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ORIGINAL RESEARCH article

Front. Chem., 29 October 2021 | https://doi.org/10.3389/fchem.2021.751054

Scrutinizing GW-Based Methods Using the Hubbard Dimer

  • 1Laboratoire de Chimie et Physique Quantiques, Université de Toulouse, CNRS, UPS, Toulouse, France
  • 2Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS and ETSF, Toulouse, France

Using the simple (symmetric) Hubbard dimer, we analyze some important features of the GW approximation. We show that the problem of the existence of multiple quasiparticle solutions in the (perturbative) one-shot GW method and its partially self-consistent version is solved by full self-consistency. We also analyze the neutral excitation spectrum using the Bethe-Salpeter equation (BSE) formalism within the standard GW approximation and find, in particular, that 1) some neutral excitation energies become complex when the electron-electron interaction U increases, which can be traced back to the approximate nature of the GW quasiparticle energies; 2) the BSE formalism yields accurate correlation energies over a wide range of U when the trace (or plasmon) formula is employed; 3) the trace formula is sensitive to the occurrence of complex excitation energies (especially singlet), while the expression obtained from the adiabatic-connection fluctuation-dissipation theorem (ACFDT) is more stable (yet less accurate); 4) the trace formula has the correct behavior for weak (i.e., small U) interaction, unlike the ACFDT expression.

1 Introduction

Many-body perturbation theory (MBPT) based on Green’s functions is among the standard tools in condensed matter physics for the study of ground- and excited-state properties. (Aryasetiawan and Gunnarsson, 1998; Onida et al., 2002; Martin et al., 2016; Golze et al., 2019). In particular, the GW approximation (Hedin, 1965; Golze et al., 2019) has become the method of choice for band-structure and photoemission calculations and, combined with the Bethe-Salpeter equation (BSE@GW) formalism, (Salpeter and Bethe, 1951; Strinati, 1988; Albrecht et al., 1998; Rohlfing and Louie, 1998; Benedict et al., 1998; van der Horst et al., 1999a; Blase et al., 2018, 2020), for optical spectra calculations. Thanks to efficient implementations, (Duchemin and Blase, 2019, 2020, 2021; Bruneval et al., 2016; van Setten et al., 2013; Kaplan et al., 2015, 2016; Krause and Klopper, 2017; Caruso et al., 2012, 2013b,a; Caruso, 2013; Wilhelm et al., 2018), this toolkit is acquiring increasing popularity in the traditional quantum chemistry community, (Rohlfing and Louie, 1999; van der Horst et al., 1999b; Puschnig and Ambrosch-Draxl, 2002; Tiago et al., 2003; Boulanger et al., 2014; Jacquemin et al., 2015b; Bruneval et al., 2015; Jacquemin et al., 2015a; Hirose et al., 2015; Jacquemin et al., 2017a,b; Rangel et al., 2017; Krause and Klopper, 2017; Gui et al., 2018; Blase et al., 2018; Liu et al., 2020; Blase et al., 2020; Holzer and Klopper, 2018; Holzer et al., 2018; Loos et al., 2020), partially due to the similarity of the equation structure to that of the standard Hartree-Fock (HF) (Szabo and Ostlund, 1989) or Kohn-Sham (KS) (Hohenberg and Kohn, 1964; Kohn and Sham, 1965) mean-field methods. Several studies of the performance of various flavors of GW in atomic and molecular systems are now present in the literature, (Holm and von Barth, 1998; Stan et al., 2006; Stan et al., 2009; Blase and Attaccalite, 2011; Faber et al., 2011; Bruneval, 2012; Bruneval and Marques, 2013; Bruneval et al., 2015; Karlsson and van Leeuwen, 2016; Bruneval et al., 2016; Bruneval, 2016; Boulanger et al., 2014; Blase et al., 2016; Li et al., 2017; Hung et al., 2016, 2017; van Setten et al., 2015, 2018; Ou and Subotnik, 2016, 2018; Faber, 2014), providing a clearer picture of the pros and cons of this approach. There are, however, still some open issues, such as 1) how to overcome the problem of multiple quasiparticle solutions, (van Setten et al., 2015; Maggio et al., 2017; Loos et al., 2018; Véril et al., 2018; Duchemin and Blase, 2020; Loos et al., 2020), 2) what is the best way to calculate ground-state total energies, (Casida, 2005; Huix-Rotllant et al., 2011; Caruso et al., 2013b; Casida and Huix-Rotllant, 2016; Colonna et al., 2014; Olsen and Thygesen, 2014; Hellgren et al., 2015; Holzer et al., 2018; Li et al., 2019, 2020; Loos et al., 2020), and 3) what are the limits of the BSE in the simplification commonly used in the so-called Casida equations. (Strinati, 1988; Rohlfing and Louie, 2000; Sottile et al., 2003; Myöhänen et al., 2008; Ma et al., 2009a,b; Romaniello et al., 2009b; Sangalli et al., 2011; Huix-Rotllant et al., 2011; Sakkinen et al., 2012; Zhang et al., 2013; Rebolini and Toulouse, 2016; Olevano et al., 2019; Lettmann and Rohlfing, 2019; Loos and Blase, 2020; Authier and Loos, 2020; Monino and Loos, 2021). In the present work, we address precisely these questions by using a very simple and exactly solvable model, the symmetric Hubbard dimer. Small Hubbard clusters are widely used test systems for the GW approximation (e.g. Verdozzi et al., 1995; Schindlmayr et al., 1998; Pollehn et al., 1998; Puig von Friesen et al., 2010; Romaniello et al., 2009a, 2012). Despite its simplicity, the Hubbard dimer is able to capture lots of the underlying physics observed in more realistic systems, (Romaniello et al., 2009a, 2012; Carrascal et al., 2015, 2018), such as, for example, the nature of the band-gap opening in strongly correlated systems as bulk NiO. (Di Sabatino et al., 2016). Here, we will use it to better understand some features of the GW approximation and the BSE@GW approach. Of course, care must be taken when extrapolating conclusions to realistic systems.

