COMPLEX-ENERGY DESCRIPTION OF MOLECULAR AND NUCLEAR OPEN

QUANTUM SYSTESMS

By

Xingze Mao

A DISSERTATION

Submited to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics – Doctor of Philosophy

Computational Mathematics, Science and Engineering - Dual Major

2020

COMPLEX-ENERGY DESCRIPTION OF NUCLEAR AND ATOMIC OPEN

ABSTRACT

QUANTUM SYSTESMS

By

Xingze Mao

Quantum systems lying close to the decay threshold experience coupling to the scattering

environment, hence they belong to a class of open quantum systems (OQSs). The study of

OQSs requires proper treatment of non-localized scattering states and resonances. The

Gamow shell model (GSM), as an extension of the traditional shell model formulated in

the complex-momentum (k) plane, can properly treat the structural and decay properties

of these threshold systems by employing the Berggren ensemble as the single-particle (s.p.)

basis. In this thesis, GSM has been used to study two types of OQSs: (i) atomic systems

such as quadrupolar anions and anions bounded by a multipolar Gaussian potential; and

(ii) light nuclei, such as lithium isotopes and their mirror partners. In atomic systems, low-(cid:96)

channels are found to be essential in defining the trajectories of resonant states near the

dissociation threshold.

In nuclear systems, a finite-range interaction has been optimized

to give a realistic description of the spectra, ranging from well-bound systems to unbound

nuclei above the decay threshold.

This dissertation is dedicated to my parents

iii

ACKNOWLEDGEMENTS

I would first like to thank my advisor Witold Nazarewicz for all his support and insightful

guidance, especially for his patience to correct all my errors and the freedom I was given to

try all my ideas when things do not go well. The principle he told me to challenge myself

every day and the high standard he held doing research will push me further in my career.

It’s been a great honor to finish my Ph.D. thesis under the guidance of Witold Nazarewicz.

Great thanks to my Ph.D. committee members, Filomena Nunes, Morten Hjorth-Jensen,

Brian O’Shea, Metin Aktulga, and Hironori Iwasaki, for the awesome guidance and cheering

me up during all my committee meetings with all the great questions.

A big thanks to K´evin Fossez, Jimmy Rotureau, Simin Wang, Nicolas Michel, Yannen

Jaganathen, and Erik Olsen for their support, scientifically and technically. They have been

of great help to my research and are always ready to put aside things in their hands to

help me whenever I brought up any questions to them. It would be way more difficult for

me to finish my research without the help from you guys. Useful discussions with Marek

P(cid:32)loszajczak and Rodolfo Id Betan are also acknowledged.

I will regret if I did not mention all the smart and energetic geniuses I met along my way,

Dan Liu, Hao Lin, Zachery Matheson, Terri Poxon-Pearson, John Bower, Thomas Redpath,

Timofey Golubev, Tenzin Rubga, Bakul Agarwal, Chunli Zhang, Maxwell Cao, Mengzhi

Chen, Tong Li, Xueying Huyan, Didi Luo, Zachary Constan.

It was a great pleasure to

work with you all on coursework, projects and non-scientific activities. Thank you all for

the accompany during this journey. This is an invaluable asset to my life. Special thanks to

Simin and Josh for helping me debug this thesis.

I would also like to express my gratitude to Kim Crosslan, Scott Pratt, Artemis Spyrou,

and Hironori Iwasaki for their support. Thank you for making this journey easier.

I was so lucky to join the MSU taekwondo and judo team and to train with all the

wonderful fellows. Besides all the physical and mental improvements, I also learned a lot

iv

from Ron’s martial philosophy and Sheehan’s candidness, especially when he shouted “Mao,

your throw is disgusting”. I would also like to express my gratitude to all my training mates,

Sarah, Kimberly, Neil, Soojin, Sam, Vici, Andrew, Igor, Jiayi, Susan, Schukou, Davina,

Smiley, Joshua, Hainite, Jun, Patrick, Alfonso, John, Pablo, James. Thank you all and it

was a good time training with you in the dojang.

Finally, I would like to express my love and gratitude to my family. I am so grateful for

the love and support of my parents. Thanks for being there with me during all the ups and

downs. I am also very lucky and fortunate to be embraced by the love and support of my

wife Huan. I would also like to thank my sister for everything, especially for saving me my

favorite snacks when I traveled back home, even though we fought a lot growing up. Love

you all.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Open quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Atomic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Nuclear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 Berggren ensemble . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Gamow states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Antibound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Berggren completeness relation . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
2
2
3

4
4
5
6

9
11
11
13
15
15
17
19

20
21
32
36

3.1 Electron-plus-molecule Model

Chapter 3 Atomic anions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Model and Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Coupled-channel equations . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Results for quadrupolar anions . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Critical quadrupolar moments . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Rotational bands in the continuum . . . . . . . . . . . . . . . . . . .
. . . . . . . .
3.3.1 Threshold trajectories for multipolar Gaussian potentials in the adia-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . .
3.3.2 Resonances of the near-critical quadrupolar Gaussian potential
3.3.3 Rotational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 Results for anions bounded by multipolar Gaussian potentials

batic limit

Chapter 4

Lithium isotopes and mirror nuclei

4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Gamow shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2
Interaction optimization . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Model space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Optimized states
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Optimized Interaction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Test against other excited states . . . . . . . . . . . . . . . . . . . . .
4.3.4 Root-mean-square radius . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Prediction for unbound nuclei: 10Li, 10N and 11O . . . . . . . . . . .
4.3.6 Continuum effects on the Thomas-Ehrman shift . . . . . . . . . . . .

. . . . . . . . . . . . . . . . 38
38
39
39
42
43
45
45
46
47
50
51
54

vi

Chapter 5

Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . 57

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

vii

46

46

46

47

49

50

53

LIST OF TABLES

Table 4.1 Energy levels used in the GSM Hamiltonian optimization. The energies
are given with respect to the 4He g.s.. The experimental values Eexp are taken
from Ref. [1]. They are compared to the GSM values EGSM.
. . . . . . . . .

Table 4.2 Central and spin-orbit strengths of the core-nucleon WS potential op-
.

timized in this work. The statistical uncertainties are given in parentheses.

Table 4.3 Groud state energies (in MeV) and widths (in keV) of 5He and 5Li
obtained from the optimized core-nucleon potential and compared to experi-
ments [2, 3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 4.4 Strengths V ST

of the two-body interaction optimized in this work. The
statistical uncertainties are given in parentheses. . . . . . . . . . . . . . . . .

η

Table 4.5 Energy levels for states not entering the optimization. The experimental
values Eexp are taken from Ref. [1]. The GSM values EGSM are shown with
the uncertainties in parentheses.
. . . . . . . . . . . . . . . . . . . . . . . .

Table 4.6 Root-mean-square proton (Rp) and neutron (Rn) radii of 6Li, 7Li/7Be,
. . . . . . . . . . . . . . . . . . . . . . . .

8Li/8B, 9Li/9C, and 11Li (in fm).

Table 4.7 Squared amplitudes of dominant configuration of valence neutrons and
protons for low-lying levels of 10Li and 10N, respectively. Energies with respect
to the one-nucleon emission threshold are shown in the parentheses for each
state. The odd proton in 10Li and the odd neutron in 10N occupy the 0p3/2
Gamow state. The tilde sign labels non-resonant continuum components.
. .

viii

LIST OF FIGURES

Figure 2.1 S.p. states in the complex-momentum plane. Bound states (b) and
antibound states (a) lie on the positive and negative imaginary-k axis, respec-
tively. Capturing states (c) and decaying resonances (d) lie symmetrically in
the third and fourth quadrants. The thick dashed line shows the deformed
contour with antibound states included in the Berggren completeness relation.
Taken from Ref. [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.1 A schematic illustration of the complex-energy electron-plus-molecule
. . . . . . . . . . . . . . . . .

model used in this work. Taken from Ref. [5].

Figure 3.2 Critical prolate electric quadrupole moment as a function of the orbital
angular momentum cutoff (cid:96)max in coupled-channel calculations in the adiabatic
limit (I → ∞). The internuclear distance is fixed at s = 1.6 a0 and the
corresponding value of Q+
0 is indicated by the dotted line.
The DIM results are marked by stars. The DIM result from [6] is denoted
by a square at (cid:96)max = 10. The convergence of the BEM results with respect
−1
to the momentum cutoff is shown for kmax = 6,8,10,and 12 a
0 . Taken from
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ref. [7].

zz,c = 6.372016 ea2

Figure 3.3 Yrast band of quadrupolar anions defined by an internuclear distance
of s = 1.6 a0, a moment of inertia of I = 104 mea2
0, and quadrupole moments
of Q−
0 on panels (a) and (b), respectively. The
BEM and DIM results are denoted with empty circles and stars, respectively,
and are almost indistinguishable for all orbital angular momentum cutoffs
considered. Taken from Ref. [7]. . . . . . . . . . . . . . . . . . . . . . . . . .

zz=−2.42 ea2

0 and Q+

zz=+6.88 ea2

Figure 3.4 Threshold trajectories (V0, r0)±

λ = 1 − 4 in the adiabatic limit. Taken from Ref. [5].

c for multipolar Gaussian potentials with
. . . . . . . . . . . . .

Figure 3.5 The lowest 0+ resonant state of the quadrupolar Gaussian potential
with r0 = a0 as a function of V0. Top: real energy and imaginary momentum.
Bottom: the channel decomposition of the real part of the norm Re(N(cid:96)). The
. . . . . . . .
critical strength V0,c is marked by arrow. Taken from Ref. [5].

ix

8

11

16

17

20

23

Figure 3.6 Trajectory of the 0+ resonant state in the complex-k plane of the
quadrupolar potential with r0 = 4 a0 as the potential strength V0 increases
in the direction indicated by an arrow. At the lowest value V0 =1.1 Ry, the
0+ g.s.
2 state associated

is bound and the state of interest is an excited 0+

with a decaying resonance. At V0 = 1.8 Ry the pole crosses the −45◦ line and
becomes a subthreshold resonance 0+
2 . At V0 = 2.857 Ry the decaying
pole reaches the imaginary-k axis and coalesces with the capturing pole with
Im(k) < 0 forming an exceptional point. The antibound states at V0 = 1.8 Ry
. . . . . . . . . . . . . . .
and V0 = 2.7 Ry are marked. Taken from Ref. [5].

d ≡ 0+

Figure 3.7 Real norms of the channel wave functions for the decaying pole 0+
d
c of Fig. 3.8. Taken from
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

shown in Fig. 3.6 and the antibound states 0+
Ref. [5].

b and 0+

Figure 3.8 Trajectories of antibound and bound 0+ states along the imaginary-k
axis as a function of V0 for the quadrupolar potential with r0 = 4 a0. With
increasing potential strength, the antibound states 0+
c become
bound states of the system 0+
3 , respectively. The open circle
marks the exceptional point of Fig. 3.6, which is the source of two antibound
states. The particular values of V0 discussed around Fig. 3.6 are marked.
Taken from Ref. [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 , and 0+

1 , 0+

a , 0+

b , and 0+

Figure 3.9 Trajectory of the 0+

d resonant state in the complex-k plane for different
values r0 of quadrupolar potential as indicated by numbers (in units of a0).
The ranges of V0 (in Ry) are: (25.6-29.0) for r0 = a0; (9.7-14.5) for r0 = 1.5 a0;
(4.8-10) for r0 = 2 a0; (1.7-4.79) for r0 = 3 a0; and (1.1-2.85) for r0 = 4 a0.
Taken from Ref. [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−
Figure 3.10 Top: trajectory of the lowest 1
1 resonant state of the quadrupolar
potential with r0 = a0 as a function of V0 in the range of (9-12.7) Ry. The
potential strength V0 increases along the direction indicated by an arrow. The
positions of the bound and antibound states at V0 = 12.34 Ry and 12.4 Ry are
marked. Bottom: real norms of channel functions for this state. Taken from
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ref. [5].

Figure 3.11 The rotational band built upon the J π = 0+

1 state of a dipole-bound
anion. The parameters V0 = 5.33 Ry, r0 = a0, and I = 103 mea2
0 have been
chosen to place the bandhead energy slightly below the zero-energy threshold,
where the rotational motion of the molecule can excite the system into the
continuum. The energy is plotted as a function of J(J + 1). Taken from
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ref. [5].

Figure 3.12 Similar to Fig. 3.11 but for rotational bands built upon the J π = 0+
1
−
1 bandheads of a quadrupolar Gaussian potential with V0 = 12.38 Ry,
and 1
r0 = a0, and for I = 50 mea2
0. Taken from Ref. [5]. . . . . .

0 and I = 100 mea2

x

25

26

28

29

30

32

33

Figure 3.13 Energy (a) and decay width (b), both in Ry, of the 3

−
1 resonance
of the quadrupolar Gaussian potential with r0 = a0 as a function of the
inverse of the moment of inertia and the potential strength. The dissociation
threshold (E = 0) is indicated. The dominant (j, (cid:96)) channel is marked in
panel (b). When the rotational energy of the molecule Ej=4
lies below/above
rot
−
1 resonance, the (4,1) decay channel is open/closed. The
the energy of the 3
−
line Ej=4
rot = E(3
1 ) (thick solid) separating these two regimes is marked, so
−
is the line Ej=2
1 ) (thick dotted) which corresponds to the threshold
energy for the opening of the (2,1) channel. The norms of the two dominant
channels (2,1) (solid line) and (4,1) (dotted line) are shown as a function of
V0 for 1/I = 0.04 m−1
. . .
e a

(c) and 0.02 m−1
e a

rot = E(3

−2
0

−2
0

(d). Taken from Ref. [5].

Figure 4.1 Energies for states of Li isotopes with respect to 4He. Red lines denote
GSM results and the black lines mark experimental values. The shaded area
represents the width of the corresponding resonance. States used for opti-
mization are marked with a (cid:70), their energies are listed in Tables 4.1 and 4.5.

Figure 4.2 Similar to Fig. 4.1 for results of mirror nuclei of Li isotopes. Energies
are given with respect to g.s. of 4He. Experimental energy of the 5/2− res-
onance in 9C was taken from Ref. [8] and the data for 11O is from Ref. [9].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.3 Spectrum for Li isotopes and their mirror partner with mass (a) A = 7,
(b) A = 8, (c) A = 9, (d) A = 10. Within each pair, the spectrum of Li
isotope and its mirror are plotted whin the same scale and different range.
The plots are shifted so that the g.s. of each pair align with each other. The
one-proton/neutron emission thresholds are also marked within each plot. . .

xi

35

48

51

55

Chapter 1

Introduction

1.1 Open quantum systems

Due to their weakly bound/unbound nature, OQSs are strongly affected by the environment.

There are many examples of OQSs across different physics areas, such as nuclear physics,

atomic and molecular physics, quantum optics, microwave resonators, and nanoscience. De-

spite some distinct features, these OQSs share many generic properties related to the presence

of resonant states, exceptional points, threshold behavior, etc.

Most isotopes far away from the beta-stability line fall into the category of OQSs. More-

over, even for those well-bound isotopes, the continuum coupling should not be overlooked

when considering excited states near the particle decay threshold. Studies on such exotic

systems offer unique opportunities to test theory, as well as understanding the important

features of continuum coupling in OQSs.

In order to describe continuum coupling properly, two problems need to be addressed.

Firstly, in practice, one needs to discretize the continuum states numerically, while ensuring

the completeness of the s.p. basis. Consequently, in larger systems involving many-body cor-

relations, computational work can become prohibitively expensive. Secondly, these unbound

states, including decaying resonances, are not square-integrable. Accordingly, to describe

OQSs, one needs to go beyond the standard Hilbert-space quantum mechanics which deals

with L2-integrable states.

This raises a challenge for the theoretical studies. One way to deal with these problems

is to extend the system Hamiltonian into the complex-energy (or momentum) plane with

1

the Berggren basis [4, 10, 11]. Since the Berggren ensemble includes bound states, decaying

resonances, and non-resonant scattering states, the continuum effects can be taken into

account properly. Based on this, several approaches, such as complex-energy electron-plus-

molecule model and GSM have been developed. In this work, we use these approaches to

study molecular and nuclear OQSs.

1.1.1 Atomic systems

In atomic physics, anions are neutral molecules that can attach an excess electron. This

valence electron is bounded by the weak Coulomb potential of the molecule due to electro-

static polarization effects, which makes anions good candidates to study physics questions

pertaining to OQSs.

Anions are difficult to identify experimentally, due to the high order polarization as

well as the weakly bound/unbound property. Therefore, to provide some guides for the

experimental study, we used a complex-energy electron-plus-molecule model to analyze the

behavior of polarized anions, where both electron motion and molecular rotational motion

are considered and coupled.

In particular, the properties of quadrupolar anions and the

molecular anions bounded by the multipolar Gaussian are discussed in Chapter 3.

