We describe the relationship between the GW approximation and various equation-of-motion (EOM) coupled-cluster (CC) theories. We demonstrate the exact equivalence of the G_{0}W_{0} approximation and the propagator theory for an electron–boson problem in a particular excitation basis. From there, we establish equivalence within the quasi-boson picture to the IP+EA-EOM unitary CC propagator. We analyze the incomplete description of screening provided by the standard similarity-transformed IP+EA-EOM-CC and the recently introduced G_{0}W_{0} Tamm–Dancoff approximation. We further consider the approximate decoupling of IP and EA sectors in EOM-CC treatments and devise the analogous particle–hole decoupling approach for the G_{0}W_{0} approximation. Finally, we numerically demonstrate the exact relationships and magnitude of the approximations in the calculations of a set of molecular ionization potentials and electron affinities.

## I. INTRODUCTION

The GW approximation, an approximation to the electron self-energy, is widely used to compute quasiparticle electron energies beyond the level of Kohn–Sham density functional theory in both molecules and materials. Although originally defined as a self-consistent self-energy approximation where the Green’s function and self-energy are iterated to consistency, it is commonly used in its one-shot formulation, denoted G_{0}W_{0}. Despite many notable successes, the GW approximation has proven hard to improve upon while retaining its favorable computational features,^{1,2} and furthermore, removing the starting point dependence of G_{0}W_{0} remains a practical challenge. For an overview of current developments, we refer to the review of Ref. 3.

In parallel, time-independent wavefunction approaches for the determination of quasiparticle energies have seen much development in molecular quantum chemistry^{4–10} and have more recently been implemented for materials.^{11–17} One such widely used set of methods is the equation-of-motion (EOM) coupled-cluster (CC) theory,^{18–20} whose variant for quasiparticle energies, i.e., the ionization potentials (IPs) and electron affinities (EAs), is termed IP/EA-EOM-CC. These methods start from the ground-state CC theory, and the EOM approximation can be related to the linear-response of the ground-state ansatz. The IP/EA-EOM-CC framework offers a simple hierarchy of approximations that become increasingly exact. Unfortunately, even at the lowest commonly employed level of approximation, namely, the single and double levels (IP/EA-EOM-CCSD), the computational cost typically exceeds that of standard numerical implementations of the GW approximation.^{21–25}

The popularity of both the GW and EOM-CC approaches has led to recent interest in examining the connections between the two. For example, G_{0}W_{0} is based on a random phase approximation (RPA) description of screening that may also be used to describe ground-state correlations, and the connection between the so-called “direct ring” version of coupled-cluster theory and the RPA ground-state has been known for many years.^{26–31} Additionally, the connection between RPA (neutral) excitation energies and EOM-CC has been established.^{32–34} Refs. 35 and 36 have analyzed and introduced a variety of useful analogies and resemblances between GW and different IP/EA-EOM-CC approximations. However, these prior works have stopped short of establishing any exact mappings between GW and EOM approaches, complicating the understanding of the relationship between the theories.

Here, we provide an analysis of the GW and IP/EA-EOM-CC theories that goes beyond previous results in the following aspects:

We demonstrate an exact equivalence between G

_{0}W_{0}and the EOM theory for an electron–boson problem in a particular excitation basis. In particular, we write down the precise time-independent Hamiltonian supermatrix whose resolvent yields the G_{0}W_{0}Green’s function. Although this expression can be deduced following similar arguments to those used in Refs. 36 and 37, it goes beyond the explicit analysis in such work, which establishes a similar equivalence only within the Tamm–Dancoff approximation (TDA) to screening.^{37,38}The frequency-independent matrix construction we demonstrate is closely related to techniques used to construct the bath in extended dynamical mean-field theory,^{39,40}and is an expression of the well-known description of GW as an electronic polaron theory.^{41}The definition of the electron–boson Hamiltonian that yields the G_{0}W_{0}Green’s function is the subject of Sec. III.We establish that, within the quasi-boson (qb) formalism, the EOM propagator starting from the ring unitary CC doubles (uCC) ground-state is exactly equivalent to that of G

_{0}W_{0}. We refer to this approach as IP+EA-qb-EOM-uCCD. In contrast, the similarity transform of conventional CC theory (referred to as IP+EA-qb-EOM-CCD) does not lead to such an equivalence. This is due to missing some of the non-TDA screening terms, as discussed in Ref. 35. Stated less precisely, the unitary formulation of EOM-CCD contains all the screening diagrams of G_{0}W_{0}, while the standard formulation of EOM-CCD does not. This analysis is worked out in Sec. IV B. Within the electron–boson approximation, employed here, we identify that the origin of this missing screening is a direct consequence of the similarity transformation, which decouples the bosonic ground-state from bosonic excitations only in the ket-space. We note that general relationships between unitary transformations and propagator theories have been identified in early work, such as in Refs. 42–44.The standard EOM-CC formulation neglects the coupling of the IP and EA sectors in the EOM propagator. This is possible due to the use of a partial decoupling transformation that is equivalent to the ground-state CC amplitude equations, as discussed in Ref. 45. We identify the analogous decoupling transformation in the quasi-boson formulation and the form of the remaining IP/EA couplings in both the standard (similarity-transform) and unitary CC languages. We refer to these two procedures as IP/EA-ST-G

_{0}W_{0}and IP/EA-UT-G_{0}W_{0}, respectively.We numerically demonstrate the equivalences and establish the magnitudes of the approximations described earlier, including the equivalence of the G

_{0}W_{0}and the uCC quasi-boson propagator (IP+EA-qb-EOM-uCCD), the size of the non-TDA contributions to the screening, including those omitted from the standard CC quasi-boson propagator (IP+EA-qb-EOM-CCD), and the contributions from the residual IP/EA coupling omitted in the standard formulation of EOM-CC theories (IP/EA-G_{0}W_{0}, IP/EA-ST-G_{0}W_{0}, and IP/EA-UT-G_{0}W_{0}). For the test systems investigated, we show that neglecting RPA screening (as in G_{0}W_{0}-TDA) or partially including RPA screening (as in IP+EA-qb-EOM-CCD) results in a decrease in the fundamental gap. We also show that the neglected residual IP/EA coupling in common EOM-CC theories (IP/EA-ST/UT-G_{0}W_{0}) has only a small effect on HOMO/LUMO quasiparticle energies.

## II. BACKGROUND ON THE GW APPROXIMATION

### A. Green’s functions and the self-energy

The central quantity in this work is the interacting one-electron Green’s function. Assuming an orbital basis, the Green’s function is represented by a frequency dependent matrix **G**(*ω*).^{46} The non-interacting one-particle Green’s function **G**_{0}(*ω*) is defined from an (arbitrary) one-particle operator $h\u0302$,

and the Dyson equation^{47} formally relates the non-interacting and interacting Green’s functions via the self-energy **Σ**(*ω*),

or, in terms of **h**,

The self-energy is thus the central quantity to approximate in a many-body calculation.

