication of two Green’s functions (by neglecting the vertexfunction)

′ ′ = − ℏ ′ ′ ′ ′P t t G t t G t tr r r r r r( , ; , ) i ( , ; , ) ( , ; , )t t (20)

If we apply the definition of the RPA polarizability (eq 16), theinverse of the dielectric response (eq 15) becomes

∫δ δ χϵ ′ ′ = + ′ ′ ′ ′−′ ′t t v t tr r r r r r r( , ; , ) d ( , ) ( , ; , )tt

1rr (21)

In the frequency domain, the screened Coulomb interaction isdefined as

∫ω ω′ ≡ ″ ϵ ″ ″ ′−W vr r r r r r r( , ; ) d ( , ; ) ( , )1(22)

Taking the Fourier transform of eq 21 and plugging the resultinto eq 22, we find W(r,r′;ω) can be rewritten as

∫ω χ ω′ = ′ + ″ ‴ ″ ″ ‴ ‴ ′W v v vr r r r r r r r r r r r( , ; ) ( , ) d d ( , ) ( , , ) ( , )

(23)

The screened Coulomb interaction W(r,r′;ω) is the effectivepotential at r′ induced by a quasiparticle at r; in general,W(r,r′;ω) is anticipated to be weaker and better behaved thanthe bare Coulomb interaction v(r,r′). The difference betweenW(r,r′;ω) and v(r,r′) is the polarizable part of the screenedCoulomb interaction Wp

∫ω ω

χ ω

′ ≡ ′ − ′

= ″ ‴ ″ ″ ‴ ‴ ′

W W v

v v

r r r r r r

r r r r r r r r

( , ; ) ( , ; ) ( , ) (24)

d d ( , ) ( , , ) ( , ) (25)

p

Wp is the “W” in the GW approximation that must now beevaluated.

II.D.3. RPA Polarizability. To calculate the RPA polarizabilityχ, the usual prescription is to perform a RPA calculation on top ofa Hartree reference state, ignoring all exchange terms. The RPAworking equation is given by33

− −= Ω⎜ ⎟⎛

⎝⎞⎠⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟A B

B AX

Y

X

Y

I

I I

I

I (26)

for excitation amplitudes XI and YI and excitation energy ΩI forRPA excited state I. A and B contain matrix elements as follows

δ δ ε ε= − − |A ia jb( ) ( )iajb ij ab a i (27)

= − |B ia bj( )iajb (28)

By diagonalizing the RPA non-Hermitian Hamiltonian, oneobtains the excitation amplitudes XI and YI and the excitationenergy ΩI for each RPA excited state I. The RPA polarizabilitycan then be computed via the following sum-overstatesexpression34

∑

χ ω ω

ω ω

= +−

=Ω +

+Ω −

−

⎜ ⎟⎡⎣⎢⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟⎤⎦⎥

⎡⎣⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤⎦⎥⎥

X

YX Y

Y

XY X

A BB A

II

( )0

0

1( )

1( )

1

I I

I

II I

I

I

II I

(29)

In terms of MOs, if the perturbing potential is of the formujb(t)(ab

†aj + aj†ab), one finds that the response function is

∑χ ωω ω

=Ω +

+Ω −

⎡⎣⎢⎢

⎤⎦⎥⎥

M M( )iajb

iajb

I

iajb

I

I I

I (30)

where

= + + +M X X X Y Y X Y Yiajb ia jb ia jb ia jb ia jbI I I I I I I I I

(31)

To express χ in a real space, one must sum over orbitals



C


∑ ∑

χ ω

ϕ ϕ ϕ ϕω ω

′

= ′ ′Ω +

+Ω −

⎡⎣⎢⎢

⎤⎦⎥⎥

M M

r r

r r r r

( , , )

( ) ( ) ( ) ( )iajb

i a j biajb

I

iajb

I

I I

I

(32)

II.D.4. Self-Energy. In practice, theGW self-energy is obtainedfrom the frequency convolution of Green’s function Gt with thescreened Coulomb interaction W

∫ωπ

ω ω ω ωΣ ′ = ′ ′ + ′ ′ ′ηω′G Wr r r r r r( , , )i

2d e ( , , ) ( , , )ti

(33)

where η is an infinitesimal positive real number. IntroducingWp(eq 25) can split the self-energy into two parts: the staticexchange part Σx and the dynamic correlation part Σc

