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dissertation entitled

Towards a Microscopic Energy Density Functional for Nuclei

presented by

Biruk B. Gebremariam

has been accepted towards fulfillment
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TOWARDS A MICROSCOPIC ENERGY DENSITY FUNCTIONAL FOR
NUCLEI

By

Biruk B. Gebremariam

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Physics and Astronomy

2010

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ABSTRACT

TOWARDS A MICROSCOPIC ENERGY DENSITY FUNCTIONAL
FOR NUCLEI

By
Biruk B. Gebremariam

In spite of their tremendous success, the limitations of current nuclear energy density
functionals (EDFS), all parameterized empirically in the form of the local Skyrme,
the nonlocal Gogny or relativistic functionals, have become apparent in the past sev-
eral years. In order to address these deficiencies, a current objective of low-energy
nuclear theory is to build non-empirical nuclear EDFs from underlying two-, three-
and possibly four-nucleon interactions and many-body perturbation theory (MBPT).
In this work, the first step towards that goal is taken by calculating the HF contri-
bution from the chiral EFT two- and three-nucleon interaction at N 2LO. The density
matrix expansion (DME) of Negele and Vautherin is a convenient method to map
the highly non-local Hartree—Fock expression into the form of a quasi-local Skyrme—
like functional with density dependent couplings. Reformulating the DME in terms
of phase space averaging (PSA) techniques, we show that the resulting DME, PSA-
DME, is more general and has a significantly better accuracy for spin-unsaturated
systems than the original DME of Negele and Vautherin. This is achieved without
compromising the accuracy of PSA-DME for spin—saturated ones. Imposing the as-
sumption of time-reversal invariance, we apply PSA-DME to the HF energy from the
chiral EFT two- and three-nucleon interaction (at NQLO) and calculate the couplings
of the emerging EDF analytically using a combination of analytical and symbolic
approaches. Subsequently, we perform preliminary analysis of these couplings and
show that their density dependence is driven by the long-range (pion—exchange) part
of the interaction. Finally, we discuss the UN EDF semi-phenomenological approach

that is attempting to utilize the results of this work.

DEDICATION

In loving memory of my aunt, Sebchiya Gebremariam, whose love and support I
still miss.

To my wife, Eden T. E103, whose love I cherish.

To my dearest friend, Misganaw A. Gashaw, “guadea!” our time-like worldlines

should cross soon!

iii

ACKNOWLEDGMENT

I am indebted to my professors, Thomas Duguet and Scott Bogner, who provided me
with all sorts of support in my research endeavor. I was lucky to tap into a much wider
knowledge base than the typical graduate student. Besides helping me connect the
major dots, the fact that they had the patience to let me explore problems in my own
way has been a wonderful experience. I am deeply grateful for all of that. I would also
like to thank the members of my guidance committee: Mark Dykman, Piotr Piecuch
and Brad Sherrill for their helpful suggestions, comments and encouragements.

Next, I would like to show my gratitude to my fellow graduate students, office-
mates and friends. I should specially mention Jacob Clifford, Morewell Gaseller,
Jeremy Armstrong, Ivan Brida, Angelo Signoracci, Liyuan J la and Rhiannon Meharc-
hand. Thanks for making me feel at home. Be it two of my three passions, science
and soccer, we always had something to discuss. As to satisfying my third passion,
literature and painting, Daniel Berhanemeskel, you played the main role there and I
owe you a big thank you.

The perfect learning and research environment that the cyclotron and physics
and astronomy department provided was one of the main ingredients for my suc-
cessful completion of this PhD research project. Starting with the highly helpful
administrative personnel and scientists to the availability of top-notch computer soft-
wares, it is really a magnificent place to do one’s research. In short, thanks! Talking
of top-notch softwares, I should specially mention Wolfram Research’s Mathematica.
Here is my appreciation for powering my innovation. I also extend my gratitude to all
members of the UNEDF collaboration, and the microscopic EDF group in particular,
for letting me be a part of this wonderful collaboration.

My unique appreciation goes to my dearest wife Eden T. Elos. Her love, support
and pragmatism have been and are very important in pulling me out of the often-

romantic world that I create around myself. A friend in need is so goes the saying.

iv

And I say, Mesge, you are indeed one true friend. I thank you for all the walks that
we had in life, be it physically through the streets of Addis or telepathically when I
am in East Lansing and you back home. Finally, thank you all my family members
in Addis and Denver, especially my mom Belaynesh Seifu, Mulumebet Asfaw and
Tekleab Hailu, and friends for all your love, good thoughts and understanding. Hey,

mom! I have arrived now.

u.

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TABLE OF CONTENTS

List of Tables .................................

List of Figures ................................

Low-energy Nuclear Physics
1. 1 Introduction ................................
1.2 Conventions and Notations ........................

Nuclear Interactions

2.1 Historical highlights ............................

2.2 Symmetry Properties of Nuclear Interactions ..............

2.3 Remarks on high-Precision Phenomenological Models .........

2.4 Chiral EFT Models ............................
2.4.1 NN part at N2LO .........................
2.4.2 NNN part at N2LO ........................

The E—term ............................
The D-term ............................
The C-term ............................
Low energy constants and parameters of the N NN interaction

at NQLO .........................

The Nuclear Many-Body Problem
3.1 Remark on ab-initio/MBPT-based methods ..............
3.2 Goldstone—Brueckner formalism .....................
3.2.1 Expansion of the ground-state wave-function and energy
Hole-line expansion for non-perturbative potentials ......
Perturbative expansion .....................
3.2.2 Choice of the one-body potential F ...............

Phenomenological Energy Density

Functionals
4.1 Phenomenological Nuclear Energy Density
Functionals ................................

4.1.1 Motivation from density functional theory ...........
4.1.2 Single- and multi-reference EDF formulations .........
4.2 Skyrme energy density functionals ...................
4.2.1 Particle-hole functional .....................
4.2.2 Particle-particle functional ....................
4.2.3 Self-consistent solution ......................

vi

20

22
22
23
26
26
27
27

29

29

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4.2.4 Existing pararneterizations .................... 39
4.2.5 Predictive power of empirical EDFs ............... 39
4.2.6 Outlook .............................. 42
5 Constructing Non-Empirical Energy Density Functionals 45
5.1 Constructing Non-Empirical Energy Density Functional ........ 45
5.1.1 Philosophy, Goals and Limitations ............... 46
5.2 The Density Matrix Expansion (DME) ................. 52
5.2.1 Basics of the DME ........................ 53
5.2.2 Existing variants of the DME .................. 55
5.3 PSA-DME ................................. 56
5.3.1 Motivation for a PSA reformulation of the DME ........ 56
5.3.2 Momentum phase-space of finite Fermi systems ........ 58
5.3.3 PSA-DME for the scalar part of the OBDM of time-reversal
invariant systems ......................... 62
5.3.4 PSA-DME for the vector part of the OBDM in time-reversal
invariant systems ......................... 67
5.3.5 k}, and isospin invariance of the resulting EDF ......... 71
5.3.6 Extension to non-time-reversal invariant systems ........ 72
Constraints on the 7r—functions ................. 76
5.3.7 Remarks on the DME of the local densities ........... 80
5.3.8 Remarks on the DME of the anomalous densities ....... 86
5.4 Accuracy of DME ............................. 91
5.4.1 Inputs to non-self-consistent tests ................ 92
5.4.2 Fock contribution from VC .................... 93
5.4.3 Fock contribution from VT .................... 99
5.4.4 Fock contribution from V1.3 ................... 106
Basic analysis ........................... 106
Further investigation of the spin-orbit exchange ........ 110
5.4.5 Hartree contribution from VC, VLS and VT ........... 115
5.4.6 Preliminary self-consistent tests ................. 118

6 Non-Empirical Energy Density IMnctional from NN interaction 123

6.1 The HF energy from an N N interaction ................. 123
6.1.1 HF contribution from a central interaction ........... 126
6.1.2 HF contribution from the spin-orbit interaction ........ 128
6.1.3 HF contribution from the tensor interaction .......... 129
6.1.4 Additional contributions to the HF energy ........... 131
6.1.5 The leading-order pairing contribution ............. 133

6.2 Application of the DME to the NN-HF energy ............. 133
6.2.1 Analytical couplings from the chiral EFT NN interaction at N2LOI36
6.2.2 Single-particle fields and equations of motion .......... 138

vii

7 Non-Empirical Energy Density Functional from Chiral EFT NNN

Interaction at N2LO 140
7.1 The Hartree-Fock energy from Chiral EFT NNN interaction at N2LO 141
Basic form of the HF energy ................... 142
HF energy from the E-term ................... 147
HF energy from the D-term ................... 147
HF energy from the C-term ................... 148

7.2 DME for the HF energy from chiral EFT N2LO 3N F in time-reversal
invariant systems ............................. 149
7.2.1 Generic forms of the 3NF energy expressions .......... 150
Generic-Form-l .......................... 151
Generic-Form-2 .......................... 151
Generic-Form-3 .......................... 152
7.2.2 The DME-coordinate system ................... 152
7.2.3 Generalized PSA-DME ...................... 153
Infinite nuclear matter limit ................... 155
7.2.4 The resulting EDF ........................ 155

Comments on the second-order truncation for spherical systems 158

7.3 Analytical Couplings from the chiral EFT NNN interaction at N2LO
for time-reversal invariant systems .................... 160
Comparison of analytical and Monte-Carlo results ....... 162

8 Semi-phenomenological EDF, Future Extensions and Conclusions 168

8.1 The semi-phenomenological approach .................. 168
8.2 Key future extensions ........................... 173
8.3 Conclusion ................................. 176
9 Appendix 178
9.1 Mathematical Formulae ......................... 178
9.1.1 Miscellaneous elementary formulae ............... 178
9.1.2 Clebsh—Gordon, Wigner 3—J and 6—J coefficients ........ 180
9.1.3 A few special functions ...................... 182
Legendre polynomials ....................... 182

Laguerre polynomials ....................... 183

Gamma functions ......................... 184

Spherical harmonics ....................... 184

Bessel functions .......................... 185

9.1.4 Three-dimensional spherical harmonic oscillator eigenfunctions 186
9.1.5 Gegenbaur expansion ....................... 187
9.1.6 Functional derivatives ...................... 188

9.2 The one-body density matrix and densities ............... 188
9.2.1 Properties of single particle states ............... 189
9.2.2 One-body density matrix ..................... 190
9.2.3 Local densities .......................... 191

9.3

9.4

9.5

9.6

9.7

9.8

9.2.4 Properties under time reversal ..................
9.2.5 Extension to anomalous contractions ..............
9.2.6 Relations among the densities ..................
Local Gauge transformation of the OBDM and local densities
9.3.1 Local Gauge transformation in many-body physics ......
Conventional formulation .....................
Second quantization formulation .................
9.3.2 Local Gauge transformation of normal densities ........
9.3.3 Local Gauge transformation of anomalous densities ......
Densities in spherical systems ......................
9.4.1 Expression for the normal densities in spherical symmetry
Scalar part of the density matrix - matter density .......
Kinetic density ..........................
The vector part of the density matrix — Spin density ......
Spin-orbit density .........................
9.4.2 Expression for the anomalous densities in spherical symmetry
pairing density ..........................
Pairing kinetic density ......................
Pairing spin-orbit density ....................
Details on the density matrix expansion ................
9.5.1 Husimi distribution and the local anisotropy Pflr”) ......
9.5.2 Wigner transform of the pq(F1, F2) up to ’12 ..........
9.5.3 Generalized PSA-DME ......................
9.5.4 Generalized PSA-DME for the scalar part of the OBDM
Recovering previous DMEs of the scalar part of the OBDM . .
Further approximation with respect to 2? ............
9.5.5 Generalized PSA-DME for the vector part of the OBDM . . .
9.5.6 Remarks on the generalized PSA-DME .............
9.5.7 The modified-Taylor series expansion ..............
Derivation of EDF from HF energy of local NN interaction ......
9.6.1 Central contribution .......................
9.6.2 Spin-orbit contribution ......................
Spin-orbit contribution in time-reversal invariant systems . .
9.6.3 Tensor contribution ........................
9.6.4 Leading-order pairing contribution ...............
9.6.5 The resulting EDF: EDF-NN-DME ...............
9.6.6 Analytical couplings from chiral EFT N N interaction at N2LO
HFB equations from EDF-NN-DME ...................
9.7.1 The mean field from EDF-NN-DME ..............
9.7.2 The Pairing field from EDF-NN-DME .............
Numerical solution of EDF-HF equations in spherical systems .
9.8.1 Full-DME in spherical systems .................
9.8.2 Exchange-only-DME in spherical systems ...........
9.8.3 Harmonic Oscillator basis expansion method .........

ix

192
193
195
197
198
198
198
200
201
202
203
203
204
205
209
210
211
211
211
212
212
214
217
218
223
224
225
228
228
232
232
234
235
236
237
237
240
243
245
247
248
249

 

IAN

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9.9

9.10

Matrix elements of the kinetic part ............... 252

Matrix elements of the central potential part .......... 253
Matrix elements of the spin-orbit part ............. 253
9.8.4 Self-consistent iterations and convergence ............ 254
The HF energy of chiral EFT NNN interaction at N2LO ....... 254
9.9.1 Remarks on the symbolic implementation ............ 254
9.9.2 HF energy from the E—term ................... 255
Direct part ............................ 255
Single-exchange part ....................... 255
Double-exchange part ...................... 255
E—term contribution for specific systems ............ 256
9.9.3 HF energr from the D-term ................... 256
Direct part ............................ 256
Single-exchange part ....................... 257
Double-exchange part ...................... 258
D-term contribution for specific systems ............ 261
9.9.4 HF energy from the C-term ................... 262
Direct part ............................ 262
Single-exchange part ....................... 263
Double-exchange part ...................... 265
C-term contribution for specific systems ............ 274
Symbolic derivation of EDF-NNN-DME for time-reversal invariance . 275
9.10.1 Generic DME ansatz ....................... 276
Key points on the DME ansatz ................. 278
Connnents on the DME ansatz ................. 280
9.10.2 The G—tensors and their analytical forms ............ 280
9.10.3 Sample DME simplification ................... 284
Angular integrations for spherical systems ........... 285
9.10.4 Contributions to EDF-NNN-DME ................ 288
Fourth order EDF from the D-term ............... 288

Fourth order EDF from the single-exchange piece of the D-like
part of the C-term ................... 289

Fourth order EDF from the double-exchange piece of the D-like
part of the C—term ................... 289

Fourth order EDF from the Rl-double-exchange piece of the
C-term ......................... 290

Fourth order EDF from the R2-double-exchange piece of the
C-term ......................... 291

Fourth order EDF from the R3-double-exchange piece of the
C-term ......................... 292

Fourth order EDF from the R4-double—exchange piece of the
C-term ......................... 292
9.10.5 Extension to deformed time-reversal invariant systems . . . . 293

X

9.11 Analytical couplings for the EDF from chiral EFT NN N interaction at

N2LO ................................... 295
9.11.1 Functional form of the couplings ................. 295
Couplings from Generic-Form-l ................. 295

Couplings from Generic-Form-2 ................. 296

Couplings from Generic-Form-3 ................. 296

9.11.2 Matching generalized PSA-DME against the DME-ansatz . . . 297
9.11.3 Application of Gegenbaur’s addition theorem .......... 298
9.11.4 Analytical and symbolic integration ............... 301
Bibliography ..................................................... 304

xi

1.1

1.2

2.1

2.2

4.1

5.1

5.2

5.3

LIST OF TABLES

Acronyms used in this work ........................

Definitions and conventions used in this work. .............

Seven Decades of Struggle: The Theory of Nuclear Forces from Ref. [22]. 10

Parameters for chiral EFT NNN interaction at N2LO, with A: =
700 [MeV]. Note that the values for c3 and c4 are from Ref. [61]. . . .

INM properties of Skyrme functionals (from Ref. [81]): saturation den-
sity pm, bulk compressibility Koo, isoscalar effective mass (m*/m)s,
Thomas-Reiche-Kuhn enhancement factor rev and energy per particle
at saturation E /A. ............................

MBPT contributions from NN and NNN interactions up to second-
order (Normal contractions) in Hugenholtz representation. ......

The first-order anomalous/ pairing diagrams, otherwise called Bogoli-
ubov contributions, from the N N and NN N interactions in Hugenholtz
represenation. ...............................

The Brink-Boeker force(B1) .......................

Full-DME and Exchange-only—DME for Brink-Boeker interaction and
several DMEs ...............................

xii

21

39

48

48

119

121

 

1.1

1.2

1.3

2.1

4.1

LIST OF FIGURES

Images in this dissertation are presented in color

(Color online) A selection of energy/ length scales in physics. .....

(Color online) Low energy static and and dynamical nuclear properties.

(Color online) The chart of nuclide and the domains of applications of
the standard nuclear structure method. The black region shows the
stable nuclei, the green lines show the traditional magic numbers and

the red curve delimits the experimentally known nuclei. From Ref. [81].

Hierarchy of nuclear forces from Chiral Perturbation Theory, classified
according to a power counting (Q/AX)”, and restricted to V S 3 for
simplicity. Three-body forces appears at next-to—next-to—leading order
(N2LO), but some of the associated low-energy constants are already
constrained by the two-body domain (black symbols) while others (gray

symbols) are to be adjusted on three-body observables. Horn ref. [81].

Illustration of the asymptotic freedom of phenomenological EDF mod-
els in the case of two-neutron separation energies. In the major shell
where empirical EDFs are adjusted on experimental data, the agree-
ment between all relativistic and non-relativistic calculations is clearly
seen. In the next major shell where no data exist, discrepancies be-
tween these models become more apparent (from J. Dobaczewski et
al. [150]). .................................

3

5

17

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

(Color online) Nuclear matter energy per particle as a function of Fermi
momentum kip at the Hartree-Fock level (left) and including second-
order (middle) and particle-particle-ladder contributions (right), based
on evolved N 3LO N N potentials and 3N F fit to E3 H and mm. Theo—
retical uncertainties are estimated by the NN (lines) and N NN (band)
cutoff variations (from Bogner et. a1. [28]) ................

(Color online) The S-wave solution of the Bethe-Goldsone equation
and the uncorrelated S-wave function ...................

The quadrupole anisotropy P§‘(R) of the local neutron momentum dis-
tribution in a selected set of semi-magic nuclei. The black, red and blue
vertical lines indicate the approximate half-radii (where the density be-
comes half of the density at the origin) ..................

(Color online) k2, k]; and the isoscalar kp extracted from a converged
self-consistent calculation of 214Pb, a neutron rich nucleus. ......

(Color online) pn(r) for 48Cr and 208Pb from a converged self-consistent
calculation using Sly4 EDF. .......................

(Color online) The parameters for Cr and Pb isotopic chains obtained
after fitting the neutron density, pn(Fl/2), with the 7r—functions as
given in Eqs. (5.67)-(5.69) .........................

(Color online) Coherence length 6 (R) for 220, 600a, 6”Ni, 104811, 120371,
212Pb (From Pillet et. al. [187]) ......................

(Color online) [,6,,(R,'r)|2 calculated with HFB-D18 for 104311, 12°Sn,
128311. Scale has been multiplied by a factor of 106 (Horn Pillet et.
al. [187]). .................................

(Color online) Comparison of 0541?, r) and CffiDMWR, r) where the
latter is computed from the 1r—functions of one of the three DMEs:
NV-DME, PSA-DME or PSA-DME—II. Upper panels: two-dimensional
integrands. Lower panels: ratios of CngME(R,r) over Cfn(R,r) for
fixed values of R. Densities are obtained from a self-consistent EDF
calculation of 208Pb with the SLy4 Skyrme EDF in the particle-hole

part and no pairing. ...........................

xiv

47

51

61

72

85

85

89

5.10

5.11

5.12

5.13

5.14

5.15

5.16

5.17

(Color online) Percentage error of EgMEhm] compared to Egan],
where the former is computed from: NV-DME, PSA-DME or PSA-
DME—II II—functions. Densities are obtained from self-consistent EDF
calculations using the SLy4 Skyrme EDF in the particle-hole channel
and no pairing. ..............................

(Color online) The same as Figure 5.10 but for two different values of
the range of the Gaussian interaction ...................

(Color online) Comparison of T5,,1(R,r) and T3133 (R, r) where the

latter is computed from NV-DME, PSA-DME, PSA-DME—II or from
PSA-DME with P2"(R) = 0 which we denote as INM—DME. Upper pan-

els: two-dimensional integrands. Lower panels: ratios of TfiAfWR, 7‘)
over Tap ,1(R,r) for fixed values of R. Densities are obtained from a

n

converged self-consistent calculation of 208Pb with the SLy4 Skyrme
EDF in the particle-hole channel and no pairing .............

(Color online) Percentage error of ETQMEhm] compared to ETF [nn]
where the former is either computed from: NV-DME, from PSA-DME
or PSA-DME-II. Densities are obtained from self-consistent EDF cal-
culations using the SLy4 Skyrme EDF in the particle-hole channel and
no pairing. Notice the different vertical scale compared to Fig. 5.10. .

(Color online) A few representative nuclei with diffuse T,f,‘,J(R, r) to-
gether with absolute Eflnn] for the corresponding isotopic chains.
Densities are obtained from a self-consistent EDF calculation using
the SLy4 Skyrme functional in the particle-hole part and no pairing. .

(Color online) Comparison of L3 5,,( R, 'r) and LS 5;? M E (R, r) where the
latter is computed from either NV—DME or PSA-DME. Upper panels:
two-dimensional integrands. Lower panels: ratios of LS 5:1 E (R, 7*) over

Lan(R,r) for fixed values of R. Densities are obtained from a con-

verged self-consistent calculation of 208Pb with the SLy4 Skyrme EDF
in the particle-hole channel and no pairing ................

(Color online) Percentage error of E5313 [7m] compared to Efshm]
where the latter is either computed from N V-DME or from PSA-DME.
Densities are obtained from self-consistent EDF calculations using the
SLy4 Skyrme EDF in the particle-hole channel and no pairing. Notice
the different vertical scale compared to Figs. 5.10 and 5.13 .......

(Color online) Ratio of the DME (Eq.(5.101)) over the exact (Eq. (5.58))
expressions of the toy nonlocal matter density ..............

XV

97

98

101

103

105

107

109

112

5.18 (Color online) Gram,(R,F) as a function of 'r for a selected set of
(R, A, N) ..................................

5.19 (Color online) Percentage error of Eg’DMEMn] with respect to Eg [nn]
for Cr isotopic chain. The upper plots show C3,, and CflDME for

NV-DME and the parameterized 7r—function which we call PI-DME.

5.20 (Color online) L85", and LngDME for NV—DME, with densities ob-
taiend from a converged self-consistent calculation of 208Pb with the

SLy4 Skyrme EDF in the particle-hole channel and no pairing .....
5.21 Comparison of Skyrme HFB and DME-based HFB codes ........

7.1 (Color online) The percentage error of the truncated Gegenbaur expan-
sion with respect to Monte-Carlo based calculation of the contribution
to E/ A in INM. Upper plots show the actual values for the calculation
based on the truncated Gegenbaur expansion. .............

8.1 (Color online) Cd" and C1“ couplings from chiral EFT NN interaction
at N2LO ...................................

8.2 (Color online) C({J couplings from the chiral N N interaction at N2LO
with error bands from naturalness requirement. ............

8.3 (Color online) C1“ couplings from the chiral N N interaction at N2LO
with error bands from naturalness requirement. ............

8.4 The saturation curves, W(p, I) of INM using the phenomenological
SLY4 functional and semi-phenomenological DME-based functionals.
Here, NZLO includes the contribution from both NN and NNN inter-
actions (From Ref. [164]). ........................

8.5 The same as Fig. 8.4 but for PNM (From Ref. [164]). .........

9.1 (Color online) R5(k, 11:2, 333,9) for a set of angles. ...........

xvi

114

117

118

119

166

170

171

172

174

Chat

10th

 

 

Chapter 1

Low-energy Nuclear Physics

1 .1 Introduction

Problems in physics are characterized by different energy or length scales as depicted
in Fig. 1.1. Low-energy nuclear physics lies well below the energy scale for quantum
chromodynamics (QCD), AQCD m 106V, and aims at describing nuclear phenomena
that occur in the energy scale of a few tens of MeV, as characterized by the typical
Fermi energy, 8;. Even though QCD establishes that nucleons, viz, protons and
neutrons have a complex structure in terms of quarks and gluons, low-energy nuclear
physics never attempts to resolve their structure as justified by SF/AQCD << 1. Its
ultimate goal is the proper description of ground- and excited-state properties of
nuclei and nuclear matter in terms of the interaction between and among the relevant
low-energy degrees of freedom: protons and neutrons. Fig. 1.2 presents the diversity

of nuclear properties one is looking after in the realm of low-energy nuclear physics.

There are several size-dependent and size-independent factors that complexify
the coherent solution of the nuclear-many body problem. For infinite nuclear mat-

ter (INM) and finite-nuclei, the existence of the so—called Coester [[1]-[7]] and Tjon

1

 

Hi ------- . , -----------
885 Nuclear 3 Atomic 5 Cosmology

 

 

 

 

 

 

 

 

 

 

 

VEV ' .' ____________ g
| QCD Neutrinos . ....... _. : Astronomy :
l ' . l Geophysrcs ;
6111‘ Z Nuclear : QHE t“; ;__: i
f-mflauon P311011 t‘] [ : "-CD-s-m -:: i : Solar Z
5 to on - ;: 2 ; I
i P m . const :: - i system
.32 g -' 5 : 1 ,
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l E i ll :: g l
1018 GeV 1] lGeV : I1T3v1K l 101“)“
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quarks + giht : i Earth. 5 length
. leptons ll CMB: '- -- - -_ . Galaxy [
Ry berg ; Classrcal Mechmncs :
Sm lhaory Particle Condensed Matter
AdS/CFT- RHIC

Figure 1.1: (Color online) A selection of energy/ length scales in physics.

lines [8] respectively, point to the fact that the nuclear—many body problem cannot
be solved successfully without allowing for many-body forces. In addition, in a sys—
tem of interacting nucleons, there exist both single—particle and collective excitations,
such as sound waves in nuclear matter and rotational/vibrational modes in finite-
nuclei, at about the same energy scale. At the same time, most nuclei (i.e. nuclei
with masses typically between 40 and 350) are intermediate between few-body and
statistical systems. This renders ab—initio techniques impractical due to computa-
tional complexity especially for systematic studies which involve hundreds of nuclei.
It also prevents the application of statistical approaches due to the smallness of the
number of constituents. Furthermore, the need to describe structure and reaction
interfaces (fission, fusion, nucleon emission at the drip-line...), the existence of a large
isospin asymmetry, and the essential role of superfluidity adds to the complexity of

the problem.

Due to these factors, a coherent understanding and description of nuclear phe-

2

 

 

«l .

a“--

L—A

 

 

[Semen

 

   

 

 

 

 

   

 

 

 
     

 

 

Spectroscopy
reflective modes 'Reaction properties
RPA, QRPA, GCM Fusion, transfer, elastic
fienvy elements Iontic behaviours
Fission, Fusion, SHE Drip lines, halos

 

'Astrophysics

r-process, NS, SN

 

 

 
  

 

Figure 1.2: (Color online) Low energy static and and dynamical nuclear properties.

nomena remains elusive, in spite of several decades of theoretical and experimental -
investigations. Still, in the last decade, theoretical nuclear physics has seen significant
progress from several fronts. Some of the main ones that are relevant to this work are
the construction of nuclear interactions within the frame of chiral effective field the—
ory (EFT) [9][[10]-[12]], the application of renormalization group techniques to soften
two- and many-nucleon interactions [[13]-[15]], the use of ab-initio approaches to cal-
culate the properties of increasingly heavier mass nuclei [23], and phenomenological
energy density functional (EDF) approaches to computationally intensive calculations
thanks to advances in computing power and numerical algorithms.

Although high-precision phenomenological two- (NN) and three-nucleon (NNN)
interactions have existed for some time [[16]-[21]] and have been successfully used
in nuclear structure and reaction calculations [23], they are inconvenient from both
theoretical and practical points of view. These interactions lack a controlled expan-
sion scheme that would provide a meaningful estimate of theoretical error bars, and
there is no clear relation between their NN and NNN parts. Additionally, these phe-

nomenological interactions lack a connection to the underlying low-energy QCD. As

3

 

 

- ~ 1
F. ,
-... r

.

D

 

 

 

 

 

u
'-~
. ,,
J .
_ l 7' ' '
‘ . ____.
- \
a ’“ ,
h . -
s
,—
P‘ V. ‘
Av <"
, ~
..
I
, a. v u I
1, '
. I
-
.—. » -
. . ....|‘.
‘ a» L
. u . . C
v- . . ‘
V ‘ ,
- _..
u
H‘ '- 'V I
“ DOA“--
I! F‘; fl‘ V
w t..-A.
_"' r . I
s
._ N. ‘.
y
p
‘\ I
~4 1"4
’ u
' .r. .
A. u" .
,.. ‘

 

 

a result, the role of chiral symmetry breaking of QCD which plays a crucial role in
determining the long-range part of nuclear interactions is not consistently treated in

such potential models [22].

From the viewpoint of nuclear structure calculations, phenomenological interac-
tions contain a strong short—range repulsive core, thereby making the nuclear many-
body problem highly non-perturbative. In general, the latter statement also holds
for chiral EFT interactions which are built with a rather high intrinsic resolution
scale, A, as this significantly couples low and high momenta [27]. Historically and in
the context of infinite nuclear matter (INM) calculation, this necessitates an infinite
re-summation of ladder diagrams, i.e. compute the Brueckner G-matrix, to obtain
a meaningful starting point for more advanced calculations based on the hole-line
expansion [[74],[30]]. The hole-line expansion method, usually at the lowest order,
has been applied to closed-shell medium to heavy nuclei but with little success [38].
In the case of light nuclei with A S 12, state of the art Green’s function Monte-
Carlo (GFMC) and no-core shell model (N CSM) calculations can be performed with
impressive results [23]. However, their large computational cost makes them inap-
plicable for beyond A > 12 region. Most recently, CC (coupled-cluster), IT-NCSM
(importance-truncated no—core shell model) and IT-CI (importance-truncated con-
figuration interaction) have been used to extend the applicability of ab—initio meth-
ods [24, 25].

Eventually, nuclear interactions necessarily depend on the resolution scale [27].
The realization of low-momentum interactions characterized by a low momentum
cut—off, A, through renormalization group techniques results in the elimination of the
non-perturbative aspects, viz, short-range repulsion and tensor forces of conventional
nuclear interactions [27]. The analysis of Weinberg eigenvalues and the calculations
of INM equation of state as well as the calculation of a selected set of finite nuclei

confirm the perturbativeness of low-momentum interactions. As a matter of fact, INM

4

—

 

 

 

 

 

I "
. H .—
’ v --
- . A
_ .
.
.
A.-
_.u.
.4
,
f‘. ..
n1
' e
n- -‘
: "‘ V
' ." a
. v
.1 ‘
J .I n
q- v< v
A: “A sh
.
“v .5‘ IA-‘
. -. ' a...
In o , ¢
\ .
- . A‘- .-

 

 

 

I
. u.
n - T‘
._ .
O.-

. C
s‘.’ ‘
k

I.
_ -.
w. .',
N. I
A».
..'. u,
'U
I 6‘

 

shows saturation already at the HF level, while the empirical saturation properties
are reproduced satisfactorily at second-order in MBPT [28]. For finite nuclei, the
energies and radii of a select set of nuclei seem to be remarkably converged at second
order with good systematics and relatively small corrections coming from particle-hole
states in the RPA [29]. Still, the application of these ab-initio methods for medium
to heavy mass nuclei involves considerable numerical complexity. In addition, the 2
accuracy of these methods is not on par with the current tool of choice for calculating
ground- and excited-state properties of medium to heavy mass nuclei, namely, energy
density functional (EDF) methods [26]. Fig. 1.3 shows the domains of application of

the stande nuclear structure methods.

~120

L100

-60N

 

 

 

I ' I

0 ' so 100 N150 ' 260 l 2750

Figure 1.3: (Color online) The chart of nuclide and the domains of applications of
the standard nuclear structure method. The black region shows the stable nuclei,

the green lines show the traditional magic numbers and the red curve delimits the
experimentally known nuclei. Ffom Ref. [81].

Currently, EDFs are completely phenomenological by construction. Modern pa—

5

rameterizations of these empirical EDFs such as the Skyrme, Gogny and their rela-
tivistic counterparts provide a fair description of bulk properties and certain spectro-
scopic features of known nuclei [26]. However, such empirical EDFs lack predictive
power away from the valley of stability or known data. In addition, the objective of
having spectroscopic quality EDFs does not seem to be attainable with current energy
functionals [26]. Consequently, an intense ongoing effort is dedicated to empirically
fitting EDFs possessing more complex analytical forms and/or enriched density de-

pendent couplings [[31]-[36]].

Along with such phenomenological approach, the quest for predictive EDFs can
be complemented by constraining the analytical form of the functional and the value
of the couplings from MBPT and the underlying low-momentum two— and three-
nucleon (N N and NNN) interactions. The present work is a step towards that goal.
In CHAP. 1, we present a brief discussion of nuclear interaction models with special
emphasis on chiral EFT. CHAP. 2 introduces the nuclear many-body problem and
the diagrammatic approaches that rely on summing a selected set of diagrams. This
is followed by CHAP. 3 where we deal with the formalism and performance of phe-
nomenological EDFs. CHAP. 4 lays out the philosophy, goals and limitations of our
approach for constructing a non-empirical EDF. After introducing the density ma-
trix expansion (DME) as the mathematical technique to make an explicit connection
between MBPT and quasi-local EDFs, we describe a new formulation of the DME
based on phase space averaging (PSA). In addition, non-self-consistent and prelim-
inary self—consistent performance tests of this newly formulated DME are given. In
the subsequent chapter, CHAP. 5, we give details of the derivation of non-empirical
EDF from a generic N N interaction, at the lowest order in MBPT (Hartree-Fock) and
the application of the result to the chiral EFT NN interaction at N2LO. CHAP. 6 dis-
cusses the contribution to the non-empirical EDF from chiral EFT N NN interactions

at N 2LO at the HF level. In addition, an on-going effort to build a universal energy

6

v‘ .
cu
: [v
\' . .
'»
- ' i.
(-
e
a.

- ‘ 9
.’
- .
.A i 5..
”D—
‘I
1..
“Jan.
T’r‘
P —
LU“
‘u—y.
—
a, l

O
.l i I .
\'\~‘.
.l \t '

 

 

. g.
..
r.T
M‘l
A
ya
cr-
A‘J
..
i..
u, -
lv--
.
.1".
.U‘
Ly-
4;“.

 

density functional (UN EDF) that incorporates the results of this work, as well as pos-
sible extensions and conclusions are discussed in the last chapter, CHAP. 7. Finally,
all relevant definitions, formulae and derivations, both analytical and symbolic, are

presented in a set of detailed appendices.

1.2 Conventions and Notations

The acronyms, notations and definitions used throughout the thesis are listed below.

Table 1.1: Acronyms used in this work.

OBDM One-body density matrix
EDF Energy density functional
DFT Density functional theory

NN interaction

NNN interaction

 

 

two-nucleon interaction

three-nucleon interaction

HF Hartree-Fock

HFB Hartree-Fock—Bogoliubov

INM Symmetric and unpolarized infinite nuclear matter
PN M Unpolarized pure neutron matter

EFT Effective field theory

RG Renormalization group

DME Density matrix expansion

MBPT Many-body perturbation theory

RPA Random phase approximation

 

.. 'TQ

I I

 

 

Table 1.2: Definitions and conventions used in this work.

Qt
\
‘H

(19;

P9

1.7

PT.

U

ST
V:

.. j‘ ~ ‘2.
af/s/ /p/9

[IR/57W;

[ST NS"
(11 [772'

[ST p/é'
a2 [7r -

[ST p/s“
a3 [7%

p/5

i

p/é'

i

r
7rp/

1'

my
1(a)
Mi?)

 

Denotes cross product
Pauli vectors: (0:, 0y. 0;), (TI, Ty, 7'2)
Unit vector along vector A, or operator A in case A is an operator

The differential solid angle with respect to A

th

Exchanges the spin coordinates of the ih and ] particles.

It IS given by P"-— - 1/2(1 + 51:52)

Exchanges the iso-spin coordinates of the i” and 3"” particles.
It IS given byPT . —— -1/2(1 + T1 T2)

Exchanges the spatial coordinates of the it“ and 3"" particles
The particle exchange operator given by Pij = PIT}. Pg P;

The spin singlet (i = 0) and triplet (2'. = 1) projectors.
These are given by flag/1 = 1/2 (1 IF 61 ~62)

The isospin singlet (i = 0) and triplet (i = 1) projectors.
These are given by 1170/1 = 1/2 (1 q: f] - 73)

NN interaction vertex of type I where I can be C-central
LS-spin orbit or T-tensor and ST can take the values
10,01, 11,00 where the first 1/0 refers to spin

and the second 1/0 refers to isospin triplet / singlet

2"" 7r—function associated with local densities such as

10(6) NU?) / ill—ib-

i‘h 7r—function associated with non-local densities
100:1, F2)/§(F13F2)/15(F17 772) /'§(F19 F2)

alSTlvrf/grf’gké) 2 4w f dr VITStr) inf/gum

arses/57.5mm) s — M We) ivrf/S‘vrf/ii

alST[7rp/37r 0/8“

fd +14st f“ (”‘1

Chapter 2

Nuclear Interactions

2.1 Historical highlights

The theory of nuclear forces started in the 19303 when Yukawa introduced the idea
that the nuclear strong force is carried by a particle with a mass approximately 200
times that of an electron [37]. Table 2.1 summarizes the major developments of the

past seven decades in the attempt to derive NN interactions from first principles.

With the conception of effective field theory (EFT)[9], it has become clear that
pion-based theories of the fifties, this time with an explicit connection with low-
energy quantum chromodynamics (QCD) [[10]-[12]], should be revived. In the last
decade, EFT has been applied successfully to the consistent derivation of NN, NNN
and many-nucleon interactions at various orders in the low-momentum expansion
scale, Q/AX, where Q is the energy scale of the low-energy physics and AX ~ 1 GeV
refers to the chiral symmetry breaking scale. Details relevant to the present work
are given in section 2.4. In parallel with these efforts to derive nucleon-nucleon
and many—nucleon interactions starting from field-theoretic approaches, various high-
precision phenomenological N N and N NN interactions have been parameterized [[16]-

[21]]. These efforts have been guided by requiring the interactions to satisfy a number

9

am

 

 

Table 2.1: Seven Decades of Struggle: The Theory of Nuclear Forces from Ref. [22].

 

1935

Yukawa: Meson Theory

 

1950’s

The “Pion Theories”
One-Pion Exchange: o.k.
Multi—Pion Exchange: disaster

 

1960’s

Many pions E multi—pion resonances:
0', p, w,
The One-Boson—Exchange Model: success

 

1970’s

Refined meson exchange models, including
sophisticated 271' exchange contributions
(Stony Brook, Paris, Bonn)

 

1980’s

Nuclear physicists discover
QCD
Quark Cluster Models

 

1990’s

and beyond

Nuclear physicists discover EFT
Weinberg, van Kolck
Back to Pion Theory!

But, constrained by Chiral Symmetry Breaking: success

 

 

of symmetries. In the following, we discuss the symmetries that are used to constrain

the form of NN interactions.

2.2 Symmetry Properties of Nuclear Interactions

While the derivation of the strong NN and many-nucleon interactions is an ongoing

effort, there are a number of symmetries that a given nucleon-nucleon interaction

should satisfy. Since one can denote the most general nucleon-nucleon interaction by

its matrix element between two—body states, we use

v(1,2) E(

-oI I I?! I
7101017202

Q2, ifii'F10141F202Q2) E v(F1k101q1, F2E202QQ), (2.1)

10

to discuss the action of the various symmetries. In Eq.(2.1), the dependence on the
momentum of the interacting particles is to allow for nonlocality of the interaction.
The following are the basic symmetry properties that a given NN interaction needs

to satisfy [38].

Hermiticity.

Invariance under an exchange of coordinates

v(1,2) = v(2,1), (2.2)
e Translational invariance
v(1,2) 2 11(7", 1:1 alql, [:2 02(12), (2.3)
o Galilean invariance
v(1.2) = WE. alql, am). (2.4)

Invariance under space reflection

’U(FE, 0'1ql, 02(12) = ’U(—F — E, alql, 02q2), (2.5)

Time reversal invariance

'00—’13, 01611, am) = 21(7“ — 13, -—01q1, —02Q2). (2.6)

Rotational invariance in coordinate space implies that the interaction is a scalar.
Additionally,

”(V-‘71:, 01111, 0292) = ”(‘77 — E, 02022. 01(12): (2-7)

11

which is due to Eqs. (2.2) and (2.5). Hence, terms in the interaction which are

linear in a,- and q,- depend only on 0 = (01 + 02)/2 and q = ((11 + (bl/2.

o Rotational invariance in isospin space which is an approximate symmetry broken
by the coulomb interaction and other isospin—breaking effects. If assumed to

hold, then

”(F/:3: 0141» 02(12) = 1’0(FE102101) + ”01(ng 02: 01)7'1 '72. (2-8)

Even after correcting for electromagnetic effects, there is a strong experimental evi-
dence that the nucleon-nucleon interaction breaks charge symmetry [39] and charge
independence [[40], [41]]. The experimental evidence comes from the difference in the
scattering lengths of pp, nn and pn systems. These values read app 2 —17.3 :1: 0.4fm,
am = —18.8 :1: 0.5fm and em = —23.74 :1: 0.02fm. In general, nucleon-nucleon in-
teractions can be classified into four classes according allowed isospin operators [42],

i.e.

0 Class I forces have only dependencies on [11, (T1 - r2)], and do not break either

charge symmetry or independence,

0 Class II forces maintain charge symmetry but are charge-independence-breaking
(CIB). They are characterized by the isotensor T12 defined by analogy to the

usual tensor 5'12 given in Eq. (2.10), and vanish for T2 = i1 (mm or pp) systems,

0 Class III forces are both charge-symmetry-breaking (CSB) and CIB, but remain
invariant under the exchange of the two nucleons, and are thus proportional to
(r21 + r22). They do not cause isospin mixing since T2 commutes with T2, and

vanish for T 2 = 0 (up) systems,

0 Class IV forces are both CSB and CIB, and are antisymmetric under the ex-

12

 

1.
. .
~i
.
.—
"1 n .
1P . 1‘
.... .-. .
’- 0'; b
b‘ ,
.. .
...
a' '
.- .
.‘ s.
_ .
v.-, ‘
. ‘ ‘ e
.. ,.
. a1 ..
3"; g
k - “
p.‘ ‘5
.
.
.
.‘1
1.. .

 

change of the two nucleons, which causes isospin mixing. They are proportional

to (r21 — 732) or (r1 (8) r2)z , and vanish for T2 = i1 systems.

The most general class-I two-body potential invariant under the fundamental sym-

metries recalled above can be decoupled into [43]

f 1 )

 

 

1
(01 '02)
211.2) = 2 «2pm 51”
p (E - 3')
6212
K 312.1? A

(7'1° T2)

where the various operators are the so—called central ll, tensor 512,7, spin-orbit (ES),

quadratic spin-orbit Q12, and 512,5 components. The operators 812,50, 312.123 and Q12

are given by

3

512,; = fi(0’1’f’)(0’2'fl—0’1'0’2, (2.9)
3 _. s

812,; = 13(0'1 ' k)(0’2 - k) — 01' 02, (2.10)
1 -. .. .. ..

Q12 = §[(01 ~L)(02 - L) + (oz-L)(01-L)], (2.11)

where all operators in (2.2) have radial prefactors, 2.7,,(r), that can be constrained from

microscopy or experimental data.

13

 

--
‘ l
. ~
‘~ -.
1
iv 0-
'u' \ 1‘
v v-
. r
. s..-
o... «v 06 i
...‘....I. ..
.. n .
J .
...L. n. n
0* . r -
n f
. “.-

 

..
.. o. v .
e- - u u.
I.' ' ‘ I
i r ,.
' ' o- . .
n
.W ’ - e
M_

 

'F‘l
, .
I.
‘ I1.
s...
Ast'
i; ‘

 

 

4 "5

1 ..
“Me

n '-

3 s.’
.1 --

 

‘
..
'9.
.
. .1
~
‘4
'\
7.
r.
1..

 

2.3 Remarks on high-Precision Phenomenological

Models

The construction of phenomenological models for nucleon-nucleon interactions pro-
ceeds by parameterizing the radial prefactors vp(r). It is well known that the long-

range (r > l/mfl) part of the interaction is given by one-pion exchange, thereby

e—mwr
7‘

 

fixing the radial form factor to the usual Yukawa form, . The phenomenolog-
ical models that have been parameterized in the last two decades [[16]-[21]] are said
to be high-precision as they are able to fit low-energy (S 350MeV) nucleon-nucleon
scattering data with a chi square per degree of freedom, x2/Ndata, close to one. Ad-
ditionally, all currently available high-precision phenomenological models are charge
dependent (CIB and CSB) and use about 40-50 parameters. The main difference

among the various phenomenological models lies in the way they attempt to capture
the intermediate- and short-range parts of the interaction.

The need to include many-body forces has been suggested by discrepancies be-
tween low-energy properties computed with two-body forces only and experimental
data, such as differential nucleon-deuteron cross-sections [[44]—[46]], triton and other
light nuclei binding energies [47], and the violation of the Koltun sum rule [48]. For
instance, the binding energies of 3H versus 4H6 computed with all available NN models
align on a so-called Tjon line that excludes the experimental point [8]. This is seen as
a necessity to use consistent N N N forces to sneak away from this Tjon line. Likewise,
the Coester line on which lies the saturation point of INM computed with NN forces
only [[1]-[7]], is another indication that N NN forces are essential to reproduce bulk

properties of nuclear matter[[49], [50]].

Phenomenological N NN potentials are available [[51]—[54]], based on mesons ex-
changes plus empirical short-range components. Using the same philosophy as phe-

nomenological NN forces, they are adjusted on binding energies and scattering observ-

14

ables of three— (and four—) body systems such as proton / nucleon-deuteron diffusion
data [[55]-[57]]. In the following section, it will be seen that chiral EFT, N NN inter-

actions appear naturally which is one of the main advantages of the EFT approach.

2.4 Chiral EFT Models

Potentials based on chiral EFT [9] exploit the separation of scales between the chiral
symmetry-breaking scale, Ax z 1 GeV, and typical momenta of low-energy processes
at play in the nuclear structure context, Q, usually about m7r z 140 MeV [[10]-[12]].
In that respect, few-nucleon processes can be treated using only nucleons and pions
as degrees of freedom, the 7r—N interaction being governed by the spontaneously
broken chiral symmetry of QCD. All other heavy mesons and nucleon resonances are
integrated out of the theory, and their effects are contained inside scale-dependent
couplings. The effective Lagrangian only depends, in this approximation, on a finite
number of low-energy constants (LECs), and can be classified using a systematic
expansion based on a power counting in terms of (Q/Ax)”, where V is called the
chiral order. At a given accuracy (Q/AX)”, only a finite number of terms in the
Lagrangian are needed in the low-momentum regime.

The leading order interaction corresponding to V = 0 is denoted by LO. There is
no contribution for V = 1, and following terms V > 1 are called (next-to-)"‘l leading-
orders (N"“1LO). This framework includes effects beyond the NN force, since three,
four-... body interactions appear naturally in the perturbative expansion, and the
hierarchy vNN >> vNNN >> UNNNN is a direct consequence of the power counting, as
shown in Fig. 2.1.

At this point, chiral interactions exist up to N3LO [[58],]59]], where most of the
NN and one-pion, two-pion and three-pion (OPE/2PE/3PE) diagrams have been

computed using various approaches [58, 59]. Improvements of such approaches may

15

consist in (i) increasing the chiral order V of the perturbative expansion, although
[power counting implies that higher contributions will be substantially smaller, as al-
ready observed in the case of OPE/2PE [62], (ii) the introduction of four-nucleon
forces arising naturally at N3LO [63], (iii) treating extra degrees of freedom explic-
itly, such as nucleon A excitations that play a role in three—body forces [[66],]68]] and
isospin breaking NN forces [68], or (iv) refining the short range phenomenological
cutoff schemes. Finally, since chiral perturbation theory is a low-momentum expan-
sion, its predictions are by essence only valid for momenta Q << AX. Several families
of chiral forces are defined depending on the values of the intrinsic high-momentum
cutoff up to which they are defined, whose values typically range between 450 and
750 MeVs. This makes chiral potentials significantly softer than phenomenological

hard-core interactions. In general, chiral EFT potentials have the general structure
VEFT : Vnn + VLAA) a (2.12)

where Vmr are due to n pion-exchanges and Va(A) refers to the contact parts which
depend on the high-momentum cutoff scale, A.

In chapter 6 and 7, we calculate the HF energy from chiral EFT N N and NNN
interactions at N2LO, with emphasis on the contribution from the finite-range parts
of the interaction VM. Hence, we now describe the chiral EFT interaction at N2LO

in some detail.

2.4.1 NN part at N2LO

At N2LO in the low-momentum expansion Q, the pion—exchange (finite-range) part

of the NN interaction can be written as

VIN : V15?) + V1?) + [4(3)

11’ 9

16

 

’0
h‘.’

 

 

 

 

 

 

 

 

 

NN diagrams NNN diagrams
LO (V = 0) --
(V = 1)
NLO (V = 2)
::: 2:: [Cl -- -- [D] -
NQLO (u = 3)
:2 [El
Nucleon line . £10) ,I'A = 0 term (no field derivatives)
------- 1r fine I £11)le = 1 term (one field derivatives)

9 £12) A = 2 term (two field derivatives)

Figure 2.1: Hierarchy of nuclear forces from Chiral Perturbation Theory, classified
according to a power counting (Q/AX)”, and restricted to V S 3 for simplicity.
Three-body forces appears at next-to—next-to-leading order (NZLO), but some of the
associated low-energy constants are already constrained by the two-body domain
(black symbols) while others (gray symbols) are to be adjusted on three—body
observables. From ref. [81].

17

us_' 'i t ‘
.1
..
v: [ ‘H_ 1'
-l.' _ 1L.
1
, .
A. 'f V’
,A-a ‘
\ r‘v’
A .
.. .
\\ u. 7
It '
. . ... .
c -I
. '.

 

3]

4|,
5

 

 

 

v... = va‘i’wx’. (2.13)

Here the superscripts denote the corresponding chiral order and the ellipses refer to
the Q4-and higher order terms which are not considered in the present work. As
can be seen, contributions due to the exchange of three-and more pions are further
suppressed. In [13) ® [(7') ® [77) space, the finite-range (pion—exchange) part of the chiral

N N interaction through N2LO takes the form1

.0

(E1, “VII-51132) = ([VC(CI) + Ti 742 VVCWH + [ VS(Q) + 7:1 ~r21Vs(q)]61 '52

where (f = If" - it: is the momentum transfer, with the relative momenta being it: =
E1 — E2 and 13’ = If; — E251? = (E1 + 1:2)/2 and K' = (131’ + lip/2 are center
of mass momenta of incoming and outgoing interacting particles respectively. The
requirement of Galilean invariance is enforced by 6U? — If ’). In passing, we remark
that the the contact part of the interaction contains terms that depend on 15' =
(13’ +13) / 2 and/ or (f. The subscripts C, S, T, LS label the form factors of central, spin-
spin, tensor and spin-orbit components of the interaction. The form factors are scalar
functions of the momentum transfer q and are such that (i) only WT gets contribution
from one pion-exchange (ii) VC, ,WC, , VT, WT, VLS, WLS get contribution from two-
pion exchange. Actual expressions and details on the contact parts of the interaction

are given in Ref. [59].

 

lThe finite-range NN spin-orbit piece is actually zero-range up to N'ZLO.

18

2.4.2 NNN part at N2LO

From a general standpoint, three-body forces can be. characterized by

_._._. - _._._. 1 - f _. ..
(klkzkalvzwlki ikzi > _ Q—F 1+E2+k3 —E 17’ ~. :31V'(k1k72k3lki"-2'”;i)1 (2.15)

where Q is the volume used in the box—normalization of the momentum basis states,
61-51 +52 +133: Ef—Eé—Eé is the Kronecker delta and 170-51 62%| “1’ #2’ .3) is a matrix element
in momentum space and an operator in spin-isospin space whose dependence on spin
and isospin degrees of freedom is not displayed. The NNN x—EF T interaction first
appears at N2LO where it is composed of three components [69] (i) the E—term (ii)

the D-term and (iii) the C-term.

