# The coupled-cluster self-energy

Christopher J. N. Coveney<EMAIL_ADDRESS>Department of
Physics, University of Oxford, Oxford OX1 3PJ, United Kingdom  David P. Tew
<EMAIL_ADDRESS>Physical and Theoretical Chemistry Laboratory,
University of Oxford, South Parks Road, United Kingdom

###### Abstract

An improved description of electronic correlation in molecules and materials
can only be achieved by uncovering connections between different areas of
electronic structure theory. A general unifying relationship between the many-
body self-energy and coupled-cluster theory has remained hitherto unknown.
Here, we present a formalism for constructing the coupled-cluster self-energy
from the coupled-cluster ground state energy. Our approach illuminates the
fundamental connections between the many-body self-energy and the coupled-
cluster equations. As a consequence, we naturally arrive at the coupled-
cluster quasiparticle and Bethe-Salpeter equations describing correlated
electrons and excitons. This deep underlying structure explains the origin of
the connections between RPA, $GW$-BSE and coupled-cluster theory, whilst also
elucidating the relationship between vertex corrections and the amplitude
equations.

The behaviour of electrons in materials and molecules has been of central
importance to the scientific community since the birth of quantum mechanics
[1]. From classifying photovoltaic devices and solar cells to accurately
predicting a binding affinity of a drug to a protein, a theoretical
description of the correlated motion of electrons is key [2, 3, 4]. Therefore,
it is necessary to continue to develop a holistic understanding of the
connections between different areas of electronic structure theory to generate
increasingly accurate and scalable correlation methods.

The many-body self-energy is a functional of the exact single-particle Green’s
function, $\Sigma\equiv\Sigma[G]$ [5]. Hedin’s equations define a formally
exact procedure for generating the exact many-body self-energy [6]. However,
there exists no known functional form for the vertex and hence no closed
functional form for the self-energy. This leads to difficulty generating well
defined procedures for finding exact solutions of Hedin’s equations even for
simple model systems [7, 8, 9, 10, 11]. The inclusion of vertex corrections to
include higher-order many-body correlation effects is neither rigorous nor
trivial. The Luttinger-Ward functional is another formally exact procedure for
constructing the self-energy functional [5, 12, 13, 14, 6, 15, 16]. In this
case, the self-energy is obtained from the functional derivative of the
Luttinger-Ward functional with respect to the exact single-particle Green’s
function. This generates the same infinite-order Feynman diagram perturbation
series for the self-energy as described by Hedin’s equations, with the
additional property that self-energies which are derivable from the Luttinger-
Ward functional must obey certain conservation laws such as particle number
[6]. However, again, no known closed form expression for this functional
exists [6, 17, 18]. The lack of a functional parametrization for the self-
energy arises due to the fact that neither Hedin’s equations nor the
Luttinger-Ward functional assume a closed form expression for the exact many-
body ground state wavefunction. However, the ‘gold standard’ coupled-cluster
(CC) parametrization provides us with exactly this. We will show how this
allows for a systematically improvable expression for an approximate self-
energy.

Recent work on the supermatrix frequency-independent formulation of the
quasiparticle equation within the $GW$ and GF2 approximations have led to
significant advances in our understanding of the nature of the many-body self-
energy [19, 20, 21, 22, 23, 24]. The relationship between the random-phase
approximation (RPA), coupled-cluster doubles (CCD) and equation-of-motion
coupled-cluster theory (EOM-CC) is also now well established [25, 26, 27, 22].
The recent work on uncovering formal and numerical similarities between the
$GW$ supermatrix and an _approximate_ algebraic self-energy from ionization
potential (IP)/electron affinity (EA) equation-of-motion coupled-cluster
singles and doubles (IP/EA-EOM-CCSD) has helped to shed light on the nature of
the $GW$ self-energy [28]. Proof of an exact equivalence between the single-
shot G0W0 approximation and quasi-boson equation-of-motion ring unitary
coupled-cluster doubles theory has also recently been presented [24]. As
recently shown in Ref. [29], uncovering the connections between coupled-
cluster theory and self-energy approaches is essential for the development of
non-divergent and scalable electronic correlation theories.

No general relationship has been introduced to define the coupled-cluster
self-energy functional and its relationship to standard Green’s function
theory. This is the main subject of this work. By taking inspiration from the
connections uncovered between RPA and EOM-CCD, the supermatrix representation
of the quasiparticle equation and the Luttinger-Ward functional, we reveal the
structure of the coupled-cluster self-energy and hence its relationship to
EOM-CC, the Green’s function formalism and Kohn-Sham density functional theory
(KS-DFT). The power of this approach rests in the formally exact coupled-
cluster ansatz for the ground state wavefunction. This approach to the
coupled-cluster self-energy provides a fundamental explanation of the
equivalence between EOM-CC theory in the minimal space of determinants and
Green’s function theory [28, 26]. It can be shown to give the exact ground
state energy as well as higher-order interaction kernels, and therefore
quasiparticle states, from Green’s function theory.