∫πω ωΣ ′ = ′ ′ ′ ′ηω′v Gr r r r r r( , )

i2

( , ) d e ( , , )tx i(34)

∫ωπ

ω ω ω ωΣ ′ = ′ ′ + ′ ′ ′ηω′G Wr r r r r r( , , )i

2d e ( , , ) ( , , )i tc

p

(35)

Note that Σx corresponds to the usual nonlocal exchangepotential, which is introduced by the Fock exchange operator inthe HF approximation

∑ ∑ μν λσΣ = − |μνλσ

μ ν λ σC C C C( )pqi

p i i qx

(36)

In eq 36 and everywhere below, we assume we are performing aGW calculation on top of a Hartree calculation. All orbitals areHartree orbitals.Σc gives rise to the correlation energy of the system, which is

not taken into consideration in a HF calculation. To evaluate theΣc contribution to the self-energy from the GW method, onesimply plugs eqs 11 and 32 into eq 37 and obtains the finalexpression for correlation part of the self-energy (in the MObasis). Plugging eq 25 into eq 35, one can express Σc in terms ofGreen’s function Gt, the RPA polarizability χ, and the bareCoulomb potential v

∫∫

ωπ

ω ω ω

χ ω

Σ ′ = ′ ′ + ′

× ″ ‴ ′ ″ ″ ‴ ′ ‴

ηω′G

v v

r r r r

r r r r r r r r

( , , )i

2d e ( , , )

d d ( , ) ( , , ) ( , )

tc i

(37)

The self-energy evaluated at the energy of Hartree orbital s canfinally be expressed as

∑ ∑ ∑

∑

εε ε η

ε ε η

Σ =| |

− Ω − −

+| |

+ Ω − +

⎛⎝⎜⎜

⎞⎠⎟⎟

ip jb iq kc

ap jb aq kcM

( )( )( )

i

( )( )i

pq sjbkc i i

a ajbkc

c H

I sH

IH

sH

IH

I

(38)

For self-consistency, we must have

ε ε ε= + Σ ( )GW GWs s

Hss s (39)

If we want to avoid a self-consistent calculation, the standardapproach is to apply the linear expansion for the real part of theΣc(ω)

ε εε ε ε

ωΣ ≈ Σ +

−ℏ

∂ Σ∂

Re ( ) Re ( )Re ( )GW

GW cc

sc

sH s s

HsH

(40)

which leads to

ε ε ε≈ + Σ + ΣZ Re ( )GW xs s

Hs

csH

s (41)

The quasiparticle renormalization factor “Zs” is given by

εω

≡ −∂ Σ

∂

−⎛⎝⎜

⎞⎠⎟Z 1

Re ( )s

csH 1

(42)

According to eqs 4 and 5, the electron affinity/ionizationpotential of a system can be obtained from the GWapproximation by taking the negative value of eq 41. At thispoint, all of the formal theory has been presented and will beapplied to a 1D Hubbard model system.

II.E. Model System and Its Affinity/Ionization Potential.TheHubbard model offers one of the simplest ways to get insightinto how the interactions between electrons can give rise toinsulating, doped, and conducting effects in a solid. In this workwe take 1D Hubbard model as our testing system, which isdescribed by the Hamiltonian

∑ ∑

∑

τ≡ + ̅ ̅ + + ̅ ̅

+ ̅ ̅

μνμ ν μ ν μ

μμ μ μ μ

μμ μ μ μ μ

⟨ ⟩

† † † †

† †

a a a a V a a a a

U a a a a

( ) ( )

(43)

where μ denotes the site of the system, τ is the hopping integralbetween neighboring sites, Vμ is the on-site energy (for site μ),andUμ is the repulsion energy between two electrons of oppositespin accommodated on each site μ. A bar indicates spin down.In practice, we first perform a normal Hartree calculation for

the model system, obtaining the Hartree orbital energy εiH and

the molecular orbital coefficientsC. The next step is to perform atime-dependent Hartree calculation within RPA to get the matrixelements for the RPA polarizability χ. Then eq 38 becomes (forthe HOMO and LUMO respectively)

