††thanks: This manuscript has been authored in part by UT-Battelle, LLC, under
contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US
government retains and the publisher, by accepting the article for
publication, acknowledges that the US government retains a nonexclusive, paid-
up, irrevocable, worldwide license to publish or reproduce the published form
of this manuscript, or allow others to do so, for US government purposes. DOE
will provide public access to these results of federally sponsored research in
accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-
public-access-plan).

# Coupled-cluster theory for strong entanglement in nuclei

Z. H. Sun  Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN
37831, USA  G. Hagen  Physics Division, Oak Ridge National Laboratory, Oak
Ridge, TN 37831, USA  Department of Physics and Astronomy, University of
Tennessee, Knoxville, TN 37996, USA  T. Papenbrock  Department of Physics and
Astronomy, University of Tennessee, Knoxville, TN 37996, USA  Physics
Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

###### Abstract

Atomic nuclei can exhibit shape coexistence and multi-reference physics that
enters in their ground states, and to accurately capture the ensuing
correlations and entanglement is challenging. We address this problem by
applying single-reference coupled-cluster theory based on spherical and
deformed reference states and the tailored coupled-cluster method. The latter
combines configuration interaction to capture static correlations with
coupled-cluster theory for dynamic correlations. We compute the atomic nuclei
12C, 28Si, and 56Ni and find that the tailored coupled-cluster method and the
single-reference approach based on a deformed Hartree-Fock state yield the
most accurate results.

##  I introduction

In the last two decades, ab initio computations of atomic nuclei have become
increasingly accurate and capable at predicting hard-to-measure properties
Navrátil and Caurier (2004); Forssén  _et al._ (2005); Nogga  _et al._ (2006);
Hagen  _et al._ (2008); Epelbaum  _et al._ (2011); Roth  _et al._ (2012); Somà
_et al._ (2014); Duguet  _et al._ (2017); Morris  _et al._ (2018); Arthuis
_et al._ (2020); Hu  _et al._ (2021). This progress is based on ideas from
effective field theory van Kolck (1994); Entem and Machleidt (2003); Epelbaum
_et al._ (2009) and the renormalization group Bogner  _et al._ (2003, 2007),
and the development and application of computationally affordable methods
Dickhoff and Barbieri (2004); Lee (2009); Tsukiyama  _et al._ (2011); Hagen
_et al._ (2014); Stroberg  _et al._ (2016) that solve the many-body
Schrödinger equation.

While these advances are based on classical computing, the advent of quantum
computing provides us with the opportunity to explore such computations
Dumitrescu  _et al._ (2018); Stetcu  _et al._ (2022); Kiss  _et al._ (2022),
to better understand entanglement Robin  _et al._ (2021); Beane  _et al._
(2019); Gu  _et al._ (2023) and correlations Sun  _et al._ (2022) in nuclei,
and to think about quantum algorithms Roggero  _et al._ (2020); Choi  _et al._
(2021); Baroni  _et al._ (2022).

Due to its conceptual and computational simplicity, coupled-cluster theory
plays an attractive role in both classical Kümmel  _et al._ (1978); Bishop
(1991); Bartlett and Musiał (2007); Hagen  _et al._ (2014) and quantum
computing Romero  _et al._ (2017); Lee  _et al._ (2019); Stetcu  _et al._
(2022); Kiss  _et al._ (2022). Approaches that combine configuration
interaction and coupled-cluster theory, such as active space methods
Stolarczyk (1994); Hino  _et al._ (2006); Kinoshita  _et al._ (2005), recently
received attention for potential use in hybrid quantum-classical algorithms
Keen  _et al._ (2021); Bauman  _et al._ (2023). This makes it interesting to
also study them in nuclear theory. Independent of advances of quantum hardware
and computing Lee  _et al._ (2022) such methods can also help us here and now
in classical computing.

Computational approaches to nuclear ground-states often start from a reference
state – a product state obtained for instance from a self-consistent mean-
field calculation – to understand the essential physics of the nucleus under
consideration and to organize how to include quantum correlations for an
accurate computation. Dynamic correlations are included via particle-hole
excitations of the reference state. They scale polynomially in number and
their inclusion yields an extensive, i.e. proportional to mass number $A$,
contribution to the binding energy Mok  _et al._ (1996); Crittenden (2013);
Ramos-Cordoba  _et al._ (2016); Benavides-Riveros  _et al._ (2017). They mix
the reference state and the high-energy particle-hole excitation which
reflects short-wavelength relative motion of particles Hollett  _et al._
(2016) and short-range behavior of nuclear force. Single-reference methods
such as the coupled-cluster (CC) method Hagen and Nam (2012); Hagen  _et al._
(2010, 2016); Morris  _et al._ (2018) and the in-medium similarity
renormalization group (IMSRG) Tsukiyama  _et al._ (2011); Hergert  _et al._
(2013, 2016); Stroberg  _et al._ (2016) are designed to capture dynamic
correlations with a computational effort that scales polynomially in mass
number and the number of single-particle states contained in Hilbert space. In
doubly-magic nuclei, coupled cluster with singles and doubles (CCSD) captures
about $90\%$ of the correlation energy, while triples contribute about $10\%$
of the CCSD correlation energy Bartlett and Musiał (2007); Hagen  _et al._
(2009). Coupled-cluster theory with singles, doubles, and triples (CCSDT)
yields about 98 to 99% of the total correlation energy. The remaining
correlation energy could be gained by including excitation beyond CCSDT;
however, the 1-2% uncertainty in the binding energy that remain at the CCSDT
truncation are often smaller than the uncertainty in the employed nuclear
Hamiltonians. Apart from benchmarking there is probably little need to solve
approximated Hamiltonians exactly.

Single-reference methods fail in systems that lack an energy gap at the Fermi
surface Horoi  _et al._ (2007); Evangelista (2018). In such a case, many-body
perturbation theory fails because of small energy denominators, and many-
particle–many-hole excitations yield considerable contributions to the
correlation energy. An alternative view is that the reference state is not the
only dominant configuration and one is dealing with a multi-reference problem
Lyakh  _et al._ (2012); Evangelista (2018). Then, one has to include static
correlations that are due to a family of (quasi-)degenerate configurations. In
nuclei, the existence of degenerate states is usually related to symmetries
broken in the mean field while quasi-degenerate configurations correspond to
situations of shape co-existence. Projection techniques allow one to capture
the energy gain from static correlations in the case of symmetry breaking Qiu
_et al._ (2017, 2018, 2019); Frosini  _et al._ (2022a, b, c); Hagen  _et al._
(2022). In such cases, static correlations do not yield extensive
contributions to the energy and usually include long wavelength physics.
Peierls and Yoccoz (1957); Frosini  _et al._ (2022b); Hagen  _et al._ (2022);
Papenbrock (2022).

In this work, we focus on static correlations that do not result from broken
symmetries. Methods such as the shell model coupled cluster Sun  _et al._
(2018, 2021) and valence space in-medium similarity renormalization group (VS-
IM-SRG) Tsukiyama  _et al._ (2012); Stroberg  _et al._ (2016) treat dynamic
correlations separately. Here, dynamic correlations are strongly renormalized
and all correlations are capturted in a single shell of single-particle
states. These methods work well in open-shell nuclei adjacent to the core with
a double magic number and have been developed to calculate a large portion of
the nuclear chart. However, suffering from the intruder state problem,
constructing effective interaction in a valence space spanned by more than two
major shells is still challenging.