The paper is organized as follows. Section 2 provides the key equations employed in MBPT to calculate removal and addition energies (or charged excitations), neutral (or optical) excitation energies, and ground-state correlation energies. In Sec. 3, we present and discuss the results that we have obtained for the Hubbard dimer. We finally draw conclusions and perspectives in Sec. 4.

2 Theoretical Framework

In the following we provide the key equations of MBPT (Martin et al., 2016) and, in particular, we discuss how one can calculate ground- and excited-state properties, namely removal and addition energies, spectral function, total energies, and neutral excitation energies. We use atomic units = m = e = 1 and work at zero temperature throughout the paper.

2.1 The GW Approximation

Within MBPT a prominent role is played by the one-body Green’s function G which has the following spectral representation in the frequency domain:

G(x1,x2;ω)=νψν(x1)ψν*(x2)ωϵν+iηsgn(ϵνμ),(1)

where μ is the chemical potential, η is a positive infinitesimal, ϵν=EνN+1E0N for εν > μ, and ϵν=E0NEiN1 for εν < μ. Here, EνN is the total energy of the νth excited state of the N-electron system (ν = 0 being the ground state). In the case of single-determinant many-body wave functions (such as HF or KS), the so-called Lehmann amplitudes ψν (x) reduce to one-body orbitals and the poles of the Green’s function εν to one-body orbital energies.

The one-body Green’s function is a powerful quantity that contains a wealth of information about the physical system. In particular, as readily seen from Eq. 1, it has poles at the charged excitation energies of the system, which are proper addition/removal energies of the N-electron system. Thus, one can also access the (photoemission) fundamental gap

Eg=INAN,(2)

where IN=E0N1E0N is the ionization potential and AN=E0NE0N+1 is the electron affinity. Moreover, one can straightforwardly obtain the spectral function, which is closely related to photoemission spectra, as

A(x1,x2;ω)=1πsgn(μω)ImG(x1,x2;ω).(3)

The ground-state total energy can also be extracted from G using the Galitskii-Migdal (GM) formula (Galitskii and Migdal, 1958)

E0GM=i2dx1lim21+it1+h(r1)G(1,2),(4)

where 1 ≡ (x1, t1) is a space-spin plus time composite variable and h(r) = − ∇/2 + vext(r) is the one-body Hamiltonian, vext (r) being the local external potential.

The one-body Green’s function can be obtained by solving a Dyson equation of the form G = G0 + G0ΣG, where G0 is the non-interacting Green’s function and the self-energy Σ is an effective potential which contains all the many-body effects of the system under study. In practice, Σ must be approximated and a well-known approximation is the so-called GW approximation in which the self-energy reads ΣGW = vH + iGW, where vH is the classical Hartree potential, and W = ɛ−1vc is the dynamically screened Coulomb interaction, with ɛ−1 the inverse dielectric function and vc the bare Coulomb interaction. (Hedin, 1965).

The equations stemming from the GW approximation should, in principle, be solved self-consistently, since Σ is a functional of G. (Hedin, 1965). Self-consistency, however, is computationally demanding, and one often performs a single GW correction (for example using G0 as starting point one builds W and ΣGW as ΣGW = vH + iG0W0, with vH = − ivcG0 and W0=[1+ivcG0G0]1vc, from which G={1G0ΣGW[G0]}1G0). This cost-saving and popular strategy is known as one-shot GW. The main drawback of the one-shot GW method is its dependence on the starting point (i.e., the orbitals and energies of the HF or KS mean-field eigenstates) originating from its perturbative nature. To overcome this problem, one can introduce some level of self-consistency. Removal/addition energies are thus obtained by solving iteratively the so-called quasiparticle equation

ω=ϵiHF+ϕiHFΣcGW(ω)ϕiHF.(5)