∑ ∑ ∑

∑

εε ε η

ε ε η

Σ =− Ω − −

++ Ω − +

×

μ ν

μ ν

μ ν μ μ μ ν ν ν

⎛⎝⎜⎜

⎞⎠⎟⎟

C C

i

C C

i

U U C C C C C C M

( )c

jbkc I i

i i

I i

a

a a

I a

p j b q k c jbkcI

homoH

homoH H

homoH H

(44)

∑ ∑ ∑

∑

εε ε η

ε ε η

Σ =− Ω − −

++ Ω − +

×

μ ν

μ ν

μ ν μ μ μ ν ν ν

⎛⎝⎜⎜

⎞⎠⎟⎟

C C

i

C C

i

U U C C C C C C M

( )c

jbkc I i

i i

i

a

a a

I a

p j b q k c jbkcI

lumoH

lumoH

IH

lumoH H

(45)

The exchange part of the self-energy Σx for the 1D Hubbardmodel is given by

∑ ∑Σ =μ

μ μ μ μU C C Cpqi

p q ix 2

(46)



D


Note that for all calculations below, we calculate only HOMOand LUMO energies and we set η = 0 in eqs 44 and 45.

III. RESULTS

We apply the theory above to 1D Hubbard models of two sizes:8-Site and 10-Site. We assume half-filling with either 8 or 10electrons, respectively. The hopping integral between neighbor-ing sites τ is set to be 1.0 for all systems, while the electronrepulsion energy U is varied. All quantities will be calculated ina.u. henceforward. Exact electron affinities and ionizationpotentials are obtained by diagonalizing the Hamiltonian withDavidson iterative diagonalization.35 To compare the behaviorsof different methods for different cases, we will plot the differencebetween the computed value and the exact answer as follows: theupper and middle panels show the results for 8-Site/8e− and 10-Site/10e−, respectively; the left and right of the top two panelsshow the electron affinity and ionization energy differencesrespectively; the lower panel shows the Z factor for LUMO andHOMO (eq 42).

III.A. Metallic System. We first study metallic systems. Inmetallic systems, every site has the same on-site energy (Vμ = 0for all sites μ).36 The electron repulsion energyU is varied from 0to 0.7. Results are shown in Figure 1. Focusing on the electronaffinity (left panel), one can see that all approximate methodsgive the exact answer whenU = 0 (of course). AsU increases, theerrors given by all the quantum chemistry methods (exceptΔCISD) increase dramatically, indicating the failure of thesemethods for strongly correlated systems. By contrast, the GWmethod, with a maximum error (∼0.01) when U = 0.7, performsfairly well (though not as well as ΔCISD, which gives almostexact results). The validation of the GW approximation for themetallic system is also strengthened by the fact that the Z factorfor both 8-Site and 10-Site is very close to 1. Similar results arefound for the ionization potential.Note that, if we want to plot the total HOMO/LUMO gap

(εlumo − εhomo or equivalently, Eion − Eaff) for such systems, thequantum chemistry methods (except ΔCISD) would stillperform very poorly; the errors for Eion and Eaff have differentsigns and subtracting these errors would yield even larger errors.

Figure 1. Errors for electron affinities and ionization potentials and the quasiparticle renormalization (Z) factor (eq 42) for the metallic system (Vμ = 0for all sites μ, τ = 1.0). All errors are relative to the exact diagonalization. The upper and middle panels show the results for 8-Site/8e−and 10-Site/10e−,respectively; the left and right of the top two panels show the electron affinity and ionization energy differences respectively; and the lower panel showsthe Z factor for LUMO and HOMO. Note that ΔCISD performs the best for metallic systems followed by the GW method.



E


III.B. Doped Impurity Systems. By lowering the energy of asingle site in the metallic system, one can model an impurity,which dopes an otherwise metallic system and can influence theoverall photoactivitiy or electrical properties of the total system.In Figures 2 and 3, we calculate results for electron repulsionenergies U between 0 and 0.7. We consider two on-site energies:Vimp = −0.5 (Figure 2) and Vimp = −1.0 (Figure 3).For the case Vimp = −0.5 (where the impurity energy is half as

large as the hopping integral τ), it can be seen in Figure 2 that theresults are roughly identical to those for themetallic case:ΔCISDgives the most accurate results and the GW method is bettercompared with other quantum chemistry methods.For the case Vimp = −1.0 (where the on-site energy is

comparable with the hopping integral), the same conclusionholds only for the ionization energies. For the affinity energies(left panel of Figure 3), the quantum chemistry approaches nowdo much better than before (and ΔCISD is again the best). Inparticular, the Hartree approximation with the exchange energycompetes withGW for the most accurate method exceptΔCISD.The ionization energy results remain very similar to those of the

former cases; that is, ΔCISD gives almost exact results and GWperforms much better than other quantum chemistry methods.In all doped systems, we note that the IP/EA-CISD methods

yield results that are even worse than direct Hartree/HFapproximations.