The interacting Green’s function determines the single-particle excitation energies and total electronic energy of the system. The excitations correspond to the poles of **G**(*ω*) and can be obtained by solving

The self-energy is commonly divided into a frequency-independent and frequency-dependent (correlation) piece,

To define **Σ**^{static}, we first specify the electronic Hamiltonian

where $a\u0302\u2020,a\u0302$ are electron creation/annihilation operators with respect to the bare electron vacuum and *p*, *q*, *r*, *s* are electron orbital labels, and we assume all matrix elements are real. In the second line, the operator $h\u0302$ is the one appearing in the definition of **G**_{0} and is related to the bare one-electron Hamiltonian by an effective one-electron operator, i.e., $h\u0302=t\u0302+v\u0302$. (For example, if $h\u0302$ is chosen to be the Kohn–Sham Hamiltonian, then $v\u0302=J\u0302+v\u0302xc$, i.e., the Kohn–Sham effective potential, which is the sum of the Coulomb and exchange-correlation potentials evaluated at the Kohn–Sham density.) Then

where **J**, **K** are the Hartree and exchange matrices associated with the density matrix of the current approximation to Green’s function, $D=\u22121\pi \u222b\u2212\u221e\mu d\omega ImG(\omega )$ (*μ* is the chemical potential to satisfy the trace condition Tr **D** = *N*, the number of electrons), i.e.,

Note that if $h\u0302$ is chosen as the Fock operator $f\u0302=t\u0302+J\u0302+K\u0302$ then **Σ**^{static} = 0. In this work, we will always assume this choice of $h\u0302$ for simplicity. It remains then to define the correlation self-energy **Σ**^{c}(*ω*), which is the principle content of the GW approximation.

As seen from the above, **Σ** depends on the Green’s function, i.e., **Σ**[**G**]. When computing it self-consistently, in the first iteration, we have **Σ**[**G**_{0}]. A fully self-consistent self-energy leads to important formal properties of the resulting Green’s function, such as conservation laws.^{48} However, we will focus on the non-self-consistent self-energy **Σ**[**G**_{0}] in this work.

### B. The random phase approximation

The GW self-energy uses some quantities from the random phase approximation (RPA), so we first define them. The RPA treats neutral electronic excitations and can be derived in several ways, including within the quasi-boson formalism discussed later. For a useful introduction, see Refs. 49 and 50.

The RPA eigenvalue problem is defined given $h\u0302$ and $V\u0302$ in Eq. (6). Assuming an orbital basis that diagonalizes $h\u0302$, with a corresponding reference Slater determinant |Φ⟩ composed of *N* occupied orbitals, the excitation energies Ω and amplitudes **X**, **Y** are obtained from

with **A**, **B** matrices defined as

where *i*, *j* label occupied orbitals, *a*, *b* label virtual orbitals, *ϵ*_{i}, *ϵ*_{a} are the occupied and virtual eigenvalues of $h\u0302$, and $\Phi ia$, $\Phi ijab$ are single and double excitations from Φ. (The above corresponds to the “direct” RPA approximation.) The Tamm–Dancoff approximation (TDA) to the RPA arises from setting **B** = 0 (and thus **Y** = 0). **X** and **Y** are normalized such that

and thus

Because the RPA approximation is defined with respect to $h\u0302$ (which is, in principle, arbitrary), it has a starting point dependence. This contributes to the starting point dependence of G_{0}W_{0}. As discussed earlier, we will choose $h\u0302=f\u0302$ in this work.

### C. G_{0}W_{0}, its supermatrix formulation, and GW

The GW approximation approximates the correlation self-energy as

where *W*_{pr,qs} are elements of the screened Coulomb interaction, itself a function of **G** and $V\u0302$. As discussed earlier, such a self-consistent self-energy must be determined iteratively, and in the first iteration, the so-called G_{0}W_{0} version of the theory, **G** is replaced by the non-interacting Green’s function and **W**_{0} is defined in terms of **G**_{0}, corresponding to a treatment of screening within the RPA. Explicitly, in the eigenvector basis of $h\u0302$,^{51–54}

where the screened interaction matrix element *W*_{njν} is defined using the RPA **X**, **Y** amplitudes,

Note that *ν* can be interpreted as a compound index with the range of the particle–hole excitation index *ia*. The self-energy expression can be written compactly as

where **d** is the diagonal matrix with entries

with sgn(*m*) = 1 if *m* is a hole index, sgn(*m*) = −1 if *m* is a particle index, and **W** has elements *W*_{npν}.

Given the explicit form of the G_{0}W_{0} correlation self-energy, we can obtain a special form of the Dyson equation where $GG0W0$ is expressed as a supermatrix inversion (here, supermatrix denotes a matrix of larger dimension than the Green’s function). First, note that for a 2 × 2 block matrix with block labels 1, 2, the identity

Then, from the matrix form of the G_{0}W_{0} self-energy, we see that $GG0W0$ arises from a 2 × 2 block matrix inversion,

If **G**_{0} is defined using **f** as chosen in this work, the supermatrix simplifies to

Since the supermatrix **H** is frequency independent, $GG0W0$ corresponds to the resolvent of **H**.

The starting point dependence of G_{0}W_{0} enters through the choice of $h\u0302$ (which defines **G**_{0} and consequently **W**_{0}) and can be removed once $GG0W0$ is determined by iterating to determine a new self-energy. The fixed point of this self-consistency is the GW approximation. However, we will not discuss self-consistency in our treatment below and instead focus on the G_{0}W_{0} approximation.

## III. CORRESPONDENCE OF G_{0}W_{0} TO THE ELECTRON–BOSON PROBLEM

We next show that the supermatrix $HG0W0$ appearing in Eq. (21) arises naturally as the matrix representation of the equation-of-motion (EOM) treatment of the charged excitations of a coupled electron–boson problem.

We define a coupled-electron boson Hamiltonian corresponding to a set of noninteracting electrons linearly coupled to a quadratic bath of bosons,

where $H\u0302e$ is the electron Hamiltonian, $H\u0302B$ is the boson Hamiltonian, and $V\u0302eB$ is the coupling. We choose^{50}

with $b\u0302\nu \u2020$ and $b\u0302\nu $ being boson creation and annihilation operators, and

We note that $b\u0302\nu \u2020$ and $b\u0302\nu $ effectively mimic fermionic particle–hole creation operators ($a\u0302a\u2020a\u0302i\u2192b\u0302\nu \u2020$, $a\u0302i\u2020a\u0302a\u2192b\u0302\nu $). However, they satisfy bosonic commutation relations and, therefore, violate Pauli’s exclusion principle. This is often referred to as the quasi-boson approximation.

We compute the excitation energies of the coupled electron–boson Hamiltonian in the EOM formalism. Namely, given a set of operators, ${C\u0302I}$, we solve the eigenvalue problem

where the supermatrix **H** and overlap **S** are given by

where |0_{F}⟩ = |Φ⟩ and |0_{B}⟩ is the boson vacuum.

