# On the exact limit of the time-dependent coupled cluster ansatz and its
approximations in the real-time equation-of-motion coupled cluster cumulant
Green’s function approach

Bo Peng<EMAIL_ADDRESS>Physical and Computational Science Directorate,
Pacific Northwest National Laboratory, Richland, Washington, 99354, USA
Himadri Pathak  Physical and Computational Science Directorate, Pacific
Northwest National Laboratory, Richland, Washington, 99354, USA  Ajay Panyala
Advanced Computing, Mathematics, and Data Division, Pacific Northwest National
Laboratory, Richland, Washington 99354, USA  Fernando D. Vila  Department of
Physics, University of Washington, Seattle, WA 98195  John J. Rehr  Department
of Physics, University of Washington, Seattle, WA 98195  Karol Kowalski
Physical and Computational Science Directorate, Pacific Northwest National
Laboratory, Richland, Washington, 99354, USA

(August 28, 2024)

###### Abstract

In this paper, we analyze the properties of the recently proposed real-time
equation-of-motion coupled-cluster (RT-EOM-CC) cumulant Green’s function
approach [J. Chem. Phys. 2020, 152, 174113]. We specifically focus on
identifying the limitations of the original time-dependent coupled cluster
(TDCC) ansatz and propose an enhanced extended TDCC ansatz ensuring the
exactness in the expansion limit. Additionally, we introduce a practical
cluster-analysis-based approach for characterizing the peaks in the computed
spectral function from the RT-EOM-CC cumulant Green’s function approach, which
is particularly useful for the assignments of satellite peaks when many-body
effects dominate the spectra. Our preliminary numerical tests focus on
reproducing, approximating, and characterizing the exact impurity Green’s
function of the three-site and four-site single impurity Anderson models using
the RT-EOM-CC cumulant Green’s function approach. The numerical tests allow us
to have a direct comparison between the RT-EOM-CC cumulant Green’s function
approach and other Green’s function approaches in the numerical exact limit.

##  I Introduction

Attosecond laser pulses exhibit a broad spectral range and relatively high
intensity, pioneering ultrafast research such as delayed photoemission,[1, 2,
3] electronic response to sudden ionization,[4, 5, 6] charge localization and
transfer in molecules,[7, 8] autoionization absorption,[9] and conductivity
control in dielectrics,[10] to name a few. Typically, in instantaneous
processes lasting up to a few femtoseconds, the electronic dynamics can be
considered free from nuclear motion, allowing theoretical treatments to focus
solely on solving the time-dependent electronic Schrödinger equation within
the Born-Oppenheimer approximation, which directly corresponds to experimental
setups. Various time-dependent electronic structure methods have been
developed in both the frequency and time domains. For example, real-time time-
dependent density-functional theory [11] often demonstrates a reasonable
balance between computational efficiency and accuracy. Nevertheless, for
dynamics of the electronic excited states that feature strong electron
correlation and often involve multiple configurations, multi-configurational
time-dependent Hartree-Fock methods [12, 13, 14] or active space self-
consistent field methods [15, 16, 17] can be employed for higher accuracy,
albeit with very demanding computational costs. Alternative high-level
approaches that exhibit modest polynomial scaling with the capability of
systematic improvement in accuracy usually focus on time-dependent coupled
cluster (TDCC) theory.[18, 19, 20] Previous efforts have demonstrated the
capability of the TDCC formulation in identifying excitation energy,[21]
computing linear response properties,[19] computing spectral functions[22] and
linear absorption spectra in ultraviolet and X-ray energy regimes,[23, 24, 25,
26] including the incorporation of relativistic wave functions,[27] finite
temperature and non-equilibrium formalism,[28, 29] reduced scaling
schemes,[30] and adaptive numerical integration.[31] Remarkably, besides
electronic dynamics, TDCC theory also applies to nuclear dynamics[32, 33, 34]
and vibrational problems.[35, 36]

On the other hand, the Green’s function (GF) approach[37] is often employed to
treat electron correlation in excited electronic states. Electron correlation
is crucial for determining and characterizing the quasiparticle (QP) and
satellite peaks observed in, for example, X-ray photoemission spectra (XPS).
To effectively capture this correlation in excited states, various theoretical
approaches have incorporated the Green’s function formulation. These include
perturbative treatments,[38, 39, 40] algebraic diagrammatic construction,[41,
42, 43, 44] dynamical mean-field theory,[45, 46] GW approximations,[47, 48,
49, 50] and ground state coupled cluster theory.[51, 52, 53, 54, 55, 56, 57,
58]

Combining TDCC theory with Green’s function formalism, Schönhammer and
Gunnarsson have demonstrated the computation of the core-hole Green’s function
from the phase factors of the TDCC wave function.[22] Building on their
formulation, we have recently developed a real-time equation-of-motion coupled
cluster (RT-EOM-CC) cumulant GF approach,[59, 60, 61, 62, 63] in which the
Green’s function formulation adopts an exponential cumulant form to build up
correlation in excited states, analogous to ground state coupled cluster
theory. Technically, the cumulant is obtained by solving coupled ordinary
differential equations of the TDCC amplitudes, providing higher-order vertex
corrections to the one-particle self-energy compared to the traditional
cumulant approximation[64, 65, 54] and the stochastic vertex
approximation.[66] Numerical results have shown the applications of our RT-
EOM-CC cumulant GF approach in reproducing the XPS of small-to-medium
molecules described by moderate basis sets.[61, 62, 63] With heterogeneous
parallel implementation and tensor algebraic techniques, RT-EOM-CC simulations
of over 2,000 spin-orbitals have been achieved.[67]