Here, we follow an alternative approach and deal with the static correlation
using the tailored coupled cluster (TCC) method Stolarczyk (1994); Hino  _et
al._ (2006); Kinoshita  _et al._ (2005); Faulstich  _et al._ (2019); Vitale
_et al._ (2020); Leszczyk  _et al._ (2022). This approach aims to combine the
simplicity of single reference coupled-cluster theory with a multi-reference
wave function. The underlying idea is as follows: In a system with two-body
interactions, the correlation energy depends only on the two-particle–two-hole
(2p-2h) amplitudes. While the numerical values of the amplitudes are informed
by 3p-3h and higher-rank amplitudes, the latter do not enter directly the
expression for the correlation energy. This makes it attractive to replace
2p-2h coupled-cluster amplitudes in the vicinity of the Fermi surface by those
from methods such configuration interaction or density renormalization group
and thereby include the effects of static correlations.

The tailored coupled-cluster method partitions the single-particle basis into
an active space (consisting of a smaller number of selected hole and particle
states) and an external space. The active space is selected to be large enough
to reflect the static correlation of the system, in which the many-body
correlations are treated exactly with full configuration interaction. The
remaining correlations from the external space are captured dynamically
through the standard coupled cluster method. As we will see, it is interesting
to compare the results of the tailored coupled-cluster method to those from a
symmetry-breaking single-reference computation.

This paper is organized as follows. In Sect. II, we introduce the single-
reference and tailored coupled-cluster theory. In Sect. III, we will study the
static correlations in a shell-model description of 56Ni and compare tailored
and symmetry-breaking coupled-cluster methods. We also study the tailored
coupled-cluster computations of 12C and 28Si using two different chiral
potentials. Finally, we summarize our results in Sec. IV.

##  II Method

The intrinsic Hamiltonian consists of one-, two-, and three-body terms

$H=\frac{A-1}{A}\sum_{i=1}^{A}\frac{p_{i}^{2}}{2m}+\sum_{i<j}\left(v_{ij}-\frac{p_{i}p_{j}}{mA}\right)+\sum_{i<j<k}^{A}v_{ijk}.$
(1)

Here $A$ is the mass number of nucleus, $m$ is the nucleon mass, $p_{i}$ is
the momentum of nucleon $i$, and $v_{ij}$ and $v_{ijk}$ are the two- and
three-body interaction, respectively. This intrinsic Hamiltonian does not
reference the center-of-mass coordinate. One normal-orders the Hamiltonian
with respect to the reference state, e.g., the Hartree Fock state

$|\Phi_{0}\rangle\equiv\prod_{i=1}^{A}\hat{a}^{\dagger}_{i}|0\rangle\ .$ (2)

— The correlation energy is defined as the difference between total energy and
the energy $E_{0}=\langle\Phi_{0}|H|\Phi_{0}\rangle$ of the reference.

$E_{\rm{corr}}=E-E_{0}$ (3)

The normal-ordered Hamiltonian with respect to the reference reads

$\displaystyle\mathcal{H}$ $\displaystyle=$ $\displaystyle
E_{0}+\sum_{pq}f_{pq}\left\\{\hat{a}_{p}^{\dagger}\hat{a}_{q}\right\\}+\frac{1}{4}\sum_{pqrs}\Gamma_{pqrs}\left\\{\hat{a}_{p}^{\dagger}\hat{a}_{q}^{\dagger}\hat{a}_{s}\hat{a}_{r}\right\\}$
(4)
$\displaystyle+\frac{1}{36}\sum_{pqrstu}W_{pqrstu}\left\\{\hat{a}_{p}^{\dagger}\hat{a}_{q}^{\dagger}\hat{a}_{r}^{\dagger}\hat{a}_{u}\hat{a}_{t}\hat{a}_{s}\right\\}.$

Here the curly brackets imply normal ordering. The operators
$\hat{a}_{p}^{\dagger}$ and $\hat{a}_{p}$ create and annihilate a nucleon in
state $|p\rangle=\hat{a}_{p}^{\dagger}|0\rangle$, respectively, and they
fulfill the canonical anti-commutation relations for fermions. The matrix
elements of the one-, two-, and three-body operators are denoted as $f_{pq}$,
$\Gamma_{pqrs}$, $W_{pqrstu}$, respectively. To avoid dealing with the three-
body terms, one usually truncates Eq. (4) at the normal-ordered two-body level
and neglects the residual three-body terms Hagen  _et al._ (2007); Roth  _et
al._ (2012). Thus, three-body forces enter in $E_{0}$, $f_{pq}$ and
$\Gamma_{pqrs}$.

Coupled-cluster theory employs a similarity transformation to decouple the
ground state from its particle-hole excitations

$\langle\Phi^{a_{1}\cdots a_{n}}_{i_{1}\cdots
i_{n}}|e^{-T}He^{T}|\Phi_{0}\rangle=0$ (5)

for $n=1,\ldots A$. Here

$|\Phi^{a_{1}\cdots a_{n}}_{i_{1}\cdots
i_{n}}\rangle\equiv\hat{a}_{a_{1}}^{\dagger}\cdots\hat{a}_{a_{n}}^{\dagger}\hat{a}_{i_{n}}\cdots\hat{a}_{i_{1}}|\Phi_{0}\rangle$
(6)

is a $n$p-$n$h excitation of the reference state, and $T$ is the cluster
operator consisting of 1p-1h, 2p-2h up to $A$p-$A$h excitations,

$T=T_{1}+T_{2}+T_{3}+\cdots+T_{A},$ (7)

and

$T_{n}=\frac{1}{(n!)^{2}}\sum_{\begin{subarray}{c}a_{1},a_{2},\cdots,a_{n}\\\
i_{1},i_{2},\cdots,i_{n}\end{subarray}}t^{a_{1},a_{2},\cdots,a_{n}}_{i_{1},i_{2},\cdots,i_{n}}\hat{a}_{a_{1}}^{\dagger}\cdots\hat{a}_{a_{n}}^{\dagger}\hat{a}_{i_{n}}\cdots\hat{a}_{i_{1}}\
.$ (8)

In these expressions (and in what follows) we labelled hole states with
indices $i,j,k,\ldots$ and particle states with $a,b,c,\ldots$. Once
decoupled, the ground-state energy is calculated as

$E_{\rm{CC}}=\langle\Phi_{0}|e^{-T}He^{T}|\Phi_{0}\rangle\ .$ (9)

It is important to note that – for normal-ordered two-body Hamiltonians – only
$T_{1}$ and $T_{2}$ contribute to the energy (9), while the remaining cluster
amplitudes $T_{n}$ with $n>2$ impact the values of $T_{1}$ and $T_{2}$ via the
decoupling Eqs. (5). Thus, any inaccuracy in coupled-cluster theory could in
principle be remedied by using more accurate $T_{1}$ and $T_{2}$ amplitudes.

In the Hartree-Fock basis, the doubles yield the dominant contribution to the
total binding. The estimate from many-body perturbation theory

$T_{ij}^{ab}\approx\frac{\Gamma_{abij}}{\epsilon_{a}+\epsilon_{b}-\epsilon_{i}-\epsilon_{j}}.$
(10)

indicates that the cluster amplitudes $T_{ij}^{ab}$ can be large when single-
particle states that are nearly degenerate with the reference. In such
situations, induced three-body terms such as

$H_{\mathrm{3b}}=\left[H,T_{2}\right]_{\rm{3b}}$ (11)

become considerable in size. This drives higher-rank cluster amplitudes and
reflects the strong entanglement in the nucleus. Single-reference coupled-
cluster theory are not expected to capture such static correlations.

The tailored coupled cluster is designed to address this shortcoming. One
splits the single-particle Hilbert space into an “active” space and “inactive”
space. The active space is selected large enough to describe the static
correlations near the Fermi surface. Figure 1 shows the partition of the
single-particle basis. The hole space is determined by a filling of shell-
model orbitals, with the particle space being the complement. Near the Fermi
surface a set of active particle and hole states is identified, while all
other orbitals are labelled as inactive.

Figure 1: Partition of the single-particle orbitals into active hole, active
particle, inactive hole, and inactive particle states. The orbitals in the
dashed rectangle around the Fermi surface (indicated by a thick dashed line
and its energy $\epsilon_{f}$) are part of the active space.