The E—term

The E-term, which is a three—nucleon contact interaction, is the simplest part of the

x—EFT 3NF at N2LO. Its expression reads

VE(IE'1E2E3IIZ;E[)= Emma-mas), (2.16)
where
E E %. (2.17)
The D-term

The D-term involves one-pion exchange plus contact interaction. Its analytical reads

 

. ~~~ ~-~ CD 014720252 02-43036";
VDkkk It, [kl E g—A AX( T-T +———‘T.T.
(”3'1“) 4—f.f£A (13+m?r 1 2 Q§+mi 2 3
03'5101'51
qf+m§r

19

where q,- = ki— —k,~’.

The C-term

The C-term of the interaction involves two-pion exchange. Its analytic form reads

 

 

 

2 -. _.
. -*-'-' ~-'-* . 01°4102'42 ma 3
V, k '- k't'” E q—A F... 0‘
C( 1 2 3! l 2 3) (2f7r (q¥+nlg)(q§+m72r) 1237-172
02 6203 {if} 05 a [3
F 7' r
((12 +mv2rl((13 +m12r) 231 2 3
03 4301 C11 08 Ta 7’3
F. , 2.19
((13 +mi)(q1 +mi) 3127.37) ( )

with

01mg,

F013 : C113 [._4 f3

ijk —

 

c, _, _, a _,
+ 2f—Zqi-qj] + %6 (3777 0;, (q,- x qj). (2.20)

Low energy constants and parameters of the NNN interaction at N2LO

Values of the various coupling constants appearing in Eqs.(2.16)-(2.20) can be found in
Table 2.2. There are several ways to extract fix the CD and CE low-energy constants,
one of which is adjusting these constants such that the binding energies of 3H and
“He from ab—initio calculations with NN and N NN interactions match experimental
values. The above statements also hold for c1. On the other hand, there is still some
controversy over which set of values is “right” for c3 and c4 with extractions from 7r—N
scattering and NN begin different with large uncertainties. Resolving these differences

is important as many quantities are sensitive to the values of c3 and c4 [60].

20

Table 2.2: Parameters for chiral EFT NNN interaction at N2LO, with
AI = 700 [MeV]. Note that the values for c3 and c4 are from Ref. [61].

 

 

 

gA 1.29
f,r [MeV] 92.400
m7r [MeV] 138.040
C] [ GeV—1] -0.760
c3 [ GeV-1] -4.780
04 [ GeV‘l] 3.960
CD -2.062
E -0.625
21

 

Chapter 3

The Nuclear Many-Body Problem

3.1 Remark on ab-initio/MBPT-based methods

Ab—initio methods for the nuclear many-body problem such as no-core shell model
(NCSM) solve the A—body problem in a given model space while quantum Monte—
Carlo methods such as Green’s function Monte-Carlo (GFMC) rely on stochastic
integration of the many-body Schrodinger equation [23]. Currently, they are able to
incorporate both NN and N NN interactions. However such methods show exponen-
tial scaling with A, thus limiting their applicability to only A S 12 [23] due to their
computation costs. In this regard, CC (coupled-cluster), IT-NCSM (importance-
truncated no—core shell model) and IT-CI (importance-truncated configuration inter-
action) should be mentioned as ab-initio methods that solve the A-body problem
approximately in the given model space. They have lower computational complexity
and thus extend the applicability of ab-initio methods to heavier nuclei [24, 25].

In contrast, MBPT-based methods rely on partial finite/infinite—order summation
of MBPT diagrams according to some organizing principle. Infinite—order summation
may be necessitated by the non-perturbativeness of the starting interaction, and may

not be necessary if one starts from perturbative low-momentum interactions [27]. The

22

non-perturbative behavior of conventional phenomenological interaction models can
be traced to [27] i.e. (i) the hard-core repulsion that makes nucleons scatter up to
very high energies and requires large basis sets, (ii) the tensor force coming from OPE
which is singular at short distances, and (iii) the presence or virtual (di—neutron) or

bound (deuteron) states.

On the other hand, vacuum nuclear interactions are strongly renormalized in the
nuclear medium. This suggests that expressing the many-body energy in terms of
an unperturbed Slater determinant coupled to an effective in-medium interac-
tion that already includes many-body correlations might be possible. That is, the
minimal set of in-medium correlations that have to be included to reach a reasonable
description of the system, i.e. infinite nuclear matter or finite nuclei, need to be in-
corporated in the definition of the in-medium interaction. This can be achieved for

simple systems in the context of Goldstone—Brueckner theory [70].

3.2 Goldstone—Brueckner formalism

As long as pairing is not explicitly included, the Hamiltonian H = t+ U can be
decomposed in terms of a one-body hamiltonian ho that has Slater determinants [<I>,~)

as eigenstates, and a perturbation hl, i.e.

H = h0+h1, (3.1)

ha 2 t+r=Zt,+Zr,,-ala,=Zené’ga, (3.2)
i ij n

h] = ’U—F. (3.3)

The quantities 6,, are the eigenenergies of ho corresponding to single-particle states app,

whereas 5,- will denote many-body eigenenergies of ho associated with unperturbed

23

 

 

 

 

Slater determinants, i.e.

A

A
5i = Z ftp [(1%) = Héj’pIO) . (3.4)
p=1

P=1

According to Gell—Mann-Low’s adiabatic theorem [71], the true ground state [60)
of H can be obtained from the adiabatic evolution of the ground state of ho from

t 2 —00 to t = 0 by gradually turning on the residual interaction [72], i.e.

 

|90)=1im._>0( U.(0,—oo)|<1>0) )

<<I>o|U.(0, —oo)|<I>0) (36)

where the adiabatic evolution operator U€(t, to) from t to to is defined in the interaction
representation starting from the Hamiltonian in the Shrodinger representation H (e, t—

to) as

 

 

.. 'h t

U(t,t0) E exp 2’2“] U€(t,t0) exp [—z,_:) ] (3.6)
[- . t

U€(t,t0) _=_ exp —1/ TH(€,7'). . (3.7)
. h t0 ‘

 

From an expansion of U6 in powers of the residual interaction and integrations over

time in Eq. (3.5), a series expansion of the ground state [80) is obtained [73], i.e.

 

50— lo

leo>=;( 1, h.)"|<1>o>u...., (3.8)

where the sum runs only over linked diagrams, i.e. where [@0) does not appear as
an intermediate state. The latter is enforced at the level of (3.5) where the denomi-

nator fixes the normalization of [90) by eliminating disconnected vacuum-to—vacuum

24

diagrams [79]. Likewise, a similar expansion of the ground-state energy E0 reads

 

1 11
E0 = 50 + Z<¢0lh1 (c h hl) [(1)0>connectede (39)
__ ,0

\—
n 0

where the sums now only runs over connected diagrams.

However, if the expansions of Eqs. (3.8) and (3.9) are truncated at a given order,
non-converging results arise if the vacuum interaction contains a non—perturbative
hard core. On the other hand, it is possible to extract a series of ladder diagrams where
a succession of interactions '0 scatters nucleons into particle states. This series can
be replaced by a reaction matrix G which resums those Brueckner’s particle-particle

ladders and can be represented by the self-consistent Bethe-Goldstone equation [[75]-

[78]]

 

C(w) = v + v C(w), (3.10)

w - ho
where w is the starting energy that corresponds to the in—medium energy of the
nucleons at the location where G is inserted, whereas the Pauli operator Q excludes
occupied states, i.e. those below the Fermi level 61? associated with the unperturbed

vacuum |<I>0), that is

Q: 2 lpp’)(pp’l- (3-11)

€p,€p/>CF
The replacement of the initial interaction by the re-summed G-matrix modifies the
short-range part of the in-medium two-body wave function, such that it is strongly
suppressed over a distance of the order of the range of the repulsive core, that is the

healing distance, or wound [70].

 

 

 

.-
Ill .1
. ,
an a
.
“ .’
\ ' “
... n J
I
.‘, .— 1
'I
-. . .
m,
r‘ '
. .. A
.
. ,,
I
a-..

Y»- '
i.
II

a“
' w
‘9' u.

..
l-. l:
._r-' \

d.‘

r17

 

 

3.2.1 Expansion of the ground-state wave-function and en-

er gy

The general idea consists in regrouping, if necessary, clusters of diagrams under G in
such a way that a converging series is obtained, i.e. a truncation at a given order
provides a result of a given precision [74]. Once the G—matrix has been computed, it
replaces all instances of v in diagrams, excluding those where successive G-matrices
are connected by a two-particle intermediate state, that is no particle-particle ladder

connecting two G-matrices must be written.

Hole-line expansion for non-perturbative potentials

While the G-matrix regularizes the hard-core repulsion, an expansion in terms of G
for the ground-state and singleparticle energies remains non-perturbative, in such
a way that the proper expansion parameter is the number of hole lines [80]. At
lowest order in the hole-line expansion, the ground-state energy E is given by the
Bruckener-Hartree—Fock (BHF) approximation. The BHF approximation consists of

a self-consistent solution of the equations

 

,, .. .. .. 1 -- 1
E0~(z][G(w)[2]) — (zylvlzj)+§ng>CF<ZJIUlmn>w_€m_€n+z-n
x (mnIG(w)|z'j), (3-12)

where e,- are the on-shell single-particle energies that are obtained by a functional
derivative of the ground-state energy and the two-body matrix elements of G (w) and v
are anti-symmetrized. Thus, the lowest order in hole lines for E0 (two hole lines) leads
not only to a term with one line in the self energy but also to a rearrangement term
containing two hole lines and coming from the functional derivative of the particle-

particle ladder propagator.

26

 

Pmurl

| 'Yjw

. IL’.

 

»a
‘-i

 

Perturbative expansion

If the starting interaction is in fact perturbative, as it will be the case for low-
momentum interactions, in—medium correlations can be treated through converging
perturbative series in powers of v for E0 and 61‘. Indeed, the ladder series from (3.10)
becomes perturbative, such that it can be truncated at a given order in intermediate
ladders. For instance, the ladder series for vlow—k is almost converged at second order
in MBPT [81]. For the relevant diagrams that appear at second order from NN and

N NN interactions, refer to Table 5.1 in section 5.1.1.

3.2.2 Choice of the one-body potential P

The proper choice of the unperturbed hamiltonian ho is crucial to have a rapidly
convergent series [82]. Several choices for the one-body field F are possible, among
which (i) a phenomenological expression that is fixed a priori, (ii) the Hartree-Fock
approximation where 6,, are eigenenergies of the Schrodinger equation associated with
the vacuum force, or (iii) a more involved approach necessary for non-perturbative
potentials, e.g. where the one-body field 1" is constructed at lowest order in the on—
shell G-matrix or includes rearrangement terms (extended Brueckner-Hartree-Fock

calculations) [[83]- [86]] .

Note that the truncation orders can be different in the series for the energy E0
and the self-energy 6,, e.g. E0 can be computed at second order while single-particle
energies are derived from a more simple (Woods-Saxon...) potential or only at first
order in 1). Still, adding more orders in the expansion of the single-particle energies
adds extra diagrams in the series for E0 such that it converges faster. Finally we
remark that description of pairing within a diagrammatic framework is possible by

defining anomalous propagators and allowing for anomalous contractions in addition

27

to the normal contractions. Refer to [87] for details.

28

Che

Ph

 

Fm

4.1

 

Chapter 4

Phenomenological Energy Density

Functionals

4.1 Phenomenological Nuclear Energy Density

Functionals

The nuclear energy density functional (EDF) approach, due to its computational
tractability, is the many—body method of choice to study medium- and heavy-mass
nuclei in a systematic manner [26]. The central element of EDF approach is the
energy density functional. Currently available realizations of the EDF approach, all
empirically constructed, vary in the way they parameterize this energy density func-
tional [26]. These include the quasi-local Skyrme, the nonlocal Gogny and relativistic

models.

4.1.1 Motivation from density functional theory

Historically, nuclear EDF based approaches were motivated by starting from effec-

tive interactions in the particle-hole and particle-particle channels and solving the

29

 

self-consistent mean-field equations [26]. Recently, the focus has shifted towards
considering the energy density functional approach as motivated from effective field
theory where the various densities of the system are the basic low—energy degrees of
freedom [35].

In parallel, the development of density functional theory (DF T) [[88]-[91]] in quan-
tum chemistry and condensed matter physics seems to have given nuclear energy den-
sity functional approaches a starting theoretical basis. DFT has been applied success-
fully to the structure of quantum many-body electronic systems (atoms, molecules,
solids...). The comparatively small computational cost of the approach makes DFT
the only feasible solution for systems with large number of electrons [92]. Instead of
the many-body wave-function, DFT takes the fermion density as the “fundamental”
variable.

The two building blocks of DFT are

0 The Hohenberg-Kohn theorem [95], which states the existence of a functional
F [p] such that the ground-state energy of a system of N particles in a one-body

external potential u(r) can be written as

Em = F[p1+ / drumpm. (4.1)

where F [p] only depends on the Hamiltonian of the interacting system, thus
is independent of the external potential u(f'). The ground- state density p00")
and energy E0 = Eu[p0] are then obtained by minimizing Eu[p] with respect
to a variation of the density p0") under the constraints that p is positive and
f df'p(f) = N. It should be noted that this existence theorem does not imply
that all the information about the ground state is contained in the electron

density p('F) [93].
0 Due to its practical difficulties, DF T is not implemented as a pure functional

3O

 

. P.
T
lv
1
E".

M

 

 

 

of the density, a la Thomas-Fermi theory [94]. Rather, one makes use of the

Kohn-Sham implementation [97], which asserts that for any interacting system,

there exists a unique local single-particle potential uK3(7"') such that the ground-

state density of the interacting system equals the ground-state density of the

auxiliary non-interacting system in the external potential U.K3(T_'), that is

2
, (4.2)

pm = pram = 2 am

 

 

expressed using the lowest N single-particle orbitals, $.01"), which are solutions
of the one—body Kohn-Sham equation
v2
[7— + mm] am = mm. (4.3)
m
where e,- are the Kohn—Sham eigenvalues.

In the Kohn—Sham scheme, F is split into
PM = Tlpl + U [pl + 5.1-rip]. (44)

where (i) T [p] is the universal (kinetic) energy functional of the non-interacting sys-
tem, (ii) U [0] is the Hartree functional depending on the two-body interaction po-
tential V(|7"',~ — Fjl), and (iii) Exc[p] is the so-called exchange-correlation functional,
including the Fock term and all remaining many-body correlations. When Eu [p] is ne-
glected, the Kohn—Sham equations reduce to the standard self-consistent Hartree ones.
Additionally, the Kohn-Sham potential is given through the condition that ground-
state energies of the interacting and non-interacting problem (U [p] = E51:(:[p] = 0) are

met for the same density p(F), i.e.

6 Ere [pl

670'? (4.5)

ukslfl 5 “(Fl +

31

While the Kohn—Sham potential is local / multiplicative, the exchange-correlation func-
tional might be highly non-local. The main difficulty for DF T practitioners lies in
the fact that no prescription is given to construct F [p], i.e. the universal exchange-

correlation part Exc[p]. Several levels of realization exist to construct

Exp] = / mum, (4.6)

and they correspond to adding more complex dependencies in the functional Eu [p].
The standard classification separates, from the most simple to the most involved level

of description [89]:

e The local density approximation (LDA), where Eu [p] only depends on the local
density, p(7"') and is matched onto the energy per unit volume of the correspond-

ing infinite homogenous system,

0 The generalized gradient approximation (GGA), where additional specific de-
pendencies on the gradient V7p(f') are added to Exc[p],

e The meta-GGA, which introduces as an additional degree of freedom the kinetic
energy density of occupied Kohn—Sham orbitals
2

as = Zlvcbm (4.7)

 

e The hyper-GOA, which takes also into account dependencies of Em[p] on single—

particle energies e,- and occupations p,- ,

o The generalized random phase approximation (RPA) which involves unoccupied
Kohn-Sham orbitals, and can be seen as the ultimate goal in terms of global

accuracy.

32

 

However, in spite of several recent developments, a rigorous connection between
nuclear EDF and DF T approaches is yet to be found [[98]—[103]]. The key aspect of
this problem is the fact that unlike the systems that are studied in condensed matter
physics and quantum chemistry (bound by external potentials), the nuclear many-
body problem involves a self-bound system. In contrast to the standard Hohenberg-
Kohn theorem which is symmetry-conserving, the nuclear Kohn-Sham potential im—
plementation of EDF approaches breaks symmetries of the Hamiltonian such as trans-
lational and rotational symmetries. Even though projection techniques can be used
to restore these symmetries, understanding its implications for DF T requires further
theoretical development. Additionally, the presence of both spin and isospin degrees
of freedom and the importance of pairing correlations need to be considered in nuclear
EDF approaches. For a related formulation of pairing within the DFT framework,
refer to Ref. [104] although the formulation corresponds to a system coupled to a

particle reservoir.

4.1.2 Single- and multi-reference EDF formulations

As mentioned in section 1.1, the fact that nuclei are self-bound fermionic systems
with both collective modes and individual excitations existing on the same energy
scale make the nuclear many-body problem a complex one. In order to handle this
problem, nuclear EDFs incorporate the assumption that these correlations can be
divided into two different classes that can be incorporated in two different steps (i)
short-range in-medium correlations which are recovered at the level of single-reference
energy density functional (SR-EDF) calculations and commonly referred to as mean-
field calculations (ii) long-range correlations that originate from collective modes and
symmetry restoration. These are handled by multi-reference energy density functional
(MR-EDF) calculations.

In SR—EDF calculations, the EDF is a functional of the normal, p,,-. and anoma-

33

H...— _ ..__.-.—-....

n—n

 

a.“

cut

 

Hi

 

lous, Hij, parts of the OBDM defined in appendix 9.2.2 and 9.2.5. In general, the

energy density in SR—EDF is given by [105]

gsnlq’ol E ESR[pij7f$ijafi;j

1 PP 1 m;
: tiJ’ pji + '2‘ Z vijklpik plj + 1 Z vijle’kik Klj
ijkl ijkl
PPP * 1 pm: -4:
Z vijklnm pli pmj pnk + Z Z vijklmn pli h‘jk h3mn 1
ijklmn ijklmn

+

QIH

(4.8)

where 2) denotes the effective interaction in the respective channel. Traditionally, SR-
EDF calculations have been referred to as self-consistent mean-field theory where one
starts from an effective two- and three—body interaction and calculates the Hartree-
Fock (HF) or Hartree-Fock-Bogoliubov (HFB) energy density. However, SR—EDF
calculations are distinctly different from mean-field calculations in that specific prop-
erties of the interaction vertices, e.g. 1135;, = v32, are not satisfied [105].

SR—EDF calculations can reproduce static collective correlations such as pairing
and deformation through the symmetry breaking of the auxiliary state |<I>0) with
respect to which the OBDM is defined. This does not hold for collective modes and
dynamical correlations, which require Multi-Reference (MR) calculations. Motivating

from Hamiltonian-based generator coordinate method (GCM) calculations [105], MR-

EDF is formulated as

g _ 230,16le foISAml‘I’o) ‘I’il ((1)0lq)1>
MR = *
20.16MB f0 f1 (‘I’ol‘l’d

 

(4.9)

where EMR[<I>0,<I>1] is the MR—EDF and the weight functions f0, f1 are determined
by symmetry consideration and/or diagonalization. If one follows the Hamiltonian
formalism, the most natural guidance for the construction of €MR[<I>0,<I>1] is pro-

vided by the generalized Wick theorem (GWT) [105] which asserts that one obtains

34

in

ft? .'

EMR[<I>(), (1)1] by replacing the SR density matrices by transition densities [105]. N ev-
ertheless, the application of this prescription to currently available EDFs encounters
several pathologies which have been traced to the occurrence of non-integer pow-
ers of the density matrix in the functional. One proposed solution [105] is the re-

parameterization of EDFs in terms of only integer-powers of the density matrix.

4.2 Skyrme energy density functionals

In the Skyrme-EDF model [26, 106], the energy density functional 8 is given as the
sum of kinetic, particle-hole, particle-particle (pairing), Coulomb and center-of-mass

correction terms, i.e.
£[p7 5: 5*] = 8km. [p] + splilp] + £pp[pa I‘E, Hf] + 6Com. [P] + 800.11). [,0] - (410)

8 is quasi-local and is expressed as the single integral in coordinate space of a local
energy density. The expressions for Skin, ECO“), and Scam, can be found in the lit-
erature [26]. They are also discussed in section 6.1.4 in relation to the application
of the density matrix expansion [[107],[170]] to the HF energ;r from a generic NN

interaction.

4.2.1 Particle-hole functional

The particle—hole part of the Skyrme-EDF resembles meta-GGA functionals in a DFT
context as it uses explicit dependencies on several local densities and currents, includ-
ing spin-orbit densities. This is crucial for the proper treatment of finite nuclei. The
functional is the most general bilinear combination of all local densities, built from
the density matrix up to second order derivatives, in such a way that 8 remains

invariant under the transformations associated with all symmetries of the nuclear

35

Hamiltonian, i.e. parity, time-reversal, rotation, translation, gauge and isospin trans-

formations [126]. The functional reads

gphlpl ZESkyrmelPl
= Z de-rApp pq pq + ApAp qupq + APT (quq _]"(1.jq)
+A33§q-§Q+A8m§q-A§q+Apv" (pqv- jq-l-jq'fi X s”)

+ 41‘7sz - 5"’)(V - 4'1) + A""( Z JZVJ:,..— seq - T‘qq)
uu
+21“. [(ZJg,)(Z Jg,,)+ ZJZVJL’H— 25"! - F0]
41

qséq’
+ Bssgq - 5”, + BSAss'q - A5", + BPVJ (9‘16 - j”, +jq ° 6 x §q’)

+BV5VS(V-s )(v-g ’)+B“(ZJZ,,J§,C—"q Tq’)
+B”[(ZJ3,,)(Z Jg,,)+ 2J3 J3], —2§" F‘I',] (4.11)

where the coupling constants AX /BX refer to the interaction between particles with
identical/different isospins, respectively. The densities that occur in Eq. (4.11) are
given in appendix 9.2.3. The coupling constants AX/BX may further depend on
densities that do not involve spatial derivatives. Historically, Eq. (4.11) was derived
starting from the HF expectation value of a Skyrme interaction [108] which contains
zero-range terms plus gradient corrections to encode finite-range effects, and is a sum

of central, spin-orbit and tensor terms. i.e.

vSkyrme(fia F) = vcent. (1313,?“ ) + ULs(I—f, F) "l" 'vtens.(fi1 F) (4‘12)

36

_. A 1 .
vcem(R,F) : t0 (1 + :roPa) 6(7") + 6 t3 (1 + 3:3Pa)

1 . 2. , - 2. . 2. :.
+§t1(1+41P.>[k%<r>+o<mk2] +42 <1+arzPa>k’-6<F>k

+p7(7_") 6(4) (4.13)
01,30?) = “42) ((27.1 + 02:2) '11:, X 60%)]? (4.14)
—o 1 -+ I a ”'1 -o —o "' _,
vtens.(r) : E te { [3 (01 k )( 2 k ) _ (01 02) km] 60‘)

+4., [3 (a. -k’) 50?) (52 .115) — (a, .52) 1}."- (sm i5] . (4.15)

In this context of viewing 8?), [p] as the HF energy from a zero-range Skyrme
force, the timeeven and time-odd terms of the coupling constants of the Skyrme
energy functional are related through the underlying parameters of the Skyrme inter-
action [106]. However, in the general EDF formulation, the time-even and time—odd
couplings are independent of each other, aside from relations dictated by local gauge
invariance. Even though this most general second-order particle—hole functional has
been known for quite some time, traditional studies concentrated only on those terms
which were deemed most important. Recently, the impact of all couplings is being

analyzed in various studies [[26], [159], [158]].

4.2.2 Particle-particle functional

Neutron-neutron and proton—proton pairing acts mostly in the spin-singlet channel
5' = 0 of the nuclear interaction, as shown by the properties of the bare NN force [117].
At the same time, it occurs mainly in the 3 wave, that is a local pairing functional.

This is usually used to justify the expression of the particle-particle functional 8”, as

841mm = /(1FA’3"Z|(5”|2. (4.16)
q

37

where usually

~

" t” (”ll
App E — 1 — . 4.17
4 i 77 psat ( )

The latter expression derives from a density-dependent delta interaction (DDDI) [127,

 

128,129,130,131]

 

vf’fi(F,f—f) E v”(ff)6(f’) E {0 (1:21)”)

1_n(_pop(fl) ] 6(F). (4.18)

It is bilinear in the pair density (3", defined in Eq. (9.99), whereas the strength {0

 

is taken to be the same for neutron-neutron and proton-proton pairing. £W[p, ,5, 5*]
enforces pairing correlations only in the T = 1 channel, as proton-neutron pairing
is usually neglected. The introduction of T = 0 pairing requires a more involved
formalism, since pairing correlations can now couple between superblocks of different
signature in the HFB equations [118, 119]. Two parameters 77 and a control the spatial
dependence of the coupling constant through the overall isoscalar density-dependent
coupling. A zero value of 77 corresponds to a pairing strength that is uniform over
the nuclear volume (“volume pairing”) while 1) = 1 corresponds to pairing strength
which is stronger in the vicinity of the nuclear surface (“surface pairing”). A value
17 = 1/2 corresponds to an intermediate situation (“mixed-type pairing”). Values

0 < 1 correspond to stronger pairing correlations at low density.

4.2.3 Self-consistent solution

After the construction of the densities pi,- and #3,,- from an auxiliary [<I>), the variation
of the EDF (Skyrme-EDF) with respect to these densities results in Hartree-Fock
Bogoliubov (HFB) equations. Refer to appendix 9.7 for a brief discussions of these
equations. One solves these equations self-consistently. For detailed discussion on

this, refer to Ref. [81].

38

 

 

p38, [fm'3] Koo [MeV] (m"‘/m)8 It.) E /A [MeV] Ref.

SLy4 0.160 229.9 0.70 0.25 -15.97 [135, 145]
3111 0.145 355.4 0.76 0.53 -15.85 [146]
m‘l 0.162 230.0 1.00 0.25 —16.07 [144]
pg, 0.145 230.0 0.70 0.25 —15.69 [144]
p3,, 0.160 230.0 0.70 0.25 —15.99 [144]
p3,, 0.175 230.0 0.70 0.25 —16.22 [144]
T6 0.161 235.6 1.00 0.00 —15.93 [147]
SKa 0.155 263.1 0.61 0.94 —15.99 [148]
T21-T26 0.161 230.0 0.70 0.25 —16.00 [142]

 

 

 

 

 

Table 4.1: IN M properties of Skyrme functionals (from Ref. [81]): saturation
density p53,, bulk compressibility Koo, isoscalar effective mass (m*/m)s,
Thomas-Reiche—Kuhn enhancement factor K,” and energy per particle at saturation

E/A.

4.2.4 Existing parameterizations

About 150 parameterizations of the Skyrme EDF have been defined so far and ad-
justed for various purposes (see [120] and references therein for the most common
parameterizations). Sample parameterizations and associated properties of INM are
shown in Table 4.1. These functionals differ in what quantities were emphasized dur-
ing the fits. For instance, T6 has an isoscalar effective nucleon mass (m*/m)3 = 1,
providing a denser single-particle spectrum, while SKa has a different isoscalar effec-
tive mass, but also a different density dependence (density—dependent term with an
exponent of 7 = 1 / 3 instead of 'y = 1/6). T21 to T26 incorporate tensor terms that

differ by their neutron-neutron couplings [142].

4.2.5 Predictive power of empirical EDFs

The discussions in the previous several sections were for the Skyrme EDF. Even
though we have not discussed Gogny and relativistic [26] realizations of the EDF, the
key points of this section regarding the predictive power of currently available EDFs

holds for all three implementations. This is due to the fact that these EDFs generally

39

provide comparable predictions, in spite of some variations for particular observables
[[26], [106]].

The application of phenomenological EDFs for a broad range of nuclear structure
problems has been a success story in the past few decades [26]. Recently, the growth
of available computational power has allowed large-scale projects, such as deformed
calculations of ground-state properties over the nuclear chart. Systematic calcula-
tions of ground-state properties, as well as some collective excitations, for all known
and theoretically predicted nuclei, are now available. Mass residuals over about two
thousand known nuclei obtained at the SR-EDF level are of the order of one MeV,
which is an accuracy sufficient for a direct comparison with experimental data [[121]
-[123]]. Such calculations also provide a reasonably good description of static proper-
ties beyond the ground-state energy, e.g. shell structure, pairing gaps, charge radii,

individual excitations or deformation.

Likewise, MR—EDF calculations have already met a lot of success, in particular
regarding the description of dynamical correlation energies, vibrational/ rotational
excitations and super-deformed bands or shape transitions [[124], [125]]. Among
other challenging areas of interest, extensive studies have for instance been dedi-
cated to [106] (i) (asymmetric) fission properties of heavy elements, (ii) the forma-
tion of superheavy nuclei, (iii) the application of dynamical approaches based on the
time-dependent HF / HF B formalism to describe nuclear fission / fusion, and (iv) collec-
tive motions through the self-consistent (quasiparticle) random phase approximation
((QlRPAl-

However, many challenges are still ahead in order to (i) further increase the overall
precision of EDF-based methods, e.g. decrease mass residuals, (ii) describe excited
states with spectroscopic accuracy (of the order of 300 keVs), as it is achieved for
sd—shell nuclei using the Shell Model [[132]-[134]], (iii) control spin and ferromagnetic

instabilities and (iv) improve the predictive power of EDFs in the unknown region

40

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TW “-0 ti ' 1 V I ' l
o-neu n separa on energies Mass Formulae
25
20 -
$ - .-'
0 s
.2,
<5
V3 5
15 5'
' data 't ‘9'! ' 0 - "'46
P cm s _-‘ data do not ex15t ,
10 1 l . | g 1 l 1 l 1 I 1 i 4_ [4L l
50 60 70 30 80 90 100 110 120 130
Neutron Number Neutron Number

Figure 4.1: Illustration of the asymptotic freedom of phenomenological EDF
models in the case of two-neutron separation energies. In the major shell where
empirical EDFS are adjusted on experimental data, the agreement between all
relativistic and non—relativistic calculations is clearly seen. In the next major shell
where no data exist, discrepancies between these models become more apparent
(from J. Dobaczewski et al. [150]).

41

of the nuclear chart. Indeed, while all empirical models constrained by experimental
data mostly agree with each other within the major shell they are adjusted in, extrap-
olations towards the nucleon drip-line do not agree with each other. This divergence
in the next major shell is seen for most standard observables such as the two-nucleon
separation energy or the pairing gap and is exemplified by Fig. 4.1.

Furthermore, empirical EDF models give rise to spurious effects. For instance,
the particle-hole effective vertex extracted from typical empirical functionals is rarely
fully antisymmetric (e.g. fractional density-dependencies). This leads to a series
of difliculties in SR— (self-interaction and self-pairing) and MR- (poles and spurious
steps) EDF calculations. Some of these issues have been identified and practical cures
have been proposed [105]. However further developments are required in order to

develop a fully satisfactory theory.

4.2.6 Outlook

Various groups are pursuing different strategies to overcome the deficiencies of phe-
nomenological EDFs and make them of spectroscopic quality. In this context, spectro-
scopic quality refers to the ability to describe and predict not only the bulk properties
such as mass and radii but also low-energy spectroscopy and collective states of nu-
clear systems far below the MeV accuracy. On the one hand is the effort to empirically

improve the analytical form and couplings of the EDFs [[31]-[36]]. This includes

0 The construction of EDFs containing beyond second order derivatives [35]. Re-
cently, the authors of Ref. [35] undertook the construction of nuclear EDF with
up to sixth order in gradients. It is possible to reduce the large number of cou-
plings significantly by the successive application of symmetry constraints such
as Galilean (gauge) invariance. Further reduction can be accomplished due if

one requires time-reversal and spherical symmetries. Furthermore, the number

42

of couplings also depends 011 whether one incorporates density dependencies 011

all or some of the couplings.

Approaches that rely on the pseudo—potential perspective, start by selectively
enriching various parts of the effective interaction. There have been several
suggestions to augment the traditional Skyrme interaction given in Eq. (4.12),

e.g. adding a spin-density dependent term [33]
1 _. . 1 _.
V = 660 +scaPaso<R>rs>6<0 + gtz‘i'u+43‘Pa{§1(R)P-9t)6(0. (4.19)

where the exponents 73 and ’73: are even integers in order for the EDF to remain
time-even. The contribution of these terms vanish in even-even nuclei. These
additions seem to remove spin and ferromagnetic instabilities [33] from conven-
tional EDFs, an improvement that must be seen in light of the fact that the
spin-isospin components of nuclear EDFs are less understood / constrained than

their scalar/isoscalar counterparts.

Systematic fitting of the nuclear EDF. This does not necessarily imply improv-
ing the form of the functional. Rather, it focuses on the application of advanced
algorithms to explore the manifold of permissible parameterizations with the use
of a large set of experimental data as a reference [106]. Traditionally, practition-
ers have taken the easier route of only using ground state properties of magic
and semi-magic nuclei to constrain the couplings. The availability of data on
nuclei far from the valley of stability have provided more stringent constraints
on the couplings, with special emphasis on the isovector properties [106] that
are less understood. The experimental data identified for this purpose include
(i) bulk properties such as binding energy and charge radii (ii) spin-orbit split-

ting in nuclei for which accurate data exists such as 40Ca, 48Ca, 90Zr or 132811

43

 

 

in addition to 16O and "me which are usually employed (iii) neutron radii (iv)
odd-even staggering of binding energies in medium to heavy nuclei (v) isotopic
shifts, deformations, excitation properties and (vi) nuclear matter saturation
properties and the equation of state of pure neutron matter. While no definite
proof exists that one can not obtain significant improvement by following this
method, recent results [109] indicate that the form of both the functional and

couplings might be too limiting to obtain predictive EDFs.

A complementary approach is one that relies less on fitting empirical functionals
to known data, but rather attempts to constrain the analytical form of the functional
and that values of its couplings from many-body perturbation theory (MBPT), based
on realistic two- and three-nucleon (NN and NNN) interactions [[110]-[154]]. This is
the path followed in this work, which is similar in spirit to OEP (orbital-dependent
energy potential or ab—initio DFT ) [115, 116]. The main techniques, results, possible

future extensions and outlooks are presented in the next several chapters.

44

Cl

Chapter 5

Constructing Non-Empirical

Energy Density Functionals

5.1 Constructing Non-Empirical Energy Density

Functional

It is commonly asserted that the nuclear many-body problem is intrinsically non-
perturbative [38]. The strong short-range repulsion, the strong tensor force from iter-
ated pion-exchange, and the presence of nearly bound states in the S-wave constitute
the main reasons as to why the nuclear many-body problem is non—perturbative [27].
However, this argument relies on the assumption that the nuclear many-body problem
is driven by an absolute, unique Hamiltonian, without making explicit reference to the

intrinsic energy or resolution scale that underlies the modeling of such a Hamiltonian.

However, recent studies have shown that the above statements need qualifica-
tion as the nuclear Hamiltonian depends on the energy resolution scale [27]. In this
context, an important recent development is the construction of low-momentum in-

teractions starting from chiral effective field theory (EFT) interactions and using

45

 

 

renormalization group (RG) methods. Even though these methods can be applied to
any interaction that originally couples low and high momentum states, chiral EFT
interactions are preferable starting points because of the consistency that character-
izes their many body-forces forces and operators as well as because of the possibility
to systematically improve their precision by going to higher chiral orders. Refer to
section 2.4 for details.

The use of low-momentum interactions simplifies the nuclear many-body problem
as it eliminates, or at least weakens, the main origins of non-perturbativeness [27].
In particular, the consistent three-nucleon interactions become perturbative as one
lowers the intrinsic momentum scale of the two-nucleon piece [28]. Calculations of in-
finite nuclear matter using MBPT in terms of low-momentum two— and three-nucleon
interactions show convergence, at least in the particle-particle channel. As Fig. 5.1
shows, including the second-order contribution from the two— and three-nucleon inter-
actions, one obtains reasonable saturation properties of infinite nuclear matter, with
weak dependence on the resolution scale [28]. Moreover, the freedom to vary the
order of the input EFT interactions and the cutoff via RG provide a powerful tool to
assess theoretical errors arising from truncations in the Hamiltonian and the chosen
many-body approximations.

All these features point to the fact that it may be possible to construct non-
empirical energy density functionals. Indeed, Hartree-Fock becomes reasonable, if not
quantitative, starting point [28], which suggests that the theoretical developments and
phenomenological successes of EDF methods for Coulomb systems may be applicable

to the nuclear case for low-momentum interactions.

5.1.1 Philosophy, Goals and Limitations

Calculations in INM [28] and the binding energies and radii of finite nuclei [29] show

that at least second-order contributions from MBPT have to be incorporated to eb-

46

 

r s ‘. e I I
9‘... .\ u

— \’ 10.2 C fill'II

sly u‘.

.. .
-.d-I-\>.Ufi .110... I

 

 

51u1r161'..;>"£F'I‘I'I'I1"Tf'-1'l
.,,.. HA=L8fm

 

 

 

5" v NN from N3LO (500 MeV) H A =20 66'
g ' 5 low k H A g 2.2 an"
._. 3NF a: to E and r ._. A = 2.3 fm "
5 ." 3H 4He '1

§ ' 2.0 < Am: <25 fm /
é-w -:
9 Empirical :
5 -15 C] saturation - ._ 1 . 1

Hartree-Fock 1’0"" 2nd order 8 pp ladders ‘

-20 1 1 l 1 l 1 l 1 1 l 1 l 1 l 1 l 1 1 l 1 l 1 l 1 l 1 l 4
0.8 1.0 1.2 1.4 1.6 0.8 1.0 1.2 1.4 1.6 0.8 1.0 1.2 1.4 1.6
-1 -l '1
RF [fm 1 1‘1: [fm l 19; [fm 1

Figure 5.1: (Color online) Nuclear matter energy per particle as a function of Fermi
momentum hp at the Hartree-Fock level (left) and including second-order (middle)
and particle-particle-ladder contributions (right), based on evolved N 3LO NN
potentials and 3N F fit to E3}! and mm. Theoretical uncertainties are estimated by
the NN (lines) and NNN (band) cutoff variations (from Bogner et. al. [28]).

tain quantitative success. Likewise, first-order treatment of pairing correlations using
low-momentum two-nucleon interaction show good agreement with experimental re-
sults [112]. On the side of the interaction (chiral EFT interactions in this case), one
needs to go up to N3LO in the chiral expansion in order to describe elastic scattering
phase shifts in the two-nucleon sector with x2 / data close to one [12]. In addition,
these interactions still contain significant coupling of low and high momentum modes
which necessitates their consistent evolution to loW-momentum to make HF a rea-
sonable starting point and obtain a convergent MBPT. Hence, a microscopic / non-
empirical calculation of the nuclear many-body problem should incorporate at least
the contribution of the diagrams shown in table 5.1 for the normal and table 5.2 for
the anomalous / pairing contributions, starting from low—momentum interactions.
Though the perturbativeness of the nuclear many-body problem when using low-
momentum interactions is quite comforting, MBPT is still numerically too expensive
for a systematic calculation of hundreds of heavy open-shell nuclei. Additionally,
the accuracy of currently favored approaches such as empirical EDFs cannot be met,

at this point, with completely non-empirical MBPT calculations. Hence, a method

47

Table 5.1: MBPT contributions from NN and N NN interactions up to second-order
(Normal contractions) in Hugenholtz representation.

 

MBPT Order

NN-interaction

NNN-interaction

 

First Order in MBPT

OO

 

Second order in MBPT

 

 

 

 

Table 5.2: The first-order anomalous/ pairing diagrams, otherwise called Bogoliubov
contributions, from the NN and NNN interactions in Hugenholtz represenation.

<30

from NN:

from NNN:

48

 

is sought to map MBPT contributions to numerically tractable forms, such as local
EDFs, with the aim of refitting some parts of the functional in a controlled and
theoretically motivated way.

In this work, we do not attempt to derive a completely non-empirical EDF. Rather,
we have a more pragmatic goal of enriching and improving current phenomenological
Skyrme EDFs by identifying and incorporating novel density dependencies arising
from missing pion physics. We further restrict the work in that only the first-order
(HF) contributions from the un-evolved chiral EFT NN + NNN interactions at N2LO
have been calculated. Subsequently, we apply the DME to the resulting nonlocal
energy functional to obtain a quasi-local Skyrme—like EDF. In practical implemen-
tations, this is to be followed by refit of the couplings, which has the added benefit
that the whole scheme can be implemented in existing codes with minimal modifi-
cation. Refer to section 8.1 for more details. With the goals and limitations of the
work in perspective, the justifications to concentrate only on the HF contribution
from non-evolved chiral interactions and subsequent application of the DME are as

follows:

0 First, it is well known that RG evolution of interactions to low-momentum
modifies only their short distance structure [15]. The input chiral interaction has
both contact and finite-range pion exchange parts, as given by Eq. (2.12). The
RC evolution modifies mostly Vd(A). However, the energy contribution from
Vd(A), at least at the HF level, is of the same form as conventional Skyrme
EDFs. Thus, refit of the Skyrme parameters should compensate for the RG
evolution of this part of the interaction. As we are primarily interested in
identifying the dominant density dependencies arising from finite-range physics,

it is justifiable to apply the DME to the energy contribution from 17,.

0 Second, inclusion of second-order contributions necessitates the development of

49

 

non-trivial extensions of the DME technique, as those expressions involve non-
localities both in space and in time [166], while the currently available DMEs
can only treat nonlocalities in space [[170],[107]]. This can be illustrated by
contrasting the contributions to the energy from the HF and second-order di-
agrams. Discarding all spin and isospin coordinates for the sake of simplicity

and considering only NN interaction, VNN ,

£HF O( del (IT-‘1 VNNUFI —F2[)p(F1,F2)p(F2,7—'.l), (5.1)

62"" cc Z/dfidfidf‘sdfl[453(F1)¢3(F2)VNN(|F1-F2|)¢)(F1)¢5(F2)
0376

>< 4:05) 42%) WWI-Fa — el) 40(5) 4446)

X Pan 9643 (1 - PM) (1 — p56)
60 + 65 — e, -— 66

 

, (5.2)

where pm, is the density matrix, defined in Eq. (9.70), in the canonical single-
particle basis of the reference HF reference state and 60, is the energy of the
single-particle level. While the HF contribution, U”, can be expressed as a
functional of p(F1, F2) only, the same cannot be said about the second-order con-
tribution, 82M, or any beyond-HF contribution. This is due to the occurrence
of energy-denominators. A satisfactory extension of the DME that can properly
handle beyond-HF contributions and in particular the energy-denominators is

yet to be invented [166].

Third, it is well known that the dominant contributions to bulk nuclear proper-
ties are of Brueckner-HartreeFock (BHF) type [38]. Operationally, this amounts
to replacing the vacuum interactions in the HF expression by a Brueckner G-
matrix, which is discussed in section 3.2. But, the G—matrix “heals” to the bare
interaction at long distances. This is usually demonstrated by studying the

behavior of the S—wave in—medium pair wave-function (at zero center of mass

50

momentum) of the Bethe—Goldstone equation in a repulsive hard-core spherical

potential [38]

__ sin(kr) sin(k7‘c) 90372)
‘4’“) — ‘7..— ‘ a. g(rc,7‘c)’ ’" Z (5'3)

$0") = 01 T < To) (5.4)

 

where To is the radius of the hard—core, k is the relative momentum of the

two-particles and

_ i 1 wdk,sin(k’r)sin(k’r)
— 27r2 rr’ kF k2—k’2 ’

 

9(7) 7") (5.5)

Figure 5.2 shows the solution of Bethe-Goldstone S—wave solution for rela-
tive momentum, k = k; / 2, and the uncorrelated two-body wavefunction,

630(7') = sin(kr)/(kr). Simple analysis shows that g(r,7") decreases rapidly

¢(X)

 

 

 

1.0_
- Uncorrelated

0.8_

0.6:
. 1 , - -* - Correlated(BG)

0.4:

0.2:
_ \ ‘ l ‘ g X =kFI‘
; 6 8 10

—0.2~ ————

Figure 5.2: (Color online) The S-wave solution of the Bethe-Goldsone equation and
the uncorrelated S-wave function.

51

with increasing 1“, with a distance scale of l/kp. One defines the healing dis-
tance, rh, which refers to the distance beyond which the the wave-function
effectively attains the unperturbed value. This is given by the approximate re-
lation kprh % 1.9 , more or less independent of the relative and center of mass
momenta [38]. Then, one can use Eq. (3.10) to show that G—matrix heals to
the bare interaction in the same manner. Hence, applying the DME to the
finite-range part of the interaction, viz, 17,, at the HF level will capture the
same contributions to the density-dependent couplings as given by the finite—
range part of the G-matrix in a more sophisticated BHF calculation. In this
way, the dominant density-dependence that arises from the finite-range of the

interactions is accounted for.

Finally, the algebra required to obtain even the starting point for the DME (viz,
DME on the HF energy from chiral EFT NN + NNN interactions at N2LO)
is so tremendous that most of the work can be done only using some form of
automation [[156], [161]]. This is especially the case if one wants to have the
complete form of the functional without any restricting assumptions regarding

time-reversal invariance and /or spherical symmetry.

This work is just the first step in the long-term project of building non—empirical

11(3lear EDF. There are several pos31ble extensmns that can be made 1n the future.

R
efer to section 8.2 for a related discussion.

5

‘ 2 The Density Matrix Expansion (DME)

T
hQ DME was originally proposed by Negele and Vautherin [170] to derive an ef-

fe -
Qtlve nuclear Hamiltonian. In the first paragraph of their paper [170], Negele and

\r

 

autherin note that the purpose of the density matrix expansion is to relate the compu-

52

tationally simple efiective interactions of 6 and Skyrme forces to the computationally

cumbersome theory derived directly from the nucleon-nucleon force.

In deriving an effective nuclear Hamiltonian, Negele and Vautherin avoided fol-
lowing the moment based expansion which were considered in earlier works [167] in
which one considers expansions of the fourier transform of a short-range interaction.
Their rational for doing that was the fact that the long range part of the nuclear
G-matrix heals to the bare one-pion-exchange-potential (OPEP), which causes con-
vergence problems for moment based expansions. Hence, they invented an expansion,
the density matrix expansion, that exactly includes the long-range OPEP tail for the

nuclear density matrix [170].

5.2.1 Basics of the DME

The central idea of the DME is to factorize a local or nonlocal density obtained from
the one-body density matrix (OBDM) by expanding it into a finite sum of terms that
are separable, usually, in the relative and center of mass coordinates, (F, it). There
are a few exceptions to the (7", If) choice as the DME-coordinates. These exceptions
are mostly relevant to the the application of the DME to the HF energy from the
chiral EFT NNN interaction at NQLO. Refer to section 7.2 for details. Adopting
(F, If) as our DME-coordinates and the notation introduced in Ref. [168], one writes

the general DME formulae

lmax

p,()=',,r~'2) z 2117(4)) 79,(R'), (5.6)
[:0

mmax
emf.) x Z 11:1,,(kr) 0,,(12) , (5.7)

m=0
"max

ache) e 2 112a.) ”..(R’). (5.8)

11:0

53

olllilX

3W2) e 211-SW) 141?), (5.9)
0:0
“max —.

9.05/2) 2 Zflmwm. (5.10)
1120

where k is a momentum scale to be determined that sets the scale for the decay in
the direction of the relative coordinate 7", II,f (k r) are the so—called n—functions that

remain to be specified, and

{731(6), 9114(6)} E {04(R))Tq(1f), Jaw/(é): WG-5), qu(§),«§'q(§),

FAR), “fill, (5‘11)

~

{7346). 90(6)} 6 {71(0), qu—i), Jam/(R): 6154(15): AMI-i), «311(6).

6(5), 6(a)), (5.12)

refer to the local anomalous densities, while gqm/Q) and Hum?) are from the set of

local normal or anomalous densities.

The DME emphasizes separability of the expansion in the relevant expansion-
coordinates above the approximation of nonlocality. That is, even for local densities
that depend on a single coordinate and hence with no nonlocality, one can talk about
an expansion in terms of the DME-coordinates as stated by Eq. (5.10). In a sense,
one is approximating the nonlocality in one of the DME-coordinates. For exam-
p18, pq(1"'1) E pq(E + F/ 2) can be expanded in terms of quantities that depend 011
If and 1" separately. In practice, however, the emphasis on separability above the
approximation of nonlocality is of limited use as most DME approaches rely on an—

alytical techniques that fail to work when there is a long-range of nonlocality in the

54

ml‘rt‘ \. v

. ‘,
lbs w r

    

.. ,0
v- T‘ 1 fl . ‘
l‘ Willi...‘ ,

,,
..-i,
.

1., .
l “-031 \, 1
..r: ml. .13) .1.

.
.’ pr -. ' u

r - n ,r; ,
.'Ls lULL‘utLuu

 

V
P'ry‘
e. W {1:16 519.
"P 7 .~ '}
3.. 9.1%.”;

l 7 lo, '
5'91”), _:r\ I

if“ ”i .
' tA“ .‘\\

T‘.‘ r
II: E ' ~
Lu“ (Jy- "1‘
A 1‘

‘,

r-
”n w» .
f“"“-‘A“ ,
A;.F‘..

P ,‘I
L.» 11111.1:
wlv
Nib—1'... I'll
‘-_, 1
.M~I
"‘1' (1‘ v
in (h
H b

(If;
.1 { i;_v.
'

expansion—coordinate.

This work concentrates mainly on the expansion of the nonlocal scalar and vector
components of the normal part of the density matrix, viz, pq(F1,F2) and §q(F1,F2).
The extension of the approach to non time-reversal invariant systems is important
for constraining the nuclear EDF for those systems. This is discussed in section 5.3.6.
The apparent need for the DME of the local densities (pq(F1/2) and L(F1/2)) that
appear in the exact HF energy of time-reversal invariant systems, justifications for
why one should avoid expanding these densities and related technical problems and
their possible solutions are discussed in section 5.3.7. The expansion of the nonlocal
anomalous densities, especially ,5(7'~'1, F2), has drawn some interest due to the need to
enrich the pairing part of the nuclear EDF. Nevertheless, unlike the nonlocal normal
densities, there are some conceptual and technical difficulties to be overcome. These
are discussed in section 5.3.8. We gauge and compare the accuracy of the various
DME approaches using non self-consistent measures. Finally, we augment this with

preliminary self-consistent tests. These tests are discussed in section 5.4.

5.2.2 Existing variants of the DME

The main problem to be solved in constructing a viable DME technique is the deter-
mination and optimization of the various 7r—functions and the identification of which
local densities occur in the expansion of the given density. The currently available
DME techniques [[170]-[173]] approach this problem in two distinct ways. On the one
hand are those methods that resum infinite order “Taylor-series” expansion terms
in a clever way, while on the other are those that start with an inspired ansatz and
paramterize and optimize the the 7r—functions phenomenologically. In the first group,
we have the original DME of Negele and Vautherin and its variants [170, 171, 173],
while in the second group we have those that are mostly based on gaussian approxi-

mations of the scalar part of the OBDM [172]. In the phenomenological optimization

55

g: war it}. 1
h ad'ilii'
rpruxiizmtv
lllilr: mm:
1135’: that ar'

an
.

41¢; XXL.» Illt
5 a ‘-"rif'.-~_i‘.1i .

TFI. ‘I ‘ Y‘:_ .
rusL DAY‘\“~~“\

’r-r-‘n 1-; ‘ .
I‘ly “““lekit
11‘ 'IY\Y\-
<1 ; i
a) X,
0" F'\
r.
v.3: PbA‘I
3'" ’~ '1‘“. 1
lf‘pdill‘u“

C—‘r, -‘
..:!’ll;_\"g ll.

,.
UV”: 1",:

‘d
In; “~‘l 1.

 

of the 7r—functions, the parameters are optimized to recover various properties of the
OBDM such as the correct local semiclassical kinetic energy density and integrated
projector identity of the OBDM (see Eq. (5.50)).