In the following, indices $i,j,k,...$ denote occupied (valence band) spin-
orbitals, $a,b,c,...$ virtuals (conduction band) and $p,q,r,...$ general spin-
orbitals. The single-particle Green’s function is defined as

$iG_{pq}(t_{1}-t_{2})=\braket{\Psi_{0}}{\operatorname{T}\left[a_{p}(t_{1})a^{\dagger}_{q}(t_{2})\right]}{\Psi_{0}},$
(1)

where $a^{\dagger}_{p}/a_{p}$ create/annihilate electrons in spin-orbital
$\ket{\phi_{p}}$, $\operatorname{T}$ is the time-ordering operator and
$\ket{\Psi_{0}}$ is the normalized, exact $N$-electron ground state [30]. The
poles of the exact Green’s function correspond to the exact ionization
potentials and electron affinities of the system. From the equation of motion
of the Green’s function, the exact ground state energy can be obtained from
the Galitskii-Migdal formula [31]. Therefore, the Green’s function
simultaneously contains both the exact single-particle spectrum and ground
state energy. From the Dyson equation, it can be shown that the IPs, EAs and
Dyson orbitals can be obtained by solution of the quasiparticle equations [32,
30]

$[f+\Sigma_{c}(\omega=\varepsilon_{p})]\ket{\psi_{p}}=\varepsilon_{p}\ket{\psi_{p}}$
(2)

where $f$ is the Fock operator and $\Sigma_{c}$ is the many-body self-energy.
Whilst the self-energy is formally frequency-dependent, this frequency-
dependence is actually present in order to capture higher-order interactions
whilst remaining in the single-particle spin-orbital basis. This is confirmed
by the fact that the same quasiparticle equation may be cast in terms of a
static, upfolded supermatrix representation. The exact self-energy may be
defined by the Dyson equation [30, 33] but this is not a universal functional
relationship. We note similarity between the quasiparticle and KS-DFT
equations in Eq. 2.

The coupled-cluster Green’s function (CCGF) is obtained by observing that the
Green’s function contains the exact single-particle spectrum. As first
proposed in Refs [34, 35], one may construct the CCGF from the corresponding
EOM-CC eigenvalue problem. In CC theory, the many-body ground state is
expressed as $\ket{\Psi^{\mathrm{CC}}_{0}}=e^{T}\ket{\Phi}$, where $T$ creates
all excitations with respect to a reference determinant $\ket{\Phi}$ as

$T=\sum_{ai}t^{a}_{i}a^{\dagger}_{a}a_{i}+\frac{1}{4}\sum_{abij}t^{ab}_{ij}a^{\dagger}_{a}a^{\dagger}_{b}a_{j}a_{i}+...$
(3)

where $\\{t^{a}_{i},t^{ab}_{ij},...\\}$ are the cluster amplitudes. The
cluster amplitudes and ground state energy are obtained by projection onto the
manifold of Slater determinants,
$\braket{\Phi_{0}}{\bar{H}}{\Phi_{0}}=E^{\mathrm{CC}}_{0};\braket{\Phi^{a}_{i}}{\bar{H}}{\Phi_{0}}=0;\braket{\Phi^{ab}_{ij}}{\bar{H}}{\Phi_{0}}=0$
and so on, where $\bar{H}=e^{-T}He^{T}$ is the similarity transformed
Hamiltonian. The manifold of excited Slater determinants are given by
$\ket{\Phi^{a}_{i}}=a^{\dagger}_{a}a_{i}\ket{\Phi_{0}}$ and so on. The ground
state energy obtained from this procedure is formally exact. Importantly, the
similarity transformed Hamiltonian, $\bar{H}$, is non-Hermitian and possesses
different left and right eigenstates.

The IPs and EAs are obtained by diagonalization of $\bar{H}$ in the basis of
all Slater determinants containing ($N\pm 1$)-electrons. In EOM-CC theory, the
supermatrix splits into separate IP and EA parts. For example, the IP-EOM-CC
eigenvalue problem can be written as the following eigenvalue problem

$\bar{H}R_{k}\ket{\Phi}=\Omega^{N-1}_{k}R_{k}\ket{\Phi}$ (4)

where the action $R_{k}\ket{\Phi}$ takes us to the target excited eigenstate,
$\ket{\Psi_{k}}$. In IP-EOM-CC theory, we take the operator to be that which
creates all determinants consisting of $(N-1)$ electrons. This corresponds to
the choice

$R_{k}=\sum_{i}r_{i}(k)a_{i}+\frac{1}{2}\sum_{ija}r^{a}_{ij}(k)a^{\dagger}_{a}a_{j}a_{i}+...$
(5)

The eigenvalues, $\Omega_{k}^{N-1}$, give the exact ionization potentials of
the system, corresponding to the poles of the exact Green’s function [34, 35,
28]. The left and right eigenstates form a complete biorthogonal set in the
$(N\pm 1)$-electron space, and can be used to construct the CCGF. However, the
exact ground state correlation energy is _not_ recovered from the CCGF and the
associated self-energy is found by inverting the Dyson equation [34, 35]. This
definition of the coupled-cluster self-energy does not reveal its relationship
to the Bethe-Salpeter kernel and neutral excitation energies. Motivated by the
fact that the CCGF does not give the exact ground state correlation energy, we
introduce an alternative definition for the coupled-cluster self-energy based
on the relations from many-particle quantum theory [5, 12, 13].