While advancing towards larger-scale RT-EOM-CC simulations to resolve the
many-body effects in the spectroscopy of more complex molecular systems,
another fundamental aspect to consider is the exactness of the introduced TDCC
ansatz in the computation of Green’s functions and its alignment with the
actual many-body physical picture of electron transitions. Our previous RT-
EOM-CC results in computing spectral functions associated with QP and
satellites, when compared with other theoretical approaches, show great
agreement with experimental results. For instance, within the single and
double excitation manifold, RT-EOM-CC results for some small molecules seemed
more accurate than those obtained with CCGF using the modest basis set.[61,
62, 63] However, in weakly correlated scenarios within a single reference
framework, CCGF can be exact in the expansion limit. It thus remains unclear
whether the same exactness can be achieved with the RT-EOM-CC approach.
Additionally, in previous treatments,[22] the TDCC ansatz was defined in the
$(N-1)$-particle space, with the incorporation between the $N$ and
$(N-1)$-particle spaces approximated through the Hamiltonian transformation
between different Fock spaces. In this paper, by explicitly considering both
the $N$ and $(N-1)$-particle spaces in the RT-EOM-CC cumulant GF approach, we
examine the quality of the previous TDCC ansatz, and propose an enhanced new
TDCC ansatz and its approximations that are based on the extended CC theory
and are capable of incorporating different Fock spaces without the
modification in the Hamiltonian. We then analyze the impact of different
ansatze on the computed Green’s functions. Moreover, we propose a scheme for
addressing the component analysis of the Green’s function computed by the RT-
EOM-CC approach, which provides a powerful tool for characterization and peak
assignment of the computed spectral functions. Our preliminary numerical test
focuses on the single-impurity Anderson model (SIAM) with a limited number of
bath sites, where high-level theoretical results and exact solutions can be
obtained to test the performance and determine the exact limits of our
proposed TDCC ansatze in RT-EOM-CC simulations.

##  II Theory

###  II.1 One-particle Green’s function

The (retarded) one-particle Green’s function at spin-orbital $c$ is defined as

$\displaystyle G_{c}^{\rm{R}}(t)$
$\displaystyle=-i\Theta(t)\langle\Psi_{c}^{(N-1)}|e^{-i(H-E_{g}^{(N)})t}|\Psi_{c}^{(N-1)}\rangle$
$\displaystyle=-i\Theta(t)e^{iE_{g}^{(N)}t}\langle\Psi^{(N)}|a_{c}^{\dagger}e^{-iHt}a_{c}|\Psi^{(N)}\rangle$
(1)

where $H$ is the electronic Hamiltonian defined on the Fock space, and
$|\Psi_{c}^{(N-1)}\rangle$ is the $(N-1)$-electron state generated by removing
one core electron at the spin-orbital $|c\rangle$ from the $N$-electron ground
state wave function $|\Psi^{(N)}\rangle$, i.e.
$|\Psi_{c}^{(N-1)}\rangle=a_{c}|\Psi^{(N)}\rangle$ with $a_{c}$ and
$a_{c}^{\dagger}$ being the fermionic annihilation and creation operators,
respectively. $E_{g}^{(N)}$ is the ground state energy of the $N$-electron
system.

Employing the bi-orthogonal coupled cluster (CC) wave function ansatze [68,
69, 70, 71, 72, 73, 74, 51, 52, 53, 75, 76, 77, 55, 78, 79] with respect to
the $N$-electron single Slater determinant $|\phi_{0}^{(N)}\rangle$

$\displaystyle|\Psi^{(N)}\rangle$
$\displaystyle=e^{T^{(N)}}|\phi_{0}^{(N)}\rangle;$ (2)
$\displaystyle\langle\Psi^{(N)}|$
$\displaystyle=\langle\phi_{0}^{(N)}|(1+\Lambda^{(N)})e^{-T^{(N)}},$ (3)

the one-particle Green’s function can be rewritten as

$\displaystyle G_{c}^{\rm{R}}(t)=$
$\displaystyle-i\Theta(t)e^{iE_{CC}^{(N)}t}\times$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\langle\phi_{0}^{(N-1)}|(1+L^{(N)})e^{-T^{(N)}}e^{-iHt}e^{T^{(N)}}|\phi_{0}^{(N-1)}\rangle.$
(4)

Here, the $N$-electron ground state energy $E^{(N)}_{g}$ is replaced by the
corresponding CC energy $E^{(N)}_{CC}$. The excitation and de-excitation CC
operators, $T^{(N)}$ and $\Lambda^{(N)}$, are defined as follows:

$\displaystyle T^{(N)}$ $\displaystyle=\sum_{n}t_{n}^{(N)}\mathbb{E}_{n};$ (5)
$\displaystyle\Lambda^{(N)}$
$\displaystyle=\sum_{n}l_{n}^{(N)}\mathbb{E}_{n}^{\dagger}$ (6)

with $t_{n}$, $l_{n}$, and $s_{n}$ representing the amplitudes, and
$\mathbb{E}_{n}$ and $\mathbb{E}_{n}^{\dagger}$ being the excitation and de-
excitation generation operators, respectively. These operators are labeled by
the compound index $n$ for single, double, or higher order excitations. For
example, for single and double excitations,

