#  Ab Initio Symmetry-adapted Emulator
for Studying Emergent Collectivity
and Clustering in Nuclei

K. S. Becker<EMAIL_ADDRESS>K. D. Launey  Department of Physics and
Astronomy,
Louisiana State University,
Baton Rouge, LA, United States  A. Ekström  Department of Physics,
Chalmers Institute of Technology,
Gothenburg, Sweden  T. Dytrych  Nuclear Physics Institute,
Czech Academy of Sciences,
Řež, Czech Republic

###### Abstract

We discuss emulators from the ab initio symmetry-adapted no-core shell-model
framework for studying the formation of alpha clustering and collective
properties without effective charges. We present a new type of an emulator,
one that utilizes the eigenvector continuation technique but is based on the
use of symplectic symmetry considerations. This is achieved by using
physically relevant degrees of freedom, namely, the symmetry-adapted basis,
which exploits the almost perfect symplectic symmetry in nuclei. Specifically,
we study excitation energies, point-proton root-mean-square radii, along with
electric quadrupole moments and transitions for 6Li and 12C. We show that the
set of parameterizations of the chiral potential used to train the emulators
has no significant effect on predictions of dominant nuclear features, such as
shape and the associated symplectic symmetry, along with cluster formation,
but slightly varies details that affect collective quadrupole moments,
asymptotic normalization coefficients, and alpha partial widths up to a factor
of two. This makes these types of emulators important for further constraining
the nuclear force for high-precision nuclear structure and reaction
observables.

ab initio SA-NCSM, nuclear collectivity, nuclear clustering, EVC, emulators,
6Li, 12C

††preprint: APS/123-QED Figure 1:  (a) Emergent symplectic symmetry in nuclei:
Contribution of the most dominant shape to the $0^{+}$ ground state of 20Ne
and its rotational band ($2^{+}$, $4^{+}$, $6^{+}$, and $8^{+}$), as well as
to excited $0^{+}$ states, pointing to a fragmented giant monopole resonance
[1]; for selected states, the deformation distribution within a shape is shown
in terms of the shape parameters, the average deformation $\beta$ and
triaxiality angle $\gamma$ (based on ab initio SA-NCSM calculations with
NNLOopt in a model space of 11 HO shells with $\hbar\Omega$=15 MeV inter-shell
distance). (b) Schematic illustration of the SA concept shown for 8Be: a
smaller model space (square) includes all possible shapes (labeled as “All”)
and yields spatially compressed wave functions (top); a larger model space
(rectangle in lower panel) accommodates, in a well prescribed way, spatially
extended modes (“SA selection”) that are neglected in smaller model spaces.
Figure from Ref. [2].

##  I Introduction

Ab initio approaches to nuclear structure and reactions (for an overview, see
Ref. [3]) aim to provide accurate predictions based on few-nucleon forces,
such as the ones derived from chiral effective field theory (EFT) [for a
review, see e.g. Ref. [4] and references therein]. To achieve this, it is
imperative to utilize high-precision nuclear forces that accurately describe
nuclear correlations, from short- to long-range correlations, as well as to
quantify uncertainties that arise from the nuclear force and the controlled
approximations in solving the many-body Schrödinger equation [5]. Such
developments employ statistical tools, including, for example, Bayesian
analysis [6], global sensitivity methods [7], and uncertainty estimates based
on regression [8, 9], that sometimes require a large number of computationally
intensive calculations which often poses a challenge.

In this paper, we seek to overcome some of these difficulties by combining the
symmetry-adapted no-core shell model (SA-NCSM) framework [1, 2, 10] with the
methodology of eigenvector continuation (EVC) [7, 11, 12]. The SA-NCSM uses a
physically relevant basis that, in manageable model spaces, achieves
descriptions of light to medium-mass nuclei, including challenging nuclear
features, such as collectivity, clustering, and related continuum effects.
Similarly, EVC further reduces the sizes of Hamiltonian matrices by mapping
them onto much smaller matrices referred to as emulators, low-dimensional
manifolds built upon a set of characteristic solutions to the many-body
Schrödinger equation. The proposed symmetry-adapted eigenvector continuation
(SA-EVC) method opens the door to calculations up through the medium-mass
region and studies of collective and clustering nuclear features that
otherwise might be computationally infeasible.

With a view toward inferring new knowledge of the nuclear forces relevant to
structure and reaction observables, we construct novel SA-EVC emulators to
study collective and clustering nuclear properties in 6Li and 12C (an emulator
for the 6Li binding energy is validated in Ref. [13]). Because this study
focuses on the method validity, we utilize SA-NCSM calculations for a single
harmonic oscillator (HO) strength $\hbar\Omega$, for which and for a specific
parameterization of the chiral potential we show that the observables under
consideration converge with the number of HO excitations, including point-
proton root-mean-square (rms) radii and $E2$ transitions. The SA-NCSM utilizes
a symplectic $\mathrm{Sp}(3,\mathbb{R})$-adapted basis and selected model
spaces111Throughout the paper, we will refer to the selected SA-NCSM model
spaces as SA model spaces. that are significantly reduced in size due to
symmetry considerations without sacrificing the physics of interest. Moreover,
we show that the set of chiral potential parameterizations used to train the
emulators has no significant effect on dominant nuclear features such as the
nuclear shape (and associated symplectic symmetry) and cluster formation,
making the SA model spaces highly suitable for this study. However, from one
parameterization to another we find that probability amplitudes of wave
functions and cluster peak distance vary slightly, affecting by a factor of
two or less collective quadrupole moments, asymptotic normalization
coefficients (ANCs), and alpha partial widths (which provide the probability
for the alpha decay among all possible decays of a state). This suggests that
these types of observables, and associated emulators, are important to inform
and construct the nuclear forces for high-precision nuclear calculations.

##  II Theoretical Methods

###  II.1  Ab Initio Symmetry-Adapted
No-Core Shell Model

Ab initio large-scale calculations [1, 2] have recently revealed a remarkably
ubiquitous and almost perfect symmetry, the $\mathrm{Sp}(3,\mathbb{R})$
symplectic symmetry, in nuclei that naturally emerges from first principles up
through the calcium region (anticipated to hold even stronger in heavy nuclei
[14]). Since this symmetry does not mix nuclear shapes, this novel nuclear
feature provides important insight from first principles into the physics of
nuclei and their low-lying excitations as dominated by only one or two
collective shapes – equilibrium shapes with their vibrations – that rotate
(Fig. 1a).