The Brueckner formulation of coupled-cluster theory (BCC) employs the same
exponential ansatz for the many-body ground state wavefunction. The difference
is that the singles amplitudes, $t^{a}_{i}$, are eliminated by construction.
The corresponding expression for the exact Brueckner ground state correlation
energy is given by

$E^{\mathrm{BCC}}_{c}=\frac{1}{4}\sum_{ijab}\braket{ij}{ab}t^{ab}_{ij}\ ,$ (6)

where $\braket{ij}{ab}\equiv v_{ij,ab}-v_{ij,ba}$ are the antisymmetrized two-
electron Coulomb integrals. Clearly, the doubles amplitudes, $t^{ab}_{ij}$,
depend on all higher order amplitudes due to the structure of the projected CC
equations. It has been shown that the BCC formalism represents an effective
Hamiltonian theory as the total energy can be written in terms of an extended
Fock operator, $F$ [36, 37, 38]. The occupied-virtual block of this extended
Fock operator is given by the $T_{1}$ equation such that convergence of the
Brueckner equations requires $F_{ia}=0$ [37].

Figure 1: The series of coupled-cluster functional derivatives obtained by
cutting lines in the Goldstone diagrams.

At zero temperature, the Luttinger-Ward theorem states that the self-energy is
given by the functional derivative of the ground state correlation energy with
respect to the Green’s function [12, 13, 14, 39, 40, 15, 41, 6]

$\Sigma^{c}(1,2)=i\frac{\delta E_{c}}{\delta G(2,1)}\ ,$ (7)

where $1\equiv(\mathbf{r}_{1}\sigma_{1},t_{1})$ is the composite spin-space-
time coordinate. Setting the Green’s function to equal time in this
expression, the explicitly static self-energy is simply the correlation
potential from KS-DFT.

Within coupled-cluster theory, the ground state energy is expressed exactly in
terms of the non-interacting Green’s function as the interaction vertex is
renormalized by the excitation amplitudes. As a result, the coupled-cluster
self-energy functional takes the form,
$\tilde{\Sigma}\equiv\tilde{\Sigma}[G_{0}]$. We choose to define the coupled-
cluster self-energy as

$\tilde{\Sigma}_{ij}[G_{0}]=i\frac{\delta
E^{\mathrm{BCC}}_{c}[\mathbf{t},G_{0}]}{\delta
G^{0}_{ji}}\Bigg{|}_{\begin{subarray}{c}\mathbf{t}=t_{\mu}\end{subarray}}=i\frac{\delta
E^{\mathrm{BCC}}_{c}}{\delta\tilde{G}_{ji}}\ ,$ (8)

where $\tilde{G}_{ji}$ is the equal-time, ‘non-interacting’ Green’s function
in the basis of the Brueckner occupied orbitals and the functional derivative
is evaluated for the exact doubles amplitudes. Diagrammatically, this
functional derivative corresponds to cutting the hole lines in the Goldstone
diagram of the Brueckner correlation energy (Figure 1). The self-energy in Eq.
8 is ‘static’, originating from the ‘upfolded’ supermatrix representation
[42]. Likewise, we may define the self-energy for the virtual states as the
functional derivative

$\tilde{\Sigma}_{ab}=i\frac{\delta
E^{\mathrm{BCC}}_{c}}{\delta\tilde{G}_{ba}}\ ,$ (9)

where $\tilde{G}_{ba}$ is the equal-time, ‘non-interacting’ Green’s function
in the basis of Brueckner virtuals. As the non-interacting propagator does not
contain virtual–occupied/occupied–virtual contributions, the self-energy for
the occupied–virtual block vanishes identically,
$\tilde{\Sigma}_{ia}=\tilde{\Sigma}_{ai}=0$. Taking the functional derivative
of the BCC correlation energy as per Eq. 8, we get (see SI)

$\displaystyle\begin{split}i\frac{\delta
E^{\mathrm{BCC}}_{c}}{\delta\tilde{G}_{ji}}=\frac{1}{2}\sum_{kab}\braket{ik}{ab}t^{ab}_{jk}=\tilde{\Sigma}_{ij}\hskip
8.53581pt.\end{split}$ (10)

This is exactly the correlation part of the generalized Fock operator
introduced in Brueckner theory for the occupied states [37, 38, 28].
Importantly, the doubles amplitudes appearing here are formally exact as they
are determined from the projected coupled-cluster equations. Similarly, the
self-energy of the virtual states is given by

$i\frac{\delta
E^{\mathrm{BCC}}_{c}}{\delta\tilde{G}_{ba}}=-\frac{1}{2}\sum_{ijc}\braket{ij}{bc}t^{ac}_{ij}=\tilde{\Sigma}_{ab}\
.$ (11)

Using these results, one may write the coupled-cluster quasiparticle equations
for the occupied and virtual states in terms of the extended Fock operator