$\displaystyle n^{(N)}=(p,q),\leavevmode\nobreak\ \leavevmode\nobreak\
|n^{(N)}\rangle=\mathbb{E}_{n}|\phi_{0}^{(N)}\rangle=a_{p}^{\dagger}a_{q}|\phi_{0}^{(N)}\rangle;$
$\displaystyle n^{(N)}=(p,q,r,s),\leavevmode\nobreak\ \leavevmode\nobreak\
|n^{(N)}\rangle=\mathbb{E}_{n}|\phi_{0}^{(N)}\rangle=a_{p}^{\dagger}a_{q}^{\dagger}a_{s}a_{r}|\phi_{0}^{(N)}\rangle,$

here the indices $p,q,r,s\cdots$ label the spin-orbitals involved in the
excitations. In Eq. (1), utilizing the commutative property
$[a_{c},T^{(N)}]=0$, it follows that

$\displaystyle|\Psi_{c}^{(N-1)}\rangle=a_{c}e^{T^{(N)}}|\phi_{0}^{(N)}\rangle=e^{T^{(N)}}|\phi_{0}^{(N-1)}\rangle$
(7)

with $|\phi_{0}^{(N-1)}\rangle=a_{c}|\phi_{0}^{(N)}\rangle$ being the
$(N-1)$-electron single determinant. Throughout this paper, we use the
subscript $(N)$ or $(N-1)$ in the operators for labeling the Hilbert space in
which the operators are defined.

###  II.2 Time-dependent equation-of-motion coupled cluster ansatz

In the previous time-dependent equation-of-motion coupled cluster (EOM-CC)
ansatz,[22, 80] the $N$-electron state $|\Psi^{(N)}\rangle$, and therefore the
$(N-1)$-electron state at $t=0$,
$|\Psi_{c}^{(N-1)}(0)\rangle=|\Psi_{c}^{(N-1)}\rangle$, were assumed to be
single Slater determinants. The time evolution of the $(N-1)$-electron state,
$|\Psi_{c}^{(N-1)}(t)\rangle$, for any time $t$, is described by the following
TDCC ansatz:

$\displaystyle|\Psi_{c}^{(N-1)}(t)\rangle:=N_{c}(t)e^{T^{(N-1)}(t)}|\Psi_{c}^{(N-1)}(0)\rangle,$
(8)

where $N_{c}(t)$ is a time-dependent normalization factor. The time-dependent
CC operator $T^{(N-1)}(t)$ in Eq. (8), defined in the $(N-1)$-particle space,
includes transitions from the occupied spin-orbitals to the hole, which are
absent in $T^{(N)}$.

Integrating the ansatz (8) into the time-dependent Schrödinger equation (TDSE)
and after some reformulations, we derive the equations-of-motion (EOMs) for
the normalization factor $N_{c}(t)$ and the CC amplitudes $t^{(N-1)}_{n}(t)$:

$\displaystyle i\partial\ln N_{c}(t)/\partial t$
$\displaystyle=\langle\Psi_{c}^{(N-1)}(0)|\bar{H}(t)|\Psi_{c}^{(N-1)}(0)\rangle$
$\displaystyle=E_{CC}^{(N-1)}(t),$ (9) $\displaystyle i\partial
t^{(N-1)}_{n}(t)/\partial t$ $\displaystyle=\langle
n^{(N-1)}|\bar{H}(t)|\Psi_{c}^{(N-1)}(0)\rangle,$ (10)

where the time-dependent similarity transformed Hamiltonian
$\bar{H}(t)=e^{-T^{(N-1)}(t)}He^{T^{(N-1)}(t)}$. The one-particle Green’s
function is then expressed as

$\displaystyle G_{c}^{\rm{R}}(t)$
$\displaystyle=-i\Theta(t)N_{c}(t)e^{iE_{CC}^{(N)}t}O(t)$
$\displaystyle=-i\Theta(t)\exp(\int_{0}^{t}-iE_{CC}^{(N-1)}(\tau)\rm{d}\tau)e^{iE_{CC}^{(N)}t}O(t)$
$\displaystyle=-i\Theta(t)\exp(-i\Delta E_{CC}(t)t)O(t).$ (11)

where

$\displaystyle O(t)$
$\displaystyle=\langle\phi_{0}^{(N)}|(1+\Lambda^{(N)})e^{-T^{(N)}}a_{c}^{\dagger}e^{T^{(N-1)}(t)}|\phi_{0}^{(N-1)}\rangle$
(12)

and $\Delta E_{CC}(t)=E_{CC}^{(N)}-[E^{(N-1)}_{CC}]_{t}$. Here,
$[E^{(N-1)}_{CC}]_{t}$ denotes the time average of $E^{(N-1)}_{CC}(\tau)$ over
the period $t$, therefore we have

$\displaystyle\exp(\int_{0}^{t}-iE_{CC}^{(N-1)}(\tau)\rm{d}\tau)=\exp(-i[E^{(N-1)}_{CC}]_{t}t).$
(13)

###  II.3 Time-dependent extended coupled cluster ansatz

It is worthwhile to notice that correlation effects in the $N$-electron state,
as described in the exact one-particle Green’s function formulation Eq.(1),
are approximated by directly considering the correlations in the
$(N-1)$-electron state as seen in Eq. (8). This distinction can also be
explicitly observed by comparing the ansatze in Eqs. (7) and (8). To enhance
the description of correlation effects, particularly by incorporating
correlations corresponding to the $N$-electron state, we propose more
generalized time-dependent extended CC (eCC) ansatz

$\displaystyle|\Psi_{c}^{(N-1)}(t)\rangle:=\tilde{N}_{c}(t)e^{T^{(N)}}e^{T^{(N-1)}(t)}|\phi_{0}^{(N-1)}\rangle$
(14)

with the initial conditions $\tilde{N}_{c}(0)=1$ and

$\displaystyle|\Psi_{c}^{(N-1)}(0)\rangle=e^{T^{(N)}}|\phi_{0}^{(N-1)}\rangle$
(15)

which reproduces ansatze (7). The difference in correlation level between the
two ansatze (8) and (14) is summarized in Figure 1. Specifically, through the
Taylor expansion of $e^{T^{(N)}}$, it is evident that the original ansatz (8)
is the leading term of the ansatz (14). Consequently, the ansatz (14) is
capable of capturing higher-order corrections to the excitation following the
generation of a hole at spin-orbital $c$, as illustrated in Figure 2.