The SA-NCSM theory [15, 10, 1] capitalizes on these findings and exploits the
idea that the infinite Hilbert space can be equivalently spanned by
“microscopic” nuclear shapes and their rotations [or symplectic irreducible
representations (irreps), subspaces that preserve the symmetry], where
“microscopic” refers to the fact that these configurations track with the
position and momentum coordinates of each particle. A collective nuclear shape
can be viewed as an equilibrium (“static”) deformation and its vibrations
(“dynamical” deformations) of the giant-resonance type, as illustrated in the
$\beta$-$\gamma$ plots of Fig. 1a [16, 1]. A key ingredient of the SA concept
is illustrated in Fig. 1b, namely, while many shapes relevant to low-lying
states are included in typical shell-model spaces (Fig. 1b, top), the
vibrations of largely deformed equilibrium shapes and spatially extended modes
like clustering often lie outside such spaces. The selected model space in the
SA-NCSM remedies this, and includes, in a well prescribed way, those
configurations. Note that this is critical for enhanced deformation, since
spherical and less deformed shapes, including relevant single-particle
effects, easily develop in comparatively small model-space sizes.

In this study, we utilize the ab initio SA-NCSM theory [1, 2, 10] that is
based on the NCSM concept [17, 18] with nuclear interactions typically derived
from the chiral EFT (e.g., [4, 19, 20, 21, 22, 23]). We use SA-NCSM model
spaces, which are reorganized to a correlated basis that respects the shape-
preserving $\mathrm{Sp}(3,\mathbb{R})$ symmetry and its embedded symmetry, the
deformation-related $\mathrm{SU}(3)$ [1, 2, 10]. We note that while the model
utilizes symmetry groups to construct the basis and calculate matrix elements,
descriptions are not limited a priori to any symmetry and can account for
significant symmetry breaking.

The SA-NCSM is reviewed in Ref. [2, 10] and has been applied to light and
medium-mass nuclei using $\mathrm{SU}(3)$\- and
$\mathrm{Sp}(3,\mathbb{R})$-adapted bases. The many-nucleon basis states of
the SA-NCSM are constructed using efficient group-theoretical algorithms and
are labeled according to $\mathrm{SU}(3)$$\times$$\mathrm{SU}(2)$ by the
proton, neutron and total intrinsic spins, $S_{\rm p}$, $S_{\rm n}$, and $S$,
respectively, and $(\lambda_{\omega}\,\mu_{\omega})$ quantum numbers with
$\lambda_{\omega}=N_{z}-N_{x}$ and $\mu_{\omega}=N_{x}-N_{y}$, where
$N_{x}+N_{y}+N_{z}=N_{0}+N$, for a total of $N_{0}+N$ HO quanta distributed in
the $x$, $y$, and $z$ directions222We follow the notations of Ref. [15]. Here,
$N_{0}\hbar\Omega$ is the lowest total HO energy for all particles (“valence-
shell configuration”) and $N\hbar\Omega$ ($N\leq N_{\rm max}$) is the
additional energy of all particle-hole excitations. Thus, for example,
$(\lambda_{\omega}\,\mu_{\omega})=(0\,0)$, for which $N_{x}=N_{y}=N_{z}$,
describes a spherical configuration, while $N_{z}$ larger than $N_{x}=N_{y}$
($\mu_{\omega}=0$) indicates prolate deformation. In addition, a closed-shell
configuration has $(0\,0)$. Indeed, spherical shapes, or no deformation, are a
part of the SA basis. However, most nuclei – from light to heavy – are
deformed in the body-fixed frame, which for $0^{+}$ states appear spherical in
the laboratory frame.

Furthermore, considering the embedding symmetry
$\mathrm{Sp}(3,\mathbb{R})$$\supset$$\mathrm{SU}(3)$, one can further organize
$\mathrm{SU}(3)$ deformed configurations into subspaces that preserve
$\mathrm{Sp}(3,\mathbb{R})$ symmetry. Each of these subspaces (symplectic
irrep, labeled by $\sigma$) is characterized by a given equilibrium shape,
labeled by a single deformation $N_{\sigma}(\lambda_{\sigma}\,\mu_{\sigma})$.
For example, the symplectic irrep
$N_{\sigma}(\lambda_{\sigma}\,\mu_{\sigma})=0(8\,0)$ in 20Ne consists of a
prolate $0(8\,0)$ equilibrium shape (static deformation) with
$\lambda_{\omega}=8$ and $\mu_{\omega}=0$ in the valence-shell $0$p-$0$h
(0-particle-0-hole) subspace, along with many other $\mathrm{SU}(3)$ deformed
configurations or dynamical deformation (vibrations), such as
$N_{\omega}(\lambda_{\omega}\,\mu_{\omega})=2(10\,0)$, $2(6\,2)$, and
$8(16\,0)$, which include particle-hole excitations of the equilibrium shape
to higher shells [1, 14, 16]. These vibrations are multiples of 2$\hbar\Omega$
$1$p-$1$h excitations of the giant-resonance monopole and quadrupole types,
that is, induced by the monopole
$r^{2}=\sum_{i=1}^{A}{\vec{r}_{i}\cdot\vec{r}_{i}}$ and quadrupole
$Q_{2}=\sqrt{16\pi/5}\sum_{i=1}^{A}r_{i}^{2}Y_{2}(\hat{r}_{i})$ operators,
respectively (for further details, see Refs. [10, 24]).

An advantage of the SA-NCSM is that the SA model space can be down-selected
from the corresponding ultra-large $N_{\rm max}$ complete model space to a
subset of SA basis states that describe static and dynamical deformation, and
within this SA model space the spurious center-of-mass motion can be factored
out exactly [25, 26]. Another benefit is the use of group theory for
constructing the basis and calculating matrix elements, including the Wigner-
Eckart theorem, which allows for calculations with $\mathrm{SU}(3)$ reduced
matrix elements that depend only on $(\lambda\,\mu)$, along with
computationally efficacious group-theoretical algorithms and data structures,
as detailed in Refs. [27, 28, 29, 30, 31]. A third advantage is that
deformation and collectivity are examined and treated in the approach without
the need for breaking and restoring rotational symmetry. The reason is that
basis states utilize the $\mathrm{SU}(3)$${}_{(\lambda\,\mu)}\supset$
$\mathrm{SO}(3)$L reduction chain that has a good orbital angular momentum
$L$, whereas all $\mathrm{SU}(3)$ reduced matrix elements can be calculated in
the simpler canonical $\mathrm{SU}(3)$${}_{(\lambda\,\mu)}\supset$
$\mathrm{SU}(2)$I reduction chain (for details, see Ref. [32, 33]). The
canonical reduction chain provides a natural reduction to the $x$ and $y$
degrees of freedom, it is simple to work with, and most importantly, provides
a complete labeling of a basis state that includes the single-shell quadrupole
moment eigenvalue that measures the deformation along the body-fixed symmetry
$z$-axis [34]. $\mathrm{SU}(3)$ reduced matrix elements calculated within this
scheme yield, in turn, matrix elements for the SA-NCSM basis by invoking the
Wigner-Eckart theorem with the appropriate
$\mathrm{SU}(3)$${}_{(\lambda\,\mu)}\supset$ $\mathrm{SO}(3)$L Clebsch-Gordan
coefficients that are readily available [32].