In tailored coupled-cluster theory, one writes the ground-state wave function
as

$|\Psi_{\rm{TCC}}\rangle=e^{T_{\rm{ext}}+T_{\rm{act}}}|\Phi_{0}\rangle\ .$
(12)

Here $T_{\rm{act}}$ is the cluster operator in the active space, and all its
indices refer to active particle and active hole states. $T_{\rm{ext}}$ is the
cluster operator in the “external” space, in which at least one index belongs
to an inactive particle or inactive hole state. Clearly, Eq. (12) can be
rewritten as

$|\Psi\rangle=|\Psi_{\rm{act}}\rangle\otimes|\Psi_{\rm{ext}}\rangle\ .$ (13)

The amplitudes $T_{\rm{act}}$ are obtained from the full configuration
interaction wave function in the active space, which we expand as

$|\Psi_{\rm{act}}\rangle=C_{0}|\Phi_{0}\rangle+C^{a}_{i}|\Psi_{i}^{a}\rangle+C^{ab}_{ij}|\Psi_{ij}^{ab}\rangle+\cdots\
.$ (14)

Here $C_{0}$, $C^{a}_{i}$, etc. are the coefficients of the reference state,
its 1p-1h, 2p-2h, and up to $A$p-$A$h excitations, respectively, and summation
over repeated indices is implied. The indices $a,b,i,j$ denote active particle
and active hole states. We can also expand

$\displaystyle|\Psi_{\rm{act}}\rangle$ $\displaystyle=$ $\displaystyle
e^{T_{\rm{act}}}|\Phi_{0}\rangle$ (15) $\displaystyle=$
$\displaystyle\left[1+T_{1}+\left(T_{2}+{1\over
2}T_{1}^{2}\right)+\cdots\right]|\Phi_{0}\rangle\ .$

Here, we have dropped the subscript “act” from the cluster amplitudes on the
right-hand-side of Eq. (15), and it is implied that all indices refer to
active single-particle states. Working in the CCSD approximation we get the
active coupled-cluster amplitudes $T_{\rm{act}}$ by comparing Eqs. (14) and
(15) giving,

$T^{a}_{i}=\frac{C^{a}_{i}}{C_{0}}\ ,$ (16)

and

$T^{ab}_{ij}=\frac{C^{ab}_{ij}}{C_{0}}-\frac{C^{a}_{i}C^{b}_{j}-C^{a}_{j}C^{b}_{i}}{C_{0}^{2}}\
.$ (17)

This can easily be extended to also include triples in the active space in
CCSDT. In these equations it is again implied that all indices of the cluster
amplitudes refer to active-space orbitals. We note that the CCSD energy
$\langle\Phi_{0}|\left(He^{T_{\rm{act}}}\right)_{c}|\Phi_{0}\rangle$ equals
the full configuration interaction energy in the active space.

The external cluster operator $T_{\mathrm{ext}}$ is calculated by solving the
usual coupled-cluster equations while keep $T_{\rm{act}}$ fixed. On the CCSD
level the decoupling equation for the external space reads

$\langle\Phi^{a}_{i}|\left(He^{T_{\rm{act}}+T_{\rm{ext}}}\right)_{c}|\Phi_{0}\rangle=0\
,$ (18)

and

$\langle\Phi^{ab}_{ij}|\left(He^{T_{\rm{act}}+T_{\rm{ext}}}\right)_{c}|\Phi_{0}\rangle=0\
.$ (19)

Here, it is understood that at least one of the displayed indices refers to an
inactive orbital. Once the $T_{\rm{ext}}$ is determined, the energy is
calculated as in eq. (9) but with $T=T_{\mathrm{act}}+T_{\mathrm{ext}}$, i.e.

$E_{\rm{TCCSD}}=\langle\Phi_{0}|\left(He^{T_{\rm{act}}+T_{\rm{ext}}}\right)_{c}|\Phi_{0}\rangle\
.$ (20)

Here, the subscript $c$ implies that only connected diagrams are included on
the right-hand side. Equations (18) and (19) show that the active space feeds
back into the amplitudes of the external space, but that the latter does not
impact the former. This is a shortcoming of tailored coupled-cluster method,
and to remedy this would require a large active space such that both static
and dynamic correlations of the exact wave function is accurately described
Melnichuk and Bartlett (2012).

In single-reference coupled-cluster theory the triples (3p-3h) amplitudes
yield about 10% of the CCSD energy Bartlett and Musiał (2007); Hagen  _et al._
(2009). Triples corrections to the tailored CCSD approximation are only
considered in the external space Lyakh  _et al._ (2011). As a result, the
triples correction is smaller than in the standard single-reference approach.
The triples amplitudes solve

$\langle\Phi^{abc}_{ijk}|\left(He^{T_{\rm{act}}+T_{\rm{ext}}+T_{3}}\right)_{c}|\Phi_{0}\rangle=0\
.$ (21)

Here, $T_{\rm{ext}}$ contains up to 2p-2h operators.

The tailored coupled-cluster method takes a cluster operator that describes
the static correlation externally from the full configuration interaction
method. In treating the dynamic correlations in the external space only, there
is no feedback to the active space. To get an accurate result, one needs to
select the active space large enough not only to guarantee the static
correlations are captured, but the wave function of the active space should
also reflect the ”true” wave function. The shortcoming of neglecting the
coupling between the active and external space has been studied by the
extended active space coupled cluster method Melnichuk and Bartlett (2012).
Here one first calculates the full configuration interaction wave function in
a large active space, and subsequently a smaller set of active orbitals is
used in the decoupling of external space in the tailored coupled-cluster
method. In this way there is indirect feedback from the external to the active
space.

##  III Results

###  III.1 Shell-model calculations

The shell-model provides us with a framework to investigate static
correlations and to what extent the tailored coupled-cluster method is able to
capture them. We consider the $0f1p$ shell and the GXPF1A effective
Hamiltonian Honma, M.  _et al._ (2005). The model space is spanned by the
$f_{7/2}$, $p_{3/2}$,$f_{5/2}$, and $p_{1/2}$ orbitals. For 56Ni, the valence
protons and neutrons fill the $f_{7/2}$ shells. To control the amount of
static correlations, we follow Horoi  _et al._ (2007), and shift the single-
particle energies of the $p_{3/2}$, $f_{5/2}$, and $p_{1/2}$ orbitals relative
to the $f_{7/2}$ shell by a constant amount $\Delta G$ from the GXPF1A
parameters. (The original GXPF1A interaction has $\Delta G=0$.) A positive
value of $\Delta G$ increases the shell gap and reduces static correlations,
while a negative $\Delta G$ reduces the shell gap and leads to a system with
stronger static correlations. Horoi  _et al._ (2007) showed that single-
reference coupled-cluster theory fails to reproduce the ground state energy
for sufficiently large negative values of $\Delta G$. This is expected because
a decrease of the shell gap introduces multi-reference physics and
destabilizes the Fermi surface.

We performed Hartree-Fock calculations for the Hamiltionian

$H^{\prime}(\lambda)=H+\lambda Q_{20}$ (22)

where $Q_{20}$ is the quadrupole operator. This yields axially symmetric
reference states. Once the reference was determined, we computed the energy
$E_{\mathrm{ref}}\equiv\langle\Phi_{0}(\lambda)|H|\Phi_{0}(\lambda)\rangle$
and also the quadrupole moment
$q_{20}\equiv\langle\Phi_{0}(\lambda)|Q_{20}|\Phi_{0}(\lambda)\rangle$, and
plotted $E_{\mathrm{ref}}$ as a function of $q_{20}$. The resulting energy
surfaces are shown in Fig. 2 for $\Delta G=0$, $-1$ and $-2$ MeV (from bottom
to top). We see that there is a quantum phase transition in 56Ni from a
spherical to a deformed ground state as $\Delta G$ is decreased from 0 to $-2$
MeV.