$\displaystyle\begin{split}F_{ij}\equiv f_{ij}+\tilde{\Sigma}_{ij}\hskip
5.69054pt;\hskip 5.69054ptF_{ab}\equiv f_{ab}+\tilde{\Sigma}_{ab}\
.\end{split}$ (12)

This definition of the self-energy decouples the IP and EA sectors. From Eq.
10, we obtain the exact ground state correlation energy by taking the trace of
the self-energy

$\frac{1}{2}\sum_{i}\tilde{\Sigma}_{ii}=\frac{1}{4}\sum_{ij,ab}\braket{ij}{ab}t^{ab}_{ij}=E^{\mathrm{BCC}}_{c}\
.$ (13)

This is identical to the form of the RPA and $GW$-BSE correlation energy,
thereby unifying both approaches [25, 38, 22]. From Eq. 12, we may find the
quasiparticle solutions by diagonalization of the extended Fock operator,
where the $N$ principle solutions give the quasiparticle energies and
wavefunctions. The advantage of this approach resides in the separation of the
IP and EA sectors, yielding a structure similar to equation-of-motion coupled-
cluster theory (Figure 2). These eigenvalues automatically correspond to
ionization energies and electron affinities, providing us with a correlated
generalized Koopman’s theorem [43]. The extended Fock operator exactly
corresponds to the $T_{2}$ transformed Hamiltonian in the minimal space of one
hole/particle (1h)/(1p) Slater determinants

$F_{ij}=\braket{\Phi_{i}}{e^{-T_{2}}H_{N}e^{T_{2}}}{\Phi_{j}}\hskip
5.69054pt;\hskip
5.69054ptF_{ab}=\braket{\Phi^{a}}{e^{-T_{2}}H_{N}e^{T_{2}}}{\Phi^{b}},$ (14)

Figure 2: Schematic of the relationship between the coupled-cluster self-
energy and EOM-CC supermatrices.

where the $T_{2}$ amplitudes are exact. The dynamical degrees of freedom of
the self-energy are simply re-expressed in terms of the static coupled-cluster
amplitudes. In the supermatrix representation, the $GW$ and GF2 approximations
couple the IP and EA sectors, implicitly containing orbital relaxation despite
giving an inexact ground state energy and spectrum [24, 28, 19]. It can be
shown that the extended Fock matrix may also be related to KS-DFT using the
similarity transformed Hamiltonian.

The two-particle Bethe-Salpeter interaction kernel is given by the functional
derivative [5, 44]

$\Xi(1,2;3,4)=i\frac{\delta\Sigma(1,3)}{\delta G(4,2)}\ .$ (15)

Depending on the relative time-ordering of the field operators, the two-
particle Green’s function may describe particle-hole or particle-particle
correlations [38, 26]. We similarly define the analogous particle-hole
interaction kernel from the CC self-energy as

$\Xi^{c}_{ia,jb}=i\frac{\delta\tilde{\Sigma}_{ij}}{\delta\tilde{G}_{ba}}\ .$
(16)

This functional derivative gives the coupled-cluster two-particle interaction
kernel from the Bethe-Salpeter equation (BSE) in the space of single excited
determinants

$\displaystyle\begin{split}i\frac{\delta\tilde{\Sigma}_{ij}}{\delta\tilde{G}_{ba}}=\sum_{kc}\braket{ik}{bc}t^{ca}_{jk}=\Xi^{c}_{ia,jb}\hskip
8.53581pt.\end{split}$ (17)

This is exactly the form of the correlated part of the $T_{2}$ transformed
interaction in the space of singly excited determinants [26, 28, 38]. It
should be noted that this kernel retains the fermionic symmetry of the many-
body wavefunction via the doubles amplitudes. Combining the functional
derivative of the HF self-energy together with Eq. 17, we generate the full
kernel and write the effective CC-BSE Hamiltonian in terms of the coupled-
cluster doubles amplitudes as (see SI) [26, 25, 22]

$~{}\bar{H}^{\mathrm{BSE}}_{ia,jb}=F_{ab}\delta_{ij}-F_{ij}\delta_{ab}+\braket{ia}{jb}+\sum_{kc}\braket{ik}{bc}t^{ca}_{jk}$
(18)

whose eigenvalues give the exciton energies. This effective Hamiltonian is
exactly the $T_{2}$ transformed Hamiltonian in the space of singly excited
determinants [28, 26, 27]

$\displaystyle\begin{split}\bar{H}^{\mathrm{BSE}}_{ia,jb}&\equiv\braket{\Phi^{a}_{i}}{e^{-T_{2}}H_{N}e^{T_{2}}}{\Phi^{b}_{j}}\
,\end{split}$ (19)

where the $T_{2}$ amplitudes are solved for exactly. This is equivalent to the
upper-left block from the excitation energy (EE)-EOM-CC treatment (Figure 2).
Through the identities derived above (Eqs 17 and 18), the relationship between
IP- and EE-EOM-CC theory is clear.