We emphasize that all basis states are kept up to some $N_{\rm max}^{\rm C}$,
yielding results equivalent to the corresponding $N_{\rm max}^{\rm C}$ NCSM
calculations. Building upon this complete $N_{\rm max}^{\rm C}$ model space,
we expand the model space to $N_{\rm max}$ by adding selected basis states to
include only the necessary vibrations of largely deformed equilibrium shapes
that lie outside this $N_{\rm max}^{\rm C}$ (such SA-NCSM model spaces are
denoted as $\langle N_{\rm max}^{\rm C}\rangle N_{\rm max}$).

###  II.2 Eigenvector Continuation Method in the Symmetry-adapted Framework

|  | SA | Complete
---|---|---|---
Nucleus |  $J^{\pi}$ |  $N_{\rm max}$ |  Dim |  $E_{\rm X}$ $[\text{MeV}]$ |  $r_{\rm rms}$ $[\text{fm}]$ |  $Q$ $[e\text{ fm}^{2}]$ |  $B(E2\uparrow)$ $[e^{2}\text{ fm}^{4}]$ |  $N_{\rm max}$ |  Dim |  $E_{\rm X}$ $[\text{MeV}]$ |  $r_{\rm rms}$ $[\text{fm}]$ |  $Q$ $[e\text{ fm}^{2}]$ |  $B(E2\uparrow)$ $[e^{2}\text{ fm}^{4}]$
6Li |  $1^{+}_{\mathrm{g.s.}}$ |  $\langle 2_{\text{All}}\rangle 8_{\text{13}}$ |  $4898$ |  – |  $2.20$ |  $-0.25$ |  $9.75$ |  $8$ |  $2\times 10^{5}$ |  – |  $2.22$ |  $-0.028$ |  $10.04$
6Li |  $3^{+}_{1}$ |  $\langle 2_{\text{All}}\rangle 8_{13}$ |  $9108$ |  $2.20$ |  $2.20$ |  $-4.12$ |  – |  $8$ |  $3\times 10^{5}$ |  $2.65$ |  $2.22$ |  $-4.21$ |  –
12C |  $0^{+}_{\mathrm{g.s.}}$ |  $6_{3}$ |  $552$ |  – |  $2.41$ |  $0$ |  $35.31$ |  $6$ |  $1\times 10^{6}$ |  – |  2.43 |  $0$ |  $35.22$
12C |  $2^{+}_{1}$ |  $6_{3}$ |  $238$ |  $5.73$ |  $2.41$ |  $+5.67$ |  – |  $6$ |  $5\times 10^{6}$ |  $3.38$ |  $2.43$ |  $+5.56$ |  –
Table 1:  Model space dimensions (labeled as “Dim”), excitation energy
$E_{\textrm{X}}$, point-proton rms radius $r_{\rm rms}$, electric quadrupole
moment $Q$, and $B(E2\uparrow)$ transition strengths from the ground state
($\mathrm{g.s.}$) to the first excited state of 6Li and 12C, calculated with
NNLOopt and $\hbar\Omega=15$ MeV in SA and complete model spaces. $\langle
2_{\text{All}}\rangle 8_{\text{13}}$ denotes an $N_{\text{max}}=2$ model space
with all symplectic irreps (complete), 13 $\mathrm{Sp}(3,\mathbb{R})$ irreps
of which extend to $N_{\text{max}}=8$; $6_{3}$ denotes 3
$\mathrm{Sp}(3,\mathbb{R})$ irreps up to $N_{\text{max}}=6$.

As introduced in Ref. [11], the EVC method utilizes the fact that if a
Hamiltonian is a smooth function of some real-valued parameters, its
eigenvectors will also be well-behaved functions of those parameters. In
practice, this means that one can use a relatively small number of known wave
functions to construct an accurate emulator well-approximated by a low-
dimensional manifold, and with it accurately predict observables for an
arbitrary chiral potential parameterization [12]. To compute these initial
wave functions from first principles, it is advantageous to use SA model
spaces that can accommodate deformation, including spatially expanded modes,
as well as medium-mass regions.

An advantage of the EVC method is that solutions are achieved by diagonalizing
matrices with sizes that are many orders of magnitude smaller than those used
in exact calculations. This results in a drastically reduced computational
time with practically no discrepancies from the exact results. EVC thus
provides a means of generating large samples of nuclear observables from
variations in the Hamiltonian parameters. This, in turn, makes computationally
intensive statistical analyses, such as sensitivity studies [7, 12], possible.
It also allows for a reduced computational load for quantifying uncertainties
of ab initio predictions.

In this study, we construct emulators capable of probing collective and
clustering features by employing the EVC method with SA model spaces. As
illustrated in Table 1, the SA-NCSM reduces the sizes of Hamiltonian matrices
by up to four orders of magnitude, or equivalently by more than $97\%$. The
application of EVC to these SA spaces results in an additional reduction of up
to 3 more orders of magnitude, or as much as $99\%$. In this combined
framework, the final size of the resulting matrices are as much as $10^{-5}$
times smaller than they would be in the corresponding $N_{\textrm{max}}$
complete spaces. As the first step, we consider a chiral EFT nucleon-nucleon
(NN) interaction truncated at next-to-next-to-leading order (NNLO), which
depends on 14 low-energy constants (LECs). It turns out that we can write the
chiral Hamiltonian as $H(\vec{c})=\sum_{i=0}^{14}c_{i}h_{i}$, where $\vec{c}$
is a vector representing a unique combination of the LECs, $h_{i}$ are the
constituent chiral potentials, $h_{0}$ is the LEC-independent part of the
chiral potential plus relative kinetic energy and the Coulomb interaction, and
$c_{0}=1$.