Figure 2: Energy of axially symmetric Hartree-Fock reference states as a
function of the quadrupole moment $q_{20}$ for 56Ni computed with the GXPF1A
interaction (bottom) and for a modified interaction where the single-particle
orbitals above the $f_{7/2}$ shell are shifted by $\Delta G$.

We performed three types of coupled-cluster computations, using (i) a
spherical, (ii) an axially symmetric reference, and (iii) tailored active
spaces. First, we take the filled $f_{7/2}$ shell as the spherical reference
state and perform CCSD and coupled-cluster singles-and-doubles and leading-
order iterative triples (CCSDT-1) Lee  _et al._ (1984) computations. The
resulting ground-state energies are shown as squares and circles, respectively
in Fig. 3 and can be compared to the exact result from full configuration
interaction. As already documented by Horoi  _et al._ (2007), spherical
single-reference coupled-cluster theory fails for sufficiently large negative
values of $\Delta G$. Second, we perform CCSD and CCSDT-1 computations using
the axially symmetric reference state of the Hamiltonian (22) using
$\lambda=0$. For $\Delta G\gtrsim-1$ MeV, the reference is spherical, while
smaller values yield an axially symmetric (and deformed) reference. The
results are shown as triangles in Fig. 3. We see that the transition region
$\Delta G\approx-1$ MeV is not well described by single-reference methods,
while coupled-cluster theory (based on an axially-symmetric reference state)
becomes more accurate in the well deformed regime. For $\Delta G=-2$ a clear
minimum develops on the energy surface (see Fig. 2) and the deformed Hartree-
Fock state captures 5 MeV more in binding than the spherical one; the deformed
CCSD captures about another 3 MeV. The total binding from the deformed CCSDT-1
is about 1 MeV away from the full configuration interaction. We estimate that
the discrepancy to full configuration interaction is mainly due to the broken
rotational symmetry. The correlation energy from deformed and spherical
coupled cluster is 3.4 and 4.5 MeV, respectively. Thus, the symmetry breaking
mean-field captures significantly more dynamic correlations.

Figure 3: Ground-state energy of 56Ni, based on the GXPF1A interaction, as a
function of the shift $\Delta G$ between the $f_{7/2}$ orbital and the
unoccupied orbitals in this nucleus. The original interaction has $\Delta
G=0$. Results from a spherical (sph.) and deformed (def.) reference state
based on the CCSD and CCSDT-1 approximations are compared to full
configuration interaction (labelled FCI).

Third and finally, we perform tailored coupled-cluster computations. We select
the $f_{7/2}$ and $p_{3/2}$ as the active space for both protons and neutrons
and find that the results are very close to the CCSD computations based on the
spherical reference, see Fig. 4. Next, we extend the active space to also
include the $p_{1/2}$ orbital. As shown in Fig. 4, this leads only to a modest
improvement. Clearly, the shell-model wave functions are highly correlated,
and full configuration interaction calculations based on the $f_{7/2}$ and
$p_{3/2}$ (or the $f_{7/2}$, $p_{1/2}$, and $p_{3/2}$) orbitals do not yield
accurate wave-function amplitudes. As a final check, we employ the extended
tailored coupled-cluster method Melnichuk and Bartlett (2012). Here, we use
the exact amplitudes from a full configuration interaction calculation and
then choose active spaces consisting of the $f_{7/2}$ and $p_{3/2}$ (and the
$f_{7/2}$, $p_{1/2}$, and $p_{3/2}$) orbitals. In these cases tailored
coupled-cluster yields significantly more accurate results, see Fig. 4.

Figure 4: Tailored coupled cluster (labelled TCC) computations (identified by
the set of quantum numbers $2j$, in square brackets, contained in the active
space) of the ground-state energy of 56Ni, calculated with GXPF1A interaction,
as a function of the shift $\Delta G$ between the $f_{7/2}$ orbital and the
unoccupied orbitals in this nucleus. The original interaction has $\Delta
G=0$. Also shown are computations using the extended (ext) tailored coupled-
cluster method. Results are compared to full configuration interaction
(labelled FCI).

Overall we see that the single-reference coupled-cluster computations that are
based on a deformed (axially-symmetric) Hartree-Fock state are quite accurate
when compared to those from tailored coupled-cluster theory when the shape of
the nucleus is spherical or deformed. Of course, the former lack rotational
invariance for small values of $\Delta G$, and this might be the price to pay
for computational and conceptual simplicity.

###  III.2 Calculations of 12C from chiral interactions

Single-reference coupled-cluster theory is efficient for nuclei with a
significant shell gap such as closed-shell nuclei or well-deformed nuclei
(which exhibit large shell gaps in a Nilsson diagram). The most challenging
nuclei are those with filled subshells that lack a clearly developed shell
gap. Examples are 12C and 28Si. These spherical nuclei are somewhat soft to
deformation and therefore difficult to compute accurately with single-
reference methods. This makes it interesting to use tailored coupled-cluster
theory.

We start with 12C, where the last four protons and four neutrons occupy the
$p_{3/2}$ orbital. However, the shell gap between $p_{3/2}$ and $p_{1/2}$ is
small, and 12C is not a closed-shell nuclei. Its structure is interesting and
complicated Navrátil  _et al._ (2000); Epelbaum  _et al._ (2012); Lovato  _et
al._ (2013).

We compute 12C using the NNLOopt nucleon-nucleon interaction Ekström  _et al._
(2013) and an oscillator basis with frequency $\hbar\omega$ and highest
single-particle energy $N_{\rm max}\hbar\omega$ (omitting the zero-point
energy). We use $N_{\rm{max}}=12$ unless stated otherwise. In a first
approach, we employ a reference state that fills the harmonic oscillator
orbitals up to and including the $p_{3/2}$ orbitals and perform coupled-
cluster computations using the CCSDT-3 approximation Urban  _et al._ (1985).
The results are shown (as a function of the oscillator spacing $\hbar\omega$)
in Fig. 5 and compared to the experiment. The interaction NNLOopt is
presumably accurate for light nuclei and this suggests that about 7 MeV of
binding energy are still lacking at the triples result. We also performed
calculations where the spherical Hartree-Fock state serves as the reference.
As can be seen in Fig. 5 the results are close to those of the harmonic
oscillator basis. Finally, we performed CCSDT-1 computations based on an
axially symmetric (and deformed) Hartree-Fock reference. These results yield a
ground state that is close to the one in the spherical harmonic-oscillator
basis and about 2 MeV lower than the CCSDT-3 results using the spherical
Hartree-Fock basis.

For the tailored coupled-cluster calculations the active space is spanned by
the $p_{3/2}$, $sd$, and $pf$ shell orbitals in the harmonic oscillator basis.
The large active space is motivated by the need to potentially include some
effects of deformation. Thus, we need to include the $pf$ shell as it has the
same parity as the occupied $p_{3/2}$. The dimension of Hamiltonian matrix in
the active space is about $2^{8}$, which is almost the limit of our current
implement of the code. The results at the singles and doubles, and at the
triples approximation – also shown in Fig. 5 – exhibit a significant gain in
the ground-state energy. This is re-assuring and points to the value of the
tailored coupled-cluster method for this nucleus. We also performed tailored
coupled-cluster calculations in the Hartree-Fock basis. This, however, was not
as successful as for the harmonic oscillator basis, because the tailored
coupled-cluster results did not improve over the CCSDT-3 results. We do not
understand why this is so.

Figure 5: Ground-state energy of 12C calculated with single-reference coupled
cluster methods based on reference states in the spherical harmonic oscillator
harmonic-oscillator (sph. HO) basis, the spherical Hartree-Fock (sph. HF)
basis, and the axially symmetric, deformed, Hartree-Fock basis (def. HF),
using CCSDT-3 and CCSDT-1 approximations. Also shown are the tailored coupled-
cluster calculations (labelled TCC) and the experimental data. The two-body
interaction is NNLOopt.