The random phase approximation (RPA) and $GW$-BSE approximation are formulated
in the space of singly excited determinants and the neutral excitation
energies are obtained from the eigenvalue problem [45, 22, 25]

$\left(\begin{array}[]{cc}\mathbf{A}&\mathbf{B}\\\
-\mathbf{B}^{*}&-\mathbf{A}^{*}\end{array}\right)\left(\begin{array}[]{c}\mathbf{X}\\\
\mathbf{Y}\end{array}\right)=\left(\begin{array}[]{c}\mathbf{X}\\\
\mathbf{Y}\end{array}\right)\mathbf{\Omega}\ ,$ (20)

where $A_{ia,jb}=\Delta^{a}_{i}\delta_{ab}\delta_{ij}+V_{ia,jb}$ and
$B_{ia,jb}=V_{ij,ab}$. Within the RPA,
$\Delta^{a}_{i}=\epsilon_{a}-\epsilon_{i}$, $V_{ia,jb}=\braket{ia}{jb}$ and
$B_{ia,jb}=\braket{ij}{ab}$. In the $GW$-BSE approach,
$\Delta^{a}_{i}=\epsilon^{GW}_{a}-\epsilon^{GW}_{i}$,
$V_{ia,jb}=\braket{ia}{jb}-W_{ia,jb}$ and
$B_{ia,jb}=\braket{ij}{ab}-W_{ij,ab}$, where $W$ is the static screened
interaction [32, 44]. The corresponding eigenstates of Eq. 20 are known as
excitons, namely correlated electron-hole pairs. It has been shown that RPA
reduces to a form of CCD where the doubles amplitudes are solved for by
keeping only the so-called ‘ring’ contractions (rCCD), and recently a similar
structure for effective doubles amplitudes has been uncovered for $GW$-BSE
[25, 22]. The relationship between RPA, $GW$-BSE and the doubles amplitudes is
obtained by identifying that the doubles amplitude is given by
$t^{ab}_{ij}\equiv\mathbf{T}=\mathbf{YX}^{-1}$ [45, 25]. Using this relation,
Eq. 20 yields the following Ricciatti equation for the doubles amplitude [25]

$~{}\mathbf{B}^{*}+\mathbf{A^{*}T}+\mathbf{TA}+\mathbf{TBT}=0\ .$ (21)

These are exactly the rCCD amplitude equations. The resulting RPA/$GW$-BSE
equations can therefore be written as [26, 22]

$\mathbf{H}^{\mathrm{RPA}}=\mathbf{A}+\mathbf{BT}\ ,$ (22)

which, using the definition of the matrix elements defined above, gives
exactly the same structure as in Eq. 18

$H_{ia,jb}^{\mathrm{RPA}}=\Delta^{a}_{i}\delta_{ij}\delta_{ab}+\braket{ia}{jb}+\sum_{kc}\braket{ik}{bc}t^{ca}_{jk}\
,$ (23)

remembering that the doubles amplitudes are determined from the rCCD equations
(Eq. 21).

Figure 3: Relationship between the coupled-cluster self-energy, RPA, $GW$-BSE
and CC-BSE.

Here, we see the connection between our formalism and RPA in a simple and
clear way. When higher-order excitation processes are neglected, as is the
case for RPA where
$R_{k}=\sum_{ai}X^{a}_{i}(k)a^{\dagger}_{a}a_{i}-Y^{i}_{a}(k)a^{\dagger}_{i}a_{a}$,
we see that the structure of our derived CC-BSE effective Hamiltonian,
$\bar{H}^{\mathrm{BSE}}$, is identical to the RPA/$GW$-BSE treatment. Our
general formalism reduces to the RPA eigenvalue problem when the effects of
the extended Fock operator are neglected and the doubles amplitude are solved
via the rCCD equations [27, 26, 38]. Likewise, the $GW$-BSE approximation can
be obtained by using the $GW$ eigenvalues for the valence and conduction bands
and the screened instead of bare interaction in the rCCD amplitude equations
[21, 19, 20]. Therefore, in RPA and $GW$-BSE, the rCCD amplitudes miss
important additional contributions from the full CC equations (Figure 3). Our
formalism provides a simple and well-defined way to improve upon both the RPA
and $GW$-BSE approximations by including three body terms via perturbative
triples amplitudes (or better) as well as the effects of the extended Fock
operator. It is clear from this analysis that the presence of the extended
Fock operator is a result of the dressing of the single-particle Green’s
function. Likewise, the doubles amplitude equations can be likened to an
iterative approach to including vertex corrections to the self-energy [46, 47,
48, 49, 50, 51, 52, 53]. This correspondence can be identified from the
structure of the doubles amplitude equations which include higher-order
correlation effects depending on the truncation of the cluster operator, $T$.
Inclusion of higher-order excitations in the cluster operator and iterative
solution of the resulting doubles amplitude equations systematically includes
higher-order correlation effects analogous to the vertex function appearing in
Hedin’s equations.