A state $\ket{\psi(\vec{c})}$ can be well-approximated as a linear combination
of known “training” wave functions
$\sum_{j}^{N_{T}}\alpha_{j}(\vec{c})\ket{\psi(\vec{c_{T}}_{j})}$, where each
$\ket{\psi(\vec{c_{T}}_{j})}$ in this study is the lowest-energy eigenvector
of $H(\vec{c_{T}}_{j})$ for a given $J^{\pi}$, $\vec{c_{T}}$ corresponds to a
training point in the LEC parameter space, and $N_{T}$ is the number of
training points. The chiral Hamiltonian matrices $h_{i}$ are constructed in
the representation of the training wave functions. These $N_{T}\times N_{T}$
matrices are used to emulate the wave function for any set of LECs $\vec{c}$
by solving the Schrödinger equation for the unknown $\alpha_{j}(\vec{c})$ as a
generalized eigenvalue problem that uses the norm matrix for the training wave
functions, $M_{ij}=\braket{\psi(\vec{c_{T}}_{i})}{\psi(\vec{c_{T}}_{j})}$.

The new features here are that we generate the emulator for the electric
quadrupole moment $Q$ by constructing the $Q$ matrix in the representation of
the training eigenvectors (as done for rms radii in Ref. [7]), and that these
are calculated using SA model spaces. The quadrupole moment is then
approximated by computing
$\braket{\psi(\vec{c})}{Q}{\psi(\vec{c})}=\sum_{ij}\alpha_{i}(\vec{c})\alpha_{j}(\vec{c})\braket{\psi(\vec{c_{T}}_{i})}{Q}{\psi(\vec{c_{T}}_{j})}$.

##  III Results and Discussions

Figure 2:  Convergence with $N_{\text{max}}$ of the quadrupole moments $Q(J)$,
point-proton rms radii $r_{\rm rms}(J)$, excitation energies $E_{\rm X}$, and
$B(E2\uparrow)$ transition strengths for the two lowest-lying states in (a)
6Li and (b) 12C. Observables are computed with the NNLO${}_{\text{opt}}$
parameterization for $\hbar\Omega$=15 MeV in SA model spaces reported in Table
1.

The results presented in this paper use the SA-NCSM in an
$\mathrm{Sp}(3,\mathbb{R})$ basis with an NN chiral potential up to NNLO as
used in [21]. The consistent treatment of NN and three-nucleon (3N) forces at
this order is feasible but outside the scope of the present study, which aims
to show the validity of the SA-EVC method. We also include the outcomes for a
specific NN parameterization, NNLOopt [21], for which the 3N forces have been
shown to contribute minimally to the 3- and 4-nucleon binding energy [21].
Furthermore, the NNLOopt NN potential has been found to reproduce various
observables, including the 4He electric dipole polarizability [35]; the
challenging analyzing power for elastic proton scattering on 4He, 12C, and 16O
[36]; neutron-deuteron scattering cross-sections [37]; along with B(E2)
transition strengths for 21Mg and 21F [38] in the SA-NCSM without effective
charges.

For the EVC calculations, we use $N_{T}=32$ training points within the
14-dimensional parameter space for NNLO. We restrict the ranges of the LECs to
lie within $\pm 10\%$ of their values for NNLOopt [21] and adopt the
regularization for NNLOopt. We sample training points using a randomly seeded
latin hypercube design, and validate the emulators for 256 points that are
different from the training points but within the same range of the LECs.

The SA-EVC results start with SA model spaces that are reduced by three to
four orders of magnitude compared to the corresponding $N_{\text{max}}$
complete model space (or, equivalently, NCSM calculations), as outlined in
Table 1. Moreover, the associated observables are in good agreement for SA and
complete model spaces, with differences that are typically comparable to
differences resulting from varying $\hbar\Omega$ (see Ref. [1], supplemental
material). Specifically, for the example of NNLOopt, we report in Table 1
excitation energies, point-proton rms radii, electric quadrupole moments, and
$B(E2\uparrow)$ transition strengths between the two lowest energy states of
6Li and 12C. We also show that for the SA spaces used to train the emulators
all of the above observables are converged with $N_{\text{max}}$ (Fig. 2).

Figure 3:  The largest SU(3) probability amplitudes (solid lines) as a
function of emulator training LECs sets for (a) 6Li $1^{+}$ ground state and
(b) 6Li $3_{1}^{+}$ state in $N_{\text{max}}=\langle 2_{\rm All}\rangle
8_{13}$ model space [all SU(3) states have $\\{S_{\rm p},S_{n},S\\}=\\{{1\over
2},{1\over 2},1\\}$], as well as for (c) 12C $0^{+}$ ground state and (d) 12C
$2_{1}^{+}$ state in $N_{\text{max}}=6_{3}$ [all SU(3) states have $\\{S_{\rm
p},S_{n},S\\}=\\{0,0,0\\}$ except for $0(1\text{ }2)$ with $\\{S_{\rm
p},S_{n},S\\}=\\{0,1,1\\}$ (orange) and $\\{1,0,1\\}$ (green)]. Results are
also shown for the NNLOopt parameterization in the corresponding $N_{\rm max}$
complete model space (labeled as “opt”).

Thus, for example, as shown in Table 1, collectivity-driven observables agree
within 0.3-2.9%, and radii agree at the sub-percent level. The largest
deviation is observed for the 6Li $1^{+}$ quadrupole moment, however, it is
important that its sign and very small magnitude are reproduced in both
calculations. Furthermore, such differences are expected to decrease in richer
model spaces; indeed, in a series of benchmark studies for light nuclei such
as 4He, 6Li, 12C, and 16O (reviewed in Ref. [2]), we have shown that the SA-
NCSM uses significantly smaller model spaces in comparison to the
corresponding large complete $N_{\rm max}$ model spaces without compromising
the accuracy for various observables (including electron scattering form
factors [39] and sum rules [35]), as well as for effective inter-cluster
potentials [31]. Ref. [2] has also shown that for light nuclei, the SA-NCSM is
in reasonable agreement with other ab initio approaches, such as
hyperspherical harmonics [40, 41], NCSM [17, 18], and quantum Monte Carlo
[42].

###  III.1 Collectivity and Clustering of Training Wave Functions

An important feature of the training wave functions is that the dominant
deformed configurations, or the SU(3) content of the states under
consideration, remain practically the same for all of the training wave
functions (Fig. 3). In addition, the SU(3) content agrees with the
probabilities obtained with NNLOopt in the corresponding $N_{\rm max}$
complete model space, also shown in Fig. 3. This ensures that the same static
and dynamical deformed modes govern the physics for all LECs sets under
considerations, thereby justifying the use of the same SA selection for all
the training wave functions.