We turn to calculations with three-nucleon forces and employed the chiral
interaction 1.8/2.0 (EM) from Ref. Hebeler  _et al._ (2011). For the coupled-
cluster calculations we want to employ the normal-ordered two-body
approximation Hagen  _et al._ (2008); Roth  _et al._ (2012). While this is
straight forward in the Hartree-Fock basis, the calculations in the harmonic-
oscillator basis require more thought and care; a naive application of the
normal-ordered two-body approximation would introduce a strong
$\hbar\omega$-dependence of the results. To avoid this we perform a spherical
Hartree-Fock calculation, and normal-order the Hamiltonian with respect to the
Hartree-Fock reference. We employ the normal-ordered two-body approximation,
discard the normal-ordered residual three-nucleon terms, and store the normal-
ordered one- and two-body matrix elements of the Hamiltonian. Coupled-cluster
calculations in the Hartree-Fock basis then use these matrix elements. For the
coupled-cluster calculations in the harmonic oscillator basis we re-normal-
order the stored matrix elements of the one- and two-body Hamiltonian with
respect to the true vacuum and transform to the harmonic oscillator basis
Djärv  _et al._ (2021); Frosini  _et al._ (2021). This then yields one- and
two-body matrix elements that contain the relevant contributions of the three-
nucleon force. Applications of this approach in the no-core shell-model
verified that the impact on the center-of-mass separation is controlled Djärv
_et al._ (2021). Figure 6 shows the ground state energy of 12C calculated with
spherical coupled-cluster theory in the Hartree-Fock and the harmonic
oscillator basis, using the CCSDT-3 approximation.

We also performed similar calculations in a deformed (yet axially symmetric)
Hartree-Fock basis, using a quadrupole constraint to map out the energy
surface. We found an oblate deformed minimum which we used as the Hartree-Fock
reference. This agrees with the dominant “triangle” configuration of $\alpha$
clusters found in Ref. Epelbaum  _et al._ (2012). In these calculations we
performed the normal-ordered two-body approximation in the deformed reference
state. The CCSDT-1 results in this basis are very close to the CCSDT-3 results
in the spherical harmonic-oscillator basis.

The tailored coupled-cluster calculations are performed in the harmonic-
oscillator basis. The active space is spanned by $p_{3/2}$ shell and the $sd$
and $pf$ shells. We verified that our calculations are converged at
$N_{\rm{max}}=12$. The tailored coupled-cluster results depend on the
harmonic-oscillator spacing and they overbind the nucleus somewhat. This
latter observation is consistent with similar results from quantum chemistry,
where small active spaces in multi-reference methods tend to overestimate the
role of static correlations Evangelista (2018). We note that the tailored
coupled-cluster results deviate about as much from experiment as the coupled-
cluster results obtained in the spherical harmonic-oscillator basis.

Figure 6: Ground-state energy of 12C calculated with single-reference coupled
cluster methods based on reference states in the spherical harmonic-oscillator
(sph. HO) basis, the spherical Hartree-Fock (sph. HF) basis, and the axially
symmetric, deformed, Hartree-Fock basis (def. HF), using CCSDT-3 and CCSDT-1
approximations. Also shown are the tailored coupled-cluster calculations
(labelled TCC) and the experimental data. The employed chiral interaction is
1.8/2.0 (EM) from Ref. Hebeler  _et al._ (2011).

###  III.3 Calculations of 28Si from chiral interactions

The nucleus 28Si, with filled $d_{5/2}$ shells in the shell model, is another
challenging nucleus. We employ the 1.8/2.0(EM) chiral potential in a model
space of $N_{\rm{max}}=12$, and show our results in Fig. 7. The single-
reference coupled-cluster calculations are again based on a spherical
harmonic-oscillator state, and on spherical and deformed Hartree-Fock states;
from these, the deformed reference yields most accurate results (and a gain of
about 3 MeV in the binding energy). We see that 28Si differs in this regard
from 12C. It is somewhat more deformed in its ground state.

Figure 7: Ground-state energy of 28Si calculated with single-reference coupled
cluster methods based on reference states in the spherical harmonic-oscillator
(sph. HO) basis, the spherical Hartree-Fock (sph. HF) basis, and the axially
symmetric, deformed, Hartree-Fock basis (def. HF), using CCSDT-3 and CCSDT-1
approximations. Also shown are the tailored coupled-cluster calculations
(labelled TCC) and the experimental data. The employed chiral interaction is
1.8/2.0 (EM) from Ref. Hebeler  _et al._ (2011).

In our tailored coupled-cluster calculation, the active space is spanned by
the $sd$ and $f_{7/2}$ shells, filled with six protons and six neutrons. This
requires us to perform large-scale shell-model calculations. The results are
clearly not converged and strongly depend on the oscillator frequency. We
understand the lack of convergence as follows: To capture quadrupole
deformation from the $d_{5/2}$ shell, we would need to also include the
$0g1d2s$ shell in the active space (because the quadrupole operator couples
the $0d1s$ and $0g1d2s$ shells), but this is not possible with current
computing resources (the shell-model dimension is beyond $10^{9}$). We also
performed similar calculation by using an active space spanned by the $p$ and
$sd$ shells, with 12 protons ans 12 neutrons in the active space. This did not
significantly change the results. The active space we employ – though large
from its computational demands – is too small to capture the deformation.

For a simpler calculation we studied 28Si with the NNLOopt interaction, using
the active space consisting of the $sd$ and $f_{7/2}$ shells. Results are
shown in Fig. 8. The single-reference coupled-cluster calculations indicate
that the nucleus exhibits stronger deformation than for the 1.8/2.0(EM)
interaction, because calculations based on a deformed reference gain about 10
MeV in binding. The NNLOopt potential is probably not as accurate in this
region of the nuclear chart as the 1.8/2.0 (EM) interaction, and – based on
our experience with single-reference coupled-cluster calculations – it
overbinds the 28Si nucleus. The tailored coupled-cluster results are better
converged than for the 1.8/2.0 (EM) interaction, although the
$\hbar\omega$-dependence is still strong. We attribute the difference between
the tailored coupled-cluster results and the calculations with the deformed
Hartree-Fock reference on the too-small active space that does not capture
deformation.

Figure 8: Ground-state energy of 28Si calculated with single-reference coupled
cluster methods based on reference states in the spherical harmonic-oscillator
(sph. HO) basis, the spherical Hartree-Fock (sph. HF) basis, and the axially
symmetric, deformed, Hartree-Fock basis (def. HF), using CCSDT-3 and CCSDT-1
approximations. Also shown are the tailored coupled-cluster calculations
(labelled TCC) and the experimental data. The two-body interaction is NNLOopt.

##  IV Summary

Single-reference coupled-cluster calculations become challenging in nuclei
that are neither strongly deformed (i.e. exhibiting rigid-rotor-like ground-
state rotational bands) nor spherical, e.g. for 12C, 28Si, and 56Ni. In such
situations a combination of full configuration interaction and coupled-cluster
methods seems promising, and the prospect of hybrid quantum-classical
computing makes this interesting to study.

We find that tailored coupled cluster methods can be alternatives to single-
reference methods in certain nuclei. They typically gain energy compared to
spherical single-reference methods and keep the symmetry of the wave function
intact. For computations of ground-state energies, symmetry-broken single-
reference approaches are quite competitive. While they require symmetry
projection (in contrast to tailored coupled-cluster), they exhibit better
convergence with respect to model-space parameters than tailored coupled
cluster methods.

###### Acknowledgements.

This work is supported by the Quantum Science Center, a National Quantum
Information Science Research Center of the U.S. Department of Energy. It is
also supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics under Award Numbers DEFG02-96ER40963, DE-SC0018223 (SciDAC-4
NUCLEI), and the SciDAC-5 NUCLEI collaboration. Computer time was provided by
the Innovative and Novel Computational Impact on Theory and Experiment
(INCITE) program. This research used resources of the Oak Ridge Leadership
Computing Facility located at Oak Ridge National Laboratory, which is
supported by the Office of Science of the Department of Energy under Contract
No. DE-AC05-00OR22725.