Figure 1: The comparison of ansatze (8) and (14) in terms of correlation
level. Figure 2: The schematic illustration of the excitation from an occupied
spin-orbital $i$ to a virtual spin-orbital $a$ following the generation of a
hole at spin-orbital $c$. The CC ansatz (8) is only able to describe the
leading component of this excitation (left panel), while the extended CC
ansatz (14) provides a higher order correction, mediated through a hole at
spin-orbital $c$ (right panel).

The new ansatz (14) bears resemblance to the double-unitary CC ansatze,[81]
which incorporates two sets of excitation operators. When the ansatz (14) is
applied to the time-dependent Schrödinger equation (TDSE), the EOMs for
$\tilde{N}_{c}(t)$ and $t^{(N-1)}_{n}(t)$ are derived as follows:

$\displaystyle i\partial\ln\tilde{N}_{c}(t)/\partial t$
$\displaystyle=\langle\phi_{0}^{(N-1)}|\bar{\bar{H}}(t)|\phi_{0}^{(N-1)}\rangle$
$\displaystyle=E_{eCC}^{(N-1)}(t),$ (16) $\displaystyle i\partial
t^{(N-1)}_{n}(t)/\partial t$ $\displaystyle=\langle
n^{(N-1)}|\bar{\bar{H}}(t)|\phi_{0}^{(N-1)}\rangle,$ (17)

where the double similarity transformed Hamiltonian for the one-particle
Green’s function is defined as:

$\displaystyle\bar{\bar{H}}(t)$
$\displaystyle=e^{-T^{(N-1)}}(t)e^{-T^{(N)}}He^{T^{(N)}}e^{T^{(N-1)}}(t).$
(18)

It is important to note that Eqs. (16) and (17) depend on solving the
$N$-electron CC equations for $T^{(N)}$, followed by the utilization of the
ordinary differential equation (ODE) integrator for the $(N-1)$-electron state
to propagate $\tilde{N}_{c}(t)$ and $t^{(N-1)}_{n}(t)$. The workflow of the
time propagation of the time-dependent $(N-1)$-electron correlated state
described using the ansatz (14) is depicted in Figure 3. Consequently, the
one-particle Green’s function $G_{c}^{\rm{R}}(t)$ can be reformulated as:

$\displaystyle G_{c}^{\rm{R}}(t)=$
$\displaystyle-i\Theta(t)\tilde{N}_{c}(t)e^{iE_{CC}^{(N)}t}O(t)$ (19)

where

$\displaystyle\tilde{N}_{c}(t)$
$\displaystyle=\exp(-i[E^{(N-1)}_{eCC}]_{t}t);$ (20)

and the modified creation operator $\overline{a_{c}^{\dagger}}$ is given by:

$\displaystyle\overline{a_{c}^{\dagger}}$
$\displaystyle=e^{-T^{(N)}}a_{c}^{\dagger}e^{T^{(N)}}=a_{c}^{\dagger}+[a_{c}^{\dagger},T^{(N)}].$
(21)

The time-dependent overlap term $O(t)$ is formulated as

$\displaystyle O(t)$
$\displaystyle=\langle\phi_{0}^{(N)}|(1+\Lambda^{(N)})\overline{a_{c}^{\dagger}}e^{T^{(N-1)}(t)}|\phi_{0}^{(N-1)}\rangle.$
(22)

Figure 3: The schematic illustration of the new TDCC ansatz (14) and how it
works to evaluate the time propagation of an $(N-1)$-electron correlated
state.

It is worth noting that the new ansatz (14) involves the product of two
exponential operators, leading to double similarity transformation (18) in
solving the EOMs (16) and (17). The double similarity transformation can
potentially increase the non-linearity and leads to instability in the
numerical propagation, thereby deteriorating the performance of the ODE
integrator. In practical implementation, since the first similarity
transformation, $\bar{H}=e^{-T^{(N)}}He^{T^{(N)}}$ is time-independent, it can
be computed upfront in the $N$-particle space before the time propagation in
the $(N-1)$-particle space, with the computational cost paid for constructing
$\bar{H}$. Alternatively, the construction of $\bar{H}$ can be entirely
avoided by employing approximate ansatze that combine the product of two
exponential operators into one.

Straightforward approximations can be obtained by utilizing the
Baker–Campbell–Hausdorff formula in (14) and truncating the expansion at
different (commutator) levels, for example:

$\displaystyle|\Psi_{c}^{(N-1)}(t)\rangle$
$\displaystyle\approx\tilde{N}_{c}(t)\times
e^{T^{(N)}+T^{(N-1)}}|\phi_{0}^{(N-1)}\rangle,$ (23)
$\displaystyle|\Psi_{c}^{(N-1)}(t)\rangle$
$\displaystyle\approx\tilde{N}_{c}(t)\times$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
e^{T^{(N)}+T^{(N-1)}+\frac{1}{2}[T^{(N)},T^{(N-1)}]}|\phi_{0}^{(N-1)}\rangle,$
(24) $\displaystyle\vdots$

The benefit of using the ansatze (23,24) in the EOMs (16 and 17) is to re-
utilize the conventional CC implementation with modest modifications in the
lists of the CC excitation operators. We will examine the numerical
performance of such approximations in model systems in the following section.