Here, we choose to start from HF spatial orbitals ϕiHF(r) and energies ϵiHF, which are corrected by the (real part of the) correlation contribution of the GW self-energy ΣcGW=ΣGWΣHF, where ΣHF = vH + ivcG is the HF (hartree plus exchange) contribution to the self-energy. ΣcGW is evaluated with GHF at the first iteration, where GHF is the self-consistent solution of GHF = G0 + G0ΣHFGHF. At the nth iteration, ΣcGW is evaluated as ΣcGW[Gn1], where Gn−1 has poles at the energies from the (n − 1)-th iteration of Eq. 5 and corresponding weights obtained from the Z factors given in Eq. 6. As a non-linear equation, Eq. 5 has potentially many solutions ϵi,νGW. The so-called quasiparticle (QP) solution ϵi,ν=0GWϵiQP has the largest renormalization factor (or spectral intensity)

Zi,ν=1ϕiHFΣcGW(ω)ωϕiHFω=ϵi,νGW1,(6)

while the satellite (sat) peaks ϵi,ν>0GWϵi,νsat share the remaining of the spectral weight. Moreover, one can show that the following sum rule is fulfilled (von Barth and Holm, 1996)

νZi,ν=1,(7)

where the sum runs over all the solutions of the quasiparticle equation for a given mean-field eigenstate i. Throughout this article, i, j, k, and l denote general spatial orbitals, a and b refer to occupied orbitals, r and s to unoccupied orbitals, while m labels single excitations ar.

In eigenvalue self-consistent GW (commonly abbreviated as evGW), (Hybertsen and Louie, 1986; Shishkin and Kresse, 2007; Blase and Attaccalite, 2011; Faber et al., 2011; Rangel et al., 2016; Gui et al., 2018), one only updates the poles of G, while keeping fix the orbitals (or weights). G is then used to build ΣGW and W. At the nth iteration, the removal/addition energies are obtained from the GW quasiparticle solutions computed from Gn−1W (Gn−1) where the satellites are discarded at each iteration. Nonetheless, at the final iteration one can keep the satellite energies to get the full spectral function (Eq. 3). In fully self-consistent GW (scGW), (Caruso et al., 2012, 2013b,a; Caruso, 2013; Koval et al., 2014), one updates the poles and weights of G retaining quasiparticle and satellite energies at each iteration.

It is instructive to mention that, for a conserving approximation, the sum of the intensities corresponding to removal energies equals the number of electrons, i.e., ϵi,νGW<μZi,ν=N. scGW is an example of conserving approximations, while, in general, the one-shot GW does not conserve the number of electrons.

2.2 Bethe-Salpeter Equation

2.2.1 Neutral Excitations

Linear response theory (Oddershede and Jorgensen, 1977; Casida, 1995; Petersilka et al., 1996) in MBPT is described by the Bethe-Salpeter equation. (Strinati, 1988). The standard BSE within the static GW approximation (referred to as BSE@GW in this work, which means the use of GW quasiparticle energies to build the independent-particle excitation energies and of the GW self-energy to build the static exchange-correlation kernel) can be recast, assuming a closed-shell reference state, as a non-Hermitian eigenvalue problem known as Casida equations:

AλBλBλAλXmλYmλ=ΩmλXmλYmλ,(8)

where Ωmλ is the mth excitation energy with eigenvector (XmλYmλ) at interaction strength λ, is the matrix transpose, and we have assumed real-valued spatial orbitals. The non-interacting and physical systems correspond to λ = 0 and 1, respectively. The matrices Aλ and Bλ are of size OV × OV, where O and V are the number of occupied and virtual orbitals, respectively, and O+ V is the total number of spatial orbitals. Introducing the so-called Mulliken notation for the bare two-electron integrals

(ij|kl)=dr1dr2ϕi(r1)ϕj(r1)vc(r1r2)ϕk(r2)ϕl(r2),(9)

and the corresponding (static) screened Coulomb potential matrix elements

Wij,kl(ω=0)=dr1dr2ϕi(r1)ϕj(r1)W(r1,r2;ω=0)ϕk(r2)ϕl(r2),(10)

the BSE matrix elements read (Maggio and Kresse, 2016).

Aar,bsλ,σσ=δabδrs(ϵrQPϵaQP)+λασσ(ar|sb)Wab,sr(ω=0),(11a)
Bar,bsλ,σσ=λασσ(ar|bs)Was,br(ω=0),(11b)

where ϵiQP are the GW quasiparticle energies, and α↑↓ = 2 and α↑↑ = 0 for singlet (i.e., spin-conserved) and triplet (i.e., spin-flip) excitations, respectively.

In the absence of instabilities (i.e., when AλBλ is positive-definite), (Dreuw and Head-Gordon, 2005), Eq. 8 is usually transformed into an Hermitian eigenvalue problem of half the dimension

(AλBλ)1/2(Aλ+Bλ)(AλBλ)1/2Vmλ=(Ωmλ)2Vmλ,(12)

where the excitation amplitudes are

(Xλ+Yλ)m=(Ωmλ)1/2(AλBλ)+1/2Vmλ,(13a)
(XλYλ)m=(Ωmλ)+1/2(AλBλ)1/2Vmλ.(13b)

Singlet (ΩmΩmλ=1,) and triplet (ΩmΩmλ=1,) excitation energies are obtained by diagonalizing Eq. 8 at λ = 1.