In addition, there is yet another classification based on whether the techniques
approximate the full quantal or semi-classical approximations of the density matrix.
While most of the existing DME techniques approximate the full quantal OBDM, the
ones that are based on Wigner-Kirkwood expansion of the single-particle propagator
fall into the second/semi—classical category [173]. Further differences appear with
regards to the choice made to fix the momentum scale k. In fact, the DME of Ref. [171]
is a variant of the original one proposed by Negele and Vautherin (NV-DME) [170]
that improves the accuracy of the expansion obtained at first order (nmax = 0) by
optimizing the momentum scale k.

In appendix 9.5.3, we recover the original DME of Negele and Vautherin using
the PSA-DME discussed in the next section and the generalized PSA-DME, while
appendix 9.5.2 contains the key points of the semi-classical Wigner-Kirkwood based

expansion of the density matrix.

5.3 PSA-DME

5.3.1 Motivation for a PSA reformulation of the DME

One of the main shortcomings of all existing DME formulations is that they are
mostly focused on the scalar part of the OBDM. For instance, Negele and Vautherin
acknowledge in their seminal paper that they were not able to design an approximation
of the vector part of the OBDM on the same level, and thus with the same accuracy,
as the one they obtained for the scalar part. This is an essential problem in view
of constraining the nuclear EDF non-empirically. Indeed, the vector part of the

OBDM is non zero in spin-unsaturated nuclei, i.e. in almost all nuclei. Moreover,

56

.1-
lr

.?ixii‘I‘J'P «

of the OED?
a the? Dixiil
2f the 0813
119:: aerate
airmen

Hence. r
37%;(1‘5 I0!’
:2: (1erng

ilfl'Ta’l‘x’lI'
. “-4 Al "'.

 

till:
L1
i-H‘r- '
J H‘J, j-‘T'
.1 .

 

..
«a. 1‘
l
,'h '-
"i4 \‘

 

all available DME techniques hold only for time-reversal invariant systems, with no
apparent extension to non time-reversal invariant systems.

These problems convinced us to formulate a DME approach that has the following
qualities: (i) the accuracy for the scalar part of the OBDM should be comparable to,
if not better than, the existing DME techniques. It should be mentioned that the
percentage error of existing DME techniques for the scalar part of the OBDM is quite
small for various measures, which should be enough to capture the correct density
dependence of the couplings in the resulting EDF. (ii) The DME of the vector part
of the OBDM should have a comparable accuracy to that of the scalar part. Except
for the DME of Negele and Vautherin [170] which performs badly for the vector part
of the OBDMI, the other techniques either do not refer to the vector part at all or
their accuracies are not gauged properly. (iii) It should readily be extended to non
time-reversal invariant systems.

Hence, we formulated a new DME technique which we call PSA-DME where PSA
stands for phase space averaging. Note that the PSA formulation of the DME is
not completely new. In fact, Negele and Vautherin start using the “local energy
approximation” technique of Ref. [174] and mention the possibility of phase space
averaging in infinite nuclear matter. For the actual derivation, they revert to a formal
Bessel-function plane-wave expansion. From a formal point of view, the PSA approach
developed below differs from that mentioned in Ref. [170] and is applied consistently
to both the scalar and the vector parts of the OBDM. For instance, in spite of the
weak angular dependence of the scalar part of the OBDM [176], the inconsistency
in the order of application of the anglaaveraging and series expansion that exists in
Ref. [170] is not an issue in the present case. Still, it is shown in appendix 9.5.3 that
our PSA—DME approach can be used to recover the original DME.

In the following, some of the key properties of the momentum phase space of

 

1Refer to section 5.4 for actual percentage errors of the. various DMEs.

57

 

a .

f vnwf-JOL"

-rr.oapana.'-I . A '1‘
.

f..“1 \,‘ -‘
UI ‘LF‘ Ill! 1

lite Fermi systems are identified with the aim of incorporating these features into
re 7r—functions with the PSA-DME approach. We implement two different strategies
. incorporate these phase space features: analytical derivation and phenomenological

)timization.

.3.2 Momentum phase-space of finite Fermi systems

he momentum phase-space distribution of quantum systems can be studied via a
ultitude of quantum phase-space distribution functions [169]. Studies using the
'igner distribution in Ref. [177] and the Husimi distribution in Ref. [178] Show that
.e local single-particle momentum distribution displays a diffuse and anisotropic
:rmi surface at the (spatial) surface of the finite system. These are peculiar features
the momentum phase-space distribution that are not present for homogeneous

stems.

The Wigner distribution function [175] is often used to approximate the phase
ace distribution of nuclei. It has been studied both analytically and phenomeno-
gically for various models applicable to nuclei (see Refs. [176], [179] , [180]). The
odels include pure harmonic oscillator with sharp and smeared occupations, har-
onic oscillator with orbital occupation from DDHF and meanfield calculations with

Woods—Saxon potential.

The analytical calculations of the various models give the same general form for
e Wigner distribution function. For the case of magic nuclei and in the absence
spin-orbit interaction, the distribution function fq(§,p‘) in a harmonic oscillator

rtential depends solely on the dimensionless parameter 5 [180]

 

 

. ‘ , I
1.: ;.~ mill la

.-u.r.'r .1 ~
fl&‘[‘r_'. Iii-ll;
r ‘.l

1) L‘\a. ‘Hl
.

'. -‘ -
llbfflll‘flll '6.

fl . ~
V 1 1 r5
.tflal“ l

4 r
. LAA .1“

?\

[r-rv~u«;=.21;u:.«i

‘- Y
The at. '
1 , .
m dlil>-.'
””1 _ .
d“ I: 'f‘ ‘1).

“All [ll-J! §[.

‘ 9

r i'wv. I
A -i “111‘ “4'1

 

 

and is given by

£41117) )KL2( (28m, , (5.14)

   

where hw = 41A‘1/3 is the oscillator size parameter, K is the principal quantum
number and L‘}; is the associated Laguerre polynomial, given in appendix 9.1.1, and
rig is the occupation probability. In Ref. [177], the authors parameterize the Wigner
distribution using the Fermi distribution function. All these studies indicate a diffuse
fermi-surface for the local momentum distribution with the diffuseness being much

pronounced around the nuclear surface.

The above model calculations are able to capture the diffuseness, but they do not
show anisotropy/ deformation of the local fermi surface. In Refs. [178] and [177], the
authors solve for the single particle wave functions in spherical Woods-Saxon potential
with no spin-orbit interaction and show that the local fermi surface is anisotropic.
This has no counterpart in the phase space distribution of infinite-fermi systems
(IN M). The anisotropy of the local single particle momentum phase space distribu-
tion can be quantified with the lowest order deformation of a spherical phase space
distribution, viz, quadrupolar deformation. In Ref. [178], the local quadrupolar de-

formation of the momentum Fermi surface (for a given isospin) is given by2
fdfi[3(é'r 'PT" - I72]Hq(F,I5)

f 6151721141317)

2 Ha- - mom)? p3.- — 1 + 0<(kz~ro>2 ). (5.15)

PM a

 

 

= lab

where Hq(f',15) is the Husimi distribution, To is a length scale used in the Husimi

distribution and k}. is a short-hand notation for the local Fermi momentum kflff)

 

2As the anisotropy is usually not large. it is not necessary (at least in this work) to go to higher
multipolas to quantify the deformation.

v

. 1 .
. -..r._-"_ {[\
,f‘un'fil ‘ l d ‘

V t I

“"WI‘[VIO~ ,
, |

\
ALE...” ‘A‘I '._

‘0» t
. 1|." , 1""
31‘“ IAll‘n.“

 

 

defined in a local density approximation through
_, 2 _, 1/3
km) E kg. = [37r pq(R)] . (5.16)

In subsequent formulae, the If dependence of kHl—f) is mostly not shown explicitly
for notational simplicity, except in formulae/ places where we have to remind its If
dependence. Equation 5.15 is computed in the basis 90,-(Fq) that diagonalizes pij,
i.e. the basis from which the Slater determinant |<I>) is built. Details on the Husimi
distribution and simplified expression of P30“) in spherical symmetry suitable for

semi-magic nuclei is provided in the appendix 9.5.1.

Fig. 5.3 shows the quadrupole anisotropy of the local neutron momentum distri-
bution calculated for a selection of semi-magic nuclei. Single-particle wave-functions
are obtained from a Skyrme—EDF calculation performed with the BSLHFB code [181]
using the SLy4 parametrization of the Skyrme EDF. The pairing terms in the EDF
were switched off. Fig. 5.3 also displays the local neutron Fermi momentum (Eq. 5.16)
in order to locate the position of the nuclear surface. In spite of pronounced shell fluc-
tuations, the result corroborates the conclusions drawn in Ref. [178]; P511?) becomes
negative just inside the surface, denoting an oblate momentum Fermi surface while,
outside this region, the local momentum Fermi surface becomes strongly prolate. In
both cases, we have taken an axis normal to the nuclear surface as the reference axis.
The next two sections show how we make use of these properties of the phase-space
distribution of finite Fermi systems to design our PSA-DME of both the scalar and
vector parts of the OBDM.

60

~ .

II-i — at“:

 

 

 

 

 

 

 

 

 

 

 

 

R [fm]

Figure 5.3: The quadrupole anisotropy P2"(R.) of the local neutron momentum
distribution in a selected set of semi-magic nuclei. The black, red and blue vertical
lines indicate the approximate half-radii (where the density becomes half of the
density at the origin).

61

'. rm,

(Mic.

 

‘ . O
K K I
...-x .
I
O i
“ll...

[9. i
... ,

A
kg)?" ‘

 

 

5.3.3 PSA-DME for the scalar part of the OBDM of time-

reversal invariant systems

In PSA-DME, there are three key steps that are used to determine the local den-
sities that occur in the expansion of the given nonlocal density and optimize the
7r—functions. These are: (i) Identifying of the nonlocality operator as an exponential
derivative operator acting on the OBDM. (ii) Performing a Taylor series expansion of
the operator about some momentum scale 1:. This is the point at which a momentum
scale is introduced in the DME, though the actual form of If; is not fixed yet. (iii)
Averaging the momentum scale over the local momentum distribution of the system

of interest.

Applying the first step, viz, extraction of the exponential nonlocality operator of

the scalar part of the OBDM, one writes

) = Z (p:(7"'20q) W(fiaq) pi:

i0

MI‘M

.. F ..
Pq(R+§’R_
_ 6 —6
= e’"(_lT2) Z @5200) 9947:1011) P211

id

(5.17)

 

_. _.

T1=1'2=R.

In the next step, one extracts a phase factor eff": in order to perform a Taylor series

expansion of the non-locality about the momentum scale 1:. Hence,

—o fl-o ._. --o" 7116—6 _i-. *4 —v
pq(R+%,R—-72:) =e‘r'k e ( 9 k) Z<Pi(7"20(1)99i(7‘109)03i

£0

7

59:}?

-o

1’1:

6 —\'7 a 1 V? —V —. 2
N 11",: #. ._1—__2._ '4 _ I‘. -1—2._.
_.e {1+r < 2 zk)+2[r ( 2 2k)[ }

>< Zenaaqwxaam , (518)

2'0

 

 

-o

r1=F2=§
Where we truncated the expansion at second order. In principle, nothing prevents to

one from including higher order terms. This is especially true in light of recent empha-

62

 

 

 

\] r" 1 J
. r. . k-
.

‘ 1 -
’ l

 

Hun

sis on the inclusion of beyond-second—order gradient terms in Skyrme like EDFs [35].
In that case, one needs to define additional local densities in addition to the ones

given in section 9.2.3.

Noting that the derivation is restricted to time-reversal invariant systems, the
next step consists of angle averaging over the orientation of 7", which is a reasonable
step as the scalar part of the OBDM has negligible dependence on the orientation of
F [176]. The final step involves averaging the dependence on the momentum scale I?
over a model phase space that characterizes the system under study. As mentioned
in section 5.3.1, we make two different choices. First, we perform the PSA with
the phase space of the locally-equivalent pure isospin infinite matter. Denoting the
function to to be averaged as 9,103), this operation amounts to setting the local phase

space distribution, fq(§, If), as
ME) = e<k— as», (5.19)

and thus

62,032) 3 [I (159411), (5.20)

E —_—3
4791:;

where GAE?) is the final result of the PSA. Prior to the application of the PSA,
Eq. (5.18) is angle averaged with respect the orientation of F. This is a valid ap-
Proximation as the scalar part of the normal component of the OBDM has a weak
dependence on the orientation of 1” [176]. The subsequent application of the PSA as
defined by Eqs. (5.19) and (5.20) on the resulting expression yields the DME of the
scalar part of the OBDM as
,1? —

_, _‘ 7.2 1 _' _.
pq(R+ > 2 H8<k2~rm<R>+gns(k‘;«r)[;,-qu<mmm)

Ml‘ir
[\DI‘ll

3 ..
+ gkfi2pq(R)[ , (5-21)

63

 

 

 

 
 

ulI I.

.r ..

‘ru or. s - Y‘l. .w. w. .

¢A .. .. p .. O t .. .

r. . . v». v! yr. » ,- x ‘ .
t a. .. ..7. ,. l; . -l
.. . i .
EJ- ‘~ « g

l.

l

..I
w

 

with

Ila/{724) E 3%? , (5.22)
Hangar) 2 3% . (5.23)
161“ 7'

The details of the derivation can be found in appendix 9.5.3, where we perform the
derivation in a more general context and recover specific cases. The reason why we do
not have IV; in the above expression is because the time-odd density, Luff), vanishes
for time-reversal invariant systems. Note that the 7r—functions given in Eqs. (5.22)-

(5.23) are completely analytical with no fit parameters.

However, the PSA did not invoke the diffuseness and anisotropy of the the phase
space of finite Fermi systems discussed in the previous section. With the step of angle
averaging over the orientation of 7", the orientations of F and I: are decoupled, implying
that the anisotropy is not going to play a key role in the subsequent approximations.

Hence, we concentrate only on the incorporation of the diffuseness.

Unfortunately, we could not find completely analytical ways of characterizing the
diffuseness. Thus, we parameterize fq(R, If). Inspired by the nature of the Wigner-
distribution of the phase space distribution as discussed in the previous, we use the
following ansatz to model the local momentum distribution

3 .2 .q2
k e—fik /AF

fq(§,lg) = C(1+ak—q2 9

(5.24)

Where a and 6 are parameters to be optimized, k} is the local Fermi momentum and
C is a. constants determined from normalization i.e. volume integral of the momentum

distribution should give 4 /37rk%3. Hence, C reads

8fi5/2 r
3fi (Ba + 23) °

64

C:

 

“I v ’
n a u
"..1‘ l
T

.‘\

 

v. .
\ \l
. - .1
“v.
MK; .v
0‘. 2 A
"v
.«
H

 

VL“

f1

Since it is of interest for later use, the average local RMS momentum is given by

 

 

112;, =(,/12) —_-3([/(5a+2*3) k; (5.26)

21'3(3c1+23)

where the average is calculated taking Eq.(5.24) for the local momentum distribution.
To finalize the determination of the scalar part 7r—functions, the following steps are
applied to Eq. (5.18): (i) Average the leading term over the local momentum distribu-
tion given in Eq.(5.24). (ii) Since the next-taleading order term is a small correction,
it is simply evaluated at the RMS momentum given in Eq.(5.26). Applying the above

prescription, one obtains the 7r—functions

Hp _ _ (a [C2272 — 2fi (28 + 30)) e —k(II;,2r2/(4(3) (5 27)
0 _ 25 (30: + 2H) '

115 = j(i.~g.2r ) (5.28)

 

  

51??”

where the actual values of the parameters a and [3 are obtained from numerical fits to
data obtained from converged self-consistent calculations of a selected set of isotopic

chains. For the optimization of these parameters and the results on the accuracies of

PSA-DME of the scalar part of the OBDM, refer to section 5.4.

Several comments are in order regarding the PSA-DME of the scalar part of the
OBDME and the 1r—functions given in Eqs. (5.22)-(5.23) and (5.27)-(5.28). To start
With, even though the phase space of finite nuclei has a marked difference from that of
IN M, the accuracy obtained using the two sets of 7r—functions is different at most by
a few percentage points, with the w—functions given by Eqs.(5.27)-(5.28) being the

better ones. This is apparent from the results of section 5.4. The reason being, unlike

the Vector part of the OBDM discussed below, the scalar part is a bulk quantity with

111081: of its contribution coming from the interior of the nucleus where, to a good

approximation, the momentum distribution resembles the one of IN M [176]. Hence,

65

 

..1

"my. - ‘
.l “.1111

fivj

.2 ‘-
1.”
'1“ _

1

v
ALI-‘1
[3’ u
'4-

[1+3

[”4

‘LV

 

this is the main reason for the comparable accuracies obtained using the two sets of

7r-functions.

From the form of the 7r—functions, it can be seen that the DME is not a naive Tay-
lor expansion of the OBDM with respect to the non-locality in 7". The 7r—functions
resum dependencies on r to all orders in some of the leading terms, such that the long
distance limit of the OBDM is reproduced. In the approach followed in Ref. [170], the
truncation of the expansion about I: to second order leaves H3 indeteminate. Specifi-
cally, the values of the coefficients of terms beyond kfir in the Taylor series expansion
of II; are undetermined. This indeterminateness gives one the freedom to optimize
11;, which can be viewed as selecting a different rearrangement and truncation of the

expansion [170].

Furthermore, the zeroth-order 7r—function 1180524“) given in Eq. (5.22) is exactly
the one found in the original NV—DME of Ref. [170]. Just as in the DME of N egele and
Vautherin, this particular PSA reduces to the first term in symmetric INM, thereby
reproducing the exact OBDM of INM. The second order 7r—function H3 (16%?) given
in Eq. (5.23) is different from

jalkl'fl)
(19%)")3 ’

 

II; = 105 (5.29)

which was obtained in Ref. [170]. However, this relates to the previous remark that
emphasized the freedom in choosing the second-order 7r-function. In conclusion, our
PSA-DME of the scalar part of the OBDM is essentially equivalent to the DME of
RBf. [170] if we choose fq(R,l-5) to be the phase space of locally equivalent neutron
or proton infinite matter. As a final remark, note that the phenomenological PSA
Whose 7r—functions are given in Eqs. (5.27) and (5.28) does not reproduce the exact
OBDM of INM. This should not be surprising as the model fq(é,g) used in that

case, Eq. (5.24), has a completely different analytical structure from that of the

66

' ’fuiul .1 ll.

irsifl I
.1.‘..u.

Valli" V'

L) 2
14‘,

Hi]

 

corresponding fq( ..’ I?) of INM (Eq. (5.19)).

5.3.4 PSA-DME for the vector part of the OBDM in time-

reversal invariant systems

Again, restricting the discussion to time-reversal invariant systems and applying the
same set of steps as for the scalar part of the OBDM, the vector part of the OBDM

can be approximated by

_. -* F " 77 * _. —o -o
.sq(R+ 5,12— 5) = Z 99,-(7‘202Q) <02l0l01>¢i(7‘101(1)10?i
270102

- - v —6 .-
=eir.ker.(_12_2_zk) 2 $3730”)
210102

x (O'QIElUl) 9915('F101(I)P?1

 

F1=F2=R
-- _, <7 _ v7 .-
26”“{1-1—7‘ [( 1 2 2—2k)} 2 (p1(T20'2(I)
10102
X <02l5l01>93i(FIUIQ) 10:11 a (5-30)
71=F2=R

 

where only the first order term in the expansion of the non-locality operator is kept.

The zero-order term in the above expansion provides the local spin density §q(§)
Which is zero for the time-reversal invariant systems. In fact, for time—reversal invari-

ant systems, the cartesian spin-current pseudotensor density Jq,pu(R') and its gradients

are the only standard local densities at hand to express the DME of the vector part of
the OBDM. Consequently, we could not express the higher-order (beyond first-order)
terms in the above expansion in a closed form in terms of the cartesian spin-current
Pseuclotensor density and its gradients. Nevertheless, section 5.4 shows that PSA-
DA’IE attains a high accuracy even at this level of approximation. Still, there is a

Possibility of studying higher-order terms in the context of the generalized Skyrme

67

 

 

 

EDF discussed in Ref. [35].

Here also we carry out the two strategies of incorporating the phase space infor-
mation: analytically and phenomenologically (parameterically). We start with the
analytical procedure which was also discussed in Ref. [170], though with no reference
to phase space of finite systems. Sticking with the first term, the authors in Ref. [170]
argued that averaging over the orientation of E and setting k = k}? should be sufficient

to provide a reasonable account of the vector part of the exact OBDM. This gives

_, "' F -' F , g 7' -‘
sq,” (R + 5, R — 5) 2 z H1(k%r) ETpJqW(R), (5.31)
where
HiU‘i‘fT) = jo(k}'»(1§)'r) - (532)

If instead one applies the same procedure as for the scalar part of the OBDM, i.e. one
performs the PSA over the locally-equivalent pure-isospin infinite matter phase-space,
as given in Eq. (5.19), one rather obtains

3151:1452) _

Hflkg‘r) : kq (R)T‘
F

(5.33)
However, and as mentioned in section 5.3.2, the local momentum phase-space
distribution of finite nuclei has a markedly different behavior than that of INM around
the nuclear (spatial) surface. Given that the vector part of the density matrix peaks
around the nuclear surface, it seems more appropriate to perform the PSA over a
diffuse and anisotropic phase space. Given that we do not have a parameter free
Way of introducing the diffuseness and the primary quantity to be averaged, emf,

couples the orientation of 1" and If, we limit ourselves to invoking the anisotropy of

the phase space. As a. completely analytical approach, we perform the PSA over a

68

deformed Fermi sphere that incorporates the information contained in the function
P§(R) discussed in section 5.3.2. We do this by averaging over a spheroidal local

momentum distribution given by

 

 

 

MRI—5) = 902’ - k2) (5.34)
where
k2 k2 2
k; = r; 3. + 5’. + '2... (5.35)
0(R)2 “(3)2 C(R)2

with a(R) and C(12) being position dependent quantities that relate to P30?) The
specific relations and various details of the derivation are given in appendix 9.5.3.
The final result differs from that in Ref. [170] only in the analytical form of Hi. The

result reads

~ ~q a
Hf(k}',r) = 3919(5)” , (5.36)
52am
where
~ -' 1/3 _.
kg. 2 (fl) 1am). (5.37)
2—1‘25’(R)

The PSA over the locally-equivalent neutron or proton infinite matter modifies

the analytical form of 11? compared to NV-DME, i.e. compare Eqs. (5.32) against

( (5.36) and (5.39)). In addition, and contrary to the scalar part of the OBDM for
which it is unimportant, taking into account the deformation of the local momentum
distribution of the finite system leads to a modification of the relevant momentum
scale 12%. In view of isolating the significance of such an effect, while preserving the

benefit of using PSA, one can set Pfllf) = 0 in Eq. 5.37.

In the second strategy, we incorporate the phase space information by parameter-

69

 

1~~v ‘ 7’
.iL

, .
(al.. A

, ‘. f
1171‘”.
I‘m -‘ ‘

u, ~r‘" "T

.. 1.1

Llf ]

 

izing the anisotropy of the Fermi surface. I11 leading order, one talks about quadrupo-
lar deformation of the Fermi Surface. Thus, the PSA is performed in a phase space

distribution with
fq(1'2',i5) = (9(5- 4:3.) (1+ a(3Cos2(6) — 1)) , (5.38)

where a is a quadrupolar deformation parameter to be optimized, 6 is the angle
between R and if and Si is the SinIntegral function. Even though the deformation
actually couples R and If, we approximate this as a coupling between F and I: to
actually obtain the final form of the 7r—function. Thus, Hi“ (R, 7') reads

_ 3 3.10627) (1

5' q
H1(kp7‘l— (1524‘) +(k%T)3

 

(—18 Si(k‘,’,r) — 61:24" c0s(k‘}+r) + 24 sin(k%r)) . (5.39)

To parameterize the deformation parameter a, we take hint from Wigner-Kirkwood
expansion of the scalar part of one-body density matrix. As explained in appendix 9.5.2,
the h2 Wigner-Kirkwood expansion of the Wigner transform of the scalar part of one-

body density matrix reads

- 52 , 52 1 ..
pwmam = as 425) — ,g—TgAvqé (A— ht) + 54—m[(vm)2 + 5(p-vrvq]
x 6”(A — 53,) + 005‘), (5.40)

Where ha, 2 H, = $172; + Vq(R) is the single particle Hamiltonian and /\ is the
Chemical potential. The origin of the deformation at this order is the (13' - V)2Vq(R)
term. It is well known that close to the nuclear surface, the self-consistent potential

that acts on the nucleons, Vq(R), and the density, pq(R), have similar profiles. Both
are usually approximated by the Woods-Saxon shape, of course with opposite signs.

Hence, we make the series of approximation [Fla/(KR) % AVq(R) o< qu(R), with

70

 

 

 

AW

... hp 1
ll.“ .Al-blll

r

l 1
Err]. .
V‘ a
-

' .v
..2. .

l
. V ‘

‘ ‘1

A
...

our final parametrization being
a. = m qu(R) + b (5.41)

where m and b are constants to be fit. If one sets a. = 0, the l—IflkZm) given in
Eq. (5.39) is recovered. In Sec. 5.4, we discuss and compare the accuracy obtained

using all of the preceding DME variants for the vector part of the OBDM.

5.3.5 It} and isospin invariance of the resulting EDF

Dealing separately with the neutron or proton OBDM in a finite nucleus, it is natural
to perform the corresponding PSA over the phase space of neutrons or protons of
the system. However, this provides 7r—functions with an explicit isospin dependence.
Even though this does not have any implication at this point, it does when we apply
the DIVIE to the HF energy of two- and three-nucleon interactions as discussed in
the next two chapters. This is because the EDF that results from the application of
the DIVI E breaks isospin invariance (but not its isospin symmetry). As mentioned in
section 2.2, there are isospin-breaking parts of the nuclear interaction, still the fact
that We get an EDF that breaks isospin invariance even when we start from one that
has that, symmetry might not be a welcome feature. A simple prescription to recover
Symmetry of isospin invariance is to replace all k} with the isoscalar kp which is

defined through
1 3
352 /

 

VV —0 —o —o

here 10(R) E pn(R) + pp(R). Fig. 5.4 shows k2, k}? and the isoscalar kp extracted
fr

0m a. converged self-consistent calculation of 214Pb. This is a neutron rich nucleus
an

d thus, the difference between the three momentum scales should be maximized.

C j. _. .. ..
onsldering the small difference between kHR), k%( R) and the isoscalar kF(R) that

71

..V-

'0

l'r'

 

 

 

1.4‘
1.2
1.0-

:5: 0.8-
E' .
“i 0.65
l
0.4-
)
0.2~

 

 

 

0°50 ' 2L0 ' 4L0 ' 6.0 ' 8L0 ' 10.0
R[fm]

Figure 5.4: (Color online) k2, k}; and the isoscalar kF extracted from a converged
self-consistent calculation of 214Pb, a neutron rich nucleus.

we see in Fig. 5.4, the prescription of replacing k} with [CF in the 7r—functions might

be a satisfactory method to recover isospin invariance in the resulting EDF.

5-3-6 Extension to non-time-reversal invariant systems

The PSA-DME approach has enabled us to obtain both analytical and parametrized

forms of the various 7r—functions that occur in the expansion of the scalar and vector

parts of the OBDM in time-reversal invariant systems. However, from a formal point
of View, PSA-DME uses the assumption of time-reversal invariance only to turn off
the tifile—odd densities. Thus, one can envision direct extension of PSA-DME to non-
time rexrersal invariant systems, where the time-odd densities such as 34R) in the
Case of 10.10"}, F2) and §q(R), TAR), PAR) in the case of §(f'1,772) start playing a vital
r Ole.

Nevertheless, it will be clear from the subsequent discussions that the appearance

72

 

 

3 l'
h".‘
r-r~'v

 

1
in“

 

of time-odd densities gives rise to various constraints that the 7r—functions have to

satisfy in order for the EDF (that results from the application of the DME to the

starting HF energy) to respect certain global and local symmetries. This requires
a systematic study of the problem. We tackle this by formulating a generic DME,
which we call the modified-Taylor series, and using it to perform a formal study of
the issues related to the extension of the DME to non-time reversal invariant systems.
The complete development of a DME for non-time reversal invariant systems with

specific analytical/parameterical 7r—functions is outside the domain of this work.

The modified-Taylor series approach was introduced in Ref. [168] and expanded
in this work. It consists of replacing the numerical coefficients in the Taylor series
expansion of the density being expanded with 7r—functions which are yet to be deter-
mined. These 7r—functions can depend on one or several variables. Illustrating the

eXpansion with the nonlocal scalar density and the nonlocal vector density, we have

 

 

 

 

 

-' -o "’ F --0 -¢
pq(7"1, 7'2) 2110(9) pq (R) + 111(9) 5 (V1 — V2) pq (r1, T2) ..
F1=F2=R
1 F 5 2 F 5 2
+ 5112(9) [(5 V1) + (5 V2) [pq(7‘1 , 7'2) (5.43)
i=fi=fi
F 7" -» _, _,
— I13(9) (— V1) (2 V2) Pq(T1, 7‘2) 5
F :17 =1?
1 2
and
35(R :1: 12:12 4: g) = ng(r2):,(f2') :t 113(9); (6. — 62);,(51, a) -
71:52:]
1 8 F _. 2 F 2 - - -
+‘H2(Q) - ' V1 + - V2 3q(7“1."2)
2 2 2 _- 5
T1—T2—
g F -* F '0 _. _. _.
_H3(Q) _ ' V1 — ' V2 Sq(r1 a 7‘2) (5.44)
2 2 F1=172=if

Wh
ere the 7r—functions are to be found analytically or optimized phenomenologically.

73

 

fl represents the variable 011 which the 71‘ functions depend. Requiring Q to be scalar,
dimensionless and depend on 1‘ implies that Q = rk. In case where Q is assumed
to depend on F, then 9 = h(F,lZ) for some scalar function h. Here, we assume the
7r-functions to be independent of the orientation of F. The choice of having four
7r—functions instead of five in Eqs. (5.43) and (5.44) is motivated by the need to get
a symmetric expansion in R + g and R — g.

Even though a definite approach with which to constrain Q and the n—functions
is be discussed, it should be mentioned that the modified-Taylor series approach is
formally applicable to all local/nonlocal and/ or normal/ anomalous densities in both
time-reversal and non time-reversal invariant systems. In line with this, the modified—
Taylor series expansion for all the densities defined in section 9.2.3 and 9.2.5 is given
in appendix 9.5.7. In the construction of this expansion for the various densities,

we have not made any reference to a constructive way of fixing the basic expansion
variable, 9, and the 7r—functions. This is where explicit connection is realized between
the modified-Taylor series and PSA-DME (and/ or other DME variants) discussed in
the previous few sections. In other words, the modified-Taylor series can be seen as a
template which can be adapted to various DMEs which in turn can be considered as
as approaches to fix 9 and the 7r—functions. However, there is a technical problem
in that, one cannot, at this point, fully express the modified-Taylor series expansion
in terms of standard local densities. Hence, we need one more layer of assumptions

to realize the explicit connection.

Both problems are solved at once by realizing that the basic quantity that one is re-
ally interested in approximating is the energy density (or energy) at the Hartree-Fock-
Bogoliubov level instead of the local densities. One starts from the exact expression
for the energy, and approximates it by replacing the exact densities (local / nonlocal)
With their counterparts as given by the modified-Taylor series expansion. Requiring

th
e reslilting expression to be a local EDF which fulfills various local and global sym—

74

 

n' ’Y".'.
T..-

‘VV‘V-
, .

[f ‘
-‘-» «1.

.,,.
at

 

.A- .

v.

 

metries results in several constraints relating the different 7r—functions. The required
local or global symmetries can be rotational invariance, parity, particle number, time—
reversal invariance, isospin invariance and local gauge invariance or its traditional
counterpart, Galilean invariance. Naturally, the validity of some of the symmetries

depends on the starting interaction.

In order to make the procedures clear, these steps are applied to typical terms
from the central and tensor exchange parts of the Hartree-Fock energy (of two nucleon
interaction). Since we are interested only in the form of the expression, numerical
coefficients and spin-isospin labels of the interaction are dropped. As discussed in

section 9.6.1, a typical term from the central part of the interaction takes the form
(@[VC|<I>) = Z / at]? d? M.('r)pq(F1,F2)pq(F2,F1), (5.45)
«1

Where [(12) is the Slater determinant HF wave function. Next, we apply the modified
Taylor series expansion of the densities, use our assumption that the 7r—functions

are independent of the orientation of F and perform angle integration f er, thereby

obtaining

<<I>|Vo|<r>) = 47.2 dedrVU") [(Hgm» p§(R)+ 012(0)) j.(R')-iq(127)

7.2
_3_
+—122Hfi(9 )HP(Q) Pq(R) (A9002) W270 ))

— % 113(0) 115(0) p.02) 5.02)] . (546)

w . . . . .
here We have truncated terms contammg beyond-second order derlvatlves. Applymg

th . . . .
e Same set of steps to a typlcal term from the tensor 1nteract1on, (<I>[VT|<I>), we obtaln

(<I>|VT|<I>) = ZdedFVT(r).§,(F1,F2)-.§‘q(ffz,F1),
q

75

= 4nZ/(11’2'drv7.(r) [(ng(§2))2§q(1§).§,(1§)

q

.2 y _. 4
+ :3- (H‘ilfllf Z qulequwrlR)

2 _, _. _. ..
+{5 3(1) WEAR) MAR) -2 (1(3))
7‘2 ~ —- -' -' -'
_ 6’ H30") H30) .§},(R) - mm] , (5-47)

with similar procedures being applicable to the remaining terms in the HF B energy.

In order to make the explicit connection between the modified-Taylor series ex-
pansion and PSA-DME (and / or other DME variants), we make use of the PSA-DME
(and / or other DME variants) to approximate the Fock energy of time-reversal invari-
ant systems and set it equal to the corresponding expression obtained from for the
modified-Taylor series expansion. In this way, the 7r—functions and their arguments Q
in Eqs. (5.46) and (5.47) that multiply the time—even densities can be fixed. Still, the
TF—functions that multiply the time-odd, local and anomalous densities are not yet

determined. This is where the relations between the 7r—functions through symmetry

and other constraints come in to the picture.

Constraints on the n—functions

Requiring the 7r—functions to be independent of the orientation of 7" and the need to
Obtain gauge (Galilean) invariant bilinear combinations of densities in the resulting
EDF iIIlpose strong constrains on the 7r—functions. The gauge transformation of the
one‘body local densities is discussed in appendix 9.3. These constraints are obtained
by applying the modified Taylor series expansion to the various local and nonlocal
densities that occur in the HF B energy of a finite-range two-body interaction and
reqlflring the resulting EDF to be gauge invariant. Dropping the arguments of the

7r\
funCtions for ease of notation, the resulting constraints read

76

 

 

”wa—

L 'v
M)?

 

(i) H3015 + 115;): 2[n’;]2 and 115012;” + 113“) = 2[Hf]2

—o

--pJ"_T.§‘. P.T_.§‘j
(11) 7’0 7rl — T6771 and 7rl 71'0 — 7r07r1

0011811? = 11‘: H5

(iv) 3; 2 I13 where ”U is either 5,151,772) or 5:1(F1J'2)

(v) 7r—functions of the pairing densities have to be real-valued functions.

One of the most important qualities of the original DME of Negele and Vautherin
is its exact treatment of unpolarized, symmetric INM at the Hartree-Fock level. Con-
straining the 7r—function DME to reproduce INM limit of the direct and exchange

parts of the energy density separately, we obtain the following two constraints on the

7r—functions

(«(02 = 1 (5.48)

p p p j1(rk) 2

(NC)2
10

 

(1192 -

1
Where k must reduce to k} = [37rzpq]3 when one goes to INM. Thus either one
has to use a fixed 19 = k}, or the parameter k should be such that it evolves to

1
k “+ k2. = [3w2pq] 3 as one goes to INM.

F11I‘ther constraint is obtained using the idempotency of the density matrix to

€XpreSS the particle number. I.e. from pg 2 p3, the particle number can be expressed

as

N. = Trpq=/d§df'lp(1?fl|2 (5.50)

Thus

a a constraint on the 7r—functions can be obtained by inserting the DME expres-

77

sion for pq(fi, F)

 

2 4 a
N, = f deF Hampqua+m€<nm<12>+ —2'"—4H5(n>(qu<R>-2TQ(R))
r2 p ~ 2
—EH3(9>Tq(R>

 

= 47r [dédr [H3(Q)2pg(§) —
7.2

12
2

— % 113(9) Ham) pqu‘i) mfé) , (5.51)

+ 113(9) H§(Q)pq(§) (A qu-é) _ 2771”?»

which is an integral constraint that can be utilized a posteriori to calculate some
parameters. In the original DME and its variants, it can be shown that the num-
ber constraint is satisfied exactly [202], while the PSA-DME breaks this constraint

slightly.

Finally, constraints on the 7r—function come from the large and small limits of r.
The W—functions should go to zero for large 7' and for small 7', the modified Taylor
series has to reduce to ordinary Taylor series expansion. In addition, we require the
7r ~functions to be such that the gradient, 6;, and the gradient squared, [57;]2, of the

densities are reproduced exactly at 7" = 0. The resulting constraints are

110(0) = 111(0) = H2(0) = 113(0) = 1,

6(0) = H'1(0)=0,

ng(0) = 0, (5.52)
lim H0(r) = lim H1(r) = lim 112(7‘) 2 lim H3(r) = 0, (5.53)

W . .
here we have dropped the denSIty label on the 1r—functlons to denote that these

CO
IlStI‘aints hold for the 7r—functions of any density. Additionally, we used only II,-

78

which refers to the it“ 7r—function from nonlocal densities, even though the constraints
are valid for 7r, (from local densities). The only exception is there is no 53 in the local

case. In the DME of Ref. [172], the authors impose a local constraint

= mi?) — 15(5), (5.54)

F=0 4

 

— [mama]

which is to be satisfied by the DME of pq(F1, F2). They refer to this constraint as
the local imposition of the correct kinetic energy density. One recovers Eq. 5.54 by
combining Eqs. (5.52), (9.108) and (9.109). It can easily be shown that 7r—functions
satisfying the small 1' limits given in Eq. (5.52) satisfy this constraint. I.e. this

particular constraint is a subset of the constraints listed in Eq. (5.52).

Concluding, the explicit relationship that we established between the modified-
TaYIor series expansion and PSA-DME (and / or other DME variants), together with
the various constraints obtained through symmetry and other subsidiary conditions
enable us to reduce the number of independent unknown w—functions significantly.
Still, the number of unknown 1r—functions is larger than the number of constraint
relations. Thus, the complete determination of all the 7r—functions requires further
parametrization of some of the 7r—functions. In practice, it may not be possible
to Satisfy all the relations among the 7r—functions and at the same time obtain a
reasonable accuracy. In that case, some of the less stringent constraints have to be
1‘EIE‘J‘ied. Since this work is confined to the development of non-empirical EDF for
timfireversal invariant systems, most, if not all, of these constraints are satisfied by

de

fault as one can simply choose the 7r—functions of the time-odd densities in such a
“’3.

y that they satisfy the constraints.

79

 

--.-..~~-_

t—Auu

 

 

5.3.7 Remarks on the DME of the local densities

The apparent need for the DME of local densities can be seen from the Hartree con-
tribution to the energy originating from the central part of a two-nucleon interaction.

Reproducing the expression derived and discussed in section 9.6.1

()4...) = / daczapqm) 12,/(a). (5.55)

where for simplicity, we have dropped the singlet-triplet label of the interaction and
numerical coefficients. Thus, if one requires a local EDF, one needs to approximate
Eq. (5.55) utilizing a suitable DME expansion for the local densities. Equivalently,

one can approximate the energy density, Eden, which is defined as

1 _. ..
Eden 5 Z7? /dQ77/)Q(Ir1)pq’('r2)a (556)

With f d9; referring to angular integration with respect to the orientation of F.

In line with this, Negele and Vautherin, in Ref. [170], approximate the energy
density given in Eq. (5.56) as

- _. 35 . 1 - - 1 .. -
goes ,5 pa ”9.1/(R)+Wnlrkr‘)[qu/(RlqulRl‘l'zpqulAPqKR)
F

1" -0 -o —o
5mm) - vmm] , (5.57)

by applying the expansion technique they devised for the nonlocal density, p(7"'1, 7‘32),
to the product of the local densities, pq(f'1) pq,(1'"2). However, subsequent numerical
teStS U183],[184]] indicated that the expansion of local densities is at the root of most

of the error propagation and enhancement in self—consistent tests of the DME. This

iS discussed in detail in section 5.4.6.

Even though the DME makes no direct reference to the range of nonlocality, as

80

mentioned in section 5.2.1, the fact that the range of nonlocality with respect to
F is very large for the local density, pq(F1/2) (where 771/2 means the argument can
be 7‘} or F2), is mentioned to be the main reason why the DME does not work as
accurately as it does for the nonlocal density, pq(F1,F2). In Ref. Bhaduri78, using
a one-dimensional harmonic oscillator model with partial occupation of the single-
particle states, the authors show that the nonlocality with respect to F of the local
nucleon density, pq(F1/2), varies on the scale of the whole system, while the scale
of the nonlocal nucleon density, pq(f'1, F2), is set by the local Fermi momentum k5.
Even in the surface of nuclei, one can see that £071,152) falls off much faster than
pq(7"’1)pq(f'2) in the relative coordinate, 1", by considering a one-dimensional surface
With an exponential decay [183]. Le. by modeling the local density with a schematic
eXponential decay function. Hence, the fact that both the local density and energy
density involved in Eq. (5.57) have a large nonlocality scale with respect to F make

the DME approximation inherently inaccurate, at least in one-dimensional problems.

Nonetheless, in problems with dimensions greater than one, the simple charac-
terization of the failure of the the DME of pq(F1/2) based on the scale of nonlocality
needs refinement. This becomes obvious when one considers closed-form analytical
9X1)? ESSions for pq(f'1, F2) and pq(1"'1/2) in various model systems. In Ref. [208], a closed
form eXpression for pq(1'"1, F2) is given for the case of an isotropic harmonically trapped
ideal Fermi gas in any dimension. The more relevant expression is the one given in

RBf- l185] for a three-dimensional oscillator with a smeared occupancy

_, _, _. 1 1 + t
pq(r1,7'2) = quR) exp [“1027‘21—_—t] a (5-58)
pq(T1/2) = W(1 — t2) 3/2 eXp [—Q2r¥/2T_—t:| (5.59)

Where t = 8—6 h”, ,3 is the inverse temperature, a2 = mw/h with the energy 8 = (N +

3/2) hw. Thus, one can argue that both the local and nonlocal nucleon densities are

81

y .
Ullznl‘

(Jr‘

4
V 1
1

. r ,. .‘
.1. it‘ (

I 1‘)"

governed by comparable scales, relegating the supposed large scale of the nonlocality

in F as an incomplete or limited explanation for the failure of the DME of pq(F1/2).

The missing piece of the explanation can be identified once the DlVIE—coordinates
are replaced in Eq.(5.59), viz, F1 /2 = 1:1" :1: 1 / 2 F. This makes the difference between
the local and the nonlocal densities to be transparent. The nonlocal density falls-
off exponentially (Gaussian fall-off) with respect to r independent of the orientation
of F, which is also the case for pq(F1, F2) extracted from a converged self-consistent
calculation of nuclei as shown in section 5.4. In contrast, the local density shows
maximally different behaviors depending on the orientation of F. In short, sitting
at a particular location in the nucleus, R, one can go to the surface or deep into
the interior of the nucleus with the same 1" but different directions of F. Thus, the
Significant dependence of pq(F1/2) on the orientation of F is partly responsible for the

failure of the DME of pq(F1/2) as DMEs invariably average over the orientation of F.

In Ref. [183], it is argued that the DME of the Hartree contribution can be avoided
by treating it exactly, especially as the exact Hartree treatment does not result in
8. Significant increase in numerical complexity. In a related work, Ref. [184], the
authors show that treating the Hartree contribution exactly removes most of the
errors in the self-consistent numerical test of the DME. This and related issues are

discusSed in section 5.4.6. Our numerical tests include both expanded and exact
Hartree treatments. In the expanded case, the 7r—functions of pq(F1/2) are fixed by

equ&ting Eq. (5.57) with the expression obtained from replacing pq(F1/2) in Eq. (5.56)
with

1 F _.

Dam/2) e «am pm) i «1(9) Viz/W?) + 57510) (— Var/1.15), (5.60)

Ml‘h

82

. 1 .. y, . I l . n
I 1 y . E

....u w u) .14 l, cl

1 . .)

.nd.

 

and truncating at second-order in the gradient

, 2

 

7' ... _. ...
a... s [75(9)] p.112»), [(1%) - —[ {(9)} qum) vp,)(R)
+1—27T 7mm)[pq(R) )q(A/) I )+ pq /(R R)qu(R) , (5.61)
which results in the 7r—functions
105 .
105 .
775(9) = mhh‘ki) , (5.64)

Where (2 = rk‘}.
The parameterized version of the 7r—functions for pq(F1/2) is inspired by the an-
alytical form of pq(F1/2) in the three-dimensional harmonic oscillator with a smeared
OCCuDancy [185] as given in Eq. (5.59). First, we fix the oscillator frequency w and

the Oscillator length b according to the Blomqvist-Molinar formula [38]

ha) = 45A‘1/3—25A‘2/3, (5.65)

197.33
b = —, 5.66
V940fiw ( )

where A is the mass number of the nucleon under consideration. The parameterized

Tr"functions are given by

p 0‘2 2 a4 4 —r2/b2
50(9) = (10+ —b— + 3 e , (5.67)
WW?) = e‘TQ/bz, (5.68)
2 2
713(9) = err/b , (5.59)
(5.70)

83

where the gradient corrections are damped with a gaussian of range b. From the short
range limit of the 7r—~functions as given in Eqs. (5.52), the leading parameter a0 = 1.
The rest of the parameters, viz, {(12, (14} are fixed by fitting the exact Eden as given by
Eq. (5.56) with Eq. (5.61), with densities extracted from a converged self-consistent

calculation of a selection of nuclear chains.

A direct justification for the form of the 7r—functions given in Eqs. (5.67)-(5.69)
comes from the fact that pq(a:) of spherical nuclei can be fit to a very good accuracy

with the ansatz

11:4
III" _r2 2
p.99 = qu(0)an5;e (b , (5.71)
n=l

where pq(0) is the value of the central density and :1: stands for 1‘1, 7'2, 7' or R. The fact
that p(0) is used instead of an additional free-parameter is due to there being local
densities that play a similar role in the DME. Fig. 5.5 shows the neutron density
Obtained from a converged self—consistent calculations of 48Cr and 208Pb and their
Corresponding fit curves Our extensive tests show that the pq(F1/2) length scale b
given in Eq. (5.66) remains uniformly valid and, perhaps not surprisingly, one can
pr 0duce the same high-quality fits to almost all nuclei. However, the parameters show
Strong Shell fluctuation, as can be seen from Fig. 5.6 which shows the parameters for

CI‘ and Pb isotopic chains.

The above discussions consider only the local nucleon density, pq(F1/2). The strong
fluctuation of the other time-even local density, .Z,(F1/2), with respect to F due to its
strong dependence on the shell structure of the particular nucleus under study, did

“0‘ permit a systematic analytical study. Hence, for the numerical tests carried out
in Section 5.4.6, a simple Taylor series approximation is used to fix the W—fllflCthnS
of j1/2

7Tb = 7r? = 713?: 1, (5.72)

Where the arguments must be extended in non time-reversal invariant systems as the

84

 

 

 

0.101
‘ _ \

3.0008.
'5: .

fl
4.30.06-

: .

G

3.10.041

5

g 1

0.021

0'000 ' 2 ' Z: ' 6‘50'2'4'6'8'1‘0

 

R [fm] R [fm]

Figure 5.5: (Color online) pn(r) for 4807" and 208Pb from a converged self-consistent
Calculation using Sly4 EDF.

      

Parameters

 

 

 

‘ VVVO’O’O\
rro’ \
2°5l —B— :12 u
. \ 3
- \,/ \
1.5] —_A- a‘
"+' as
0.54 *0" 4 1 +++_++_+_.,.+.+-+-
++-+-o--+-‘-\+-o--+-+-+-+-+-+-+-+-+'+'* l
'OOS“ ‘ J)
\o—o—o—O-o-o-oor' H
‘ 1 ~ AH. .
.1.5‘ , ° Whhh‘fl/
2'0 - 2‘8 T316 ' 4'4 - 5'2 - 9.6 T 194 71127 120 - 118 I
N N

Figure 5.6: (Color online) The parameters for Cr and Pb isotopic chains obtained
after fitting the neutron density, pn(F1/2), with the 7r—functions as given in
Eqs. (5.67)-(5.69).

85

time-odd local densities jq(F1/2), .§’q(F1/2) , Tq(F1/2) and Rq(-F1/2) do not vanish. In this
context, the discussion in section 5.3.6 is relevant. Finally, it should be mentioned that
these expansions of the local densities can be avoided once the Hartree contribution
to the HF energy from two-nucleon interaction is treated exactly. The situation is
different and more complex for the HF contribution from the chiral EFT three-nucleon
interaction at N2LO. However, with a particular choice of DME-coordinates, we avoid

the expansion of local densities altogether. This is discussed in section 7.2.

5.3.8 Remarks on the DME of the anomalous densities

Currently, there are several simple effective interactions that are being used in the
pairing channel to perform HFB and related calculations. From a practical point
0f view, the simplicity of the effective interactions is necessitated by the numerical
COmplexity that one would have to overcome in order to perform a 3D (deformed)
HF B calculation for deformed nuclei. Still, the accuracy of current pairing part of
C"Trent functionals calls for further improvements and constraints on the form and
CDuplings of the functional [26].

Similar to what is being implemented in the case of particle-hole part, two com-
plementary approaches, viz, phenomenological parameterizations and non-empirical
construction of the functional are being undertaken [112]. The non-empirical ap—
proach tries to address the role of the the bare NN + NNN interaction and their
finite“ranges by successive addition of MBPT contributions in the pairing channel.

Recently, the first step towards non-empirical pairing functional for nuclei has been
taken in Ref. [112] where the first systematic calculation of pairing gaps in semi-magic
mldei is carried out. By fixing the normal self-energr contributions with conventional

Skyrme functionals, using low-momentum NN interaction and accounting only for the

contribution of 130 partial wave to the pairing gaps, the results show that it is indeed

the leading order (Bogoliubov diagram) that contributes the bulk of the pairing gaps

86

in finite nuclei. However, including N NN interaction in the treatment degrades the
agreement between theoretical and experimental results leaving plenty of room for

coupling to collective fluctuations.

One caveat of these works is the fact that they are limited to spherical nuclei, due
to the aforementioned numerical complexity to perform 3D HFB calculations. The
nonlocal contribution to the total energy from the pairing part can be seen from the
form of the leading order (Bogoliubov diagram). Reproducing the expression given

in Eq. (9.272) for the spin-singlet, isospin-triplet channel from the central part of the

interaction,

(tn/31(5),... or Z / dfi ng V810) |,3,(7~1,7=‘2)|2, (5.73)
q

Where one notes that the nonlocal pairing density has the same role as the role of

Pq(F1, F2) in the particle-hole part of the functional.

From a formal point of view, the modified-Taylor series DME can be directly
aPDIied to the anomalous densities, thereby approximating the exact leading-order

nonlocal pairing functional with a local one. For instance, the nonlocal pairing density

can be expanded as

 

 

~ -o 77 __, 7? ~ ~ —0 r F " -' ~ —o —o
pQ(R It 5, R 3F E) % Hg(Q)pq(R) :l: Hf(fl)§ ' (V1 — V2) q(T1,T2) .,
Tl=d2zR
+ ‘2‘ 112(9) 5 ' V1 + 5 V2 pq(T1, 7‘2) _,
1=®=R

- 4 —o F 4 ~ --0 _'

—H§(9) (5 ° V1) (5 ' V2) Pam) 7‘2) 1 (5-74)

F1=72=R

Where the 1r—functions are yet to be specified.