To start, we choose the operator basis ${C\u0302I\u2020}$ = ${a\u0302i,a\u0302a,a\u0302ib\u0302\nu \u2020,a\u0302ab\u0302\nu}$ (we will refer to these indices as {*h*, *p*, *hb*, *pb*}). If we view the bosons as generating particle–hole excitations, then this basis is analogous to the standard singles and doubles bases used to describe charged excitations in quantum chemistry treatments. Note that this set is not closed under the action of $H\u0302B$, i.e., $[H\u0302B,C\u0302I\u2020]$ can generate an operator outside of the span of ${C\u0302I\u2020}$, because $[b\u0302\mu b\u0302\nu ,b\u0302\nu \u2020]=b\u0302\mu $, but we are missing operators such as $a\u0302ib\u0302\nu ,a\u0302ab\u0302\nu \u2020$ in ${C\u0302I\u2020}$. (This is related to the problem of missing time-orderings in GW-TDA.) Evaluating the matrix elements, we obtain *S*_{IJ} = sgn(*I*)*δ*_{IJ}, where sgn(*I*) = 1 for the hole space {*h*, *hb*}, and sgn(*I*) = −1 for the particle space {*p*, *pb*}. The non-zero matrix elements of **H** are

where the sign of *H*_{m,pν} depends on whether it contributes to the particle or hole space; sgn(*m*) = −1 for the hole space and +1 for the particle space.

Multiplying Eq. (26) with −**S**^{−1} from the left, we obtain the supermatrix,

with **Δ** the diagonal matrix with blocks

$HG0W0-TDA$ is precisely the supermatrix identified by Bintrim and Berkelbach in Ref. 37; inserting this in the definition of the Green’s function [Eq. (21)] leads to the approximation to the self-energy and Green’s function termed GW-TDA in Ref. 37.

The G_{0}W_{0}-TDA supermatrix is clearly similar to the G_{0}W_{0} supermatrix but differs in the elements involving bosonic indices. For example, the diagonal matrix **d** in Eq. (21) is replaced by the non-diagonal energy matrix **Δ**, while the screened interaction **W** is replaced by the bare coupling **V**. To obtain the G_{0}W_{0} supermatrix, we first diagonalize the bosonic problem, i.e., rotate to the eigenbasis of $H\u0302B$. Although such a unitary transformation does not affect the spectrum of $H\u0302eB$, it does affect the EOM supermatrix eigenvalues when an incomplete operator basis is used. To diagonalize $H\u0302B$, we use a Bogoliubov transformation,^{50}

where we have used the parentheses $(b\u0302,b\u0302\u2020)$ to denote the operator basis in which $H\u0302B$ is being expressed. We recognize $ABBA$ as the RPA matrix in Eq. (9). We thus obtain the diagonalized form

where

With respect to the new $b\u0304$ eigenbasis, we find that

with $ERPAc$ the (direct) RPA correlation energy

and *W*_{pq,ν} is the matrix element in the G_{0}W_{0} self-energy defined in Eq. (16).

We can now define the EOM supermatrix with respect to the ${C\u0302I\u2020}$ operator basis introduced above. Unlike in the case of G_{0}W_{0}-TDA, the operator basis is now closed under the action of $H\u0302B$ because the non-particle number conserving bosonic terms are removed in the diagonalization. The non-zero matrix elements of the supermatrix are (after multiplication with −**S**^{−1}),

and in matrix form

Now consider the case starting in the eigenvector basis of $f\u0302$. Then, **d**_{hb,hb}, **d**_{pb,pb} are diagonals (and **F**_{h,p} = **F**_{p,h} = 0). We see that this is identical to the G_{0}W_{0} supermatrix defined in Eq. (22) for the choice of $h\u0302=f\u0302$, showing that the G_{0}W_{0} Green’s function (starting from a Hartree–Fock **G**_{0}) corresponds precisely to that of an electron–boson problem in a particular excitation basis.

## IV. INSIGHTS FROM AND RELATIONSHIPS TO EQUATION-OF-MOTION COUPLED-CLUSTER THEORY

### A. Background on EOM-CC and definition of the IP+EA/EOM-CC approximation

In IP/EA-EOM-CC theory, one first computes the coupled-cluster ground-state $|\Psi \u3009=eT\u0302|\Phi \u3009$, where $T\u0302$ is the cluster excitation operator. At the singles and doubles levels, one has $T\u0302=T\u03021+T\u03022=\u2211iatiaa\u0302a\u2020a\u0302i+14\u2211ijabtijaba\u0302a\u2020a\u0302b\u2020a\u0302ja\u0302i$. Then we can define

where $H\u0304$ is termed the similarity transformed Hamiltonian. Note that $H\u0304$ is not Hermitian and thus has different left and right eigenvectors. The right ground-state is |0_{F}⟩, and we denote the left ground-state as $\u30080\u0304F|$. The amplitudes satisfy

The above means that the ground-state and neutral excitations are decoupled only in the ket space.

$H\u0304$ can, in principle, be used in an EOM formulation as in Eq. (27). Namely, we can take the operator basis ${C\u0302I\u2020}={a\u0302i,a\u0302a,a\u0302a\u2020a\u0302ia\u0302j,a\u0302i\u2020a\u0302aa\u0302b}$, and define the corresponding supermatrix elements,

The standard EOM-CC formulation, however, corresponds to defining separate supermatrices for the IP and EA sectors. We do so by dividing the operator basis above into IP and EA parts, ${C\u0302I\u2020IP}={a\u0302i,a\u0302a\u2020a\u0302ia\u0302j}$ and ${C\u0302IEA}={a\u0302a\u2020,a\u0302a\u2020a\u0302b\u2020a\u0302j}$. We then construct a supermatrix for the IP parts and EA parts separately,

and the overlap matrix is $SIJIP-EOM-CCSD=\u27e80F|[C\u0302IIP,C\u0302J\u2020IP]|0F\u27e9=\delta IJ$ and $SIJEA-EOM-CCSD=\u27e80F|[C\u0302I\u2020EA,C\u0302JEA]|0F\u27e9=\delta IJ$. The solutions of the two eigenvalue problems then yield the IP and EA excitations.

We see there are some similarities and some differences in the construction of IP/EA-EOM-CC supermatrices and the G_{0}W_{0} supermatrix. Of course, one difference is the replacement of the electronic Hamiltonian and excitation operators by the electron–boson Hamiltonian and electron–boson operator basis used in the quasi-boson formalism of G_{0}W_{0}. However, there are other important differences, principally the use of the similarity transformed Hamiltonian $H\u0304$ and the neglect of the particle–hole coupling in the standard IP/EA-EOM-CC supermatrixes. To isolate the impact of these latter two choices, we can formulate an EOM-CC formalism in the quasi-boson approximation, which corresponds to starting from the electron–boson Hamiltonian. We now proceed with this analysis.

### B. Quasi-boson CC and exact equivalence of equation-of-motion ring unitary coupled-cluster theory and G_{0}W_{0}

To obtain a quasi-boson formulation of EOM-CC (referred to as qb-EOM-CC in the following), we first define the ground-state CC problem as that of finding the ground-state of $H\u0302B$. Then the CC ansatz is

where $T\u0302B$ is a pure boson excitation operator $T\u0302B=12\u2211\mu \nu t\mu \nu b\u0302\mu \u2020b\u0302\nu \u2020$. The similarity transformed boson Hamiltonian is

and the amplitudes satisfy

It can be shown that $t\mu \nu =[YX\u22121]\mu ,\nu $,^{28} and using the correspondence *ν*, *μ* → *ia*, *jb*, these amplitudes can be identified with those generated in the standard ring CC doubles (D) approximation to the amplitude equations of Eq. (39).^{28,30} Similarly to the fermionic CC formulation, Eqs. (46) and (45) signify that the bosonic ground-state is decoupled from the bosonic excitations only in the ket-space.