III.C. Molecular (Semiconducting/Insulating) Systems.To simulate a “molecular” Hubbard model, we construct analternating Hamiltonian, whereby Veven is significantly loweredrelative toVodd = 0. In doing so, we expect that orbitals with lowerenergies will be doubly occupied and well separated from virtualorbitals. Thus, by creating an energy gap, we should be simulatingclosed-shell insulators and/or semiconductors (depending onthe gap size).For “molecular” systems with a small energy gap (Veven =

−0.5), the electron repulsion energy U is varied from 0 to 0.8. Asshown in Figure 4, the performances of the quantum chemistrymethods are now far improved compared with the metallic case.Although ΔCISD remains the best, the Hartree and HFapproximations compete with the GW method for accuracyand sometimes they prevail. As in Figures 2 and 3, the GW error

Figure 2. Errors for electron affinities and ionization potentials and the quasiparticle renormalization (Z) factor for the metallic system for the dopedsystem (with V5 = −0.5 for the 8-Site system and V6 = −0.5 for the 10-Site system; for all other sites, Vμ = 0). τ = 1.0. All errors are relative to the exactdiagonalization. ΔCISD is still the most accurate method and GW consistently outperforms other wave function methods.



F


turns over as U gets larger; the GW error is maximized at someintermediate value of U.It should be pointed out that, so far, the IP/EA-CISDmethods

have not provided any results better than the direct HF orHartree approximation. In fact, wave function corrections toHartree/HF calculations only increased the error in Figure 4.However, one must be careful not to quickly extrapolate from thedata above. For “molecular” systems with an energy gap equal tothe hopping integral (Veven = −1.0), Figure 5 demonstrates thatour results are completely different from the previous case (Veven

= −0.5). In particular, now, IP/EA-CISD methods areconsistently more accurate than the direct HF or Hartreeapproximation. For the 8-Site system, IP/EA-CISD(HF) yieldsthe overall best results apart from ΔCISD. For the 10-Sitesystem, ΔCISD is followed by ΔSCF and GW, which are betterthan the other approaches.Lastly, unlike previous cases, according to Figure 5, the

Hartree approximation yields remarkably large errors. As onewould expect, the quality of the Hartree reference simply

degrades as the system becomes more molecular (and exactexchange becomes critical).Finally, we consider the final, extreme “molecular” case

whereby the on-site energy difference is now twice as large as thehopping integral (Veven =−2.0). As Figure 6 shows, asU increasesfrom 0 to 1.2, GW and ΔSCF are quite accurate, though slightlyworse than ΔCISD. By contrast, the HF related methodsperform worse than GW, and the Hartree related methods areeven worse: the Hartree orbital energies are no longer any good.

IV. DISCUSSION

The results above (in section III) have compared the morechemical (wave function) versus the more physical (GW)methods for calculating electron affinities and ionizationenergies. Of all the methods considered excluding ΔCISD,GW is usually the most reliable approach when we consider abroad range of systems. This preliminary conclusion might be abit surprising from the quantum chemistry point of view. Afterall, on the one hand, with regards to scaling, GW is limited by

Figure 3. Errors for electron affinities and ionization potentials and the quasiparticle renormalization (Z) factor for the doped system (withV5 =−1.0 forthe 8-Site system and V6 = −1.0 for the 10-Site system; for all other sites, Vμ = 0). τ = 1.0. All errors are relative to the exact diagonalization. Clearly,ΔCISD again gives the most accurate results. GW performs better for ionization potentials than other quantum chemistry approach, but the Hartreeapproximation with exchange energy performs equally well for the electron affinity.