However, both the technical and conceptual problems that need to be settled in
01‘der to obtain accurate 7r-functions for the anomalous densities seem to be signifi-

cantly harder than for the normal densities. To start with, the size of the nonlocality

87

with regards to F of the nonlocal pairing density, 5,,(R :l: g , R IF ’5), in finite nuclei
is still under discussion [[186],[187]]. It is commonly characterized by the coherence
length, 6, of nucleonic Cooper pairs. Arguments based on infinite nuclear matter and
the local density approximation (LDA) seem to suggest that f as given by Pippard’s

relation [188]
R2 A F

__ 5.75
7rm*A’ (a )

f:

is of the size of the nucleus. In Eq. (5.75), m" is the effective mass and A is the pairing
gap. If the supposed large spatial extension of the nucleonic Cooper pairs in finite
nuclei were to held without any modification, a DME approach which does not rely
on any assumed short-range nonlocality needs to be invented. In fact, the stronger
versions of these arguments stipulate that the existence of a small parameter ro/fi,
Where r0 is the interaction range, implies that pairing in nuclei should be insensitive
130 the details of the nuclear interaction. This is supposed to justify phenomenological

Paralneterizations using a local functional with no derivative corrections [189].

Practical calculations in finite nuclei paint a moderately different and favorable
picture- Indeed, results from the recent non-empirical pairing calculations [112] sug-
gest t311%?“ some details of the interaction may be important. Furthermore, the intuitive
arguments that one builds starting from INM regarding such quantities as the size
0f the nucleon cooper pairs require precise qualification. In Ref. [187], the authors

StUdY the neutron correlation length, 5,,(R), having defined it as

 

fdr 1"4[)5,,(I_f,F)|2
fdrr2[5n(R,1:)[2’

 

MR) = (5.76)

Where the subscript n denotes that the quantities are extracted for neutrons. The
Strong position (density) dependence of the correlation length can be seen from

Fig. 5.7. In addition, the authors extract [fin(R,7‘) I2 which is shown in Fig. 5.8.

88

 

14

 

 

 

 

 

 

 

 

F.
2115:“? 5.7: (Color online) Coherence length 5 (R) for 220, 600a, 6”Ni, 104572, 12oSn,
5 (From Pillet et. 51. [187]).

89

It can be seen that the nonlocality in F is in general in the range of 2 — 3 fm. In
contrast, the correlation length is much larger except close to surface where it attains
values in the range of 2 —— 3 f m. Nonetheless, the basic quantity that is approximated
by the DME is [ 5,,(R, 1") [2, which reduces the significance of the large values of the
coherence length. In Ref. [190], the authors conduct a related study of a slab of in-
finite nuclear matter, confirming the smallness of the local correlation length at the

surface and its largeness inside the slab.

       

8

A6

5.4

[I .
2 , .
0 . ,
0246 024.3000“H

r(fm) r(fm) mm

Figure 5.8: (Color online) [5,,(R, 7‘)[2 calculated with HFB-D18 for 10487), 120811,
12sign. Scale has been multiplied by a factor of 106 (From Pillet et. al. [187]).

Hence, these realistic studies of pairing in finite nuclei point to the possibility
of developing a DME for the anomalous densities. In Ref. [191], a leading-order
semi—classical expansion of the anomalous density based on the Thomas-Fermi ap-

Pl‘oximation is given as
545.5) s Chases/245'). (5.77)

Where jo denotes the spherical bessel function of order zero and 0 stands for constants
Characterizing the pairing field strength and the local Fermi momentum. In Ref. [187],

it is Shown that even at this level of approximation, there is a qualitative agreement

90

between the exact and the corresponding DME approximation. In Ref. [38], the
Wigner-Kirkwood h—expansion of the Bloch propagator of a superfluid system is
derived (up to 112). Leaving aside any further approximation that might be required,
performing inverse laplace transform of the h—expanded Bloch propagator should
recover the gradient corrections to the leading-order expression given in Eq. (5.77).
In relation to the DME of anomalous densities, further works along these lines include
working out the analytical expressions for the inverse laplace transform, recovering

the expressions for the n—functions and extensive systematic accuracy tests.

5.4 Accuracy of DME

The accuracy of a particular DME can be tested by comparing the exact density
with its DME—approximation. But, our main objective is approximating the HF
energy density and energy from two— and three-nucleon interactions. As discussed in
section 6.1 for the two-nucleon case and section 7.1 for the three nucleon case, the HF
energy expression involves a bilinear or trilinear combination of densities extracted
from the OBDM. The numerical tests we conduct in this work concentrates on how
well the DME approximates various contributions to the two—nucleon HF energy, in
both non self-consistent and self-consistent HF calculations3.
Prior to going in to the details, the following remarks are in order: (i) We consider
only time-reversal invariant systems, (ii) Primarily, we concentrate on the DME of
pq(F1, F2) and §q(F1,F2). In the two-nucleon HF energy, these nonlocal densities oc-
cur in the Fock/exchange contribution from the central, tensor and spin-orbit pieces,
While the picture is both different and significantly more complex in the case of three-

mlCleon interaction (refer to section 7.2). The fact that we concentrate mainly on the

Fock contributions is because the DME is inherently inaccurate for the Hartree con-

 

3 . . .
_ In this section, we make repeated references to the HF energy from the generic ave-nucleon
Intel‘action derived in the next chapter.

91

tribution from N N interactions [[l83],[l84]]. This is indeed confirmed in section 5.4.6,
where we advocate treating the Hartree terms exactly. (iii) In a related note, the
accuracy of the DME in reproducing the various contributions to the HF energy from
three-nucleon interaction (chiral EFT three-nucleon interaction at N2LO) has not
been tested. However, we use analytical approaches that ensure that the approxima-
tions used in these calculations are exactly the same as the ones we use for the HF
energy from N N interactions. (iv) Besides the two PSA-DMES (the analytical and
parameterized), we include the original DME of Ref. [170] for the accuracy test, as
all the other DMEs [ [171]- [173]] concentrate only on pq(Fl, F2) and give comparable

accuracy.

The three DMEs that we test in the following several sections are labeled as PSA-
DME, PSA-DME—II and NV—DME. For PSA-DME, the 7r—functions are given in
Eqs. (5.22), (5.23) and (5.36), PSA-DME—II (the parameterized version) is the one
with w—functions as given in Eqs. (5.27), (5.28) and (5.39) and NV—DME refers to
the original DME of Negele and Vautherin with w—functions as given in Eqs. (5.22)
(the same as PSA), (5.29) and (5.32).

5.4.1 Inputs to non-self-consistent tests

The generic form of the central, spin-orbit and tensor interactions in the different
Spin-isospin channels are discussed in section 9.6.1, 9.6.2 and 9.6.3. The radial form
factors in the present calculations are either a gaussian or a renormalized Yukawa

according to Ref. [192]). Specifically we use

()0 e-'2/“2 ,
va(7) = (5.78)

329[(ra""7r"'erfc("—]71 — TA) — (T -> —T)] a

92

 

independently of the spin/isospin-singlet/triplet channel, (S, T), and with co = 50
MeV, a = 1.5 fm, m, = 0.7 fm’l. The momentum cut-off A is set equal to 2.1 fm‘1

while erfc is the complementary error function which is defined as

e7'fc(;1:) = % foo (It (42. (5.79)

It must be stressed that none of these interactions are realistic two-nucleon interac-
tions, but rather schematic representatives. Still, they are reasonable form factors as
the objective of this section is to gauge the accuracy of the DME variants against a
reasonable reference point that is not itself meant to provide useful or realistic results.
Finally, note that neutron density matrices and local densities used in the following
sections have been obtained, for all semi-magic nuclei of interest, through spherical
self-consistent EDF calculations employing the SLy4 EDF parameterizations with no

pairing.

5.4.2 Fock contribution from VC

In time—reversal invariant systems, the expression of the Fock contribution to the
energ from the central part of the two-nucleon interaction contains a bilinear product
of non-local matter densities as well as a bilinear product of non-local spin densities.
Since the latter also appears as part of the tensor contribution to the Fock energy,

we postpone the discussion regarding the spin-density product to section 5.4.3.

Before comparing the Fock energy to its DME counterpart, we first conduct a
m Ore stringent test 011 the energy density in which the integration over the angle of

I“ has already been performed, i.e. we compare the integrand

_. 1 _. _' _. ...
C:n(R1T) E E[(iQrPralrlyr'z)0,.(7‘2J‘1) , (5.80)

93

 

to its DME counterpart

—o 1'2

Ohm“? 1'2" 2 HP 2 1? 12 up I." n” ii 1A R
nn ( J“) — 0( r") Pnl )Pn( ) + 0( Fr) 2( F’)Pn( ) Pn( )
3 4

.. 3 _.
- 7.71(1?) + 3A}1‘2p12(R)) 7 (5'81)

where the latter depends on which variant of the DME has been adopted. We de-
note such integrands as energy densities throughout this section. Strictly speaking,
it is necessary to multiply them by the interaction to obtain the dimension of an
energy density. Still, we postpone the folding with the interaction to the second mea-
sure introduced below. In the definition of CngMWR, r), we have truncated terms
with beyond-second order gradients, in line with current phenomenological imple-
mentations of Skyrme EDF. In addition, a consistent account of such fourth-order
derivatives in the EDF would require to go also to fourth order in the DME itself,
which is not addressed in this work. This is an important point that underlines our
philosophy that the primary purpose of the DME method is not to reproduce the fine
details of the OBDM, but rather to reproduce as best as possible the energy density
and the total energy at a given order in the expansion. The latter two are precisely

what are gauged in this work, whereas no tests dedicated to the reproduction of the

OBDM by itself are performed.

The parameters of PSA-DME—II 7r—functions of the nonlocal matter density, as.

defined by Eq. (5.27), read
a = —0.4130, )3 = 1.2430. (5.82)

We obtained these values fitting Eq. (5.80) with Eq. (5.81) using densities extracted

fiOIn a converged self-consistent calculations of a selected set of nuclei.

94

 

NV-DME PSA-DME PSA-DME-II Exact

9
E6
L‘.
I"3
0 M 7“) mi Mi
0 3 6 3 6 9

9 0 3
R [fm] R [fm] R [fm]

OUQO

 

 

  
 

— NV-DME
— — PSA-DME
' " ‘ PSA-DME-II

R = 3.0[fm]

 

 

 

 

\
R = 6.0[fm] \ /

 

 

 

 

' o i i 3 (1 i 2 3
r [fm] r [fm]

Figure 5.9: (Color online) Comparison of 05,,(R, r) and thDM E (R, r) where the
latter' lS computed from the 7r- —functions of one of the three DMEs: NV~DME,
PSA-DME or PSA-DME—II. Upper panels. two-dimensional integrands. Lower
Pa-nels: ratios of CF51) M E (R 7*) over 05,,(R, 7‘) for fixed values of R. Densities are
0h tained from a self-consistent EDF calculation of 208Pb with the SLy4 Skyrme
EDF in the particle-hole part and no pairing.

95

 

Figure 5.9 shows " that all the three DMEs provide comparably good profile-
reproduction of the integrand C5,(R.7‘) within the typical range of nuclear inter-
actions (7‘ ~ 2 fm). Beyond such a non locality, the quality of the reproduction
deteriorates significantly, with that of PSA-DME deteriorating slightly faster. In ad-
dition, one sees from the lower panels of Fig. 5.9 that the quality of the reproduction
decreases as one goes to the nuclear surface, i.e. for R Z 4 fm. This is slightly
improved by taking into account the diffusivity of the local momentum distribution
when designing the PSA-DME for the scalar part of the OBDM, as shown by the
better accuracy of PSA-DME-II. In general, PSA-DME—II stays much more close to
one than the other two DMEs, with slight overestimation in the range of nuclear inter-
actions. Note also that, although the plots are provided for two sample nuclei, more

systematic tests have been performed over several semi-magic isotonic and isotopic

chains that support such conclusions.

Coming to the energy itself, i.e. to the integrated product of the interaction 110(7‘)

with the central energy density, we compare

Eghm] : 47r/dR (17‘ 7‘2 11(7(r) GEAR, T) , (5.83)
Eg’mwhm] = 4Tr/dR (17‘7‘2 ’UC(T‘) CHF,’PME(R, 7'). (5.84)

Figure 5.10 shows the relative error obtained from the three DME variants com-
pared to the exact Fock contribution for both the Gaussian and the renormalized-

Yukawa radial form factors and for three semi-magic isotopic chains.

Let us start with Fig. 5.11 that shows that the dependence of the accuracy on
the range of the (Gaussian) interaction used is significant, i.e. about a factor of 1.5

hatween a = 1.0 fm and a = 1.5 fm. As can be expected from the two—dimensional

 

 

4 . . . . . . . . .~ «
Note that for semi-maglc spherical nuclei used 1n this work, the energy densities C,’,,,(R. r) and
F. - , _.
CrinD M E (R, 1‘) only depend on the magmtude of R.

96

 

 

Gaussian + NV-DME - - — - Yukawa + NV-DME

 

 

 

 

 

 

------------ Gaussian + PSA-DME -—---— Yukawa + PSA-DME
------- Gaussian + PSA-DME-II -—-- Yukawa + PSA-DME-II
8.0-
In:
e 6.0
L:
a.)
o O
3: 4 0 Cr [ N1
E z‘\ "x
8 I, \\ \\~
I.- 2 0‘ I ‘\ ’/’~\\\ ] ‘\\ ’
a: \\/’ ~ \‘v”
0.0 .1 .. ..............
18 2'6 3'4 4'2 5'0 5'8 24 3'2 4'0 48
N N
I-
o
In
In
(a
35
a 4.0+ Sn * Pb
H
5 ’,/”‘\‘
2 2.01 ‘~~~--- ,x— - -_/ \~——-
5: a,” ..............
0.0 ----- \~~‘p' """ -_.-T

 

 

 

 

48 5'6 6'4 7'2 8'0 8'8 94 102 1i0 lfii 126 134
N N

Figure 5.10: (Color online) Percentage error of EgM E [7171] compared to Eg[7m],

Where the former is computed from: NV—DME, PSA-DME or PSA-DME—II
Y\‘functions. Densities are obtained from self-consistent EDF calculations using the
SLY 4 Skyrme EDF in the particle—hole channel and no pairing. '

97

 

a = 1.0[fm] + NV-DME — - — -a = 1.5[fm] + NV-DME

 

 

 

 

------------ a = 1.0[fm] + PSA-DME --‘--- a = 1.5[fm] + PSA-DME
------- a = 1.0[fm] + PSA-DME-II - ---a = 1.5[fm] + PSA-DME-II
6.01
h
8 4.01
l-
a
Q 4
if
a 2.01 Cr
0
G)
In
0
m ”"’\\ 'a—.
0.0 /l’ ......... “A\\ ’IA””__---______::~-
I” ............ :.::,_~’{:— ----------
18"'2'6fi1'3'4" '4'2"'5'o"'5's'
N

Figure 5.11: (Color online) The same as Figure 5.10 but for two different values of
the range of the Gaussian interaction.

98

;,»,
6r \ .
1

‘iav D

~41
' 1
I29 6

1
V ‘0.

-"T‘ if]

k '1
a
‘v

Y
Anlfll:
v7.“
.10,

o.
:J—a.

 

'II'

316--

arm
1

 

density profiles in Fig. 5.9, the accuracy decreases as the range of interaction increases.
which holds for all available DME techniques [[170]-[173]]. This stresses that the local
quasi-separability of the OBDM with respect to 7" and if underlining the DME, which
is exact in IN M, deteriorates with increasing non-locality 'r in finite nuclei. As long as
the hypothesis of quasi-separability is well realized within the range of the interaction,

the DME can be quantitatively successful.

On average, the error obtained with PSA-DME and NV—DME are similar as can
be seen in Fig. 5.10, i.e. about 6—8% for the three isotopic chains and for both for
the Gaussian and the renormalized-Yukawa interactions. While PSA-DME—II gives a
better accuracy with a percentage error between —0.5 and 2. Similar improvement
over that of Ref. [170] is reported in Refs.[[171],[172]]. PSA-DME—II, while it is much
better than PSA-DME and NV-DME, it shows a gradual drift of the sign of the error,
from overestimation to underestimation of the exact value, as the interaction range

increases.

5.4.3 Fock contribution from VT

We now turn to the Fock contribution coming from the tensor part of the NN inter-
action. Such a contribution involves bilinear products of non—local spin densities As
a matter of fact, two terms with different analytical structures emerge such that the

exchange tensor energy-density reads

 

ram-‘2'») a T.f;,.<é,r>+T.§.,2<fi,r>. (5.85)
. _. 1
T,f,,,1(Rr) : 117? er§u(f1,fi2)-§,,(f'2,7'~‘1), (5.86)
"' 1 rrru ... —.
T512(R9r) E a; er Z :3 8,14,01,73)
. W

X Sun/(7:727 F1) , (5.87)

99

.
. .
u 1
.K c
I
.4
.‘v,
’-
~A..L’~.

2.. ‘ 1
{4"
"r4

 

._.. .
1 4

 

where T,f;,J(R, I) also appear in the central contribution to the Fock energy. The two
DME counterparts, which eventually depend on which variants of the DME is being

adopted, read

.2

——[I1f(k;tv >121 ....(V R)Jn,,...(fi).

11 11:1:

INF i:(J..,(,..12).]...(1‘2’)+J..,,.,.(I"2')J..W(R‘)

TF,DA!E(1'§, 7')

nn,l

7'2

F. 11E “’
Tn? (RJ')

71 ,

+Jn,pu(R)J nun (2)111)

and reduce for spherical systems to

”I

nn,1

TF ”Mai, r) —% [11f (12:2.r)]2j;,(1‘2’) - HR), (5.88)

“I
o

TF‘DA'!E(1§3 7‘)

11712

(5.89)

One recovers a pattern which is seen when deriving the empirical Skyrme EDF from
an auxiliary Skyrme effective interaction. That is, the central part of the inter-
action only produces the so-called symmetric bilinear tensor terms proportional to
Jn,p,,(R) Jn,,w(R) while Tfigh I E (R, r) that contains asymmetric bilinear tensor terms
proportional to Jn,,w(R) Jn,.,,,(R) solely comes from the tensor interaction [196]. This
can be easily traced back to the spin-space coupling that characterizes the tensor
operator. Since the numerical tests are presently carried out for spherical systems,
we are only concerned withT m, F1(R, 7') andT TF’DME(R, 7'). For spin-unsaturated nu-
clei, Tan1(R,r) IS highly localized around the nuclear surface as seen in Fig. 5.12 for
208Pb. The same figure shows the progressive and significant improvement that the

PSA approach brings to the DME of the vector part of the OBDM. This is realized
in both PSA-DME and PSA-DME—II. The optimal parameters that we obtained for

100

 

 

0 7.. IL ..I. 0. q 1.. l 1 l hlllllllll I
9 40 xi. 0 9 6 3 Av .v-al... ... . A- ~.v3¥..¢~\w-\CA~ ......fi .. ... ... ...... .
...-b- .- -=h— .- u -\ nut “L1 ..4 at. ....H 51 ..

 

INM-DME PSA-DME

   

 

   

9
‘6
3
NV-DME
0 ------------ INM-DME
3 6 9 0 3 6 9 """" '
R [rm] R [rm] PSA DME
_ _ _ — PSA-DME-II

   

‘g 1.5 m R = 4.0[fm]

 

 

 

2.0 -

I \

 

E/Exact
I-1 H
e in

M
P .
u-

 

 

D

 

P
e

r [fm] 1' [fm]

Figure 5.12: (Color online) Comparison of Tnle(R, r) and T33 E (R, 7") where the
latter is computed from NV—DME, PSA-DME, PSA-DME—II or from PSA-DME
with P;(R) = 0 which we denote as INM-DME. Upper panels: two-dimensional
integrands. Lower panels: ratios of T333 E (R, 1*) over Tfl‘fiR, r) for fixed values of
R. Densities are obtained from a converged self-consistent calculation of 208Pb with
the SLy4 Skyrme EDF in the particle-hole channel and no pairing.

101

PSA-DME—II of £1071, 1:72), as defined by Eq. (5.39), read
m = 2.543, b = —0.0799. (5.90)

Within the typical range of nuclear-interactions, N V—DME falls off much faster than
both PSA-DMES. Less importantly, NV—DME also introduces artificial and pro-
nounced structures in a region that corresponds to the tail of the interaction. Both of
these drawbacks are rectified progressively by the PSA-DME approach. While most
of the improvement is already brought by the spherical PSA (P2(R) = 0, which is
the same for both PSA-DME and PSA—DME—II), an even better accuracy is obtained
by incorporating the quadrupolar deformation P2(R) of the local momentum Fermi
distribution. The overestimation of T5,,1(R, r) at very small 7' seen for all DMEs in
the lower panels of Fig. 5.12 corresponds to a region where the integrand is small and
where its weight is further reduced in the integrated energy by the r2 phase-space

factor.

Coming to the energy itself, i.e. to the integrated product of the interaction 'vT(r)

with the tensor energy density, we compare

Eflnn] = 47r/dR dr 1‘2 vT(r) T,f;,(R, r), (5.91)
Eg’DMEhm] = 47r/dR dr 7‘2 117(7) 1:3,ME(R, r) , (5.92)
which for spherical nuclei reduce to the contribution from T5“ and T5,?” Figure 5.13

shows the relative error of the three DMEs compared to the exact Fock contribution,
for both the Gaussian and the renormalized-Yukawa radial form factors and for three
semi-magic isotopic chains. For both types of interaction, the percentage error of N V-

DME easily reaches 40%. This is in contrast to PSA-DME and PSA-DME—II whose

percentage errors are typically within 21:10% for most parts of the three isotopic

102

 

Gaussian + NV-DME - - —- Yukawa + NV-DME

 

 

 

 

 

 

------------ Gaussian + PSA-DME ------ Yukawa + PSA-DME
------- Gaussian + PSA-DME-II — - -- Yukawa + PSA-DME-II
60.0-
I-
2 40.0
I-
0
q, J
M
3
1: 20.0J
0
O
l-
0
a. l
0.0-
18
“ 60.01
G
l-
8
0 4000
no
3
:1
8 20.0(
In
0
n-
0.01

 

 

 

 

 

48 56 64 72 80 88 94 102 110 118 126 134
N N

Figure 5.13: (Color online) Percentage error of EgMEhm] compared to E; [7m]
where the former is either computed from: NV-DME, from PSA-DME or
PSA-DME—II. Densities are obtained from self-consistent EDF calculations using
the SLy4 Skyrme EDF in the particle-hole channel and no pairing. Notice the
different vertical scale compared to Fig. 5.10.

103

chains. This can be traced to the fact that, while both NV-DME and PSA-DMEs
overestimate the reference quantity for small 7“ (typically less than 1 fm), NV-DME
decreases much faster with 1', thereby overcompensating for its initial overestimation.
In contrast, PSA-DMES stays close to the exact value for a much larger range of 7‘

values

There exist short sequences of isotopes for which the percentage error shows a
considerable increase. The fact that all three DMES display such a feature suggests
that the problem is independent of the specific form of the 11:7 function used. To iden-
tify the source of the problem, Fig. 5.14 shows T "11(R, r) for three nuclei displaying

a sudden loss of accuracy. One notices that T F (R, 1') extends over larger intervals

nn,1
in R and 1" than for 20st (see Fig. 5.12). This corresponds to the fact that the
selected nuclei are nearly spin-saturated and generate very small Eff: [nn] in absolute
value, as seen from the lower panels of Fig. 5.14. As a result, the relative inaccuracy
of any DME becomes large and the percentage error increases suddenly. Of course,
the resulting error in the total EDF remains very small as the corresponding tensor
contribution is anyway negligible, i.e. the local spin-orbit density J;(R) is close to

zero in nearly spin-saturated nuclei. Therefore the sudden losses of relative accuracy

are not as worrying as Fig. 5.13 initially suggests.

In conclusion, the use of PSA techniques has allowed us to bring the DME appli-
cable to the bilinear product of non-local spin densities on the same level of accuracy
as for terms depending on the scalar part of the OBDM. One could certainly work
even harder to bring down the overall DME percentage error. This could be achieved
by (i) allowing for additional parameters in the H—functions to be optimized on a set
of reference calculations. However, our extensive optimization of the two-parameters
of Hf have convinced us that one cannot remove the sudden loss of relative accuracy
discussed above for spin-saturated nuclei. As already stated, this is not a problem in

the end as the corresponding contribution to the energy is negligible anyway. (ii) One

104

Exact - 64Cr Exact - 114Sn

    
 

20 28 36 44 52 60

Sn
40_0 . — Gaussian
£5. , """ Yukawa ‘
V20.0 l

 

 

 

 

50 58 66 74 82 96 104 112 120 128

N N
Figure 5.14: (Color online) A few representative nuclei with diffuse T5,,1(R, 7‘)
together with absolute E; [rm] for the corresponding isotopic chains. Densities are
obtained from a self-consistent EDF calculation using the SLy4 Skyrme functional
in the particle-hole part and no pairing.

105

can go to higher orders in the DME, consistently for both the scalar and the vector
parts of the OBDM. This should however be done within the frame of the generalized
Skyrme EDF proposed in Ref. [35].

5.4.4 Fock contribution from VLS
Basic analysis

We now turn to the spin-orbit contribution to the F ock energy. Unlike for the central
and tensor forces, such a contribution involves both the scalar and the vector parts
of the OBDM. In this case, we first compare the spin-orbit energy density

2'

Lsf,(1'2',r) = Z1; d9.§,.(fi,52).rxmaxi-8,51), (5.93)

to its DME counterpart

4 1 - Z . - -
LSS.D“’E<R,1~> = gflflkiidd Z 6“"‘3anuavg(ns<k2r>pn<8)).
11.u.;‘3=:r

which eventually depends on which variants of the DME is being adopted and that

reduces for spherical systems to
, -‘ 1 ~ -0 -v -¢ _.
LSS.D"’E(R,1~) = 6 113% ~24) r2Jn(R) - V11 (”Sumpam) ° (5.94)

Note that we have truncated terms with more than two gradients in LSfif’fiR, r).
The numerical tests shown in the present section actually use PSA-DME only (no
PSA-DME—II) with the quadrupolar deformation parameter set to zero (PQ"(R) = 0).
we still label the results as PSA-DME. It will be seen that these simplifying choices

have no bearing on the discussion at hand.

Figure 5.15 shows that PSA-DME significantly overestimates (in absolute values)

106

 

NV-DME PSA-DME
J11 '.

   

 

 

 

 

 

 

9 ‘ 9
E“ 6
1.13. 5 3

0 . ~ . ~ 0

0 3 6 9 0
R[fm]

2.0]

1.5“

—NV-DME
——PSA-DIVIE

 

DME/ Exact
rd
’9

P
u-

 

0.0 . . .
3.0 A y
2.5 1 I \

t

92.0 I \

 

DME/Exa
r-I 1-1
c u:

l

I

I

l

\

\

\\
B

/

/

I

l

I

l

\

\

\

0 5, R = 6.0[fm j R = 6.5[fm]

 

 

 

0.0 - . . . . .
0 l 2 3 0 1 2 3
1' [fm] 1' [fm]

Figure 5.15: (Color online) Comparison of LS:,,(R, r) and LS:;,DME(R, 'r) where the
latter is computed from either NV-DME or PSA-DME. Upper panels: _,
tWo-dimensional integrands. Lower panels: ratios of LSEKWR, r) over LS:,,(R, r)
f0r fixed values of R. Densities are obtained from a converged self—consistent
Calculation of 208Pb with the SLy4 Skyrme EDF in the particle—hole channel and no
Pairing. 107

the maximum peak of LS:,,(R,7') at the nuclear surface. In addition, oscillations
at larger 1', i.e. in the tail of the two-nucleon interaction, are not captured by PSA—
DME. In contrast, NV-DME reproduces relatively well the density profile Lan(R, r),
in particular as for the main peak at the nuclear surface. This suggests that the
significant improvement for PSA-DME over NV—DME as to reproducing the tensor
energy density does not carry over to the spin-orbit energy density. The previous
assertions are supported by tests carried over several isotonic and isotopic chains.
Looking for possible improvements, we tested that including truncated higher-order
terms associated with the action of 618' on (1 /4Apn — Tn + 3/5k}2pn), when going
from Eq. 5.93 to 5.94, does not improve the accuracy of PSA-DME.

Coming to the energx itself, i.e. to the integrated product of the interaction 111,5(7')

with the spin-orbit energy density, we compare

EELS-[7171] = 47r/derr2 vL3(7')Lan(R,1‘) , (5.95)

1111

Efglqnn] = 47r/derr217L3(r)r2LSDME(R,r) . (5.96)

Figure 5.16 shows the percentage error obtained for three isotopic chains. In agree-
ment with the analysis done for the spin-orbit energy density, the percentage error
of PSA-DME is impractically large and negative, in the range of -15% to -50% for
the two schematic interactions used. In contrast, NV-DME provides a much better
accuracy with percentage errors within :1: 10% for most studied isotopes. Last but not
least, one notes that the spikes in the percentage errors already discussed in section
5.4.3 arise for the same isotopes and relate to the vanishing non-local spin density in

near spin-saturated nuclei.

108

 

Gaussian + NV-DME — - — - Yukawa + NV-DME

 

 

 

 

 

 

 

 

------------ Gaussian + INM-DME ~----- Yukawa + INM-DME
._. 25.0-
2 ,4- r.
s 0 0 I ‘\‘ I \'7—‘-——h.. >\ If;
at) / \\ ' \\\ 7
E -25.0- ~. \J Ct ........ ' I """"" Niki.“
2 I .\_ .4 ............. \_\ ........... , \ _____
o I \. --.,~ ......... \.\ 7 \ ,
°" -50 0 ' . h: ’ f . \. . '''''' I .
' 18 26 34 42 so 58 24 32 40 48
N N
1... 25.0' 1 "s
2 /’ \-\
g _\_ ’ _-- -~-
O 0'0 ... \/ x—Q‘
393,111 ‘r\ Pb
u \'-, _. \
5 -25.0- ........... -’ YES“ l I?" ~ ----------------
U “I | '. x \ .......
In x ................. -~. _
0 -—-~ »' ‘ ~. -------
A. -50 0 . . ._._'._._._'. - . t__._.'_._.- .
° 48 56 64 72 80 88 94 102 110 118 126 134
N N

Figure 5.16: (Color online) Percentage error of EESMEUm] compared to Efs[nn]

where the latter is either computed from NV—DME or from PSA-DME. Densities
are obtained from self—consistent EDF calculations using the SLy4 Skyrme EDF in
the particle-hole channel and no pairing. Notice the different vertical scale
compared to Figs. 5.10 and 5.13.

109

Further investigation of the spin-orbit exchange

The results of the previous section show that N V—DME is better suited than PSA-
DME to reproduce the spin-orbit contribution to the Fock energy. This can be c011-
founding in light of the better accuracy obtained using PSA-DME to reproduce the
tensor contribution to the Fock energy. We can infer from Fig. 5.12 that NV—DME
underestimates the main peak of the nonlocal spin density while the latter is well cap-
tured by PSA-DME. It is thus puzzling to find the opposite for the Fock spin-orbit
energy density. In the following we employ a toy model of the OBDM of finite nuclei

to show that this is due to a fortuitous cancelation of errors.

Having already a handle on the non-local spin density §q(F1, F2), we focus on the
term it multiplies in the spin-orbit energy density, i.e. 7" x 62pq(771,772), which we
first approximate by 7" x 61'? pq(7"'1, F2) thanks to the weak dependence of the non-local
matter density on the orientation of 7" [176]. Hence, and focusing arbitrarily on

neutrons, we want to compare the two quantities

GE 2 612014117?) 3 (5'97)
V8 (118(kpr)pn(§)) , (5-98)

GDME =

where the latter is independent of whether NV—DME or PSA-DME is used. To do
so, we employ the toy model we discussed in section where the expressions for the
local and nonlocal neutron densities are given by Eqs. (5.58) and (5.59). From these

equations, one easily obtains

6w}? + g, 8‘ — g) = exp[—1/4a2r21———i:] [mm] (5.99)
" " 405—1/21a2 2____

110

The corresponding PSA-DME reads

 

.. ,1. .. F j Mr 7'2 1+t . 2 n ..
,1;

such that, given the definition of WAR), one can easily obtain

 

V711 [1W ‘2‘7‘)p(§)] = 10( ’2‘7')6§pn(R) (5.102)
and show that
" GDME( 9, F) . [ 2 21+t]
Gnu-0 R,r E _, = "17‘ ex 1 4a 7‘ — .
‘( 9 GE(R,f) 30(F) P / 1—t

In order to study Gratio quantitatively, we fix the inverse oscillator length, (1, using
the Blomqvist and Molinari formula, i.e. a = (0.90/11/3 + 0.70). In subsequent
discussions, we take reasonable combinations of A and N although we show that the

conclusions of the present section are independent of the actual value of A.

Before analyzing the behavior of Gmtio(R,1—‘), it is worth noticing that the toy
nonlocal matter density is exactly separable in relative and center-of-mass coordi-
nates. Such a separability being one inherent, usually only approximate, aspect of
the DME, we expect the latter to work well in the present case [185]. Computing
the same ratio as in Gran-(,(R, 7‘) without the gradient operators, we do indeed obtain
the good performance of the DME as is visible in Fig. 5.17. Note in particular that
the ratio is independent of the value of R. Such a result proves that the toy model
provides a situation comparable to the one studied in Sec. 5.4.2, i.e. the DME of the
scalar part of the density matrix performs well. Such a performance sets the stage in

View of qualifying the results obtained below for Gran-(,(R, 7").

In order to identify the short distance behavior of Gran-AR, 7"), we perform a Taylor

111

 

 

'A=120 N=80

 

 

 

.\

 

 

 

j

'A=200 N=120

 

 

 

A = 120 N = 60
g 4.0 ‘ R = 0.0 [fm]
a ----- R = 2.5 [fm]
\ --—-R=4.9[fn1]
E 2.01
G
\
0.0 - - -
A = 200 N = 100 l
g 4.0! R = 0.0 [fm]
g3 ----- R = 2.9 [fm]
\ —-—— R=5.8 [fm]
E” 2.01
a
0.0 a - \
0.0 1.0 2.0 0.0
r [fm]

k

\

 

110 2.0
r [fm]

Figure 5.17: (Color online) Ratio of the DME (Eq.(5.101)) over the exact
(Eq.(5.58)) expressions of the toy nonlocal matter density.

112

series expansion in 7'

 

(5.103)

_. kn? ,‘2 1 t
Gratio(Refl % 1 + (_ [l 1:2)72

6 4(1—t)

Looking close to the surface of the nucleus, one can neglect k}? / 6 in comparison
with the second term of Eq. (5.103). Defining Germ,.(R,f) E Gmti0(R,f') — 1, one

obtains

4 .2
G.,.....(R,F) a» (0.326 — 0—03) 7'2. (5.104)

711/3

Equation 5.104 is valid around the nuclear surface. Inside the nucleus, one cannot
neglect the first term 162:2 / 6 of Eq. (5.103). This is irrelevant as the spin-orbit energy
density is concentrated around the nuclear surface. Figure 5.18 bears our expectation
i.e. overestimation of GE by GDME around the nuclear surface for a wide range of R,
A and N values. It can also be seen that there is a gradual and systematic shift from
slight underestimation to overestimation as one moves from inside the nucleus to the

nuclear surface.

Keeping the results shown in Fig. 5.17 as a reference, we conclude that the applica-
tion of the gradient operator on the scalar part of the density matrix deteriorates the
quality of the DME that overestimates the exact results, in particular as one goes to
the surface of the nucleus where the exchange spin-orbit energy density is maximum.
Combined with the good approximation of the vector part of the density matrix, such
a semi-quantitative analysis explains the overall overestimation (in absolute value) of
the exchange spin-orbit energy provided by PSA-DME (see Fig. 5.16). Contrarily,
the underestimation of the vector part of the density matrix by N V—DME provides a
fortuitous, but rather accurate, cancelation of errors such that the nonlocal spin-orbit
energy density is much better reproduced overall (see Fig. 5.16). Even though we can
be satisfied with such a situation in the short term future and advocate the use of

the NV-DME variant for the spin-orbit contribution to the Fock energy, it would be

113

 

 

 

 

 

 

A = 120 N = 60
R=0.0[fm] A=120 N=80
22.0 ----- R = 2.6 [fm] ’
‘a —-—R = 5.1[fm] ‘/' ‘/,,
Ul- _____/’ "‘/
0.0 -
A = 200 N = 100
R = 0.0 [fm] A = 200 N = 120
32.0 ----- R = 3.0 [fm] / I
a? —-—R=6.1 [fm] ,/’ ‘/’
L51- _‘/.-% ’_/—/
0.0
0.0 1.0 2.0 0.0 1.0 2.0
r [fm] 1' [fm]

Figure 5.18: (Color online) Gmm(R, F) as a function of r for a selected set of
(1%, A, N).

114

more satisfying 011 the long run to design a suitable DME for the gradient of the
scalar part of the density matrix that can be combined with the improved PSA-DME
for the vector part.This loss of accuracy for the spin-orbit part does not have any
impact on the application of the DME to the HF energy from chiral EFT NN +
N NN interaction at N2LO. This is due to the fact that the N N spin-orbit interaction
that we have at N2LO is zero-range/contact, thus does not require the application
of the DME. In the NNN case, the problem does not seem to be relevant. Refer to

section 7.2 for details.

5.4.5 Hartree contribution from VC, VLS and VT

The numerical results given in section 5.4.6 confirm the View that the DME should be
applied only to the Fock part of the HF energy. However, for the sake of completeness,
we gauge the accuracy of the DME when applied to Hartree contributions. As shown
in the next chapter, the Hartree contribution from the tensor part of the two-nucleon
interaction vanishes for time—reversal invariant systems. For central and spin-orbit

parts, the exact integrands for the profile comparison are

Craig: T) E 41—17;]er pn(Fl) pn(7-“2) v (5.105)
_. 1 _,
LSEH(R, 7') E :1"; fdflr P1471) F ' 41,1072) , (5.106)

while the corresponding DME expressions read

0.52”“ .3 [.g(6)]2p,(é)pq,(a_’_[.((m]25p,(é).epq,(5)

2 -O —O
+ {inseam pq(R)Ap,/(R) + pawn/2(8)] . (5107)
1.3.7.0“ z gri’mfiwvmfi). (5.108)

where the 7r—functions in Eq. (5.107) are fixed in two ways: (i) The first set (NV-
DME) are given in Eqs. (5.62)-(5.64). (ii) The second set consists of the parameterized
functions given in Eqs. (5.67)-(5.69) and whose optimized parameters are: a2 = 0.850
and (1.1 = 0.3000. Eq. (5.108) results after fixing the 7r—functions according to
Eq. (5.72) (simple Taylor series expansion). The corresponding integrated energy

contributions are given as

Eg[nn] = 47r dedrr2 210(T)C,g,(R,r), (5.109)
Eg’DME[nn] = 47r dedrr2 120(1‘) CflDME(R,r), (5.110)
Efshm] = 47r dedrr2210(r) Lan(R,r), (5.111)
EggDMEMn] = 47r dedrr2 110(7')LS£I,’,DME(R,T), (5.112)

Fig. 5.19 shows that for large 1' values, Eq. (5.107) does not reproduce the correct
profile of Eq. (5.81) for both sets of 7r—functions. The two DMEs (sets of 7r—functions)
have opposite effect in that region. In contrast, one can achieve an accurate repro-
duction of the integrated contribution, Eq. (5.109), with Eq. (5.110), when the range
of the interaction is short. Furthermore, the plots contrast the accuracy of the two
sets of 7r—functions, with the parameterized version performing significantly better
as the range of the interaction increases, though with a decreasing overall accuracy.
Even though this decrease in accuracy is a general trend for all DME approximated
quantities, the deterioration is more significant in this case than the Fock contribu-
tions. Perhaps this is due to the wrong prediction of the exact profile (Eq. (5.81))
with Eq. (5.81). Fig. 5.20 compares the profile of the exact Hartree contribution from
the spin-orbit part of the interaction to its DME approximation. One can see the
DME fails to properly capture the profile, with a very large error resulting in the

integrated contributions.

116

Percentage error

 

 

10.0 ‘ ————————— -— " a = 1.0 [fm] + NV-DME

-20.0‘

 

 

1 .17:- ’ '1 -----------
------------ a = 1.5 [fm] + NV-DME
0.0
, ------------------- Yukawa + NV-DME
-10.0* Cr ———- a=[l.0 fm]+PI-DME

..... — a = [1.5 fm] + PI-DME

— — — — Yukawa + PI-DME

..—
——-
___________
..................
-—----——-— -----

 

 

18" 2'6 3'4" ' '4'2' 5'0 58
N

Figure 5.19: (Color online) Percentage error of Eg’DME[nn] with respect to Eghm]
for Cr isotopic chain. The upper plots show 03,, and CflDME for NV—DME and the
parameterized 7r—function which we call PI-DME.

117

- Exact NV-DME
10 ' 10

I [fill]

 

R [fin] R [fin]

Figure 5.20: (Color online) L35... and L5,?“ for NV-DME, with densities

obtaiend from a converged self-consistent calculation of 208Pb with the SLy4 Skyrme
EDF in the particle-hole channel and no pairing.

5.4.6 Preliminary self-consistent tests

In practical terms, one of the important benefits of the DME approximations is
the fact that existing Skyrme HFB codes require minimal modifications to be used
with EDFs obtained from the DME. Fig. 5.21 contrasts how a code is implemented
for Skyrme HFB against one for the DME based functional. As can be seen the
main change is in replacing the eventually constant Skyrme couplings, with density—

dependent couplings obtained from the DME.

We carried out a limited set of self-consistent test to gauge the accuracy of the
DME, in both full- and exchange-only-DME. In full—DME, both the Hartree and
Fock contributions are approximated with the DME, while in exchange-only—DME,
only the Fock contributions are approximated with the DME while the Hartree ones
are treated exactly. In addition to confining the test to time-reversal invariant and

spherically symmetric systems, there are several simplifying choices that we made.

118

Roam Potentials Kohn—Sham Potentifls

A A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Skyrme DME
energy HFB energy HFB
t0, t1: tV Alp], BUM
Orbital: lid Occupation t’e Orbital. and Occupation I's
(a) Skyrme-HFB code. (b) DME-based HFB code.

Figure 5.21: Comparison of Skyrme HFB and DME—based HFB codes.

These are

e We used the Brink-Boeker [193] force which has only a central component with

gaussian form factors. The actual form and parameter values are
4 2 2
Ve(r) = Z a,- e—(T12/“al +b,-e‘(’"12/“b> , (5.113)

i=1

with pa = 0.7fm and ab = 1.4fm. The magnitude and range parameters in

the four spin-isospin channels read

Table 5.3: The Brink-Boeker force(B1)

 

 

113113 11311; 11:11“; 113115

 

a,- 389.5 389.5 801.6 801.6
b,- -140.6 -140.6 -3.82 -3.82

 

Even though one usually adds a zero-range spin-orbit part to this interaction,
this is not done in this work as our target is to compare the DME approxima-
tions, and a zero-range interaction is treated exactly in the DME. A complete
self-consistent test of DME approximations should make use of modern N N in-

teractions that have central, spin-orbit and tensor components. In this regard,

119

building a local chiral interaction that. has a low-momentum cutoff, and thus

gives sensible HF results, will be useful.

0 we calculate only the total energy, and its components such as kinetic, Hartree
and Fock contributions of closed-shell nuclei: 16O, 40Ca, “Ca and 90Zr. The
fact that we do nOt resolve the single-particle spectra (such as spin-orbit split-
ting) prevents us from assessing the DME of 81,071, 8'2) for spin-unsaturated 48Ca
and 90Zr. Specifically the impact of the significant improvement in the DME
of 82,61,772) brought by our PSA-DME is not yet gauged with self-consistent
calculations. Additionally, we do not calculate the corresponding exact HF re-
sults. Rather, we use results from Ref. [184] when we need to refer to exact HF

results.

Derivations related to the self-consistent numerical test are given in Appendix 9.8.
For faster convergence of the calculations, we implemented both Broyden’s [194] and
imaginary-time methods. Table 5.4 lists the results of the self-consistent calculations.
At this point, we remind the reader that there are four densities that appear in the
exact HF energy of time-reversal invariant systems: pq(f'1, F2), §q(F1,F2), pq(F1/2) and
.2071”) (which appears only if the given NN interaction has a finite-range spin-orbit
part). The second column show how the 1r—functions of these densities are fixed.
Obviously, pq(1"'1/2) and LG} /2) are not expanded in exchange-only-DME calculations.

We start with the nonlocal densities. First, we have the labels N V-full and NV-
exc-only. The full and exchange-only labels should be self-explanatory. NV refers to
fixing the 7r—functions of pq(F1,F2) and §q(F1,F2) according to the original DME of
Negele and Vautherin (Ref. [170]). PSA-II-exc-only use the parameterized versions of
the 7r—functions of pq(f'1, F2) and 5.1071372). The parameters that we used are the ones

that we optimized for the non self-consistent test, while for pq(F1/2), the parameters

and 7r—functions are as discussed in the previous section ( 5.4.5). For the 7r—functions

120

—o

of Jq(F1/2), we use Taylor series as was done in Ref. [170].

Table 5.4: Full-DME and Exchange-only-DIVIE for Brink-Beeker interaction and

several DMEs

 

 

 

 

 

 

 

EM kin Dir Exch

lBO NV —full -6.204 13.948 - -
NV-exe—only -5.600 13.474 18.839 -37.914

PSA-II—full —7.932 15.417 - -
PSA-II—exc—only ~5.635 13.338 18.513 ~37.487

40Ca NV-full -8.526 16.822 - -
NV-exc—only -7.516 15.793 22.567 -45.878

PSA-II-full -10.359 17.575 - -
PSA-II—exc-only -7.539 15.583 22.075 -45.198

48Ca NV-full -7.447 16.678 - —
NV-exc—only -6.625 15.762 21.334 -43.803

PSA-II-full -9.304 17.57 - -
PSA-II—exc-only -6.646 15.529 20.884 -43.062

902r NV-full -9.339 18.778 - -
NV-exc—only -8.388 17.320 24.322 -50.041

PSA-II—full -11.543 19.038 - -
PSA-II-exc-only -8.389 17.040 23. 700 -49.140

 

A complete self-consistent test should include the exact self-consistent HF calcula-
tion and the calculation of several other quantities such as matter, proton and neutron
radii, proton and neutron densities. For the single-particle energies, the balance of
Hartree to Fock contributions need to be assessed [184]. Still, our preliminary test is

consistent with the main conclusion of Ref. [184], viz, the full—DME gives an excess

121

of 1 MeV per particle binding energy compared to exchange-only-DME, irrespective
of the performed DME. The exact HF calculations given in Ref. [184] show that the
error in the exchange-only-DME (compared to the exact HF of Ref. [184]) are much
smaller than that of the full-DME. Consequently, one can obtain a significant reduc-
tion of the error in DME approximations by treating the Hartree contribution exactly
as exact treatment of the Hartree contribution does not add to the numerical com-
plexity of the problem. Comparing the two DMES, it can be seen that the difference
between the exchange-only versions of NV-DME and PSA-DME—II is marginal, while
for the case of the the full-DME version, there is a significant difference with NV-
DME being much closer to the exact HF results reported in Ref. [184]. This must be
due to the strong parameter dependence and self-consistent error enhancement as the
non—self~consistent percentage error from PSA-DME—II (which we called PI-DME) of
the Hartree contribution, given in section 5.4.5 is less than that of NV—DME. This

requires further investigation.

122

Chapter 6

N on—Empirical Energy Density

Functional from NN interaction

In this chapter, we calculate the HFB energy from a generic two-nucleon interaction
that contains central, spin-orbit and tensor components. Furthermore, we apply the
DME to the HFB energy to obtain a local EDF. Analytical couplings of the particle-
hole (HF) part of the resulting EDF are calculated using the finite-range part of the
chiral EFT two-nucleon interaction at NZLO. Following the usual convention, we
represent momentum transfers with q. To avoid ambiguity, the isospin coordinates of

the particles are labeled with T.

6.1 The HF energy from an NN interaction

Starting with a two-body interaction, the Hartree-Fock energy is given, in an arbitrary

basis, by

l . . *
ENN = E: Z (iTJT’|Vlkr”lT"’)<PkT/’ir plr’”jr’ ‘ Pir”’ir pie/91'): (6.1)

5H .../Tile"!

123

 

where (irjr’ | V lkr”lr’”) are non-antisymmetrized matrix elements of the two-body

interaction and p I denote the one—body density matrix, as defined in Eq. (9.70).

iTjT

Because isospin is presently assumed to be a good quantum number, the density

matrix is diagonal in isospin space, phi/,4, = pl,- (ST/IT. Thus Eq. (6.1) becomes

. 1 . . . T TI 1 , , . T Tl
EN” = 5: Z (ITJT'I VlleT’)/)kiplj - 5: Zbrir'l VlkT’lTlpl-ipkf (6-2)

ijkl "I ijkl "I

Using the completeness relationship in the two-body Hilbert space

E Z [I F10171;F202T2> (F101T1;F202T2|dF1dF2 = 11 , (6.3)
”162 717-2
and the definition of the density matrix in h") 69 I6) 8) I?) space as given in Eqs. (9.70),

the HF energy can also be written as

4
1 .
NN— — ‘1“ 1 132"...“

I; —- 2 22::‘/{:I:[ ctr,(73¢71717,g727§|l/’ ITJCK§T3T4CT4TQ)

or 1:1

X p(F303T3, F101T1)P(F404T4, 71.20272)
1. 4 ......~132....(1)_...(2)_...
: ETI'ITTQ H dr,(r1r2|V lT3T'4)/) (7'3, 7'1)p (7'4, 7‘2), (6.4)
i=1

where V18” = V(1 — P12), with P12 being the particle exchange operator defined in
Table 1.2. A matrix notation is used in the second equation and the traces, denoted by
Tr, denote summation over the spin and isospin indices of “particle 1” and “particle
2”. The quantity MOO-3,71) is defined in Eq. (7.12). As discussed in section 7.1 in
relation to a similar calculation for the chiral three-nucleon interaction at NQLO, this
notation makes the direct Mathematica implementation of the equations transparent.

Refer to that section for details on this notation 1.

In this chapter, we eventually qualify all results for the finite-range (pion-exchange)

 

1The calculations for the NN case are relatively much simpler than for the NNN case. Conse-
quently, the complete HF + DME calculation was carried out manually in this case.

124

part of the chiral EFT two-nucleon interaction at NZLO, which is discussed in sec-
tion 2.4.1. As the HF energy from the contact part is already in a quasi-local (Skyrme—
like) form, it does not require the application of the DME. Thus, we do not discuss
it any further. The actual expression for the contribution to the EDF from the HF

energy of the contact part can be found in Ref. [153].

The application of the DME to the HF energy requires expressing the HF energy
in terms of the scalar/vector-isoscalar/isovector components of the OBDM in IF) 8)
IE) (8 I?) space. This is due to the fact that the DME, as formulated in section 5.2,
is most intuitively expressed in that space. For its formulation in momentum space,
consult Ref. [154]. Hence, we need to perform inverse-Fourier transformation of the

chiral interaction given in Eq.(2.14). This results in

(R‘r' (v: if?) = ([VC(T) + T1 .T21/VC(T)] + [ v5.6) + T, .T21I11'5(T)]5, ~02
+ [W(T) + T1 .T,WT(T)] s, 8.51, - 5,. + % [v,,,.(T)
+ T1 -T21;V,,,(T)] (s, + 5,) . (A. a 53)) 6(F- 1'”) 6(R — 1?!) ,

(6.5)

where R’, R denote the center of mass coordinates, f", r are the relative coordi-
nates and 6,. and FR refer to gradients with respect to 7" and R respectively. The
{Vc(r), Wc(r), ...} denote the inverse-Fourier transform of the respective form factors

given in momentum space [153]

 

 

d" ._._.
W") = / (2:)36“""%(q) forz'=C.s.T, (6.6)
i d” .~~_,. _, ,
= E (273)36” (q-F)V,-(q) for1=LS. (6.7)

One should note that chiral interactions come with a regulator that cuts off high-

125

momentum components. It should be noted that we have not included regulators
in Eqs. (2.14), (2.16), (2.18) and (2.19). The commonly used regulators result in a
non-local interaction since (TH-cg |V| E172) which can also be written as (11" WIT) is
replaced with f(k’/A)(E’|V|i5)f(k/A) for some momentum scale A, where f(k/A) —>
0 for k >> A and f(k/A) z 1 for 1: < A. In contrast, Eq. (6.5) shows that the
I7“) <8) I5) <8 I?) space representation of the interaction given in Eq. (2.14), viz, without
the regulator, is diagonal and depends on the gradient \"7 with respect to 1". The
spin-orbit part in Eq. (2.14) is actually the only term that depends on 6,3. This
dependence on V7,}. is usually remarked by referring to the interaction as quasi-local.
In order to obtain a local interaction, one could use a regulator that suppresses large
momentum transfers instead of large relative momenta. In any case, we neglect
the regulator since we work at the HF level which samples only the low-momentum

spectrum of the single-particle Hilbert space. This argument will remain valid as long

as the local Fermi momentum kp << A.