###  II.4 Cluster analysis of $G_{c}^{R}$

It is important to note that the inclusion of $O(t)$ in the formulation of
$G_{c}^{R}(t)$ causes the Fourier transform of $G_{c}^{R}(t)$ to take a
convolution form, i.e.,

$\displaystyle G_{c}^{\rm{R}}(\omega)$ $\displaystyle\stackrel{{\scriptstyle
t\rightarrow\infty}}{{=}}-i\int_{-\infty}^{\infty}\mathcal{F}\\{O(t)\\}(u)\delta(\omega-u-\Delta
E)\text{d}u$ $\displaystyle=-i\mathcal{F}\\{O(t)\\}(\omega-\Delta E)$ (25)

where

$\displaystyle\Delta
E=\lim_{t\rightarrow\infty}\left\\{E_{CC}^{(N)}-[E^{(N-1)}_{CC/eCC}]_{t}\right\\}$
(26)

approaches a constant scalar in the large $t$ limit (due to
$\lim_{t\rightarrow\infty}[E^{(N-1)}_{CC/eCC}]_{t}=E^{(N-1)}_{CC}$, see also
Figure 4b,d below). $\delta(\omega-u-\Delta E)$ is a Dirac delta function, and
$\mathcal{F}\\{O(t)\\}$ is the Fourier transform of $O(t)$:

$\displaystyle\mathcal{F}\\{O(t)\\}(\omega)=$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\langle\phi_{0}^{(N)}|(1+\Lambda^{(N)})\overline{a_{c}^{\dagger}}\mathcal{F}\\{e^{T^{(N-1)}(t)}\\}(\omega)|\phi_{0}^{(N-1)}\rangle.$
(27)

Furthermore, the expansion of $e^{T^{(N-1)}(t)}$ is given by

$\displaystyle e^{T^{(N-1)}(t)}$
$\displaystyle=e^{\sum_{n}t_{n}^{(N-1)}(t)\mathbb{E}_{n}}$
$\displaystyle=\prod_{n}\left(1+t_{n}^{(N-1)}(t)\mathbb{E}_{n}\right)$
$\displaystyle=1+\sum_{n}\tilde{t}_{n}^{(N-1)}(t)\mathbb{E}_{n}$ (28)

where the relationship between $\tilde{t}_{n}^{(N-1)}(t)$ and
$t_{n}^{(N-1)}(t)$ can be determined through cluster analysis of Eq. (28). For
example, if the CC operators only include the singles and doubles, we have:

$\displaystyle\tilde{t}^{(N-1)}_{(p,q)}(t)$
$\displaystyle=t^{(N-1)}_{(p,q)}(t)\leavevmode\nobreak\ ,$ (29)
$\displaystyle\tilde{t}^{(N-1)}_{(p,q,r,s)}(t)$
$\displaystyle=t^{(N-1)}_{(p,q,r,s)}(t)+t^{(N-1)}_{(p,r)}(t)t^{(N-1)}_{(q,s)}(t)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ -t^{(N-1)}_{(p,s)}(t)t^{(N-1)}_{(q,r)}(t).$ (30)

Therefore, in Eq. (27):

$\displaystyle\mathcal{F}\\{e^{T^{(N-1)}(t)}\\}(\omega)=\delta(\omega)+\sum_{n}\mathcal{F}\\{\tilde{t}^{(N-1)}_{n}\\}(\omega)\mathbb{E}_{n}.$
(31)

From Eqs. (II.4) and (31), the component analysis of the $G_{c}^{R}(\omega)$
becomes straightforward.

##  III Numerical results and discussion

In this paper, we focus on evaluating the performance of our newly proposed
ansatz for computing the one-particle Green’s function in its exact limit.
Particularly, we compare it with the performance of the previous ansatz. To
this end, we employ the SIAM as our test framework within the RT-EOM-CC
approach to compute the exact one-particle impurity Green’s function. The SIAM
Hamiltonian is expressed as follows:

$\displaystyle H_{\rm SIAM}=H_{\rm imp.}+H_{\rm bath}+H_{hyb.},$ (32)

where

$\displaystyle H_{\rm
imp.}=\sum_{\sigma}\mu_{c}c^{\dagger}_{\sigma}c_{\sigma}+Uc^{\dagger}_{\uparrow}c_{\uparrow}c^{\dagger}_{\downarrow}c_{\downarrow}$
(33)

describes the impurity-site with potential $\mu_{c}$ and the Coulomb
interaction $U$ between electrons with opposite spins ($\sigma=\uparrow$ or
$\downarrow$) at the impurity site,

$\displaystyle H_{\rm bath}=\sum_{i=1,\sigma}^{N_{\rm
b}}\mu_{d,i}d^{\dagger}_{i,\sigma}d_{i,\sigma}$ (34)

characterizes the non-interacting bath sites with potentials $\mu_{d}$’s, and

$\displaystyle H_{\rm hyb.}=\sum_{i=1,\sigma}^{N_{\rm
b}}V_{i}\big{(}c^{\dagger}_{\sigma}d_{i,\sigma}+d^{\dagger}_{i,\sigma}c_{\sigma}\big{)}$
(35)

describes the coupling between the impurity site and the bath sites due to the
hybridization. In subsequent tests, we focus on the three-site and four-site
SIAM configurations, setting $N=4$, $U=3.0$ a.u., $\mu_{c}=-1.5$ a.u.,
$V_{i}=0.5$ a.u. $\forall i$, and $\mu_{d,i}\in[-1.0,1.0]$ a.u.