G


diagonalizing a matrix of orderNoccNvirt ×NoccNvirt. On the otherhand, the IP/EA-CISD methods require the diagonalization ofcomparatively larger matrices of order NoccNvirt

2 × NoccNvirt2 (for

EA-CISD). Thus, GW is able to provide improved results withless computational effort.To understand this strong performance byGW, we investigate

a simplified version of IP/EA-CISD, that is, fixing a single MO(HOMO for the ionization potential and LUMO for the electronaffinity) and making corrections on top of this single orbital.Taking the 8-Site/8e− “molecular” system (Veven = −1.0) as anexample, we compare the result of this so-called IP-CISD(HF/homo) or EA-CISD(HF/lumo) result against HF, IP/EA-CISD(HF), and GW results in Figure 7. For the electron affinity,EA-CISD(HF/lumo) provides a result very similar to that bydirect HF but with some small improvement. For the ionizationenergy, IP-CISD(HF/homo) yields a noticeable correction andthe result is much closer to the full IP-CISD(HF) energy. Inother words, a single-orbital approximation for the hole does notdegrade the answers significantly. Thus, there are clearlyHamiltonians for which quantum chemistry wave function

approaches can outperform GW with the same computationalcost. (IP-CISD(homo) and EA-CISD(lumo) require the samecomputational cost as GW.)That being said, in general, GW does outperform a more

expensive algorithm. Obviously, the strength of GW is that theGW self-energy includes a summation of Coulomb interactionsby diagonalizing the RPA matrix and constructing the fullresponse matrix (which leads to the screening potential W). Bycontrast, the IP/EA-CISD methods include only one bareCoulomb interaction through variational calculations. Thus, theresults above reflect the relative value of these two approaches,and the GW approach would appear to be more robust overall.Given that theGW performs so well above for a range of model

Hubbard Hamiltonians, it is worthwhile to test its convergencewith respect to the number of RPA states included. Consideringeq 37, we see that the GW self-energy directly depends on theRPA polarizability χ, the polarizability can be written as a sumover states (eq 32), and all excited states contributions to the self-energy will decay inversely proportional to the energy of thatexcited state. It is well-known thatGW converges poorly with the

Figure 4. Errors for electron affinities and ionization potentials and the quasiparticle renormalization (Z) factor for “molecular”Hubbard models (Veven= −0.5, Vodd = 0, τ = 1.0). All errors are relative to the exact diagonalization. Here, though ΔCISD is always the most accurate, the simple Hartreeapproximation with the exchange energy performs better than other methods. TheGW approximation is still reasonable, which can also be shown fromthe Z factor values.



H


number of virtual orbitals,37,38 but we would like to test thisconvergence for different Hamiltonians.With this background in mind, in Figure 8 we plot the error

electron affinity given by GW as a function of the number ofsinglet states included in the calculation for two different cases:(i) 8-Site/8e− metallic system with U = 0.5 and (ii) 8-Site/8e−

molecular system with U = 0.5 and Veven = −2.0. On the onehand, for the metallic system, a strong GW correction can beobtained by including just one RPA excited state; for thesesystems and this U value, we find one low lying state that isreasonably separated energetically from higher excited states andyields a strong correction. On the other hand, for the molecularsystems, the error decreased very slowly (almost linearly). Inother words, one must include all RPA states to maintain theaccuracy of the GW method for a molecular system.39,40 Giventhe large cost of the full GW calculation, it will be interesting tocompare relative (as opposed to absolute) ionization potentialsand electron affinities for real molecules as a function of thenumber of states included. This work is ongoing.

Finally, a few words are in order about self-consistency. Toavoid self-consistency, one can use eq 41. Alternatively, one caniterate the GW equations to satisfy eq 39. We have performed asimple eigenvalue self-consistent GW calculation (evGW)41 andwe find that results given by evGW are actually very close to ourperturbative GWmethod with the linear expansion (eq 41). Thissuccess of the perturbative GW approximation is consistent withthe fact that the Z-factor is very close to 1 (larger than 0.9) andvalidates the results presented in Figures 1−6.We have found, however, that the Z-factor values for orbitals

other than HOMO and LUMOwere usually far away from 1withcorrespondingly large errors compared to exact attachment−detachment energies.

V. CONCLUSION

In this paper, we have systematically compared the performanceof GW against various wave-function-based methods for thecalculation of ionization potentials and electronic affinities forseveral 1D Hubbard models. Three different types of systems

Figure 5. Errors for electron affinities and ionization potentials and the quasiparticle renormalization (Z) factor for “molecular”Hubbard models (Veven=−1.0,Vodd = 0, τ = 1.0). All errors are relative to the exact diagonalization. AlthoughΔCISD remains themost accurate, theHartree approximation nowperforms the worst. Excluding ΔCISD, IP/EA-CISD(HF) yields the overall best results for the 8-Site system whereas ΔSCF and GW become the bestfor the 10-Site system,.