Likewise, if we want to describe the particle-particle (2p) or hole-hole (2h)
two-particle correlations, via a different time-ordering of the two-particle
Green’s function, we may define these kernels as

$\displaystyle\begin{split}\Xi^{c}_{ab,cd}=i\frac{\delta\tilde{\Sigma}_{ac}}{\delta\tilde{G}_{db}}\hskip
7.11317pt;\hskip
7.11317pt\Xi^{c}_{ij,kl}=i\frac{\delta\tilde{\Sigma}_{ik}}{\delta\tilde{G}_{lj}}\
.\end{split}$ (24)

For the particle-particle and hole-hole effective interaction we have

$\displaystyle\begin{split}i\frac{\delta\tilde{\Sigma}_{ac}}{\delta\tilde{G}_{db}}=\sum_{i<j}\braket{ij}{cd}t^{ab}_{ij}\hskip
7.11317pt;\hskip
7.11317pti\frac{\delta\tilde{\Sigma}_{ik}}{\delta\tilde{G}_{lj}}=\sum_{a<b}\braket{ij}{ab}t^{ab}_{kl}\
.\end{split}$ (25)

These are exactly the relationships found when solving the particle-particle
RPA problem [26] and correspond to the projection of the $T_{2}$ similarity
transformed interaction in the minimal basis of 2p or 2h determinants [54].
The equations derived here demonstrate the equivalence between the BSE, ground
state CC and EOM-CC theory in a minimal basis of determinants. It is now clear
why only the $T_{2}$ transformed Hamiltonian has appeared in previous work on
connecting coupled-cluster to RPA [25, 22].

We see that the exact ground state correlation energy is obtained by taking
the trace over the interaction kernels

$\displaystyle\begin{split}\frac{1}{2}\operatorname{tr}\mathbf{\Sigma}=\frac{1}{4}\operatorname{tr}\mathbf{\Xi^{c}}=\frac{1}{4}\sum_{ijab}\braket{ij}{ab}t^{ab}_{ij}=E^{\mathrm{BCC}}_{c}.\end{split}$
(26)

This is similar to the elegant proof of the relationship between the RPA and
CCD ground state correlation energies [25]. However, here we have shown that
this relationship holds more generally. Importantly, it is the CC ansatz which
provides us with a master equation for the doubles amplitudes and therefore
the _exact_ ground state correlation energy.

The relationship to EOM-CC theory can now be made explicit (Figure 2). In
regular IP-EOM-CC, the similarity transformed Hamiltonian is diagonalized in
the basis of 1h and 2h1p determinants. By neglecting the off-diagonal
couplings between the 1h and 2h1p sectors in the IP-EOM-CC approach, we are
left with exactly the resulting expressions obtained for the self-energy and
its higher-order functional derivatives. However, in our approach, formally
the doubles amplitudes are solved for exactly. Similarly, we see the
equivalency for neutral excitations from EE-EOM-CC theory. By neglecting the
off-diagonal couplings between the singly (1p1h) and doubly (2p2h) excited
determinants, we are again left with exactly the expressions obtained from the
CC-BSE and third-order functional derivative of the self-energy. The higher-
order relaxation effects encoded in the off-diagonal elements of the EOM-CC
supermatrices are not captured in this framework.

In summary, we have introduced a natural definition of the CC self-energy that
unifies elements of CC theory with the Green’s function formalism. By
appealing to the CC similarity transformed Hamiltonian, we present a closed
form functional expression for the self-energy which is systematically
improvable via the CC amplitude equations. This allows us to derive a
correlated extension of Koopmann’s theorem and correlated exciton states from
the formally exact ground state energy. We have discovered that the EOM-CC
treatment naturally emerges as a consequence of our formalism, providing us
with a simple way to improve over $GW$-BSE and RPA for ground state and
excitation energies. The connections uncovered in this work directly connect
IP- and EE-EOM-CC theory by identifying the relationship between the single-
particle self-energy and Bethe-Salpeter kernel. We hope that our contribution
will stimulate renewed work on novel electronic structure theories for
correlated excited state phenomena.