1.1.2 Nuclear systems

In low-energy nuclear physics, the development of next-generation experimental facilities

(including Facility for Rare Isotope Beams (FRIB) at Michigan State University) will al-

low more rare isotopes, which inhabit remote regions of the nuclear landscape, to become

accessible [12]. Properties of these isotopes are at the forefront of nuclear structure and

reaction research, which provide unique opportunities to study OQS phenomena. In thresh-

old regions around particle drip lines strong continuum coupling effects are present, which

result in exotic nuclear properties such as nuclear halos [13, 14, 15], presence of unexpected

intruder states [16, 17, 18], clusterization [19, 20], appearance of new ‘magic’ numbers, and

2

two-nucleon decay [21, 22, 23, 24].

In this thesis, we are interested in halo systems, in which the valence nucleons are

impacted by the continuum environment. The halo phenomenon was first discovered in

11Li [25]. Another example is 11Be, where continuum effects play a significant role in forming

the halo structure as well as the inversion of the ground state (g.s.) parity [26, 27, 28, 29].

Halo systems are often studied within phenomenological models, which assume the

presence of large cluster substructures [30, 18]. In this work, we use the GSM to reveal how

weakly bound (or unbound) nuclear states are formed and affected by many-body correla-

tions. Specifically, the lithium isotopes and their mirror partners have been studied with an

effective Hamiltonian optimized to selected low-lying nuclear states. These results will be

discussed in Chapter 4.

1.2 Outline

This thesis is on the application of a complex-energy configuration interaction approach

to the nuclear and atomic OQSs.

In Chapter 2, a complex-momentum Berggren basis is

introduced. Chapter 3 presents the work on atomic anions, including both quadrupolar

anions and anions bound by multipolar Gaussian potentials. In Chapter 4, we study the

lithium isotopes and their mirror partners. Finally, this thesis concludes in Chapter 5 with

a summary of our results and an outlook for future studies.

3

Chapter 2

Berggren ensemble

In well-bound systems, which can be viewed as closed quantum systems, the wave functions

of the low-lying states are spatially localized and can thus be expanded in the harmonic

oscillator (HO) basis, which decays asymptotically.

In OQSs, however, coupling to the

environment becomes non-negligible and non-localized continuum states must be considered.

An approach that goes beyond the Hilbert space is needed. In our work, we use a more

general basis, named Berggren basis [10], to study OQSs.

In this chapter, Gamow states, being one of the most important features in OQSs, will

be first introduced. Berggren completeness relation, used in all the calculations through this

thesis, is then illustrated.

2.1 Gamow states

Gamow states [31, 32], also known as resonant or Siegert [33] states, were introduced for the

first time by George Gamow in 1928 to describe the phenomenon of α decay. In the quasi-

stationary formalism, Gamow states have only outgoing wave function in the asymptotic far

region

and complex energy:

u(k, r)r→∞ ∝ C+H +

(cid:96)η(kr),

˜En = En − i

Γn
2

,

4

(2.1)

(2.2)

where the real part En corresponds to the mean energy of the state and the imaginary part

can be associated with the decay width Γn. The decay width is related to the decay half-life

by the usual relation:

T1/2 =

ln2
Γ

.

(2.3)

Similar to bound states, resonances are poles of the scattering matrix in the complex-

momentum plane, reflecting the properties of the binding potential. In the complex-momentum

plane, bound states lie on the positive imaginary-k axis kn = iκn (κn > 0), while decaying

resonances lie in the fourth quadrant with kn = κn − iγn (κn > 0, γn > 0), as shown in
Fig. 2.1. Capturing resonances are located symmetrically in the third quadrant. Decaying

resonances with κn < γn are referred to as subthreshold resonances [34, 35, 36, 37]. While

they can not be observed experimentally, their presence can impact the near-threshold struc-

ture as well as the corresponding observables. Although the positive-energy Gamow states

are not square-integrable in real space, they can still be normalized through complex-scaling

method [38, 39] by choosing a proper integration path in the complex plane.

2.2 Antibound states

Antibound states [40, 41, 42, 43], also known as virtual states, lie in the negative imaginary

axis of the complex-momentum plane with kn = −iκn(κn > 0). With real and negative en-
ergy similar to bound states, virtual states lie on the second Riemann sheet of the complex

energy plane and their wave functions are not localized. The negative imaginary momentum
leads to exponentially diverging wave function in the space, u(r) ∼ eκnr. A physical inter-
pretation of antibound states is usually difficult as the exponential increasing wave function

cannot support a state. Therefore, similar to subthreshold resonances, antibound states

cannot be measured. However, antibound states can reveal themselves with increased cross

section near the threshold [44, 45, 46, 47].

5

2.3 Berggren completeness relation

Due to the exponential growth and exponential decay, resonances can not be described prop-

erly in the Hilbert-space. Within the real-energy scheme, resonances can be either extracted

from the real-energy continuum level density or obtained by joining the bound-state solution

in the interior region with the asymptotic solution using the R-matrix approach [48, 49]. Pro-

jected subspace has been used to artifically separate the bound/resonant and non-resonant

scattering parts in the shell model embedded in the continuum. With the advantages to de-

scribe observables such as elastic/inelastic cross-sections [50], this approach is limited when

considering several particles in the non-resonant scattering continuum.

The rigged Hilbert-space (RHS) formulation provides a good description for resonances

by offering a unified treatment of bound, resonance and scattering states. By using the

regularization method with a Gaussian convergence factor, Berggren normalized the Gamow

staes and proved the Berggren completeness relation [10] with the continuum states included

in a complex-plane s.p. basis:

(cid:88)

n

un(En, r)un(En, r

(cid:48)

) +

(cid:90)

L+

u(E, r)u(E, r

(cid:48)

)dE = δ(r − r

(cid:48)

),

(2.4)

un(En, r) are the normalized wave functions for discrete resonant states, including both

bound states and decaying resonances. The second term consists of the non-resonant con-
tinuum states along an arbitrary scattering contour L+ with the wave function of u(E, r).
The Berggren completeness relation for each partial wave can be seen more clearly in

the momentum space: (cid:88)

n∈(b,d)

|˜un(cid:105)(cid:104)un| +

(cid:90)

L+ |u(k)(cid:105)(cid:104)u(k)| dk = 1,

(2.5)

where b and d stands for bound states and decaying resonances, respectively. L+ is the
scattering contour in complex-momentum plane. The tilde symbol indicates the time-reversal

6

operation. As we show in Fig. 2.1, one can draw a contour L+ that starts from the origin,
extends to the fourth quadrant, then comes back to the real axis and finally extends to the
infinity along the real axis. As a result, the scattering states along the contour L+, bound
states on the positive imaginary axis plus all the decaying resonances between the real axis
and the contour L+ form the Berggren completeness relation. Contours with different shapes
are equivalent as long as all the resonant states between the real axis and the contour are

included. Antibound states can also be included in the generalized completeness relations
with slightly deformed contour L+.
bound states of energies higher than antibound states are present, they must be excluded

It is to be noted that in the unlikely situation that

from the sum in Eq. (2.5), see Fig. 2.1.

7

Figure 2.1: S.p. states in the complex-momentum plane. Bound states (b) and antibound
states (a) lie on the positive and negative imaginary-k axis, respectively. Capturing states
(c) and decaying resonances (d) lie symmetrically in the third and fourth quadrants. The
thick dashed line shows the deformed contour with antibound states included in the Berggren
completeness relation. Taken from Ref. [4].

In our applications, the contour L+ is defined by three points: kpeak in the fourth
quadrant, kmid, and cutoff momentum kmax on the real axis. The resulting three segments

along the contour are usually discretized with (N1,N2,N3) Gaussian-Legendre points. kmax

and N need to be sufficiently large to ensure the completeness of the basis as well as the

convergence of results. While the bound states are normalized in the standard way, decaying

resonances are normalized using the exterior complex scaling method [38, 39]. The scattering

states are normalized to the Dirac-delta function, (see Ref. [4]).

8

Chapter 3

Atomic anions

Since the critical moment µc required to bind an extra electron by a point-dipole was first

determined by Fermi and Teller [51, 52, 53, 7], extensive theoretical as well as experimental

studies [54, 55, 56, 57] have been carried out to investigate dipolar anions. However, this

is not the case for other multipolar anions. As the higher-order multipolar potentials are

even shallower than the dipolar potential, molecular anions associated with neutral cores

with no dipole moments have been more challenging to find experimentally. One interesting

question is to identify the limit of existence for higher-order multipolar anions. To determine

the critical quadrupole moment needed to bind an excess electron, we carried out exploration

of quadrupolar anions with a linear charge configuration.

The property of polarized anions above the dissociation threshold is also an interest-

ing aspect to investigate. For example, the study of dipolar anions in Ref. [58] suggests

the presence of different behavior in the strong-coupling (subthreshold) and weak-coupling

(above threshold) regimes. One might wonder whether such a pattern would be present in

quadrupolar anions, where the binding potential is more localized and deformed.

While the binding of multipole-bound anions is fragile, low-energy resonances in such

systems are expected to be less sensitive to details of the short-range molecular potential as

the spatial extension of the valence electron is huge. This situation resembles universal behav-

ior, independent of the details of the interaction, exhibited by other weakly-bound/unbound

quantum systems, such as nuclear halos, cold atomic gases near a Feshbach resonance, and

helium dimers and trimers, see, e.g., Refs. [59, 60, 61, 62, 63, 64, 65, 66, 67, 68].

In all

of those cases, simple arguments based on scale separation and effective field theory cap-

9

ture the essential physics [69, 70, 71, 72, 73, 74, 64]. However, due to the similarity of

single-electron and rotational energy scales, coupling between the valence electron motion

and molecular rotational motion might become complicated above the dissociation thresh-

old [75, 76, 77, 78, 79, 80]. Therefore, to investigate generic properties of multipole-bound

anions, in this work we include a nonadiabatic coupling between electronic and molecular

motion in our complex-energy electron-plus-molecule model, and simulate the short-range

multipole potential with a Gaussian form factor.

In this chapter, we present the framework of our complex-energy electron-plus-molecule

model as well as the corresponding results for these two types of anions: (i) quadrupolar

anions; and (ii) anions described by multipolar Gaussian potentials.

10

3.1 Electron-plus-molecule Model

3.1.1 Model and Hamiltonian

Figure 3.1: A schematic illustration of the complex-energy electron-plus-molecule model used
in this work. Taken from Ref. [5].

As shown in Fig. 3.1, a polarized anion system can be schematically described as an electron

moving in the potential generated by a multipole molecule. Since the attached electron is as-

sumed to be rather far from the core, the spin-orbit interaction is neglected. Similar to other

molecular structure problem, if vibrational motion is considered, the average asymptotic po-

tential experienced by the valence electron will just be that dominated by the average dipole

moment of the polar systems. Vibrational motions can be separated out and are not included

11

electronmolecule molecularframejrθin this model [81]. The electron-plus-molecule Hamiltonian can be written as [82, 58]:

ˆH =

ˆj2
2I

+

ˆp2
e
2me

+ V (r).

(3.1)

The first term is the rotational energy of the molecule with angular momentum ˆj and moment

of inertia I. The second term represents the kinetic energy of the electron of mass me and

linear momentum ˆpe. V (r) defines the effective interaction (potential) between valence

electron and the molecule with r being the position vector pointing from the molecule to

the valence particle.

Quadrupolar anion potential

To simulate the potential (electrostatic field) V (r) in a quadrupolar anion, we consider a

linear distribution of point charges (with the amount of charge q) separated by a distance

of s. Two configurations of (q,−2q, q) and (−q, 2q,−q) correspond to a prolate and oblate
shape, respectively, with the middle charge residing in the center of the molecule and q > 0.

Considering the cylindrical symmetry along the molecule axis (z axis), a quadrupole moment
of the considered configuration is Q±

zz = ±2qs2.

The potential V (r) can be expressed through a multipole expansion:

V (r, θ) =

Vλ(r)Pλ(cosθ),

(3.2)

where Vλ(r) is the radial part for each multipolarity λ. For the linear charge distribution

λ

(±q,∓2q,±q) considered here, the form of Vλ(r) can be explicitly written as [7]:

(cid:88)

 1

r

r> − 1
( r<
)λ 1
r>
r>

Vλ(r) =

e

4π
0

Q±
s2

for λ = 0,

for λ = 2, 4, 6...

(3.3)

with r>=max(r, s) and r<=min(r, s). The angular part of the potential Pλ(cosθ) is given

12

by a Legendre polynomial of order λ, with θ being the angle between the direction of the

valence electron r and the symmetry axis of the molecule, see Fig. 3.1.

Multipolar Gaussian potential

In multipolar Gaussian anions, the interaction between the molecule and the valence electron

V (r) can be modeled by a short-range potential of the axially-deformed Gaussian form with

multipolarity λ:

(cid:18)

(cid:19)

r2
2r2
0

V (r) = −V0 exp

−

Pλ(cos θ),

(3.4)

where V0 is the potential strength, and r0 is the potential range.

3.1.2 Coupled-channel equations

The total angular momentum of the system ˆJ is given by the sum of the angular momenta

of the molecule ˆj and the valence electron ˆ(cid:96). Although the potential V (r) is deformed in

the intrinsic frame, the whole system (electron + molecule) is rotationally invariant in the

laboratory reference frame. Therefore, ˆJ commutes with the Hamiltonian ˆH and the wave

function can be written as:

ΨJ (r) =

(cid:88)

uJ
c (r)ΘJ

c (ˆσ),

(3.5)

c

where c labels all possible channels (j, (cid:96)) for a given J, uJ

c (r) is the radial channel wave

function, and ΘJ

c (ˆσ) is the angular channel wave function. The eigenstates Eq. (3.5) are

also labeled by means of the parity quantum number π; hence, in the following we use the

spectroscopic notation J π

n , where n = 1 marks the lowest J π-state, n = 2 – the next one,

and so on.

To properly describe the nonadiabatic coupling between the electronic and molecular

motion, we solve the coupled-channel equations:

(cid:20) d2

dr2 −

jc(jc + 1)

I

(cid:96)c((cid:96)c + 1)

r2

−

+ EJ

13

(cid:21)

uJ
c (r) =

(cid:88)

c(cid:48)

V J
cc(cid:48)(r)uJ

c(cid:48)(r),

(3.6)

which are obtained by inserting the wave function (3.5) into the Schr¨odinger equation. V J

cc(cid:48)(r)

is the channel-channel coupling potential, and can be evaluated by rewriting the multipolar

potential V (r) in the laboratory frame [82, 58]. In this case, the motion of the electron is

weakly coupled to the rotation of the molecule. The adiabatic, or strongly-coupled, limit

corresponds to an infinite moment of inertia (I → ∞) where the rotational band of the
system collapses to the bandhead energy. In this section devoted to atomic systems, we will

be using Rydberg units (energy expressed in Ry and distance in Bohr radius a0).

One way to solve the coupled-channel equations (3.6) is by means of the direct inte-

gration method (DIM). While well-bound state can be obtained fast with quite arbitrarily

chosen starting point, a reasonable initial guess is required to ensure convergence for weakly

bound and unbound states [7]; this can be difficult for those exotic systems whose compo-

nents are not well known in advance. Also, higher-multipolarity potentials require a larger

number of channels, which makes this method computationally demanding.

An alternative to the DIM is to use the basis expansion technique based on the Berggren

ensemble. With the basis generated using the diagonal part of the potential Vcc [58] for each

channel, this Berggren expansion method (BEM) provides a faster way of solving Eq. (3.6).

Moreover, in this method, one can obtain all the eigenstates at once by diagonalizing the

complex-symmetric Hamiltonian.

Finally, one needs to identify resonances from the non-resonant scattering background.

This can be done by analyzing the internal wave function of an eigenstate or using the fact

that resonances do not depend on a detailed choice of the contour [4]. Moreover, as a further

test, the resonant states obtained in BEM are used in the DIM as an initial guess, and it is

checked that the BEM results are reproduced.

14

3.2 Results for quadrupolar anions

In this section, we present the results for the quadrupolar anions with a focus on two aspects.

One is the critical quadrupolar moments that binds the anion. The other is the coupling

scheme between the molecule and valence electron in both above and below the dissociation

threshold, with a focus on the transition between two regimes.

3.2.1 Critical quadrupolar moments

To test the precision of our model, we benchmark the DIM and BEM as applied to quadrupo-

lar anions. Our results corrresponding to the adiabatic limit can be compared with the
analytical results of Ref. [83] for the critical electric quadrupole moment Q±

zz,c = ±2q±

s,cs.

The internuclear distance s is fixed at 1.6 a0 as in Ref. [6]; this value is close to the internu-

−
clear distance in CS
2 (s = 1.554 a0 [84]). The corresponding critical quadrupole moments
obtained analytically are Q−

zz,c = 6.372016 ea2
0.