I


have been studied: metallic, impurity, and molecular (semi-conducting/insulating) systems.

Although ΔCISD gives almost exact results for all cases, themost striking conclusion from the data above is that the GW

Figure 6. Errors for electron affinities and ionization potentials and the quasiparticle renormalization (Z) factor for “molecular”Hubbard models (Veven=−2.0, Vodd = 0, τ = 1.0). All errors are relative to the exact diagonalization. Note that theGWmethod, though slightly worse thanΔCISD, outperformsother methods even in this very insulating case.

Figure 7. Errors for electron affinities and ionization potentials for the 8-Site/8e−molecular system (Veven = −1.0). Results given by HF, IP/EA-CISD(HF), and GW are taken from Figure 5. Here we compare against IP-CISD(HF/homo) and EA-CISD(HF/lumo). Note that by freezing oneorbital, the electron affinity changes significantly but the ionization potential does not.



J


method surpasses other wave function methods in two extremelimits. On the one hand, for the metallic case with reasonablylarge U, GW performs significantly better than wave functionmethods (except ΔCISD). On the other hand, for the molecularsystem with the largest HOMO/LUMO gap, GW also gives avery accurate result. For systems between these two extremecases, the advantages of GW becomes less obvious and the wave-function-based methods behave better.As DFT specialists may well appreciate, the Hartree

approximation yields a small improvement over HF for boththe impurity systems and molecular systems with small energygaps (Vμ ≪ τ). For the insulator case with a large HOMO/LUMO gap, the Hartree approximation dramatically fails tomake accurate predictions but can be used as a reasonablereference for GW. Altogether, these results emphasize thecomparative advantages of the GW method versus traditionalquantum chemistry approaches across a large range ofHamiltonians (even if GW is not always the optimal method).Looking forward, it is clear why GW is currently being applied toa variety of molecular systems.40,42−44

■ AUTHOR INFORMATIONCorresponding Author*Q. Ou. E-mail: [email protected]. Phone: (215)290-1022.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe thank Timothy Berkelbach and Jefferson Bates for veryhelpful conversations and Kuang Yu for very helpful references.We also thank our JPC reviewers for a very careful reading of themanuscript and suggestions for improvement. This work wassupported by NSF CAREER Grant CHE-1150851. J.E.S. alsoacknowledges a Cottrell Research Fellowship, a Camille-HenryDrefyus award, and a David and Lucile Packard Fellowship.

■ REFERENCES(1) Dreuw, A.; Head-Gordon, M. Single Reference Ab Initio Methodsfor the Calculation of Excited States of Large Molecules. Chem. Rev.2005, 105, 4009.(2) Szabo, A.; Ostlund, N. S.Modern Quantum Chemistry: Introductionto Advanced Electronic Structure Theory; Dover Publications: Mineola,NY, 1996.(3) Ashcroft, N. W.; Holt, N. D. M. Solid State Physics; Rinehart andWinston: New York, 1976.