## References

  * Navrátil and Caurier (2004) P. Navrátil and E. Caurier, “Nuclear structure with accurate chiral perturbation theory nucleon-nucleon potential: Application to ${}^{6}\mathrm{Li}$ and ${}^{10}\mathrm{B}$,” Phys. Rev. C 69, 014311 (2004).
  * Forssén  _et al._ (2005) C. Forssén, P. Navrátil, W. E. Ormand, and E. Caurier, “Large basis ab initio shell model investigation of ${}^{9}\mathrm{Be}$ and ${}^{11}\mathrm{Be}$,” Phys. Rev. C 71, 044312 (2005).
  * Nogga  _et al._ (2006) A. Nogga, P. Navrátil, B. R. Barrett, and J. P. Vary, “Spectra and binding energy predictions of chiral interactions for ${}^{7}\mathrm{Li}$,” Phys. Rev. C 73, 064002 (2006).
  * Hagen  _et al._ (2008) G. Hagen, T. Papenbrock, D. J. Dean, and M. Hjorth-Jensen, “Medium-mass nuclei from chiral nucleon-nucleon interactions,” Phys. Rev. Lett. 101, 092502 (2008).
  * Epelbaum  _et al._ (2011) Evgeny Epelbaum, Hermann Krebs, Dean Lee, and Ulf-G. Meißner, “Ab Initio Calculation of the Hoyle State,” Phys. Rev. Lett. 106, 192501 (2011).
  * Roth  _et al._ (2012) Robert Roth, Sven Binder, Klaus Vobig, Angelo Calci, Joachim Langhammer, and Petr Navrátil, “Medium-Mass Nuclei with Normal-Ordered Chiral $NN\mathbf{+}3N$ Interactions,” Phys. Rev. Lett. 109, 052501 (2012).
  * Somà  _et al._ (2014) V. Somà, A. Cipollone, C. Barbieri, P. Navrátil, and T. Duguet, “Chiral two- and three-nucleon forces along medium-mass isotope chains,” Phys. Rev. C 89, 061301 (2014).
  * Duguet  _et al._ (2017) T. Duguet, V. Somà, S. Lecluse, C. Barbieri, and P. Navrátil, “Ab initio calculation of the potential bubble nucleus ${}^{34}\mathrm{Si}$,” Phys. Rev. C 95, 034319 (2017).
  * Morris  _et al._ (2018) T. D. Morris, J. Simonis, S. R. Stroberg, C. Stumpf, G. Hagen, J. D. Holt, G. R. Jansen, T. Papenbrock, R. Roth, and A. Schwenk, “Structure of the lightest tin isotopes,” Phys. Rev. Lett. 120, 152503 (2018).
  * Arthuis  _et al._ (2020) P. Arthuis, C. Barbieri, M. Vorabbi, and P. Finelli, “Ab initio computation of charge densities for sn and xe isotopes,” Phys. Rev. Lett. 125, 182501 (2020).
  * Hu  _et al._ (2021) Baishan Hu, Weiguang Jiang, Takayuki Miyagi, Zhonghao Sun, Andreas Ekström, Christian Forssén, Gaute Hagen, Jason D. Holt, Thomas Papenbrock, S. Ragnar Stroberg, and Ian Vernon, “Ab initio predictions link the neutron skin of 208Pb to nuclear forces,” arXiv e-prints , arXiv:2112.01125 (2021), arXiv:2112.01125 [nucl-th] .
  * van Kolck (1994) U. van Kolck, “Few-nucleon forces from chiral Lagrangians,” Phys. Rev. C 49, 2932–2941 (1994).
  * Entem and Machleidt (2003) D. R. Entem and R. Machleidt, “Accurate charge-dependent nucleon-nucleon potential at fourth order of chiral perturbation theory,” Phys. Rev. C 68, 041001 (2003).
  * Epelbaum  _et al._ (2009) E. Epelbaum, H.-W. Hammer, and Ulf-G. Meißner, “Modern theory of nuclear forces,” Rev. Mod. Phys. 81, 1773–1825 (2009).
  * Bogner  _et al._ (2003) S. K. Bogner, T. T. S. Kuo, and A. Schwenk, “Model-independent low momentum nucleon interaction from phase shift equivalence,” Physics Reports 386, 1 – 27 (2003).
  * Bogner  _et al._ (2007) S. K. Bogner, R. J. Furnstahl, and R. J. Perry, “Similarity renormalization group for nucleon-nucleon interactions,” Phys. Rev. C 75, 061001 (2007).
  * Dickhoff and Barbieri (2004) W.H. Dickhoff and C. Barbieri, “Self-consistent green’s function method for nuclei and nuclear matter,” Prog. Part. Nucl. Phys. 52, 377 – 496 (2004).
  * Lee (2009) Dean Lee, “Lattice simulations for few- and many-body systems,” Prog. Part. Nucl. Phys. 63, 117 – 154 (2009).
  * Tsukiyama  _et al._ (2011) K. Tsukiyama, S. K. Bogner, and A. Schwenk, “In-Medium Similarity Renormalization Group For Nuclei,” Phys. Rev. Lett. 106, 222502 (2011).
  * Hagen  _et al._ (2014) G. Hagen, T. Papenbrock, M. Hjorth-Jensen, and D. J. Dean, “Coupled-cluster computations of atomic nuclei,” Rep. Prog. Phys. 77, 096302 (2014).
  * Stroberg  _et al._ (2016) S. R. Stroberg, H. Hergert, J. D. Holt, S. K. Bogner, and A. Schwenk, “Ground and excited states of doubly open-shell nuclei from ab initio valence-space hamiltonians,” Phys. Rev. C 93, 051301 (2016).
  * Dumitrescu  _et al._ (2018) E. F. Dumitrescu, A. J. McCaskey, G. Hagen, G. R. Jansen, T. D. Morris, T. Papenbrock, R. C. Pooser, D. J. Dean, and P. Lougovski, “Cloud quantum computing of an atomic nucleus,” Phys. Rev. Lett. 120, 210501 (2018).
  * Stetcu  _et al._ (2022) I. Stetcu, A. Baroni, and J. Carlson, “Variational approaches to constructing the many-body nuclear ground state for quantum computing,” Phys. Rev. C 105, 064308 (2022).
  * Kiss  _et al._ (2022) Oriel Kiss, Michele Grossi, Pavel Lougovski, Federico Sanchez, Sofia Vallecorsa, and Thomas Papenbrock, “Quantum computing of the ${}^{6}\mathrm{Li}$ nucleus via ordered unitary coupled clusters,” Phys. Rev. C 106, 034325 (2022).
  * Robin  _et al._ (2021) Caroline Robin, Martin J. Savage, and Nathalie Pillet, “Entanglement rearrangement in self-consistent nuclear structure calculations,” Phys. Rev. C 103, 034325 (2021).
  * Beane  _et al._ (2019) Silas R. Beane, David B. Kaplan, Natalie Klco, and Martin J. Savage, “Entanglement suppression and emergent symmetries of strong interactions,” Phys. Rev. Lett. 122, 102001 (2019).
  * Gu  _et al._ (2023) Chenyi Gu, Z. H. Sun, G. Hagen, and T. Papenbrock, “Entanglement entropy of nuclear systems,” arXiv e-prints , arXiv:2303.04799 (2023).
  * Sun  _et al._ (2022) Z. H. Sun, C. A. Bell, G. Hagen, and T. Papenbrock, “How to renormalize coupled cluster theory,” Phys. Rev. C 106, L061302 (2022).
  * Roggero  _et al._ (2020) Alessandro Roggero, Chenyi Gu, Alessandro Baroni, and Thomas Papenbrock, “Preparation of excited states for nuclear dynamics on a quantum computer,” Phys. Rev. C 102, 064624 (2020).
  * Choi  _et al._ (2021) Kenneth Choi, Dean Lee, Joey Bonitati, Zhengrong Qian, and Jacob Watkins, “Rodeo algorithm for quantum computing,” Phys. Rev. Lett. 127, 040505 (2021).