0 and Q+

zz,c = −2.35152 ea2

In the DIM, the parameter that controls the accuracy of calculations is the orbital

angular momentum cutoff (cid:96)max that determines the size of the channel basis. For (cid:96)max = 12,
the DIM gives a critical oblate quadrupole moment of Q−

0. In the BEM,

zz,c = −2.35162 ea2

in addition to (cid:96)max, the momentum cutoff kmax needs to be fixed. By taking a real contour
−1
0 , one obtains Q−
L+ discretized with 80 points, and kmax = 12 a
critical oblate quadrupole moment can be approached closely with both methods because

zz,c = −2.35164 ea2

0. The

it corresponds to a configuration of the attached electron that is well localized around the

two positive charges at the center of the molecule. Thus, the electron is expected to be

primarily in low-(cid:96) orbitals. For the prolate quadrupole moment, the situation is different.

Here, the attached electron, attracted by the external positive charges, is less bound and

higher-(cid:96) partial waves are expected to play a more important role.

Indeed, as shown in

Fig. 3.2, the DIM and BEM results do not reproduce the analytical value as precisely as for

the oblate configuration. For (cid:96)max = 14 and kmax = 12 a

−1
0 , we obtained Q+

zz,c = 6.398 ea2
0

15

zz,c = 6.3984 ea2

and Q+
with (cid:96)max (and kmax) is slower than for Q−

0 with the DIM and BEM, respectively. While the convergence of Q+

zz,c

zz,c, DIM and BEM results are fairly consistent

for (cid:96)max = 14 and kmax = 12 a

−1
0 , and our results are in agreement with the DIM results of

Ref. [6].

Figure 3.2: Critical prolate electric quadrupole moment as a function of the orbital angular
momentum cutoff (cid:96)max in coupled-channel calculations in the adiabatic limit (I → ∞). The
internuclear distance is fixed at s = 1.6 a0 and the corresponding value of Q+
zz,c = 6.372016 ea2
0
is indicated by the dotted line. The DIM results are marked by stars. The DIM result from [6]
is denoted by a square at (cid:96)max = 10. The convergence of the BEM results with respect to
the momentum cutoff is shown for kmax = 6,8,10,and 12 a

−1
0 . Taken from Ref. [7].

In realistic molecules, the effect of Pauli blocking at short distances [85, 86, 87] re-

duces the binding in the oblate configuration; hence, in general, the prolate configuration

16

is more likely to bind electrons. Thus, while our oblate configuration results are useful for

benchmarking purposes, their physical interpretation should be dealt with caution.

3.2.2 Rotational bands in the continuum

zz=+6.88 ea2

zz=−2.42 ea2

Figure 3.3: Yrast band of quadrupolar anions defined by an internuclear distance of s =
1.6 a0, a moment of inertia of I = 104 mea2
0 and
Q+
0 on panels (a) and (b), respectively. The BEM and DIM results are denoted
with empty circles and stars, respectively, and are almost indistinguishable for all orbital
angular momentum cutoffs considered. Taken from Ref. [7].

0, and quadrupole moments of Q−

In a previous study on dipolar anions [58], the yrast band has been predicted to disappear

above the dissociation threshold, which implies a transition for anions going from below to

17

above the threshold. Below the detachment threshold, the motion of the attached electron

is strongly coupled to the rotational motion of the molecule. Above the threshold, however,

the electron becomes weakly coupled and moves almost independently.

Compared with the dipolar potential (∝ 1/r2), the quadrupolar potential has a faster
asymptotic falloff (∝ 1/r3) that may affect the structure of the delocalized resonant states.
In order to see the structure of resulting rotational bands, the binding energy for Q−
zz =
−2.42 ea2
function of J(J + 1).

0 is plotted in Fig. 3.3(a) and Fig. 3.3(b), respectively, as a

zz = +6.88 ea2

0 and Q+

Here we use the same parameters as in the previous section (s = 1.6 a0 and I =

0). The contour L+

104 mea2
zero and is defined by the three points: (0.3,−10−5), (0.6, 0), and (6,0) (all in a

is assumed to be identical for all partial waves:

c

it starts at

−1
0 ). The

three resulting segments are discretized with 30, 30, and 40 scattering states, respectively to

ensure convergence. The specific values of Qzz has been chosen so that the binding energy

approaches zero for a total angular momentum J ≈ 2, 3 at (cid:96)max=4.

The BEM and DIM results are practically indistinguishable for all the values of (cid:96)max con-

sidered. Perfect rotational behavior is predicted for both prolate and oblate configurations,

even above the dissociation threshold. This is confirmed by the collapse of all eigenenergies

to the same bandhead energy in the adiabatic limit (I → ∞). At the maximal orbital an-
gular momentum cutoff (cid:96)max considered, the states in the lowest-energy (yrast) band are all

dominated by the (cid:96) = 0 channel at about 99.7% and 87.9%, for the oblate and prolate con-

figuration, respectively. Unlike in the dipolar case, rotational bands of quadrupolar anions

persist in the continuum.

18

3.3 Results for anions bounded by multipolar Gaussian

potentials

In this section, we investigate the generic near-threshold behavior of multipole-bound anions

at the transition between the subcritical (below dissociation threshold) and supercritical

(above dissociation threshold) regimes. The main objective of this part is to show the role

of low-(cid:96) partial waves in shaping the properties of low-lying states. We assume that the

molecular potential has a Gaussian form given by Eq. (3.4). However, we wish to emphasize

that the particular choice of the radial form factor is not important as it represents an a priori

unknown short-range behavior. One can view this particular realization as a regularized zero-

range interaction. To study the threshold behavior of the system we investigate the pattern

of resonant poles as a function of four parameters: the strength and range of the Gaussian

form factor, the multipolarity of the potential, and the molecular moment of inertia.

19

3.3.1 Threshold trajectories for multipolar Gaussian potentials in

the adiabatic limit

Figure 3.4: Threshold trajectories (V0, r0)±
in the adiabatic limit. Taken from Ref. [5].

c for multipolar Gaussian potentials with λ = 1−4

±
λ,c mark the limit between the sub-
As mentioned earlier, the critical multipole moments Q

−
λ,c = −Q+
critical and supercritical regimes. One may notice that Q

λ,c for odd-multipolarity

potentials, but there is no such relation for even multipolarity potentials. As discussed ear-

lier, there are two critical values of the quadrupole moment for a quadrupole-bound anion

(λ = 2): Q+

−
−
2,c (cid:54)= −Q+
2,c.
2,c (oblate), and Q
2,c (prolate) and Q

As the usual −1/rλ+1 radial dependence of multipolar potentials is replaced in our work

20

0.00.20.40.60.81.0Potentialranger0(unitsofa0)0.00.20.40.60.81.0PotentialstrengthV0(Ry)-4-20λ=1(a)unbound-8-40λ=2(b)unbound135-12-60λ=3(c)unbound135-24-120λ=4(d)unboundV+0−V−0by the Gaussian form factor, the dissociation threshold is obtained at the critical trajectories
of (V0, r0)±

c . Fig. 3.4 shows such trajectories obtained in the adiabatic limit for the J π = 0+
1

g.s. of anions with multipolarities λ = 1 − 4.

These results are obtained with a Berggren contour defined by the points kpeak =

(0.5,−0.1), kmid = 1.0, and kmax = 14.0 (in units of a
by 40 Gauss-Legendre points. To ensure convergence, we took (cid:96)max = 4 for λ = 1, 2, 3 and

−1
0 ), with each segment being discretized

(cid:96)max = 8 for λ = 4, 5.

As one would expect, the absolute value of the critical potential strength |V0,c| required
to bind an excess electron decreases with the range r0 and for a fixed range |V0,c| increases
with multipolarity. Also, as noted in previous studies [7, 83, 88, 89], for even multipolarities,

the value of |V0,c| for negative-V0 potentials (“prolate”) is larger than that for positive-V0
potentials (“oblate”).

It is interesting to note that at the threshold the wave functions are dominated by the

(cid:96) = 0 component. Dividing the intrinsic wave function into the inner region (r < R) and

outer region (r > R) contributions, where R is the distance at which the molecule potential

becomes practically unimportant, one can show [90, 91, 92] that the probability of finding

the electron in the outer region approaches one at the dissociation threshold if the (cid:96) = 0

component is present in the intrinsic wave function. This has been practically demonstrated

in our work on quadrupole-bound anions [7] in the context of the scaling properties of root-

mean-square (rms) radii.

3.3.2 Resonances of the near-critical quadrupolar Gaussian po-

tential

In order to study the role of low-(cid:96) partial waves on the structure of multipole-bound anions,

one has to recognize the impact of (cid:96) = 0 partial waves on resonant states near threshold [92].

In our coupled-channel formalism, resonant states appear through the mixing of different

channels. To study general features of near-threshold resonances, we consider three states of

21

the quadrupolar potential in the adiabatic approximation. Namely, we investigate: (i) the

J π = 0+

1 g.s. dominated by the (cid:96) = 0 partial wave; (ii) an excited J π = 0+

d state dominated

−
by the (cid:96) = 2 channel; and (iii) the lowest J π = 1
1 state, which is primarily (cid:96) = 1 and

without contribution from (cid:96) = 0. The quadrupolar case discussed here is characteristic of

other multipolar potentials.

(i) Resonant states dominated by the (cid:96) = 0 channel

The g.s. of the quadrupolar potential is computed with the BEM, using the extended contour
L+ defined by the points: k = (0, 0), (−0.1,−0.4), (0.1,−0.4), (2, 0), and (14, 0) (all in a
each segment being discretized with 40 Gauss-Legendre points. By considering the contour

−1
0 ),

that extends into the third quadrant of the complex-momentum plane, antibound states can

be revealed [4, 93, 94].

22

Figure 3.5: The lowest 0+ resonant state of the quadrupolar Gaussian potential with r0 = a0
as a function of V0. Top: real energy and imaginary momentum. Bottom: the channel
decomposition of the real part of the norm Re(N(cid:96)). The critical strength V0,c is marked by
arrow. Taken from Ref. [5].

Fig. 3.5(a) shows the energy and momentum of the 0+

1 state for different values of the

potential strength V0. For large values of V0, the g.s. is bound (Re(E)<0) and has a positive

imaginary momentum. As the potential strength decreases, the energy of the g.s. moves

up and approaches the E = 0 threshold at V0,c = 8.7 Ry. For V0 < V0,c the lowest 0+ state

becomes antibound (Re(E) < 0, Im(k) < 0). With the complex-energy scheme, the norm

for the wave function also has a complex form. In our work, the sum of the real part of the

norm for all channels is normalized to 1. The real part of the norm for each channel can

have values beyond the range of 0 to 1. As illustrated in Fig. 3.5(b), the contributions N(cid:96) to

23

0.00.20.40.60.81.0PotentialstrengthV0(Ry)0.00.20.40.60.81.0-0.10.00.1Re(E)(Ry)λ=2,Jπ=0+1,r0=a0(a)Im(k)Re(E)789100.00.40.81.2Re(Nl)(b)V0,c(cid:22)=0(cid:22)=2(cid:22)=4-0.40.00.4Im(k)(units ofa−10)the complex norm of the wave function from different (cid:96)-channels ((cid:96) = 0, 2, 4) vary smoothly

when crossing the threshold. The norm is largely dominated by the (cid:96) = 0 component. At

the critical strength, the (cid:96) > 0 contributions to the norm vanish, cf. discussion in Sec. 3.3.1.

This indicates that the presence of near-threshold antibound states indeed impacts the near-

threshold structure of weakly-bound systems [95, 96, 97, 98, 35].

(ii) Resonant states dominated by a (cid:96) (cid:54)= 0 channel

We now consider the evolution of an excited state of a wider quadrupolar potential with

r0 = 4 a0. At V0 =1.1 Ry, the lowest 0+ state is bound and the second J π = 0+

2 state is

a decaying resonance, see Fig. 3.6. Figure 3.7(a) shows the channel decomposition for this

second 0+

2 state. It is seen that its configuration has the predominant (cid:96) = 2 component.

24

Figure 3.6: Trajectory of the 0+ resonant state in the complex-k plane of the quadrupolar
potential with r0 = 4 a0 as the potential strength V0 increases in the direction indicated by
an arrow. At the lowest value V0 =1.1 Ry, the 0+ g.s. is bound and the state of interest is an
excited 0+
2 state associated with a decaying resonance. At V0 = 1.8 Ry the pole crosses the
−45◦ line and becomes a subthreshold resonance 0+
2 . At V0 = 2.857 Ry the decaying
pole reaches the imaginary-k axis and coalesces with the capturing pole with Im(k) < 0
forming an exceptional point. The antibound states at V0 = 1.8 Ry and V0 = 2.7 Ry are
marked. Taken from Ref. [5].

d ≡ 0+

As the potential gets deeper, the pole crosses the −45◦ line at V0 ≈ 1.8 Ry and becomes

a subthreshold resonance labeled as 0+

d . At V0 = 2.7 Ry, a rapid transition to a configuration

dominated by the (cid:96) = 4 partial wave takes place, which is indicative of a level crossing in the

25

1.00.00.10.20.30.4-0.4-0.3-0.2-0.10.00+20+b0+c1.1 RyV02.7 Ry1.8 Ry2.857 Ry=2,r0=4a0+dRe(k)(units ofa−10)Im(k)(units ofa−10)complex-k plane. At V0 = 2.857 Ry the decaying pole arrives at the imaginary-k axis and

coalesces with the symmetric capturing pole forming an exceptional point [99, 100, 101]. At

still larger values of V0, the exceptional point splits up into two antibound states moving up

and down along the imaginary-k axis as shown in Fig. 3.6. A similar situation was discussed

in Refs. [43, 102] in the context of electron-molecule scattering and optical lattice arrays,

respectively.

Figure 3.7: Real norms of the channel wave functions for the decaying pole 0+
Fig. 3.6 and the antibound states 0+

c of Fig. 3.8. Taken from Ref. [5].

b and 0+

d shown in

In the range of V0 corresponding to the trajectory 0+

2 → 0+

d shown in Fig. 3.6, there

26

0.01.0-0.50.00.51.01.5Re(N(cid:31))λ=2(cid:31)=0(cid:31)=2(cid:31)=4Jπ=0+1.01.52.02.53.0V0(Ry)-0.50.00.51.0Re(N(cid:31))0+b0+c1.82.70+d0+(a)(b)(c)(cid:31)=2(cid:31)=2(cid:31)=0(cid:31)=0(cid:31)=4(cid:31)=4appear antibound states in the threshold region. Their trajectories along the imaginary-k

axis are shown in Fig. 3.8 and their channel decompositions are given in Fig. 3.7(b) and

(c). As V0 increases, the antibound states 0+

a , 0+

b , and 0+

c emerge as bound physical states

of the system labeled as 0+

1 , 0+

2 , and 0+

3 , respectively. The lowest antibound state 0+

a has

a dominant (cid:96) = 0 configuration, similar to that of Fig. 3.5. At low values of V0, the wave

function of the antibound state 0+
b

is predominantly (cid:96) = 2. As seen in Fig. 3.6, this state

appears close to the decaying pole 0+

d at V0 ≈ 1.8 Ry and the crossing between these two
poles in the complex-k plane is seen in their wave function decompositions. Following the

b acquires a large (cid:96) = 0 component. The antibound state 0+

crossing, the state 0+
an (cid:96) = 4 configuration. At V0 ≈ 2.7 Ry, this state interacts with 0+
changes to (cid:96) = 2. One can thus see that the presence of antibound states results in the

d and its configuration

c begins as

particular shape of the 0+

d -pole trajectory in the complex-k plane.

27

Figure 3.8: Trajectories of antibound and bound 0+ states along the imaginary-k axis as
a function of V0 for the quadrupolar potential with r0 = 4 a0. With increasing potential
1 , 0+
strength, the antibound states 0+
2 ,
and 0+
3 , respectively. The open circle marks the exceptional point of Fig. 3.6, which is the
source of two antibound states. The particular values of V0 discussed around Fig. 3.6 are
marked. Taken from Ref. [5].

c become bound states of the system 0+

a , 0+

b , and 0+

28

0.00.20.40.60.81.00.00.20.40.60.81.001234PotentialstrengthV0(Ry)-0.4-0.20.00.20.4Im(k)(unitsofa−10)λ=2r0=4a00+10+20+30+a0+b0+c0+d1.82.72.857Figure 3.9: Trajectory of the 0+
d resonant state in the complex-k plane for different values
r0 of quadrupolar potential as indicated by numbers (in units of a0). The ranges of V0 (in
Ry) are: (25.6-29.0) for r0 = a0; (9.7-14.5) for r0 = 1.5 a0; (4.8-10) for r0 = 2 a0; (1.7-4.79)
for r0 = 3 a0; and (1.1-2.85) for r0 = 4 a0. Taken from Ref. [5].