Specifically, we find that one SU(3) irrep dominates the dynamics of each
state at the $50$-$60\%$ level, with several additional configurations each
contributing from $1\%$ to $20\%$ depending on the LECs set. Moreover, when
the basis states are further organized into $\mathrm{Sp}(3,\mathbb{R})$
irreps, we find that a single symplectic irrep – which contains the dominant
SU(3) configurations – contributes at practically the same level from one
training wave function to another. For example, the $(2\,0)$ symplectic irrep
in 6Li accounts for $83$-$88\%$ of each $1^{+}$ training wave function,
whereas the $(2\,0)$ contributes at the $85$-$88\%$ level in the case of the
$3^{+}$, out of thirteen available different irreps. Similarly, the
probability of the $(0\,4)$ irrep in each of the 12C training ground states is
between $80$-$88\%$, and between $82$-$94\%$ for the first $2^{+}$ states.
This is a strong indicator that the emulators are trained on wave functions
that retain the symmetry-preserving and symmetry-breaking patterns that are
observed in nuclei [1] and that the SA model spaces used in this study are
sufficient to capture nuclear collectivity. Indeed, the fact that the
$\mathrm{Sp}(3,\mathbb{R})$ symmetry remains a near perfect symmetry for each
of the training wave functions, retaining the same shape from one wave
function to another, further supports the use of SA selections in the EVC
method, or otherwise, the SA model spaces would need to be re-examined.

Another important feature of the training wave functions is that cluster
formation is largely unaffected by the choice of interaction parameters. To
study this, we project the 6Li states onto the $\alpha+d$ system, following
Ref. [43]: we use a ground state for each cluster that is renormalized to the
most dominant SU(3) configuration, and we adopt $R$-matrix theory to match the
amplitude of the cluster wave function and its derivative to those of the
exact Coulomb eigenfunctions at large distances. We note that we are primarily
interested in the effect of the LECs on the correlations in the training wave
functions; hence, we fix the threshold energy to the experimental one. For the
${}^{3}S_{1}$ partial wave, we observe about $20\%$ variations in the
calculated asymptotic normalization coefficients ($C_{0}=1.45$-$2.07$ fm-1/2)
around their average value and $10\%$ variations in the spectroscopic factor,
namely, $SF=0.75$-$0.90$(Fig. 4a). This tracks with the $\pm 10\%$ variation
in the LECs. For comparison, the NNLO${}_{\textrm{opt}}$ ANC for this
particular channel is $C_{0}=1.77$ fm-1/2 with $SF=0.87$. Interestingly, the
height of the second peak, which is located near the nuclear surface and
informs the probability of cluster formation, remains fixed for all the
parameterizations and coincides with the one for the NNLO${}_{\textrm{opt}}$
case, only its position slightly varies with the LECs.

Figure 4: $\alpha+d$ (a) ${}^{3}S_{1}$-wave and (b) ${}^{3}D_{3}$-wave as
functions of the relative distance $r$, computed from the 6Li training wave
functions for SA model spaces reported in Table 1. The spread of the curves is
given by the $\pm 10\%$ variation in the LECs. The case for
NNLO${}_{\textrm{opt}}$ is shown in black.

While the ${}^{3}D_{3}$ spectroscopic factors ($SF=0.73$-$0.92$, with $0.90$
for NNLOopt) vary approximately at the $15\%$ level (Fig. 4b), which is
practically the same as for the ${}^{3}S_{1}$ partial wave, $\alpha$ widths of
the $3^{+}$ state range from $\Gamma_{\alpha}=6.34$ keV to 14.05 keV, which is
about $\pm$40% from $\Gamma_{\alpha}=9.81$ keV calculated for this particular
channel with NNLOopt (similarly to the ANCs, we use the experimental threshold
energy). We note that the NNLOopt values for $C_{0}$ and $\Gamma_{\alpha}$ are
reported for a single channel without taking excitations of the clusters into
account (e.g., see Ref. [44]) and should not be compared directly to
experiment. Of particular interest for this study is that the LECs sets induce
a change in both the location and magnitude of the peak, to which the
probability for alpha decay is typically sensitive to.

To summarize, the behavior of the surface peaks in both channels and the
nuclear shapes of the $1^{+}$ and $3^{+}$ states in 6Li (as well as the shapes
of the $0^{+}$ and $2^{+}$ states in 12C) are relatively consistent. This
suggests that the terms of the nuclear potential that are independent of the
LECs, including parts of the long-range interaction, are largely responsible
for cluster formation, along with the development of the nuclear shape
[equivalently, almost perfect $\mathrm{Sp}(3,\mathbb{R})$ symmetry]. In
contrast, the LECs, which capture the unresolved short-ranged interactions
between nucleons, fine-tune collective and clustering features, and affect the
associated observables by only a factor, namely, 1.4 for the $1^{+}_{\rm
g.s.}$ ANCs, 2.2 for the $3^{+}_{1}$ alpha width, and 1.4 for the $3^{+}_{1}$
quadrupole moment in 6Li. Similarly, the quadrupole moment for the $2^{+}_{1}$
in 12C is affected by a factor of 2.1. While the clustering features are
explored in this study for the training points only, the SA-EVC approach – the
validation of which is discussed next – enables uncertainty quantification of
such collective and reaction observables if the probability distributions for
the LECs are available.

###  III.2 Validation of the SA-EVC

Figure 5:  Exact vs. SA-EVC observables in 6Li (blue circles) for the
quadrupole moment $Q$ of (a) the $1^{+}$ ground state and (b) the first
excited $3^{+}$ state, as well as (c) for the excitation energy $E_{\rm X}$ of
the $3^{+}$ state, in $\langle 2_{\text{All}}\rangle 8_{13}$ SA model spaces
and for $\hbar\Omega=15$ MeV. Also shown is the agreement between the exact
and emulated values to guide the eye (red line), and experimental results
(vertical green line) where available. Insets show $5\%$-regions surrounding
reported experimental data [45] or the NNLOopt result where data is not
available [a $50\%$-region is used for the very small $Q$ in (a)].

To validate the SA-EVC approach, we show that for the quadrupole moments of
the 6Li $1^{+}$ ground state and first excited $3^{+}$ state, as well as for
the $3^{+}$ excitation energy, the emulators provide very accurate results
compared to the exact outcomes (Fig. 5). The average relative errors over all
$256$ validation LECs sets are respectively $6.91\times 10^{-2}$, $7.70\times
10^{-4}$, and $1.20\times 10^{-4}$. It is clear that any deviations of the
emulators from the expected values are negligible, especially considering
that, as mentioned above, the SA selection reduces the Hamiltonian dimension
by more than $97\%$, and the EVC projection by an additional $99\%$ or more.

It is worth noting that the average error for the ground state quadrupole
moment is two orders of magnitude larger than that of the $3^{+}$ state. We
note that $Q(1^{+})$ of 6Li is very similar in nature to the deuteron
quadrupole moment. The extremely small value in both nuclei results from a
small mixing of an $L=2$ component into the ground state of 6Li (and of the
deuteron), which is not collective in essence like, e.g., the quadrupole
moments of the $3^{+}$ state in 6Li or the $2^{+}$ state in 12C (discussed
below). Indeed, the results of Fig. 5a reflect the high sensitivity of the
underlying NN interaction (and likely 3N forces [46]) to the $L=2$ mixing in
the ground state wave function.