  * Baroni  _et al._ (2022) A. Baroni, J. Carlson, R. Gupta, Andy C. Y. Li, G. N. Perdue, and A. Roggero, “Nuclear two point correlation functions on a quantum computer,” Phys. Rev. D 105, 074503 (2022).
  * Kümmel  _et al._ (1978) H. Kümmel, K. H. Lührmann, and J. G. Zabolitzky, “Many-fermion theory in expS- (or coupled cluster) form,” Physics Reports 36, 1 – 63 (1978).
  * Bishop (1991) R. F. Bishop, “An overview of coupled cluster theory and its applications in physics,” Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta) 80, 95–148 (1991).
  * Bartlett and Musiał (2007) Rodney J. Bartlett and Monika Musiał, “Coupled-cluster theory in quantum chemistry,” Rev. Mod. Phys. 79, 291–352 (2007).
  * Romero  _et al._ (2017) Jonathan Romero, Ryan Babbush, Jarrod R. McClean, Cornelius Hempel, Peter Love, and Alán Aspuru-Guzik, “Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz,” arXiv e-prints , arXiv:1701.02691 (2017).
  * Lee  _et al._ (2019) Joonho Lee, William J. Huggins, Martin Head-Gordon, and K. Birgitta Whaley, “Generalized Unitary Coupled Cluster Wavefunctions for Quantum Computation,” J. Chem. Theory Comput. 15, 311–324 (2019).
  * Stolarczyk (1994) Leszek Z. Stolarczyk, “Complete active space coupled-cluster method. Extension of single-reference coupled-cluster method using the CASSCF wavefunction,” Chem. Phys. Lett. 217, 1–6 (1994).
  * Hino  _et al._ (2006) Osamu Hino, Tomoko Kinoshita, Garnet Kin Lie Chan, and Rodney J. Bartlett, “Tailored coupled cluster singles and doubles method applied to calculations on molecular structure and harmonic vibrational frequencies of ozone,” J. Chem. Phys. 124 (2006), 10.1063/1.2180775.
  * Kinoshita  _et al._ (2005) Tomoko Kinoshita, Osamu Hino, and Rodney J. Bartlett, “Coupled-cluster method tailored by configuration interaction,” J. Chem. Phys. 123, 0–6 (2005).
  * Keen  _et al._ (2021) Trevor Keen, Bo Peng, Karol Kowalski, Pavel Lougovski, and Steven Johnston, “Hybrid quantum-classical approach for coupled-cluster green’s function theory,”  (2021).
  * Bauman  _et al._ (2023) Nicholas P. Bauman, Bo Peng, and Karol Kowalski, “Coupled cluster downfolding techniques: a review of existing applications in classical and quantum computing for chemical systems,” arXiv e-prints , arXiv:2303.00087 (2023), 2303.00087 .
  * Lee  _et al._ (2022) Seunghoon Lee, Joonho Lee, Huanchen Zhai, Yu Tong, Alexander M. Dalzell, Ashutosh Kumar, Phillip Helms, Johnnie Gray, Zhi-Hao Cui, Wenyuan Liu, Michael Kastoryano, Ryan Babbush, John Preskill, David R. Reichman, Earl T. Campbell, Edward F. Valeev, Lin Lin, and Garnet Kin-Lic Chan, “Is there evidence for exponential quantum advantage in quantum chemistry?” arXiv e-prints , arXiv:2208.02199 (2022).
  * Mok  _et al._ (1996) Daniel K. W. Mok, Ralf Neumann, and Nicholas C. Handy, “Dynamical and nondynamical correlation,” The Journal of Physical Chemistry 100, 6225–6230 (1996), https://doi.org/10.1021/jp9528020 .
  * Crittenden (2013) Deborah L. Crittenden, “A hierarchy of static correlation models,” The Journal of Physical Chemistry A 117, 3852–3860 (2013), pMID: 23547909, https://doi.org/10.1021/jp400669p .
  * Ramos-Cordoba  _et al._ (2016) Eloy Ramos-Cordoba, Pedro Salvador, and Eduard Matito, “Separation of dynamic and nondynamic correlation,” Phys. Chem. Chem. Phys. 18, 24015–24023 (2016).
  * Benavides-Riveros  _et al._ (2017) Carlos L. Benavides-Riveros, Nektarios N. Lathiotakis, and Miguel A. L. Marques, “Towards a formal definition of static and dynamic electronic correlations,” Phys. Chem. Chem. Phys. 19, 12655–12664 (2017).
  * Hollett  _et al._ (2016) Joshua W. Hollett, Hessam Hosseini, and Cameron Menzies, “A cumulant functional for static and dynamic correlation,” The Journal of Chemical Physics 145, 084106 (2016), https://doi.org/10.1063/1.4961243 .
  * Hagen and Nam (2012) Gaute Hagen and Hai Ah Nam, “Computational aspects of nuclear coupled-cluster theory,” Progress of Theoretical Physics Supplement 196, 102–110 (2012).
  * Hagen  _et al._ (2010) G. Hagen, T. Papenbrock, D. J. Dean, and M. Hjorth-Jensen, “Ab initio coupled-cluster approach to nuclear structure with modern nucleon-nucleon interactions,” Phys. Rev. C 82, 034330 (2010).
  * Hagen  _et al._ (2016) G. Hagen, G. R. Jansen, and T. Papenbrock, “Structure of ${}^{78}\mathrm{Ni}$ from first-principles computations,” Phys. Rev. Lett. 117, 172501 (2016).
  * Hergert  _et al._ (2013) H. Hergert, S. K. Bogner, S. Binder, A. Calci, J. Langhammer, R. Roth, and A. Schwenk, “In-medium similarity renormalization group with chiral two- plus three-nucleon interactions,” Phys. Rev. C 87, 034307 (2013).
  * Hergert  _et al._ (2016) H. Hergert, S. K. Bogner, T. D. Morris, A. Schwenk, and K. Tsukiyama, “The in-medium similarity renormalization group: A novel ab initio method for nuclei,” Phys. Rep. 621, 165 – 222 (2016).
  * Hagen  _et al._ (2009) G. Hagen, T. Papenbrock, D. J. Dean, M. Hjorth-Jensen, and B. Velamur Asokan, “Ab initio computation of neutron-rich oxygen isotopes,” Phys. Rev. C 80, 021306 (2009).
  * Horoi  _et al._ (2007) M. Horoi, J. R. Gour, M. Włoch, M. D. Lodriguito, B. A. Brown, and P. Piecuch, “Coupled-cluster and configuration-interaction calculations for heavy nuclei,” Phys. Rev. Lett. 98, 112501 (2007).
  * Evangelista (2018) Francesco A. Evangelista, “Perspective: Multireference coupled cluster theories of dynamical electron correlation,” The Journal of Chemical Physics 149, 030901 (2018), https://doi.org/10.1063/1.5039496 .
  * Lyakh  _et al._ (2012) Dmitry I. Lyakh, Monika Musiał, Victor F. Lotrich, and Rodney J. Bartlett, “Multireference nature of chemistry: The coupled-cluster view,” Chemical Reviews 112, 182–243 (2012), pMID: 22220988, https://doi.org/10.1021/cr2001417 .
  * Qiu  _et al._ (2017) Yiheng Qiu, Thomas M. Henderson, Jinmo Zhao, and Gustavo E. Scuseria, “Projected coupled cluster theory,” The Journal of Chemical Physics 147, 064111 (2017).