2.2.2 Correlation Energies

Our goal here is to compare the BSE correlation energy EcBSE obtained using two formulas, namely the trace (or plasmon) formula (Rowe, 1968; Ring and Schuck, 1980) and the expression obtained using the adiabatic-connection fluctuation-dissipation theorem (ACFDT) formalism. (Furche and Van Voorhis, 2005; Toulouse et al., 2009, 2010; Hellgren and von Barth, 2010; Angyan et al., 2011; Heßelmann and Görling, 2011; Colonna et al., 2014; Maggio and Kresse, 2016; Holzer et al., 2018; Loos et al., 2020). The two approaches have been recently compared at the random-phase approximation (RPA) level for the case of Be2, (Li et al., 2020), showing similar improved performances at the RPA@GW@PBE level with respect to the RPA@PBE level and an impressive accuracy by introducing BSE (BSE@GW@HF) correction in the trace formula. Here we would like to get more insights into the quality of these two approaches.

The ground-state correlation energy within the trace formula is calculated as

EcTr@BSE=Ec,Tr@BSE+Ec,Tr@BSE=12mΩmTr(A)+12mΩmTr(A),(14)

where AσσAλ=1,σσ is defined in Eq.11a and   Tr denotes the matrix trace. We note that the trace formula is an approximate expression of the correlation energy since it relies on the so-called quasi-boson approximation and on the killing condition on the zeroth-order Slater determinant ground state (Li et al., 2020 for more details). Note that here both sums in Eq. 14 run over all resonant (hence real- and complex-valued) excitation energies while they are usually restricted to the real-valued resonant BSE excitation energies. Thus, the Tr@BSE correlation energy is potentially a complex-valued function in the presence of singlet and/or triplet instabilities.

The ACFDT formalism, (Furche and Van Voorhis, 2005), instead, provides an in-principle exact expression for the correlation energy within time-dependent density-functional theory (TDDFT). (Runge and Gross, 1984; Petersilka et al., 1996; Ullrich, 2012). In practice, however, one always ends up with an approximate expression, which quality relies on the approximations to the exchange-correlation potential of the KS system and to the kernel of the TDDFT linear response equations. In this work, therefore, we use the ACFDT expression within the BSE formalism and we explore how well it performs and how it compares to the trace Eq. 14.

Within the ACFDT framework, only the singlet states do contribute for a closed-shell ground state, and the ground-state BSE correlation energy

EcAC@BSE=1201dλTr(KPλ,)(15)

is obtained via integration along the adiabatic connection path from the non-interacting system at λ = 0 to the physical system λ = 1, where

K=Ãλ=1Bλ=1Bλ=1Ãλ=1(16)

is the interaction kernel, (Angyan et al., 2011; Holzer et al., 2018; Loos et al., 2020) Ãar,bsλ,σσ=ασσλ(ar|sb), and

Pλ=Yλ(Yλ)Yλ(Xλ)Xλ(Yλ)Xλ(Xλ)0001(17)

is the correlation part of the two-body density matrix at interaction strength λ. Here again, the AC@BSE correlation energy might become complex-valued in the presence of singlet instabilities.

Note that the trace and ACFDT formulas yield, for any set of eigenstates, the same correlation energy at the RPA level. (Angyan et al., 2011). Moreover, in contrast to density-functional theory where the electron density is fixed along the adiabatic path, (Langreth and Perdew, 1979; Gunnarsson and Lundqvist, 1976; Zhang and Burke, 2004), at the BSE@GW level, the density is not maintained as λ varies. Therefore, an additional contribution to Eq. 15 originating from the variation of the Green’s function along the adiabatic connection should, in principle, be added. However, as commonly done within RPA (Toulouse et al., 2009, 2010; Angyan et al., 2011; Colonna et al., 2014) and BSE, (Holzer et al., 2018; Loos et al., 2020), we neglect this additional contribution.

3 Results

As discussed in Sec. 1, in this work, we consider the (symmetric) Hubbard dimer as test case, which is governed by the following Hamiltonian

Ĥ=tσ=,ĉ1σĉ2σ+ĉ2σĉ1σ+Un̂1n̂1+n̂2n̂2.(18)

Here n̂1σ=ĉ1σĉ1σ (n̂2σ=ĉ2σĉ2σ) is the spin density operator on site 1 (site 2), ĉ1σ and ĉ1σ (ĉ2σ and ĉ2σ) are the creation and annihilation operators for an electron at site 1 (site 2) with spin σ, U is the on-site (spin-independent) interaction, and − t is the hopping kinetic energy. The physics of the Hubbard model arises from the competition between the hopping term, which prefers to delocalize electrons, and the on-site interaction, which favors localization. The ratio U/t is a measure for the relative contribution of both terms and is the intrinsic, dimensionless coupling constant of the Hubbard model, which we use in the following. In this work we consider the dimer at one-half filling.

3.1 Quasiparticle Energies in the GW Approximation

We test different flavors of self-consistency in GW calculations: one-shot GW, evGW, partial self-consistency through the alignment of the chemical potential (pscGW), where we shift G0 or GHF in such a way that the resulting G has the same chemical potential than the shifted G0 or shifted GHF, (Schindlmayr, 1997), and scGW. In the one-shot formalism, we also test two different starting points: the truly non-interacting Green’s function G0 (U = 0) and the HF Green’s function GHF. These two schemes are respectively labeled as G0W0 and GHFWHF in the following.