Similar to 6Li, the SA-EVC emulated $2^{+}_{1}$ quadrupole moment and
excitation energy for 12C are in very close agreement to the exact results
(Fig. 6). Namely, the average relative errors are given by $1.02\times
10^{-4}$ and $6.72\times 10^{-5}$, respectively. Compared to the average
errors reported above for the $3^{+}_{1}$ quadrupole moment and excitation
energy for 6Li, we find eight and two times improvement in the emulator’s
predictions for 12C, respectively. The reason is likely related to the much
smaller SA selection in 12C and the stronger collective nature observed in the
low-lying states of 12C. Specifically, in 6Li the SA-EVC uses thousands of
basis states, whereas in 12C only hundreds of basis states (see Table 1). We
therefore expect the mixing of configurations to exert a more noticeable
effect on 6Li than on 12C. The result is that the eigenvectors of 12C vary in
fewer directions than those of 6Li, suggesting that more training points for
6Li may be beneficial to improve errors. While this warrants further study,
this speaks to an advantage of merging the SA and EVC frameworks.

Figure 6:  The same as in Fig. 5 but for (a) the quadrupole moment $Q$ and (b)
the excitation energy of the first $2^{+}$ state in 12C, calculated in $6_{3}$
SA model spaces and for $\hbar\Omega=15$ MeV. Insets show $5\%$-regions
surrounding reported experimental data [47].

##  IV Conclusions

We have for the first time combined the framework of the SA-NCSM with the EVC
procedure into the SA-EVC method for studies of collective and clustering
observables. This builds upon earlier SA-NCSM explorations that have shown
that an $\mathrm{Sp}(3,\mathbb{R})$-adapted model space selection can
successfully capture nuclear collectivity while significantly reducing the
sizes of Hamiltonian matrices [1]. Here, we show that excitation energies,
point-proton rms radii, electric quadrupole moments and $E2$ transitions in
the two lowest-lying states of 6Li and 12C calculated with the specific
parameterization NNLOopt for $\hbar\Omega$=15 MeV in SA model spaces are in
reasonable agreement with those calculated in the corresponding
$N_{\text{max}}$ complete model space (or equally, to NCSM outcomes). We also
show that these observables are converged with $N_{\text{max}}$ for the SA
selections under consideration.

Further, we demonstrate that SA-EVC emulators trained on SA model spaces are
capable of accurately predicting such observables as the LECs are varied,
while further reducing the dimensions of operator matrices by an additional
$2$-$3$ orders of magnitude. Combined with the initial reduction provided by
the SA-NCSM, the emulator matrices have a dimension as much as $10^{-5}$ times
smaller than the corresponding $N_{\textrm{max}}$ complete model spaces. They
are small enough to perform linear algebra operations using a single CPU
thread on a standard laptop without difficulty. Moreover, the SA-EVC approach
will be critical for nuclei beyond the lightest systems; thus, e.g. in 20Ne,
the complete $N_{\rm max}=8$ model space has dimension of $1.52\times
10^{11}$, while the ab initio SA-NCSM solutions are achieved when using 112
million basis states for $J^{\pi}=0^{+},2^{+},4^{+}$. This can be further
reduced to emulators of dimension $10^{2}$ especially given the predominance
of a single symplectic irrep in the ground-state rotational band of this
nucleus. Comparing the emulator results to exact calculations performed in the
same SA spaces, we find that the average relative errors are typically
$10^{-4}$. A larger error ($\sim 10^{-2}$) is found for the quadrupole moment
of the 6Li ground state, which is highly sensitive to the $L=2$ admixture and
hence to the underlying nuclear force, as discussed in the text. A future
study that utilizes larger training sets may provide further insight.

In addition to validating the SA-EVC procedure, we show that the symmetry
patterns and clustering features in the emulator training wave functions do
not respond strongly to variations in the LECs. Across all of the training
wave functions, there is a single nuclear shape (approximate symplectic
symmetry) that accounts for $81$-$94\%$ of the total probability. Furthermore,
the dominance of important $\mathrm{SU}(3)$ configurations is preserved from
one training wave function to another. Projecting the training wave functions
for 6Li onto the $\alpha+d$ system, we find that the likelihood of cluster
formation in both the ${}^{3}S_{1}$\- and ${}^{3}D_{3}$-wave channels is
largely unaffected by the choice of LECs. Spectroscopic factors, ANCs and
$\alpha$-widths extracted from the cluster wave functions all vary within
relatively narrow ranges around their average values, ranges that track
reasonably well with the $10\%$ variation of the LECs. This suggests that the
part of the nuclear potential that is independent of the LECs and is
practically the same for all chiral potentials (up to the regularization and
related cutoffs employed) provides the dominant features of the wave function,
such as $\mathrm{Sp}(3,\mathbb{R})$ symmetry patterns and clustering
formation, while varying the LECs and associated unresolved short-range
interactions has an effect on, e.g., collective quadrupole moments, asymptotic
normalization coefficients (ANCs), and alpha partial widths up to a factor of
two.

In order to better understand the relationships between collectivity and
clustering explored in this study, and how both relate to the underlying
nuclear forces, sensitivity analyses are required. As we enter the era of
high-precision nuclear physics, this is also an important step towards
constructing accurate interactions, with quantified uncertainties. We note
that properly accounting for clustering features is important for the ab
initio modeling of nuclear reactions, and related process from fusion to
fission. The SA-EVC method provides a clear and now verified framework for
generating the huge number of chiral parameterizations required for such
analyses. Hence, the door is now open to perform ab initio calculations with
quantified uncertainties that emerge from the interaction and the controlled
many-body approximations, from exotic light nuclei up to medium-mass isotopes,
as well from spherical to highly enhanced collective and clustering modes.

###### Acknowledgements.

We acknowledge invaluable discussions with Jerry P. Draayer, George
Rosensteel, David Rowe, and Daniel Langr. This work was supported in part by
the U.S. National Science Foundation (PHY-1913728, PHY-2209060), the European
Research Council (ERC) under the European Unions Horizon 2020 research and
innovation program (Grant agreement No. 758027), the Czech Science Foundation
(22-14497S). KSB greatly appreciates the financial support of a research
fellowship from the Louisiana Board of Regents. This work benefited from high
performance computational resources provided by LSU (www.hpc.lsu.edu), the
National Energy Research Scientific Computing Center (NERSC), a U.S.
Department of Energy Office of Science User Facility operated under Contract
No. DE-AC02-05CH11231, as well as the Frontera computing project at the Texas
Advanced Computing Center, made possible by National Science Foundation award
OAC-1818253.