The dependence of the 0+

d -pole trajectory on the potential range is illustrated in Fig. 3.9.

For potentials with longer ranges, pole trajectories appear closer to the origin. Due to

the numerical stability issue, the contour used can not go below −0.4 a
momentum. Consequently, a wider potential with r0 = 4 a0 is chosen so that the whole

for imaginary

−1
0

complex-momentum trajectory can be revealed. In all the cases shown, a transition from

decaying to subthreshold resonances takes place. These poles have large widths and are

expected to impact the structure of the low-energy scattering continuum.

29

0.00.20.40.60.81.00.00.21.00.00.20.40.6Re(k)(unitsofa−10)-0.6-0.4-0.20.0Im(k)(unitsofa−10)λ=2,Jπ=0+dr0=4a0r0=3a0r0=2a0r0=1.5a0r0=a0(iii) Resonant states without a (cid:96) = 0 component

−
Here we discuss the lowest J π = 1
1 state, which is primarily (cid:96) = 1 with a small admixture of

the (cid:96) = 3 channel. This case closely follows the discussion of Ref. [43] for p-wave scattering

from short-range potentials.

−
1 resonant state of the quadrupolar potential with
Figure 3.10: Top: trajectory of the lowest 1
r0 = a0 as a function of V0 in the range of (9-12.7) Ry. The potential strength V0 increases
along the direction indicated by an arrow. The positions of the bound and antibound states
at V0 = 12.34 Ry and 12.4 Ry are marked. Bottom: real norms of channel functions for this
state. Taken from Ref. [5].

The corresponding trajectory of this state in the complex-momentum plane is shown

30

0.00.20.40.60.81.00.00.20.40.60.81.0-0.20.00.20.40.6Re(k)(a−10)-0.2-0.10.00.10.2λ=2,Jπ=1−1r0=a0(a)12.3412.3412.4012.40-0.050.000.050.100.150.20Re(E)(Ry)0.00.20.40.60.81.0Re(N(cid:26))(b)(cid:26)=1(cid:26)=3Im(k)(unitsofa−10)−
in Fig. 3.10(a). At larger values of V0, the 1
1 state is bound. As V0 decreases, this state

crosses the dissociation threshold and becomes a narrow decaying resonance. The trajectory

of the capturing resonance, symmetric with respect to the imaginary-(k) axis, is not shown.

As discussed in Ref. [43], the exceptional point appears at the origin at V0,c. Close to the

threshold, the bound state and the antibound state are located symmetrically to the origin.

For the p-wave dominated state, the transition from the subcritical to the supercritical regime

is smooth, i.e., the wave function amplitudes hardly change with V0, see Fig. 3.10(b). This

is because the contributions from antibound and bound poles cancel each other out.

In

this case, the structure of the low-energy continuum is not expected to be affected by the

presence of threshold poles.

The situation presented in Fig. 3.10 is rather generic for p-wave dominated resonant

poles.

Increasing the potential range moves the pole trajectory closer to the real-k axis.

Consequently, states containing no s-wave component are likely to appear as isolated narrow

resonances. For odd-multipolarity potentials, a (j = J, (cid:96) = 0) component of a J π state

becomes large as the dissociation threshold is approached, see Sec. 3.3.2. On the other hand,

for even-multipolarity potentials, odd-J states cannot have an s-wave component, as the

molecule’s angular momentum j must be even, and narrow near-threshold resonances can

appear.

31

3.3.3 Rotational motion

Figure 3.11: The rotational band built upon the J π = 0+
1 state of a dipole-bound anion. The
parameters V0 = 5.33 Ry, r0 = a0, and I = 103 mea2
0 have been chosen to place the bandhead
energy slightly below the zero-energy threshold, where the rotational motion of the molecule
can excite the system into the continuum. The energy is plotted as a function of J(J + 1).
Taken from Ref. [5].

To describe multipole-bound anions, one has to take into account the nonadiabatic coupling

between the rotational motion of the molecule and the s.p. motion of the electron. Whether a

multipole-bound anion can exhibit rotational bands depends on multipolarity. For instance,

it was shown in Ref. [58] that rotational bands of dipolar anions do not extend above the

dissociation threshold while a similar study for quadrupole-bound anions [7] demonstrated

that the rotational motion of the anion is hardly affected by the continuum.

32

3--0.02-0.010.000.014+2+0+1-5-detachment thresholdangular momentum Jenergy (Ry)λ=1Figure 3.12: Similar to Fig. 3.11 but for rotational bands built upon the J π = 0+
1 and
−
1
1 bandheads of a quadrupolar Gaussian potential with V0 = 12.38 Ry, r0 = a0, and for
I = 50 mea2

0. Taken from Ref. [5].

0 and I = 100 mea2

Figure 3.11 illustrates the case of a rotational band built upon the subthreshold J π = 0+
1

state of the dipolar Gaussian potential. It is seen that the rotational band is not affected

when the zero-energy threshold is crossed below J = 4.

In the realistic calculation for

dipole-boudn anions [58], it was found that rotational band does not extend beyond the

detachment threhshold, indicating a transition between strong-coupling regimes and weakly-

coupling regimes when cross the detachment threshold. Our result indicates that the presence

of the two coupling regimes predicted to exist in realistic calculations for dipole-bound an-

ions [58] must be due to difficulties in imposing proper boundary conditions at infinity for

the dipolar potential (∼ r−2) when the rotational motion of the molecule is considered nona-

diabatically [82]. Since in the present work the radial part of the dipolar pseudopotential is

replaced by a Gaussian function, the outgoing boundary condition can be readily imposed.

33

-0.20.00.20.4I=50mea20I=100mea20angular momentum Jenergy (Ry)3-4+2+0+1-5-detachmentthresholdλ=2We now investigate the impact of the molecular rotation on the energy spectrum of the

anion. By definition, changing the moment of inertia of the molecule is expected to have

a larger effect on states dominated by channels with large j, but in practice, such channels

are unlikely to dominate at low energies. As an illustrative example, we study the 3

−
1 state

of the quadrupolar (λ = 2) Gaussian potential. Figure 3.13(a,b) shows, respectively, the

−
energy and decay width of the 3
1 resonance as a function of the potential strength and the

inverse moment of inertia.

A similar result is obtained for the quadrupolar case shown in Fig. 3.12 for two rotational

bands built upon the J π = 0+

1 and 1

−
1 bandheads. The existence of rotational bands extend-

ing above the dissociation threshold is consistent with the findings of Ref. [7] employing the

realistic quadrupolar pseudopotential. The results for higher-multipolarity potentials follow

the pattern obtained for the dipolar and quadrupolar cases; hence, they are not shown here.

34

−
1 resonance of the quadrupo-
Figure 3.13: Energy (a) and decay width (b), both in Ry, of the 3
lar Gaussian potential with r0 = a0 as a function of the inverse of the moment of inertia
and the potential strength. The dissociation threshold (E = 0) is indicated. The dominant
(j, (cid:96)) channel is marked in panel (b). When the rotational energy of the molecule Ej=4
rot
−
1 resonance, the (4,1) decay channel is open/closed.
lies below/above the energy of the 3
−
The line Ej=4
1 ) (thick solid) separating these two regimes is marked, so is the line
−
Ej=2
rot = E(3
1 ) (thick dotted) which corresponds to the threshold energy for the opening
of the (2,1) channel. The norms of the two dominant channels (2,1) (solid line) and (4,1)
(dotted line) are shown as a function of V0 for 1/I = 0.04 m−1
(d).
e a
Taken from Ref. [5].

(c) and 0.02 m−1
e a

rot = E(3

−2
0

−2
0

35

0.020.04(a)0.020.041/I(units of   m  −1ea−20 )(4,1)(2,1)(b)0.00.40.8Re(N)(c)(4,1)(2,1)1/I=0.048101214V0(Ry)0.00.40.8(d)1/I=0.020.00.40.8EΓE=0E=0At large values of V0 when the 3

−
1 resonance lies close to the threshold, its wave function is

primarily described in terms of two channels with (j, (cid:96)) = (2, 1) and (4, 1) with the dominant

(2,1) amplitude, see Fig. 3.13(c,d). At a finite value of I, as the energy of the resonance

increases, a transition takes place to a state dominated by the (4,1) component that is

associated with a reduction of the decay width. This transition can be explained in terms

−
of channel coupling. At very low values of 1/I, the energy E(3
1 ) lies above the rotational

4+ state of the molecule. As the moment of inertia decreases, the 4+ member of the g.s.

rotational band of the molecule moves up in energy, and at some value of I it becomes

−
1 ) resonance, i.e., Ej=4
degenerate with the energy of the E(3

−
rot = E(3
1 ). At still higher

values of 1/I, the (4,1) channel is closed to the electron emission. As seen in Fig. 3.13(b),

the irregular behavior in the width of the resonance can be attributed to the (4,1) channel

closing effect [103]. A second irregularity in Fig. 3.13(c,d), seen at large potential strengths,

corresponds to Ej=2

−
rot = E(3
1 ). As the resonance approaches the threshold, its tiny decay

width can be associated with the (0,3) channel. Due to its higher centrifugal barrier, (0,3)

channel contributes around 1% to the total norm in the threshold region.

3.4 Summary

In the chapter, two types of molecular anions approximated by a nonadiabatic electon-

plus-molecule model were studied using the complex-energy BEM within the coupld-channel

formalism, including both quadrupole-bound anions and anions bounded by multipolar Gaus-

sian potentials.

In the qudrupole-bound anions, the quadrupolar potential was generated by a linear

distribution of point charges, where the critical quadrupole moments calculated using both

BEM and DIM are compared with the analytical results. Rotational band for quadrupolar

anions were also predicted to extend above the detachment threshold.

Anions bounded by a multipolar Gaussian potential are expected to describe general

36

trends of near-threshold resonant poles for multipolarities λ ≥ 2. By calculating the thresh-
old lines for anions of different multipolarity, we predicted that within this model, higher-λ

anions can exist as marginally-bound open systems. The role of the low-(cid:96) channels in shap-

ing the transition between subcritical and supercritical regimes has been explored. We

demonstrate the presence of a complex interplay between bound states, antibound states,

subthreshold resonances, and decaying resonances as the strength of the Gaussian potential

is varied. In some cases, we predict the presence of exceptional points. The fact that anti-

bound states and subthreshold resonances can be present in multipolar anions is of interest as

they can affect scattering cross sections at low energy. For Gaussian potentials, the outgoing

boundary condition can be readily imposed. Consequently, the rotational band of the anion

is not affected when the zero-energy threshold is reached. This indicates that the presence of

two coupling regimes of rotation predicted to exist in realistic calculations for dipole-bound

anions [58] must be due to specific asymptotic behavior of the dipolar pseudo-potential in the

presence of molecular rotation. The non-adiabatic coupling due to the collective rotation of

the molecular core can give rise to a transition into the supercritical region. We also predict

interesting channel-coupling effects resulting in variation of an anion’s decay width due to

rotation.

In summary, by looking systematically at the pattern of resonant poles of multipole-

bound anions near the electron detachment threshold we uncover a rich structure of the

low-energy continuum. These simple systems are indeed splendid laboratories of generic

phenomena found in marginally-bound molecules and atomic nuclei. It’s also noted our work

has promoted some experimental explorations of dipolar and quadrupolar anions [104, 105].

Besides providing guidance for experimental searching for the weakly bound anions, the study

on the generic properties of resonant states near the threshold can also help understand OQSs

in different areas.

37

Chapter 4

Lithium isotopes and mirror nuclei

4.1

Introduction

In the light-nuclei region of the nuclear landscape, the imbalance between proton and neu-

tron numbers can reach high values, which is susceptible to continuum effects due to the

presence of low-lying decay channels. Wave functions of such systems often “align” with the

nearby threshold and are expected to have substantial overlaps with the corresponding decay

channels. Among them, lithium isotopes are of particular interest as they reveal rich phe-

nomena, including the binary cluster (α+d) of 6Li [106, 107], the anitbound (virtual) state

of 10Li [44, 45, 46, 47], as well as the spatially extended halo structure of 11Li [18, 25, 108].

However, such OQSs pose many challenges for nuclear theory, since drip line nuclei can

not be described in a typical HO-based configuration interaction method. As a result, for

instance, the theoretical descriptions of 11Li are usually based on a few-body approximation,

including Faddeev equation [109] and other similar techniques [110, 111, 112, 113]. In this

case, 11Li is described as a loosely bound three-body system (9Li core and two valence

neutrons), due to its Borromean property [18, 25, 108], in which none of the two-body

subsystems is bound.

However, the core polarization effects can be large for some nuclei, where a cluster

approximation might not be sufficient, therefore, a more elaborated approach is required.

To this end, we adopt GSM based on the BEM technique introduced in Chapters 2 and 3,

which can give a comprehensive description of the interplay between many-body correlation

and continuum coupling.

38

In this work, we study the lithium isotopes (6−11Li) with a realistic residual interaction

including uncertainty estimation of the Hamiltonian parameters as well as predicted spectra.

Another interesting aspect existing only in nuclear OQSs is the asymmetry between

proton and neutron thresholds. This can result in different asymptotic behavior of proton

and neutron wave functions resulting in the Thomas-Ehrman effect (TEE) [114, 115]. To

study the TEE, the mirror partner of lithium isotopes have also been studied in this work.

4.2 Gamow shell model

4.2.1 Hamiltonian

The lithium isotopes and their mirror partners are studied in terms of valence nucleons

coupled to the 4He core. This is justified by the strongly bound nature of 4He, whose first

excited state is 20.21 MeV above the g.s. [1]. In this picture, the Hamiltonian is given as a

sum of an s.p. core-nucleon potential and effective two-body interactions among the valence

nucleons. In the intrinsic frame of the Cluster Orbital Shell Model (COSM) [116], where

the nucleon coordinates are defined with respect to the center of mass of the core, the GSM

Hamiltonian is expressed as:

n(cid:88)

i

(cid:20) p2

i
2µi

(cid:21)

+ Ucore(i)

+

n(cid:88)

i=1,i<j

(cid:20)

Vi,j +

(cid:21)

pipj
Mcore

(4.1)

H =

where n denotes number of valence nucleons, µi and Mcore are the reduced mass of valence

nucleon and core, respectively, and Ucore, Vi,j are the core-nucleon potential and nucleon-

nucleon interaction, respectively. The last term in Eq. (4.1) is the recoil term in the COSM

coordinates, which is used to eliminate the energy contribution from the center-of-mass

motion. In GSM, to account for the many-body correlations, Slater determinants are built

upon the discretized s.p. Berggren basis states of each shell to serve as the many-body basis

within which the complex-symmetric H is diagonalized [4].

39

Core potential

The core-nucleon potential is taken as a Woods-Saxon (WS) field, with a central and spin-

orbit term and Coulomb field for protons.

Ucore(r) = V0f (r) − 4V(cid:96)s

1
r

df (r)

dr

(cid:96) · s + UCoul(r),

(4.2)

where f (r) = −(1 + exp[(r − R0)/a])−1. The WS radius and diffuseness parameters were

taken from Ref. [117]: R0(n) = 2.15 fm, R0(p) = 2.06 fm, a(n) = 0.63 fm, and a(p) = 0.64

fm. The Coulomb potential is generated by a spherical Gaussian charge distribution with

the radius Rch = 1.681 fm [118]. The strength of the central and the spin-orbit part of the

potential, represented as V0 and V(cid:96)s, respectively, are optimized in the work with respect to

the selected states.

Two-body interaction

The effective two-body interaction is constructed based on the finite-range Furutani-Horiuchi-

Tamagaki (FHT) force [117, 119, 120], which has been shown to successfully describe struc-

ture and reactions involving light nuclei [121, 117, 122, 123, 124].

The FHT interaction contains central (c), spin-orbit (LS), tensor (T ) and Coulomb

terms:

V = Vc + VLS + VT + VCoul.

(4.3)

The central and tensor parts are both a sum of three Gaussian functions with different

ranges representing the short, intermediate and long ranges of nucleon-nucleon interaction.

The spin-orbit part is a sum of two Gaussian functions [117]. In order to be applied in the

present GSM formalism, the interaction is rewritten in terms of the spin-isospin projectors

40

ΠST :

Vc(r) =V 11

c f 11

c (r)Π11 + V 10

c f 10

c (r)Π10

+ V 00

c f 00

c (r)Π00 + V 01

c f 01

c (r)Π01,

VLS = (L · S)V 11

LSf 11

LS(r)Π11,

VT (r) = Sij[V 11

T f 11

T (r)Π11 + V 10

T f 10

T (r)Π10],

(4.4)

(4.5)

(4.6)

with the seven interaction strengths V ST

η

(η = c, LS, T ) in different spin-isospin channels,

remaining to be adjusted to experimental data and f ST

η

are the unitless form factors [117].

r ≡ rij stands for the distance between the nucleon i and j, ˆr = rij/rij, L is the relative
orbital angular momentum, S = (σi + σj)/2, and Sij = 3(σi · ˆr)(σj · ˆr) − σi · σj.