6.1.1 HF contribution from a central interaction

A local two—nucleon central interaction can be split into four different spin-isospin
channels as

V's-j : l/cijh‘) (SO-:1 — F3) 6(F2 - F4) H02. HT]. , (6.8)

where i, j€{0, 1} denote the singlet and triplet channels, I/Cij(r) is the form factor
and Hat. and HT]. are the spin and isospin projection operators defined in Table 1.2.
For the chiral interaction given in Eq. (6.5), the form factors of the central interaction

in the different channels is given by

vg“(r) = v.(T) -— 3ll”(.(’l‘) + V..(r) — 3w,(T) , (6.9)

V8 (r) W) + W) — 3v..(r) — 3W.(r), (6.10)

126

v3. (T) = no) + l"l"7,.(1') + mm + W..(r) , (6.11)

V80(r) = W(r) — 31'1“},(1‘) — 3V3(r) + 91V,(r) , (6.12)

where Vc(r), Wc(r), are the coordinate space form factors given by Eq. (6.6)- (6.7).

Starting from Eq. (6.4), the HF energy in the four different channels can be derived
by replacing V with the corresponding interaction given in Eq. (6.8). The details can

be found in appendix 9.6.1. For the spin-triplet and isospin-singlet channel, one has

I . 1 _, _, 3 _, _, 3 _, _, _, _,
E35110] = g;/ dr, dT2VciO(T)[§/)T(7‘1)Pr(7“2) + §pT(r1,r2)pT(r2,r1)
1,, _, _, _, 1 _, _, _, _, _, _,
+534“) ' 5r(7'2) + §ST(7‘1T7“2) ' 36(7‘2TTI)] T (6-13)
While for the spin-singlet, isospin-triplet-channel
NN _ l —: -: 01 _. T -. - ~ ~
EC l01l — 82/ d71d72 Vc (T)[PT(T1)PT(7‘2) + Pr(T1TT2)Pr(7‘2T7‘1)

1 _, _, 1 _, _, 1 _, __, _, _,
+ g2] (1T1 (1712 V810") [5PT(T1)Pr(T2) + —p,(r1,r2)pT(7‘2,T1)

The HF energy for the triplet-triplet and singlet-singlet channels read

1 .. - _. - _. - - ..
EgNllll = gZ/drldmvcill'r)[3Pr(rllpr(7‘2) — 3PT(7‘1=7‘2)PT(T2T7‘1)

—O

+§T(7'l) ' §T(F2) _ §T(F1T'F2) ' §T(F2’Fl)]

+ 6;] d’fidfé Vél(7')[gPT(F1)Pr(F2) — ng(F1.7‘72)pT(F2T7-‘i)
1_, _, _, _, 1_, _, _, _, _, _,
+5.46)- .6.) —§(>()] (6.15)

and

The corresponding expressions for the finite-range part of the chiral NN interaction
at N 2L0 can be found by utilizing Eqs. (6.9)—(6.12) in the place of the generic form
factors V” (1‘). In time reversal invariant systems, the symmetry properties of the one-
body density matrix discussed in section 9.2.4 can be used to simplify the expressions.
In particular, those terms that depend on the local spin density vanish as §q(7"') = 0
in this case. Finally, it should be mentioned that the channel by channel expressions
for the central interaction agree with the unpolarized and symmetric infinite nuclear

matter limit of the same expression given in [170].

6.1.2 HF contribution from the spin-orbit interaction

A given quasi-local two-nucleon spin-orbit interaction can be split into its spin-isospin-

singlet-triplet channels as

. 2.. 2' i. _. _, _. _‘ _. _. _.
V139 = -—§VLJS(T) 6(r1 — r3)6(r1 — r3) V - (01 + 02) Ilai HTj' (6.17)

The spin-orbit orbit interaction vanishes in the spin-singlet channels lxsmgm) =
1/\/2(|T1)—|11)) as S2 IXsingzet> = 0, with two-body spin operator 5'. = (31+62)/2.

For the chiral interaction given in Eq.(6.5), the spin-orbit form factors read

Vigil") = 2V1.50“) - 6WLS(7‘) T (6.18)
V2.66) = 2163(7) + 2WLS(T). (6.19)

128

The HF contribution from the spin-orbit interaction in the spin-triplet and isospin-

singlet channel is given by

.0

1 -o -o r -o _. —o —o —o -9 - -o

E53110] = :1- Z/ (17‘, (Ir-2 V2g(r)r- I: T(r1)p;(r2) + 8,.(1‘1) X ]T(r2)
i —o —o —o -—o a -o d —o —o
+1 2/ (17', (173 V1330) [.s,('r1,r2) . r x Vgpf(r2,r1)

+pT(F1,T_“2)FX 62 ' TTf-(F2,F1):l , (6.20)

while in the spin—triplet and isospin-triplet channel, the result reads

—o

1 _. _. _. ~ .. _. _. _. . ..
Elllelll = ‘2‘ Z/ drl (”2 V1366)” [J,(r1)p,(r2) + 3r(7‘1) X Jr(7“2)]

an.
-o

1 4 ... _’ ... _, _, ‘3 _.
+ Z Z] dTTdTTvgarV' [Jr(r1)/2T(r2)+ sm) X “ml
i —p -O "’ d q -. -' ._. -.
— 21:, /d7‘1dT2VL‘§-(r)[BdrnrzlrxV20T("2~7‘1)

+ pT(F1,F2) 7? X fig ' .Tf-(Fg, F1) ] . (6.21)

The actual derivation is given in appendix 9.6.2.

6.1.3 HF contribution from the tensor interaction

A local two-nucleon tensor interaction in the four spin-isospin channels is given by

V1? 2 V71;j(r)6(f'1 — F3) 6(7-3 — 705121-10’. HT]. , (6.22)

129

 

where 512 is the tensor operator given in Eq. (2.10). The tensor interaction acts only
in the spin-triplet channels. This becomes obvious once 512 is written as

$12 = 6 (§ - f)2 — 2 S2, where the total spin operator, 5;, has zero expectation value
in the spin-singlet state. Hence, concentrating on spin-triplet channels, the chiral NN

interaction at N2LO given in Eq.(6.5) has the following components

V7100) = éVTU') — W’TU) , (6.23)
v;‘(T-) = -;—V,(T) + éWfi-r) . (6.24)

The HF contribution to the energy in the spin-triplet and isospin-singlet channel is

given by

_‘ _. 3T 1 TV _, ... —-o —o -' —’
ENNllol: 211;: Z / drl (1T2 VTI‘O(r) I: 7,-2 8T.#(r1)81-’,V(T2) — STU‘I) . 371(7‘2)
1' pu

3r 1‘ _, _, _. _, _. _. _, _. _. _.
—71'/§—uSTTH(TlaT2) 8?,U(r23r1) — 87(7‘1’7'2) . 71(7‘2’7‘1) ’

while for the spin-triplet and isospin-triplet channel

37' 73, _, _. _. _. _. _.
E¥”{111=—2 Z Z / dr. dr2v1(w~)[——,—“ s.,.(r.)s...(r2) — 3.6.) - 3.62)

1[31' r _, _. _. _. _. _. ..
——:2 u S’Tp(7‘1 T2)3Tu(1‘2,r1) + s,(r1,r2) . 37(r2,r1)]

37' 71, _, _. _. _. _. _.
+—41:WZ/dT1dT2L/TII(T) [Tp23‘r.p(r1)31",u(7‘2) — ST( 1) ' .‘7-(7‘2)

 

3r 7‘ _, _, _. _. _. _. _.
_ —%_VsT-H(T17T2) Sit/(r'brl) + 37(7‘127'2) ° ‘ ;(7‘2,7‘1)].

(6.26)

Once more, the relevant expressions for finite-range part tensor part of the chiral NN

interaction at N2LO can be obtained by making use of Eqs.(6.23)-(6.24). For time-

130

reversal invariant systems, those terms that depend 011 the local spin density, £107),
vanish. This recovers the expression derived in Ref.[170]. Refer to appendix 9.6.3 for

details.

6.1.4 Additional contributions to the HF energy

In addition to the contributions to the HF energy that come from the starting N N
interaction, there are several additional terms that are due to the kinetic energy, the
center of mass correction and the coulomb interaction. The simplest is the uncorre-

lated kinetic energy associated with the reference product (HF) state

2
Em = h— : / dT’T,(T*). (6.27)

Since (HF) meanfield solutions are localized in space, translational invariance of the
actual nuclear hamiltonian is broken. Consequently, one needs to correct for the
center of mass motion, which can be done by defining an intrinsic Hamiltonian [38].
In addition to the the starting Hamiltonian, the intrinsic Hamiltonian contains a

correction term ECM which reads

. _ (q’chzwl‘I’) _ (‘I’I(Zkfikl2|‘1’>
ECM — 2Am _ T 2Am (6'28)

 

where POM = 2k 13'), is the sum of single-particle momentum operators and A is the

number of nucleons and the |<I>) is the reference independent particle or quasi-particle

131

state. Expressing EC)” in terms of densities, one obtains
E _ h2 2 / dF (r) h? Zr, // Jada/1' (.3 ‘3 ('3!)
C“ — 2Am 7’ 2Am (T r 9" JT’ '
,_. ~I
+2Am 2; f/drdr p,( r, I‘IT)T /(r,r) (6.29)
+ mm 2, f / df'df" c616") g,(T=*,F').

 

 

 

Several comments are at play concerning such a correction to the the HF energy
(i) the first term is a one—body center of mass correction with an overall effect of
rescaling the kinetic energy term, Ekin (ii) the second term is really local and is zero
if the single-particle states have a good parity, which we assume to be the case (iii)
the non-local third term is the so-called two-body center of mass correction and is
often omitted; if single-particle states have a good parity, the two coupled densities
are labeled with opposite parities (iv) the non-local fourth term contributes to the
pairing energy appears if one considers a reference independent quasi-particle state.
It has never been considered in practical calculations [26]. If the single particle states
have a good parity, the two coupled densities are labelled with opposite parities. In
its full generality, such a term generates neutron-proton pairing. These nonlocal (the

third and fourth) terms will be neglected in our application of the DME.

The last correction arises from the Coulomb repulsion among the protons. It
has both direct and exchange parts. The nonlocal exchange contribution is usually
approximated with the slater approximation [155]. This is due to the fact that for the
long-range Coulomb interaction, the simple slater approximation seems to perform
at least as good as the DME techniques discussed in the previous section. Hence, we

write the contribution from the Couloumb interaction as the HF energy reads

so,“ _= —eW//d1"’pfifl_pp£ll)—(223(;)1/3/de:/3(77) . (6.30)

132

6.1.5 The leading-order pairing contribution

The leading-order pairing contribution is obtained by calculating the expectation
value of the interaction in a Bogoliubov quasi-particle vacuum. At this point, we
enforce several restrictions: (i) we neglect proton—neutron pairing, hence no isospin-
singlet contribution. This is justified in most cases as the Fermi energies of protons
and neutrons are quite different for most nuclei [38]. (ii) only central interaction is
considered, as it is the 1So channel that exhausts most of the pairing contribution in
nuclei, i.e. the contribution of other partial waves are negligible [117]. Leaving the

details to appendix 9.6.4, the spin-singlet isospin—triplet contribution reads

1 _. _. ~ _. ..
(¢|V31|<I>)m,-r = :1- : / drld'rg V310)|p7('r1,r2)|2 , (6.31)

while for the spin-triplet and isospin-singlet channel, we have
1 _, _, :3, 4 _, ~ _, _,
((blvéll (I) >pair = Z Z / drl (1T2 V8710) 87(T1, 7'2) ' .97-(7'1, T2) . (6.32)

The Coulomb interaction has an important effect on proton pairing gaps [110]. Specif-
ically, its repulsive nature reduces proton pairing gaps (anti-pairing effect). To cal-
culate these contributions, one simply replaces the form factors, V6110“) and V810),

with the corresponding Coulomb interaction form factor.

6.2 Application of the DME to the NN-HF energy

In this section, we apply the DME to the HF B energy derived in the previous section
to obtain a local EDF. In section 5.4.6, we have verified that the DME of the Hartree
contribution is the main source of the descripancy between the DME-approximation

and exact HF, thereby advocating the exact treatment of the Hartree contribution.

133

Still, for the sake of completeness, we apply the DME to all contributions of the
HFB energy: Hartree, Fock and Bogoliubov. As the HF B energy is derived for a
generic two-nucleon interaction, we perform the derivation of a local EDF using the
modified-Taylor series detailed in appendix 9.5.7. As explained there, all available
DMEs, including PSA-DME developed in this work, can be mapped in to this formal

expansion.

Since, the starting point is the strict Hartree, Fock and Bogoliubov contributions
(diagrams), the energy functional is intrinsically a bilinear functional of p and K, i.e.

5[p,n,n*] = 5;; + 552 + 5;; + 55.2, + 5”” (6.33)

coul ’

where the right hand side corresponds to the uncorrelated kinetic energy, the particle-
hole (HF), the particle-particle (Bogoliubov/ B), the center of mass and Coulomb
corrections. The p and n exponents denote genuine, original dependence on the
density matrices. To recap the steps for the application of the DME, first we replace
the densities in the HFB energy with their formal expansion given in appendix 9.5.7.
This is followed by the simplification of the expression using the angle independence of
the 7r— functions and the relations among the 7r—functions discussed in section 5.3.6.
After neglecting terms with beyond second-order gradients, the particle-hole part of
the EDF takes exactly the same form as given by Eq. (4.11), where in this case, the

A/ B couplings are functionals of the 7r—functions and the starting interaction.

Through the DME, finite-range contributions of the starting interactions are en-

coded into density-dependencies of the EDF couplings. For instance,

App 2 3 fair?"2 [V31(r) ((7r3(kp7'))2 + (H8(kpr))2) + 3Vél(r) ((7rf,’(kpr))2

— ( Ham)?” , (6.34)

134

where we used the isoscalar kp as the DME length scale and suppressed the I? de-
pendence of kp for brevity. It should be noted that due to the density dependence of
the couplings, the usual integration by parts that is used in traditional Skyrme EDFs
to reduce the number of independent terms can not be applied here. For instance,
in conventional Skyrme EDFs, it is possible to convert the pAp term of the EDF
into Vpr , thereby reducing the number of terms. Generally speaking, this is not

possible in the previous case.

The couplings depend on the central, spin-orbit and tensor parts of the interaction

as follows

Central ——+ {App, A”, APT , APAP, AVPVP, AJJ , ASAS, AVsoVs,
Bpp’ B” , BPT , BPAP, BVPVP ’ BJJ 3 BsAs ’ BVsoVs }
Spin-Orbit -—> {ApVJ, AVPJ, BPVJ, BVPJ}
Tensor —> {AJJ, A” , A3133, AVSVS, AVSOVS

?

BJJ , BJJ , BsAs , BVsVs , BVsoVs } .

The complete expression of the couplings is given in appendix 9.6.5.

In the particle—particle (pairing) channel, the application of the DME to the pair-
ing contribution results in a functional that is more complex than the usual phe-

nomenological forms, given in Eq. (4.16). It reads

epp[p,n,n*] a 5; / diz' [Ac war + prg) (Am) — we)
+Ac m1?) (AMI?) — 4mm)
+Ajj Z j:,pu(1-?:) jT-#V(§):| 3
[JV

(6.35)

135

where the A” terms originate from spin-triplet pairing while the rest originate from
spin-singlet pairing. The actual expressions for all couplings is given in appendix 9.6.5.
The EDF that results from the correction terms, namely, uncorrelated kinetic

energy, center of mass and Coulomb corrections are simply given by

 

1‘12 —. -»
6': = 'fiZ/(IRTT(R)’
h2T .. -+
555, = —2Am Z/dRT.(R).
535111 = / dI—f {CPP pPpP + CpApppApp + CVpr 6:01) ' 6p],
3 2 3 1/3 " 43 "
—Ze (g) [dRpp/ (R), (6.36)

where as noted in section 6.1.4, we have neglected the third and fourth terms of
Eq. (6.29) while its second term vanishes due to the assumption of good parity for
the single-particle states. For the Coulomb correction, we have applied the DME to
the direct piece, while leaving intact the Slater approximation for the exchange part.
The application of the DME to the Hartree contribution is given just for the'sake of
completeness. In fact, even the exchange contribution from the Coulomb interaction
can not be treated accurately due to the long-rangedness of the Coulomb interaction.
One can expect the DME of the direct part to be much worse. Still, we give the

values for the C couplings in appendix 9.6.5.

6.2.1 Analytical couplings from the chiral EFT NN interac-
tion at NQLO

There are three steps necessary to obtain the analytical calculation of the couplings of
the local EDF derived in the previous section. First, we have to restrict the discussion

to time—reversal invariant systems as the analytical forms of the 7r—functions for time-

136

odd densities are not completely determined yet. This is discussed in section 5.3.6 in
detail. The next two steps involve (i) fixing the interaction which in this case is the
finite-range part of the two-nucleon chiral EFT interaction at NQLO. The respective
use of the three-nucleon interaction is the subject of the next chapter. (ii) Specifying
the 7r—functions. This can be PSA-DME, or any of the other available DMEs. In fact,
using different DMEs results in different couplings, which is mentioned in section 8.2
as a way to estimate the error of the DME couplings. In our case, we calculate the

couplings for PSA-DME. A similar calculation can easily be done for the original

DME of Ref. [170].

The derivation is discussed in appendix 9.6.6. As the final expressions are too
lengthy, we discuss only the skeleton expressions of the couplings. For the more on
the couplings, consult section 8.1 for a relevant discussion. The lengthy analytic
expressions for the DME couplings tend to obscure their underlying structural sim-
plicity. Therefore, it is more illuminating to display the couplings in “skeleton form”.
Each coupling Ct“) is given by the sum of the LO (n = 0), NLO (n = 1), and N2LO

(n = 2) contributions

2
cm) = 20m 2'6 {pt/mm), . . .}. (6.37)

n=0

where the dimensionless variable u E kp/m,r and t = {0, 1} is the isoscalar/isovector
index. The fact that we express the couplings in terms of isoscalar/isovector no-
tation instead of proton-neutron is for conformance with the notation used in the
derivations related to the three-nucleon interaction. Refer to [156] as to why the
isoscalar/isovector notation is more convenient in that case, and Eqs. (9.72)-(9.75) for

the simple algebra relating isoscalar/isovector notation with that of proton-neutron

137

notation. Now, each coupling can be written as
Cgf,),(u) = Z cry-Rn, t, u)fj(n, u) (6.38)
i

where (lg-001,13, u) are rational polynomials in u and .73-(11, u) are functions which may
exhibit non-analytic behavior in u due to the finite-range of the NN interaction. In
the skeleton expressions listed below, we use a more compact notation where the

dependence of the a’s on u, t, and n is not explicitly shown:

0 L0 couplings

 

C(i) = as) + a?) log(1 + 4u2) + a?) arctan(2u) (6.39)

o NLO couplings

 

: 2
C”) = of,” + a?) log(1+ 2u2 + 2uv1+ 23)]

 

+ agi)\/1 + u2 log(1 + 2n2 + 2u\/1 + u2) (6.40)
o NZLO couplings
C“) = as) + all” log(1 + U2) + agi) arctan(u) (6.41)

6.2.2 Single-particle fields and equations of motion

In appendix 9.7, we give the derivation of the single-particle fields and HF B equa-

tions of motion that result from the variation of the Skyrme-like EDF given in

138

Eqs. (4.11),(6.35) and (6.36). Additionally, we give similar derivation for the case
where the DME is applied only to the exchange part of the HF energy. As discussed
in 5.4.6, all numerical tests are carried out for the case of spherical symmetry. Hence,
we give the most simplified single-particle fields and equations of motion that result
when spherical symmetry is imposed. Furthermore, the numerical methods used to
solve the self-consistent spherical HF equations are also discussed. All this can be

found in appendix 9.8.

139

Chapter 7

Non-Empirical Energy Density

Functional from Chiral EFT NNN
Interaction at N2LO

In this chapter, we calculate the HF energy from the chiral EFT NNN interaction
at NZLO. Next, PSA-DME, formulated and discussed in chapter 4, is generalized in
such a way that it becomes applicable to the NZLO chiral EFT NNN HF energy. This
is followed by the application of this generalized PSA-DME to obtain a local EDF and
the analytical calculation of the couplings. Additionally, we make several references
to the actual symbolic implementation of the calculation. Again, following the usual
convention, we represent momentum transfers with q. To avoid ambiguity, the isospin

coordinates of the particles are labelled with r.

140

7.1 The Hartree-Fock energy from Chiral EFT N NN

interaction at N2LO

The consistent application of an MBPT calculation starting from a chiral EFT in-
teraction, at a given order, requires utilizing all the components of the interaction:
two- and many-nucleon interactions. As discussed in section 2.4.2, the leading three-
nucleon interaction appears at N2LO in the chiral expansion and it has three main
pieces: the three—nucleon contact which is referred to as the E—term, the one-pion ex-
change plus contact (D-term) and the two-pion exchange which is called the C-term.
In this section, we calculate the HF energy from these pieces of the NNN interaction

and apply the DME to obtain a quasi-local EDF.

Unlike the NN case, the algebra required to arrive at our final target, namely, a
quasi-local EDF, is so complicated that one can simply rule out a manual derivation.
This is due to the tremendous size of the algebra required in both layers of the
problem. Firstly, we have to derive the exact HF energy in terms of the scalar / vector-
isoscalar/isovector parts of the OBDM. This has to be followed by the application of
the DME to obtain the final quasi-local EDF.

However, the whole problem displays several features that make it amenable to
symbolic automation [151]: (i) it involves many similar and repetitive algebraic steps
(ii) most of it does not involve numerical computation, and (iii) the part of it that
seems to require numerical computation, such as multidimensional integrals, can be
performed using a combination of analytic expansion and symbolic integration. In
the following section, the HF energy from chiral EFT NNN interaction at N2LO
is expressed in a form that makes the symbolic implementation transparent. The

complete symbolic derivation is discussed in Ref. [156].

141

Basic form of the HF energy

A three-nucleon interaction can in general be decomposed as a sum of three terms
V3N E V12 + V23 + V13, (7-1)

where 17,5 is symmetric in nucleon i and j. Specifically, for the chiral EFT three-
nucleon interaction at N2LO, Vi]- depends on momentum transfers (j;- and fl} and, in
general, on the spin-isospin coordinates of the three nucleons. Refer to section 2.4.2

for details. Starting with the HF energy from a three-nucleon interaction

. 1 . . ~ . .
(V3513?) 2 —6— :(ZJICIWAI(1+ P13P12 + P23P12)(1— P12)ll]k> , (7.2)
ijk
a few basic algebraic manipulations are in order to express the HF energy in terms

of only one of the three 17,-]- operators, e.g. I723, as

. 1 . . ~ . .
(‘65?) = 5 Z fulfill/230 — 2P13 — P23 + 2P12P23)|’1Jk)a (7-3)
ijk
where sz denotes the exchange operator (of particles 1 and m) defined in Table 1.2
whereas i, j andk denote occupied HF single-particle states. Note that for ease of
notation, we are using the single-particle basis that diagonalizes the one-body density

matrix of the HF Slater determinant.

One can identify three groups of terms in Eq. (7.3): direct, single-exchange and

1

double-exchange terms . The direct term corresponds to the expectation value of

V23, the single-exchange term to the expectation value of I723(—2 P13 — P23) and the

 

1This should not be confused with one- and two-pion exchanges contribution to the three-nucleon
interaction.

142

double-exchange term to that of 2 127231912ng

7(ir 1 .. " ..
my!) a ‘Q‘ZQJklléalUkl. (7.4)
ijk.
0353’") 2 5 Emmet—21913mutate). (7.5)
ijk
main 2 Z<ijkl1723P12P2;.lz'-jk>. (7.6)
ijk

As the derivation of the Skyrme-like quasi-local EDF from the exact HF energy
requires the application of the DME, we need to express the HF energy in the IF) (8)
IE) ® IF) single-particle basis. Hence, we perform inverse-Fourier transformation of
the interaction. This transformation leaves the spin-isospin dependencies untouched.
Furthermore, just as in the case of the NN interaction, the fact that the calculation
is confined to the HF contribution enables us to neglect the regulator so long as
hp << A, the momentum cutoff scale. The absence of the regulator makes the
interaction local in coordinate space and simplifies the form of the interaction in

IF) 8) lo) 69 Ir) space. Confining the discussion to the spatial dependence, we have

-OI -OI -OI -OI

(F1772F3IV23I'7‘1T27‘9 = (5071 - 7‘1)5(F2 — 772') 5(773 r 7‘3)

X V2303 — F1, F3 - F1), (7-7)

where

1
(2r)6

 

v23(r2 — F1. F3 — r.) s [dadaeiiwirfi’eta-(i341)vixen). (7.8)

At this point, we do not actually perform the integrals over the momentum co-
ordinates in Eq (7.8), except for the E—term of the interaction which is a trivial
three-nucleon contact interaction, thereby yielding simple delta functions as shown in

Eq. (9.347). Rather, Eq. (7.8) is used as it is, resulting in fifteen-dimensional integrals

143

in Eqs. (7.13)-(7.15). As discussed in section 7.2, the application of the DME prior

to the actual multi-dirnensional integrations is crucial.

The next target is to rewrite Eqs. (7.4)-(7.6) in a form transparent for Mathemat-
ica implementation. We illustrate the steps required to achieve that with Eq. (7.6),

for which we have

MW") 2 :(ijkll/ggPlgP23lz'jk)
ijk
3 3
= Z Z 2 f Hng, Hdfil (ijlc|Fllol'r1'F/202'r2'FI3oérg)
ijk m=1 n=1

0f..03 T1,..T3

.. I I a I I —» I I " 1- or -+ .. _.
X (rllolr1 ”2027'2 rl3o3r3ll/23Pf’2 P23 lrlolrl rgogrg r3o3r3)

><(71.1017’1775027'2 71.3037'3lpffzpziilijkla (7-9)
where we used completeness relations in the three Hilbert space

3
E E f H dr, |r101r1 rgogrg r3o3r3) (rlolrl T202T2 r303r3| = 11 , (7.10)
01 ..03 1'1 ..T3 i=1

and ("I E (in (In. We split the particle-exchange operator such that the coordinate
part acts on the wave-functions while the spin-isospin piece is taken care of along

with the interaction. Let X:- represent (Fioiri) such that the one—body density matrix

reads as

9(Xjan) —=— QffiUjTjstUka) E Z ‘pfffkak Tk)‘19i(7?j 03' 7'1), (7-11)

I

where the sums is over occupied single-particle HF states. Making use of this, we

define another quantity, which we call the auxiliary density matrix, as

07X)“, Xk) E etfim’n’, From), (7.12)

144

where ie {1, 2, 3}. Basically, the spin—isospin coordinates of this quantity are those of
the i‘" particle. Applying the steps demonstrated in Eq. (7.9) and using Eqs. (7.12),
(7.7)-(7.8), one can express the direct, single-exchange and double-exchange parts of

the three-nucleon interaction HF energy as

‘ ( ir 1 -o —0 -¢ -' a ' _.
(VS/['1 l = 2 Tr1Tl‘2rpf3 [/ (17'1d7'2d7‘3 91(X1) 920(2) 930(3)
X V2303 - F1, F3 — F1)] , (7.13)
(VII/PIX) = - TriTI‘2Tr3 [/ 61771617726177“; 91(X3.)?1) 92032) 03031, X3)

XV2:5(F2 — 77197-33 — F1) :57]

1 _. _. _. _. _. —. . _. —.
— 5 rI‘I‘1TI‘2rIlr3 [/ drl(17‘2dT3 91(X1) 92(X39 X2) 93(X21X3)

XV23(F2 — F1, F3 — 71.013537] , (7.14)
(Mffvmxl = ITITF2TT3/df'id7—‘tzdf3 [01(X2,X1)02(23,X2)Q3(X1,X'3)

XV2;3(F2 — F1, F3 — F1) Pég-PIOJ] , (7.15)

where gi(Xj) E Qi(X-.j, X3) and Tr,- refers to tracing over spin and isospin coordinates
of the ith particle. The key to understand the form of these equations is the splitting of
the particle exchange operator, performed in Eqs. (7.9), that results in the spin-isospin
coordinates of each particle to be grouped in a single auxiliary density matrix. These
are the basic equations that are implemented directly in Mathematica. In Ref. [156],
it is shown that the implementation of these equations is transparent, viz, directly
interpretable to the language that Mathematica understands. This would not have
been the case without the trick used to group spin-isospin coordinates of each particle
in a single auxiliary density matrix, Eq. (7.12).

The following sections state the contributions to the HF energy of time-reversal

invariant systems from the E-, D— and C-terms of the chiral EFT NNN interaction

145

at N 2L0. The complete expressions where the assumption of time-reversal invariance
is relaxed are given in appendix 9.9. Even for time—reversal invariant systems, some
of the expressions are too long. In those cases, the expressions are relegated to the
same appendix, where we also give the corresponding results for INM and PN M (pure

neutron matter).

Prior to delving in to the details of the expressions, the following observations
can be made regarding the HF energy: (i) Each term in the energy expression should
contain three local/nonlocal densities. (ii) As discussed in appendix 9.2.4, the various
local and non-local densities that result from the one-body density matrix have specific
timereversal properties. Hence, considering that energy is a time-reversal invariant
quantity, there can be no term that contains one/ three time-odd densities. Note that
at the level of exact HF, the only time-odd density that we have is the local spin
density, s'q(F). Nonetheless, the application of the DME extends this set to include
all the time-odd densities that are discussed in appendix 9.2.4. (iii) The fact that
the starting interaction is isospin invariant makes the energy isospin invariant as well.
Therefore, there can be no term in the energy expression that contains one/ three
isovector densities. (iv) For each part of the interaction, there are the direct, single-

and double-exchange contributions as given by Eqs. (7.13)-(7.15).

Finally, we remark that the HF expressions and the resulting quasi-local EDF from
the chiral EFT NN N interaction at NZLO are given in terms of isoscalar-isovector no-
tation instead of proton-neutron notation. In Ref. [156] where we discuss the Math-
ematica implementation, it is shown that the isovector-isoscalar notation is better
suited to the implementation. Finally, keeping in mind that the NNN chiral EFT
interaction at N2LO does not have isospin invariance breaking terms, the isospin in-
variance of the energy expressions of both exact HF and quasi-local EDF become

transparent in isoscalar-isovector notation.

146

HF energy from the E-term

The actual operator structure and analytical form of the E-term of the chiral EFT
NNN interaction at N2LO are given in Eq. (2.16). The HF energy that results from

it for time-reversal invariant systems is given as

<v3i‘t‘E'TR‘> = —,%E/dr(p3<o—po<op¥m). (7.16)

As expected due to its complete zero—range character, the HF energy from the E-
term is already in the form of a local EDF. Consequently, it does not require the
application of the DME. It should be noted that the E—term has a direct counterpart
in the Skyrme interaction, Eq. (4.12). In most phenomenological Skyrme EDFs, the
density dependence is in the form of p30") where *7 takes a fractional value [26], while
the one that results from the E-term has 7 = 1 .

Three-nucleon contact interaction requires the three nucleons to have the same
coordinate F, which implies that three neutrons or three protons cannot interact via
a contact interaction, due to Pauli’s exclusion principle. Indeed, Eq. (9.346) shows
that ([51153me vanishes for pure neutron matter and likewise for pure proton matter

(though pure proton matter can not exist as it is not energetically stable).

HF energy from the D-term

The operator structure and analytical expression of the D-term of the chiral EFT
NNN interaction at NQLO are given in Eq. (2.18). Its contribution to the HF energy
is nonlocal due to the finite-range pion-exchange part of the interaction. In time-
reversal invariant systems, the HF energy contribution from the D-term takes the

form

 

— CD 1 l q - 7 (133qu
VHF,D,TRI : 9A _ [dyd-o / ' 1-0 1(13.(T3-—72) ‘ 3 3
< 3” l 4f; m, 16 ’2 T3 (27r)3(q3e q§+mf,2,
14

7

x —3 (5.53 /)()(7_'T2)P0(7_"2e 773)/)u(7:3~ 72,2)

+ 2 6m 00(772)Pl(r‘2 fi;)p1("-‘3~772)

+ 5..., p1(7_"2)/’0(772: some» F2)

— 3p0(F2)sg(7"72. 779630132772)

+ 3 mam. H 7%)
+ parasite. rashes: F2)

_ 6.... .va p0(1=:2)s$(r2. Fair-1’63. F2)

+2p.(e>s€(ae>sY<FmF2)

— 2 cm" ew’a" p1(F2)sg(F2, F3)s‘i’(F3, F2) ] . (7.17)

The multi-dimensional integrals that occur in the HF energy are tackled only after
the application of the DME. As can be seen, each term in Eq. (7.17) is composed of
one local and two nonlocal densities. This can be traced to the fact the D-term of

the interaction combines a contact term and pion-exchange interaction.

HF energy from the C-term

The operator structure and analytical form of the C-term of the chiral EFT NNN
interaction at N2LO are given in Eq. (2.19). The HF energy from the C-term can be
grouped into two categories: a D-like term and remaining terms (which we call R-
part). This grouping originates from the operator structure of F SE given in Eq.(2.20).
. . ~ C m2 C
The D-like term is assocrated w1th 005 [—4 -%21 + 2 fig”,- - 6}] whereas the R—part
7r 7T
relates to fieamrg 6k - (d;- + 6}). For time-reversal invariant systems, the HF energy
71'
contribution from the C-term is relegated to appendix 9.9.4 due to its length. Rather,

out of line with the previous sections, the contribution of the C-term to the HF energy

148

in the particular case of symmetric IN M is given by

 

 

2
(v3H1J‘.C,IIVI\I) = (gfé') IE/dFldT—':2d773/wdq3(l(z361(I2-(T2_’ll61(13.(r3—T1)
7r 7T
31 ‘11 2
' - — . +2_‘CI2'(13)
((13 + "1%)(13, + mi) f7? f1?

(- 2 53111 P0(F1)PO(F3s 772)!)00—“2: Fa)
+ 55171 10001 773)P0(F2a F1)/)0(773, 73))

C4 V V ,6 7 _. _. _. _. _. _.
‘2? 65171 63272 (122Q32 P0(7“1,T3)P0(T2a7‘1)p0(7"3a7‘2)] - (7-18)

17

Even though the full complexity of the expression for time-reversal finite systems,
let alone non-time-reversal systems, cannot be appreciated by analyzing Eq.(7.18),
one can still make a few observations. To start with, the complete reduction of the
expression to a local EDF requires calculating twelve dimensional integrals, after the
application of the DME. This is due to there being three position and two momentum
coordinates, while a local EDF allows only one position coordinate in the energy
density. This is in contrast to the E—term contribution which is already in a local
EDF form and the D-term where one has to calculate five dimensional integrals. The
details of the DME technique used that preceeds the multi-dimensional integrations
and the specific analytical and symbolic approaches used for the multi-dimensional

integration are discussed in the next several sections.

7 .2 DME for the HF energy from chiral EFT N2LO

3NF in time-reversal invariant systems

In this section, restricting the discussion to time-reversal invariant systems, we ap-

ply the DME to the HF energy from chiral EFT 3NF at N2LO. In contrast to the

149

application of the DME to the HF energy from N N interactions, the task at hand is
complicated by several factors. Firstly, the usual choice of DME-coordinates as the
relative and center of mass coordinates, of the three—nucleons in this case, turns out
to be of little use. Secondly, the HF energy from local N NN interactions in general
depend on three-coordinates. This is also the case for N NN chiral EFT interaction at
N 2LO. Hence, there are two-nonlocality coordinates which should be integrated out
in the final energy density. Finally, the complexity of the starting HF energy expres-
sion, even for the case of time-reversal invariant systems, renders manual derivation
impractical. In order to mitigate these complexifying factors, it is imperative that we
organize the HF energy expression in a systematic manner.

Even though one has a large number of seemingly different terms coming from HF
contribution of the C and D-terms of the NNN chiral EFT interaction, it is possible
to group these terms into three generic classes. Note that the E-term results in a
local EDF without the application of the DME, and thus we do not refer to it in
this section2. This systematic organization of the HF energy is done with the aim of
identifying the optimal DME-coordinate system. What makes this systematic organi—
zation possible is the fact that, after the expansion of each local / nonlocal density, the
DME requires only angle integrations irrespective of the nature of the densities in-
volved. Certainly, each density has its own 7r—functions, but this does not undermine

the previous statement.

7.2.1 Generic forms of the 3N F energy expressions

There are only three generic forms that appear in the HF energy from chiral EFT 3N F
at NzLO. These are listed in Eqs.(7.19), (7.20) and (7.21). In listing these expressions,
the conventions used are that (i) all numerical and constant coefficients have been

dropped, (ii) §”1,c"2,§“3 can be any of the scalar/ vector, isoscalar/isovector densities,

 

2
Its contribution to the final contributions will be simply added at the end.

150

and TS‘I‘Q’ 3

(iii) the tensors 7211412 may not depend 011 111,112, or 113, then that

173
272
particular index 18 dropped from T. This is the case when the corresponding density

is a scalar density, 100/1- For instance, if c“? is a scalar density, then the tensor

orT;1 3. This
171Mfi272 5111

unconventional notation has been chosen to treat the scalar and vector densities with

T will not depend 011 #2- Hence, it will take the fo1m T51

the same routine in Mathematica.

Generic-Form-l

The generic form of this group of terms is

I51 71
VHF, ,Gl : deIdF2dF3 \/dq_-'2dq_:3 eiq"2.(F2—F1)eiq"3.(F3—F1) (12 q3
< 3” l «13+ m§><q§ + mfi)

#1“ 2” 3 1 W2 3 B2 q72 #1#2#3
“[6 Tfim + C2 ‘12 (13773171 + C3(12‘11731‘1152'12

X CI11(7-317 773)§H2 (F21 72.1)CM3 (F31 F2) 1 (719)

 

where c1, c2 and c3 are either zero or one. The double-exchange of the C-term (both

D-like and R—part) is the sole origin of this type of terms.

Generic-Form—2

The generic form of this group of terms is

 

c +0 7 - 7
(WILEG2) : deIdF2dF3 /dq-'2d(zselq2- (7‘2— 7'1) 6703 (1‘-3 7'1) q2q3( 12 2g? qg)
((12+m 1r)(QB+m1r)

13“)

xrf‘l“2"3[<“1r(1)<”2(F2.F3)<“3<aa) , (7.20)

where cl and c2 are either zero or one. The single-exchange of the C—term is the origin

of this type of terms.

151

Generic-Form—3

The generic form of this group of terms is

<V3‘ VHF, G3) =/d7'2(17'3 /(1q3 eiq3. .(1‘3-7‘2)_____ 2'qu3 Tllfi1p2p3
(13 + m3,

X[CI‘1('F2)<”2(7723773)<"3(773,F2) . (7.21)

The D—term is the only origin of this type of terms.

7 .2.2 The DME-coordinate system

Most of the complexifying factors in the application of DME to the HF energy from
the chiral EFT NNN interaction at N2LO can be mitigated by the proper choice
of a DME-coordinate system. There are several qualities that we require from a
viable DME-coordinate system. As discussed in section 5.3.7, the expansion of local
densities is problematic. Hence, the DME-coordinate system should reduce the need
for the expansion of the local densities. The other quality required of the coordinate
system relates to the need to have analytical expressions for the couplings of the
resulting EDF. This is a requirement in so far as there is no apparent reduction in
the accuracy of the whole approximation. Finally, and related to the previous point,
the coordinate system should be such that the amount of mathematical manipulation
required to arrive at the final result is manageable.

These requirements rule out the usual relative and center of mass coordinates of
the three-nucleon system. Likewise, the three-body Jacobi coordinates are found to

be non-optimal. Rather, the coordinate system that we use is

F1 = 1‘], (7.22)

which implies that we are expanding about the coordinate of the first particle, F1.
While this choice is directly applicable to generic forms given in Eqs. (7.19) and (7.20),
for that of Eq.(7.21), one simply sets F1 = F2 which results in :32 = 0. The main ad-
vantages of this coordinate system are:(i) It allows exact integration of the factors
resulting from the interaction, with any approximation being confined to the expan-
sion of the densities. This implies that this work can be modified and/ or extended
by simple modification of the expansion of the densities. This point is discussed in
section 9.10.1. (ii) The chosen coordinate system enables one to apply the DME only

to the nonlocal densities that occur in the HF energy expression.

7 .2.3 Generalized PSA-DME

Using the coordinate system specified in Eq. (7.22), the non-local densities that appear
in Eqs. (7.19)-(7.21) are of the form ("1 (771,771 + 553), (“2(F1 + 552,771) and (“30"; +
:33, F1 + 52). These non-local densities can be scalar/ vector, isoscalar/isovector. The
generalized PSA-DME aims at approximating each of these non-local densities in
terms of local densities, dependent on F1, and 7r—functions which can depend on 552
and/ or :33.

A detailed discussion on the generalized PSA-DME of these nonlocal densities is
given in appendix 9.5.3. Reproducing the main results, the PSA-DME of the nonlocal

densities that occur in the HF energy from chiral EFT three-nucleon interaction at

NQLO take the form

.. a s a 1'2 1 .. _.
pq(r1, r1 + $2) = H8(k%xg)pq(rl) + -6—2 flaking) (2 qu(r1) — Tq(7‘1)
3 ..
was), (7.23)

153

with the same form holding for pq(f'1,fl +1173). From the time-reversal property of
the scalar part of the OBDM stated in Eq. (9.85), pq(f'1 + fgfl) = p§('f"1,'F1 + :82).
Hence, their DME can be obtained from Eq. (7.23). The DME of pq(F'1 + :32, F1 + 53),

which involves two nonlocality coordinates, .1772 and 13. is given by

_. 1

pawn-amen -—- H5<:~7«x>(p.<a)+ X 7m )+§(X-V71)2pq(m)

+— (5211”(Ilc":z:)((a2 —a+ ;)qu(7"1)— M723)

3
+ 5183’ WA )) (7.24)
where the coordinates 53' and X" are given by if = :32 — f3 and X = (1 — a) :32 +

(7553. For a discussion on the DME-coordinate optimization parameter a, refer to
appendix 9.5.3. The same form holds for its time-reversal counterpart pq(F1 + .733, 7‘] +

:32). The vr-fimctions that occur in both Eqs. (7.23) and (7.24) are given by

 

. kq
US$19) = 33122;?!) = “3(1'62‘71). (7-25)

The corresponding expansions for the vector part of the OBDM are given by

Sq,u(F1,77+fg) 2 2H§( (kp$2) )2: 113211.] QPV (T1) (7.26)
sq,,,(1=1+ .72, 7+ :73) 2 7717‘ (1.7.3:) 2 $171147.) , (7.27)
11:1:

where the 7r—function occurring in Eqs. (7.26) and (7.27) is given by

- - " is”.
II'“(k",y) E (Mi—‘1). (7.28)
l I qu
F

is}. as given in Eq. (5.37). A similar expansion holds for sq,,,(7"'1,f' + 753). Using

the time-reversal property of the nonlocal vector density as given in Eq. (9.85), the

154

expansions for sq,.,(7_"1 + .752, F1), 5%,,(771 + :33, 771) and sq.,,(7""1 + :33, F1 + £2) can easily

be generated from Eqs. (7.26) and (7.27).

Infinite nuclear matter limit

The generalized PSA-DME, just like the PSA-DME developed in section 5.3, is exact
in INM. Since §q(7"1 + :32, F1 + 553) = O. in spin-unpolarized INM, we consider only

the scalar part of the density matrix. In INM, it is given by

 

Pq(7‘laT1 + 332) = Bylaw), (7-29)
kpx'z
—-o -o —o —o - kq -' — -. -0
pm. +w2,rl+x3) = 3]“ ”$3 ”:2” q(r1), (7.30)

kj’plfg — {if-2|

which can be recovered exactly from Eqs. (7.23) and (7.24) by noting that fipqh’)
and qu(1'°') vanish in INM and 7,,(7‘) = gkgpqo") . This implies that in the application
of the DME to the HF energy from chiral EFT NNN interaction at N2LO results in
a local EDF which will reproduce the exact HF energ7 with no discrepancy. The
previous statement holds as long as the DME is the only approximation in the whole

set of steps followed to obtain a local EDF. As discussed in section 7.2.2, this is one

of the benefits of the adopted DME-coordinate.

7.2.4 The resulting EDF

The application of the generalized PSA-DME given in the previous section to the
exact HF energy from the chiral EFT three-nucleon interaction at N2LO, followed
by a set of mathematical steps that mostly involve angular integrations, results in a
local EDF. Yet again, the complexity of the algebra required to arrive at the final
simplified form of the EDF rules out manual derivation. Hence, we automate the
derivation using Mathematica. The main ingredients of the symbolic derivation of

the EDF are given in appendix 9.10, while the complete derivation can be found in

155

Ref. [161].

In the symbolic derivation of the EDF, the analytical PSA-DME expressions of the
nonlocal densities given in Eqs. (7.23), (7.24), (7.26) and (7.27) are replaced with
their symbolic counterparts. Even though this might seem an irrelevant technical
detail of the actual implementation, it is important in the following respect. We have
mentioned that, once nonlocal densities are approximated by their DME counterparts,
the DME-coordinates F1, 552, f3 allow for the exact simplification of all components
of the exact HF expression. This implies that any disagreement between the exact
HF energy and the corresponding EDF can be reduced by further optimization of
the DMES of these nonlocal densities. In line with this, we develop a very general
symbolic DME ansatz and one can consider the generalized PSA-DME discussed in
the previous section as a specific realization of this symbolic DME ansatz. Refer to
appendix 9.10.1 for detail. In this way, future improvements to the DME can be

automatically implemented in the current approach.

There are several strong points about this symbolic derivation: (i) The couplings of
the EDF are functionals of the 7r—functions. Consequently, by fixing the 7r—functions
according to some analytical DME scheme, one can generate the corresponding cou-
plings of the EDF. In the next section, we discuss how we obtain the analytical
couplings for our choice of the w—functions according to the generalized PSA-DME.
(ii) The automation of the whole task enables us to keep all the higher-order terms
(up-to-sixth order) in the EDF. Note that only even orders occur in th EDF i.e. the
occurrence of terms with one, three or five derivatives is precluded by the requirement
of rotational invariance, as energy is a scalar quantity. The complete EDF, includ-
ing all the higher-order terms, is reported in the Mathematica files accompanying
Ref. [161]. The usual truncation of EDFs at second-order in gradients is mostly due
to the complexity of numerical techniques to solve an equation of motion with beyond

second—order gradient terms. Recently, there have been several efforts geared towards

156

incorporating higher order terms in the EDF [35]. In appendix 9.10.4, we report the

EDF by including terms up to fourth-order gradients.

At this point, the result of this work is being utilized along with phenomenological
extensions. This is discussed in section 8.1. As terms with at most second-order
gradients are the ones being used in the referred work, here, we report the form of
the EDF by truncating it at second-order, with the added assumption of spherical

symmetry,

8mm 2 fdf’{ Cpg p307) + C'pop¥ POO?) 72%”) + Cpgro 703(7) 7'00?)
+ CPiTO p10?) 700*) + CPOPITI pom p107) TM“)
+ cponoVPo pom 6pm - vpom
+ CPOV’HV"1 700(7’) 7701(7) - 7701(7)
.1. CPIVPOVPI p1(7_“) 6p0(f’) - 6pm?) + CpgApo 7030:) Ape“)
+ cPiAPo pm Apom + c7071“1 pom pm mm
+ c7013 pom 71.3.7.7?) + cm"12 pom Jim - Jim
+ cleoJl pm Jim - Jim + cpoWoJo pm 77200") - JIM
+ CPOVPI 11 p0(f) 7mm - J] (J) + C”1‘7”0Jl pom 7pm”) - j} (7“)
+ CPIVPIJO pom 6101(7) - Jim + Cpgw0 723(7) 7 ° Jém
+ cpiwo pm 7 - Jim + CPOPIWI pm pm 6 - Jim } ,

(7.31)

where C‘1‘2‘3 are the couplings of the EDF. Note the explicit isotopic symmetry of

the functional as each term contains an even number of isovector densities.

157

Comments on the second-order truncation for spherical systems

Even though all higher order terms in the resulting EDF (up to sixth order) can be
used in the future, currently only up to second—order terms are being considered in the
UNEDF implementation of this work as described in section 8.1. Numerical tests per—
formed in section 5.4 confirm that the higher order terms generated by the DME are
much smaller than the leading order terms. Still, it is important to ask which terms
will be missing when truncating the resulting EDF at second-order. This is especially
important for the tensor and spin-orbit pieces of the functional as current phenomeno-
logical EDFs Show significant deficiencias in that respect [[109],[159],[158]]. It is also
well known that the two-pion exchange part of the three-nucleon interaction plays a
significant role in the spin-orbit splittings of atomic nuclei [154], further increasing
the importance of reproducing the exact HF energy with the DME approximation.
Consequently, we analyze what the practical second-order truncation entails for
terms in the exact HF energy that contain the nonlocal spin density, 50/107,, 7‘3), as
these are the possible origins of tensor and spin-orbit terms in (the resulting EDF.
The assumption of time-reversal invariance sets §0/1(7") = 0, meaning we have to
consider only nonlocal spin densities. In addition, the analysis is done when the
DME adopted is the generalized PSA-DME, instead of the more general symbolic
DME ansatz described in appendix 9.10.1. The key difference between the two, for
the purpose of the following analysis, is the fact that the DME of the nonlocal spin
density does not involve the gradient, V7), JV, and the laplacian, AJV, corrections in the
generalized PSA-DME while the symbolic DME ansatz contains such terms. Noting

that the isoscalar/isovector label does not matter in the following analysis

0 Generic-Form-l (Eq. (9.406)) - here either one, two or all three of the nonlocal
densities, viz, §“1(F1,f'3), <”2(F2,F1), §”3(7"'3, F2) can be nonlocal 3'0/1. Starting

with one of them being nonlocal 530/1, the terms that are not considered during

158

second-order truncation include fourth-order terms only such as Vp0/1 - j()/1 70/1.
A similar analysis for the more general symbolic DME ansatz shows that there
are neglected sixth-order terms in this case, as the DME of nonlocal 530/1 include
corrections from gradient and laplacian of jg) /1. When two of the nonlocal
densities are nonlocal Sig/1, there are again only fourth-order terms such as
70/1 ..}-(.1/1 - .IIVI. The case is much simpler when all the nonlocal densities are
the nonlocal 50/1. The contribution of these terms to the EDF vanishes, as
one cannot form a scalar by picking three spin-orbit densities from the set
containing just two elements: (j), .71). Hence, the truncation does not introduce
any missing terms when all the nonlocal densities are nonlocal 50/1. A similar,
but more complex, analysis can be done for the case of the symbolic DME

ansatz.

Generic-Form—2 (Eq. (7.20)) - here either one or both of the two nonlocal den-
sities, viz, (”2 (F2, F3), (”3 (F3, F2) can be nonlocal 5’0“. Starting with one of the
them being nonlocal 530/1, there are no terms that are not considered due to the
second order truncation. This can be realized from the fact that the local den-
sity (”1 (F1) is not expanded and the only gradient terms come from the DME of
the other nonlocal density. However, we have only up to second order gradients
in the expansion of the other nonlocal density, of which only the first order term
(along with f of the nonlocal 3'0 /1) contributes to the EDF. This is due to the
requirement of rotational invariance. Hence, there are no terms that contribute
to the tensor/spin—orbit part of the EDF and are beyond second-order in gradi-
ents. When both nonlocal densities are nonlocal 5'0 /1, the only EDF terms that
result are of the form p0/1 jfl/l - .IEJ/l, which contain only second-order gradi-
ents. Consequently, the second-order truncation does not result in any missing

tensor/spin-orbit terms from terms of Generic-Form—2.