###  III.1 Energy fluctuations of $(N-1)$-electron states

Figure 4: The energy fluctuation of the $(N-1)$-electron states of the three-
and four-site SIAMs in the RT-EOM-CC simulations employing the ansatze (8) and
(14). The reference state in the RT-EOM-CC simulations is a three-electron
state (see the insets) where one electron resides at impurity site and two at
the bath sites. In both cases, the time-average of the time-dependent
$(N-1)$-electron energies, $[E_{CC}^{(N-1)}]_{t}$, converge to the stationary
$(N-1)$-electron CC energies (denoted by the dashed lines), $E_{CCSD}^{(N-1)}$
of $-3.713370$ a.u. and $E_{CCSDT}^{(N-1)}$ of $-3.925962$ a.u., respectively,
for the three- and four-site SIAMs.

We first evaluate the energy fluctuation of the $(N-1)$-electron state in the
RT-EOM-CC simulations. For a three-site SIAM, as depicted in the inset of
Figure 4b, since the highest level of excitations are double excitations, the
exact propagation of the non-equilibrium $(N-1)$-electron state
$|\Psi_{c}^{(N-1)}\rangle$ can be accurately performed at the coupled cluster
with singles and doubles (CCSD) level. Figure 4a illustrates the energy
fluctuations of the $(N-1)$-electron state in RT-EOM-CCSD simulations using
both the previous ansatz (8) and the new ansatz (14). In both ansatze, the
CCSD operators are defined as:

$\displaystyle T^{(N/N-1)}$ $\displaystyle=T_{1}^{(N/N-1)}+T_{2}^{(N/N-1)},$
$\displaystyle\Lambda^{(N)}$
$\displaystyle=\Lambda_{1}^{(N)}+\Lambda_{2}^{(N)}.$ (36)

Notably, the two RT-EOM-CCSD simulations have different energy starting points
due to the choice of the ansatz. The starting energy in the simulation using
ansatz (8) is the Hartree-Fock energy of the $(N-1$-electron state, while the
starting energy with ansatz (14) is considered to be that of a non-stationary
$(N-1)$-electron correlated state. Additionally, as shown Figure 4b, although
the time-dependent energy curves of the $(N-1)$-electron state vary between
the two simulations, the time-averaged energies eventually converge to the
same stationary $(N-1)$-electron CCSD energy of the half-filled three-site
SIAM, irrespective of the ansatz used.

Figure 4c,d display the energy fluctuations in the RT-EOM-CCSDT simulations of
the four-site SIAM. Due to the additional bath site compared to the three-site
model, the CC operators in the exact limit are expanded as follows:

$\displaystyle T^{(N)}$
$\displaystyle=T_{1}^{(N)}+T_{2}^{(N)}+T_{3}^{(N)}+T_{4}^{(N)},$
$\displaystyle T^{(N-1)}$
$\displaystyle=T_{1}^{(N-1)}+T_{2}^{(N-1)}+T_{3}^{(N-1)},$
$\displaystyle\Lambda^{(N)}$
$\displaystyle=\Lambda_{1}^{(N)}+\Lambda_{2}^{(N)}+\Lambda_{3}^{(N)}+\Lambda_{4}^{(N)}.$
(37)

Although the energy fluctuations are not as uniform as in the three-site
model, the similar converging behavior of the time-averaged energies is
observed, albeit over a slightly longer duration. It is worth noting that,
with the selected parameters, the impact of quadruple excitations on the exact
ground state of the four-site SIAM becomes negligible, therefore the exact
ground state can be well approximated by the CCSDT wave function with an
energy deviation $<2.0\times 10^{-7}$ a.u.

###  III.2 $G_{c}^{R}$ computed through RT-EOM-CC with different ansatze

Figure 5: The impurity Green’s functions, $G_{c}^{R}(t)$ and
$G_{c}^{R}(\omega)$, of the three-site SIAM ($N=4$) computed using the RT-EOM-
CCSD approach employing the two ansatze (8) and (14). In the ansatz (14), the
$N$-electron CC amplitude, $T^{(N)}$, is obtained from the $N$-electron CCSD
calculation of the three-site SIAM. In the RT-EOM-CCSD simulation, the Runge-
Kutta-Fehlberg approach, RK45, and its implementation in SciPy[82] were used
in numerically solving the ODEs (10) and (17) for obtaining $G_{c}^{R}(t)$
with $t\in[0,75]$ a.u. and $\Delta t=0.05$ a.u. Both $G_{c}^{R}(t)$ and
$G_{c}^{R}(\omega)$ are normalized to be between $[-1,1]$. Figure 6: The
impurity Green’s functions, $G_{c}^{R}(t)$ and $G_{c}^{R}(\omega)$, of the
four-site SIAM ($N=4$) computed using the RT-EOM-CCSDT approach employing the
two ansatze (8) and (14). In the ansatz (14), the $N$-electron CC amplitude,
$T^{(N)}$, is obtained from the $N$-electron CCSDTQ calculation of the four-
site SIAM. In the RT-EOM-CCSD simulation, the Runge-Kutta-Fehlberg approach,
RK45, and its implementation in SciPy[82] were used in numerically solving the
ODEs (10) and (17) for obtaining $G_{c}^{R}(t)$ with $t\in[0,75]$ a.u. and
$\Delta t=0.05$ a.u. Both $G_{c}^{R}(t)$ and $G_{c}^{R}(\omega)$ are
normalized to be between $[-1,1]$.