In contrast, the Bogoliubov transformation of $H\u0302B$ in Sec. III is a unitary (canonical) transformation. We can write the unitary operator as $e\sigma \u0302B$ with $\sigma \u0302=12\u2211\mu \nu \sigma \mu \nu (b\u0302\mu \u2020b\u0302\nu \u2020\u2212b\u0302\mu b\u0302\nu )$ and $\sigma \u0302B\u2020=\u2212\sigma \u0302B$. This leads to a bosonic version of the unitary coupled-cluster (uCC) ground-state theory,

The corresponding bosonic unitary transformed Hamiltonian $H\u0304uCCB$ and amplitude equations satisfy

and

In the bosonic uCC, the bosonic excitations are decoupled from the ground-state in both the bra and ket spaces, and we can further choose $\sigma \u0302B$ such that the Hamiltonian is fully diagonalized, as we did in the analysis of G_{0}W_{0} above. The bosonic uCC amplitude equations correspond to a direct ring-version of the standard uCCD amplitude equations.

Although the bosonic CC and bosonic uCC theories create the same ground-state of $H\u0302B$, i.e., the RPA ground-state, up to a normalization factor *N*_{0},

the different form of $H\u0304CCB$ and $H\u0304uCCB$ lead to different EOM approximations. To see this, we first extend the above analysis to the full electron–boson Hamiltonian $H\u0302eB$. This is simple because if $|\Psi CCB\u3009$, $|\Psi uCCB\u3009$ are eigenstates of $H\u0302B$ then $|0F\Psi CCB\u3009$, $|0F\Psi uCCB\u3009$ are eigenstates of $H\u0302eB$ with the same eigenvalue. We can similarly define the similarity and unitarily transformed electron–boson Hamiltonians,

Then $H\u0304uCCeB$ is identical to the Bogoliubov transformed Hamiltonian of the above analysis of G_{0}W_{0} theory. Consequently, within the electron–boson picture, the IP+EA-qb-EOM-uCCD is precisely equivalent to the G_{0}W_{0} theory.

In contrast, if we define an IP+EA-qb-EOM formulation using the similarity transformed electron–boson Hamiltonian $H\u0304CCeB$ (denoted as IP+EA-qb-EOM-CCD), the non-number conserving nature of $H\u0304CCB$ means that the G_{0}W_{0} operator basis is not closed under the action of $H\u0304CCB$. Therefore, although some non-TDA screening contributions are included in this treatment, others are not, and this may be viewed as an expression of the missing time-orderings in the screening terms in IP/EA-EOM-CCSD, which was diagrammatically analyzed in Ref. 35. We examine the numerical consequences of this in Sec. V A.

### C. Decoupling of particle and hole subspaces

A second source of differences between the G_{0}W_{0} approximation and the IP/EA-EOM-CC treatments arises from the neglect of the IP and EA couplings in the IP/EA-EOM-CC supermatrices. Reordering the $HG0W0$ supermatrix, we write

and we see that **C** is the coupling between the IP and EA sectors. Analogous to the IP/EA-EOM-CC treatment, we could choose to neglect the coupling matrix and diagonalize the IP and EA blocks of the supermatrix separately. We will term this the IP/EA-G_{0}W_{0} approximation.

This, however, is not a perfect analogy, because if the ground-state CC amplitude equations are satisfied up to some level of excitation, then some elements of **C** are removed in the IP+EA-EOM-CC supermatrix. (This has been discussed in the context of unitary coupled-cluster theory in Ref. 43.) For example, if $C\u0302\u2020EA=a\u0302a$ and $C\u0302IP=a\u0302i\u2020a\u0302j\u2020a\u0302b$, then

and if $H\u0304$ is constructed with CCSD amplitudes, this term is 0 [see Eq. (39)]. In other words, the ground-state amplitude equations mean that the IP+EA-EOM-CCSD supermatrix is (partially) decoupled between the particle and hole sectors, and IP/EA-EOM-CCSD approximations only neglect the residual coupling. Identifying boson labels in ${CI\u2020}$ with the excitations *ia*, the supermatrix in Eq. (41) has the structure

(For notational economy in this section, we reuse the symbols **F**, **W**, etc., even though their values are different in different supermatrices. **D** is, in general, non-diagonal.) IP/EA-EOM-CCSD corresponds to neglecting only the **C**_{hb,pb} coupling (in this case, the lower 2 × 2 block of couplings does not affect the eigenvalues of the supermatrix, although it does affect the left eigenvectors).

To understand the quality of this approximate decoupling, it is instructive to pursue a similar decoupling of the electron–boson Hamiltonian in the G_{0}W_{0} setting. Here we assume that we have already performed the unitary Bogoliubov transformation to diagonalize $H\u0302B$, i.e., we are using $H\u0304eB$, whose supermatrix is $HG0W0$. As established, this is equivalent to **H**^{IP+EA-qb-EOM-uCCD} in the electron–boson setting, but we will primarily view this as a decoupling of the G_{0}W_{0} theory in this section. We can then require that our correlated vacuum $|0F0\u0304B\u3009$ satisfy amplitude equations that set parts of the coupling matrix **C** to zero. In the standard CC theory, such amplitude equations are the same as the ground-state amplitude equations, but appear as additional amplitude equations in the current setting. We can decouple only the upper-block interactions, similar to IP/EA-EOM-CC theory, using a pure excitation operator $T\u0302eB$ and a similarity transform,

yielding the ST-G_{0}W_{0} supermatrix

Diagonalizing the IP and EA blocks (evaluated in the ${C\u0302I\u2020IP}$ and ${C\u0302IEA}$ bases) separately yields what we term the IP/EA-ST-G_{0}W_{0} approximations. Alternatively, we can use a unitary operator $\sigma \u0302eB=T\u0302eB\u2212T\u0302eB\u2020$ and require

leading to the symmetrical block structure

Diagonalizing the IP and EA blocks separately leads to the IP/EA-UT-G_{0}W_{0} approximations.

The decouplings above do not correspond to a simple matrix decoupling of the Hamiltonian because the transformations are performed at the second-quantized level (i.e., in terms of infinite matrices) before being projected down into the finite operator basis. Consequently, the eigenvalues of the G_{0}W_{0}, UT-G_{0}W_{0}, and ST-G_{0}W_{0} supermatrices are all different from each other. Because we are using second-quantized transformations, the particle–hole decoupled Hamiltonians contain different interactions from $H\u0304eB$ even in the purely electronic sector. This is a type of renormalization of the electronic Green’s function. In the Baker–Campbell–Hausdorff (BCH) expansion of these Hamiltonians, the exact expansion of the commutators leads to higher particle–electronic interactions. To control the complexity, we use an approximate BCH expansion that retains the same interaction form as the original $H\u0304eB$ Hamiltonian,

where the subscript 1 indicates that only up to 1-particle electronic terms after normal ordering with respect to |0_{F}⟩ are retained. This approximation includes mean-field-like contributions from the two- and higher-particle electronic interactions generated in the decoupling transformation. The BCH expansion for the similarity transformed (ST) Hamiltonian naturally truncates at third order and two-particle interactions. Because the G_{0}W_{0} operator basis does not contain more than single electron holes or particles, the terms dropped in the ST expansion do not contribute to the ST-G_{0}W_{0} supermatrix. Numerical results for the particle–hole decoupling transformations are presented in Sec. V B.