## Data Availability Statement

The raw data supporting the conclusions of this article will be made available
by the authors, without undue reservation.

## Author Contributions

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

## Conflict of Interest

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

## References

  * Dytrych  _et al._ [2020] T. Dytrych, K. D. Launey, J. P. Draayer, D. J. Rowe, J. L. Wood, G. Rosensteel, C. Bahri, D. Langr, and R. B. Baker, Physics of nuclei: Key role of an emergent symmetry, Phys. Rev. Lett. 124, 042501 (2020), arXiv:1810.05757.
  * Launey  _et al._ [2021] K. D. Launey, A. Mercenne, and T. Dytrych, Nuclear dynamics and reactions in the ab initio symmetry-adapted framework, Annu. Rev. Nucl. Part. Sci. 71, 253 (2021).
  * Johnson  _et al._ [2020] C. W. Johnson, K. D. Launey, _et al._ , From bound states to the continuum, J. Phys. G 47, 23001 (2020), arXiv:1912.00451.
  * Bedaque and van Kolck [2002] P. F. Bedaque and U. van Kolck, Effective field theory for few-nucleon systems, Annu. Rev. Nucl. Part. Sci. 52, 339 (2002), arXiv:nucl-th/0203055.
  * Tews  _et al._ [2022] I. Tews _et al._ , Nuclear forces for precision nuclear physics: A collection of perspectives, Few-Body Systems 63, 10.1007/s00601-022-01749-x (2022), arXiv:2202.01105.
  * Furnstahl  _et al._ [2015] R. J. Furnstahl, N. Klco, D. R. Phillips, and S. Wesolowski, Quantifying truncation errors in effective field theory, Physical Review C 92, 10.1103/physrevc.92.024005 (2015).
  * Ekström and Hagen [2019] A. Ekström and G. Hagen, Global sensitivity analysis of bulk properties of an atomic nucleus, Phys. Rev. Lett. 123, 252501 (2019), arXiv:1910.02922.
  * Sargsyan  _et al._ [2022] G. H. Sargsyan, K. D. Launey, M. T. Burkey, A. T. Gallant, N. D. Scielzo, G. Savard, A. Mercenne, T. Dytrych, D. Langr, L. Varriano, B. Longfellow, T. Y. Hirsh, and J. P. Draayer, Impact of clustering on the ${}^{8}\mathrm{Li}$ $\beta$ decay and recoil form factors, Phys. Rev. Lett. 128, 202503 (2022), arXiv:2107.10389.
  * Stroberg  _et al._ [2021] S. R. Stroberg, J. D. Holt, A. Schwenk, and J. Simonis, Ab initio limits of atomic nuclei, Phys. Rev. Lett. 126, 022501 (2021).
  * Launey  _et al._ [2016] K. D. Launey, T. Dytrych, and J. P. Draayer, Symmetry-guided large-scale shell-model theory, Prog. Part. Nucl. Phys. 89, 101 (review) (2016), arXiv:1612.04298.
  * Frame  _et al._ [2018] D. Frame, R. He, I. Ipsen, D. Lee, D. Lee, and E. Rrapaj, Eigenvector continuation with subspace learning, Phys. Rev. Lett. 121, 032501 (2018), arXiv:1711.07090.
  * König  _et al._ [2020] S. König, A. Ekström, K. Hebeler, D. Lee, and A. Schwenk, Eigenvector continuation as an efficient and accurate emulator for uncertainty quantification, Physics Letters B 810, 135814 (2020), arXiv:1909.08446.
  * Djärv  _et al._ [2022] T. Djärv, A. Ekström, C. Forssé n, and H. T. Johansson, Bayesian predictions for a = 6 nuclei using eigenvector continuation emulators, Phys. Rev. C 105, 10.1103/physrevc.105.014005 (2022), arXiv:2108.13313.
  * Rowe [1985] D. J. Rowe, Microscopic theory of the nuclear collective model, Reports on Progr. in Phys. 48, 1419 (1985).
  * Dytrych  _et al._ [2007] T. Dytrych, K. D. Sviratcheva, C. Bahri, J. P. Draayer, and J. P. Vary, Evidence for symplectic symmetry in ab initio no-core-shell-model results for light nuclei, Phys. Rev. Lett. 98, 162503 (2007), arXiv:0704.1108.
  * Rowe [2013] D. Rowe, The fundamental role of symmetry in nuclear models, AIP Conf. Proc. 1541, 104 (2013), arXiv:1304.6115.
  * Navrátil  _et al._ [2000] P. Navrátil, J. P. Vary, and B. R. Barrett, Properties of 12C in the Ab Initio Nuclear Shell Model, Phys. Rev. Lett. 84, 5728 (2000), arXiv:nucl-th/0004058.
  * Barrett  _et al._ [2013] B. Barrett, P. Navrátil, and J. Vary, Ab initio no core shell model, Prog. Part. Nucl. Phys. 69, 131 (2013).
  * Epelbaum  _et al._ [2002] E. Epelbaum, A. Nogga, W. Glöckle, H. Kamada, U.-G. Meißner, and H. Witala, Three-nucleon forces from chiral effective field theory, Phys. Rev. C 66, 064001 (2002), arXiv:nucl-th/0208023.
  * 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), arXiv:nucl-th/0304018.
  * Ekström  _et al._ [2013] A. Ekström, G. Baardsen, C. Forssén, G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, W. Nazarewicz, _et al._ , An optimized chiral nucleon-nucleon interaction at next-to-next-to-leading order, Phys. Rev. Lett. 110, 192502 (2013), arXiv:1303.4674.
  * Epelbaum  _et al._ [2015] E. Epelbaum, H. Krebs, and U. G. Meißner, Precision nucleon-nucleon potential at fifth order in the chiral expansion, Phys. Rev. Lett. 115, 122301 (2015).
  * Schiavilla  _et al._ [2021] R. Schiavilla, L. Girlanda, A. Gnech, A. Kievsky, A. Lovato, L. E. Marcucci, M. Piarulli, and M. Viviani, Two- and three-nucleon contact interactions and ground-state energies of light- and medium-mass nuclei, Phys. Rev. C 103, 054003 (2021), arXiv:2102.02327.
  * Launey  _et al._ [2020] K. D. Launey, T. Dytrych, G. H. Sargsyan, R. B. Baker, and J. P. Draayer, Emergent symplectic symmetry in atomic nuclei: Ab initio symmetry-adapted no-core shell model, Eur. Phys. J. Spec. Top. 229, 2429 (2020), arXiv:2108.04900.