In Ref. [117], the FHT interaction was used in the GSM description of bound and

unbound nuclei with A ≤ 9. While a good energy reproduction was achieved, the systematic
statistical study of the parameters carried out in Ref. [117] demonstrated that some of the

terms in the FHT interaction were sloppy, i.e., not well constrained. According to this, we

used a simplified version of the FHT interaction where the central V 10

c

, V 01

c

, and tensor

V 10
T

terms have been considered. This choice is also justified by the Effective Field Theory

(EFT) arguments [125, 126, 127, 128, 129].

Indeed, in the EFT expansion of the bare

nucleon-nucleon interaction, these three terms appear at leading order, whereas the other

terms present in the original FHT interaction correspond to higher orders of EFT. However,

we have observed that adding the central term V 00

c

improves the overall description of the

nuclei considered in this work and hence we have also included it in Vi,j.

As it is customary in shell model studies [130, 131], a mass-dependent interaction-scaling

factor of the form (6/A)α is introduced for the two-body interaction to effectively account

for the missing three-body forces [132, 133], where the factor is multiplied to all the two-

body interaction terms in Eq. 4.3. We found that the value α = 1/3 gives a very reasonable

description of experimental energies.

41

Finally, the Coulomb interaction between valence protons is treated by incorporating its

long-range part into the basis potential and expanding the short-range two-body component

in a truncated basis of HO states [134, 135].

4.2.2 Interaction optimization

In order to provide predictions, the interaction parameters have been optimized through a

fitting process, which consists in the minimization of the χ2 penalty function

(cid:18)

Nd(cid:88)

i=1

Oi(p) − Oexp

i

δOi

(cid:19)2

χ2(p) =

,

(4.7)

where Nd is the number of observables and p is a vector of parameter used. O(p) and Oexp
are the calculated value for observables and experimental values, respectively. δOi is the
adopted error that has been estimated by normalizing the penalty function to the number
of degrees of freedom Ndof = Nd − Np at the minimum χ2(p0) [136]. In the case of a single
type of data and assuming that experimental and numerical errors are negligible, this can be

i

χ2(p0)/Ndof [117].
achieved through a global scaling of the initial adopted errors δOi → δOi
This renormalization guarantees that χ2 ∼ 1, which is necessary for calculated statistical
uncertainties to be meaningful [137].

(cid:112)

Interactions used in this model are linear in strength parameters, which enables us to

calculate the first derivative (Jacobian J) exactly using Hellmann-Feynman theorem [138],

(cid:12)(cid:12)(cid:12)(cid:12)p0

.

Jiα =

∂Oi
∂pα

1
δOi

The covariance matrix C can be approximately expressed in terms of Jacobian J:

−1
C (cid:39) (J T J)

The uncertainty for each parameter is given by the diagonal elements in the covariance

42

(4.8)

(4.9)

matrix C, and the correlation between parameters α and β is given by [117]

Cαβ(cid:112)CααCββ

cαβ =

.

(4.10)

In the situation where the Jacobian matrix is noninvertible or has a very small deter-

minant, the Gauss-Newton method used in the minimization of χ2 becomes unstable as the

Jacobian matrix must be inverted. This typically happens when some parameters are sloppy,

i.e., not well constrained by observables. To stabilize the calculation, the matrix inversion is

replaced by its pseudo-inverse, derived from the singular value decomposition (SVD) of the

Jacobian matrix [117].

Only well-established states are used for interaction optimization to give a solid basis

for this work, where both bound states and weakly bound states are both included. Weakly

bound states, such as g.s. of 11Li, are included so that the optimized interaction can be used

for extrapolation of states where continuum coupling is important.

The previous work [117] provides a good starting point for the core potentials and two

body interactions, which are fitted to the nucleon-4He phase shifts and selected states of

nuclei with mass number A ≤ 9, respectively. Only four parameters of two-body interaction
were well-constrained through statistical analysis and thus adopted in this work. An addi-

tional tuning of four strengths of the core potential was found to improve the optimization

quality further.

In this work, four strengths of the WS potential and four parameters of the two-body

interaction are thus simultaneously optimized to reproduce fifteen energy levels of lithium

isotopes and their mirror partners.

4.2.3 Model space

The calculations were performed in a model space which includes s1/2, p3/2 and p1/2 partial

waves for both protons and neutrons. Since the optimization involved energies only, for the

43

sake of speeding-up the optimization and for better stability, we used a deeper WS potential

to generate the basis, in which the 0p3/2 and 0p1/2 poles are bound. A real contour was
then used to describe the non-resonant continuum space. The contour L+ for Berggren
basis was divided into 3 segments with kpeak = 0.25 fm−1, kmid = 0.5 fm−1, and the cutoff
momentum kmax = 4 fm−1. Discretizing each segment with 10 points using the Gauss-

Legendre quadrature guaranteed the convergence of results.

To calculate resonance’s width, one has to generate a basis based on a much shallower

basis-generating WS potential, in which the 0p3/2 and 0p1/2 poles are decaying resonances.

In this case, a complex contour defined by a complex value of kpeak is employed.

It is

to be noted that the calculation of the width is more demanding than that of energy. A

higher discretization of (20,20,20) is needed for the contour. Due to the Coulomb repulsion,

the mean-field used to generate the s.p. basis for proton-rich nuclei varies with proton

number. The contour was adjusted separately for each system to assure that the Berggren

completeness relation is met. To ensure the numerical stability, the chosen contour should

neither lie too close to the Gamow poles nor lie too far from the real-k axis. In this work,
kpeak is chosen to lie slightly (0.05 fm−1) below the position of the 0p3/2, 0p1/2 poles, but with
the imaginary part greater than −0.2 fm−1. The calculations were repeated with several

slightly different values of kpeak to assure the full convergence of results.

As in any configuration interaction models, the dimension of the Hamiltonian matrix

grows quickly with the number of active particles. In the context of the GSM, it increases

more quickly than in the conventional shell model due to the presence of discretized scat-

tering states. A more adapted basis is thus needed. The natural orbitals are defined as

the eigenvectors of the one-body density matrix [139], and have widely used in no-core shell

model [140]. In this work, we truncated the model space by working with natural orbitals

which provides an optimized set of s.p. states [117, 141, 139]. Natural orbitals are first

calculated in a truncated configuration space which permits only three valence particles in

the continuum shells. A truncation is then performed on the s.p. basis by keeping only

44

natural orbitals for which the modulus of the occupation number is greater than a certain
value (10−6). Finally, a new set of Slater determinants was constructed and served as the

basis for the diagonalization of the Hamiltonian matrix using the Davidson method [142].

To check the accuracy of this truncation procedure in the case of the largest systems,

computation of the energy was also performed using the Density Matrix Renormalization

Group (DMRG) method [143, 144]. The DMRG allows performing calculations without

truncation on the s.p. basis and without restrictions on the number of particles in the

continuum. In this approach, the many-body Schr¨odinger equation was solved iteratively in

tractable truncated spaces which were increased until numerical convergence was reached.

We have checked that, in all cases discussed in this work, the GSM results were in good

agreement with those of DMRG.

4.3 Results

4.3.1 Optimized states

To optimize our interaction, we used experimentally well-established levels, shown in Ta-
ble 4.1 and marked with (cid:70) in Figs. 4.1 and 4.2 respectively. The experimental data are

taken from Ref. [1] unless otherwise specified. All energies are considered relative to the g.s.

energy of 4He.

As can be seen in Table 4.1, very good consistency with experimental results has been

achieved for the fifteen selected states. The rms deviation from experimental values is
160 keV. The largest discrepancy is obtained for the 1/2− states of 7Li and 7Be, with a

deviation of 340 keV from the data. All other states are within the range of 170 keV from

experiment.

We also want to point out that three states lying close to the threshold are also well re-

produced, including: (i) the 0+ state of 6Li which is 140 keV below the threshold of 4He+n+p;

(ii) the g.s. of halo nucleus 11Li lying 369 keV below the two-neutron emission threshold;

45

Table 4.1: Energy levels used in the GSM Hamiltonian optimization. The energies are given
with respect to the 4He g.s.. The experimental values Eexp are taken from Ref. [1]. They are
compared to the GSM values EGSM.
Nucleus State Eexp (MeV) EGSM (MeV) Nucleus State Eexp (MeV) EGSM (MeV)

6Li

7Li

8Li

9Li

11Li

1+
0+
3/2-
1/2-
2+
1+
3/2-
1/2-
3/2-

-3.70
-0.14
-10.95
-10.47
-12.98
-12.00
-17.05
-14.35
-17.41

-3.72
-0.10
-11.02
-10.14
-13.14
-11.93
-16.90
-14.50
-17.48

7Be

8B

9C

3/2-
1/2-
2+
1+
3/2-
1/2-

-9.30
-8.88
-9.44
-8.67
-10.74
-8.52

-9.36
-8.53
-9.60
-8.50
-10.85
-8.59

(iii) and the 2+ state of 8B being 140 keV away from one proton decay.

4.3.2 Optimized Interaction

The optimized values of the parameters for the WS potentials and the two-body interaction

are displayed, along with their statistical uncertainties, in Tables 4.2 and 4.4, respectively.

As one can judge from the small parameter uncertainties in Tables 4.2 and 4.4, the GSM

Hamiltonian is rather well constrained. As expected, the central term V 00

c has the largest

uncertainty of ∼12%.

Table 4.2: Central and spin-orbit strengths of the core-nucleon WS potential optimized in
this work. The statistical uncertainties are given in parentheses.
Neutrons Protons
42.1 (4)
39.5 (2)
10.7 (2)
11.1 (5)

Parameter
V0 (MeV)
V(cid:96)s (MeV·fm2)

Table 4.3: Groud state energies (in MeV) and widths (in keV) of 5He and 5Li obtained from
the optimized core-nucleon potential and compared to experiments [2, 3].

Nucleus EGSM Eexp
5He
0.798
5Li
1.69

0.74
1.6

ΓGSM Γexp
640
648
1230
1300

46

Table 4.4: Strengths V ST
uncertainties are given in parentheses.

η

of the two-body interaction optimized in this work. The statistical

parameter
V 01
(MeV)
c
V 10
(MeV)
c
V 00
(MeV)
c

V 10

T (MeV·fm−2)

value

-9.425 (70)
-8.309 (90)
-8.895 (1130)
-22.418 (970)

Compared with the core-nucleon potential parameters obtained by fitting proton-4He

and neutron-4He phase shifts in Ref. [117], the strength parameter of core potentials for both

protons and neutrons are shallower, which push p-shell poles up further into the continuum.

Moreover, the spin-orbit strength is larger than that in Ref. [117], which separates the 0p3/2

and 0p1/2 poles further apart. Consequently, the 0p3/2 pole appears at similar energy as in

Ref. [117] and the 0p1/2 pole lies high above in the continuum. To assess the quality of the
newly obtained interaction, the energy and width for the g.s. (3/2−) of both 5He and 5Li are

calculated and shown in Table 4.3. These values are indeed very close to the experimental

values.

The new core potential is not optimal in describing 0p1/2 resonant states of 5He and 5Li,

whose physical importance decrease with its large decaying width. But this does not extend
to heavier nuclei. For example, the 1/2− state of 7Li and 7Be are mainly determined by the

0p1/2 pole and the obtained results are still quite close to the experimental data.

4.3.3 Test against other excited states

To check the quality of our optimized interaction, states not included in the optimization

were calculated and compared with experiment. Results are shown in Figs. 4.1 and 4.2. To

distinguish from the states used for optimization, states not entering the optimization are

shown together with their energies displayed; the widths are marked by the shaded boxes.

The corresponding energies and statistical uncertainties of predicted states are listed in

Table 4.5. As one can see, the optimized interaction not only allows for a good reproduction

47

Figure 4.1: Energies for states of Li isotopes with respect to 4He. Red lines denote GSM
results and the black lines mark experimental values. The shaded area represents the width
of the corresponding resonance. States used for optimization are marked with a (cid:70), their
energies are listed in Tables 4.1 and 4.5.

48

0.00.20.40.60.81.00.00.20.40.60.81.06Li7Li8Li9Li10Li11Li−16−14−12−10−8−6−4−202Energy(MeV)1+3+0+2+2+-1.571.782.23-1.510.61.673/2−1/2−7/2−5/2−5/2−-6.04-3.50-1.73-6.3-4.34-3.49GSMexp2+1+3+1+-10.59-8.9-10.73-9.773/2−1/2−5/2−-12.64-12.751+2+1−2−-16.22-16.55-15.5-16.78-16.54-15.553/2−−16.5−16.010LiGSMexp1999exp20151+2+1+2+1+2+of experimental energies for the fifteen selected states but also gives satisfactory results

for excited states not included in the fit. For instance, the calculated 3+ state in 6Li at

−1.57 MeV is only 60 keV below the experimental energy. The experimental widths for the
second 5/2− state in 7Li (89 keV) and 5/2− state in 9Li (88 keV) are very reasonable: the

calculated values are 22 and 62 keV, respectively.

The 5/2− and 7/2− excited states in 7Be are slightly (>300 keV) above the correspond-
ing experimental values, whereas the position of the 3+ state in 8B and 5/2− state in 9C
are well reproduced. It is also worth to mention that the second excited state 5/2− of 9C is

in good consistency with the R-matrix analysis and continuum shell model calculations in

Ref. [145]. The calculated width for 5/2− state is 340 keV and very close to the value 673 ±

50 keV extracted from experiment [146].

Table 4.5: Energy levels for states not entering the optimization. The experimental values
Eexp are taken from Ref. [1]. The GSM values EGSM are shown with the uncertainties in
parentheses.
Nucleus State Eexp (MeV) EGSM (MeV) Nucleus State Eexp (MeV) EGSM (MeV)

6Li
7Li
8Li
9Li
10Li

3+
7/2−
3+
5/2−
2+
1+

-1.51
-6.3
-10.73
-12.75
-16.78
-16.54

-1.57(2)
-6.04(2)
-10.59(2)
-12.64(2)
-16.55(5)
-16.22(5)

7Be
8B
9C
10N

7/2−
3+
5/2−
1−
2−

−4.73
-7.12
-7.14
-8.84
-7.94

−4.47(2)
-7.11(2)
-7.12(5)
-8.93(6)
-8.46(6)

In general, we do not expect the same quality of data reproduction for all excited states

due to the fact that the higher partial waves with (cid:96) ≥ 2, which may contribute to the wave
functions of those states, are not included in the model space. The estimated statistical

uncertainties on the predicted energies are small: in most cases they are in the range of 20

– 60 keV.

49

Table 4.6: Root-mean-square proton (Rp) and neutron (Rn) radii of 6Li, 7Li/7Be, 8Li/8B,
9Li/9C, and 11Li (in fm).

Nucleus State Rp/Rn(fm) Nucleus State Rp/Rn(fm)

6Li
7Li
8Li
9Li
11Li

1+
3/2−
2+
3/2−
3/2−

2.22/2.21
2.12/2.28
2.12/2.48
2.12/2.53
2.12/2.81

7Be
8B
9C

3/2−
2+
3/2−

2.26/2.09
2.57/2.14
2.62/2.09

4.3.4 Root-mean-square radius

One of the key features to identify halo nucleus is the rms radius. To give an estimation, Ta-

ble 4.6 shows rms proton/neutron radius for lithium isotopes and their mirror partners [147]:

< r2 >=

2

Nval + 2

< r2

core > +

Nval

Nval + 2

< r2

i >,

(4.11)

where the first and second term corresponds to the contribution from both the core and

valence particles, respectively, with Nval being the number of valence protons/neutrons. The

experimental value of 1.67 fm is used for the rms radius of both proton and neutron in the

4He core [148]. Valence protons and neutrons are treated as point-like particles, whose sizes

are not taken into account.

Since the corrections, such as spin-orbit effects and core swelling [147], are not included

in our calculation, we do not intend to compare our results directly against to the experi-

mental charge radii.

By comparing with other lithium isotopes, a dramatical increase of the rms neutron

radius of 11Li can be seen from Table 4.6, which indeed indicates the halo structure in 11Li.

Since all considered Li isotopes have only one well-bound valence proton, the proton radii

are almost constant for each Li isotopes.

As to the proton-rich side, a sharp increment can be seen clearly for the rms proton

radii for 8B and 9C as compared to those of 6Li and 7Be, suggesting that both 8B and 9C

50

Figure 4.2: Similar to Fig. 4.1 for results of mirror nuclei of Li isotopes. Energies are given
with respect to g.s. of 4He. Experimental energy of the 5/2− resonance in 9C was taken
from Ref. [8] and the data for 11O is from Ref. [9].

are indeed good candidates for proton halo nuclei due to their weakly bound properties

[149, 150, 151].