## V. NUMERICAL INVESTIGATIONS

We now describe numerical experiments to illustrate our above analysis, in particular, with respect to the different treatments of screening in G_{0}W_{0}, the relationship between G_{0}W_{0} and EOM-CC theories, and the effectiveness of approximate particle–hole decoupling. Each experiment corresponds to constructing a different Hamiltonian supermatrix, which we diagonalize to report the poles, focusing mainly on the quasiparticle HOMO and LUMO energies. In the case of G_{0}W_{0}, this means that the quasiparticle poles reported include the contributions of the full self-energy matrix, i.e., we do not employ the common diagonal approximation (similarly to Refs. 37 and 55). Because we are interested in how the different approximations for the construction of the supermatrices affect quasiparticle energies relative to G_{0}W_{0}, G_{0}W_{0} quasiparticle energies are chosen as reference values in the following.

The working equations for all methods were generated using WICK^{56} and the methods were implemented using PySCF.^{57,58} The ground-state direct ring CCD amplitudes appearing in IP+EA-qb-EOM-CCD obtained from the RPA **X**, **Y** matrices, while the direct ring unitary CC doubles Hamiltonian $(H\u0304uCCeB)$ is constructed based on canonical transformation theory.^{59,60} The required commutator expressions are given in Subsection 3 of the Appendix. The IP/EA decoupling amplitude equations in the similarity transformed approach are given explicitly in Subsection 2 of the Appendix, while the unitary transformation decoupling equations were solved within the canonical transformation approach. A list of abbreviations for the variety of EOM approaches is provided in Subsection 4 of the Appendix. All molecular structures were taken from the GW100 test set,^{61} and we used the def2-TZVP^{62} basis in all calculations. All calculations used the Hartree–Fock Green’s function as **G**_{0}.

### A. G_{0}W_{0}, G_{0}W_{0}-TDA, IP+EA-qb-EOM-(u)CCD

We first investigate the effect of different treatments of RPA screening on the quasiparticle energies given by G_{0}W_{0} and G_{0}W_{0}-TDA and the two EOM-CC theories for the electron–boson Hamiltonian, denoted IP+EA-qb-EOM-CCD and IP+EA-qb-EOM-uCCD. We show the G_{0}W_{0} HOMO/LUMO quasiparticle energies, the fundamental gap *E*^{Gap}, and the differences from G_{0}W_{0} [reported as mean absolute errors (MAE)] in Table I.

. | G_{0}W_{0}
. | G_{0}W_{0}-TDA
. | IP+EA-qb-EOM-CCD . | IP+EA-qb-EOM-uCCD . | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Molecule . | HOMO . | LUMO . | E^{Gap}
. | ΔHOMO . | ΔLUMO . | ΔE^{Gap}
. | ΔHOMO . | ΔLUMO . | ΔE^{Gap}
. | ΔHOMO . | ΔLUMO . | ΔE^{Gap}
. |

He | −24.301 | 22.401 | 46.702 | 0.143 | −0.025 | −0.168 | 0.076 | −0.013 | −0.089 | 0.000 | 0.000 | 0.000 |

Ne | −21.362 | 21.197 | 42.559 | 0.605 | −0.077 | −0.682 | 0.332 | −0.040 | −0.372 | 0.000 | 0.000 | 0.000 |

H_{2} | −16.308 | 4.404 | 20.712 | −0.027 | −0.006 | 0.021 | −0.009 | −0.003 | 0.006 | 0.000 | 0.000 | 0.000 |

Li_{2} | −5.165 | 0.018 | 5.183 | −0.056 | −0.068 | −0.012 | −0.024 | −0.034 | −0.010 | 0.000 | 0.000 | 0.000 |

F_{2} | −16.274 | 0.753 | 17.027 | 0.790 | −0.208 | −0.998 | 0.431 | −0.106 | −0.537 | 0.000 | 0.000 | 0.000 |

SiH_{4} | −13.082 | 3.341 | 16.423 | 0.055 | −0.107 | −0.162 | 0.034 | −0.053 | −0.087 | 0.000 | 0.000 | 0.000 |

LiH | −7.949 | 0.123 | 8.072 | 0.112 | −0.009 | −0.121 | 0.062 | −0.004 | −0.066 | 0.000 | 0.000 | 0.000 |

CO | −14.99 | 1.094 | 16.084 | 0.220 | −0.087 | −0.307 | 0.131 | −0.042 | −0.173 | 0.000 | 0.000 | 0.000 |

H_{2}O | −12.789 | 3.114 | 15.903 | 0.464 | −0.058 | −0.522 | 0.265 | −0.029 | −0.294 | 0.000 | 0.000 | 0.000 |

BeO | −9.788 | −2.097 | 7.691 | 0.366 | −0.050 | −0.416 | 0.233 | −0.026 | −0.259 | 0.000 | 0.000 | 0.000 |

MgO | −7.863 | −1.506 | 6.357 | 0.968 | 0.132 | −0.836 | 0.562 | 0.071 | −0.491 | 0.000 | 0.000 | 0.000 |

H_{2}CO | −11.206 | 1.822 | 13.028 | 0.446 | −0.191 | −0.637 | 0.249 | −0.094 | −0.343 | 0.000 | 0.000 | 0.000 |

CH_{4} | −14.637 | 3.650 | 18.287 | 0.102 | −0.076 | −0.178 | 0.064 | −0.038 | −0.102 | 0.000 | 0.000 | 0.000 |

SO_{2} | −12.827 | −0.483 | 12.344 | 0.353 | −0.045 | −0.398 | 0.203 | −0.020 | −0.223 | 0.000 | 0.000 | 0.000 |

MAE | 0.336 | 0.081 | 0.390 | 0.191 | 0.041 | 0.218 | 0.000 | 0.000 | 0.000 |

. | G_{0}W_{0}
. | G_{0}W_{0}-TDA
. | IP+EA-qb-EOM-CCD . | IP+EA-qb-EOM-uCCD . | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Molecule . | HOMO . | LUMO . | E^{Gap}
. | ΔHOMO . | ΔLUMO . | ΔE^{Gap}
. | ΔHOMO . | ΔLUMO . | ΔE^{Gap}
. | ΔHOMO . | ΔLUMO . | ΔE^{Gap}
. |

He | −24.301 | 22.401 | 46.702 | 0.143 | −0.025 | −0.168 | 0.076 | −0.013 | −0.089 | 0.000 | 0.000 | 0.000 |