  * Verhaar [1960] B. J. Verhaar, A method for the elimination of spurious states in the nuclear harmonic oscillator shell model, Nucl. Phys. 21, 508 (1960).
  * Hecht [1971] K. T. Hecht, Use of SU(3) in elimination of spurious center of mass states, Nucl. Phys. A 170, 34 (1971).
  * Draayer  _et al._ [1989] J. P. Draayer, Y. Leschber, S. C. Park, and R. Lopez, Representations of U(3) in U($N$), Comput. Phys. Commun. 56, 279 (1989).
  * Langr  _et al._ [2019] D. Langr, T. Dytrych, K. D. Launey, and J. P. Draayer, Accelerating many-nucleon basis generation for high performance computing enabled ab initio nuclear structure studies, The International Journal of High Performance Computing Applications 33, 522 (2019).
  * Oberhuber  _et al._ [2021] T. Oberhuber, T. Dytrych, K. D. Launey, D. Langr, and J. P. Draayer, Transformation of a nucleon-nucleon potential operator into its su(3) tensor form using gpus, Discrete & Continuous Dynamical Systems-S 14, 1111 (2021).
  * Dytrych  _et al._ [2021] T. Dytrych, D. Langr, J. P. Draayer, K. D. Launey, and D. Gazda, Su3lib: A c++ library for accurate computation of wigner and racah coefficients of su(3), Computer Physics Communications 269, 108137 (2021).
  * Mercenne  _et al._ [2022] A. Mercenne, K. Launey, T. Dytrych, J. Escher, S. Quaglioni, G. Sargsyan, D. Langr, and J. Draayer, Efficacy of the symmetry-adapted basis for ab initio nucleon-nucleus interactions for light- and intermediate-mass nuclei, Computer Physics Communications 280, 108476 (2022).
  * Draayer and Akiyama [1973] J. P. Draayer and Y. Akiyama, Wigner and Racah Coefficients for SU(3), J. Math. Phys. 14, 1904 (1973).
  * Bahri and Draayer [1994] C. Bahri and J. P. Draayer, SU(3) Reduced Matrix Element Package, Comput. Phys. Commun. 83, 59 (1994).
  * Carvalho  _et al._ [1986] J. Carvalho, R. Le Blanc, M. Vassanji, D. Rowe, and J. McGrory, The symplectic shell-model theory of collective states, Nuclear Physics A 452, 240 (1986).
  * Baker  _et al._ [2020] R. B. Baker, K. D. Launey, S. Bacca, N. N. Dinur, and T. Dytrych, Benchmark calculations of electromagnetic sum rules with a symmetry-adapted basis and hyperspherical harmonics, Phys. Rev. C 102, 014320 (2020), arXiv:2003.05865.
  * Burrows  _et al._ [2019] M. Burrows, C. Elster, S. P. Weppner, K. D. Launey, P. Maris, A. Nogga, and G. Popa, Ab initio folding potentials for nucleon-nucleus scattering based on no-core shell-model one-body densities, Phys. Rev. C 99, 044603 (2019), arXiv:1810.06442.
  * Miller  _et al._ [2022] S. Miller, A. Ekström, and K. Hebeler, Neutron-deuteron scattering cross-sections with chiral $nn$ interactions using wave-packet continuum discretization, arXiv preprint arXiv:2201.09600 (2022), arXiv:2201.09600.
  * Ruotsalainen  _et al._ [2019] P. Ruotsalainen, J. Henderson, G. Hackman, G. H. Sargsyan, Launey, _et al._ , Isospin symmetry in $B(E2)$ values: Coulomb excitation study of ${}^{21}\mathrm{Mg}$, Phys. Rev. C 99, 051301 (2019), arXiv:1811.00774.
  * Dytrych  _et al._ [2015] T. Dytrych, A. C. Hayes, K. D. Launey, J. P. Draayer, P. Maris, J. P. Vary, D. Langr, and T. Oberhuber, Electron-scattering form factors for 6Li in the ab initio symmetry-guided framework, Phys. Rev. C 91, 024326 (2015), arXiv:1502.03066.
  * Kievsky  _et al._ [2008] A. Kievsky _et al._ , A high-precision variational approach to three- and four-nucleon bound and zero-energy scattering states, J. Phys. G 35, 063101 (2008), arXiv:0805.4688.
  * Bacca  _et al._ [2002] S. Bacca, M. A. Marchisio, N. Barnea, W. Leidemann, and G. Orlandini, Microscopic calculation of six-body inelastic reactions with complete final state interaction: Photoabsorption of $\text{ }{\text{ }}^{6}\mathrm{H}\mathrm{e}$ and $\text{ }{\text{ }}^{6}\mathrm{L}\mathrm{i}$, Phys. Rev. Lett. 89, 052502 (2002).
  * Carlson  _et al._ [2015] J. Carlson, S. Gandolfi, F. Pederiva, S. C. Pieper, R. Schiavilla, K. E. Schmidt, and R. B. Wiringa, Quantum monte carlo methods for nuclear physics, Rev. Mod. Phys. 87, 1067 (2015), arXiv:1412.3081.
  * Dreyfuss  _et al._ [2020] A. C. Dreyfuss, K. D. Launey, J. E. Escher, G. H. Sargsyan, R. B. Baker, T. Dytrych, and J. P. Draayer, Clustering and $\alpha$-capture reaction rate from ab initio symmetry-adapted descriptions of ${}^{20}\mathrm{Ne}$, Phys. Rev. C 102, 044608 (2020), arXiv:2006.11208.
  * Hupin  _et al._ [2015] G. Hupin, S. Quaglioni, and P. Navrátil, Unified description of ${}^{6}\mathrm{Li}$ structure and deuterium-${}^{4}\mathrm{He}$ dynamics with chiral two- and three-nucleon forces, Phys. Rev. Lett. 114, 212502 (2015).
  * Tilley  _et al._ [2002] D. Tilley, C. Cheves, J. Godwin, G. Hale, H. Hofmann, J. Kelley, C. Sheu, and H. Weller, Energy levels of light nuclei a=5, 6, 7, Nuclear Physics A 708, 3 (2002).
  * Filin  _et al._ [2021] A. A. Filin, D. Möller, V. Baru, E. Epelbaum, H. Krebs, and P. Reinert, High-accuracy calculation of the deuteron charge and quadrupole form factors in chiral effective field theory, Phys. Rev. C 103, 024313 (2021), arXiv:2009.08911.
  * Kelley  _et al._ [2017] J. Kelley, J. Purcell, and C. Sheu, Energy levels of light nuclei a=12, Nuclear Physics A 968, 71 (2017).