(4) Chakraborty, A.; Pak, M. V.; Hammes-Schiffer, S. Development ofElectron-Proton Density Functionals for Multicomponent DensityFunctional Theory. Phys. Rev. Lett. 2008, 101, 153001.(5) Elward, J. M.; Hoja, J.; Chakraborty, A. Variational Solution of theCongruently Transformed Hamiltonian for Many-Electron SystemsUsing a Full-Configuration-Interaction Calculation. Phys. Rev. A: At.,Mol., Opt. Phys. 2012, 86, 062504.(6) Chan, K. T.; Neaton, J. B.; Cohen, M. L. First-Principles Study ofMetal adatom Adsorption on Graphene. Phys. Rev. B: Condens. MatterMater. Phys. 2008, 77, 235430.(7) Refaely-Abramson, S.; Sharifzadeh, S.; Govind, N.; Autschbach, J.;Neaton, J. B.; Baer, R.; Kronik, L. Quasiparticle Spectra from aNonempirical Optimally Tuned Range-Separated Hybrid DensityFunctional. Phys. Rev. Lett. 2012, 109, 226405.(8) Sinanog ̌lu, O. Many-Electron Theory of Atoms and Molecules. I.Shells, Electron Pairs vs Many-Electron Correlations. J. Chem. Phys.1962, 36, 706−717.(9) Stollhoff, G.; Fulde, P. On the Computation of ElectronicCorrelation Energies within the Local Approach. J. Chem. Phys. 1980, 73,4548−4561.(10) Saebø, S.; Pulay, P. Fourth-Order Møller-Plessett PerturbationTheory in the Local Correlation Treatment. I. Method. J. Chem. Phys.1987, 86, 914−922.(11) Pardon, R.; Graf̈enstein, J.; Stollhoff, G. Ab initio Ground-StateCorrelation Calculations for Semiconductors with the Local Ansatz.Phys. Rev. B: Condens. Matter Mater. Phys. 1995, 51, 10556−10567.(12) Maslen, P.; Head-Gordon, M. Non-iterative Local Second OrderMøllerPlesset Theory. Chem. Phys. Lett. 1998, 283, 102−108.(13) Flocke, N.; Bartlett, R. J. A Natural Linear Scaling Coupled-Cluster Method. J. Chem. Phys. 2004, 121, 10935−10944.(14) Aryasetiawan, F.; Gunnarsson, O. The GW method. Rep. Prog.Phys. 1998, 61, 237.(15) Hedin, L. On Correlation Effects in Electron Spectroscopies andthe GW Approximation. J. Phys.: Condens. Matter 1999, 11, R489.(16) Rohlfing, M.; Louie, S. G. Electron-Hole Excitations and OpticalSpectra from First Principles. Phys. Rev. B: Condens. Matter Mater. Phys.2000, 62, 4927−4944.(17) Inkson, J. C.Many-Body Theory of Solids: An Introduction; PlenumPress: New York, 1984.(18) Fetter, A. L.; Walecka, J. D. Quantum Theory of Many-ParticleSystems; Dover Publications: Mineola, NY, 2003.(19) Hedin, L. New Method for Calculating the One-Particle Green’sFunction with Application to the Electron-Gas Problem. Phys. Rev.1965, 139, A796−A823.(20) Brinkman,W.; Goodman, B. Crystal Potential and Correlation forEnergy Bands in Valence Semiconductors. Phys. Rev. 1966, 149, 597−613.

Figure 8. Errors for electron affinities for 8-Site/8e−metallic system (U = 0.5) and molecular system (U = 0.5, Vμ =−2.0) with respect to the number ofsinglet states included in the calculation (eq 32). Including no RPA states is the Hartree result, and including the maximum number of states gives theGW result. Note that, on the left panel, including the first excited RPA state yields a pretty good result whereas, on the right panel, all RPA states need tobe included due to the slow convergence.



K

mailto:[email protected]