4.3.5 Prediction for unbound nuclei: 10Li, 10N and 11O

10Li

Several experiments [44, 45, 46, 47] and theoretical calculations [152, 93] have indicated

that the structure of the g.s.

in 10Li may correspond to a neutron in a virtual (cid:96) = 0 state

above a 9Li core. However, the presence of such a virtual state near the threshold has

not been confirmed in a recent experiment [153] with higher statistics, see the theoretical

analysis [154, 155]. This indicates that the situation in 10Li is still not well understood.

We wish to note, however, that a virtual state in 10Li cannot be associated with an

energy level of the system; the appearance of such a state in the complex-momentum plane

51

0.00.20.40.60.81.00.00.20.40.60.81.07Be8B9C10N11O−10−8−6−4−2Energy(MeV)3/2−1/2−7/2−5/2−-4.47-2.3-4.73-2.572+1+3+-7.11-7.12GSMexp3/2−1/2−5/2−-7.12-7.141−2−2+1+-8.93-8.46-8.12-7.9-8.84-7.943/2−5/2+manifests itself through a low-energy enhancement of the n+9Li cross-section. For that

reason, we limited our calculations to resonant states in 10Li that can be interpreted as

experimentally-observable resonances.

The computed 2+ g.s. and first excited state 1+ of 10Li lie at 0.35 and 0.68 MeV above
the n+9Li threshold, respectively. The next excited states are the degenerate 1− and 2−

states at 1.05 MeV. Due to the strong coupling to the continuum in these states, using the

WS potential of the GSM Hamiltonian to generate the basis would make the numerical

computation unstable. To achieve the numerical stability, the computations were performed

by a deeper WS potential. We have checked that our predicted energies do not depend on the

choice of the WS used for the construction of the basis: in all cases considered, the variation

of the energy did not exceed 1 keV. On the other hand, the computed width associated with

the states is of the order of a few hundred keV and consequently not stable. For that reason,

we do not show them in Fig. 4.1.

To shed light on the structure of 10Li, Table 4.7 lists the squared amplitudes of the

dominant neutron configurations for the four low-lying states of 10Li. The positive parity

states are primarily made from the 0p3/2 and 0p1/2 resonant shells. The negative parity states

contain one neutron in the 1s1/2 shell. The contribution from the non-resonant continuum

space to the low-lying states is very small.

Experimentally, Ref. [156] observed two positive-parity states at 0.24 MeV and 0.53 MeV

above the n+9Li threshold [156], with the lower state assigned to be 1+. This spin assignment

contradicts the results in Ref. [157] where the lowest positive parity state was assigned to
be a 2+, see the inset in Fig. 4.1. In our prediction, the computed position of 1−, 2− are in

agreement with the observation of a negative-parity state at 1.05 MeV from Ref. [153].

Unbound 10N

10N is the unbound mirror partner of 10Li, lying beyond proton dripline. The spectrum of

10N is not experimentally known with certainty.

In Fig. 4.2, we show the tentative level

52

Table 4.7: Squared amplitudes of dominant configuration of valence neutrons and protons
for low-lying levels of 10Li and 10N, respectively. Energies with respect to the one-nucleon
emission threshold are shown in the parentheses for each state. The odd proton in 10Li and
the odd neutron in 10N occupy the 0p3/2 Gamow state. The tilde sign labels non-resonant
continuum components.

configuration

(0p3/2)4(0p1/2)1
(0p3/2)3(0p1/2)2

(0p3/2)4(1s1/2)1

(0p3/2)4((cid:103)s1/2)1
(0p3/2)3(0p1/2)1((cid:103)s1/2)1
(0p3/2)2(0p1/2)2((cid:103)s1/2)1

(0p3/2)3(0p1/2)1(1s1/2)1

(0p3/2)2(0p1/2)2(1s1/2)1

10Li

10N

2+ (0.35 MeV)

1+ (0.68 MeV)

2+(2.73 MeV)

1+(2.95 MeV)

0.84
0.10

1−(1.4 MeV)

0.81
0.06

2−(1.4 MeV)

0.81
0.10

1−(1.92 MeV)

0.78
0.05

2−(2.39 MeV)

0.72

0.14

0.07

0.73

0.14

0.07

0.44
0.29
0.09
0.06
0.04
0.03

0.37
0.35
0.07
0.07
0.03
0.03

assignments used in Ref. [1]. According to Refs. [158, 159], the g.s. of 10N is most likely a 1−

state in the energy range from 1.81 to 1.94 MeV. In a more recent work [160], they observed

two low-lying negative-parity states but they were not able to assign J π values.

Due to the presence of the Coulomb barrier, the 1s1/2 single-proton state is a broad

resonance rather than a virtual state [9, 161]. To capture this state, a complex contour is

used with kpeak = (0.25,−0.05) fm−1. Our calculations for 10N predict the g.s. to be a 1−
state with (E, Γ) = (−8.93, 0.9) MeV that lies 1.92 MeV above the 1p threshold. The first
excited state is predicted to be a 2− state with Γ = 0.3 MeV slightly below the value quoted

in Ref. [160]. This result is consistent with the recent Gamow coupled-channel analysis of

Ref. [161]. We also predict an excited 1+ state with Γ=0.3 MeV, lying 2.9 MeV above the

9C+p threshold, which is consistent with Refs. [162, 163, 164]. A second positive-parity state

with J π = 2+ is also predicted at 0.81 MeV with a width of 0.36 MeV.

Table 4.7 shows the squared amplitudes of the dominant proton configurations for the

four low-lying states of 10N. Similar to 10Li, the positive parity states are primarily made

from the 0p3/2 and 0p1/2 resonant shells. The dominant configurations of negative parity

states contain one (cid:96) = 0 proton, which can either be in the 1s1/2 shell or in a non-resonant

53

continuum state.

Unbound 11O

11O is a 2p-emitter as the mirror partner of the 2n-halo nucleus 11Li. With one more proton

above 10N, 11O is more unbound, which makes it more challenging to study experimentally.

The first observation of 11O was done recently with two-neutron knockout reactions of 13O

beams at NSCL [9]. A broad peak with a width of 3.4 MeV was observed which was inter-

−
−
2 , 5/2+
preted in terms of four overlapping resonances:J π = 3/2
1 , 3/2

1 , 5/2+

2 [9, 161]. The

−
1 and 5/2+
3/2

1 states are 4.16 MeV and 4.65 MeV above 9C+2p threshold and the widths

are 1.3 MeV and 1.06 MeV respectively.

−
In this work, we predicted the g.s. 3/2
1 at 4.85 MeV with a width of 0.13 MeV and a

first excited state 5/2+

1 at 5.03 MeV with the width of about 1 MeV. These predictions are

consistent with the Gamow coupled-channel calculations of Refs. [9, 161].

4.3.6 Continuum effects on the Thomas-Ehrman shift

The Thomas-Ehrman shift [114, 115] reflects the energy shift between the mirror pair of

nuclei primarily due to the Coulomb repulsion. To study the effect of particle continuum

due to different positions of particle thresholds in mirror partners, we compared the level

schemes of Li isotopes and their mirror partners with mass number A = 7, 8, 9, 10, from

well-bound states to unbound resonances high above the threshold. Results are shown in

Fig. 4.3. Within each pair, the ground states agree.

This agreement holds for excited states as long as both are bound, as can be seen for
the 3/2−,1/2− and 7/2− states of A = 7 nuclei in Fig. 4.3(a). Moving up to higher energy,
the 5/2− state of 7Li and 7Be are both above the one-nucleon emission threshold. The 5/2−

level of proton-rich nuclei 7Be is lower than that of the neutron-rich partner 7Li.

A similar trend can also be seen in results for the 8Li/8B and 9Li/9C pairs in Fig. 4.3(b,c).

As discussed in Sec. 4.3.4, 8B and 9C are both likely to be halo nuclei having large spatial

54

Figure 4.3: Spectrum for Li isotopes and their mirror partner with mass (a) A = 7, (b)
A = 8, (c) A = 9, (d) A = 10. Within each pair, the spectrum of Li isotope and its mirror
are plotted whin the same scale and different range. The plots are shifted so that the g.s.
of each pair align with each other. The one-proton/neutron emission thresholds are also
marked within each plot.

55

..2..7Li7Be02463/2−1/2−7/2−5/2−6Li+n6Li+p8Li8B0122+1+3+7Li+n7Be+p9Li9C012343/2−1/2−5/2−8Li+n8B+p10Li10N-2-1012+1+1−/2−1−2−2+1+9Li+n9C+p(a)(b)(c)(d)Energy (MeV)extensions.

The 10Li/10N pair is the most interesting one as both nuclei lie above the particle

threshold. As seen in Table 4.7, the effect of the very low 9C+p threshold in 10N on the
negative-parity states 1− and 2− containing the s-wave proton is huge: it results in a rather

dramatic shift of both negative parity states when going from 10Li to 10N that gives rise to

a different structure of low-lying resonances in these nuclei.

56

Chapter 5

Conclusion and perspectives

This thesis is devoted to the study of OQSs with the complex-energy methods and proper

description of continuum coupling. By studying both atomic anions and lithium isotopes

(including their mirror partners), this work addresses important problems in the field of

OQSs in atomic physics and nuclear physics.

In the atomic domain, quadrupolar anions and anions bounded by the multipolar Gaus-

sian potentials were simulated using the complex-energy electron-plus-molecule model. The

coupling between the rotational motion of the molecule and the valence electron near the

dissociation threshold has been studied. By analyzing the rotational bands extended above

the dissociation threshold, we predicted that spatial charge distribution does not affect the

coupling between the molecule’s rotational motion and electron’s motion.

As for anions bounded by multipolar Gaussian potential, effects of the low-(cid:96) channels

on the trajectory of resonant states in the complex-momentum plane are revealed. By

increasing potential strength, a resonance with a moderate contribution from s-wave can

become a subthreshold resonance, and then an antibound state that eventually becomes a

bound state. This demonstrates that antibound states and subthreshold resonances, which

can not be observed in experiments, can profoundly affect the structure of OQSs.

In the nuclear physics area, the properties of lithium isotopes and their mirror partners

have been studied with an optimized FHT interaction. This interaction has been devel-

oped by fitting energies to fifteen well-established states of lithium isotopes and their mirror

partners, with an rms deviation from experiments of 160 keV. Statistical analysis of the

interaction parameters indicates the statistical uncertainty less than 12%. Calculations for

57

other excited states demonstrate the good predictive power of the new interaction. Predic-

tions have been make for the spectra of very exotic nuclei 10Li, 10N and 11O. A very large

Thomas-Ehrmann effect has been predicted for 10Li/10N pair.

In summary, in this work, I applied the complex-energy method to study OQSs, from

the simple one-body system (atomic anions) to the complicated many-body systems (lithium

isotopes and their mirror partners). The success of depicting lithium isotopes highlights the

ability of GSM to describe both well-bound nuclei and exotic nuclei systematically. With

interactions extracted from well-bound systems and systems near the threshold, reliable

predictions can be made for exotic isotopes lying above the threshold. More solid and

insightful works are expected to help understand other exotic nuclei. One possibility for

future work is to extend this work to other isotopic chains such as boron and beryllium

isotopes. By extracting interaction information from well-established states of each isotopic

chain, extrapolations can be made for the more exotic boron and beryllium isotopes. Also,

since boron and beryllium isotopes have more valence protons than the lithium isotopes,

proton-neutron pairing is expected to play a more important role. By comparing different

interactions adapted to each isotopic chain, a better understanding of the interactions is

expected.

Besides 4He, other nuclei, such as 22O, can also serve as the core, with which heavier

isotopes like oxygen, fluorine isotopes are accessible. This work focused on the spectra of

the studied nuclei and also explored the size of halo nuclei. Other anticipated applications

include two-nucleon radioactivity, clustering, and isospin violation in dripline nuclei.

With the current high-performance computing facilities, this work was able to extend

calculations for a nucleus core with up to seven valence nucleons. Calculations with more

valence nucleons in a larger space are still inaccessible. Algorithms that can better take

advantage of the modern computational facilities, such as GPUs, are also expected to benefit

the understanding of OQSs.

58

APPENDIX

59

List of Publications

1. K. Fossez , X. Mao, W. Nazarewicz, N. Michel, W. R. Garrett, and M. P(cid:32)loszajczak.

Resonant spectra of quadrupolar anions. Phys. Rev. A, 94:032511, 2016. (discussed

in Chapter 3)

• Performed the calculation for rotational bands.
• Contributed to the rotational bands part of the draft.

2. X. Mao, K. Fossez, W. Nazarewicz. Resonant spectra of multipole-bound anions.

Phys. Rev. A, 98:062515, 2018.(discussed in Chapter 3)

• Developed a C++ code for the Gaussian potential part.
• Performed the calculations, analyzed results, and produced figures.
• Wrote the first draft of the manuscript.

3. X.Mao, J.Rotureau, W.Nazarewicz. Gamow shell model description of Li isotopes

and their mirror partners. arXiv:2004.02981, 2020. (discussed in Chapter 4)

• Carried out all the calculations, including the interaction optimization and GSM

studies of individual nuclei .

• Developed Python scripts to generate all the plots.
• Selected experimental data and researched literature.
• Wrote the first draft of the manuscript.

60

BIBILOGRAPHY

61

BIBLIOGRAPHY

[1] http://www.nndc.bnl.gov/ensdf, 2015.

[2] D. R. Tilley, C. M. Cheves, J. L. Godwin, G. M. Hale, H. M. Hofmann, J. H. Kelley,
C. G. Sheu, and H. R. Weller. Energy levels of light nuclei A = 5, 6, 7. Nucl. Phys. A,
708:3, 2002.

[3] http://www.tunl.duke.edu/nucldata/, 2015.

[4] N. Michel, W. Nazarewicz, M. P(cid:32)loszajczak, and T. Vertse. Shell model in the complex

energy plane. J. Phys. G, page 013101, 2009.

[5] X. Mao, K. Fossez, and W. Nazarewicz. Resonant spectra of multipole-bound anions.

Phys. Rev. A, 98:062515, 2018.

[6] W. R. Garrett. Critical electron binding to linear electric quadrupole systems. J.

Chem. Phys., 128:194309, 2008.

[7] K. Fossez, X. Mao, W. Nazarewicz, N. Michel, W. R. Garrett, and M. P(cid:32)loszajczak.

Resonant spectra of quadrupolar anions. Phys. Rev. A, 94:032511, 2016.

[8] G. V. Rogachev, J. J. Kolata, A. S. Volya, F. D. Becchetti, Y. Chen, P. A. DeYoung,
and J. Lupton. Spectroscopy of 9C via resonance scattering of protons on 8B. Phys.
Rev. C, 75:014603, 2007.

[9] T. B. Webb, S. M. Wang, K. W. Brown, R. J. Charity, J. M. Elson, J. Barney, G. Cer-
izza, Z. Chajecki, J. Estee, D. E. M. Hoff, S. A. Kuvin, W. G. Lynch, J. Manfredi,
D. McNeel, P. Morfouace, W. Nazarewicz, C. D. Pruitt, C. Santamaria, J. Smith,
L. G. Sobotka, S. Sweany, C. Y. Tsang, M. B. Tsang, A. H. Wuosmaa, Y. Zhang, and
K. Zhu. First observation of unbound 11O, the mirror of the halo nucleus 11Li. Phys.
Rev. Lett., 122:122501, 2019.

[10] T. Berggren. On the use of resonant states in eigenfunction expansions of scattering

and reaction amplitudes. Nucl. Phys. A, 109:265, 1968.

[11] T. Berggren and P. Lind. Resonant state expansion of the resolvent. Phys. Rev. C,

47:768, 1993.

[12] A. B. Balantekin, J. Carlson, D. J. Dean, G. M. Fuller, R. J. Furnstahl, M. Hjorth-
Jensen, R. V. F. Janssens, Bao-An Li, W. Nazarewicz, F. M. Nunes, and W. E. Or-
mand. Mod. Phys. Lett. A, 29:1430010, 2014.

[13] J. Al-Khalili. An Introduction to Halo Nuclei. Springer Berlin Heidelberg, Berlin,

Heidelberg, 2004.

[14] K. Riisager. Nuclear halo states. Rev. Mod. Phys., 66:1105, 1994.

62

[15] A. S. Jensen, K. Riisager, D. V. Fedorov, and E. Garrido. Structure and reactions of

quantum halos. Rev. Mod. Phys., 76:215, 2004.

[16] V. Tripathi, S. L. Tabor, P. F. Mantica, C. R. Hoffman, M. Wiedeking, A. D. Davies,
S. N. Liddick, W. F. Mueller, T. Otsuka, A. Stolz, B. E. Tomlin, Y. Utsuno, and
A. Volya. 29Na: Defining the edge of the island of inversion for Z = 11. Phys. Rev.
Lett., 94:162501, 2005.