  * Qiu  _et al._ (2018) Yiheng Qiu, Thomas M. Henderson, Jinmo Zhao, and Gustavo E. Scuseria, “Projected coupled cluster theory: Optimization of cluster amplitudes in the presence of symmetry projection,” The Journal of Chemical Physics 149, 164108 (2018).
  * Qiu  _et al._ (2019) Y. Qiu, T. M. Henderson, T. Duguet, and G. E. Scuseria, “Particle-number projected bogoliubov-coupled-cluster theory: Application to the pairing hamiltonian,” Phys. Rev. C 99, 044301 (2019).
  * Frosini  _et al._ (2022a) M. Frosini, T. Duguet, J.-P. Ebran, B. Bally, T. Mongelli, T. R. Rodríguez, R. Roth, and V. Somà, “Multi-reference many-body perturbation theory for nuclei,” The European Physical Journal A 58, 63 (2022a).
  * Frosini  _et al._ (2022b) M. Frosini, T. Duguet, J.-P. Ebran, and V. Somà, “Multi-reference many-body perturbation theory for nuclei,” The European Physical Journal A 58, 62 (2022b).
  * Frosini  _et al._ (2022c) M. Frosini, T. Duguet, J.-P. Ebran, B. Bally, H. Hergert, T. R. Rodríguez, R. Roth, J. M. Yao, and V. Somà, “Multi-reference many-body perturbation theory for nuclei,” The European Physical Journal A 58, 64 (2022c).
  * Hagen  _et al._ (2022) G. Hagen, S. J. Novario, Z. H. Sun, T. Papenbrock, G. R. Jansen, J. G. Lietz, T. Duguet, and A. Tichai, “Angular-momentum projection in coupled-cluster theory: structure of 34Mg,” arXiv e-prints , arXiv:2201.07298 (2022), arXiv:2201.07298 [nucl-th] .
  * Peierls and Yoccoz (1957) R. E. Peierls and J. Yoccoz, “The collective model of nuclear motion,” Proceedings of the Physical Society. Section A 70, 381–387 (1957).
  * Papenbrock (2022) T. Papenbrock, “Effective field theory of pairing rotations,” Phys. Rev. C 105, 044322 (2022).
  * Sun  _et al._ (2018) Z. H. Sun, T. D. Morris, G. Hagen, G. R. Jansen, and T. Papenbrock, “Shell-model coupled-cluster method for open-shell nuclei,” Phys. Rev. C 98, 054320 (2018).
  * Sun  _et al._ (2021) Z. H. Sun, G. Hagen, G. R. Jansen, and T. Papenbrock, “Effective shell-model interaction for nuclei ”southeast” of ${}^{100}\mathrm{Sn}$,” Phys. Rev. C 104, 064310 (2021).
  * Tsukiyama  _et al._ (2012) K. Tsukiyama, S. K. Bogner, and A. Schwenk, “In-medium similarity renormalization group for open-shell nuclei,” Phys. Rev. C 85, 061304 (2012).
  * Faulstich  _et al._ (2019) Fabian M. Faulstich, Mihály Máté, Andre Laestadius, Mihály András Csirik, Libor Veis, Andrej Antalik, Jiří Brabec, Reinhold Schneider, Jiří Pittner, Simen Kvaal, and Örs Legeza, “Numerical and theoretical aspects of the dmrg-tcc method exemplified by the nitrogen dimer,” Journal of Chemical Theory and Computation 15, 2206–2220 (2019), pMID: 30802406, https://doi.org/10.1021/acs.jctc.8b00960 .
  * Vitale  _et al._ (2020) Eugenio Vitale, Ali Alavi, and Daniel Kats, “Fciqmc-tailored distinguishable cluster approach,” Journal of Chemical Theory and Computation 16, 5621–5634 (2020), pMID: 32786911, https://doi.org/10.1021/acs.jctc.0c00470 .
  * Leszczyk  _et al._ (2022) Aleksandra Leszczyk, Mihály Máté, Örs Legeza, and Katharina Boguslawski, “Assessing the accuracy of tailored coupled cluster methods corrected by electronic wave functions of polynomial cost,” Journal of Chemical Theory and Computation 18, 96–117 (2022), pMID: 34965121, https://doi.org/10.1021/acs.jctc.1c00284 .
  * Hagen  _et al._ (2007) G. Hagen, T. Papenbrock, D. J. Dean, A. Schwenk, A. Nogga, M. Włoch, and P. Piecuch, “Coupled-cluster theory for three-body Hamiltonians,” Phys. Rev. C 76, 034302 (2007).
  * Melnichuk and Bartlett (2012) Anna Melnichuk and Rodney J. Bartlett, “Relaxed active space: Fixing tailored-cc with high order coupled cluster. i,” The Journal of Chemical Physics 137, 214103 (2012), https://doi.org/10.1063/1.4767900 .
  * Lyakh  _et al._ (2011) Dmitry I. Lyakh, Victor F. Lotrich, and Rodney J. Bartlett, “The ’tailored’ CCSD(T) description of the automerization of cyclobutadiene,” Chem. Phys. Lett. 501, 166–171 (2011).
  * Honma, M.  _et al._ (2005) Honma, M., Otsuka, T., Brown, B. A., and Mizusaki, T., “Shell-model description of neutron-rich pf-shell nuclei with a new effective interaction gxpf 1,” Eur. Phys. J. A 25, 499–502 (2005).
  * Lee  _et al._ (1984) Yoon S. Lee, Stanislaw A. Kucharski, and Rodney J. Bartlett, “A coupled cluster approach with triple excitations,” The Journal of Chemical Physics 81, 5906–5912 (1984).
  * Navrátil  _et al._ (2000) P. Navrátil, J. P. Vary, and B. R. Barrett, “Large-basis ab initio no-core shell model and its application to ${}^{12}\mathbf{C}$,” Phys. Rev. C 62, 054311 (2000).
  * Epelbaum  _et al._ (2012) Evgeny Epelbaum, Hermann Krebs, Timo A. Lähde, Dean Lee, and Ulf-G. Meißner, “Structure and rotations of the hoyle state,” Phys. Rev. Lett. 109, 252501 (2012).
  * Lovato  _et al._ (2013) A. Lovato, S. Gandolfi, Ralph Butler, J. Carlson, Ewing Lusk, Steven C. Pieper, and R. Schiavilla, “Charge form factor and sum rules of electromagnetic response functions in ${}^{12}\mathbf{C}$,” Phys. Rev. Lett. 111, 092501 (2013).
  * Ekström  _et al._ (2013) A. Ekström, G. Baardsen, C. Forssén, G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, W. Nazarewicz, T. Papenbrock, J. Sarich, and S. M. Wild, “Optimized chiral nucleon-nucleon interaction at next-to-next-to-leading order,” Phys. Rev. Lett. 110, 192502 (2013).
  * Urban  _et al._ (1985) Miroslav Urban, Jozef Noga, Samuel J. Cole, and Rodney J. Bartlett, “Towards a full ccsdt model for electron correlation,” The Journal of Chemical Physics 83, 4041–4046 (1985).
  * Hebeler  _et al._ (2011) K. Hebeler, S. K. Bogner, R. J. Furnstahl, A. Nogga, and A. Schwenk, “Improved nuclear matter calculations from chiral low-momentum interactions,” Phys. Rev. C 83, 031301 (2011).
  * Djärv  _et al._ (2021) T. Djärv, A. Ekström, C. Forssén, and G. R. Jansen, “Normal-ordering approximations and translational (non)invariance,” Phys. Rev. C 104, 024324 (2021).
  * Frosini  _et al._ (2021) M. Frosini, T. Duguet, B. Bally, Y. Beaujeault-Taudière, J.-P. Ebran, and V. Somà, “In-medium k-body reduction of n-body operators,” The European Physical Journal A 57, 151 (2021).