The G0W0 self-energy (in the site basis) and removal/addition energies are already given in Ref. (Romaniello et al., 2012) for the Hubbard dimer at one-half filling. For completeness we report them in Supplementary Appendix S1, together with the renormalization factors, which are discussed in Sec. 3.1.1.

Starting from GHF, which reads

GHF,IJ(ω)=12(1)(IJ)ω(t+U/2)+iη+1ω+(tU/2)iη,(19)

where I and J run over the sites, the (correlation part of the) GHFWHF self-energy is Σc,IJGW(ω)=ΣIJGW(ω)δIJU/2 with

Σc,IJGW(ω)=U2t2h1ω(t+h+U/2)+iη+(1)IJω+(t+hU/2)iη,(20)

where h=4t2+4Ut. Here we used the following expression for the polarizability P = − iGG with elements

PIJ(ω)=(1)IJ41ω2t+iη1ω+2tiη(21)

to build the screened interaction W = vc + vcPW, whose only non-zero matrix elements read

WII,JJ(ω)=UδIJ+(1)IJU2th1ωh+iη1ω+hiη(22)

due to the local nature of the electron-electron interaction. The quantities defined in Eqs 1922 can then be transformed to the bonding (bn) and antibonding (an) basis (which is used to recast the BSE as Eq. 8) thanks to the following expressions:

bn=1+22,an=122.(23)

Therefore, the one-shot removal/addition energies read

ϵ1,±=+h2+U2±(h+2t)2+4tU2/h2,(24a)
ϵ2,±=h2+U2±(h+2t)2+4tU2/h2,(24b)

with the quasiparticle solutions being ϵbnQP=ϵ1, and ϵanQP=ϵ2,+, which correspond to the bonding and antibonding energies, respectively. As readily seen in Eqs 24a, 24b, in addition to the quasiparticle, there is a unique satellite per eigenstate given by ϵbnsat=ϵ1,+ and ϵansat=ϵ2,. Moreover, the closed-form expression of the renormalization factors (Eq. 6) reads

Zbn/anQP=th2+2ht+2U2+h(h+2t)2+4tU2/hh3+4h2t+4ht2+4tU2h2(h+2t)2+4tU2/h(25)

and Zbn/ansat=1Zbn/anQP.

The evGW and scGW calculations were performed numerically using the meromorphic representation of G, following Ref. (Puig von Friesen et al., 2010) with some slight modifications (Supplementary Appendix S2 for more details). At each iteration, the solution of the Dyson equations for G and W (Sec. 2.1) produces extra poles. In order to keep the number of poles under control in scGW, the poles with intensities smaller than a user-defined threshold (set from 10–4 to 10–6 depending on the ratio U/t) are discarded and the corresponding spectral weight is redistributed among the remaining poles.

In Figure 1, we present the spectral function of G (Eq. 3) for different values of the ratio U/t (U/t = 1, 5, 10, and 15) and using GHF as starting point. We consider three GW variants: GHFWHF, evGW, and scGW. For U/t ≲ 3, all the schemes considered here provide a faithful description of the quasiparticle energies. For larger U/t, GW (regardless of the level of self-consistency) tends to underestimate the fundamental gap Eg (Eq. 2), as shown in the upper left panel of Figure 2. GHFWHF and evGW give a very similar estimate of Eg, whereas the quasiparticle intensity Zbn/anQP defined in Eq. 25 is quite different and overestimated by both methods, at least in the range of U/t considered in Figure 2 (center left panel).

FIGURE 1
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FIGURE 1. Spectral function of G (Eq. 3) as a function of (ωμ)/t (where μ = U/2 is the chemical potential) at various values of the ratio U/t (U/t = 1, 5, 10, and 15) for different levels of theory: exact (black), GHFWHF (red), evGW (blue), and scGW (green). All approximate schemes are obtained using GHF as starting point.

FIGURE 2
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FIGURE 2. Fundamental gap (Eg), quasiparticle weight factors (Zbn/anQP), and ground state energy (E0) as functions of U/t obtained from one-shot GW (dashed red line), evGW (dashed-dotted blue line), scGW (dotted green line) using GHF (left) or G0 (right) as starting point. The black curves are the exact results.

The main effects of full self-consistency are the reduction of Eg (see upper left panel of Figure 2), and the creation of extra satellites with decreasing intensity (see upper panel of Figure 1). For small U/t, the fundamental gap is similar to the one predicted by other methods while for increasing U/t the agreement worsen and Eg is grossly underestimated. The quasiparticle intensity is very similar to the one predicted by GHFWHF. Concerning the position of the satellites, we observe that the one-shot GHFWHF scheme gives the most promising results. Numerical values of quasiparticle and first satellite energies as well as their respective intensities in the spectral functions presented in Figure 1 are gathered in Table 1.