[17] Y. Kondo, T. Nakamura, R. Tanaka, R. Minakata, S. Ogoshi, N. A. Orr, N. L. Achouri,
T. Aumann, H. Baba, F. Delaunay, P. Doomenbal, N. Fukuda, J. Gibelin, J. W. Hwang,
N. Inabe, T. Isobe, D. Kameda, D. Kanno, S. Kim, N. Kobayashi, T. Kobayashi,
T. Kubo, S. Leblond, J. Lee, F. M. Marqu´es, T. Motobayashi, D. Murai, T. Murakami,
K. Muto, T. Nakashima, N. Nakatsuka, A. Navin, S. Nishi, H. Otsu, H. Sato, Y. Satou,
Y. Shimizu, H. Suzuki, K. Takahashi, H. Takeda, S. Takeuchi, Y. Togano, A. G. Tuff,
M. Vandebrouck, and K. Yoneda. Nucleus 26O: A barely unbound system beyond the
drip line. Phys. Rev. Lett., 116:102503, 2016.

[18] J. S. Al-Khalili and J. A. Tostevin. Matter radii of light halo nuclei. Phys. Rev. Lett.,

76:3903, 1996.

[19] M. Freer. The clustered nucleus–cluster structures in stable and unstable nuclei. Rep.

Prog. Phys., 70:2149, 2007.

[20] M. Freer, H. Horiuchi, Y. Kanada-En’yo, D. Lee, and U. Meißner. Microscopic clus-

tering in light nuclei. Rev. Mod. Phys., 90:035004, 2018.

[21] C. Dossat, A. Bey, B. Blank, G. Canchel, A. Fleury, J. Giovinazzo, I. Matea,
F. de Oliveira Santos, G. Georgiev, S. Gr´evy, I. Stefan, J. C. Thomas, N. Adimi,
C. Borcea, D. Cortina Gil, M. Caamano, M. Stanoiu, F. Aksouh, B. A. Brown, and
L. V. Grigorenko. Two-proton radioactivity studies with 45Fe and 48Ni. Phys. Rev. C,
72:054315, 2005.

[22] B. Blank, A. Bey, G. Canchel, C. Dossat, A. Fleury, J. Giovinazzo, I. Matea, N. Adimi,
F. De Oliveira, I. Stefan, G. Georgiev, S. Gr´evy, J. C. Thomas, C. Borcea, D. Cortina,
M. Caamano, M. Stanoiu, F. Aksouh, B. A. Brown, F. C. Barker, and W. A. Richter.
First observation of 54Zn and its decay by two-proton emission. Phys. Rev. Lett.,
94:232501, 2005.

[23] J. Giovinazzo, B. Blank, M. Chartier, S. Czajkowski, A. Fleury, M. J. Lopez Jimenez,
M. S. Pravikoff, J.-C. Thomas, F. de Oliveira Santos, M. Lewitowicz, V. Maslov,
M. Stanoiu, R. Grzywacz, M. Pf¨utzner, C. Borcea, and B. A. Brown. Two-proton
radioactivity of 45Fe. Phys. Rev. Lett., 89:102501, 2002.

[24] B. Blank and M. P(cid:32)loszajczak. Two-proton radioactivity. Rep. Prog. Phys., 71:046301,

2008.

[25] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoshikawa, K. Sugimoto, O. Ya-
makawa, T. Kobayashi, and N. Takahashi. Measurements of interaction cross sections
and nuclear radii in the light p-shell region. Phys. Rev. Lett., 55:2676, 1985.

63

[26] D. L. Auton. Direct reactions on 10Be. Nucl. Phys. A, 157:305, 1970.

[27] F. Ajzenberg-Selove, E. R. Flynn, and O. Hansen. (t, p) reactions on 4He, 6Li, 7Li,

9Be, 10B, 11B, and 12C. Phys. Rev. C, 17:1283, 1978.

[28] R. Ringle, M. Brodeur, T. Brunner, S. Ettenauer, M. Smith, A. Lapierre, V.L. Ryjkov,
P. Delheij, G.W.F. Drake, J. Lassen, D. Lunney, and J. Dilling. High-precision Penning
trap mass measurements of 9,10Be and the one-neutron halo nuclide 11be. Phys. Lett.
B, 675(2):170, 2009.

[29] H. Esbensen, B. A. Brown, and H. Sagawa. Positive parity states in 11Be. Phys. Rev.

C, 51:1274, 1995.

[30] M. V. Zhukov, B. V. Danilin, D. V. Fedorov, J. M. Bang, I. J. Thompson, and J. S.
Vaagen. Bound state properties of Borromean halo nuclei: 6He and 11Li. Phys. Rep.,
231:151, 1993.

[31] G. Gamow. Zur Quantentheorie des Atomkernes. Z. Physik, 51:204, 1928.

[32] R. W. Gurney and E. U. Condon. Quantum mechanics and radioactive disintegration.

Phys. Rev., 33:127, 1929.

[33] A. F. J. Siegert. On the derivation of the dispersion formula for nuclear reactions.

Phys. Rev., 56:750, 1939.

[34] L. P. Kok. Accurate determination of the ground-state level of the 2He nucleus. Phys.

Rev. Lett., 45:427, 1980.

[35] A. M. Mukhamedzhanov, B. F. Irgaziev, V. Z. Goldberg, Yu. V. Orlov, and I. Qazi.
Bound, virtual, and resonance S-matrix poles from the Schr¨odinger equation. Phys.
Rev. C, 81:054314, 2010.

[36] A. M. Mukhamedzhanov, Shubhchintak, and C. A. Bertulani. Subthreshold resonances
and resonances in the R-matrix method for binary reactions and in the Tgrojan horse
method. Phys. Rev. C, 96:024623, 2017.

[37] S. A. Sofianos, S. A. Rakityansky, and G. P. Vermaak. Subthreshold resonances in

few-neutron systems. J. Phys. G, 23:1619, 1997.

[38] B. Gyarmati and T. Vertse. On the normalization of Gamow functions. Nucl. Phys.

A, 160:523, 1971.

[39] B. Simon. The definition of molecular resonance curves by the method of exterior

complex scaling. Phys. Lett. A, 71:211, 1979.

[40] R. G. Newton. Scattering Theory of Waves and Particles. Springer-Verlag, New York,

2nd edition, 1982.

[41] H. M. Nussenzveig. Causality and Dispersion Relations. Academic, New York, 1st

edition, 1972.

64

[42] J. R. Taylor. Scattering Theory: The Quantum Theory on Nonrelativistic Collisions.

John Wiley and Sons, Inc., New York, 1st edition, 1972.

[43] W. Domcke. Analytic theory of resonances, virtual states and bound states in electron-

molecule scattering and related processes. J. Phys. B, 14:4889, 1981.

[44] M. Zinser, F. Humbert, T. Nilsson, W. Schwab, Th. Blaich, M. J. G. Borge, L. V.
Chulkov, H. Eickhoff, Th. W. Elze, H. Emling, B. Franzke, H. Freiesleben, H. Geissel,
K. Grimm, D. Guillemaud-Mueller, P. G. Hansen, R. Holzmann, H. Irnich, B. Jonson,
J. G. Keller, O. Klepper, H. Klingler, J. V. Kratz, R. Kulessa, D. Lambrecht, Y. Leifels,
A. Magel, M. Mohar, A. C. Mueller, G. M¨unzenberg, F. Nickel, G. Nyman, A. Richter,
K. Riisager, C. Scheidenberger, G. Schrieder, B. M. Sherrill, H. Simon, K. Stelzer,
J. Stroth, O. Tengblad, W. Trautmann, E. Wajda, and E. Zude. Study of the unstable
nucleus 10Li in stripping reactions of the radioactive projectiles 11Be and 11Li. Phys.
Rev. Lett., 75:1719, 1995.

[45] M. Thoennessen, S. Yokoyama, A. Azhari, T. Baumann, J. A. Brown, A. Galonsky,
P. G. Hansen, J. H. Kelley, R. A. Kryger, E. Ramakrishnan, and P. Thirolf. Population
of 10Li by fragmentation. Phys. Rev. C, 59:111, 1999.

[46] H.B. Jeppesen, A.M. Moro, U.C. Bergmann, M.J.G. Borge, J. Cederk¨all, L.M. Fraile,
H.O.U. Fynbo, J. G´omez-Camacho, H.T. Johansson, B. Jonson, M. Meister, T. Nilsson,
G. Nyman, M. Pantea, K. Riisager, A. Richter, G. Schrieder, T. Sieber, O. Tengblad,
E. Tengborn, M. Turri´on, and F. Wenander. Study of 10Li via the 9Li(2H,p) reaction
at REX-ISOLDE. Phys. Lett. B, 642:449, 2006.

[47] H. Simon, M. Meister, T. Aumann, M.J.G. Borge, L.V. Chulkov, U. Datta Pramanik,
Th.W. Elze, H. Emling, C. Forss´en, H. Geissel, M. Hellstr¨om, B. Jonson, J.V. Kratz,
R. Kulessa, Y. Leifels, K. Markenroth, G. M¨unzenberg, F. Nickel, T. Nilsson, G. Ny-
man, A. Richter, K. Riisager, C. Scheidenberger, G. Schrieder, O. Tengblad, and M.V.
Zhukov. Systematic investigation of the drip-line nuclei 11Li and 14Be and their un-
bound subsystems 10Li and 13Be. Nucl. Phys. A, 791:267, 2007.

[48] A. M. Lane and R. G. Thomas. R-matrix theory of nuclear reactions. Rev. Mod. Phys.,

30:257, Apr 1958.

[49] A.M. Lane. Isobaric spin dependence of the optical potential and quasi-elastic (p, n)

reactions. Nucl. Phys., 35:676, 1962.

[50] N. Michel, J. Oko(cid:32)lowicz, F. Nowacki, and M. P(cid:32)loszajczak. First-forbidden mirror

beta-decays in A = 17 mass region. Nucl. Phys. A, 703:202, 2002.

[51] E. Fermi and E. Teller. The capture of negative mesotrons in matter. Phys. Rev.,

72:399, 1947.

[52] J. E. Turner. Minimum dipole moment required to bind an electron – molecular
theorists rediscover phenomenon mentioned in Fermi-Teller paper twenty years earlier.
Am. J. Phys., 45:758, 1977.

65

[53] T. Klahn and P. Krebs. Electron and anion mobility in low density hydrogen cyanide

gas. I. Dipole-bound electron ground states. J. Chem. Phys., 109:531, 1998.

[54] C. Desfran¸cois, H. Abdoul-Carime, and J. P. Schermann. Ground-state dipole-bound

anions. Int. J. Mol. Phys. B, 10:1339, 1996.

[55] R. N. Compton and N. I. Hammer. Multipole-Bound Molecular Anions. Elsevier, 1st

edition, 2001.

[56] K. D. Jordan and F. Wang. Theory of dipole-bound anions. Annu. Rev. Phys. Chem.,

54:367, 2003.

[57] J. Simons. Molecular anions. J. Phys. Chem. A, 112:6401, 2008.

[58] K. Fossez, N. Michel, W. Nazarewicz, M. P(cid:32)loszajczak, and Y. Jaganathen. Bound
and resonance states of the dipolar anion of hydrogen cyanide: Competition between
threshold effects and rotation in an open quantum system. Phys. Rev. A, 91:012503,
2015.

[59] J. von Stecher, J. P. D’Incao, and C. H. Greene. Signatures of universal four-body

phenomena and their relation to the Efimov effect. Nature Phys., 5:417, 2009.

[60] M. R. Hadizadeh, M. T. Yamashita, Lauro Tomio, A. Delfino, and T. Frederico. Scaling

properties of universal tetramers. Phys. Rev. Lett., 107:135304, 2011.

[61] M. R. Hadizadeh, M. T. Yamashita, Lauro Tomio, A. Delfino, and T. Frederico. Uni-
versality and scaling limit of weakly-bound tetramers. AIP Conf. Proc., 1423:130,
2012.

[62] R. Lazauskas and J. Carbonell. Complex scaling method for three- and four-body

scattering above the break-up thresholds. Few-Body Syst., 54:967, 2013.

[63] A. Kievsky, M. Gattobigio, and E. Garrido. Universality in few-body systems: from

few-atoms to few-nucleons. J. Phys. Conf. Ser., 527:012001, 2014.

[64] S. K¨onig, H. W. Grießhammer, H. W. Hammer, and U. van Kolck. Nuclear physics

around the unitariry limit. Phys. Rev. Lett., 118:202501, 2017.

[65] R. ´Alvarez-Rodr´ıguez, A. Deltuva, M. Gattobigio, and A. Kievsky. Matching universal

behavior with potential models. Phys. Rev. A, 93:062701, 2016.

[66] A. Deltuva. Universality in fermionic dimer-dimer scattering. Phys. Rev. A, 96:022701,

2017.

[67] M. A. Shalchi, M. T. Yamashita, M. R. Hadizadeh, T. Frederico, and Lauro Tomio.
Neutron-19C scattering: Emergence of universal properties in a finite range potential.
Phys. Lett. B, 764:196, 2017.

[68] G. A. Miller. Non-universal and universal aspects of the large scattering length limit.

Phys. Lett. B, 777:442, 2018.

66

[69] E. Braaten and H. W. Hammer. Universality in few-body systems with large scattering

length. Phys. Rep., 428:259, 2006.

[70] C. A. Bertulani, H. W. Hammer, and U. van Kolck. Effective field theory for halo

nuclei: shallow p-wave states. Nucl. Phys. A, 712:37, 2002.

[71] P. F. Bedaque, H. W. Hammer, and U. van Kolck. Narrow resonances in effective field

theory. Phys. Lett. B, 569:159, 2003.

[72] H. W. Hammer and L. Platter. Efimov states in nuclear and particle physics. Annu.

Rev. Nucl. Part. Sci., 60:207, 2010.

[73] H. W. Hammer and R. J. Furnstahl. Effective field theory for dilute Fermi systems.

Nucl. Phys. A, 678:277, 2000.

[74] H. W. Hammer, C. Ji, and D. R. Phillips. Effective field theory description of halo

nuclei. J. Phys. G, 44:103002, 2017.

[75] D. R. Herrick and P. C. Engelking. Dipole coupling channels for molecular anions.

Phys. Rev. A, 29:2421, 1984.

[76] D. C. Clary. Photodetachment of electrons from dipolar anions. J. Phys. Chem.,

92:3173, 1988.

[77] D. C. Clary. Vibrationally induced photodetachment of electrons from negative molec-

ular ions. Phys. Rev. A, 40:4392, 1989.

[78] E. A. Brinkman, S. Berger, J. Marks, and J. I. Brauman. Molecular rotation and the

observation of dipole-bound states of anions. J. Chem. Phys., 99:7586, 1993.

[79] S. Ard, W. R. Garrett, R. N. Compton, L. Adamowicz, and S. G. Stepanian. Rotational
states of dipole-bound anions of hydrogen cyanide. Chem. Phys. Lett., 473:223, 2009.

[80] W. R. Garrett. Non-Born-Oppenheimer approximation for very weakly bound states

of molecular anions. J. Chem. Phys., 133:224103, 2010.

[81] W. R. Garrett. Excited states of polar negative ions. J. Chem. Phys., 77:3666, 1982.

[82] K. Fossez, N. Michel, W. Nazarewicz, and M. P(cid:32)loszajczak. Bound states of dipolar
molecules studied with the Berggren expansion method. Phys. Rev. A, 87:042515,
2013.

[83] A. Ferron, P. Serra, and S. Kais. Finite-size scaling for critical conditions for stable

quadrupole-bound anions. J. Chem. Phys., 120:8412, 2004.

[84] P. J. Linstrom and W. G. Mallard. NIST Chemistry WebBook. National Institute of

Standards and Technology, Washington, 1st edition, 2016.

[85] H. Abdoul-Carime and C. Desfran¸cois. Electrons weakly bound to molecules by dipolar,

quadrupolar or polarization forces. Eur. Phys. J. D, 2:149, 1998.

67

[86] H. Abdoul-Carime, J. P. Schermann, and C. Desfran¸cois. Multipole-bound molecuar

negative ions. Few-Body Syst., 31:183, 2002.

[87] W. R. Garrett. Quadrupole-bound anions: Efficacy of positive versus negative

quadrupole moments. J. Chem. Phys., 136:054116, 2012.

[88] J. P. Neirotti, P. Serra, and S. Kais. Electronic structure critical parameters from

finite-size scaling. Phys. Rev. Lett., 79:3142, 1997.

[89] V. I. Pupyshev and A. Y. Ermilov. Bound states of multipoles. Int. J. Quant. Chem.,

96:185, 2004.