Ne | −21.362 | 21.197 | 42.559 | 0.605 | −0.077 | −0.682 | 0.332 | −0.040 | −0.372 | 0.000 | 0.000 | 0.000 |

H_{2} | −16.308 | 4.404 | 20.712 | −0.027 | −0.006 | 0.021 | −0.009 | −0.003 | 0.006 | 0.000 | 0.000 | 0.000 |

Li_{2} | −5.165 | 0.018 | 5.183 | −0.056 | −0.068 | −0.012 | −0.024 | −0.034 | −0.010 | 0.000 | 0.000 | 0.000 |

F_{2} | −16.274 | 0.753 | 17.027 | 0.790 | −0.208 | −0.998 | 0.431 | −0.106 | −0.537 | 0.000 | 0.000 | 0.000 |

SiH_{4} | −13.082 | 3.341 | 16.423 | 0.055 | −0.107 | −0.162 | 0.034 | −0.053 | −0.087 | 0.000 | 0.000 | 0.000 |

LiH | −7.949 | 0.123 | 8.072 | 0.112 | −0.009 | −0.121 | 0.062 | −0.004 | −0.066 | 0.000 | 0.000 | 0.000 |

CO | −14.99 | 1.094 | 16.084 | 0.220 | −0.087 | −0.307 | 0.131 | −0.042 | −0.173 | 0.000 | 0.000 | 0.000 |

H_{2}O | −12.789 | 3.114 | 15.903 | 0.464 | −0.058 | −0.522 | 0.265 | −0.029 | −0.294 | 0.000 | 0.000 | 0.000 |

BeO | −9.788 | −2.097 | 7.691 | 0.366 | −0.050 | −0.416 | 0.233 | −0.026 | −0.259 | 0.000 | 0.000 | 0.000 |

MgO | −7.863 | −1.506 | 6.357 | 0.968 | 0.132 | −0.836 | 0.562 | 0.071 | −0.491 | 0.000 | 0.000 | 0.000 |

H_{2}CO | −11.206 | 1.822 | 13.028 | 0.446 | −0.191 | −0.637 | 0.249 | −0.094 | −0.343 | 0.000 | 0.000 | 0.000 |

CH_{4} | −14.637 | 3.650 | 18.287 | 0.102 | −0.076 | −0.178 | 0.064 | −0.038 | −0.102 | 0.000 | 0.000 | 0.000 |

SO_{2} | −12.827 | −0.483 | 12.344 | 0.353 | −0.045 | −0.398 | 0.203 | −0.020 | −0.223 | 0.000 | 0.000 | 0.000 |

MAE | 0.336 | 0.081 | 0.390 | 0.191 | 0.041 | 0.218 | 0.000 | 0.000 | 0.000 |

The differences between G_{0}W_{0}-TDA and G_{0}W_{0} capture the contribution of non-TDA screening to the quasiparticle energies. We see that the neglect of non-TDA screening results in a more positive HOMO and a less positive LUMO energy (with a few exceptions) and consequently a smaller fundamental gap. The non-TDA terms have a greater effect on the HOMO than the LUMO quasiparticle energies. We observe a MAE for G_{0}W_{0}-TDA of 0.336 eV (HOMO) and 0.081 eV (LUMO), and the MAE for Δ*E*^{Gap} is 0.390 eV.

IP+EA-qb-EOM-CCD includes a subset of the non-TDA screening contributions, and this results in a decrease of the MAE for the HOMO and LUMO quasiparticle energies to 0.191 and 0.041 eV, respectively, and 0.218 eV for Δ*E*^{Gap}. However, deviations from G_{0}W_{0} of more than 0.5 eV (HOMO, MgO) are still observed, highlighting the inability of the similarity transformation to capture some important effects of RPA screening.

IP+EA-qb-EOM-uCCD yields numerically identical quasiparticle energies to G_{0}W_{0}. This demonstrates the equivalence of the bosonic unitary transformation in IP+EA-qb-EOM-uCCD to RPA screening, as already shown theoretically in Sec. IV B.

In Table II, we show more detailed data on the quasiparticle energies from HOMO-2 to LUMO+2 for F_{2} and MgO. From the deviations from G_{0}W_{0}, it is clear that neglecting parts of RPA screening does not result in a uniform shift in the quasiparticle IPs or EAs, e.g., the deviations for G_{0}W_{0}-TDA for the HOMO and HOMO-1 of MgO are 0.968 and 0.203 eV, respectively. Interestingly, we see particularly large deviations for the HOMO-2 of MgO. This appears to be related to the overall small quasiparticle weights (i.e., the total norm in the single-particle sector for excitation *n*, $\Vert Rhn\Vert 2+\Vert Rpn\Vert 2$) of 0.31 (G_{0}W_{0}-RPA), 0.54 (G_{0}W_{0}-TDA), and 0.39 (IP+EA-qb-EOM-CCD). This means that there are large contributions from the bosonic sectors of the Hamiltonian supermatrix, which is the part that is treated differently in all the approaches.

. | F_{2}
. | MgO . | ||||
---|---|---|---|---|---|---|

G_{0}W_{0}
. | ΔG_{0}W_{0}-TDA
. | ΔIP+EA-qb-EOM-CCD . | G_{0}W_{0}
. | ΔG_{0}W_{0}-TDA
. | ΔIP+EA-qb-EOM-CCD . | |

HOMO-2 | −20.773 | −0.267 | −0.106 | −25.309 | 2.567 | 2.058 |

HOMO-1 | −19.863(2×) | 0.896(2×) | 0.501(2×) | −8.444 | 0.203 | 0.157 |

HOMO | −16.274(2×) | 0.790(2×) | 0.431(2×) | −7.863(2×) | 0.968(2×) | 0.562(2×) |

LUMO | 0.753 | −0.208 | −0.106 | −1.506 | 0.132 | 0.071 |

LUMO+1 | 15.778 | −0.270 | −0.120 | 1.088(2×) | −0.026(2×) | −0.015(2×) |

LUMO+2 | 15.828 | −0.254 | −0.157 | 2.606 | −0.092 | −0.053 |

. | F_{2}
. | MgO . | ||||
---|---|---|---|---|---|---|

G_{0}W_{0}
. | ΔG_{0}W_{0}-TDA
. | ΔIP+EA-qb-EOM-CCD . | G_{0}W_{0}
. | ΔG_{0}W_{0}-TDA
. | ΔIP+EA-qb-EOM-CCD . | |

HOMO-2 | −20.773 | −0.267 | −0.106 | −25.309 | 2.567 | 2.058 |

HOMO-1 | −19.863(2×) | 0.896(2×) | 0.501(2×) | −8.444 | 0.203 | 0.157 |

HOMO | −16.274(2×) | 0.790(2×) | 0.431(2×) | −7.863(2×) | 0.968(2×) | 0.562(2×) |

LUMO | 0.753 | −0.208 | −0.106 | −1.506 | 0.132 | 0.071 |

LUMO+1 | 15.778 | −0.270 | −0.120 | 1.088(2×) | −0.026(2×) | −0.015(2×) |

LUMO+2 | 15.828 | −0.254 | −0.157 | 2.606 | −0.092 | −0.053 |

### B. Particle–hole decoupling

We now investigate the approximate particle–hole (IP/EA) decoupling of G_{0}W_{0} (inspired by the analogous particle–hole decoupling of EOM-CC) as discussed in Sec. IV C. The MAE deviations from G_{0}W_{0} of the HOMO and LUMO quasiparticle energies for the IP/EA-G_{0}W_{0} (neglecting any IP/EA coupling), similarity transformed decoupled IP/EA-ST-G_{0}W_{0} [Eq. (58)], and unitary transformed decoupled IP/EA-UT-G_{0}W_{0} [Eq. (60)] are shown in Table III.