TABLE 1
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TABLE 1. Numerical values of quasiparticle energy ϵanQP and satellite energy ϵansat (anti-bonding components) and respective intensities (ZanQP and Zansat) for the spectral functions presented in Figure 1. Energies are relative to the chemical potential μ = U/2. All spectral functions presented in Figure 1 are symmetric with respect to μ, which means that ϵbnQP/sat=ϵanQP/sat and ZbnQP/sat=ZanQP/sat.

We notice that a similar analysis for H2 in a minimal basis has been presented in Ref. (Hellgren et al., 2015) with analogous conclusions.

For the sake of completeness, we also report in the bottom left panel of Figure 2 the total energy calculated using the Galitskii-Migdal formula (Eq. 4). Since the Galitskii-Migdal total energy is not stationary with respect to changes in G, one gets meaningful energies only at self-consistency. However, for the Hubbard dimer, we do not observe a significant impact of self-consistency, as one can see from Figure 1 by comparing the total energy at the GHFWHF, evGW, and scGW levels. For each of these schemes which correspond to a different level of self-consistency, the Galitskii-Migdal formula provides accurate total energies only for relatively small U/t (≲ 3).

If we consider GHF as starting point and we define the chemical potential as μ=(ϵanQP+ϵbnQP)/2, then the alignment of the chemical potential has no effect on the spectrum, this means that GHFWHF and pscGW are equivalent.

3.1.1 G0: A Bad Starting Point

In the following we will illustrate how the starting point can influence the resulting quasiparticle energies. The Green’s function obtained from the one-shot G0W0 does not satisfy particle-hole symmetry, the fundamental gap is underestimated (top right panel of Figure 2) yet more accurate than GHFWHF (top left panel of Figure 2), the quasiparticle intensity relative to the bonding component is close to the exact result up to U/t ≈ 16 (center right panel of Figure 2), while overestimated for the antibonding components. Moreover, we note that the intensities of the two poles of the bonding component crosses at U/t = 24. This means that if we sort the quasiparticle and the satellite according to their intensity at a given U/t, the nature of the two poles is interchanged when one increases U/t, which results in a discontinuity in the QP energy. Meanwhile, the total number of particle is not conserved (N < 2). For G0W0 we found a small deviation from N = 2 for small U/t (e.g. N = 1.98828 at U = 1), which becomes larger by increasing the interaction (e.g. N = 1.55485 for U/t = 10). Instead, starting from GHF the particle number is always conserved. We checked that for the self-consistent calculations the total particle number is conserved, as it should.

Considering G0 as starting point in evGW, we encounter the problem described in Ref. (Véril et al., 2018), namely the discontinuity of various key properties (such as the fundamental gap in the top right panel of Figure 2) with respect to the interaction strength U/t. This issue is solved, for the Hubbard dimer, by considering a better starting point or using the fully self-consistent scheme scGW. Note, however, that improving the starting point does not always cure the discontinuity problem as this issue stems from the quasiparticle approximation itself. Full self-consistency, instead, avoids systematically discontinuities since no distinction is made between quasiparticle and satellites. Unfortunately, full self-consistency is much more involved from a computational point of view and, moreover, it does not give an overall improvement of the various properties of interest, at least for the Hubbard dimer, for which GHFWHF is to be preferred. For more realistic (molecular) systems, it was shown in Ref. (Berger et al., 2020). that the computationally cheaper self-consistent COHSEX scheme solves the problem of multiple quasiparticle solutions.

3.2 Bethe-Salpeter Equation

For the Hubbard dimer the matrices Aλ and Bλ in Eq. (8) are just single matrix elements and they simply read, for both spin manifolds,

Aλ,=ΔϵGW+λU2,Bλ,=λU24tUh2+1,(26a)
Aλ,=ΔϵGWλU2,Bλ,=λU24tUh21,(26b)

while Ãλ,=λU. We employ the screened Coulomb potential given in Eq. 22 at ω = 0 for the kernel, and the GW quasiparticle energies from Eqs 24a and 24b to build the GW approximation of the fundamental gap ΔϵGW=ϵanQPϵbnQP. For comparison purposes, we also use the exact quasiparticle energies [see Eq. (C3) of Ref. (Romaniello et al., 2012).], which consists in replacing ΔεGW by the exact fundamental gap Eg=16t2+U22t. In such a case, one is able to specifically test how accurate the BSE formalism is at catching the excitonic effect via the introduction of the screened Coulomb potential.

We notice that, within the so-called Tamm-Dancoff approximation (TDA) where one neglects the coupling matrix Bλ between the resonant and anti-resonant parts of the BSE Hamiltonian (Eq. 8), BSE yields RPA with exchange (RPAx) excitation energies for the Hubbard dimer. This is the case also for approximations to the BSE kernel which are beyond GW, such as the T-matrix approximation. (Romaniello et al., 2012; Zhang et al., 2017; Li et al., 2021), and it is again related to the local nature of the electron-electron interaction. Hence, to test the effect of approximations on correlation for this model system we must go beyond the TDA.