[90] T. Misu, W. Nazarewicz, and S. ˚Aberg. Deformed nuclear halos. Nucl. Phys. A, 614:44,

1997.

[91] K. Riisager, A. S. Jensen, and P. Møller. Two-body halos. Nucl. Phys. A, 548:393,

1992.

[92] K. Yoshida and K. Hagino. Role of low-l component in deformed wave functions near

the continuum threshold. Phys. Rev. C, 72:064311, 2005.

[93] R. M. Id Betan, R. J. Liotta, N. Sandulescu, and T. Vertse. A shell model represen-

tation with antibound states. Phys. Lett. B, 584:48, 2004.

[94] N. Michel, W. Nazarewicz, M. P(cid:32)loszajczak, and J. Rotureau. Antibound states and

halo formation in the Gamow shell model. Phys. Rev. C, 74:054305, 2006.

[95] K. Rohr and F. Linder. Vibrational excitation in e − HCl collisions at low energies. J.

Phys. B: Atom. Molec. Phys., 8:200, 1975.

[96] K. Rohr and F. Linder. Vibrational excitation of polar molecules by electron impact I.
Threshold resonances in HF and HCl. J. Phys. B: Atom. Molec. Phys., 9:2521, 1976.

[97] K. Rohr. Interaction mechanismes and cross sections for the scattering of low-energy

electrons from HBr. J. Phys. B: Atom. Molec. Phys., 11:1849, 1978.

[98] W. D. Heiss and R. G. Nazmitdinov. Spectral singularities and zero energy bound

states. Eur. Phys. J. D., 63:369, 2011.

[99] W. D. Heiss. The physics of exceptional points. J. Phys. A: Math. Theor., 45:444016,

2012.

[100] M. M¨uller and I. Rotter. Exceptional points in open quantum systems. J. Phys. A,

41:244018, 2008.

[101] J. Oko(cid:32)lowicz and M. P(cid:32)loszajczak. Exceptional points in the scattering continuum.

Phys. Rev. C, 80:034619, 2009.

[102] S. Garmon, M. Gianfreda, and N. Hatano. Bound states, scattering states, and resonant

states in PT -symmetric open quantum systems. Phys. Rev. A, 92:022125, 2015.

68

[103] K. Fossez, W. Nazarewicz, Y. Jaganathen, N. Michel, and M. P(cid:32)loszajczak. Nuclear

rotation in the continuum. Phys. Rev. C, 93:011305(R), 2016.

[104] Gaoxiang Liu, Sandra M. Ciborowski, Cody Ross Pitts, Jacob D. Graham, Allyson M.
Buytendyk, Thomas Lectka, and Kit H. Bowen. Observation of the dipole- and
quadrupole-bound anions of 1,4-dicyanocyclohexane. Phys. Chem. Chem. Phys.,
21:18310, 2019.

[105] Gaoxiang Liu, Sandra M. Ciborowski, Jacob D. Graham, Allyson M. Buytendyk, and
Kit H. Bowen. The ground state, quadrupole-bound anion of succinonitrile revisited.
The Journal of Chemical Physics, 151:101101, 2019.

[106] M.A.K. Lodhi. Cluster model wave function of 6Li. Nuclear Physics A, 97:449, 1967.

[107] P. G. Roos, D. A. Goldberg, N. fS. Chant, R. Woody, and W. Reichart. The α-d and

τ -t cluster structure of 6Li. Nucl. Phys. A, 257:317, 1976.

[108] I. Tanihata, H. Hamagaki, O. Hashimoto, S. Nagamiya, Y. Shida, N. Yoshikawa, O. Ya-
makawa, K. Sugimoto, T. Kobayashi, D. E. Greiner, N. Takahashi, and Y. Nojiri.
Measurements of interaction cross sections and radii of He isotopes. Phys. Lett. B,
160:380, 1985.

[109] R. Crespo, J. A. Tostevin, and I. J. Thompson. Structure signatures in proton scat-

tering from 9,11Li. Phys. Rev. C, 54:1867, 1996.

[110] S. N. Ershov, B. V. Danilin, J. S. Vaagen, A. A. Korsheninnikov, and I. J. Thomp-
son. Structure of the 11Li continuum from breakup on proton target. Phys. Rev. C,
70:054608, 2004.

[111] D. Baye, E. M. Tursunov, and P. Descouvemont. β decay of 11Li into 9Li and a deuteron

within a three-body model. Phys. Rev. C, 74:064302, 2006.

[112] P. Descouvemont. Microscopic three-cluster study of light exotic nuclei. Phys. Rev. C,

99:064308, 2019.

[113] Y. Kikuchi, T. Myo, K. Kat¯o, and K. Ikeda. Coulomb breakup reactions of 11Li in the

coupled-channel 9Li+n+n model. Phys. Rev. C, 87:034606, 2013.

[114] J. B. Ehrman. On the displacement of corresponding energy levels of C13 and N13.

Phys. Rev., 81:412, 1951.

[115] R. G. Thomas. An analysis ofthe energy levels of the mirror nuclei, C13 and N13. Phys.

Rev., 88:1109, 1952.

[116] Y. Suzuki and K. Ikeda. Cluster-orbital shell model and its application to the He

isotopes. Phys. Rev. C, 38:410, 1988.

[117] Y. Jaganathen, R. M. Id Betan, N. Michel, W. Nazarewicz, and M. P(cid:32)loszajczak. Quan-
tified Gamow shell model interaction for psd-shell nuclei. Phys. Rev. C, 96:054316,
2017.

69

[118] I. Sick. Precise root-mean-square radius of 4He. Phys. Rev. C, 77:041302, 2008.

[119] H. Furutani, H. Horiuchi, and R. Tamagaki. Structure of the second 0+ state of 4He.

Prog. Theor. Phys., 60:307, 1978.

[120] H. Furutani, H. Horiuchi, and R. Tamagaki. Cluster-model study of the T = 1 states

in A = 4 system. Prog. Theor. Phys., 62:981, 1979.

[121] K. Fossez, J. Rotureau, N. Michel, and W. Nazarewicz. Continuum effects in neutron-

drip-line oxygen isotopes. Phys. Rev. C, 96:024308, 2017.

[122] K. Fossez, J. Rotureau, N. Michel, Q. Liu, and W. Nazarewicz. Single-particle and

collective motion in unbound deformed 39Mg. Phys. Rev. C, 94:054302, 2016.

[123] M. D. Jones, K. Fossez, T. Baumann, P. A. DeYoung, J. E. Finck, N. Frank, A. N.
Kuchera, N. Michel, W. Nazarewicz, J. Rotureau, J. K. Smith, S. L. Stephenson,
K. Stiefel, M. Thoennessen, and R. G. T. Zegers. Search for excited states in 25O.
Phys. Rev. C, 96:054322, 2017.

[124] K. Fossez, J. Rotureau, and W. Nazarewicz. Energy spectrum of neutron-rich helium

isotopes: Complex made simple. Phys. Rev. C, 98:061302, 2018.

[125] C. Ord´o˜nez and U. van Kolck. Chiral lagrangians and nuclear forces. Phys. Lett. B,

291:459, 1992.

[126] P. F. Bedaque and U. van Kolck. Effective field theory for few-nucleon systems. Annu.

Rev. Nucl. Part. Sci., 52:339, 2002.

[127] P. F. Bedaque, H. W. Hammer, and U. van Kolck. Narrow resonances in effective field

theory. Phys. Lett. B, 569:159, 2003.

[128] I. Stetcu, J. Rotureau, B. R. Barrett, and U. van Kolck. Effective interactions for light

nuclei: an effective (field theory) approach. J. Phys. G, 37:064033, 2010.

[129] P. Capel, V. Durant, L. Huth, H.-W. Hammer, D. R. Phillips, and A. Schwenk. From
ab initio structure predictions to reaction calculations via EFT. J. Phys.: Conf. Ser.,
1023:012010, 2018.

[130] B. A. Brown and W. A. Richter. New “USD” Hamiltonians for the sd shell. Phys.

Rev. C, 74:034315, 2006.

[131] L. Huth, V. Durant, J. Simonis, and A. Schwenk. Shell-model interactions from chiral

effective field theory. Phys. Rev. C, 98:044301, 2018.

[132] A. P. Zuker. Three-body monopole corrections to realistic interactions. Phys. Rev.

Lett., 90:042502, 2003.

[133] S. R. Stroberg, H. Hergert, S. K. Bogner, and J. D. Holt. Nonempirical interactions

for the nuclear shell model: An update. Annu. Rev. Nucl. Part. S., 69:307, 2019.

70

[134] G. Hagen, M. Hjorth-Jensen, and N. Michel. Gamow shell model and realistic nucleon-

nucleon interactions. Phys. Rev. C, 73:064307, 2006.

[135] N. Michel, W. Nazarewicz, and M. P(cid:32)loszajczak. Isospin mixing and the continuum

coupling in weakly bound nuclei. Phys. Rev. C, 82:044315, 2010.

[136] R. T. Birge. The calculation of errors by the method of least squares. Phys. Rev.,

40:207, 1932.

[137] J. Dobaczewski, W. Nazarewicz, and P.-G. Reinhard. Error estimates in theoretical

models: A guide. J. Phys. G, 41:074001, 2014.

[138] R. P. Feynman. Forces in molecules. Phys. Rev., 56:340, 1939.

[139] I. J. Shin, Y. Kim, P. Maris, J. P. Vary, C. Forss´en, J. Rotureau, and N. Michel. Ab

initio no-core solutions for 6Li. J. Phys. G, 44:075103, 2017.

[140] Ch. Constantinou, M. A. Caprio, J. P. Vary, and P. Maris. Natural orbital description

of the halo nucleus 6He. Nucl. Sci. Tech., 28:179, 2017.

[141] L. Brillouin. La m´ethode du champ self-consistent. Act. Sci. Ind., 71:159, 1933.

[142] G. J. G. Sleijpen and H. A. van der Vorst. A Jacobi–Davidson iteration method for

linear eigenvalue problems. SIAM J. Matrix Anal. Appl., 17:401, 1996.

[143] J. Rotureau, N. Michel, W. Nazarewicz, M. P(cid:32)loszajczak, and J. Dukelsky. Density
matrix renormalisation group approach for many-body open quantum systems. Phys.
Rev. Lett., 97:110603, 2006.

[144] J. Rotureau, N. Michel, W. Nazarewicz, M. P(cid:32)loszajczak, and J. Dukelsky. Density
matrix renormalization group approach to two-fluid open many-fermion systems. Phys.
Rev. C, 79:014304, 2009.

[145] G. V. Rogachev, J. J. Kolata, A. S. Volya, F. D. Becchetti, Y. Chen, P. A. DeYoung,
and J. Lupton. Spectroscopy of 9C via resonance scattering of protons on 8B. Phys.
Rev. C, 75:014603, 2007.

[146] K. W. Brown, R. J. Charity, J. M. Elson, W. Reviol, L. G. Sobotka, W. W. Buhro,
Z. Chajecki, W. G. Lynch, J. Manfredi, R. Shane, R. H. Showalter, M. B. Tsang,
D. Weisshaar, J. R. Winkelbauer, S. Bedoor, and A. H. Wuosmaa. Proton-decaying
states in light nuclei and the first observation of 17Na. Phys. Rev. C, 95:044326, 2017.

[147] G. Papadimitriou, A. T. Kruppa, N. Michel, W. Nazarewicz, M. P(cid:32)loszajczak, and
J. Rotureau. Charge radii and neutron correlations in helium halo nuclei. Phys. Rev.
C, 84:051304, 2011.

[148] I. Sick. Form factors and radii of light nuclei. J. Phys. Chem. Ref. Data, 44:031213,

2015.

71

[149] G. W. Fan, M. Fukuda, D. Nishimura, X. L. Cai, S. Fukuda, I. Hachiuma, C. Ichikawa,
T. Izumikawa, M. Kanazawa, A. Kitagawa, T. Kuboki, M. Lantz, M. Mihara, M. Na-
gashima, K. Namihira, Y. Ohkuma, T. Ohtsubo, Zhongzhou Ren, S. Sato, Z. Q. Sheng,
M. Sugiyama, S. Suzuki, T. Suzuki, M. Takechi, T. Yamaguchi, and W. Xu. Density
distribution of 8Li and 8B and capture reaction at low energy. Phys. Rev. C, 91:014614,
2015.

[150] G.A. Korolev, A.V. Dobrovolsky, A.G. Inglessi, G.D. Alkhazov, P. Egelhof, A. Estrad´e,
I. Dillmann, F. Farinon, H. Geissel, S. Ilieva, Y. Ke, A.V. Khanzadeev, O.A. Kiselev,
J. Kurcewicz, X.C. Le, Yu.A. Litvinov, G.E. Petrov, A. Prochazka, C. Scheidenberger,
L.O. Sergeev, H. Simon, M. Takechi, S. Tang, V. Volkov, A.A. Vorobyov, H. Weick,
and V.I. Yatsoura. Halo structure of 8B determined from intermediate energy proton
elastic scattering in inverse kinematics. Phys. Lett. B, 780:200, 2018.

[151] R. Han, J. X. Li, J. M. Yao, J. X. Ji, J. S. Wang, and Q. Hu. Effects of pairing

correlations on formation of proton halo in 9C. Chin. Phys. Lett., 27:092101, 2010.

[152] I. J. Thompson and M. V. Zhukov. Effects of 10Li virtual states on the structure of

11Li. Phys. Rev. C, 49:1904, 1994.

[153] M. Cavallaro, M. De Napoli, F. Cappuzzello, S. E. A. Orrigo, C. Agodi, M. Bond´ı,
D. Carbone, A. Cunsolo, B. Davids, T. Davinson, A. Foti, N. Galinski, R. Kanungo,
H. Lenske, C. Ruiz, and A. Sanetullaev. Investigation of the 10Li shell inversion by
neutron continuum transfer reaction. Phys. Rev. Lett., 118:012701, 2017.

[154] A. M. Moro, J. Casal, and M. G´omez-Ramos. Investigating the 10Li continuum through

9Li(d,p)10Li reactions. Phys. Lett. B, 793:13, 2019.

[155] F. Barranco, G. Potel, E. Vigezzi, and R. A. Broglia. 9Li(d, p) reaction as a specific
probe of 10Li, the paradigm of parity-inverted nuclei around the N = 6 closed shell.
Phys. Rev. C, 101:031305, 2020.

[156] H. G. Bohlen, A. Blazevic, B. Gebauer, W. Von Oertzen, S. Thummerer,
R. Kalpakchieva, S.M. Grimes, and T.N. Massey. Spectroscopy of exotic nuclei with
multi-nucleon transfer reactions. Prog. Part. Nucl. Phys., 42:17, 1999.

[157] J. K. Smith, T. Baumann, J. Brown, P. A. DeYoung, N. Frank, J. Hinnefeld, Z. Kohley,
B. Luther, B. Marks, A. Spyrou, S. L. Stephenson, M. Thoennessen, and S. J. Williams.
Selective population of unbound states in 10Li. Nucl. Phys. A, 940:235, 2015.

[158] R. Sherr and H. T. Fortune. Energies within the A = 10 isospin quintet. Phys. Rev.

C, 87:054333, 2013.

[159] H. T. Fortune. Mirror energy differences of 2s1/2 single-particle states: Masses of 10N

and 13F. Phys. Rev. C, 88:024309, 2013.

[160] J. Hooker, G.V. Rogachev, V.Z. Goldberg, E. Koshchiy, B.T. Roeder, H. Jayatissa,
C. Hunt, C. Magana, S. Upadhyayula, E. Uberseder, and A. Saastamoinen. Structure
of 10N in 9C+p resonance scattering. Phys. Lett. B, 769:62, 2017.

72

[161] S. M. Wang, W. Nazarewicz, R. J. Charity, and L. G. Sobotka. Structure and decay

of the extremely proton-rich nuclei 11,12O. Phys. Rev. C, 99:054302, 2019.

[162] D.R. Tilley, J.H. Kelley, J.L. Godwin, D.J. Millener, J.E. Purcell, C.G. Sheu, and H.R.

Weller. Energy levels of light nuclei A=8,9,10. Nucl. Phys. A, 745:155, 2004.

[163] S. Aoyama, K. Kat¯o, and K. Ikeda. Resonant structures in the mirror nuclei 10N and

10Li. Phys. Lett. B, 414:13, 1997.

[164] A. L´epine-Szily, J. M. Oliveira, V. R. Vanin, A. N. Ostrowski, R. Lichtenth¨aler,
A. Di Pietro, V. Guimar˜aes, A. M. Laird, L. Maunoury, G. F. Lima, F. de Oliveira San-
tos, P. Roussel-Chomaz, H. Savajols, W. Trinder, A. C. C. Villari, and A. de Vismes.
Observation of the particle-unstable nucleus 10N. Phys. Rev. C, 65:054318, 2002.

73