. | IP/EA-G_{0}W_{0}
. | IP/EA-ST-G_{0}W_{0}
. | IP/EA-UT-G_{0}W_{0}
. | |||
---|---|---|---|---|---|---|

Molecule . | ΔHOMO . | ΔLUMO . | ΔHOMO . | ΔLUMO . | ΔHOMO . | ΔLUMO . |

He | 1.178 | −0.333 | −0.007 | −0.002 | 0.000 | −0.006 |

Ne | 2.059 | −0.375 | 0.077 | −0.004 | 0.086 | −0.013 |

H_{2} | 1.162 | −0.125 | −0.023 | −0.004 | −0.016 | −0.002 |

Li_{2} | 0.598 | −0.094 | −0.008 | −0.007 | −0.016 | −0.005 |

F_{2} | 2.083 | −2.120 | 0.094 | −0.091 | 0.134 | −0.067 |

SiH_{4} | 1.257 | −0.207 | −0.020 | −0.016 | −0.017 | −0.011 |

LiH | 0.845 | −0.048 | −0.026 | −0.002 | −0.002 | −0.001 |

CO | 1.803 | −1.011 | −0.024 | −0.063 | 0.039 | −0.023 |

H_{2}O | 1.866 | −0.216 | 0.040 | −0.009 | 0.090 | −0.008 |

BeO | 1.920 | −0.131 | 0.090 | −0.020 | 0.097 | −0.004 |

MgO | 1.845 | −0.801 | 0.341 | −0.006 | 0.222 | −0.029 |

H_{2}CO | 1.904 | −0.999 | 0.052 | −0.066 | 0.065 | −0.043 |

CH_{4} | 1.549 | −0.15 | −0.026 | −0.013 | 0.005 | −0.009 |

SO_{2} | 2.032 | −1.483 | 0.028 | −0.076 | 0.060 | −0.042 |

MAE | 1.579 | 0.578 | 0.061 | 0.027 | 0.061 | 0.019 |

. | IP/EA-G_{0}W_{0}
. | IP/EA-ST-G_{0}W_{0}
. | IP/EA-UT-G_{0}W_{0}
. | |||
---|---|---|---|---|---|---|

Molecule . | ΔHOMO . | ΔLUMO . | ΔHOMO . | ΔLUMO . | ΔHOMO . | ΔLUMO . |

He | 1.178 | −0.333 | −0.007 | −0.002 | 0.000 | −0.006 |

Ne | 2.059 | −0.375 | 0.077 | −0.004 | 0.086 | −0.013 |

H_{2} | 1.162 | −0.125 | −0.023 | −0.004 | −0.016 | −0.002 |

Li_{2} | 0.598 | −0.094 | −0.008 | −0.007 | −0.016 | −0.005 |

F_{2} | 2.083 | −2.120 | 0.094 | −0.091 | 0.134 | −0.067 |

SiH_{4} | 1.257 | −0.207 | −0.020 | −0.016 | −0.017 | −0.011 |

LiH | 0.845 | −0.048 | −0.026 | −0.002 | −0.002 | −0.001 |

CO | 1.803 | −1.011 | −0.024 | −0.063 | 0.039 | −0.023 |

H_{2}O | 1.866 | −0.216 | 0.040 | −0.009 | 0.090 | −0.008 |

BeO | 1.920 | −0.131 | 0.090 | −0.020 | 0.097 | −0.004 |

MgO | 1.845 | −0.801 | 0.341 | −0.006 | 0.222 | −0.029 |

H_{2}CO | 1.904 | −0.999 | 0.052 | −0.066 | 0.065 | −0.043 |

CH_{4} | 1.549 | −0.15 | −0.026 | −0.013 | 0.005 | −0.009 |

SO_{2} | 2.032 | −1.483 | 0.028 | −0.076 | 0.060 | −0.042 |

MAE | 1.579 | 0.578 | 0.061 | 0.027 | 0.061 | 0.019 |

Simply neglecting particle–hole couplings as in IP/EA-G_{0}W_{0} gives a MAE of 1.579 eV for the HOMO and 0.578 eV for the LUMO quasiparticle energies. These large deviations highlight the importance of particle–hole coupling in G_{0}W_{0} (similar results are obtained for G_{0}W_{0}-TDA in Ref. 37). The approximate decoupling procedures, IP/EA-ST-G_{0}W_{0} and IP/EA-UT-G_{0}W_{0}, significantly reduce the deviations to a MAE of 0.061 eV (HOMO)/0.027 eV (LUMO) and 0.061 eV (HOMO)/0.019 eV (LUMO), respectively. The remaining deviation is due to the approximate nature of the decoupling from the truncation of the cluster expansion at the singles/doubles level and (in the case of the unitary transform) the neglect of some of the new electronic interactions generated by the unitary transformation (Sec. IV C).

Deviations from G_{0}W_{0} for the HOMO-2 to LUMO+2 quasiparticle energies are shown for F_{2} and MgO in Table IV. We see that the particle–hole coupling contributions vary significantly depending on the quasiparticle, e.g., in IP/EA-G_{0}W_{0}, the deviations for the LUMO and LUMO+1 of F_{2} are −2.120 and −0.355 eV, respectively; for IP/EA-ST-G_{0}W_{0} and IP/EA-UT-G_{0}W_{0}, the dependency is less pronounced. We see large deviations for the HOMO-2 of MgO (2.309 eV IP/EA-G_{0}W_{0}, 1.174 eV IP/EA-ST-G_{0}W_{0}, and 1.049 eV IP/EA-UT-G_{0}W_{0}). As already discussed in Sec. V A, the quasiparticle weight in this case is small, highlighting a strong interaction of the electron/hole with the quasi-bosons.