3.2.1 Neutral Excitations

In Figure 3, we report the real part of the singlet and triplet excitation energies obtained from the solution of Eq. 8 for λ = 1. For comparison, we report also the exact excitation energies obtained as differences of the excited- and ground-state total energies of the Hubbard dimer obtained by diagonalizing the Hamiltonian (18) in the Slater determinant basis {1,1,1,2,1,2,2,2} built from the sites [Ref. (Romaniello et al., 2009a) for the exact total energies]. For the singlet manifold, this yields, for the single excitation Ω1 and double excitation Ω2, the following expressions:

Ω1=12U+16t2+U2,Ω2=16t2+U2,(27)

while the unique triplet transition energy is

Ω1=12U+16t2+U2.(28)

FIGURE 3
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FIGURE 3. Real and imaginary parts of the singlet (solid) and triplet (dotted) neutral excitations, Ω1 and Ω1, as functions of U/t: exact (black), BSE with exact quasiparticle energies and WHF (gray), BSE@GHFWHF (red).

Of course, one cannot access the double excitation within the static approximation of BSE, (Strinati, 1988; Romaniello et al., 2009b; Loos and Blase, 2020), so only the lowest singlet and triplet excitations, Ω1and Ω1, are studied below.

Using one-shot GHFWHF quasiparticle energies (BSE@GHFWHF) produces complex excitation energies (see right panel of Figure 3). We find the same scenario also with other flavors of GW (not reported in the figure), such as scGW. The occurrence of complex poles and singlet/triplet instabilities at the BSE level are well documented (Holzer et al., 2018; Blase et al., 2020; Loos et al., 2020) and is not specific to the Hubbard dimer. For example, one finds complex poles also for H2 along its dissociation path, (Li and Olevano, 2021), but also for larger diatomic molecules. (Loos et al., 2020). For U/t > 12.4794, the singlet energy becomes pure imaginary, the same is observed for the triplet energy for 7.3524 < U/t < 12.4794. These two points corresponds to discontinuities in the first derivative of the excitation energies with respect to U/t (Figure 3). The BSE excitation energies are good approximations to their exact analogs only for U/t ≲ 2 for the singlet and U/t ≲ 6 for the triplet. Using exact quasiparticle energies instead produces real excitation energies, with the singlet energy in very good agreement with the exact result; the triplet energy, instead, largely overestimates the exact value. This seems to suggest that complex poles are caused by the approximate nature of the GW quasiparticle energies, although, of course, the quality of the kernel also plays a role. Indeed, setting W = 0 but using GW QP energies, BSE yields real-valued excitation energies. It would be interesting to further investigate this issue by using the exact kernel together with GW QP energies. This is left for future work.

3.2.2 Correlation Energy

For the Hubbard dimer, we have EHF = − 2t + U/2, and the correlation energy given in Eq. 15 can be calculated analytically. After a lengthy but simple derivation, one gets

EcAC@BSE=U2+t22U22U(2t+3U){ΔϵGW12(t+U)[U2+2(t+U)ΔϵGW]U(2t+3U)+2(t+U)ΔϵGW}t+2U2U(2t+3U)3t+4U2t+3U+tUΔϵGWatanUU(2t+3U)2ΔϵGW(t+U)+[U2+2(t+U)ΔϵGW][U(2t+3U)+2(t+U)ΔϵGW].

Results are reported in Figure 4 and are compared with the exact correlation energy (Romaniello et al., 2009a)

Ec=16t2+U22+2t.(29)

FIGURE 4
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FIGURE 4. Real and imaginary parts of the BSE@GHFWHF correlation energy as a function of U/t at various levels of theory: total (dotted blue line) and singlet-only (dashed green line) Tr@BSE, AC@BSE (dot-dashed magenta line), RPA (triple-dotted orange line), GM (double-dot-dashed red line), and exact (solid black line). For comparison also the BSE@exact (Tr@BSE, double-dotted dark grey line; AC@BSE, dot-dashed light grey line) correlation energies are shown. Discontinuities in the first derivative of the energy (corresponding to the appearance of complex poles) are indicated by open circles.

The AC@BSE correlation energy does not possess the correct asymptotic behavior for small U, as Taylor expanding Eq. 29 for small U, we obtain

EcAC@BSE=U232t5U396t2+323U46144t3+O(U4),(30)

while the exact correlation energy behaves as

Ec=U216t+U41024t3+O(U6).(31)

Moreover, we found that the radius of convergence of the small-U/t expansion of EcAC@BSE is very small due to a square-root branch point for U/t ≈ − 2/3.

In the case of the trace formula Eq. 14, the singlet and triplet contributions behave as

Ec,Tr@BSE=U232t7U3128t2+99U42048t3+O(U5),(32a)
Ec,Tr@BSE=U232t+7U3128t2157U42048t3+O(U5),(32b)

which guarantees the correct asymptotic behavior for the total Tr@BSE correlation energy

EcTr@BSE=U216t29U41024t3+O(U5),(33)

and cancels the cubic term (as it should).