159

o Generic-Form-3 (Eq. (7.21)) - a similar analysis shows that the same conclu—
sion as in the case of Generic-Form-2 holds. In other words, the second-order

truncation does not result in any missing tensor/spin—orbit terms.

Concluding, the above analysis shows that, for the generalized PSA-DME, truncating
at second-order keeps most, if not all, of the important spin-orbit / tensor terms of the
resulting EDF. In fact, the few missing higher-order terms are fourth-order terms
from Generic-Form—l. These terms can be expected to be absorbed in the part of the
couplings that are phenomenologically fit. Refer to section 8.1 for detail. The above

analysis can simply be extended to non spin-orbit / tensor terms of the EDF.

7 .3 Analytical Couplings from the chiral EFT N NN
interaction at NQLO for time-reversal invariant
systems

In this section, the general analytical structure of the couplings is discussed. As
discussed in section 5.3.5, we use the isoscalar kp instead of It} in order to obtain
isospin preserving EDF. Starting from Generic-Form—l, given in Eq. (9.406), it can
be seen that the application of the DME to the nonlocal densities results in couplings

of the form
6312‘?) ~ / (1:72 (1:733 dag dag, F1(kp,§:'2,f3,(j'2,(]§), (7.32)

where 023% denotes the couplings obtained from the application of the DME to
Generic-Form—l type terms. In this equation, F1 is in general separable in (1'2 and (73,
but not in .752 and 2133. This is due to the fact that the sole origin of the momentum

transfer coordinates, (1'2 and (75, is the interaction where they already occur in separable

160

form, while :32 and :33 remain coupled even after the application of the DME. This
in turn is because of the 7r—functions associated with (”3 (F1 + :iig, F1 + 23,), as given
in Eqs. (7.25) and (7.28). A similar analysis shows that the couplings from Generic-
Form-2 given in Eq. (7.20) take the same form as Eq. (7.32). The form of the couplings

from Generic-Form-3 is simpler and it reads

02538 ~ / dadaF3(kp.a.a). (7.33)

In general, direct, exact and analytical integrations of couplings of the form
Eq. (7.32) that require twelve-dimensional integration is a hopeless task. Even for
couplings of the form given in Eq. (7.33) and a few other cases Where there is a com-
plete separation between terms dependent on (552, (72) and (1753, (jig), in general we have a
product of three spherical Bessel functions in addition to exponential and polynomial
prefactors for each group, i.e. for those dependent on (152,62) and those that depend
on (:33, (73). In general there is no known analytical method to calculate these types of
integrals, further complicating the problem. One can thus envision doing numerical
Monte-Carlo integrations, thereby resulting in non-analytical/numerical couplings.
However, the resulting lack of elegance and inconvenience for systematic study of the
couplings convinced us to invent a combination of analytical and symbolic methods to
integrate these couplings analytically. The symbolic and analytical procedures used
for the analytical calculation of these couplings are discussed in appendix 9.11. In
this regard, Gegenbaur’s addition theorem which is detailed in appendix (9.1.5) and

tested in appendix(9.11.3) plays a significant role.

The simple analytical structure that emerges for all the couplings of the EDF
reads

C‘1‘2‘3[u] = C]<1<2C3 [u] + 031‘2‘31u11n14u2 + 1] + C;:1<2q3[u] arctan[2u], (7.34)

161

C q. S" . . . . . .
where 0,1 2 3 are polynonnal functions of u. The variable u, wlnch IS also used in

section 6.2.1, is a dimensionless quantity given by

7.»..(1‘7)

m,r

 

. (7.35)

it

where we explicitly show the R dependence of kp(R) to emphasize that the couplings
are functions of density/ position. Note that in writing Eq. (7.34) for C‘1‘2C3[u], we
have dropped coefficients that appear in front of 0:1‘2C3. These coefficients are given
in terms of the interaction parameter, m,r and kp(§). This can be seen from the
requirements of the dimension of the couplings. For instance, 0”” needs to have
M eV ( fm)9 dimension. Finally, we remark that the variable u stays predominantly
within u e [1.0, 2.0]. For this, we used kp z 1.4 fm‘1 inside the nucleus and half that
value out in the surface, while m,r z 0.7 fm’1 . For the actual analytical forms of

the couplings, consult Mathematica files of Ref. [161].

Comparison of analytical and Monte-Carlo results

We mentioned that the combination of analytical and symbolic approaches have en-
abled us to calculate the couplings of the EDF analytically. The calculation of the cou-
plings from Generic-Form—l and Generic-Form—2, given by the generic form Eq. (7.32),
relies on Gegenbaur’s addition theorem of Bessel functions followed by the symbolic
approach that we invented for this purpose. In contrast, the integration of the cou-
plings from Generic-Form—3, denoted by the generic form Eq. (7.32), do not require
the use of Gegenbaur’s addition theorem. Rather, they are calculated directly using

the symbolic approach. Refer to section 9.11 for details.

Gegenbaur’s addition theorem is exact only if it is not truncated at a finite or-
der. Since we are forced to truncate the expansion at a finite order, this calls in to

question the accuracy of the couplings which are calculated using this approach. In

162

appendix 9.11.3, we discuss the accuracy of a truncated Gegenbaur’s addition theo—
rem, where we show that including at least the first five terms of the expansion gives
a practically exact expansion. In this section, we test the impact of this truncation
by comparing the local EDF that results after the application of the DME with the
exact HF energy from chiral EFT N NN interaction at N2LO. We perform the test for

symmetric INM, where in principle we should have exact agrement.

The exact INM HF energy contribution from the chiral EFT NN N interaction,

given in appendix 9.9, can be written in terms of Eqs. (7.29) and (7.30) as

. 3
(VJLF‘JM‘) = 76E] (1717236.). (7.36)

which is already in local form and does not need any DME. For the D-term, we have

3
.. P - _.
(“SIREN“) = /drch(.)1NM/)ll(rl)a (7-37)
where
P3 _ 279A CD j1(kF$3)j1(kF-’B3)

 

 

2
[(1553 (171367353 ‘13 . (7.38)

C =
D,INAI 512 7T3 ngx (1% + m12r kaL‘3 kF$3

Note that, in this formula 15,: = kph—"3) = [3/2n2p0(F1)]1/3, using the isoscalar den-
sity. This coupling can be integrated analytically without the application of Gegen-

baur’s addition theorem and thus we do not discuss it further. The C-term contributes

HF.C‘,INM HRC‘DIXJNM rHF.CD2X.INM
(V ) = (V >+ (V )

3N 3N 3N
+ (vQPLFJnXJNM) (7.39)

... (’8 3 _.
2 d7‘1 CCJNM P001) a (7-40)

163

where

P8 vii (’8 vii
CCJNM E C(fDlrJgV'M + CC'DQIJNM + CRZIJNM' (741)

We have separated the three different contributions to the coupling in Eq. (7.41). As
discussed in section 7.1, the C-term of the chiral EFT three-nucleon interaction at
N2LO has what we call the D-like and R pieces. In the coupling shown in Eq. (7.41),
the first two terms of the couplings are from the single- and double-exchange parts of
the D-like piece, while the last one is from the double-exchange part of the R—piece.

These are given by

 

7% 54 9A
CCDIXJNIU -10247I'6 2f
6 7

513171 q21q31 01777.3, C3 _. _.
2 2 —4 + 2—(12 ' (13
((12 + "13.)(73 + m3.) f3 f1?
j1(krli"3 — 52]) j1(’€F|53 — f2|)
kFlf3 — 552] (9154553 — 552]

2
) / (fl—:2 £133 (152 (lg—:3 €262.52 81-6333

 

 

 

(7.42)

3 2
p 81 9A ~~~~rr 11:?
CC(i)2x,1NM = 1024176 (2f ) f (1332 C1333 (1‘12 d(13 6 q2"’2 6 (,3. ‘3
1r
[3 ‘7
x 56171 (121931 ( 4017713r + 203 .. s)
— —(12 ' (13
((1% + "”3013 + "13.) f3 f3
x J'IUCFJI2) j1(kF$3) j1(kF|53 — 52])
(9171132 kF$3 kFlfs — 1332]

 

 

 

 

. (7.43)

 

p3 _ 243 57.4 2C4 _. —. —. -. 7.7.5 127.5
CR'ZXJNM = —10247T6 2ft 712— (11172 dedq2 dq3 8 2 26 3 3

31 ’71 B2 “'2
Q2 Q3 Q2 (13 68171116732721!

((1% + m3)(t1§+ m3.)
leko332) j1(kF333) j1(kFlf3 '" 5‘2”
(@1132 ($271133 (CPI-”173 — 552]

 

 

(7.44)

As can be seen from Eqs. (7.42), (7.43) and (7.44), all of them require the appli-
cation of Gegenbaur’s addition theorem. This is due to the occurrence of j1(kp|f2 —

f3|)/(kp]fg — 553]) in these terms that are not separable in 552 and E3. The numer-

164

ical test discussed in appendix 9.11.3 shows that we can truncate the Gegenbaur
expansion of this term at fifth order as
j1(/.~,.~(3'3 — .721) /144 r(3/2) 5 3 , ,
.. _. z + — ' 1 k :1? 1.? ~17.
kFll‘s—l‘zzl 7T #7133132 gm 2)J’+l( I 2)]“+1( f 3)

x CZ/2(cos(6)) , (7.45)

 

 

where 6 is the angle between :52 and $3 This is followed by analytical integration
of the couplings. Furthermore, one notes that Eq. (7.42) requires the application
of two Gegenbaur expansions while Eq. (7.43) and (7.44) require the application
of only one Gegenbaur expansion. To assess the accuracy of the couplings/EDF in
Eq. (7.39) when calculated with the truncated Gegenbaur expansion, we compare
the result with the case when the couplings are calculated with the essentially-exact

Monte-Carlo integration (without Gegenbaur expansion).

In Fig. 7.1, we show the percentage error of the Gegenbaur-based calculation
with respect to the Monte—Carlo ones, for the contribution of the three terms of
Eq. (7.39) to the energy per particle of INM as a function kp. For each of the
three terms, we have two curves where the insets show the actual contribution to the
energy per particle when the couplings (Eqs. (7.42)- (7.44)) are calculated analytically
with the truncated Gegenbaur addition theorem, at fifth order, and the lower curves
(main curves) represent the percentage errors. The constants of the chiral EFT
three-nucleon interaction that are used in this particular calculation are specified in

Table 2.2, with he = 197.327 [MeV fm].

The results show that, the percentage error resulting from truncating the Gegen-
baur expansion is less than 0.5% over the range of physically interesting kp values.
In fact, for the double-exchange from the D—like and R pieces, the percentage error
shows a strong fluctuation between 0 and 0.5%. For the single-exchange from the D-

like term, the percentage error shows a steady increase from 0 to about 0.3%, which

165

 

 

 

 

 

$100.0- 5 0.0--— a 3 0.01“.
> . > . \ 3 . s
g 80.0- g -5.0- \ g .501 \
I—a ) I—l ‘
g 60.0- =-10.0< \ 2.100
30 ' 3’0 ‘ \ 0 ‘ \
:3 40.0: 5-15.0] \ gin-15.0:
a. 20.0- 2-20.03 $9-200] \
. . x
S 0.0- S - . . - . N . a . . .
o 0.75 1.25 1.75 O 0.75 1.25 1.75 °‘ 0.75 1.25 1.75
.1 .1 -1
k1,. [fm ] anm ] kF[fm ]

P
UI
‘—
“——

 

Percentage error

 

 

 

-0.5

0.75 ' 1.25 ' T75
k1,. [fm'I]

Figure 7.1: (Color online) The percentage error of the truncated Gegenbaur
expansion with respect to Monte-Carlo based calculation of the contribution to E/ A
in INM. Upper plots Show the actual values for the calculation based on the
truncated Gegenbaur expansion.

166

is not surprising as we needed to apply Gegenbaur’s addition theorem twice in that
case. Considering the unavoidable numerical errors/flucuations in the Monte-Carlo
calculation and the smallness of the percentage errors, obtained we can conclude that
the truncation of Gegenbaur’s addition theorem at fifth order provides a practically
exact truncation. At this point, one should realize yet another reason for the need to
automate the whole calculation. I.e. the application of Gegenbaur’s addition theo-
rem replaces each term in the couplings with about five terms when truncated at fifth
order. For instance, in the integration of Eq. (7.42), the single Monte—Carlo integra-
tion is replaced with about 25 integrals due to the double—application of Gegenbaur’s
theorem. Hence, even though it enables us to obtain completely analytical couplings,
Gegenbaur’s addition theorem comes with a tricky overhead: about two orders of
magnitude increase in the number of integrals to be calculated. Finally, we remark
that the conclusion of this section, viz, the truncated Gegenbaur’s addition theorem
enables us to calculate the couplings in a practically exact manner, holds for all other
couplings as the truncated Gegenbaur expansion is the only “approximation” that
goas into the calculation of the couplings. In the next chapter, besides the possible
future extensions and conclusions, we perform preliminary analysis of the couplings
and the ongoing semi-phenomenological approach that is attempting to make use of

this work.

167

Chapter 8

Semi-phenomenological EDF,

Future Extensions and Conclusions

8.1 The semi-phenomenological approach

Based on the arguments discussed in section 5.1.1, we advocate a semi-phenomenological
approach in which the phenomenological Skyrme functional is to be augmented with
the DME-functional. Here, DME-functional refers to the EDF that we obtained
from the application of the DME to the Fock energy contributions of finite-range
N N and the complete HF of finite-range NNN chiral EFT interactions at N2LO. In
this scheme, the Hartree contributions from the NN part are to be treated exactly.
Finally, the phenomenological Skyrme parameters are to be re—fit to IN M and finite
nuclei properties, leaving those couplings/terms that Originate from the DME intact.
Actually, the so—called phenomenological Skyrme parameters can also considered to

have originated from the contact part of the chiral EFT N N and N NN interactions.

The generic structure of EFT interactions given in Eq. (2.12), VEFT = Vw+Vct(A)-.
shows a clean separation between long- and short-distance physics. Consequently,

each DME coupling at the HF level can be decomposed as the sum of a density-

168

independent, A-dependent piece, which are subsequently re—fit, coming from the con-
tact terms of the EFT NN and NNN (E-term) interaction (VCt(A)) and a density-

dependent, A-independent piece coming from the finite—range pion exchanges

cf“? = cf1‘2(A;v..)+Cfl‘2(1‘2’;v..), (8.1)

631“?“3 = cf1‘2‘3(A;v..)+cfl‘2‘3(fi;v.), (8.2)

where (1 C2 and <19 (3 are bilinear and trilinear combinations of densities that occur

in the EDF [[153], [160]].

In this sense, the re—fit parameters can be viewed as containing the effects of
the HF contribution from the contact interactions, Vet, plus higher order effects that
would arise in a more sophisticated Brueckner—HartreeFock or 2nd-order MBPT cal-
culations. In this regard, through the loose connection of the refit Skyrme parameters
to the EFT contact terms, the EFT concept of naturalness might provide useful the-

oretical constraints for the fitting procedure [163].

The following several plots show sample 6:1 <2. As can be seen from fig. 8.1, the
novel density-dependence is controlled by the long-range parts of the NN interaction.
Therefore, it is not surprising to see that the density profile of the couplings shown in
the figures is driven by the LO term (one-pion exchange) since the NLO and N2LO
interactions are of shorter two-pion exchange range. Even though the couplings in
fig. 8.1 seem to satisfy the hierarchy requirement, viz, LO > NLO > NQLO, it is
not guaranteed that this will always hold. This is because we are including only HF
contributions to the couplings, with our focus being primarily on finite-range pieces.
The fact that the hierarchy might not be maintained should not seem to be a big
problem as HF amounts to comparing the LO, N LO, NQLO potentials (which are not

observables) and therefore are they not required to obey any hierarchy.

Fig. 8.2 and 8.3 show the CO” and C1” couplings with +/— error bands as de-

169

400

CH [MeV-fms]

0.
c>

[MeV-fms]
'3

I";
H

-150

C

-200

 

 

Illllrffi l I I I l Tllllll
— LO ‘

T=0 ——- NLO a

_ NNLO .

 

 

 

 

 

 

 

 

 

 

lllllllllllllllllll

 

0

OJ. (115 (12

p [fm'3]

(105

Figure 8.1: (Color online) GP and Cl“ couplings from chiral EFT N N interaction at

N2LO.

170

500

 

 

   
   
   

 

 

 

 

 

 

 

 

 

I I I I I I I II I I I I I I I II
1 A = 500 MeV
SFR — SLy4_Tse1f
400 ——- SLy5_T
SkM*
w"; 300 IT $31.21
«.7. _ --- TZA
% 200 —
E _
i: o
100 —
0 ':'.'-L—'.'—.' é-'_-'-_'—;'—_':';—'_'—_'—i __
- I I I I I I] II I I I I I l I II
0.01 0.1
-3
P [fm ]

Figure 8.2: (Color online) 65’ couplings from the chiral NN interaction at N2LO
with error bands from naturalness requirement.

171

 

 

 

   

 

   

 

 

 

 

 

 

50 TiT TIIIWII I I IIIIIII I
r ................................ 7
0
,_ -50 _______
“a
“—1
> -100
a)
2-150
2 _, — SLy4_T self
0 -200 --- SLy5_T _
_ SkM* _
250 -- T22
- — _ -—-- T44 -
_ASFR—SOOMeV ___ TZA _
_ I l I I l I I II I I I I I I III I
300 0.01 0.1
-3
P [fm ]

Figure 8.3: (Color online) (31” couplings from the chiral NN interaction at N2LO
with error bands from naturalness requirement.

172

termined from the naturalness requirement [163], compared with the corresponding
phenomenological values. The error bands cover all phenomenological values in the
density region of interest. Thus, the main conclusion that can be made at this stage
is that the DME couplings are close to phenomenology, but with a novel density-
dependence, as long as one allows for natural-sized contact terms. For a detailed

discussion, refer to Ref. [153], while for the corresponding discussion on 6:19:31 refer

to Ref. [160].

The first calculations following the semi-phenomenological approach advocated
in this section are underway [164]. Figs. 8.4 and 8.5 shows one of the exploratory
“results” regarding the saturation curve, W(p, I), of INM and PNM. Here, I = (pn —
pp) / p. The parameters of the DME—based functional used for the saturation curves
are not optimized, rather they are simple educated guesses. Preliminary indication
from this study is the Skyrme functional that is augmented with the DME functional
is more flexible in that it relaxes some of the interdependencies that one observes in

phenomenological functionals [164].

8.2 Key future extensions

In this section, we revise the main directions in which this work can be extended in

the future. These are

0 Extensive self-consistent test of the PSA-DME. As discussed in section 5.4, our
tests can be judged to be extensive only for non self-consistent ones. Along with
the self-consistent tact, the invention of a local N N and N NN chiral interaction

that is soft enough to be used for these tests is important.

0 Studying the impact of the different DMEs on the couplings of the resulting
EDF. Note that the non-self consistent tests that we performed in this work are

using schematic interactions and it will be beneficial to extend this and perform

173

W(p)

 

 

 

 

.. ’ " SLY4

50 _
: SLY4—”.0
F SLY4+LO+NLO

w ’—
~ SLY4+LO+NLO+N2LO
;
L-

30 >—
)-
f
p

20 ~

I0 7
I
F 1 L 1 l n i I I 1 I 1 4
1 0.1 0.2 0.3
F'

- IO *-

f

 

 

Figure 8.4: The saturation curves, W(p, I ) of INM using the phenomenological
SLY4 functional and semi-phenomenological DME-based functionals. Here, N2LO
includes the contribution from both NN and NNN interactions (From Ref. [164]).

 

 

 

 

 

 

W(p, 1:1)
- - - SLY4
100 : SLY4+LO
~ SLY4+LO+NL0
80 : SLY4+LO+NLO+N2LO
)-
w i—
L
40 ...
b O
)-
20 r
J L J l L J l A A A A 1 A I l I l l l l l l L I 1 L I A 1
0.] 0.2 0.3 0.4 0.5 0.6

Figure 8.5: The same as Fig. 8.4 but for PNM (From Ref. [164]).

174

p

 

 

extensive comparison of the actual couplings that result from the application of
different DMEs. This should provide a better estimate of the associated DME

errors /uncert ainities.

o Non-self—consistent and self-consistent tests of generalized PSA-DME. This is
important to gauge the accuracy of the DME approximation that we made to
the HF energy from the chiral EFT NNN interaction at NQLO. Even though we
have shown in appendix 9.5.3 that the approximations that we used to obtain
the generalized PSA-DME are equivalent to the ones used for PSA-DME, the
existence of more than one non-locality coordinate may change the relative
accuracy of generalized PSA-DME with respect to PSA-DME. Furthermore,
the effect of DME-coordinate optimization parameter a in Eq. (7.24) should be

investigated.

0 Calculation of Bogoliubov contribution from NN + NN N , extension of PSA-
DME or its variants for pairing densities. Furthermore, the required renormal-

ization should be designed along with the DME.

0 From the interaction side, the extension should include the contributions from
the N3LO component of chiral EFT interactions. It should be noted that, even
limiting the calculation at the HF level, the four-nucleon interaction which first

appears at this order will make the extensions more complex. In principle,

 

one needs to incorporate the contribution from the four-nucleon interaction.
Nevertheless, current estimations of its effect on nuclei, at least in light-nuclei,
suggest that it can be ignored safely. For instance, estimates in 4He show
that the additional binding energy it provides is of the order of a. few hundred

keV [165].

0 Extension of the DME scheme to approximate higher-order contributions as

175

 

discussed in section 5.1.1.

0 Analysis of self-interaction and self-pairing issues that arise in the context of

the DME [105].

8.3 Conclusion

This work is a part of a long-term project to develop nuclear EDFs starting from
many—body perturbation theory and the underlying two- and three—nucleon interac-
tions [[110]-[154]]. This is necessitated by the fact that empirical EDFs lack solid mi-
croscopic foundations and often result in uncontrolled (i.e., parameterization-dependent)
predictions away from known data.

We used the DME as a tool to explicitly build microscopic physics associated with
long-range pion-exchange interactions into existing Skyrme functionals in the form of
novel density dependencies. An important component of this endeavor is the improved
PSA-DME and its NNN counterpart, the generalized PSA-DME, which are crucial
if we want to provide microscopic guidance to the description of spin-unsaturated
nuclei. The rich spin/isovector dependence of pion-exchange interactions gives us
hope that their inclusion via the DME will give valuable microscopic constraints on
the isovector properties of the EDF. Moreover, it is comforting that these constraints
are coming from the best-understood part of nuclear interactions.

The EDF obtained as a result of the present work contains only HF physics such
that further correlations must be added to produce any reasonable description of
nuclei. In the short term, such an addition is being implemented empirically by
adding the DME couplings to empirical Skyrme functionals and performing a refit of
the Skyrme constants to data [164]. While this is a purely empirical procedure, it is
motivated by the well-known observation that a Brueckner G-matrix differs from the

starting vacuum NN interaction only at short distances. Therefore, one can interpret

176

 

the refit to data as approximating the short-distance part of the G-matrix with a zero-
range expansion through second order in gradients. Eventually though, it is the goal
of the UN EDF (universal energy density functional) project to design a generalized
DME that is suited to higher orders in perturbation theory and move closer and closer

to complete microscopy.

177

 

Chapter 9

Appendix

9.1 Mathematical Formulae

In this section, we list the Mathematical definitions, relations and formulae that have
been used in the rest of the work. Only the relevant mathematical relations and
formulae are listed, and for a more extensive list, refer to classic references such as

[[152],[151]].

9.1.1 Miscellaneous elementary formulae

In various parts of this work, we use the following general linear coordinate transfor-
mation. Starting with two coordinates (f1, :52), we define a new coordinate system
(2?, 2?) as

f=f1—f2 X=(1—a)f1+af2, (9.1)

where the unspecified parameter a is a real number satisfying a 6 [0, 1] . The corre-

sponding gradient operators are given by

6,1, = a6 —(1—a)€71

1'1 2?

178

VX __— V_.1'1 + 61‘21 (93)

with a = 1/ 2 recovering the usual center of mass and relative coordinates.

In the derivation of local densities, detailed in appendix 9.4 and other parts of the

work, the following elementary results are important.

 

 

... _., ., A 2
VV (7' '7') 17le = :2“, (9.4)
,. A, 7'; 6,37" —Ti7'j ..
(9;,(7' r) = 2-7( 7‘3 )k (95)
ii
-—1- d0~(7.A”)(7 B') — 12.4.1? (96)
47f r 7‘ 7' — 3 .

The manual derivation of the HF energy from a generic two-nucleon interaction

involves a modest amount of spin-isospin algebra. First, the Pauli matrices are given

They satisfy the commutation and anticommutation relations

0'in - O'jO'i = i2€ijk0k7 (9.8)

030,- + 03-0,-

which can be used to prove

(A-3)(§-5) .—. (X-B)I+ia-(A'x§),

cos(a.) + i(7’i - ('7') sin(a,) ,

179

(9.10)

(9.11)

for any two vectors A, B and A = (1A. Additional, elementary relations are

5..., = 0 , (9.12)

2051'. a 050,, = 221' for all A’. (9.13)

9.1.2 Clebsh—Gordon, Wigner 3—J and 6-J coefficients

Representing Clebsch-Gordon, Wigner 3-J and 6—J coefficients by

. . . J1 J2 J J1 J2 J
(Jm I J17711J2m2> , , , (9.14)
m1 mg m 771.1 771.2 m

respectively, the following is a list of the relations that are important for different

parts of this work.

 

1 2" 1
2317721500772)? = 2] :1 , (9.15)
for all m and m, such that m = 7m + a.
. . . - - . J1 J2 J
(J1miJ2m2lJml = (“llmfllfi2 v 2] + 1 (9-16)

7711 7712 m

E ( 1),, J1 J5 J6 J4 J2 J6 J4 J5 J6
m4m5m6 m1 m5 m6 —m.1 772.2 7716 7714 -m5 m6
J1 J2 J3 J1 J2 J3

= , (9.17)
j4 js jfi m 1 "2.2 m;;

where a = 3'.) + 7m + J}, + ms, + 3'6 + m6 .

180

J1J2J ,---J1JJ2

 

 

 

 

 

 

 

 

= (_1)11+J2+J (9.18)
7711 mg 771 ml m 7712
L l 1 l L 1 1
Z(2L+1)f(l,L) = ——(—1)’ l(l+1). (9.19)
L 1 1 z 0 0 0 V5
where
—l 2111—13 if L = 1+ 1;
f(l,L) = —(l+1) fl iszl—l;
. 0 otherwise.
I l 1
: (-1)'-mz m’ 5... _m, (9.20)
mi m; 0 \/l(l+ 1)(2l+ 1) l’ 1
l l 1 (l—m1)(l+m1+1)
= —1 "ml 6 9.21
m: m; 1 ( ) \/ 21(1 + 1)(21 + 1) "’l""‘z"‘ ( )
l l 1 _
:1)-ll+m +1 (1+ ml)( )(l ml + 1) m .4” [+1 (9.22)
mi ml, _1 21 (1+1) (21+ 1) z~ 1
1 1 J x/E \/l(l+1)(2l+1)

181

9.1.3 A few special functions
Legendre polynomials

Starting with the associated Legendre differential equation, for integer l and m,

51— [(1—1132)P,"‘($)] + [l(l+1) — m2 JPNJT) = 0, (9.24)

 

d2? 1—2:2

where a: e R (the set of real numbers), P,"‘(a:) is the associated Legendre polynomial.

The associated Legendre polynomials satisfy

—m m (l — m)! m
Pt (17) = (-1) WP: (ll (9-25)

For m = 0, the differential equation given in Eq. (9.24) can be reduced to

(ll—1:2)i2—dI-Z—(2£2 — 22:52}? + l(l+1)H(;I:) = 0, (9.26)

where H(:L') is Legendre polynomial of order I. The first few Legendre polynomials

are

Petr) = 1. (9.27)
p101.) = 1;, (9.28)
102(2) = $13924). (9.29)

The Legendre polynomials are orthogonal over the. range (—1, 1) and satisfy

2 -

= m Omn . (9.30)

/ da: Putz) Pmo)

182

Additionally, Pz(1) = 1 for any I. The derivative of Legendre polynomials satisfy the

following properties

 

 

 

 

 

 

 

(1P 1 l 1
’(x) = ( + ) , (9.31)
:L' I=1 2
"2 — ldP .
l [(73) = 2‘P,(;r.) — 131-1(2). (9.32)
71 d1):
Laguerre polynomials
The Laguerre polynomials are solutions of the Laguerre differential equation
(12 Ln(:r) (11),,(93)
a: (13:2 + (1— ) d2: Ln(:r) — 0. (9.33)
The can also be defined using Rodrigues formula
0 , _ x—a ex dn -—.r n+0
Ln(:L) — n! dx" (6 :1: ), (9.34)

where L303) is the associated Laguerre polynomial. The Laguerre polynomials are

recovered by setting a = 0

 

Ln(:1:) 2 L200). (9.35)
The first few associated Laguerre polynomials are given by
L3(;r) = 1, (9.36)
L?(:r) = —x + a + 1, (9.37)
..2 ' 2 . 1
13(2) = 32— — (a+2):1: + (“Jr )5” ). (9.38)

183

Gamma functions

The Gamma function, which appears from the extension of the factorial with a down-

ward shift of the argument by 1, is given by

r(.r) = / dttT-le-fi (9.39)
0

where :1: is a complex number with a positive real part. In this work, we need only

the Gamma function for positive integer arguments, which is given by

I‘(n) = (n — l)!. (9.40)
Spherical harmonics

Spherical Harmonics are eigenfunctions of angular momentum operators L2 and L2,

and are given by

 

 

We. 9) = \/(2’4:(1,)ff g)?” 1D("(00899) (9.41)

with their orthonormality relation being give by

11' 27r
/ d6 (131/ma, a) Y,';"(9, a) = (SH/6 . (9.42)
0

m m I
0

There are various relations satisfied by spherical harmonics. and which are of interest

to this work. These are

 

Z Kmr(f',)Y;"-Il (7") = 471- 131 (,1! . 1A,) 9 (943)

"II

184

where the sum extends over all allowed values of 112.).

ram

6 (f9) 131,49] 2 ; ___—— Yz’"(v") + fir) V7Yz’”(7‘%

07'

where f (r) is any function dependent only 011 7“.

ml . 1 , .
V11)”. (1) = -,: Z f(l. L) (um mlLM>Y£’(r).

LM
where
—z 2% ifL=l+1;
f(l.L)= —(z+1),/,,’—_1 ifL=l—1;
0 otherwise.

 

 

K7116, ¢) = ¢<2l+1)(l_ m)! Bm(COSB) eimgb,

 

 

47r(l+m)!
03,1109, 0) F (155,1) me. 0) ,
—m , _ m (l — m)! m
P. (x) — (—1) ((+m)! P. (x).

Bessel functions

Bessel functions are solutions of Bessel’s differential equation

2 .
$2.1 Jam) + deaa)

_. 2 _ 2 =
(11.2 (1.17 + (.1: a )Jc,(a:) 0,

 

(9.44)

(9.45)

(9.46)

(9.47)

(9.48)

(9.49)

where a is a complex number and Ja(:r) is Bessel. function of order a. In most

physical problems with spherical symmetry, 0: takes half-integer values, a = n + 1/ 2 .

185

Consequently, one defines the spherical Bessel functions of integer order, n, as

. (71 n 1 (I "sin(.r) r
Jn(I) E ‘27'In+1/2(l') E (—$) (I E'l') fl? - (9.00)

The first few spherical Bessel functions are

 

jotr) = (9.51)

 

 

 

11(1) =

 

 

12(1’) =

l
E ' ECO

13(4) = (13") 5‘“ NE}— —)C—(—°: ), (9.54)
(

105 45 3:) sin( :13) (105 10) cos(1)
+ 1 — .

 

 

 

331(1’) =

2:3 :r :17

9.1.4 Three-dimensional spherical harmonic oscillator eigen-

functions

The isotropie three-dimensional harmonic oscillator is described by Schrodinger’s

equation

h 1
—-2-EA + 5771922 7‘2 (b,,),,,(r, 19,99) = 6n1¢n1,,,(r,6,.¢) (9.56)

The wave-function is separable in the radial and polar coordinates as

 

 

12.. ...
¢1l_1n,,(7‘,9,¢) : :_(r) Y1 (6,96) a (9'57)
where
1 2(n — 1)! ”2 1+1 _ .. 1+1 2
R1, = — .- 1/21; / .,
[(7‘) r \/31"(n+l+1/2)3 fl 6 ""1 (r)
a: = 137"?

186

7714.)
n
e...) = hw(2n+l—1/2). (9.58)

Note that there are two conventions in use regarding the possible values of integer n:

n 2 0 and n 2 1. The latter is used in this work.

9.1.5 Gegenbaur expansion

Gegenbaur’s addition theorem of bessel functions of the first kind reads

 

 

TV ml/y

JV(T) _ 2” F(:’) 2 (V + H) Ju+n($) Ju+u(y )CV(COS(0))’ (9'59)
p=0

where u > 0 and for all values of :17, y and 6 (the angle between :E and 37). The variable

 

r is given by r = \/.T2 + y2 — 2:1:ycos(6). I‘(V) is the Gamma function and C: refers

to Gegenbaur polynomials. The first few Gegenbaur polynomials are given by

 

05’ = 0,
Cf” = 3cos(6),
3/2 _ _E 15 COS2(6)
C2 _ 2 + 2 a
03/2 _ _15 003(6) + 35 cos3(0) .
2 2
(9.60)
A formula related to Eq. (9.59) is
J_,,(7‘): 2”F( (II—___) V
W D—I (u + 1)4)1_.,_..( >449) 0.144(6)), (9.61)

p=0

and it holds only when Iyeiwl < |;L‘|. Combining Eqs. (9.59) and (9.61), one obtains

the relevant expansion for Bessel functions of the second kind and those of the mod-

187

ified Bessel functions. For details, refer to [195]. Gegenbaur’s addition theorem is a
key ingredient for the analytical calculation of the EDF couplings obtained from the
application of the DME to the HF of chiral EFT three-nucleon interaction at N2LO.

Refer to section 9.11 for details.

9.1.6 Functional derivatives

A functional maps functions into a number. Analogous to the derivative of functions,

one defines the functional derivative of a functional, F [f (13)], with respect to f (:17) as

 

 

6F . 0F[f(:l/) + 8591-33)] - F[f(y)l.

= lzm€_.

5f(117) E (9.62)

The functional derivative satisfies several relations which are analogous to the ones

satisfied by the derivatives of functions. For instance, if F and G' are two functional

of fix).

6(F G) 6(F) 6(0)

 

 

5m) 2 CW + FW’ (963)
while for F[f(:l:)] = I: (11' [f(:r)]".
6(F) _ TL ’13 n—l
W) — [11)] - (964)

9.2 The one-body density matrix and densities

The basic quantity in the EDF approach for nuclei is the OBDM and the various
nonlocal and local densities that can be extracted from it. An extensive discussion of

these basic quantities is given in this section.

188 '

9.2.1 Properties of single particle states

The wave function of a particle having spin S is a spinor of rank 25', i.e. is composed of
25' + 1 components. The particles constituting the nucleus are protons and neutrons
which have spin, S = g. The single particle states are assumed to have a good isospin
projection, but mix spin states We use a = i%, q = :l:% to designate the spin, isospin
and i, j, k... the remaining quantum numbers of the single-particle states respectively.

Thus the single-particle states can be designated as

I 2'4) --— XI 2'04). (9.65)

In spinor notation, the single particle spinors are given by four real functions 1,91 , ..., 1,9,,
, _, 996(FU = +6 (I) 991,i(FQ) + i 992,1(‘F9)

(7129) = 994(7‘9) = _‘ 1 = _ fl . (9.66)
1,9,0“ 0 = -5 (I) 993,i(7‘ (I) + 2 9911,1(7‘9)

The orthonormality and closure relationships are given as

/ (1179-9: (.41) 44-974) = 5.6%,, (9.67)

2 (01(F0q) ,9,()="a'q’) = 6(F— F’)6w,6qq, . (9.68)

It is important to characterize the single-particle properties under time reversal. The
time-reversal operator is given by T = iayKo where K 0 denotes the complex conjuga-
tion operator and cry is a pauli spin matrix. Thus the time—reverse of the single-particle

states is

(T90). (F09) = 20901-5559). (969)

where 6 = — a. For an extensive discussion, refer to Ref. [81].

189

 

9.2.2 One-body density matrix

The one-body density matrix can be written in (Faq) space as
pq(Fa.F’0’) = ((PIC (7 a’cq)(roq)l<1>=Z 9021 7" ’0’)q 991900) 937,-, (9-70)

where N?) defines the many-body wave function and p2, = (<1) | CI c_,- | (D) defines the
density matrix in the basis {Ci/1,9,}. Since the single—particle states have definite
isospin quantum number, the density matrix is diagonal in isospin subspace. The
density matrix can be separated in its sealar/vector-isoscalar/isovector parts

, 1

p844? 0') = 1 [42866) 6..) + 40(66').6..I

+ (—1)1/2“’(p1(f’,7"’) (5M; + .9'1(F,F').6wr)] , (9.71)

where the scalar-isoscalar, scalar-isovector, vector-isoscalar and vector-isovector parts

are respectively

po(.7'"’) = qu(r0r”)0 )6OJ=ZZ¢:(T’09) 99) ”19)/2).

aa’ q 09 ii

= EPA-‘17?) (9°72)
p.626") = Z pq('F0,F'0’)6 .. =ZA(4) 214994) 46606;);
I ij

= z 9607‘"), (9.73)
56(11‘”) = Z pq(Fa.F’0’)5’.,/a = Z 2 99217" ”0(1) 991(FUQ) 50/. p3.-
ii

00 q 00’ q

= Z 6;,(177’), (9.74)
§1(F,F') = Z pq(_'0,-f’0')a :2, A( (q) .2]- 99,-( 7‘ ")0q 1,91-(r0q)00 I 0p],-

00’ q aa'q

190

Z A(q) 6;,(71 7"), (9.75)

1-
where A(q) = (—1)? q . The extraction of the scalar/vector—isoscalar/isovector parts
from Eq.(9.71) can be done easily using properties of the Pauli matrices given in

Eqs. (9.12)-(9.13).

9.2.3 Local densities

Working in neutron / proton representation and taking derivatives up to second order,

 

the following local densities that can be formed from pq(7"', 7"") and §q(7"', F’)

pq(7‘) = ZwJ 1(1(7'9) p.J-. (976)
76(7) = ZVwJ-(Fq) -V<p.-(Fq)pZ-’J-. (9.77)
ji
86.40"”) = Zeflf'quifqufJ-a (9-78)
jam = —%Z(¢}(FQ)V6 94(Fq)—Vu<p}(Fq)¢1(Fq)) 422-, (9.79)
ji

«1.11.09 = -i Z Z 5,“... 9H5?» [Vuaa 904779)] 923- (9-80)
110 ji

me = 2 742394) (a. 7.499)] pig- (9.81)
879.90?) = -" ;Z 2 5111/0 (991(Fq)[vv 00 994772)]
— (v. 423671)] a. 42.64)) p23 (9.82)

Jq.,...(i~) = —5 Z(¢}<Fq) [4.7.9.66] — [7. am») a. 44174))72, (9 83)

F4140") = - Z< [V°5¢}(F9)]V1199.(F9) + [V1.¢}(Fq)] V 0491(Fq))p.-J

191

 

These are the matter density, kinetic density, spin density, current density, spin-orbit
density, the spin kinetic density, the cartesian spin-orbit tensor density and the tensor
kinetic density. These local densities, except Jj,’ are all real. It can easily be shown

that in the case of time-reversal invariant systems, Jq’ and JJ, are equal to each other.

9.2.4 Properties under time reversal

According to Eq.(9.69), the scalar pT(F,f”) and vector §T(7",7‘") parts of the time

reversed density matrix pT(-F0, F’a’) are [196]

pflfif’) = 93621:”) = 96073?) (9-85)

§J(F,'F’) = 43(71):") = —.§;,(7=",1=') (9.86)

Thus, under time reversal, the transformation of the local densities is

pz‘m = pm. {(7) =49). 63(7) -—— 4(7). 1% = —3’.(a (9.87)

J36) = 4(7). fife) = - 3(7), Fm = —F‘(a, (9.88)

where one can simply count the number of time-odd operators used to define the
density to obtain its transformation property under time-reversal. For time-reversal

invariant systems, one sets all the densities equal to their time-reversed counterparts

93627”) = 9457‘") = (19”.?) (9-89)
.6} (11%”) = 61,037"): ——8’q(i”,f’), (9.90)

and the time-odd densities vanish. I.e.

61(4) = 0. 12,9“):0,

fim = 0 F'.(r'~)=0. (9.91)

The rest pq(f') , Tq(f) , and .1710?) are time-even densities.

9.2.5 Extension to anomalous contractions

The normal density is insufficient for the explicit treatment of pairing correlations.

Thus in addition to the normal density, 6,1, = one), , one introduces pairing tensor,

I
referred to as the anomalous density, Fall, = ClICI. The generalized one-body density

matrix is defined as

(‘I’lfil‘m (‘PIRI‘N (MCI/61W) (‘I’ICHCII‘N

-<<I>|(%|<I>>* 1—<<I>lf>l<1>>* (914,619) (914,699)
(9.92)
where l’ andl are elements of the single particle configuration space and |<I>) is a
quasi—particle vacuum [38]. The operator p“ is hermitean (p‘l = p”) and R. is skew
T

anti symmetric k = —k. Here kT refers to the trans ose of it , not the time
P

reverse of Fe. Two important relations hold for )6 and F; . These are
fife = ftp“. (9.93)

Using the above two relations, it can be easily seen that the generalized matrix R is
idempotent i.e. R2 = R. In fact, R is also Hermitean. The pairing tensor can be

transformed into the pairing density matrix, )5. This is given in (Faq) space as

-'III

6 Foqn‘ aq E 26’ (I) 6.1-; (2,7,, (I) = 25’H(F0’(],'F’5"(1’). (9.94
7 aq’ (I

We still assume that the density matrix is diagonal in isospin subspace. In pairing

terms, this assumption means that there is no proton-neutron pairing. This implies

193

 

that the pairing density [3(Faq, F ’ 0’ q’ ) can be written in the form [1,, (Fa, F’a’) (5 /

qq'

The pairing density matrix can be resolved into its scalar/isoscalar, vector/isovector

parts in exactly the same way as the normal density matrix. Thus, appropriate

contractions in spin and isospin space yield

2 pq( (F0 "’ a’ )6 =222699.(7’ 69) 996(F69)77" 75..

00 ,q aq ij

M7"? F’) 1 (9.95)

q

2 6q(Fa,F’6’)6JJ/A(9) = Z A(9) Z 26 94(F’69)91(F69)K§.,
09 if
2 A(9)f)q("1F’) . (996)
)6,(Fa,r “’ a’) —2 2 26’ 991(F’6’9) 930690. a 63.7
2 6",(7', 7") , (9.97)

q
2 6,60, r'a’) (7.7. 4(4)

aa’q

2 Mg) 2 26' 4247071) 44904) 4.7..
ii

00’, q

2 s,(". F’) (9.98)

(1

Working in neutron/ proton representations, and going directly to the local densities,

the following is the list of the local pairing densities that can be formed by taking

derivatives up to second order

= 2 2 26 9.- (F69) 99j(FUQ) 773%. (9-99)
2 2 :2: 2a ,(mq) VqJJ-(Faq) R3,, (9.100)
= 2 2:: 26’910"9)6<l611|6)w(799)= 0, (9.101)

194

39470:.) :

T947“) :

J9~u(7?) :

J9.uv(fl =

F9470?) :

2 26 (997(F69)V9 '99)(F69) - VJJ999(F69)¢)(F69))H§.JJ (9-102)

oij

Z Z 26’ 6 9.6m) - (alone) 6 ¢J(F69) 79-3., (9.103)
ij

0’0"

i _, —-o[—I I "‘
_§ :2 :20 577W (ca-(7' a q)VJ,(0I0(J|0>99j(T0(1)

ua 0.0! ij

4.996279) <a'loala) 992767107939. (9.104)
__;'_ Z Z 26' (9.66%)) v). <6’I69l6) 9.6m)

— VJ. 9.6671) (diam) 9.6071) ) , (9.105)
$2 ._ 257(9. (a'lalo>9j(mq)v99i(7'aq)

+ V), 991-(Fc’7'q)6- (a’lé’la) ap,(7"aq)) 7117-. (9.106)

Jl

One notes that most of these anomalous local densities are not used in current

empirical parameterizations of nuclear EDFs (see section 4.2.2). In fact, only the

local pairing density, fig, is used. As discussed in section 5.3.8, the application of

the DME to the anomalous part of the OBDM results in these local densities. Thus,

these densities may be useful in future non-empirical construction of the particle-

particle/ pairing part of nuclear EDFS.

9.2.6 Relations among the densities

One can establish a number of relations among the various local and nonlocal densities

defined in the previous section. We start with those relations which are important in

the derivation of the DME of the scalar/ vector components of the normal part of the

OBDM. These are

[(V2 + V0)pq(7", F’)]

= V2pq(F) — 2Tq(F) (9.107)

 

 

 

 

 

 

 

 

sz’ 2
Wm? r') = $172M)— W) -19 m
H (9.109)
2 $6M?) +134?) (9.110)
[(v2 + v”) Mm F) 4 = stmh“) — 2T,,,.,(F) (9.111)
[Vt-91,407,?) F—F’ : [Vim/AF fl] *4
= évpsu,q(1r)+w,w,q(r). (9.112)

Most of these relations were initially given in Ref. [197]. Here, we extended the
list by deriving additional relations which are found be useful in the derivation of
the generalized PSA-DME, discussed in section 9.5.3. We illustrate the derivation of
these relations by taking Eq. (9.107) as an example. Starting with the left hand side
of Eq. (9.107)

[(v2 +v'2)p.,(r,f~)]

9

 

 

~=r [<V2 + V”) Zepflfll) 991(7‘9) 1023]

4 .. ~
J1

r=r

= 2 p23 12,-(ravage + «pg-(fa) 1729391)],
jz' L

= Z 1033'
ji

 

= VIM?) — 29,0"), (9.113)

where we used the definitions of pq(f’) and Tq(f") given in Eqs. (9.76) and (9.77). The

same set of relations holds for the anomalous densities as differentiation is the only

196

operation that is applied in the derivations. Hence, we avoid repeating these relations
for the anomalous densities. However, under general circumstances discussed in [81],

the nonlocal anomalous density, fiq(~f', F’) satisfies

[MO-:1 FI) : 159(F’sflv (9114)

~

which results in LU?) = 0. For the nonlocal anomalous spin density, we have

= —§'q(i~",r*). (9.115)

9.3 Local Gauge transformation of the OBDM and

local densities

Neglecting relativistic effects, the nuclear EDF must be invariant under Galilean
transformation, while the usual justification for the requirement of a locally gauge
invariant EDF is the fact that the underlying nuclear interaction is expected to be
gauge invariant. This seems to hold only for local interactions. Still, it is shown
in Eq.(9.130) that Galilean transformation is a special kind of local gauge transfor-
mation. In addition, the locally gauge-invariant bilinear combination of densities,
which is what we are interested in, do not depend on whether one invokes Galilean
invariance or local gauge invariance. Thus for the sake of generality, we discuss the
local gauge transformation of the normal and anomalous parts of the OBDM and
densities. The ultimate objective is to obtain the bilinear combination of densities

that are invariant with respect to local gauge transformation.

197

9.3.1 Local Gauge transformation in many-body physics

In many-body physics, one can formulate local gauge transformation in two equivalent
ways (i) local gauge transformation of the many-body wave-function which is an intu-
itive generalization of the local gauge transformation of single-particle wave-functions
especially in the case of Hartree-Fock approximation (ii) local gauge transformation
in second quantization formalism which is very general and can be applied in any

many-body approximation.

Conventional formulation

In the usual formulation, local gauge transformation is applied to the N -body Hartree-

Fock wave-function |<I>) in space, spin and isospin coordinate space

A
<I>(:1:1,172,...,1t,,,t) = exp{i§:¢(;rj)} <I>(:c1,;r2,...,:r,,,t), (9.116)
j=1

where x,- = (fl, 0,, q.) and (9(13) 2 45(1‘}, t) is an arbitrary, differentiable real function
of the position 7" and time t. In general 975(13) are independent of spin and isospin

coordinates, and in the static picture, they do not depend on t.

Second quantization formulation

For generalization of local gauge transformation to other many-body approximations
where there is no explicit conservation of particle number, one has to formulate local
gauge transformation in second quantization. Since the state vectors in second quanti-
zation are elements of abstract Hilbert space or Fock space, local gauge transformation
must be performed by an abstract unitary operator [198]. When the transformed-
state vector is projected in to the N-particle subspace, the transformed wave-function
in the subspace is equal to the projection of the original-state vector times the ap-

propriate space and time dependent phase factors. In second quantization the state

198

 

vector |<I>), which is not necessarily an eigenstate of particle number, is represented

in Fock space by the column vector
|<I>) 2 col {|<I>0), |<I>1), |<I>"), ...}, (9.117)

where |<I>i) refers to i-particle component of the state-vector. A local single-particle
operator 95, which is assumed to be diagonal in isospin space, can be represented in

second quantization as an operator in Fock space

2 2: / dfl'FalqlcblFU-qua*(F019)a(Fa-2q). (9.118)

”1‘72 ‘1
Defining the unitary operator U as
U = exp 2': Z / dr<Faqu¢>IFazq>a*(ralq)a(razq) . (9119)
“1‘72 ‘1

the local gauge transformation of the state-vector is given by [199]
|<I>)' = U|<I>). (9.120)

One can easily verify that this gives back the previous formulation when applied to an
N -particle Hartree-Fock wave-function. Once the unitary operator U is defined, local
gauge transformation can be carried out by transforming the creation and annihilation

operators

(1'(Faq) :2 Ua(f"aq)Ul = (”WU") (1(F'0q)

al,(f'0q) = Ua*(r~'aq)U* = eWWaMFOq). (9.121)

199

9.3.2 Local Gauge transformation of normal densities

The local gauge transformation of the normal part of the OBDM is

p; (17101, F202) 2 exp {i(¢9(f‘]) — 99030} pq('F101,1-"202). (9.122)

When the various local densities involved in the EDF are calculated from the 10-

cally gauge-transformed density matrix Eq.(9.122), one obtains the following relations

[200]
10W) = (90’) (9-123)
999 = rq<9+2L<9-vo<9+p.(9(v¢(9)2, (9124)
3529(6) 2 62,907), (9.125)
1.3.99 = j...99+p.<9v..¢(9. (9.126)
T4,.99 = T....99 + 2 Z J......99v..999 + 3,..99 (W99). (9.127)
J5....99 = J,,,,,.(.-)+s,:,(.=)v,,(.-), (9.128)

Fé,u(f) 21701474) +VV¢ (F) Evil-9% 7:) 991a (7") +Zvu¢( flJunF)

+ v.,<b( 7'") Z J, ,.,,( (9129)

Horn the previous relationships, the only bilinear combinations of local densities

which are invariant under local gauge transformation are

9) [2.99949 — 31.99 5,199

99 p.996 . Q99 +5.99 - 9‘7 x 5,99,
(iii) v9.99 J99“) — s.99 v x ,- ,99

(iv v9.99 £999 + .999 x 5,199

9) sq99 412/99 — 2., J...,..99 J,x,..,.99

200

(Vi) Z}; J0.llll(f) Zn Jq’,uu(f) + :14; ”AIM/10?) 'Jqlwi/(f') _ 2§Q(,—‘) ' 132/(F)

Galilean invariance is a special case of the local gauge invariance for which the

phase in Eq. (9.116) is given by

999 = $ (9.130)

where 13' is the linear momentum of the boost transformation [200]. Transformation

properties of 7,0") and 3,0") allow one to interpret {795(7") as a velocity field,

6'99 — 5 6999 (9.131)

m

which shows that the flow of matter obtained through the local gauge transformation

is irrotational, 6 x 17 = 0 [200}.