We then proceed to examine the computed $G_{c}^{R}$ from the RT-EOM-CC
approach using different ansatze. Figure 5 presents the impurity Green’s
function of the three-site SIAM, computed by the RT-EOM-CCSD approach
employing ansatze (8) and (14). For comparison, the exact $G_{c}^{\rm R}$
curves computed by Eq. (1) and exact diagonalization are also provided. As
observed in Figure 5, the RT-EOM-CCSD simulation employing ansatz (14)
successfully reproduces the exact $G_{c}^{R}(t)$ curves, while the simulation
using the ansatz (8) only captures a rough envelop of the exact time-
propagation, missing the fine structure (i.e. the wiggles) of the exact
curves. Consequently, after the Fourier transform, the curve obtained with
ansatz (14) accurately reproduces both the main and satellite peaks, while the
curves with ansatz (8) only capture the main peak at $\omega\sim 0.1$ a.u.

The performance difference between the two ansatze (8) and (14) is further
highlighted in Figure 6 when employing the RT-EOM-CCSDT approach for computing
$G_{c}^{R}(t)$ of the four-site SIAM. Notably, since the $G_{c}^{R}(t)$
obtained using ansatz (8) does not strictly adhere to the envelope of the
exact $G_{c}^{R}(t)$, its Fourier counterpart, $G_{c}^{R}(\omega)$, exhibits a
slight shift in the main peak position. This shift is evident in Figure 6c,
where the orange curve obtained with ansatz (8) deviates slightly from the
exact main peak position at $\sim-0.2$ a.u.

###  III.3 Approaching the exact limit of $G_{c}^{R}$ in RT-EOM-CC simuations

Figure 7: The deviation of the time-dependent impurity Green’s functions,
$G_{c}^{R}(t)$, from their exact values of the four-site SIAM ($N=4$) computed
by different theoretical approaches. The exact values are obtained from the
exact diagonalization of the Hamiltonian, and can also be equivalently
obtained through CCSDTQ-GF calculations in the time domain. All the RT-EOM-CC
simulations employ the ansatz (14), where the $N$-electron CC operators are
obtained from converged CC ground state calculations with the convergence
criteria of the energy change less than $10^{-6}$ a.u. and the norm of the CC
amplitude change less than $10^{-7}$. The Runge-Kutta-Fehlberg approach, RK45,
and its implementation in SciPy[82] were used in numerically solving the ODEs
(10) and (17) for obtaining $G_{c}^{R}(t)$ with $t\in[0,37.5]$ a.u. and
$\Delta t=0.05$ a.u. Figure 8: The deviation of the impurity Green’s
functions, $G_{c}^{R}(t)$, from the exact solution of the four-site SIAM
($N=4$) computed by the RT-EOM-CCSDT/CCSDTQ (with different time steps) and
CCSDT-GF approaches. The exact solution is computed by the CCSDTQ-GF approach.
All the RT-EOM-CC simulations employ the ansatz (14), where the $N$-electron
CCSDTQ operators are obtained from converged CCSDTQ ground state calculations
with the convergence criteria of energy change less than $10^{-6}$ a.u. and
the norm of the CC amplitude change less than $10^{-7}$. The Runge-Kutta-
Fehlberg approach, RK45, and its implementation in SciPy were used in
numerically solving the ODEs (10) and (17) for obtaining $G_{c}^{R}(t)$ with
$t\in[0,37.5]$ a.u. with different choices of time step.

To evaluate how closely the RT-EOM-CC approaches reach the exact limit, we
closely examine the $G_{c}^{R}(t)$ curves. Figure 7 illustrates the deviation
of the $G_{c}^{R}(t)$ obtained from the RT-EOM-CC simulations from the exact
values, employing ansatz (14) across all simulations. The ground state CC
approaches are explicitly mentioned in the Figure, because ansatz (14) also
includes the $N$-electron CC operators obtained from the converged CC ground
state calculations. Besides, we also include the CCSD-GF and CCSDT-GF results
for comparison. Here, the CCGF values are obtained by subsituting the CC left
and right wave functions,
$\langle\phi_{0}^{(N)}|(1+\Lambda^{(N)})e^{-T^{(N)}}$ and
$e^{T^{(N)}}|\phi_{0}^{(N)}\rangle$, for $\langle\Psi^{(N)}|$ and
$|\Psi^{(N)}\rangle$ in Eq. (1), respectively.

The simulations at the CCSD level, whether RT-EOM-CCSD or CCSD-GF, exhibit the
largest deviations from the exact values among the employed approaches.
Furthermore, the numerical stability of the employed ODE integrator becomes
concerning when $t>35$ a.u. Within the RT-EOM-CC framework, there is an
improvement of roughly one to two orders of magnitude from the RT-EOM-
CCSD/CCSD results to the RT-EOM-CCSDT/CCSDT results, and the ODE integration
becomes more stable for $t>35$ a.u. as observed in Figure 6. Adding the higher
order excitations, the transition from RT-EOM-CCSDT/CCSDT results to RT-EOM-
CCSDT/CCSDTQ results shows negligible change, even though the latter could be
considered as approaching the exact limit of the RT-EOM-CC simulation for the
four-site SIAM. By contrast, improvements in the CCGF approaches are
consistent when increasing the excitation level to enhance the CC wave
function. For instance, the CCSD-GF results are comparable in quality to the
RT-EOM-CCSD/CCSD results, whereas the CCSDT-GF results clearly outperform the
RT-EOM-CCSDT/CCSDTQ results, and the CCSDTQ-GF results numerically match the
exact solution.

We attribute the deviation of the supposedly exact RT-EOM-CCSDT/CCSDTQ results
to the discretization error caused by the finite size of the time step. To
validate this hypothesis, we re-ran the RT-EOM-CCSDT/CCSDTQ simulations with
varying time steps, and the results are depicted in Figure 8, with the CCSDT-
GF results included for comparison. It is evident that as the time step
decreases, the deviation of $G_{c}^{R}(t)$ is consistently reduced, achieving
and surpassing the accuracy level of the CCSDT-GF results for the four-site
SIAM.