The trace formula is strongly affected by the appearance of the imaginary excitation energies: as shown in Figure 4 where we plot the real and complex components of the BSE@GHFWHF correlation energy as functions of U/t at various levels of theory, irregularities (i.e., discontinuities in the first derivative of the energy) appear at the values of U/t for which the triplet and singlet energies become purely imaginary. The ACFDT expression, instead, is more stable over the range of U/t considered here with only a small cusp on the energy surface at the singlet instability point after which the real part of EcAC@BSE behaves linearly with respect to U/t. Overall, however, the correlation energy obtained by the trace formula is almost on top of its exact counterpart over a wide range of U/t, with a rather small contribution from the triplet component, i.e., Ec,Tr@BSEEc,Tr@BSE. For comparison purposes, the RPA correlation energy, which is obtained from the trace or ACDFT formula using BSE@GHFWHF with W = 0 in the BSE kernel, is also reported in Figure 4. Both formulas yield the same correlation energies as expected, and they show no irregularities thanks to the fact that BSE excitation energies are real-valued at the RPA level. Also correlation energies obtained using BSE@exact (also shown in Figure 4) do not show irregularities for the same reason. Moreover, they show a visible upshift with respect to the corresponding AC@BSE@GHFWHF and Tr@BSE@GHFWHF results, which worsens the agreement with the exact correlation energy. Finally, we observe that both expressions for the correlation energy (at BSE@GW level) produce better results than the Galitskii-Migdal Eq. 4, as one can see from Figure 4, in particular at large U/t.

4 Conclusion

In this work we have used the symmetric Hubbard dimer to better understand some features of the GW approximation and of BSE@GW. In particular, we have found that the unphysical discontinuities that may occur in quasiparticle energies computed using one-shot or partially self-consistent GW schemes disappear using full self-consistency. However, full self-consistency does not give an overall improvement in term of accuracy and, at least for the Hubbard dimer, GHFWHF is to be preferred.

We have also analyzed the performance of the BSE@GW approach for neutral excitations and correlation energies. We have found that, at any level of self-consistency, the excitation energies become complex for some critical values of U/t. This seems related to the approximate nature of the GW quasiparticle energies, since using exact quasiparticle energies (hence the exact fundamental gap) solves this issue. The BSE excitation energies are good approximations to the exact analogs only for a small range of U/t (or U/t ≲ 2 for the lowest singlet-singlet transition and U/t ≲ 6 for the singlet-triplet transition), while the strong-correlation regime remains a challenge.

The correlation energy obtained from these excitation energies using the trace (or plasmon) formula has been found to be in very good agreement with the exact results over the whole range of U/t for which these energies are real. The occurrence of complex singlet and triplet excitation energies shows up as irregularities in the correlation energy. The ACFDT formula, instead, is less sensitive to this. However, we have found that the AC@BSE correlation energy is less accurate than the one obtained using the trace formula. Both, however, perform better than the standard Galitskii-Migdal formula. Finally, we have studied the small-U expansion of the correlation energy obtained with the trace and ACFDT formulas and we found that the former, contrary to the latter, has the correct behavior when one includes both the singlet and triplet energy contributions. Our findings point out to a possible fundamental problem of the AC@BSE formalism.

Although our study is restricted to the half-filled Hubbard dimer, some of our findings are transferable to realistic (molecular) systems. In particular: 1) a fully self-consistent solution of the GW equation cures the problem of multiple QP solutions, avoiding in the process the appearance of discontinuities in key physical quantities such as total or excitation energies, ionization potentials, and electron affinities; 2) a “bad” starting point (G0 in the case of the Hubbard dimer) may result in the appearence of multiple QP solutions; 3) potential energy surfaces computed with the trace formula and within the ACFDT formalism may exhibit irregularities due to the appearence of complex BSE excitation energies; 4) for the Hubbard dimer at half-filling, the trace formula has the correct asymptotic behavior (thanks to the inclusion of singlet and triplet excitation energies) for weak interaction, contrary to its ACFDT counterpart. It would be interesting to check if it is also the case in realistic systems.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author Contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Funding

This study has been partially supported through the EUR grant NanoX no ANR-17-EURE-0009 in the framework of the “Programme des Investissements d’Avenir” and by the CNRS through the 80|Prime program. PR and SDS also thank the ANR (project ANR-18-CE30-0025) for financial support. PFL also thanks the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. ∼863481) for financial support.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fchem.2021.751054/full#supplementary-material

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Keywords: hubbard dimer, multiple quasiparticle solutions, GW, bethe-salpter equation, trace formula, adiabatic-connection fluctuation-dissipation theorem

Citation: Di Sabatino S, Loos P-F and Romaniello P (2021) Scrutinizing GW-Based Methods Using the Hubbard Dimer. Front. Chem. 9:751054. doi: 10.3389/fchem.2021.751054

Received: 31 July 2021; Accepted: 28 September 2021;
Published: 29 October 2021.

Edited by:

Patrick Rinke, Aalto University, Finland

Reviewed by:

Gianluca Stefanucci, University of Rome Tor Vergata, Italy
Maria Hellgren, Sorbonne Universités, France

Copyright © 2021 Di Sabatino, Loos and Romaniello. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: S. Di Sabatino , disabatino@irsamc.ups-tlse.fr