9.3.3 Local Gauge transformation of anomalous densities

The local gauge transformation of the anomalous part of the OBDM reads

fiq’(F101, F202) 2 exp {2(¢(F1) + $03)) } fiq(F101,F202). (9.132)

The local gauge transformation of the various local anomalous densities are calcu-
lated from the locally gauge-transformed anomalous part of the OBDM as given in

Eq.(9.132). Thus one obtains

r3399 = 6209999 (9.133)

73407) = 626“?) [74?) ‘1' 662(7) ' (61/347112 F2) + 6”3‘10?“ 71.2))

 

+ p999 (69(9)?) (9134)

201

S" (1") = 6'20“?) 8 qk(F)— — 0, (9.135)

 

q.1
17,9") = 6’20‘217‘71 (9.136)
fit?) = 61291;)[73] (79+V9m (filamfiz)+62.§9,(F1.F2))
132:.
+§,99(%(79)2], (9.137)
7111(6) = emquklffl- (9.138)

The above relationships yield the following locally gauge invariant bilinear combina-

tion of densities in the pp-channel

9) 93.99)?

(ii) (94.7))?

(iii) fiq(f')(Af);(7“) - 4%.;(79)

(iv) .99) - (A999 — 421399)

v) |32<9|2

(vi) Zuni“ q ,4, jq up , noting that complex conjugates of (iii) and (iv) are also gauge
invariant combinations. One can verify that starting from a local / semi-local interac-
tion, such as the standard Skyrme given in Eq. (4.12), only the bilinear combination

of normal and anomalous densities identified in this section occur in the HFB energy.

9.4 Densities in spherical systems

Under spherical symmetry, which also implies time-reversal invariance, the various
densities defined in the previous section take simplified forms. In spherical symme—

try, the HFB quasiparticle wave functions in the traditional representation take the

202

form [81]

('N m
U;’;zj(111~1(,-.«q) _—. —'———U"J I) Z 11(r) (Inu— é-O'ljnl)|0') (9.139)
"1’0
ml ([k] “lg-(711),”!
km“ (1(1) = —:Y( (f‘)20(lml— ;0|jm)|0). (9.140)
"110

9.4.1 Expression for the normal densities in spherical sym-

metry

Using the specific form of the quasiparticle wavefunctions given in Eqs.(9.139) and

(9.140), the full density matrix reads

[1,997, F’a’) = Z VJIW‘V (r09) )W’JQIWWU’Q)

nljm

V 7
= Z K—L—jv )2; Y ml ’)’20 (lml'éé ’|jm)-L:(—)-

nljm

m l
x ZY( 1( r)(2a (lmfialjm), (9.141)

71110

where we have made use of the fact that the radial parts of the quasiparticle wave
functions and Clebsch-Gordon coefficients are real. In the following sections, the most

simplified forms of the various normal densities are given (for spherical symmetry).

Scalar part of the density matrix - matter density

The simplification of the nonlocal matter density proceeds by making use of the

relations stated in appendix Eq.(9.15) and Eq.(9.43). Thus

 

2' 1V: V:
[MU-'27:”) : Z ]+ Ij(r I) Ij(r)191(’l:'f’). (9142)

47r 7" r

203

 

where P) is Legendre polynomial of order I. The above expression is manifestly
symmetric in F, 1"", viz. pq(7"',1'”) = p,,(f",1"'). For the local part, one simply sets

F" = F in Eq.(9.142) to obtain

 

qufl_ :3 2j4+1[ Vylljffllj . (9.143)

nlj

To derive the local gradient and laplacian densities, one simply operates on Eq.

(9.143) using the respective operators. Hence,

 

 

 

 

69.99 = Z 2j4:1§-,.-[ 14.39)] 1‘ (9.144)
nlj
AMT”) : Z 214:1 (86:2 + 3:) [an;(1")]2 (9.145)

nlj

Kinetic density

For the kinetic density, one can proceed in two different ways: (i) Operating on the
simplified form of the nonlocal matter density as given by Eq.(9.142) (ii) Operating on
the quasiparticle wave functions (lower component) as given in (9.140) and employing
angular momentum algebra. The first path is followed for the sake of simplicity.

Starting with the nonlocal kinetic density,

 

 

 

 

 

 

 

 

 

 

 

r’ r

 

 

 

1' . '2 , ~
V99 1 9) 19'9" , ,) Z 6111’ — 0112'? 77 1.,

(9.146)

where P,’ and P,” denote first and second derivatives of the Legendre polynomials.

Using this same notation and simplifying the above expression one obtains Even

though the above expression looks complicated, it is handy to use when one wants

to extract the local value. With the nonlocal kinetic density as given in Eq. (9.146)

and setting 1‘" = F, one can simplify the expression much further. Making use of the

relations stated in Eq.(9.5) and Eq.(9.31), one obtains

 

 

1,9) = Z 217;: [(143139) — V""( ——))+ I“ :11)[V1§’11(r)12 -

. 7'
711]

The vector part of the density matrix - Spin density

The nonlocal spin density is given by

 

§,,(F',F') = Z’pq(Fa,F’a')(0'|5|a)
T
: Z anlj(r’) 1311' () 1;, Y m’tl )(lml,%0_ OJIjm> Zyml(
nljm mla

1
x (ln1,§0|jn1)466' (6'|6|6) .

In the orthonormal coordinate system defined according to

60:521 Gil ::F—(é..ri“7y)

(9.147)

A)

1')

(9.148)

(9.149)

 

 

and using

I 1
<5’l011l5l = «3501145091 (9-150)

one can write the nonlocal spin density as

 

71)an7‘ m’: m
M = $2 ,(1 3;” Z 1119')Y1’9=)(21+1)
"(1m mlaml’a’
X (_1)2l+2m+130-2 2 J 2 J
m, a ——m m, 0’ -m
l 1 1
x 2 2 . (9.151)
6 11 0’

Making use of the 3j—6j symbols relation stated in appendix 9.1.2, one obtains

 

11. .1 . l/nq.'( 7") n ' m, 1
W() = 193me ’;, ’3 Z 11;"! Y1""1~()(—1)“t+
nlj m m,
l l
l l 1 l l 1 )(
x , . (9.152)
é % j m, m,’ 11)

One can plug in the algebraic values for the 3j and 6j symbols in the above equation
and do further simplification. To proceed further than that, it is imperative to choose
a coordinate system in which one of the vectors (7‘,r ’) is along the z—axis. Let 7"

ml, :0

be the one along the z-axis. This implies that 6’ = 0 and (9’ = 0. Thus, only Ylm

contributes. Making use of the relations in Eq.(9.20)-Eq.(9.23) and simplifying,

 

 

 

Sq,0(77,7?l) = 0
....1 . 111 . V1311” V1310") -1, 1.. 1 1
s,,19,r) =—— 223—1)] (21+1) 1' 1 Y, (1)1», (6:019:01
nlj
><3/4+l(l+1)—j(j+1)
fil(l+ 1)(21+ 1)

206

V" (1') V" (r)

 

 

 

81.102?) = iZ(—1)21(21+1) "Ii, "'3 Y‘(7‘)Y°*(6’ =0,¢’=0)
nlj
X3/4+l(l+1)— 1(1+1).
([21(1+1)(21+1)

One obtains the prefactor 7'. after properly summing the exponents of —1 which takes
the form (—1)“ where a 2 2J' + 3/2 or a : 2J' + 1/2. Writing the components in

1‘, y, z coordinate space, one obtains

sq,z(7"',7'"") = 0 (9.153)

saw") = [Z (7")"‘J()P1<cos<)>11°<0'=o,1'=0>

nlj

xcos(¢)(3/4+l(l+1—) j(j+1))

 

1
l\/(1+ 1)(21+ 1)

T, (1
31.1121") =—\/4:D21)"3”)31(cos(1)mo(1'=o,1:0)

nlj

xsin(cb )(3/4+l(l+ 1)—J (J+1))

 

(9.154)

 

 

 

(9.155)

 

1
1\/(1+ 1)(21+ 1) °

The above result can be used to show that the nonlocal spin density is in the direction
of 7" x 7” . This can also be shown to be true from a different perspective : using the
properties of nonlocal spin density under time-reversal and symmetry arguments. In
spherical systems, the general form of the nonlocal spin density can be constrained
as follows. There are only three vectors available for the construction of any vector

physical quantity. These are 7", 7“" and 7" x 7"". Thus
§(7",7"’) = sa(r,7",0)7" + sb(7‘,7",0)7""’ + sc(r,7",0)7"x 7’", (9.156)
where the sa(r, 7". 0,) 91,3(7 7" ,0) and 36(7‘, 7" ,0) are scalar functions dependent 011 the

magnitudes of" 7‘, r’and 0 which IS the angle between 7‘ "and" 7" In spherical systems,

207

 

the nonlocal spin density should satisfy
.,§',,(F 7'") = .§'q(—f', «~F’). (9.157)

This condition can be satisfied only by setting all except sc(r, 7") to zero. Thus the
nonlocal spin density is proportional to F X 1"". To further constrain the form, let us
invoke the property of nonlocal spin density under time-reversal invariance. Under

time reversal

#015") = —§'*(r,r') = —§q(r',F), (9.158)

from which one recovers that §q(f') = 0. Using the above property, one can easily

show that the nonlocal spin density has only an imaginary component. Thus
(1(7‘37’”) = if’x F’sq(r,r’,6). (9.159)

This result has been verified by the derivation in Eq.(9.153)

: _ Vigil? anj; (7‘) 0 1311(C036)
sq(r,r, 6) — —(/4—17}- 2 Y (O O)—sin0

23+1)(3/4+l(l+1)j(j+1))
l(l+1)\/2l+1'

 

 

(9.160)

In ref. [170] a similar expression is given for the nonlocal spin density with sq(r, r’, 6)

being
1 K? 7" V:. r
sq(r,r',0) = :t— 2 if]: ) if: )P,'(cos6), (9.161)

21r ,
n1]

 

where :1: is for j = l :t 1 / 2, P,’ is the derivative of Legendre polynomial P, and 6' is
the angle between 7" and 7”. One can show that Eq. (9.161) reduces to Eq. (9.160)

by using the relations Eq.(9.46) given in the appendix. Obviously, in time—reversal

208

 

invariant systems, the local part of the spin density is zero. I.e.

3:,(F) = 0. (9.162)

Spin-orbit density

Starting with the definition of the local spin-current tensor

 

Jthi/(f‘) : “% (V); ““ VL) Sq,V(F,F,) _, _,,, (9.163)
and making use of Eq.(9.159), we can write the local spin-orbit tensor as
.. Ta
Jq.pu( r) = — rsq(r) Z 6W0 7. (9.164)

0

One can write JQWU‘) as a sum of pseudoscalar, (antisymmetric) vector and (sym-

metric) traceless pseudotensor parts

_, 1 9 1 z 9 1 ,
Jq.w( 7.) : 35W JéO)(r) + E Zéwk Jé‘lgh‘) + JéizuU‘), (9.160)

k=1:

where the three components read

ngm = EdeqWW) (9.166)
pu
153m) = Zewkttm (9.167)

pu
1

J96) = Jam— 3

_. 1 . _.
6,”, JéO)(r) — 5 26“” J5,1),)(r). (9.168)
k=:r.

Combining the above results, it can easily be shown that both the pseudoscalar and
the pseudotensor parts are zero i.e. J (0)07) = 0 and J(2)(f') = 0. Thus one needs

to simplify only the vector part. Even though one can perform a series of angular

209

 

 

 

momentum coupling operations to obtain the most simplified form for the vector part
of the local spin-current tensor in spherical systems, a simple physically motivated
derivation is given in Ref. [201]. In spherical systems j;( F) must be proportional to

F. Thus

K'1
\_/
‘3 l‘31

= ([15), (9.169)

resulting in

’5 ' "'_-_. J—fi
IL-

 

L(F)= r 2(2j+1)[j(j+1)—l(l+1):](V,,,,())2. (9.170)

9.4.2 Expression for the anomalous densities in spherical sym-

metry

For the anomalous part, starting with the anomalous density matrix as defined in the

traditional representation [81]

       

pq (7'0” Fla!) = _ Z 26Unqu[k](T—r 0)anjq[k]* («I6 I)
nljm
U
2 )(lm,’% a’ljm) 26’—1”—(-—)
nljm T
x Zle'O" [)(ml— éaljm) 26. (9.171)
mla

For the following anomalous densities, we follow exactly the same mathematical steps
as their normal counterparts. Thus only the results are stated. The two changes are
(i) overall sign becomes opposite to that of the normal densities (ii) one of the lower

part of the quasiparticle wavefunction is replaced with its upper part. Thus ringing

210

 

 

the change in the respective densities, one obtains the following results.

pairing density

The nonlocal part of the pairing density reads

 

Vq
fig“? _ _ :4 23.71. +1 nlj‘(,r 70’) Unlj(r)P P('f° 'f’) , (9172)

7'
nlj

while for the local part, one simply sets 7‘" = “F in Eq.(9.172) to obtain

‘3‘](7?) : —Z 2.74:1 anj( 12:1] ”(7) . (9.173)

 

The local pairing gradient and laplacian densities are given by

 

 

 

 

_ 2.7 + 1 8 anzgl )UZUU') .
_. —Z M 0r r2 7‘ (9.174)
111]
~ 2j+1 02 2 .. "(r)U" ()
qum = _Z 4” (W + ;)V ’9 r2“? . (9.175)

1in
Pairing kinetic density

The local pairing kinetic density reads

= 222;: [<99 - ——> (___)

ll(+1)

 

 

V9.1 )Uz..( >] (9.176)

Pairing spin-orbit density

The pairing spin-orbit density is given by

fir):—

 

3
7220141)j([(j+1)-l(l+1)-4 14:1,()U:'.,,<-> (9.177)
nlj

211

9.5 Details on the density matrix expansion

In this part of the appendix, we derive and discuss the generalized PSA-DME. First,
we start with brief discussion of the Husimi distribution and the derivation of the
quadrupolar deformation, P30"). This is followed by few remarks and derivations
related to the Wigner transform of the pq(F1, F2) up to 52. Subsequently, we derive
the generalized PSA-DME, from which we recover all the special cases such as the
PSA-DME discussed in section 5.3, the original DME of Ref. [170] and its subsequent
generalizations [202]. Finally, we give the formal modified-Taylor series expansion,

discussed in section 5.3.6, of all the local densities.

9.5.1 Husimi distribution and the local anisotropy P51 (7")

The Husimi distribution is one of the many quantum phase-space distribution func-
tions. It possesses the key property of positive definiteness [[203],[169]] and is defined

as

g .. %5'(F*71)—#2(F‘1) ..
11.6.17) N jam-(rune 0 dn pa. (9.178)

 

where N E 1/(1r3/‘4rg/2) and To is a chosen parameter. In the following, we use the
HF‘ single-particle wave functions. The occupation probability of a given spherical
shell pqnj’ is one or zero, except for open-shell semi-magic nuclei where the so-called
filling approximation provides the valence shell with a partial occupation. Modifying
the derivation to include pairing (HFB) can be done through the proper formulation
of the Wigner distribution as given in Ref. [204], as the Husimi distribution is a

COarse-graining of the Wigner distribution using gaussian phase space factors[169].

To derive Eq. (5.15) for the quadrupolar local anisotropy of the momentum Fermi

212

surface P2”('F') we start from the definition

(13' 36;.- 2— *2 H, '1,
P.2"(F) —:- f ’ [ (11.721161?) (T ’37, (9.179)

 

and make use of the relations

2'. -1 s , ,_. _. J ...
[(1131) e WWI—"1’: (27r)“h"V{-V0(F1’— v1), (9.180)
_ 1 F _,...12
312(11)~6r—*' k" ‘2
6 ~ (7‘1 7'1) + O(( Fro) ). (9.181)

Through direct application of the above relations, one obtains

[@152 1147113) x (279355 ZIF¢1(Fq)|ZP§-’i + 0((k%7'0)2)1

/ (190.13)?qu w (270312"Zlcvwrrqn2pz.+0066?)

which, plugged into Eq.(9.179), gives

 

PM = [ 37321617) lep..-1]+O((k%ro)2)-

Further simplifications can be performed for spherical systems, using single-particle
wave-functions essentially the same as the ones given in Eq. (9.140). However, note
that we are working in the HF picture. Using the angular momentum relations in

section 9.1.2, one obtains

 

 

 

 

213

 

Ill ‘2
+ ZFUJ )(L—nlj2(r)) pqnfl,

7in

where F (l, j) is given in Eq. (9.1.3) and pm”l is the occupation probability of the
(71,1, j) shell with q labeling the protons/ neutrons. Plugging these intermediate results

into Eq. (9.182) yields the expression of P2(F) as

P2(r : 7,15)" 227r +1[(g_3V.?_z_-_;( ))2_z(z2:21)(v,3;( ))] pq,,,-,(9.182)

where Tq(F) is the kinetic density as defined in Eq. (9.77).

 

 

 

9.5.2 Wigner transform of the pq(F1, F2) up to it?

In the parameterized PSA-DME of 5,,(F1, F2) , we used the Wigner transform of pq(-F1, F2)
up to h2 to motivate the form of Eq. (5.41). In this section, we derive Eq. (5.40).
We restrict the derivation to the HF approximation. Refer to [204] for a recent work
on the formalism of the Wigner distribution in systems with pairing (HFB). One can

write the scalar part of the normal part of the OBDM as

0377,?)
p( W2 :99. T2q7‘1(I)9()\q—€i)=2LE_1.,\q [_(Ta—l (9.183)

where G is the unit step function, Aq is the fermi energy, 6,- is the single-particle energy,
,3 = it and L“1 refers to the inverse Laplace tranform. C5(F1, F2) is the single-particle

propagator which, in the HF approximation, reads

C”(F F =2 99,- (F)2q to, (qu)e '6‘?" 9(/\q — 6,). (9.184)

214

 

 

 

 

Since the Wigner transform of the right and left hand sides should be equal, one has

0? 1‘2',
M]. (9.185)

[Adj-Eel?) : QLELAQ [ d

The derivation of the Wigner transform of the density matrix up to h2 can be ob-
tained by working out the inverse Laplace transform up to h2 of the single particle

propagator. It reads [38]

 

, )2 .. ~ 2 2 —. _. ..
Cf(R,p‘) : e—li(%I—H+W(R))(1+251(—AVQ(R)+§3(VV(I(R))2

13

3—m

—p(- 6) mm) + 0014)). (9.186)

2 .
Defining the single-particle Hamiltonian hq = “ail—MA + V, where Vq IS the self-

consistent HF potential, one can use the following relations

7+ioo [3(N hq )
, . — _ q
/ did 6 ’ ll — 6(/\q — hw),
'7

 

 

563T]: ""38d(i""'(‘1'"” = 6'eq— hit),
for integer n to obtain
Lfiixq Be; - ugiwquaf ___ 00F 22:7_Vq(§)),
Lgiiq [@752 '3 12“?!” mm); = gmu?)
“(Ag—gt gun),
Lag [gfi‘iw (vi/Amy] = ‘ giggmmf
x 6”()\q — 21); (Km),

215

 

_ ml .1“ 2+1"(fi)) _. ~ 2 ~ h? 1 ~ 2 , ~
133—”VI 247726 £7 (1 (p-V) mun] : -24m(p-V) ”(12)
5" A ”2 V I?
X (q—‘Qg- q( ))
(9.187)

Plugging into Eq.(9.185), we obtain the Wigner transform of pq(F1, F2) up to h?

a if I
pWK.q(Rvm :e(’\q - hilt”) — g—AVQ‘S (“\q — hft')
m (9.188)
)1? 1
+ 2—477— [(VVq)2 + 505‘- V)21/,,]6”(Aq — hip) + 00'1“),

where hfv 2 HQ = — V q(R) and the derivatives of the dirac-delta functions

m
are performed with respect to Aq. Even though it is not the main target of this
section, one can calculate the inverse Wigner transform of Eq.(9.188) to obtain the
density matrix up to order 152. This effectively gives the extended Thomas Fermi
approximation to pq(f"1,f'2). We can call EFT-DME. The important relations one has
to use are

m a :Bkz. 6

fz—ZITF 5(1: —- kp) and —— —— (9.189)

- _h, =
W" W) 0A 8Aq0k}’

2 (12
his
where Ag = 7,111:- + V q(R). Since the derivatives act on the dirac-delta functions,

one has to perform integration by parts using the relation

n

n n a
[(1136 (A!) —h(fv) F(paT:/\q) : (— 1) 87" F(p,T A (”'11:va (9190)

where F (p, 7‘, Aq) is any well behaved function and pp is the value of 1) that satisfies

the equation Aq — hfv = 0. Applying these mathematical relations, one essentially

216

 

 

obtains EFT-DME of p,,(7"'1,1"72)

_, _, Ath33j1(kZ.r) 1

 

 

 

pq(7'la 7'2) = 37f? k? T‘ + 127r2 A”? [Joflifl _ ki'l‘jfikfild]
F
1 Vk" 2 . . . .
24”; HF) [30(kgm) — 4kr}.r]1(k‘,’,r) + kg‘2r2]2(kz.r)]
F
1 1 " kqfikq F F k0 - k0. qu 2° k9 R4
___247T2EV F F-; ; —3 Fr]1( F7)+ ,1.~7‘J2( FT) +O( )

(9.191)

In Ref. [173], the authors make angle-averaging (with respect to the orientation of F)

approximations, followed by the expansion of the Fermi momentum up to ’12

1/3 -1/3 2
1 (W) Ap
kq=(37r2 ) +—(37r2 ) [——" -2—" , 9.192
p pq 72 pa pg ,0, ( )

and K6193)? 2 kgAkg to obtain

 

- ~ _ ~ 3310924") T2 . ‘q 3.1047")
pETI‘.q(Ra T) _ pQ(R) kg”, + '72qu 30(kFr) 6 kg”,
7'2 (qu)2f - q j1(k%r)
_ ET [4J0(LFT) — 9w] . (9.193)

9.5.3 Generalized PSA-DME

In section 5.3, we discussed PSA-DME of the scalar and vector components of the
normal part of the OBDM of time-reversal invariant systems. In that derivation, the
chosen set of DME—coordinates (R, 7") were integral parts of the derivation. In addi-
tion, we had a single nonlocality coordinate, namely, 1". Here, we give a generalized
formulation of PSA-DME for time-reversal invariant systems where we relax these
two constraints. This allows us to recover the DME of the nonlocal densities that
occur in the HF energy of both NN and NNN interactions. It is to be noted that

the three nonlocal densities that occur in the expression for the HF energy from the

217

chiral EFT NN N interaction at N2LO are of the form 9,071, F1 + 1’2), gq(F1,F1 + 1‘73)
and <q(-f‘] + f2, 1“} + :33), while for the HF energy from the two-nucleon interaction,
the nonlocal densities are of the form <q(7"1,172). In this notation, the N NN case uses
the coordinates defined in Eq. (7.22). Furthermore, (q can be either pq or 5;. As can
be seen from the coordinate dependence, the nonlocal densities that occur in the N N
case are particular cases of the ones that occur for the NNN. In fact, all the nonlocal
coordinate dependencies can be generated from the general case §(F1 + :32, F1 + :33).
For instance, the usual DME-coordinates (R, 7") used in the NN case is recovered for
$2 = —:i?3 = F/ 2 with F1 playing the role of the center of mass coordinates. Hence,
we give the PSA-DME for pq(7"'1 + (132,?1 + :32) and 82’0"] + 132,71 + :32), which we call

generalized PSA-DME, and generate more specific cases from it.

9.5.4 Generalized PSA-DME for the scalar part of the OBDM

In order to obtain the generalized PSA-DME for pq(7'“'1 + 52, F1 + 52), we essentially
follow the same set of steps as we did in section 5.3.3. One starts by extracting the
exponential nonlocality operator before introducing a phase factor and performing a
Taylor series expansion up to second order

615-(52—53) 6:82.(62—i§)+53.(v3+ii5)

pq(Fl + £27 Fl + £3) : pq(F29 F3)

 

F2=F3=Fl
(9.194)

22
m‘)
{'51
[31
l
0;:
r——fi
p—a
+
5.3
N
A
<1
:0
|
S
g»-
v
+
H
r...
A
<1
a
+
Q
>1
v

 

F2=F3=Fl
(9.195)

Note that for the PSA-DME developed in section 5.3.3, the next approximations

involve angle averaging with respect to the orientation of the relative coordinate and

218

averaging with respect to I: over a Fermi sphere. For details of the logical arguments
in favor of performing these approximations, refer to that section. Here also, we apply
exactly the same approximations. First, We define a new coordinate system as given

in Eq. (9.1)

H1

= 52 _ 53, (9.196)

X’ = (1—a);f7'2+afg, (9.197)

where ac [0, 1]. The essence of this parameter will be clear later in this section.
Angle averaging with respect to the orientation of the relative coordinate, 12', en—
tails performing },Lfl f (195 where ()5 denotes the orientation of 53'. Let us apply these

approximations to Eq. (9.195) term by term.

0 The Leading term gets simplified as

 

 

3 -' de i111 ... .. "'
—fi / dk / —e* (p.17-29,) = H8(k%x)pq(r1),(9-198)
4W3kp )1.)ng 47‘ F2=F3=F1
where
' kqfll‘) j1(kq|52-5‘73|)
11” «0.. = M I“ r = F * . .1
0(kFT) 3 16%;]: p(7‘1) 3 [£24235 _f3l (7'1) (9 99)

o The linear (first-order correction) term has two origins. The first one is from

—ik - (532 — £3) and the second is from :32 - V2 + 553 - V3. Hence

 

_W /“ q (“C / 4 6* ["k ' $]pq(r-2,»r3) = L1(k2~$)pq(7‘1),

(9.200)

219

 

 

 

where

 

 

j1(k;’ L‘) , _. .. J1(kflf2 — f3”
L Icq.." : — Irq : _ ' kq‘ —‘/"' 9
1( P 1) 3J0(1 1‘)+ 9J 112.1. 3J0( 1"“ 13') + 1:24:32 -J'33|
(9.201)

For the simplification of the second linear term, first note that according to
Eqs. (9.196)-(9.197), the operator in this term, viz, f2 . V72 + 53 - 63 simplifies
to f- (a 62 — (1 — (1)63) + X - (F2 + 63). Let us designate

V7,, E a9. — (1 — a) 63. (9.202)

Thus,

ffizgfr A'I/erP iIx‘-1‘(f2 oV2+l3' V3) 2 /dQI: /de€iE'f[i-I°€a

+x.(v.+03)]

 

= j0(k.z:))? - ($72 + 63),

which implies,

3 / ~/d95 15,—. (a .. .. )
, dk —e" z-V +f-V Ff-
47r3k3r3 Wish-,3; 47f 2 2 3 3 Pq( 2 .3)

 

 

1"‘2="3=”1

with

 

.' q, - .9 ~ __ -1
L2(I.:‘;.a.~) = 3’10““) — 391””“1‘2 “D. (9.204)

kid: _ lama—m

o The second-order correction term is generated by the operator

220

.1
fl.

 

[ii-2 - (62 —— H?) + it}, - (Vi, + 2'13] 2 / 2 , where we are not showing the phase factor
explicitly. Using Eqs. (9.196)-(9.197), this operator reduces to
[i’ (—2'I:f+ 6,1) + X - (62 + 63)]2/2. The relations

 

/% 5%] ‘19? eif'f’w-Exrfia) 2: 0, (9.205)

19,; d9: ~ _. -» ~
/(__47T W—f (3“ 1(j‘. ° Ix’)(X ' (V2 + V3» 2 0, (9.206)
and Eq. (9.5) can be used to simplify the second-order correction as

3 ‘* dad? 'T H. .., F T
= _——./ q dk/— "ml/2W1 zk+Va.)+X'(V72+‘73)l2
“<1.

 

 

 

 

 

My}: , _.F 47r
X Pq(F2= 11.3) 7
F2=F3=F1
3 ' qur _. _. _, .. .. _. ..
= 2J1I5q; )[X - V1]2pq(r1) + L1(k%13) -V1pq(r1) + L:3(k‘}.ar) pq(’l‘1)
F
1:2 ‘ kqa: _, _,
+419.” )Aapqem) . (9.207)
2 kpx ? _.. _2
12—1‘3—1‘1
where
L kq. _ 3(kg‘x)2 ' kQ. ' ..q 2 8
A we) — 62,... m Fr) — 620w). (9. 0 )
A“ 5 Va. (9.209)

The last term involving Aa requires further simplification. Expanding the oper-

ator and using relation Eqs. (9.108)- (9.109) and the definition of kinetic density

221

 

 

 

given in Eq. (9.77)

Aap0(f:2= F3)

 

®=fi=fi

2
: (av-2 — (1 — (I)V3> /)q(f:23 F5)

(1:2

2
(1 - (1)2
2
— 2a(1— a)Tq(7"'1)
20.2 - 2a + 1

 

 

 

172:173 :51

: — (qu(7_"1) —‘ 27(1(F1) +126 ' 511071))

’

: qu(F1) _ Tq(Fl) 1

2

(qum) — 2mm) — 2'26 5.971))

(9.210)

where we used the fact that 311071) 2 O for time-reversal invariant systems.

Collecting all the contributions, the complete generalized PSA-DME for pq(f'1 +

:32, F1 + 553) in time-reversal invariant systems takes the form

with

Pq('F1 + 53:2, F1 + :33)

118 (A321) 2
Hakim) =
A1(kz.1') '2

A2(kg‘1') Z

 

+§ (X' - amen]

2

6

5

swim) _ 3 j1(k%lf2 — 433')

= US$2.33) [pq(F1) + A1071”)? - 61%(771)

l _. ..
+ $— 11309111?) [(02 — a + §)qu(7'l) - 79(r1)

3 ..
+ assets/2.91)] .

 

 

 

3“ — a

kfix kqplfz — 553'
3j1(’€%1‘) = j1(kj{.|f2 —il"73|)

111.2: 191.);2'2 — :53) ’

L1(k%.’17)

1 —— m1 k": 2

L kq‘, L qu‘,.
1+ 1(p-13)+ 2( fl) zl+0((k%7)2).

”5(ki'viv)

222

(9.211)

(9.212)
(9.213)
(9.214)

(9.215)

 

 

 

As a. can take any value between zero and one, it can be considered as an optimization
parameter that is to be used to select the best DME-coordinates. Refer to the next.

section for some detail on related works.

Recovering previous DMEs of the scalar part of the OBDM

At this point, it should be emphasized that the approximations that are used up
to now are exactly the same as the ones that are used in section 5.3.3 when we
derived PSA-DME for pq(f’1,f'2) using (11, 7") as the DME-coordinates. Hence, we can
recover the PSA-DME of pq(I-f+ 772,11- 7‘72), i.e., Eq. (5.21) by setting 532 = —i"3 =
F/2, F1 2 if and a = 1/2. In this case, X = 0, f = Fand Eq. (9.211) reduces to

-+ F .. F _. 7‘2 1 —. _.
p.92 + 5. R. — 5) 2 Hakim) p.119 + gnaw) [$qu) — TAR)
+ 3921141334 , (9.216)
Q

where US$21“) and 113(1324‘) are as given in Eqs. (9.212) and (9.213), with if being
replaced with F. In obtaining Eq. (9.216), we considered only the leading order
contribution to A2(k}:r). The only difference between that of Eq. (9.216) and the
corresponding expression in Ref. [170] is the fact that flaky) = 105 j3(k%r)/(k}r)3

in the original DME. As can be seen from the series expansion,

 

jl(k‘[]~‘7) ~ ,q , 2
My ~ 1+O((AF7) ), (9.217)
j3(k%7‘) ~ .9 .. 2

105 ———(k%r)3 ~ 1+0((1,.,) ), (9.218)

the two 7r—functions are similar in their leading order. Due to the (kg-r)2 prefac-
tor that we have in the second-order correction, the difference between the second-
order correction terms of PSA-DME and the original DME appears in terms beyond
(DH/1121)"). As noted in Ref. [170], this difference in the higher-order terms should not

223

 

 

 

 

be surprising due to the the ambiguity of DME correction terms beyond O((k‘}r)2).

Hence, we have effectively recovered the DME of Ref. [170].

In Ref. [202], the authors generalize the original DME of [170] as

, 2
pq(R + bF, R - (1 — b) 7") 2 Ila/91.7“) pq(R) + %H§(k%r) [(b2 — b + 1/2) qu(R)
_. 3 _.
_ Tq(R) + gk%2pq(R)] 9 (9.219)

with the same 7r—functions as given in Ref. [170]. Parameter b = 1/2 recovers the

usual relative and center of mass coordinates. To obtain this expansion from our

generalized PSA-DME, Eq. (9.211), one sets
F1 = 12', =3, = bf", and 533 = —(1 —b)f’, (9.220)

which implies a? = F and if = 0. Hence, Eq. (9.211) reduces exactly to Eq. (9.219)
with parameter a playing the role of parameter b. Optimizing parameter b, the
authors of Ref. [202] note that b = 0 which amounts to expanding about one of the
particles, instead of the center of mass, seems to give the best accuracy for molecular
systems. This further enforces the view that optimization of parameter a can result

in increased accuracy of the DME.

Further approximation with respect to if

In the generalized PSA-DME of pq(7"'1 + f2, 7"} + :33), our angle averaging with respect
to the orientation of if is a well-supported step in that the scalar part of the OBDM
is known to have a weak angular dependence on the orientation of the relative co-
ordinate [176]. If we stretch the argument and assume that the dependence on the

orientation of the other non-locality coordinate, X, is weak, we can average over the

224

 

 

 

 

orientation of X . In this case, Eq. (9.211) reduces to

_. _. _. _. _, .132 .. 1 X2 _,
pq(r1 + 1:2, 71+ 1:3) 2 Hakim) pq('r1) + —6— H§(k}';;r) [(a2 — a + -2- + 7) qu(r1)
—o 3 ( —o
4.00 + gb'f'2A2U'f-F)PATH]a (9221)

whose simplicity makes investigating its accuracy a worthy step. As mentioned in
the previous section, parameter a may be optimized to reduce some of the inaccuracy

that may result from averaging over the orientation of X .

9.5.5 Generalized PSA-DME for the vector part of the OBDM

The generalized PSA-DME for the vector component of the OBDM involves a signif-
icantly less algebra than that of the scalar component as we stop at the linear order
in the Taylor series expansion. it involves exactly the same approximations as the
ones that we used in section 5.3.4. Extracting the non-locality operator and a phase

space factor, followed by Taylor series expansion of the operator

—o —¢

SqJ/(Fl + $2773 + £3) : eiE-(EZ—fg) 852-(62—ig)+53'(63+i§) Sq.u(f2a 773)

 

1V2=""3=""1
2:: eiE'(52_53) [I + :32 - (V72 — 2'19.)

—0 ._.

+113 ' (V3 + 1(1)] 8%,,(772, 713) (9.222)

f'2‘-"*3='""1

 

where we truncated the expansion at first order. Since §q(f'1) = 0 for time-reversal
invariant systems, the contribution from the leading term vanishes. Likewise, the
contributions from the linear 2232/3 - 1: terms vanish. Using Eq. (9.112), the definition

of the cartesian spin-orbit density, 114,”, given in Eq. (9.83), one writes Eq. (9.222)

225

 

as

.s-,,_,,(r~'1 + .32. F1 + .133) z ie“"<f2—f3> Z 2,. Jq.,.,,(r’-1). (9.223)

p.
The final step involves performing the PSA over a deformed sphere that characterizes
the local momentum distribution. Let us start from a spheroid in momentum space

defined by the equation

 

 

 

= 1. (9.224)

For ease of notation, we write (1(1-Z) as a and C(12) as c in the following. We constrain
the position-dependent quantities a and c by requiring that the spheroid has a given

volume and quadrupole moment, viz,

 

4 . 4
Vq E gash}? = §w3a2c, (9.225)

_. 2 (—a2 + 02)
P570?) 2 2a2 + c2 . (9.226)

The II—function is obtained via the integration over the phase space of interest
Hf ': —q3/ (”C 62.1“]; . i (9.227)

Carrying out the integration over the volume V,, encompassed by the spheroid given
in Eq. (9.224) can be done by using a stretched coordinate system from the transfor-

mation

~

I? E (k,,ky,k,) _. 113' E (kg/($311.), (9.228)
C

226

 

 

 

 

such that one finally obtains

 

 

 

3,901 + .132. F, + 2'3) 2 'i Hid-2.1;) Z 1:,,.1.,,,,.,(7‘-3), (9.229)
p
where
~~ .1133 "ma—q
Hakim) E 3J‘1q1'l) = 3111 Pl] 3"), (9.230)
kpr k; 132—rd
and
~ 2 2P" " ”3
kg. 2 ( + 2(5)) 1:}. (9.231)
2-P:§’(R)

Setting P303.) = 0, which consists of performing the PSA over INM phase-space,
results in the same II—function with is}. replaced by k3,. For spherical systems, one
can simplify the expression further by writing Jq,w,(ff) as a sum of pseudoscalar,
vector and (antisymmetric) traceless tensor parts given in Eq. (9.165). Since in these
systems, both the pseudoscalar and the tensor parts vanish, one obtains

i

2 nfulgr) .13 x L071). (9.232)

gq (F1 + f2,'1—‘1 + £3) 1"
Hence, using Eq. (9.229), we have a DME for any type of nonlocal coordinate depen-
dence. For example, 3(F1,1"'1 + 553) is obtained by simply setting :32 2: 0, while setting

7"} = If and f2 = —j:'3 = ”/2, Eq. (9.229) reduces to

—o

9,,(1‘2' + 3,1? — g) 2 i n-;‘(1.s‘;.-r) Z r,,J,,,,,,(1‘?'), (9.233)

p

where 1'1“" 1.” r is as given in E . 9.230 , with :1? replaced with 1". Due to the possible
1 F Cl

dependence of the accuracy of the DME on the specific coordinates used, one cannot

227

claim or expect the same accuracy in expanding, for instance, .§'q('f'1,f'1 + 1172) and

am + 772,11 — 772).

9.5.6 Remarks on the generalized PSA-DME

As can be seen from Eqs. (9.212), (9.213) and (9.230), the final results of PSA-
DME of pq(7“'1 + :32, F1 + 553) and §',,(F1 + :32, F1 + 553) are not separable in :32 and :33.
In contrast, all nonlocal densities that involve only two of the coordinates such as
pq(7"'1, 772), pq(7"'1, F1 + £2), led to a completely separable expansions. This can leave
the perception that the objective of having a separable approximation, which is what
the DME proposes to achieve, is not yet met.

However, pq(7"'1 + 52, F1 +53) and 52,071 +552, F1 +553) appear in the HF energy from
the chiral EFT NNN interaction at NZLO where 1'32 and 53 are part of the interaction
form factors. Refer to section 7.1. In fact, the interaction does not depend on F1.
Thus, all terms that depend solely on 52 and :33, whether they are separable in these
two coordinates or not, can in principle be integrated out with the interaction form
factors. The actual direct analytical integration of such terms is very difficult, if not
impossible. Refer to section 9.11 for details on how we solve this problem. Leaving
the technicalities for the relevant sections, it should be clear at this point that a local
EDF will rasult from the application of the generalized PSA-DME of pq(7“'1+a':'2,1"'1+53)

and §q(7'"1 +f2,1"'1 +5233) to the HF energy of the chiral EFT NNN interaction at NzLO.

9.5.7 The modified-Taylor series expansion

As discussed in section 5.3.6, the modified—Taylor series approach provides a formal
framework to extend the applicability of the density matrix expansion to non-time
reversal invariant systems. It is obtained by replacing the coefficients of the Taylor

series expansion of the densities with 7r-functions. The dimensionless variable 9 is

228

used to denote the possible argument of the 7r—functions. The modified Taylor series

expansion of the densities that. appear in the exact HF energy from a generic NN

interaction reads

 

 

 

_. 7" _. 1" _. ..
pq(R i '2') N 778(9) 1’9”?) i “am 5 VIZ/MU?)
1 F 9 2 _.
+5799) (5 - vi.) p.02) (9234)
-o 4 4" 9 d 5' F d -° _.
9.,(12 i ) m we) sq(R) 1: M9) (5 v3 3.,(12)
1 q " .. 2 a
+§n;(9) (a R) 5721(3), (9235)
—o -o F . —' F -o —. -o
“(R i 5) = «((9)309 i «((9) (5 - v3 Jq(R)
1 ~ F 2 2.. ..
+57%» 5 . VR AR), (9236)
-* -* 7:. d -‘ -‘ " 1? —o -o —o
Jq(R i 5) “ 7r5(9) AR) i «Km-2- VR q(R)
1 f F 2 -+ _.
+57% (Q) E V}; Jq(R). (9.237)
_. F .. F p .. p F -o -' _.
pq(R 2i: ‘2‘, R 3F '2‘) % H0(Q)pq(R) Ii: H1(Q)§ V1 — V2 pq(r1 T2) ..
Fl="2=R
1 p F _. 2 F 9 2 fl _,
'i' §H2(§2) 5 V1) + (5 ‘ V2) pq(T1 , 7‘2)
i=2=fi
77 .. F .. _, _,
415(9) (5 - v1) (5 v2)pq(r1.r2) , (9.238)
51:52:}?
-0 " F " F 9 -* d s F a d " " "’
Sq(R It 5, R :F 5) = H0(Q)9q(R) Ii: H1(Q)§ ( 1 — V2)Sq( 1, 7‘2) _,
"1=I-'2=R
~ F 2 4 2
._n.m)[(§ .) + ( v2 )2(... , F.) _
~1:‘2=R

 

229

and for the pairing densities

~ 77 , ~ _. - F _. ~ _.
pq(Ri") z 7r(“SD/MU?)3'37Ti)(9)§ vhf/MU?)
1 , F - K s
+§ng(o)(§ . v3) pqm) (9.240)
1 .. F - . F - _~_, ..
9.02%) ~ 7r.<sz>s.(mivrl(m(§ v1.)s.(12)
1 g F .. - q
+§7r2(o) 5 - v squz) (9.241)

 

 

 

 

~ - F . F -
1 - F .. 2 F .. 2
+ — 113(9) V] + — V2 fiq(Fl 3 F2)
2 2 2 -. _-. -R
1" 2—
p F - F _. 1 _, _,
— 113(0) - V1 — V2 pq(71 , 7‘2) , (9.242)
2 2 -' _-.° .1?
7‘1—12—
:. -o 7? -‘ F q _, “ 9 F _' T. :0 —o —o
we i 5,12 :1: 5) 2 11.191941?) i H1(9)§ - ( 1 — v2)e.(n , r2) _.
“1:72:12
1 .~ .. 2 F .. 2 ~
+ - H"(9) - V1 + - V2 511(71 , F2)
2 2 - _- _fi
1"?—
3 F -. F ~ .. e
—H3(Q) — ' V1 — V2 .Sq(71, 72) , (9.243)
2 2 1=Q=fi

At this point, the modified-Taylor series expansions of the densities can not be written
in terms of the local densities defined in appendix 9.2.3 and 9.2.5. Implementing the

steps explained in section 5.3.6, one obtains an equivalent expansion, this time with

explicit local densities. These are

pqu'i i g > ~ «ampm 1
+ :- ng’m) (g - 63)? pq(1'2') (9.244)
5.0% i g“) a WWW?) iwf<n)§-VRF.<R)
+ :— 7.3m) (g “3) 25:11?) (9245)

Hsmmm > 2+: mm 123-M) + 54-11392) (qu0?)

_. F ~ I"
/)q(Ri§.R:F§) ~
7.2
-2T(l(fi)) — -1-2-H§(§2)TQ(R), (9.246)
2 F 2 F ., 2 , é. _. 2 g. 2
3MB: :1: 5,12 ; 5) s 113(0) equu?) :tzH‘1(Q) erWuz) + in: (Q) (Aswm)
. 2 _3
47343)) — 1’15 §(Q)Tq‘,,(R), (9.247)
_. 17 1' ... .. v F _. ._. _.
JQ(R i i) z 7r(J)(Q).7(1(R) i “{(Q)§ VRJq(R)
1 -.- “ .. 29 _.
+§W$(9)(%'VR) AR), (9248)
-o -o F " -° -' " 7:. -0 -o -o
Jq(R 1*: 5) % 7r91(9)Jq(R) i ”“535 (VR)Jq(R)
1 ~ ‘* ~ 2~ -+
+§7F2J(Q)(‘;"VR) AR), (9249)
and for the pailing densmeb
., " F - ~ -‘ ~ F —0 ~ —o
Pq(R i 5) ”1 778(9) (09(3) i WW5 VRPq(R)
1 p F 2 K r
+57%”) §°Vn Pq(R) (9.200)
:. 7" ~ .3 7" ~ 2. ~
69(3 i 5) % «062)de i7T1(Q)§'VRSq(R)
1 “ .. 2 9
+§7r§(Q) (avg) 93(3), (9.251)
~ “ F ‘* F -+ . _, " .. 7‘2 ~
MR 2: 5,12 2: 5) z 118(9) MR) iznflmr MR) + firm) (qum)
2
2 r ~ 2 ~
4902)) fingqum), (9252)
~ ~ 7" ~ 7" I: ~ ; ~ 7‘2 % ~ ~
Sq.u(R i 5: R :F 5) % “MUS“ q.u AR) i7H1.(Q)7‘quw/(R) + EH25?) (Asun?)
~ ... 7‘2 ~ _9
4233412)) 1511.362)?" ,,,(R) (9.253)

231

9.6 Derivation of EDF from HF energy of local

N N interaction

In this section, we give detailed derivation of the HF energy from a generic local

two-nucleon interaction and the EDF that results after the application of the DME.

9.6.1 Central contribution

We demonstrate the derivation of the contribution to the HF energy from the central
part of N N interaction by deriving the corresponding expression for the spin-triplet,
isospin-triplet channel. Making use of the spin/isospin projection operators given in
Table 1.2, the projection of a local central interaction in this channel reads

V51 = (1+13102)(1+ “Ian/0%). (9.254)

tb-IH

Plugging this in Eq. (6.2), using the definition of the OBDM (Eq. (9.70)) and its

scalar/isoscalar—vector/isovector decomposition, we obtain

1 _. .. .. .. .. _. .. ..
Emu] = g / dn .5. V519) [90(71)P0(7‘2) — po(v-1,r2)po(r2,n)
1 .. _. .. _.
+ Z (5 00(7‘1) 60102 + 250(71) 00102)
”1‘72
1 _, 1_, _, _,
X (5 pO( 2) 60102 + 530(7‘2) 00201)
.. .. 1s .. a _.
- (”202(5/9001113) 60102 + 250(7‘1,7‘2) 00102)
l _, _, - 1_, _, _, _,
X (5 p0(7"2, Tl) 00102 + 58003: T1)'00201)
1 _, 1 _ _, 1 r 1 Lg ..
+ [(5 p051) + E(—1)2 P1(T1))(§Po(7'2)+ 5(4)? p.99)

1 fl _, 1 1.., 1 _.
X (5p0(r2, 7‘1) + §(_1)2 pl(r297‘l))
1 .. _. _. .. _. _. _. ..
+ ‘13 Z Z (/)()(l‘1)/)()(I‘2) 60102 + 3()(r1)oaalo2 X 8()(7‘2)°00201
q 0102

+101 (r1) ,0] (F2) 60102 + S—l(Fl)-50102 X 31(71‘2)-50‘201)

1 _, _, _. ..
_ E Z Z (100(7‘177‘2) P0("'2a T1) 50102

‘1 ”1‘72

+s‘6(fi,r2).50152 X ‘57)(F23F1l5‘7201 + p1(F1’3F2)p1(F2’F1)60102

+3-l(F17 71:2).60102 X 51(F21F1)50201) ] ' (9'255)

To find the final, most simplified form of the above expression, we make repeated

use of the relations listed in Eqs. (9.12)-(9.13) to obtain

1 _, _, 9 _, _, 9 _, _, 3 _. ..
EgNllll = g/drl (1T2 V610) [190(7‘1)PO(7'2)— 1| 100(7‘11T2)|2+1P1(7‘1)P1(7‘2)
3 _, _, . 3_, _, _, _, 3_, _, _, _, _, _,
— 1| 101(7‘1~.7‘-2)|2 + 130(7‘1).30(7‘2)— 130(7'1,7‘2)-30(7'2a7”1)
1_, _, _, 4 l_, _, _, _, _, _, r
+ 181(7‘1).81(T2) — 181(7'1,7'2).81(I‘2,7'1):l . (9.206)

In terms of the proton and neutron densities

1 _. _. _. _. .. _. .. _.
ECNNllll : gZ/drl dV‘2 VC1'1(7')I:3pq(rl)pq(r2) _ 3pq(rla7'2)pq(r217'l)
‘7

+55%) - 9(3) — 5.15.52) - W()] (9257)

1 -o —o 3 -0 “
+g E fdrl drz V8.10?) I: pq(T1)pq(7‘2) “ pq(7'1 72)pq(72 T1)
1 a —o —o 1 -o -o —-o -o —v -o r
+-2—sq(r1) - 9.,(1‘2) — §Sq(7‘1 7‘2) 5‘(’2.T1) (9 208)

Similar derivations can be done for the other three channels.

233

9.6.2 Spin-orbit contribution

For the spin-orbit interaction, we demonstrate the derivation for the spin-triplet,
isospin-singlet channel. Starting with Eq. (6.2) and the projection of the spin-orbit

part of the interaction in this channel, we can write the Hartree contribution as

i - - .. a ~ _. s _.
Efslyullol = ‘1: [6171 d7'2 V133 ('7‘) T' [Pq(7'2)V1X3c7(7‘1,7‘2')|r~2/=r~1
q
+ §q(:.l) X 62p¢i(772, 772,)IF2,=172 ]
i a _, a _, l-o a a .~ a
= 7:; [drldy~2vgg(r)r. [pq(7‘2)(§VX 'q('r1)+z.]q(r1))

_, a la _. .7 a
+8.,(7‘1) X (“V/Jam) + WWW]

VP‘IIH

Z / (1771 (1742 Vgg(r)7='. [mm/5(a) + -r'-'- 5.,(7‘1) 69 32%
q

i

2 v x awn/2.03) + gqum) x am] (9259)

Noting that
/ cm die V1.30") .7. v a [mm mm] = o, (9.260)

where 6 = 61 — 62, the Hartree contribution from the spin-orbit interaction in the

spin-triplet, isospin-singlet channel reads
ENN _1 l.-'1-:V10.. -+ "-* 1' 1* -' 7.7 21
LS.d2'r[10] — Z Z ”1072 15(7)?" q("1)/)(i(72) + v5q(7‘1) X L702) - (9- 6 l
q
The Fock contribution in the spin-triplet, isospin singlet channel reads

Efiéi’pllm = §(V5 — V. + v7 — v8), (9.262)

234

Where expressing the V,- s in terms of the density matrix

Vs, = Z Z: / (17"1d7"2ng(7 )7 x V2pq(7",202 F101) -(01|0|03) pq( F103 F202)
010203 q

V6 = Z Z/dfldflVfig )I">< §1p(;(7"103,I"202)-(01|5|0:3)pq(1"202,7"101)
010203 q

V7 2 Z Z/drl d7‘2VL5( )rxV2pq (7204,7101) (a2lala4)p —(7"101,7"2a2)
010204 q

V8 2 Z Z / d7"1d7"2 V112- (7‘) 7"x {71/75 (F101, F202) - (02IEIU4) pq (7"204,7"101).
010204 q

(9.263)

The manipulation of the above four expressions involves repeated application of

spin-traces. Finally, one obtains

i _. _. .. _. _. _. -‘ .. _.
ELS,e;rci10i = .51 Z/ dT1d1‘2 V23”) [80(7‘177‘2) ' 7' ® V2pq(7'2,7'1)
q

—o

+Pq(7_‘ia7"2)"? ‘33 V2 ‘ -§q(F2,F1)] - (9264)

Spin-orbit contribution in time-reversal invariant systems

To recover the expression given in [170] for the contribution from the Hartree-Fock
energy in spin-orbit interaction, it suffices to show that

_ 2:q f dFl d7"2 VngO‘) pq(I“‘1,f'2)I'" X {72 ‘ 55(F1, 772) can be simplified as

= " Z/ (1'71(1F2V£2(7‘)pq(7"1,772)77x V2 ' $707143)
‘1

Z Z] dFl d7"2 V359“) [$70717le ' 'FX fi2Pa(7-‘ie7—"2)
q
—- pq-(7‘1,F2).§'q(r1,7‘2) - 62 x 7‘"
1 dVL _, _. .. _, _. _. _.
" Z] drld; 2 — rd“: )‘ 97(T1J2)3q(7‘127'2) ‘7‘ X 7‘

235

= Z/ (1F1(lT-::2 V320") 87,071, F2) ‘ 7? X figqu-"l, F2) . (9.265)
r1
Thus taking the above result, adding it to the terms coming from the spin-triplet
isospin-triplet one and assuming 1032(7) = VL‘;(7‘) 1 one obtains

-O

_ 1 a 4 ~ ~ ~
Egg ._. E /d7‘1dr‘2VLs(r)[7‘- (mm)
+7.2 2; gm» 7x mama],
‘1

(9.266)

which is exactly the same expression as given in [170] for time-reversal invariant
systems. Note that there is a factor of two difference between our expression and
the expression in Ref. [170] which is due to a factor of two difference between the

spin-orbit interaction used in our derivation and in Ref. [170].