## References

  * Schrödinger [1926] E. Schrödinger, An undulatory theory of the mechanics of atoms and molecules, Physical Review 28, 1049 (1926).
  * Grancini  _et al._ [2013] G. Grancini, M. Maiuri, D. Fazzi, A. Petrozza, H.-J. Egelhaaf, D. Brida, G. Cerullo, and G. Lanzani, Hot exciton dissociation in polymer solar cells, Nature Materials 12, 29 (2013).
  * Dimitriev [2022] O. P. Dimitriev, Dynamics of excitons in conjugated molecules and organic semiconductor systems, Chemical Reviews 122, 8487 (2022).
  * Mollaamin and Monajjemi [2012] F. Mollaamin and M. Monajjemi, DFT outlook of solvent effect on function of nano bioorganic drugs, Physics and Chemistry of Liquids 50, 596 (2012).
  * Hedin [1965] L. Hedin, New method for calculating the one-particle Green’s function with application to the electron-gas problem, Physical Review 139, A796 (1965).
  * Stefanucci and Van Leeuwen [2013] G. Stefanucci and R. Van Leeuwen,  _Nonequilibrium many-body theory of quantum systems: a modern introduction_ (Cambridge University Press, 2013).
  * Verdozzi  _et al._ [1995] C. Verdozzi, R. Godby, and S. Holloway, Evaluation of $GW$ approximations for the self-energy of a Hubbard cluster, Physical Review Letters 74, 2327 (1995).
  * Von Friesen  _et al._ [2009] M. P. Von Friesen, C. Verdozzi, and C.-O. Almbladh, Successes and failures of Kadanoff-Baym dynamics in Hubbard nanoclusters, Physical Review Letters 103, 176404 (2009).
  * Romaniello  _et al._ [2009] P. Romaniello, S. Guyot, and L. Reining, The self-energy beyond $GW$: Local and nonlocal vertex corrections, The Journal of Chemical Physics 131, 154111 (2009).
  * von Friesen  _et al._ [2010] M. P. von Friesen, C. Verdozzi, and C.-O. Almbladh, Kadanoff-Baym dynamics of Hubbard clusters: Performance of many-body schemes, correlation-induced damping and multiple steady and quasi-steady states, Physical Review B 82, 155108 (2010).
  * Romaniello  _et al._ [2012] P. Romaniello, F. Bechstedt, and L. Reining, Beyond the $GW$ approximation: Combining correlation channels, Physical Review B 85, 155131 (2012).
  * Kohn and Luttinger [1960] W. Kohn and J. Luttinger, Ground-state energy of a many-fermion system, Physical Review 118, 41 (1960).
  * Luttinger and Ward [1960] J. M. Luttinger and J. C. Ward, Ground-state energy of a many-fermion system. ii, Physical Review 118, 1417 (1960).
  * Luttinger [1961] J. Luttinger, Analytic properties of single-particle propagators for many-fermion systems, Physical Review 121, 942 (1961).
  * Potthoff [2014] M. Potthoff,  _Making Use of Self-Energy Functionals: The Variational Cluster Approximation_ (e-print, 2014).
  * van Setten  _et al._ [2013] M. J. van Setten, F. Weigend, and F. Evers, The $GW$-method for quantum chemistry applications: Theory and implementation, Journal of Chemical Theory and Computation 9, 232 (2013).
  * Pokhilko  _et al._ [2021] P. Pokhilko, S. Iskakov, C.-N. Yeh, and D. Zgid, Evaluation of two-particle properties within finite-temperature self-consistent one-particle Green's function methods: Theory and application to $GW$ and GF2, The Journal of Chemical Physics 155, 024119 (2021).
  * Martin  _et al._ [2016] R. M. Martin, L. Reining, and D. M. Ceperley,  _Interacting electrons_ (Cambridge University Press, 2016).
  * Bintrim and Berkelbach [2021] S. J. Bintrim and T. C. Berkelbach, Full-frequency $GW$ without frequency, The Journal of Chemical Physics 154, 041101 (2021).
  * Bintrim and Berkelbach [2022] S. J. Bintrim and T. C. Berkelbach, Full-frequency dynamical Bethe–Salpeter equation without frequency and a study of double excitations, The Journal of Chemical Physics 156, 044114 (2022).
  * Scott  _et al._ [2023] C. J. Scott, O. J. Backhouse, and G. H. Booth, A moment-conserving reformulation of $GW$ theory, The Journal of Chemical Physics 158 (2023).
  * Quintero-Monsebaiz  _et al._ [2022] R. Quintero-Monsebaiz, E. Monino, A. Marie, and P.-F. Loos, Connections between many-body perturbation and coupled-cluster theories, The Journal of Chemical Physics 157, 231102 (2022).
  * Monino and Loos [2023] E. Monino and P.-F. Loos, Connections and performances of green’s function methods for charged and neutral excitations, arXiv preprint arXiv:2305.16959 (2023).
  * Tölle and Kin-Lic Chan [2023] J. Tölle and G. Kin-Lic Chan, Exact relationships between the $GW$ approximation and equation-of-motion coupled-cluster theories through the quasi-boson formalism, The Journal of Chemical Physics 158 (2023).
  * Scuseria  _et al._ [2008] G. E. Scuseria, T. M. Henderson, and D. C. Sorensen, The ground state correlation energy of the random phase approximation from a ring coupled cluster doubles approach, The Journal of Chemical Physics 129, 231101 (2008).
  * Berkelbach [2018] T. C. Berkelbach, Communication: Random-phase approximation excitation energies from approximate equation-of-motion coupled-cluster doubles, The Journal of Chemical Physics 149, 041103 (2018).
  * Rishi  _et al._ [2020] V. Rishi, A. Perera, and R. J. Bartlett, A route to improving rpa excitation energies through its connection to equation-of-motion coupled cluster theory, The Journal of Chemical Physics 153, 234101 (2020).