(21) Lipari, N. O.; Fowler,W. B. Effect of Electronic Correlation on theEnergy Bands of Insulating Crystals. Application to Argon. Phys. Rev. B1970, 2, 3354−3370.(22) Strinati, G.; Mattausch, H. J.; Hanke, W. Dynamical CorrelationEffects on theQuasiparticle Bloch States of a Covalent Crystal. Phys. Rev.Lett. 1980, 45, 290−294.(23) Strinati, G.; Mattausch, H. J.; Hanke, W. Dynamical Aspects ofCorrelation Corrections in a Covalent Crystal. Phys. Rev. B: Condens.Matter Mater. Phys. 1982, 25, 2867−2888.(24) Stan, A.; Dahlen, N. E.; van Leeuwen, R. Fully Self-ConsistentGW Calculations for Atoms and Molecules. http://arxiv.org/pdf/1512.04556v1.pdf, 2015.(25) Stan, A.; Dahlen, N. E.; van Leeuwen, R. Levels of Self-Consistency in the GW Approximation. J. Chem. Phys. 2009, 130,114105.(26) Bruneval, F. Ionization Energy of Atoms Obtained from GW Self-Energy or from Random Phase Approximation Total Energies. J. Chem.Phys. 2012, 136, 194107.(27) Bruneval, F.; Marques, M. A. L. Benchmarking the Starting Pointsof the GWApproximation for Molecules. J. Chem. Theory Comput. 2013,9, 324−329.(28) McClain, J.; Lischner, J.; Watson, T.; Matthews, D. A.; Ronca, E.;Louie, S. G.; Berkelbach, T. C.; Chan, G. K.-L. Spectral Functions of theUniform Electron Gas via Coupled-Cluster Theory and Comparison tothe GW and Related Approximations. http://arxiv.org/pdf/cond-mat/0610330.pdf 2006,.(29) Gutzwiller, M. C. Effect of Correlation on the Ferromagnetism ofTransition Metals. Phys. Rev. Lett. 1963, 10, 159−162.(30) Hybertsen, M. S.; Louie, S. G. Electron Correlation inSemiconductors and Insulators: Band Gaps and Quasiparticle Energies.Phys. Rev. B: Condens. Matter Mater. Phys. 1986, 34, 5390−5411.(31) Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L. Density-Functional Theory for Fractional Particle Number: DerivativeDiscontinuities of the Energy. Phys. Rev. Lett. 1982, 49, 1691−1694.(32) Golubeva, A. A.; Pieniazek, P. A.; Krylov, A. I. A New ElectronicStructure Method for Doublet States: Configuration Interaction in theSpace of Ionized 1h1h and 2h1p2h1pDeterminants. J. Chem. Phys. 2009,130, 124113.(33) Casida, M. In Recent Advances in Density Funtional Methods;Chong, D. P., Ed; World Scientific: Singapore, 1995; Vol. 1; pp 155−192.(34) Furche, F. On the Density Matrix based Approach to Time-Dependent Density Functional Response Theory. J. Chem. Phys. 2001,114, 5982−5992.(35) Davidson, E. R. Super-Matrix Methods. Comput. Phys. Commun.1989, 53, 49−60.(36) It is well-known that an infinite half-filled Hubbard chain is alwaysinsulating.45 Thus, formally, any “metallic” features of this Hubbardchain exist only because of the chain’s finite size. That being said, themain conclusions presented here continue to hold if we consider a chainwith less than half filling.(37) Bruneval, F.; Gonze, X. Accurate GW Self-Energies in a Plane-Wave Basis Using Only a Few Empty States: Towards Large Systems.Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 78, 085125.(38) Deslippe, J.; Samsonidze, G.; Jain, M.; Cohen, M. L.; Louie, S. G.Coulomb-Hole Summations and Energies for GW Calculations withLimited Number of Empty Orbitals: A Modified Static RemainderApproach. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87, 165124.(39) Steinbeck, L.; Rubio, A.; Reining, L.; Torrent, M.; White, I.;Godby, R. Enhancements to the GW space-time method. Comput. Phys.Commun. 2000, 125, 105−118.(40) Lischner, J.; Sharifzadeh, S.; Deslippe, J.; Neaton, J. B.; Louie, S. G.Effects of Self-Consistency and Plasmon-Pole Models on GWCalculations for Closed-Shell Molecules. Phys. Rev. B: Condens. MatterMater. Phys. 2014, 90, 115130.(41) Faber, C.; Boulanger, P.; Duchemin, I.; Attaccalite, C.; Blase, X.Many-Body Green’s Function GW and Bethe-Salpeter Study of theOptical Excitations in a Paradigmatic Model Dipeptide. J. Chem. Phys.2013, 139, 194308.

(42) Tamblyn, I.; Darancet, P.; Quek, S. Y.; Bonev, S. A.; Neaton, J. B.Electronic Energy Level Alignment at Metal-Molecule Interfaces with aGW Approach. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84,201402.(43) Rostgaard, C.; Jacobsen, K. W.; Thygesen, K. S. Fully Self-Consistent GW Calculations for Molecules. Phys. Rev. B: Condens.Matter Mater. Phys. 2010, 81, 085103.(44) Marom, N.; Ren, X.; Moussa, J. E.; Chelikowsky, J. R.; Kronik, L.Electronic Structure of Copper Phthalocyanine from G0W0 Calcu-lations. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 195143.(45) Lieb, E. H.; Wu, F. Y. Absence of Mott Transition in an ExactSolution of the Short-Range, One-BandModel in One Dimension. Phys.Rev. Lett. 1968, 20, 1445−1448.



L

http://arxiv.org/pdf/1512.04556v1.pdf

http://arxiv.org/pdf/1512.04556v1.pdf

http://arxiv.org/pdf/cond-mat/0610330.pdf

http://arxiv.org/pdf/cond-mat/0610330.pdf


comparison between gw and wave-function-based...

Documents