. | F_{2}
. | MgO . | ||||
---|---|---|---|---|---|---|

ΔIP/EA-G_{0}W_{0}
. | ΔIP/EA-ST-G_{0}W_{0}
. | ΔIP/EA-ΔUT-G_{0}W_{0}
. | ΔIP/EA-G_{0}W_{0}
. | ΔIP/EA-ST-G_{0}W_{0}
. | ΔIP/EA-UT-G_{0}W_{0}
. | |

HOMO-2 | 2.978^{a} | −0.025 | −0.003 | 2.309 | 1.174 | 1.049 |

HOMO-1 | 1.833^{a}(2×) | 0.115(2×) | 0.166(2×) | 2.144 | 0.105 | 0.072 |

HOMO | 2.083(2×) | 0.094(2×) | 0.134(2×) | 1.845(2×) | 0.341(2×) | 0.222(2×) |

LUMO | −2.120 | −0.091 | −0.067 | −0.801 | −0.006 | −0.029 |

LUMO+1 | −0.355 | −0.037 | −0.040 | −0.151(2×) | −0.028(2×) | −0.001(2×) |

LUMO+2 | −0.367 | −0.058 | −0.056 | −0.130 | −0.022 | −0.023 |

. | F_{2}
. | MgO . | ||||
---|---|---|---|---|---|---|

ΔIP/EA-G_{0}W_{0}
. | ΔIP/EA-ST-G_{0}W_{0}
. | ΔIP/EA-ΔUT-G_{0}W_{0}
. | ΔIP/EA-G_{0}W_{0}
. | ΔIP/EA-ST-G_{0}W_{0}
. | ΔIP/EA-UT-G_{0}W_{0}
. | |

HOMO-2 | 2.978^{a} | −0.025 | −0.003 | 2.309 | 1.174 | 1.049 |

HOMO-1 | 1.833^{a}(2×) | 0.115(2×) | 0.166(2×) | 2.144 | 0.105 | 0.072 |

HOMO | 2.083(2×) | 0.094(2×) | 0.134(2×) | 1.845(2×) | 0.341(2×) | 0.222(2×) |

LUMO | −2.120 | −0.091 | −0.067 | −0.801 | −0.006 | −0.029 |

LUMO+1 | −0.355 | −0.037 | −0.040 | −0.151(2×) | −0.028(2×) | −0.001(2×) |

LUMO+2 | −0.367 | −0.058 | −0.056 | −0.130 | −0.022 | −0.023 |

^{a}

Different energetic quasiparticle orders for IP/EA-G_{0}W_{0} compared to G_{0}W_{0}. The quasiparticle notation corresponds to the energetic order of G_{0}W_{0}.

## VI. CONCLUSION AND OUTLOOK

In this work, we established a variety of exact relationships between the G_{0}W_{0} approximation and equation-of-motion (EOM) coupled-cluster theory within the quasi-boson formalism. The starting point was the exact equivalence between the G_{0}W_{0} propagator and that of a particular electron–boson Hamiltonian supermatrix. From there, we could demonstrate the equivalence to a quasi-boson version of EOM-unitary coupled-cluster theory and elucidate the differences, both theoretically and numerically, between the standard EOM-similarity transformed coupled-cluster theory, as well as the recently introduced G_{0}W_{0}-TDA approximation. These relationships also motivated a particle–hole decoupling transformation of G_{0}W_{0}, which we formulated and numerically explored.

The precise relationships established here between the G_{0}W_{0} approximation and various quantum chemistry theories, including CC theory, will be useful in future directions. For example, we did not focus on computational costs in this work. However, the time-independent formulation of G_{0}W_{0} opens up the incorporation of a wide variety of computational quantum chemistry techniques. In particular, the IP+EA-qb-EOM-UCCD formulation of G_{0}W_{0} could provide a competitive alternative to standard G_{0}W_{0} implementations when combined with commonly employed acceleration techniques such as density fitting. It also suggests theoretical extensions to areas not traditionally treated using the GW formalism, such as multireference problems. The identification of unitary CC as the correct starting point to include all RPA screening in the quasiparticle energies also suggests new avenues to improve purely electronic theories of quasiparticle energies.

## ACKNOWLEDGMENTS

We thank T. Berkelbach and S. Bintrim for generously sharing their G_{0}W_{0}-TDA implementation. We thank T. Berkelbach and A. Sokolov for their helpful comments on the article. This work was supported by the U.S. Department of Energy through Award No. DE-SC0019390. J.T. acknowledges funding through a postdoctoral research fellowship from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Grant No. 495279997.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Johannes Tölle**: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (lead); Writing – original draft (equal); Writing – review & editing (equal). **Garnet Kin-Lic Chan**: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (lead); Resources (lead); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: SPIN-FREE WORKING EQUATIONS

In this appendix, some of the working equations of this paper are given in spin-free (sf) form. For this, a singlet spin adaptation is used for the bosonic part,^{63}

and fermionic creation and annihilation operators are replaced by

*V*_{pqμ} as

and *W*_{pqμ} as

where *s* refers to singlet excitations.

#### 1. Working equations IP+EA-EOM-CC

The direct ring-CCD amplitudes are denoted as $t\nu \mu rCCD$. Denoting the quasiparticle amplitudes by *R*, then for IP+EA-qb-EOM-CCD, one finds the following *σ* vector expressions:

#### 2. Similarity-transformed amplitude equations for IP/EA decoupling

The singles amplitudes are obtained from

and the doubles amplitudes from

#### 3. Commutator expressions $H\u0304uCCeB$

The commutator expressions required for the construction of $H\u0304uCCeB$ are

and

with

#### 4. Abbreviations

A list of abbreviations for the different equation-of-motion (EOM) approaches considered in this work is provided in Table V.

Abbreviation . | Description . | Equations . |
---|---|---|

G_{0}W_{0} | G_{0}W_{0} supermatrix | (37) |

G_{0}W_{0}-TDA | G_{0}W_{0} supermatrix within the Tamm–Dancoff approximation | (29) |

IP+EA-EOM-CCSD | Similarity-transformed CC EOM using combined IP/EA operator basis | (41) |

IP/EA-EOM-CCSD | Similarity-transformed CC EOM using separate IP/EA operator basis | (42) and (43) |

IP+EA-qb-EOM-uCCD | IP+EA-EOM-uCCD in the quasi-boson approximation | (52) |

IP+EA-qb-EOM-CCD | IP+EA-EOM-CCD in the quasi-boson approximation | (51) |

IP/EA-G_{0}W_{0} | G_{0}W_{0} without particle–hole coupling | (53) |

IP/EA-ST-G_{0}W_{0} | Singles–doubles similarity transformation for particle–hole decoupling for G_{0}W_{0} | (58) |

IP/EA-UT-G_{0}W_{0} | Singles–doubles unitary transformation for particle–hole decoupling for G_{0}W_{0} | (60) |

Abbreviation . | Description . | Equations . |
---|---|---|

G_{0}W_{0} | G_{0}W_{0} supermatrix | (37) |

G_{0}W_{0}-TDA | G_{0}W_{0} supermatrix within the Tamm–Dancoff approximation | (29) |

IP+EA-EOM-CCSD | Similarity-transformed CC EOM using combined IP/EA operator basis | (41) |

IP/EA-EOM-CCSD | Similarity-transformed CC EOM using separate IP/EA operator basis | (42) and (43) |

IP+EA-qb-EOM-uCCD | IP+EA-EOM-uCCD in the quasi-boson approximation | (52) |

IP+EA-qb-EOM-CCD | IP+EA-EOM-CCD in the quasi-boson approximation | (51) |

IP/EA-G_{0}W_{0} | G_{0}W_{0} without particle–hole coupling | (53) |

IP/EA-ST-G_{0}W_{0} | Singles–doubles similarity transformation for particle–hole decoupling for G_{0}W_{0} | (58) |

IP/EA-UT-G_{0}W_{0} | Singles–doubles unitary transformation for particle–hole decoupling for G_{0}W_{0} | (60) |