  * Lange and Berkelbach [2018] M. F. Lange and T. C. Berkelbach, On the relation between equation-of-motion coupled-cluster theory and the $GW$ approximation, Journal of Chemical Theory and Computation 14, 4224 (2018).
  * Coveney and Tew [2023] C. J. N. Coveney and D. P. Tew, A regularized second-order correlation method from Green’s function theory, Journal of Chemical Theory and Computation (2023).
  * Fetter and Dirk [1971] A. Fetter and J. Dirk,  _Quantum Theory of Many Particle Systems_ (McGraw-Hill, 1971).
  * Galitskii and Migdal [1958] V. M. Galitskii and A. B. Migdal, Application of quantum field theory methods to the many-body problem, Sov. Phys. JETP 7, 18 (1958).
  * Hybertsen and Louie [1986] M. S. Hybertsen and S. G. Louie, Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies, Physical Review B 34, 5390 (1986).
  * Mahan [2000] G. D. Mahan,  _Many-particle physics_ (Springer Science & Business Media, 2000).
  * Nooijen and Snijders [1992] M. Nooijen and J. G. Snijders, Coupled-Cluster approach to the single-particle Green’s function, International Journal of Quantum Chemistry 44, 55 (1992).
  * Nooijen and Snijders [1993] M. Nooijen and J. G. Snijders, Coupled-Cluster Green’s function method: Working equations and applications, International Journal of Quantum Chemistry 48, 15 (1993).
  * Scuseria [1994] G. E. Scuseria, An alternative expression for the perturbative triples correction to Brueckner coupled-cluster doubles theory, Chemical Physics Letters 226, 251 (1994).
  * Scuseria [1995] G. E. Scuseria, On the connections between Brueckner–coupled-cluster, density-dependent Hartree–Fock, and density functional theory, International Journal of Quantum Chemistry 55, 165 (1995).
  * Scuseria  _et al._ [2013] G. E. Scuseria, T. M. Henderson, and I. W. Bulik, Particle-particle and quasiparticle random phase approximations: Connections to coupled-cluster theory, The Journal of Chemical Physics 139, 104113 (2013).
  * Backhouse  _et al._ [2020] O. J. Backhouse, M. Nusspickel, and G. H. Booth, Wave function perspective and efficient truncation of renormalized second-order perturbation theory, Journal of Chemical Theory and Computation 16, 1090 (2020).
  * Phillips and Zgid [2014] J. J. Phillips and D. Zgid, Communication: The description of strong correlation within self-consistent Green's function second-order perturbation theory, The Journal of Chemical Physics 140, 241101 (2014).
  * van Setten  _et al._ [2015] M. J. van Setten, F. Caruso, S. Sharifzadeh, X. Ren, M. Scheffler, F. Liu, J. Lischner, L. Lin, J. R. Deslippe, S. G. Louie, C. Yang, F. Weigend, J. B. Neaton, F. Evers, and P. Rinke, $GW$ 100: Benchmarking $G_{0}W_{0}$ for molecular systems, Journal of Chemical Theory and Computation 11, 5665 (2015).
  * Barbieri and Carbone [2017] C. Barbieri and A. Carbone, Self-consistent green’s function approaches, An Advanced Course in Computational Nuclear Physics: Bridging the Scales from Quarks to Neutron Stars , 571 (2017).
  * Phillips [1961] J. Phillips, Generalized Koopmans’ theorem, Physical Review 123, 420 (1961).
  * Rohlfing and Louie [2000] M. Rohlfing and S. G. Louie, Electron-hole excitations and optical spectra from first principles, Physical Review B 62, 4927 (2000).
  * Ring and Schuck [2004] P. Ring and P. Schuck,  _The nuclear many-body problem_ (Springer Science & Business Media, 2004).
  * Schindlmayr and Godby [1998] A. Schindlmayr and R. W. Godby, Systematic vertex corrections through iterative solution of Hedin’s equations beyond the $GW$ approximation, Physical Review Letters 80, 1702 (1998).
  * Molinari [2005] L. G. Molinari, Hedin’s equations and enumeration of Feynman diagrams, Physical Review B 71, 113102 (2005).
  * Ness  _et al._ [2011] H. Ness, L. Dash, M. Stankovski, and R. Godby, $GW$ approximations and vertex corrections on the Keldysh time-loop contour: Application for model systems at equilibrium, Physical Review B 84, 195114 (2011).
  * Lani  _et al._ [2012] G. Lani, P. Romaniello, and L. Reining, Approximations for many-body Green’s functions: insights from the fundamental equations, New Journal of Physics 14, 013056 (2012).
  * Kutepov [2017] A. L. Kutepov, Self-consistent solution of Hedin’s equations: semiconductors and insulators, Physical Review B 95, 195120 (2017).
  * Harkov  _et al._ [2021] V. Harkov, A. I. Lichtenstein, and F. Krien, Parametrizations of local vertex corrections from weak to strong coupling: Importance of the Hedin three-leg vertex, Physical Review B 104, 125141 (2021).
  * Kutepov [2022] A. L. Kutepov, Full versus quasiparticle self-consistency in vertex-corrected $GW$ approaches, Physical Review B 105, 045124 (2022).
  * Mejuto-Zaera and Vlček [2022] C. Mejuto-Zaera and V. Vlček, Self-consistency in $GW\Gamma$ formalism leading to quasiparticle-quasiparticle couplings, Physical Review B 106, 165129 (2022).
  * Gauss and Stanton [1995] J. Gauss and J. F. Stanton, Coupled-cluster calculations of nuclear magnetic resonance chemical shifts, The Journal of Chemical Physics 103, 3561 (1995).