###  III.4 Computing $G_{c}^{R}$ using approximate ansatze in RT-EOM-CC
simulations

Figure 9: The deviation of $G_{c}^{R}(t)$ of the four-site SIAM computed by
RT-EOM-CCSDT with different approximate ansatze to the ansatz (14).

In the previous sections, we demonstrated the performance differences between
ansatze (8) and (14) and explored how to approach the exact limit of ansatz
(14) in the RT-EOM-CC simulations of $G_{c}^{R}(t)$ for the three- and four-
site SIAMs. However, as discussed in Section II.3, there may be challenges
related to tensor operations and method implementation associated with the
extended TDCC ansatz’s algebraic structure and the tensor transition between
the $N$\- and $(N-1)$-particle spaces. These challenges could, in the worst-
case scenario, necessitate a redesign of the entire computational structure
and library, adding another layer of complexity to realistic simulations.

To efficiently utilize the existing computational infrastructure, we introduce
approximate ansatze, such as (23) and (24), in the RT-EOM-CC simulations to
assess their efficacy. Figure 9 compares the performance of ansatze (8), (23),
and (24) in the RT-EOM-CCSDT simulations of the four-site SIAM to that
employing the ansatz (14). Here, we treat ansatz (8) as an approximation of
ansatz (14) by assuming $T^{(N)}=0$. The results show no significant
performance difference between ansatze (8) and (23). However, incorporating
the single commutator between $T^{(N)}$ and $T^{(N-1)}$ in the exponent to
generate ansatz (24) leads to a marked improvement. It’s noteworthy that the
single commutator introduces higher order corrections in the exponential
operators, such that

$\displaystyle
e^{T^{(N)}+T^{(N-1)}+\frac{1}{2}[T^{(N)},T^{(N-1)}]}|\phi_{0}^{(N-1)}\rangle=e^{T^{(N)}}e^{T^{(N-1)}}|\phi_{0}^{(N-1)}\rangle$
(38)

for the four-site SIAM, explaining the minimal deviations observed with ansatz
(24) in Figure 9.

###  III.5 Component analysis of $G_{c}^{R}$

Figure 10: The component analysis of the impurity Green’s functions,
$G_{c}^{R}(t)$ and $G_{c}^{R}(\omega)$, of the three-site SIAM. The inset in
(c) shows the labeling of the spin-orbitals in the three-site SIAM.

Finally, we demonstrate the component analysis on the $G_{c}^{R}$ of the
three-site SIAM employing the approach described in Section II.4. The analysis
implicitly assumes the large $t$ limit, which is justified by the observation
in Figure 4b, where Eq. (26) essentially represents the energy difference
between $N$\- and $(N-1)$-electron stationary states. Figure 10 presents the
component analysis of the $G_{c}^{R}(t)$ and $G_{c}^{R}(\omega)$ using Eqs.
(II.4)-(31). As observed, in addition to the main QP peak (at $\sim 0.1$
a.u.), significant contributions to the satellite peaks between -0.5 and 0.0
a.u. are provided by the single excitations from the impurity site to the bath
site ($0\rightarrow 4$), from one bath site to another ($3\rightarrow 5$), and
the double excitation ($0,3\rightarrow 4,5$).

##  IV Conclusion and outlook

In the paper, we have analyzed and examined a series of TDCC ansatze in the
RT-EOM-CC simulations for computing the one-particle Green’s function. Unlike
the previously used TDCC ansatz used in RT-EOM-CC simulations, we introduced a
new TDCC ansatz that features an extended CC form$-$the product of the
exponential CC operators from $N$ and $(N-1)$-particle spaces. Preliminary
analysis and simulations on simple SIAMs demonstrate that, compared to the
previous TDCC ansatz, the new one is capable of approaching the exact limit by
incorporating higher order excitations in the $(N-1)$-electron CC exponential
operator and by using small time steps. By employing the BCH expansion of the
new ansatz and truncating at different commutator levels, we have also
introduced some approximate TDCC ansatze to the RT-EOM-CC simulations. The
approximate ansatze feature a single exponential algebraic structure that
potentially balances the complexity of implementation with accuracy.
Additionally, we have formalized a recipe for analyzing the components of the
computed Green’s function in RT-EOM-CC simulations, paving the way for larger-
scale and efficient implementations and detailed spectral function analysis
for complex molecular systems in the near future. Future work will focus on
incorporating the double-unitary CC ansatze[81] into the RT-EOM-CC framework,
extending RT-EOM-CC to compute the nonequilibrium Green’s function,[28, 29]and
improving numerical aspects including more stable ODE integrator[83] and
robust interpolation and extrapolation techniques[84, 85].

##  V Acknowledgements

This material is based upon work supported by the “Transferring exascale
computational chemistry to cloud computing environment and emerging hardware
technologies (TEC4)” project, which is funded by the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences, the Division of
Chemical Sciences, Geosciences, and Biosciences (under FWP 82037). F.D.V and
J.J.R. acknowledge the support from the Center for Scalable Predictive methods
for Excitations and Correlated phenomena (SPEC), which is funded by the U.S.
Department of Energy (DoE), Office of Science, Office of Basic Energy
Sciences, Division of Chemical Sciences, Geosciences and Biosciences as part
of the Computational Chemical Sciences (CCS) program at Pacific Northwest
National Laboratory (PNNL) under FWP 70942. B.P. also acknowledges Dr. Niri
Govind for the fruitful discussion during the preparation of the manuscript.

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