9.6.3 Tensor contribution

The derivation of the Hartree contribution from the tensor part of the NN interaction

is trivial due to the specific operator structure, viz,

(0102IEI. ‘ é;- 52 ' grl030'4> = <01|61|03> ' gr <02|52|04> ' 57-, (9.267)

(0102' 51 ‘ 32 [0304) = (01'61IO'3> ‘ (02'32'0’4> . (9.268)
For example, in the spin—triplet, isospin—singlet channel,

35“},(7‘3) ' 7: "(7(772) ‘ f — 537071) ° $7072) (9269)

 

, 1 _. _.
57%?”10] = Z 2/ (17‘1 dT‘Q VIEOU‘)
q

 

1This assumption is essential to obtain the form given in [170] as the authors use the same 7‘
dependence in both channels.

236

The derivation of the Fock contribution involves a significant number of spin-traces.
Due to the similarity with the derivation given in section 9.6.1 for the central piece,

we do not repeat the derivation.

9.6.4 Leading-order pairing contribution

Restricting the derivation to a central interaction and to the spin-singlet, isospin-

triplet channel, the leading-order pairing contribution reads

(<1>|Vc°‘l<l>>pm-r = g: 2: KM (z'CIJ'qlVO1 quzqw (9270)

q ijkl

Thus,

1 _, _, _
(cplvgl|q>)p,,, = {3‘2 122 / (mam/31m

1 .. ..-
><2(:,1 —Pq (7202,7"100‘3—Pq (730277101)

1 ~ 1
_2__p‘l* (r202,r101)?;pq (T1019T202)

1 1
_E; ”q (1302,7103)20—gfiq(rq2.0’117:lo_'2)

1 , i i_ 1 .. i i_
+—Pq 0202,7101)_Pq(7‘102,7"201) - (9271)
20' 0'1 20'1

After resolving into scalar/isoscalar, vector/isovector parts and simplifying one gets

<<I>IV°ll<I>> £12] an (172 )Ipqm. 1w. (9272)

9.6.5 The resulting EDF: EDF-NN-DME

In section 5.3.6, we discussed the basic steps that are involved in the derivation

of a local EDF from the exact HF energy of a generic NN interaction through the

237

application of the modified-Taylor series (or any other DME). These steps are best
exemplified by the simplifications that we carried out to obtain Eqs. (5.46) and (5.47).
\Ve apply essentially the same step on the exact HF B given in section 6.1 and arrive at
a local EDF which we call EDF-NN-DME. It has three components: the particle-hole
EDF (given in Eq. (4.11)), the particle-particle EDF (given in Eq. (6.35)) and the
additional terms of the EDF (given in Eq. (6.36)). The couplings of the particle-hole

part of the EDF, in terms of the notation defined in Table 1.2, are given by

AP = 3a3P1[<ws)2+n(w>21 -afPP[(«s>2—(na21
AP = (— 3i3a2PPP+f —3a§“>[n5(ns+n§)1
APPP = (3—1—3a + 332321133533] + (3.1233 — gamma]
APPPP = —<3—13a§*PP+3-33a2PPP)[(vrr>2]
APP = gamma (1132] +3af“[<«§)2— (1132]
APW = :aé‘su[7r57r{— H3113]
AVPP = —:a3222[«f«J—n3n31
APAP = —3—12a§°1[7r37r3+n3r13] +(géagn—éagll)[7rgww§— H3113]
+333“ [23333 _ 113113]
3% = —3a3“[(«a21+§a3“[n5( §+na1
AVsoVs : (3__12aC01_ 3_12 acn 8+éa§11)[(flf)2]
APP = (— 1—13a3PP—3—13a2P“+3a3P{newsman-s1
A” = —g-a§‘11[113‘113]
BPP = (7323 33+ 1—13a1PPP>[(«3>2
+<ns>21 + (332—3311 + 713a?2°)[(vr8)2 - (11:92]
B"T = (— —3—32( 1010—§1§(z€01+%a€“+3—12ag'00 )[H8(H‘°+H§)]

238

B233” (3—33a E33 + 33-30333 + 3—330 5” + 33—1309” )[5553]
+ (535610 + 61133301 63433011 _ 6143200 )[ HGHP]
BPPP —(3—33a5PP + 333a? + 3335“ + 333(15PP )[(+5)2)
BPP (333a5PP — 333+ 5PP)[<+5)2 +< n52 1
+(3—33c5 - 3—13af°°)[(7r5)3—(n5)3]
BPW %a3510[7r33 7:3 —ngnP]+ +513 3511533 7r1 ngnf]
BPPP _; 33103731 753 HPHf] _ {335351153537 + H3113]
BPPP (334550 — 3—33a5PP)[75 5+ H5115)
+(333-a5“ - 313M — H5115)
+<— 3—33a5PP + 35° - 3135“ + 333331333333]
+(— 3—33a + 3—33 50 + 3—33a — 15633333133333.3333
BPPPP -(Z a5PP+ 3 3a5PPn< 1))
+(—1—3—6 a3T104+ 136 aT“) [I13 (II; + HP 3)]
BVsoVs (—&a€30 +16_14acr01 _ 6-1—4 (1011 + 611(33700
+323a5PP3 +— 35% 1))
B“ ($125310 3—1:ag01 _ 3335511 + $5500 _ gag“) + 35310
+305“ -3331 )[n5(n5+n5))
3” (3a5PP —3a5 “)[H5H51
and for the partlcle-partlcle part of the EDF
AP = 3—33-a5PP[n5)2,
APP = I:730,3P“’1[I13n.3],
A” = iag’lflngnfl. (9.273)

In the EDF that results from the additional terms, given in Eq. (6.36), only the

couplings from the Coulomb piece are unspecified. These are given by

CPP = 03‘ «(3)2, (9.274)

CPAP =

0Vpr

where in these equation a? refers to using the Coulomb interaction to compute these

couplings.

9.6.6 Analytical couplings from chiral EFT NN interaction
at N2LO

In this section, we derive the analytical couplings of the particle—hole EDF (given in
Eq. (4.11)) for time-reversal invariant systems starting from the finite-range part of
chiral EFT NN interaction at N2LO (Eq. (6.5)). In line with the exact treatment
of the direct part advocated in section 5.3.7, the contributions to the couplings that
come from the Hartree part of the HF energy are not included. Furthermore, in
conformance with the notation used in the NNN case, we use isoscalar/isovector
notation instead of proton/ neutron notation.

The starting point for the derivation is the expression for the couplings expressed
as a functional of the 7r—functions. Note that the Fock contributions that we are
interested in correspond to those terms that contain solely HE, i.e. no 7rf, for any

density (. We illustrate the derivation taking the calculation of App as an example

APP = 3 fdrr2[V(91(r)(H8(I-cpr))2—3V313+1(r)(1'18(kpr))2)],(9.275)

where the interaction vertices V3910) and V3310) are given by Eq. (6.10) and (6.11).

240

The starting chiral interaction is in momentum space and hence

1 . ._._.
App : fdr 7,2 diezq-r [I/(Ql(q) (H8(kF7.))2

 

16n2

— 3 V519,) (H3(L-,.~7-))2)] . (9.276)

The subsequent step requires specifying the 7r—functions which can be fixed according
to any viable DME approach. In our case, we use PSA-DME with the 7r—functions
given by Eqs. (5.22), (5.23) and (5.36). Next, we perform the integration with respect
to r first. This is actually an important step to see that the integrals do not actually

diverge. In contrast, in the NN N case, it is easier to perform the integrations first

 

with respect to the momentum coordinates. Refer to appendix 9.11.4 for a related

discussion. In performing the r integrals, we define

11(9) 2 [Mme(1159))? (9277)
12(9) 2 /r4clv~jo(qr)H8(r)H3(r), (9.278)
13(9) 2 /r2drj0(qr)(nf(r))2, (9.279)
[4(a) s /r4drj0(q-r)(nf(r))2, (9.280)

where (i = q / kp. Upon inserting the PSA-DME 7r—functions, these integrals become

119;) = 1293:1392) = 2207“ — 12g + 16) 9(2-9), (9281)
14(9) = 2% (2 — r22) 8(2 — (i), (9.282)

 

where 9 denotes the unit step function. What remain are one-dimensional integrals

with respect to the momentum coordinate, q. At this point, the couplings take the

241

 

form

pp_
Ct —

pAp _
C, —

JJ _
Ct —

where

q2dq Fi’m) [MW 1....) + 212(9/kr)]

— .3
7d,. F

2
k. [cqu r:'(q)12(q/k.~)
7T F

1 .

._ch
I

4wk%

 

 

/ q2dq 13(q/kp) (1+ 295%) I1W9)

 

— M; f (M; r1391) lug/a),

1“?"(0) = W(Q) i=0

= warm) t=1

with 2' E {C, S, T, LS}. These exchange-force form factors are given by

Vé(q)

W501)
V5101)
st (q)
Vflq)

W235 (9)

= VC(q) + u«’c(q) + 3Vs((1) + 3WS(‘I)
+q2VT(q) + (12WT((1),

= wc(q) + 3W3(q) + <12WT((1),

= VC((1) + Wc(q) — Vs(q) — Ws(q),

= Wc(q) - I'll/5(0),

: VT(q) + I’VTW):

= — WT(q).

(9.283)
(9.284)

(9.285)

(9.286)

(9.287)

(9.288)
(9.289)
(9.299)
(9.291)
(9.292)

(9.293)

The remaining one-dimensional integrals are calculated after plugging the chiral EFT

NN interaction form factors (at NQLO) given in Ref. [12]. The complete expressions

for the couplings are too lengthy to reproduce here. Consult the Mathematica files

242

of Ref. [161]. Here, we list the contribution to the couplings from the LO finite-range
piece. As given in Ref. [12]. the only LO finite-range piece is a one-pion exchange

term. Therefore we have

I
2 + 772.27

 

W7?” = —( 9‘“ )2(] (9.294)

M

where all other components (VC, etc.) are zero. With 21. E lap/mm the non-zero

couplings from the finite-range LO potential are:

 

 

2
_ 9A 2 , 4 6 _ 2
Ag) — —W{(—21 + 498a + 64a — 16a ) — 12u(35 + 4n )arctan(2u)
3
+472 (7 + 16u2(8 — 913)) log(1 + 41(2)}, (9.295)
856; = 2A???) (9.296)
Amp — — 35934 {(—3 + 72u2 + 4224) — 60 u arctan(2u)
<0) " 3072 fgmguf’ ‘ '
1
+ 4—u_2 (3 + 54112 - 72124) log(1 + 4122)} (9.297)
B53" = 2.4%)” (9.298)
A
Ag) 2 —4Af0)p (9.299)
Bg’g, = 2A5, (9.300)
2 g3, 5 + 12a2 4 2
A1 = { —1 4 } .
(0) 48f1¥m12r (1 + 421.2)2 + U2 Og(1 + u ) (9 301)
2 2
13(0) = 221(0) (9.302)

which can easily be put under the form of Eq. (6.39).

9.7 HFB equations from EDF-NN-DME

The general formalism of HFB equations is discussed in Ref. [81]. Just like HF

equations, they are solved self—consistently. In coordinate space, HFB equations for

243

—0

general nonlocal “mean field”, h"(f'a, F’ 0’), and pairing field, A4070, r’ 0'), take the

form
h'q(Fa, F'a’) A"(F(7, 7"" 0') Ug(f"a'q) U307}; q)
/ F"; = E:
o’ —A"*(FU, F’a') —h’q*('f'0, F’a’) VL"(7""’0’q) V’f’(f"0q)

(9.303)
where U30” 0’ q) and Vflf‘" 0’ q) represent the upper and lower components of the
quasi-particle wave functions. E3 is the quasi-particle energy and h’q(f' a, 1" ’ 0’ ) is
defined as

h.'q(Fa, 1“" 0') E h.q(Fa, F'a') — X’ (500/ 5(F— f"). (9.304)

In Eq. (9.304), X1 is the chemical potential which is calculated from particle number
constraint at each stage of the self-consistent iteration [205]. In configuration space,

the mean and pairing fields are given by

68
65

1]

Starting from a local HFB energy density, the mean and pairing fields become 10-
cal in coordinate space. This is shown explicitly in the next sections where we de-
rive hq(f'0, 1"" 0’), and pairing field, A"(Fa, F’ 0’) for EDF—NN-DME discussed in the
previous section. These derivations involve just repeated applications of functional

derivative which is briefly discussed in appendix 9.1.6.

244

T‘l

 

 

9.7 .1 The mean field from EDF-NN-DME

The derivation of h"(f'0,f"' 0’) in (F,a,q) space proceeds by taking the functional

 

derivative
6 s”. + 5"" + 5’?”
hj-l- E ( A 6pq (DUI-+1771) ’ (9.307)
if

where from the coordinate representation of the mean field, one has the configuration

representation

h}.- = Z jdfidf’g 99;('F101(1)hq(F101:71202)99i(7-‘202(1)a (9308)
“102

with 99,-(Faq) denoting the spin up/ down components of the basis 92,:(Fq). Since the

energy functional is quasi-local, it results in a local field of the form
llq(F10'1, F202) E 60?] — F2) hq(F1, 0'1, 02). (9.309)

This field acts on the spin up and spin down components of the wave function through

 

 

Z h"(F1;0102)¢,-('F102q) = [11" 99,-](77101q), (9.310)
(72
and it is given by
hwm) = [4‘7 - 3,07)? + Um + §,(F).a — 2mm 9 + (7' Km]
—\‘7’- [C'.,(~F)-5]V7 — 2 [mm a W + W 8 mm]
—%[6.D,(F)afi + 5.61347?) - 6] 2,021). (9.311)

where a shorthand notation for the tensor product has been used

A ® B = 2 AW, By“. The various components of the are given by

u [.1

, _ (A—l) 2 p7 pT ,
87m — 2A... h +A ,).,(mB 72.0). (9312)

0,0) = 2APPpQ(F) + A” mm + APAPA/qu + A [ApAppM]

 

—26- [vam + AP“? 3 13m — fi-[AVP’LM]
+2Bpppqm + Bcrqm + BPAPqum + A [WM/2m]

-22 [3mm] + 8”“? - far) — fi-[BVPJ £702]
+62/d7—y P120“) _ 82(§_)1/3p1/3(,:'), (9.313)

lF—F’l 71' p

8.7m = 281835.22) — 274/1““?- 22.09] — AJJT....<7=)
—2AJ~7F.,,.,(F) + A‘“A3Asq,u(F) + A [A3A85.,..,(1=)]

— QAVSOVS A8q_u(7:) + 2 61/023 Va [APVJ jq/3(F)]
OB
— 2 51/06 [141va Vajq.t3(f)]
00
+28“ $77,110:) — 2VV[BVWS6 ' 52(5)] — 3'” Tau/("7)

—QBJqu'J/(7_') + BSASASQ,”(F) + A [BSA886,U(7_‘)]

— QBVSOVS AquUT) + 2 61/03 Va [BPVJ $74303]
03

— Z Ema [BVPJ Vajti.fi(fl] 1 (9-314)
06

Am) = 47427.37) + quaatAPV’VowU‘71
GB

_ 2 61/03 Va [AVPJ Sq,i3(7‘.’)]

m3

_2BP7' jqr/(F') + 2 gym; [BPVJ Va-Sci.i3(f')]
03

_ 2 6m); Va- [BVPJ Sad?”
OH
’22

.., . _.I I.
Am fdr 2,: Jq’-V(r )’ (9310)
q

246

Cqm = —A“sq.u(r*> — B'”.s-q...(0. (9.316)
I)... = 2A“J.,.,,..(0 — Z a... v. [MW/1m]
+ z [22122.09]
+22” [J.,,,,,,(r*) + 6.4.2400] + 2B""JWW(F)
-— Z ..,... V. [BM/2.7m] + Z ..,... [va vapqm]
+ 2;” [Jmm + 6,,” Z Jq,.,..]a, (9.317)

Dam = —2AJ-is...m — zBJJ's....(o. (9.318)

 

9.7 .2 The Pairing field from EDF-NN-DME

The derivation of the pairing field proceeds by starting with the variational

A3]. = W‘ (9.319)

U

The pairing field A” in coordinate space is defined through [81]

A?) E Z [(17-31dF299:(F101(I)993(77202Q)Aq(77101s77202)- (9-320)
”102
It is local
A(I(F10'1, 77202) = 6(771— 7'72) Aq(1—"1;0’1.0'2). (9.321)

and has the structure

~
-o

Aq(f';01,02) = U,,(F;01,02) + 6 - Dq('f";01,02)6 + Aq('f';01,02) - 6, (9.322)

247

 

where the field components read

0.2.01.0.) = 27123230 5.5.1.2 + 2A!” (A209 — 42(7)) 3.3.1.,
+ 2 A [AW (3,0)] 5160,32
+3 2 (62(aglaulal) + 01(61IO’VI5’2)) V7,,[Ajj jq,,,,,(f')] ,
W (9.323)
15,07; 0., 02) = 812323.41?) a. 5.152, (9.324)

Aq‘p(77;0'1,0’2) = ZZ<52<02|0V|01) + 01(51|0V|0’2>) Ajjjq,uu(7_‘) . (9.325)

V

9.8 Numerical solution of EDF-HF equations in

spherical systems

For the preliminary self-consistent tests of the DME discussed in section 5.4.6, we
performed self-consistent calculation of the HF equations. This calculation was done
with the assumption of spherical symmetry, which also implies time-reversal invari-
ance. As the starting EDF, we took two different cases: (i) EDF-NN-DME with
the Bogoliubov contributions turned off. This is what we call full-DME. (ii) In the
second case, EDF-NN-DME is changed in such a way that the Hartree contributions
to EDF—NN-DME are replaced with their exact counterparts, with the Bogoliubov
contribution still turned off. This is what we call exchange-only-DME. Since the Bo-
goliubov contribution is turned off in both calculations, we refer to both calculations

as EDF-HF calculation.

In both full-DME and exchange-only-DME, the spherical self-consistent HF equa-

tions take the form
hqwfifl = fiq99z'(77(1)a (9-326)

248

where h" is the single particle Hamiltonian given by

... ...

m = —V-Bq(r)V + Uq(r) — trig-6 x a. (9.327)

The only difference between the two is in the actual values of the field components:

8.09. Um and Wm

9.8.1 Bill-DME in spherical systems

 

The components of h" for the case of full-DME are given by

(A — 1) 'r
qu = my + Am PW?) + Bp pm, (9328)

mm = 2.4/”pm + A" mm + ApApqum + A [AW/Mm]
-29. [AV/Wm] + AW - fqm - WWW
HEW/m + Emma + mam/2.7m + A [BPAppm]
—2€7- [BVPVPV'pq] + BM? - Jim — {HEW £109]
+62/dw ”PM — e2(g)‘/3p:/3(o. (9329)

IF- F’l

W. = AJJLm — V [Ammm + Awfipm — .4”qu

 

+3“ng — 6 [BM/am] + BVPJW.<0 - BJJJEW,

(9.330)

where 6 and A operators that occur in the fields probe only the radial part as we
are dealing with spherical systems. It should be noted that (i) all the local densities
depend only on the magnitude of 7" (ii) all the derivative operators are not meant to
act on the wave functions, they only act on the densities. All the couplings such as

A” and 8"” are as defined in section 9.6.5.

249

 

9.8.2 Exchange-only-DME in spherical systems

In this case, the field components read

_ (‘14— 12) pT pT
B.(r9 — 2.4... n +A p.,<79+B p.09 (9331)

v.09 = [dc [2V5"“”’(IF—F’I)pq(7"’)+2Vé5‘7'””(IF—F’l)pa(f")
+/d1‘~" [vggppur —r'))(r'-7=').j;(r')

+ VZ‘E‘WUF- F’|)(F’ — B) 47.06)

‘l‘ 2AM) 900:) + APT 79(7?) + APAP AM?)

 

+3 [ApAppqm] _ 2:7. [216mm,] + we . m

—6 [AVPJJ 7,99] + 23%.,(29 + B‘" 7,01) + 8””qu (F)
A

[épAppq(f)] — 26 [BVPV‘Ofi/y] + BPVJV “q(.)
—6O[BW jam] + ere/did I:p( ‘3) T e2(%)%p§(fl. (9.332)

d?” VZ§””( 7‘ - 7" ’l)pq(7" )+ Vi’fé’ppflr- 7609.70”) 7'“

+A’“J.<v9 - 6 [Ampqc'i] + AVPJ6pqo~9 — NM)

£31
n
\
1 t—'=—-I

+BJ’J209 — 6[B B” p.09] + BVP’6pav9 — 12”ng (9.333)

In this case, the couplings [431% and 133‘ (for any bilinear combination of local den-
sities, 99.») are obtained from the corresponding Aff and B? couplings given in ap-
pendix 9.7 by setting Hartree contributions to zero. Note that, according to our
notation, Hartree contributions are expressed exclusively in terms of 7r (the small let-
ter 7r—functions) while the Fock ones are expressed in terms of II (the capital letter
7r — functions).

For the numerical result reported in this work, we evaluate the 3D-integrals, which
reduce to 2D for spherical systems, directly using Gauss-Legendre integration tech-

nique. An alternative is to solve the corresponding Helmholtz problem [206].

250

 

9.8.3 Harmonic Oscillator basis expansion method

In the numerical solution of the single particle HF equations, the spherical harmonic
oscillator basis expansion method is used. Coupling the spin and angular momentum

to a total angular momentum, the basis are given by

 

.3 R7110.) m A 7-
9511039) = r Q11/2J'(”‘)X1/2a (9334)
where
p = (nljmr) refers to the single particle quantum numbers
571/230“) = Z (lsmflnsljmm 109%}? (9.335)

mlms

with R4,,(7') is defined in appendix 9.1.4. In the basis expansion method, one expands

the HF single particle states,|<pa), in terms of the basis states, I96“)

I2.) = Z 0;: 12.). (9.336)
p

where C“: are the expansion coefficients. The basic steps in a single iteration of the HF
self-consistent calculations is (i) the calculation of the matrix elements of the single
particle hamiltonian (ii) the diagonalization of the hamiltonian matrix to identify the
spectrum (iii) updating the single-particle fields or expansion coefficients using one of
the available schemes to drive the calculation towards convergence. The form of the
matrix elements for each of the three parts (the kinetic, central and spin-orbit parts)
of the single particle hamiltonian listed in Eqs.(9.327) is given below. Of course, the

calculation is performed self-consistently.

251

Matrix elements of the kinetic part

By the kinetic part, we are referring to the 6 - Bq(7‘)6 term of the single particle

field. The corresponding matrix elements read

 

. . = , 2_2_L __ , __ . _ _ ._
(CZ/IV Bq(r)V|o,,) [drr r (aquU‘) (9r 7‘ + Bq(r) (01‘? + r) 7‘ )
X 6H, 6jj’ 6mm, 61'1" " (9'337)

where p’ = (n’l’j’m’T’) and p = (nljm‘r). One can be tempted to use the exact

relation one has in spherical harmonic oscillator basis for kinetic energy operator

 

T=—WflmA
(271+l—1/2) ifnzn',
. hw
(¢,,IITI¢,.) = ~2— (flV(N+z+1/2) if In -—n’| = 1 N = min(n,n’),

0 if |n’—n| > 1.

In order to make use of this relation, one writes (prllfi - Bq(r)§|qb,,) as

(9,207 - 3.109691.) = (cm/IWBQUH .69)..) + <¢.I|3q(r)A|¢p>,
(9.338)

where one can write (¢“/|Bq(r)A|gb,,) as

(éyllBNMIép) = Z<¢>flIIBq(7‘)|¢./)(MAW)

= ——Z<¢,.IIBq(r)I<2u)<¢ulT|¢u>- (9339)

V

Finally, using the exact kinetic relation in Eq.(9.339) and plugging in Eq.(9.338),

one obtains a simplified formula for the matrix element. This is exact in the ideal

252

case of both no truncation of the basis states and a box of infinite size. In practical
calculations, one has to truncate the number of basis states and also use a finite—
sized box. These truncations make the use of Eq.(9.339) numerically unstable and
erroneous? This is the case especially when the inverse-effective mass term, Bq(r),
is very different from 1 inside the nucleus. For cases where the inverse-effective mass
remains more or less the same as the inverse bare nucleon mass, using Eq.(9.337) or

Eqs.(9.338) and (9.339) give the same results for the matrix elements.

Matrix elements of the central potential part

The matrix element for the central potential part of the single particle field, Uq(r),

reads

 

Ru!

5”! (SD-I 5 Id I (9.340)

mm 11' ’

<¢,./|Uq(r)l<bp> = / 717-72_§£U,(.~)

where again where )1’ = (n’l’j’m’T’) and p = (nljmr).

Matrix elements of the spin-orbit part

The matrix element of the spin-orbit part of the single particle field, 3W - 6 x ('7', reads

3' " _. . . . 3 R ' R"
x 6”! (51.1.1 5mm, 5”! , (9.341)

where again 21’ = (n’l’j’m’r’) and p = (7107717).

 

2This must be due to the practical violation of the completeness relation and the use of a finite
box size. It can be shown numerically (by increasing the box size) that effect of the later is minimal.

253

9.8.4 Self-consistent iterations and convergence

As can be seen from the results of the matrix elements for the three parts of the
single-particle fields, the hamiltonian couples basis states only within a single I — j
block. Hence, in the actual numerical solution of the HF equations, one diagonalizes
each I — j block independently. Of course, the other parts of the calculation will
involve all the relevant l — j blocks. To drive the calculation towards convergence, we
implemented both Broyden’s method [194] and Imaginary-time method in separate
calculations. After convergence, the results of the two methods usually agree to three
decimal points, and hence the results reported in section 5.4.6 have been obtained

using both methods.

9.9 The HF energy of chiral EFT NNN interaction

at N2LO

Here, we give a few remarks on the symbolic derivation of the HF energy of chiral
EFT NNN interaction at N 2L0 and give the complete expression for non time-reversal
invariant systems. The corresponding simplified expressions for IN M, PNM and time-

reversal invariant systems are also stated.

9.9.1 Remarks on the symbolic implementation

The details of the symbolic derivation of the HF energy from the chiral EFT is
discussed in Ref. [156]. In addition to automating a tremendous amount of spin—
isospin and other algebraic steps, we have demonstrated that the approach can be
generalized to treat nonlocal interactions such as the quasi-local Skyrme interactions.
There are several extensions of the symbolic derivation that can be made in the

future: (i) One can envision expanding the work in such a way that first-order pairing

254

correlations (due to the N N N interaction) are treated along with the HF part, viz,
performing HFB (Hartree-Fock—Bogoliubov) calculations. Combining this extension
with proton-neutron mixing, one can have a start- ing Skyrme-like functional that
can be used to handle proton-neutron pairing correlations as discussed in Ref. [207].
(iv) Implementing a similar scheme to treat four-nucleon interactions can also be one

area of extension.

9.9.2 HF energy from the E-term
Direct part

The direct part, which comes from the E—term, reads
. ~ - 1
(V3 '3‘“) = 5 E / drpo09 7909. (9.342)

Single-exchange part

The contribution from the single-exchange part, which originates from the E-term,

reads

(vamp = 2B [07(37239 + 32.09pm + 3po09 Bum-Bum

—po09 B109 - 3109 + 4p109st'o09 - 51(9) (9343)

Double-exchange part

The contribution from double-exchange part of the E-term reads

., 1 2 . 3 _.
043?“) = 1—6E / .1. [371309 + 72.097139 + 9p..<9so(9-so<9

+1000") 51(0-5'1m + 2p1(79§1(7‘)'«?o(7‘) . (9-344)

255

E—term contribution for specific systems

In symmetric IN M, the HF energy from the E-term reduces to

<vz:t"~“”“> = — 1% B / mm. (9.345)

In unpolarized PNM (pure neutron matter), the HF energy from the E—term vanishes

which is due to Pauli exclusion principle. In time-reversal invariant systems, the HF

energy contribution from the E-term takes the form

wit”) = —-,% E / 0(7):;09 — pom/£09 . (9.347)
9.9.3 HF energy from the D-term
Direct part

The contribution from the direct part of the D-term reads

 

 

urban- —9A CD 1 iq (r —r ) __‘I___23‘13
, , : ___ 2 3 3 2
(V337 ) 4f2 f2A -2- /(172dr3 [1) (277) 3d(13( (12 +772?
X72002) 3302) 109). (9.348)

In symmetric IN M, there is no contribution from this term. Likewise, for time-reversal

invariant systems where .§'0/1(F) = 0, the contribution from this term vanishes.

256

Single-exchange part

The contribution from the single-exchange part of the D-term reads
/HF.D,l;r __ —!}A CD 1 3., .-(r -r) Q3343,
(lm ) _ mm4/d12dr3/(2i) ——3dq3 c. 3 3 2 m

x[— p0(r3,r2)51(72)31(12,71)
‘750’1730(F3 F2)31('F21F3)3?(F2)
—p1(F2.F3)30(F3 F2)81(F2) - 91(F3 F2)30(F2.F3)31(F2)

+i€“”7 33(7‘},f};)s’l‘(f'3,f'2)3‘§3(F2) - p1(F2.F3)33(F31F2lsflF2)

1 1 —o -o -o -o -o
_ 590(F2)31(F3.F2)31(F2aF3) — 5637 PO(T2)P1(7'3a7'2)P1(7'zs7'3)
1 a w'u -* a -' " W “ '*
+56 W6 3 P(1(7‘2)-91(7'317'2)31(F27T3)
3 _. _.
71131339333. F2) 3(7‘217‘3)

_. 3 _. .. '3 ..
+po(7'2)3[1 (‘7'317‘2)3'17(7'2.7‘3) 2 .37 90(72)P0(F31F2)00(F21F3)
+ (5137- P0(F2)/)1(F3. F2)fl1(F2. F3)

3 . _. _. _. . .. _.
+ 5 60"” (““3" p0(r2) 83(1'3, 7'2) 3502, 7‘3)

— €07” CW” 100(F2) 3?('F3. F2)3i)(F2a F3l] - (9349)

For symmetric IN M, the expression simplifies to

 

.‘ — CD 1 q?
VHI,D.1:1¢,INM : 914 /d7‘ “d /_<_2_1__) ___d iQ3. .(r3— r2)
<31» ) 4f2 f2A——;r 4 7‘2 T3 (13 e q3+m,2,
X [—3 5 p0(7‘2)p()(7'3, 7‘2)/)()(7‘2, 733)] , (9.350)

while for unpolarized PNM, one has

 

2. , — CD 1 1 ._. _. .. (1.2
VHI,D.1:.PNM : 9.4 —/d"3d“’-/ d" lQ3.(r3—r2) 3
< J” l 41.2 f3A.4 ’2 ’3 (2733 7‘38 q§+m3
3
x [— 2 p..<r.)p..(r3 ,—~. 13.32. m] (9.351)

257

 

1. 7‘. LN . Z " It. 1‘ V l‘. .3," (7) 7 - : » I E. L.
In t1n1e—rewrsal 111v11‘1a11t systems 115111 9 /1 1 0, one obt 11115

 

 

 

1 -~ — Cu (1 (1
LINENDIJKIRI : (1A 1/(1 l-o f1) 1 1113..(r3—r2)3 5
( 1.1 > 4f? f2A1: 4 ”(7‘ (271)3“136 (73—sz
>< [—3 E511 00(72)P0(7:1 772)/}o(7"'2 11.73)

1 _. _. _. _. _.
+ 5 5117 p0(7‘2)/)1(7‘3, 7’2)p1(7‘2, 7‘3)

+ _ CW 518” p0(1='2)sg(r3,112113031125)

1217121911713,712191012211) . (9.352)

Double-exchange part

The contribution from the double-exchange part of the D-term reads

 

 

— CD 1 qfi q
VIIF,D,21' : 9A /d I—o [__2_1_) __gd iq3. .(r3— r2)______ 3
< 3N ) 4f; LEA: T6 T2( T3 (13 6 q, + 7112

x [3 (5,37 po('r‘72)po(7_"21‘7:31)P0(77317:2)

— 5111/11(772)Po(7-"21F3)P1(7731772)
+ 3 p(1(7"2)s€(f72, 773193973: 7‘2)

— 3 cm" 6MB" P0(7"2)33(7_"21773)33;(F31772)

- 10007219130721 73)«917(7‘_:117—"2)

+e"7”€wiiupo(7—"2)S 1"(7217‘3)1'(7'317‘2)
+6/70(7_"217‘31)-‘>’g(712)33(7711”7:72) " 35137{70(73):773)~‘7g(7_‘:2)33(7?3v71:2)
+3511 0003 72)93(7‘2)- 0(73 7‘2) — 2101072»'773)-9‘g(7-"72)51’1(F3a71:2)
+5131P1(F217"?1)Sg(772)81y(7731772) — 137P1(511F2)8g(772)81(F217‘31)
_. 23 (~12! 612113114 CW 331 (121932 (113, 1121933 (1;, 13,)

258

—23 ("13" 6"?" ”‘4 6"4W291;1(F22 F3)9o2(F2)~9g3(’-'32 F2)
+i6 ("1‘63" "2’3“ ("47” 3,17‘71(24)9g2(F5-, F2)-9:)‘3(F22F5)
+ 2' 6‘1"" e"2“3"4 6"47”531(F2)-9l1‘2(F3-.F2)'9‘1‘3(F2=F5)
+ 2'. ("1‘3" e"‘2“3“4 ("47" 531(F2 F3) 962(F2) 903(F3a F2)

22 #13" "2*‘3Hd‘47" .“1( r2) ’1‘2(r3,r.-2) "3(‘F22'F3)
+ 3557 p1(F2)p0(F32 F2)/)1(F2e F3)

— 5m p1(‘Fz)/)1(F32 F2)PD(F22 F3)
—P1(F' )Sg(7‘227‘3)5i(7'3772)

+ (071/ fwdu p1(F2)33(‘F22 F3).9:’(F3,F2)

 

+ 3 P1(F2)‘9g(F32 F2)5il(F2a 713)

— 3 6M" 6w" p1(F2)83(F32 Fz)'9‘i'°(F22 F3)

— 2po(F2a'F3)9ij(F2)32(F3~F2) + 6x37 00(F2 7 39) i3('F2)31(F32F2)

+ 363,. p0(r3. r2)9’13(F-2)81(F3 F2) + 6101(F2a F3)5f(F2)33(F32 F2)
— 3 63,, p1(F2, F3)3'[13(F2)33(F3, F2) — 5,37 201(F3, F2)«9'?(F2)33(F2a F5)
+ i 621193” €u2213l‘4 6‘4"" 5:1(F2)s’;2(f'3, 213.933 (F2, F3)

+ 2: #19" €“2"3"4 6‘47" 351 (7‘2, 7%)8’1‘2 (F2)8i'3 (F3, F2)

_ 22 fulfill 6222324 2242'” s’;1(2='2)s’;2(r3, 2333332, 53)

_ 2'3 6‘15" €p2p3p4 3‘4?" 3‘1” (@352 (F3, F2)s’1'3(f‘2, F3)

— 2'3 6‘1"?” e"2"3"4 6‘47" 3:1(F22 F3)3i‘2(F2)=9$3(F32 F2)

+26 21116” (#22324 2242" 3’1‘1(r2)sp2(773 r2) $303,273) . (9.353)

In symmetric IN M, the contribution reduces to

 

 

‘ ‘ 'x ' ’92! CD 1 (13
VHP’D'Z ,IMI : l (11" jd; ciq3-(r3-12) __
(W > 4!: LEA-.216 ”2 3 1‘ q§+méi

259

 

X I: 3 [)()(7—':2)/)0(7-':2. F3)p0(7-':5, 7;) :l , (9.354)

while for unpolarized PNM, one has

 

 

‘ . ) V _ CD . (1:2
VHP.D.2.r.I NM :- 941/ l—o d /_(_2_1__)1 iq3. .(F3—r2)
< 3N > 4f? PA 16 ( r2 T3 3( (13 e (13 + m2
X [ 3pyz(‘f72)p1.('72,F3)pn(7:35,7::2)] - (9355)

For time-reversal invariant s ’steins, usin the relations 5’0 1 r = 0 and 0 1 772,173 =
/ /

p0/1(f"3, F2), one obtains

— CD 1 1
VHF.D,2.r.,TRI = 9A _ fd—o d” f
< 3N > 4f; LEA; 16 r2 r3 (27r 3

x [35m puma/Mg, F3)Po(7-"3, F2)

B 7
£153 653.6342) 93%
(1% + mgr

 

 

 

+ 5137' P1(F2)P0(F2, 7130/9101?» 7.32)
+3p.(2=r2)s€(r='2, mswm)
— 3 6m" 6”" po(7—"2)83(‘F2a fi;)8‘6(7::3, F2)
—P0(’2)3 a[(73 7‘3)9‘1("3~7‘2)

+ 6m" 6w" po(F2)3?(F2a F3)8°1"(773, 72'2)
+2p1(F2)-95(F2,738193,?»

— 2 607” CLUB” p1(F2)83('F2, F3)8T (F3, F2) . (9.356)

260

 

 

 

D-term contribution for specific systems

Combining the results obtained for the D-term, the contribution to the HF energy of

symmetric INM reads

 

 

 

', — CD 1 1 -- .. ~ (1.2
( 3” l 4 f3 133A. 16 ””3 (2703 W (13+mgr
X [ ‘3po(772).00(7::29F3)PO(F3aF2)]- (9-357)

Combining the results obtained for the D-term, the contribution to the HF energy of

unpolarized PNM reads

 

 

 

<VHF.D.PNM> ___ _QA CD i fdf. (17.3/ 1 drf ei-(73.(F3—F2) (I;
3” 4f: LEA. 16 2 3 (2703 “ (1% +m£
X [ _3 pn(772)pn(772» 7.;?3)p11(7?3v7?2)] a (9358)

where pn(F) and pn(F, 1"" ) refer to the local and non-local parts of neutron matter
density. In time-reversal invariant systems, the HF energy contribution from the

D—term takes the form

 

 

— CD 1 1 -~ ~ ~ (1%.7
MHF,D,TRI : gA _ [d-od-o / d-O zq3.(r3—r2) 3 3
< 3” l 4f: ffiA. 16 T2 ’"3 (2w)3 q” q§+m£

x l46.5.po(2='2>po<fi2,r3>po(fa,F2)
+ 2 5m {10(F2)101(F2, 773)!)1073: F2)

+ 5,37 p1(F2)p0(F2, 71'3)/)1(”F3a 7:32)

— 3 [2007983012,ers-3%. F2)

+ 3 6m" 6W3” p0(-F2)38(F2. f3)s‘(j( 173, F2)
+ p0('f'2)s‘13(7"'2, mews, F2)

071/ (yd-£311

_ 6 p0(F2)S?(F2, F3)S‘i)(7737 772)

261

 

+wreaks-”('52.fis>sY(W2>

- 2 6m” 6”" P1(Fiz)83(7‘72, F3)s‘f(F3, 'le - (9359)

9.9.4 HF energy from the C-term

The HF energy from the C-term of the chiral EFT N2LO 3N F can be grouped into
two groups: a D-like term and remaining terms (which we call R-part). This grouping
originates from the operator structure of F3: given in Eq.(2.20). The D-like term is
associated with 605 [ —4 fig + 2 igrcj; (13] whereas the R—part relates to 2260377,: Ek-
((f,- + (L). In the following, the HF energy from the various parts of the C-term are

given.

Direct part

The contribution from the D-like piece of the direct part is

 

 

2
_ ~ -. l 1 ,s _. s .- s ..
<%Ig‘,CD,dn> : (g; ) 5 deIdF2(l7?3 [quadfiezq21r2—rl) ezq3.(r3—r1)
1r
'7 2
c —4 +2— . ‘
(r13 +m2xq§ +m§> l L? Le"? ‘13

XPO(F1)3?(F2)SY(F3)-

(9.360)

For both symmetric INM and time-reversal invariant systems, this contribution van-

ishes. The R-part contribution from the direct part vanishes
(Vgfim’i’dir) = 0. (9.361)

Hence, the direct part vanishes for spin-unpolarized INM/PNM and time-reversal

invariant systems.

262

Single-exchange part

The contribution from the D-like piece of the single—exchange part reads

 

2
1 1 _-~ 7 .~ ~ ~
<v3‘kF'CD""> = (2%) Z/ ”1“”1’3/ Wdedaewm-rnelm-m
1f
7 2
—4——+2— -.
(q%+m2.>(q§+m2.>l f: 3‘” ‘13

X [—Po(773»7'"'1)3i}(772)3i(7?1f3)
_z'em 8‘13(1'"'2)s’0‘(7"3, F1)s‘f(f"1,773)
— P1071, F3) sf('f’2)83(f'3, F1) — 101073: F1) 3?(F2)33(F1a 7:3)

—i 6"” 8f(7’2)-95(F3, 771)8‘i’(771.7"3) - 91071, 773)«9lj(7-"2)83(F3a F1)
1 ,3 1

—‘2'00(771)31(7733F2)3i(72,f35) - 55.87 P0(F1)pl('F3’F2)p1(F2’F3)
1 V w u -* -' ‘* w " 4

+56” 6 7 po(7‘1)8’1‘(7‘3a7‘2)31(7‘2ar3)
3

—O

_ 5 90(7‘1)86 (7’3, 7‘2)33(T22 7‘3)

_. .. _. _. _. 3 .. _. .. _. _.
+00(7'1)3?(7‘337‘2)3i(7‘2a7‘3) — 55m Po(T1)Po(T3aT2)/)0(T2a7"3)

+ 6x37 pO(F1)p1(F3a F2)Pl (772’ F3)

3 _. _, _, _. ..
+ 5 wwem" p0(r1)sf)‘(r3, r2)8‘6’(7‘2: 7‘3)

— Gym/6M7” p0(F1)8lll(T-:'3, F2)SLIU(F2, 7.33)] . (9.362)

For INM, this expression reduces to

 

 

 

2
‘ ‘ I 2 1 —o —o —o 1 —-o —o 1". '7 —F ._. F —-F
(VB’S'CD’IXJNM) = (79;) X fd‘l‘1dT-2d7‘13/(2 )6dQ2dq3e‘l2-('2 1)e'Q3~(3 1)
1r 71'
fl 7 2

(1293 617". 03 3 -

. . -4 +2—r '0

(q§+m2,)<q§+m§> l f3 L!” “l

3 .. _. .. .. _.
>< [— 5 537- P0(7‘1)Po(7‘3a 7‘2)l00(7‘2a 73)] 9 (9-363)

263

while for unpolarized PNM, one has

 

1 1 -~ 7 ~ ~ ~ ~
(LrLIIF, ,CD 1)(. .PNM) : (g; )24 _/drld7—: 2(17-03 / (2 )6 déTquij elq2.(12—r1) ezq3.(r3—r1)
7r 71’
. ‘7 2
(12(13 4017",. 203 _. _.
X . —(
(13+ ".3331. n23.) l 7.3— + I: ’2 “l
3 _, _. _. _. ..
X [— 5 6L”? pn('rl)pn(7'3v T"2)pn(7'2e 7.3)] - (9364)

For time-reversal invariant systems, it becomes

 

 

21 1 ,3 _, _, ._. .4 _.
< 3N > Q—fn 71 7‘2 7‘3 (27rls (12 Q3? 8
2
+2—(I2'Q3
X(c1§ +m {Wig +7713) 4f2 f3
36
x[- 5 3.3.3.1 (7'1)100(73172)100(r2ar3)
1
+ 2 5m PO(T1)P1(737 7‘2)p1(7‘2, 7 3)
+51000031(7‘337‘2)31(7‘2,7‘3) — §PO(7‘1)30(7‘33T2)"’0(r2’r3)
3 1"” w -y -o —o —o u; —o a
+§€ld 6 ‘7 100(7‘1)Sg(7‘377‘2)30(7‘2,7'3)
1 u w u _. .. -' «2 ~ -'
— 53" e 3 po<r1131<r3m2>sl (23,731] . (9.365)

The contribution from the R—part of the single-exchange piece reads

1 ._. _. _. ._. _. ..
(Mg/Foam) = (29;)2 ;—:2- : /df'1dr2dr3/qufidfie'qz-(rr’l)8333-03—71)
31 B2 71
20. " —0 ' I B2 o_ 7' .7. (-i T ,7"
.z....m.[a<m( ,,. ...3.. 1

‘ ‘ -o —o w —0 —o - 7 —o —o —o —o
+0327.2 6"”71 s’l‘(r3,r1)80(r1,r3) + 63272" 311(7‘3, r1)sg(r1, 3)

 

+ 67172 €Wfi2 85(773: FI)ST(F11 71:3) _ 613272 Cllwflrl 85(F39 F1)S‘ij(lfol 1 F3)
—6‘32"?"331(7‘3,F1)s'1’(7"1,F3) + 6‘3272V.9Y1(F3aFilsmflafis)

‘3. “' 7 -o -o -o -o - _. ... [3 .. _.
— €32'2U.S'01(T3,T1)811/(7'1,T3) - 25713-2 po(7"3,7"1)812(7‘1,7‘3)

264

 

. - _. ._. "1' ... _, . - _, .., 13‘ _. _.
_ 2 0.33272 p0(7‘3, 7'1)311 (711’ 7‘3) + 107172 p103: 7'1)302(7'1, 7.3)

- ‘ —o —-o 7 —o —o - -o —o ‘3 —o —o
—‘10i32—32 {)1("337'1)301(7'1~7'3) — 153,132P0U13730312035371)
- ‘ —o —o A! —o —-o - ‘ —o -o 3 —o —0
"10332132 p0("‘137'3)311("3- 7'1) + '1 07132 p1(r1,r3)302(r3,r1)
- —o —-o A! a —o ' "f —-0
— '2 (5,3272 p1(r1, r3)301(r3, 71)) — (”71/31 3'1’(1‘1)

33....7_... 3....‘3'.._.
(— 812(7‘3, 7‘2)812(T2, T3) + 362(7'3: T2)'912(r23 TB)

+ 513203, 7_"3)""32(f"33 F2) — 5,3272 35033 7723111072373)

+31 5117232 p0(7"3, F2)s’l‘(F2, F3) — 2'6”??? p1(f'3, F2)35(7"2, F3)
+ i e“72"32 p0(7"'2, F3)8‘1‘(7_‘°33 52) — “”7232 P1 (71:23 7.'3)'~‘>‘ii(7-'33 F2)
—533232 /)0(7_':237T3)301(7-"33772) + 53232 Po("-"33F2)P1(F23773) ] -

(9.366)

The contribution of the R-part of the single-exchange piece vanishes for both spin—

unpolarized INM/PNM and time-reversal invariant systems.

Double-exchange part

The contribution from the D-like piece of the double-exchange part reads

HF,CD,2
(V333 x)

2
1 1 ~-o co -o ---o :0 --0
(9A) [df'ldfizdig / qu‘gdq‘g 6'02-(r2-T1)c'03-('3-’1)
7T

 

 

‘27; IE
5 7 2
(12(13 Clmw C3 —o ...
. . - +2— « ~(-
(33 +m£><q§ +3333) i f3? f3” “i

X [ 353573 100(F23 F1)P0(7:33 F2)P0(F1~. F3)
— 5m [10(7123 ‘F1)/)1(7_:33 fi2)/)1(7-"137-‘3)
+ 3 00(T2- 7‘1)"i)(7‘33 7‘2)33("13 7'3)

— 3613116va [100-323 7‘"! )85(F33 72:2)3ii(f013 F3)

265

— [)0(F2~ F1)8’;1’(7—"3,F2)8Y(F12 72:5)
+6"”"6 “7"po(F22F1)-9’ {(F52F2)8‘1’(F1e7-‘3)
+6/)0(F_.r 3.)s3(72 7751)‘ ,"(F3, F2)
— 3o 2, pm m~2§< “'2 F1)83(F?s-. 62)
+ 3022/10(F:s, F2) (3(F2, F1)b 3071: F3)
- 2p1(F1,F3)-9g(722 7'1)SY("'32T2)
+ a p.102,63)s€(62,ms’1(7'3, 62)
- 5,37 p1(F32 F2)-93(F2e F1)SY(F1.» 7T3)
+ 35,31 Puffs, 72.1091075: 7701010712 71'3)
— 5,32 Puma F3)P1(F2v 7_"1)701(7-"'32 F2)
+ 3p1('F2,F1)-9€(F32 F2)SY(F1, F3)
— 36“”"6W" 7010722 72.0533013, 7:93:10???“
— p1(F2, F1).9'13(F3,F2)33(F1, F3)
+ EWUEWU p1(F2,T-"1)8’1‘(7-‘332 F2)SBJ(F12 73)
— 2 p0(-F1, F3).s"13(F2, F1)31 (F3, F2)
+ 63., p0(F1, F3)si’(7'2, F1)s}'(7'~’3, 72)
+ 35132 P0073, F2)8§3(F22 F0319}, F3)
+ 6p1(F1,F3)8'?(F22F1)83(F32F2)
_ 3d37 p1 (F1, F3)sf(F2, F1)83(F3, F2)
- 5,32 .01 (F3, £2)5i3(F22 F1)33(F17 7723)
+13 [901(r1,r3)s02(72,r1)so3(r3,7‘2)
— % sf1(F1, F3)552(F2,F1)S’113(F32 F2)
+ 37162.63)s:‘2(62,61)s$3<m62>
1 #1 _.

--30 (71,7:3)8#2(7‘2271)313(F-‘59'F2)J

3

266

 

 

X [_Ep3plp4(Li/147262,;72V + €;3/13;146;42;r4u€7p1u __ €13p3u4 6114;11u6'w2u

_ 6.7/13Il46/11114V6311211] ] . (9367)
For symmetric INM, this reduces to

2
.‘( v, 1 _, __, _, 1 —o —o 1" 1“. —F I... F —F
(lGESCD'Z‘JM’) = (#29?) 1—6[drldmdrg/-——(27r)6dq2dq3€q2-(2 1)(/1343 1)
7r

13 “r 2
x —4—,—- + 2—(12-q3]
(2% + mime? + mi) f: f3

x [35732PolF1,FilP0(F22F1)PO(7-:3,73)]2 (9.368)

 

while for unpolarized PNM, one has

2
‘ , 1 —' —o —o 1 4 —-0 '_° _' -_. 'F 7 _-'
(VJLF‘CDJX‘PNM) = (Ty?) -1_6 /d7'1(17‘2dr3 f—(2 )sdq2dq3 e'q2'lr2 ’lle"13-(’3 r1)
1r 7f
7 2
(1503 €1m1r 63.. ..
. -4 +2—. ' .
(22+m2)(q§+m3,>l f3 f£q2 “l

X I: 36137 7012(771, 71:3)pn(7?2a Fl)pn(773$ 772)] ° (9-369)

 

 

For time-reversal invariant systems, the expression given in Eq.(9.367) can be simpli-
fied only slightly. The reason is the phase factor eii2'lf2"F1)ei§3'(F3‘Fll prevents one
from treating F1 on equal footing as F2 and F3, i.e. even though one can interchange
F2 and F3 and recover the same expression, the same can not be said of F1 and F2 or F1
and F3. This is further compounded by the fact that the HF energy of the C-term’s
double exchange involves invariably three non-local scalar / vector densities. Another
simple interpretation of this is, most if not all of the available symmetries in the co-
ordinates have already been utilized in Eqs. (7.13)-(7.15). Hence, the corresponding

expression reads

 

‘2
1 1 .2 2 2 .2 2 2
Vl‘if2CD.2X.TRl : 9.4 _ [die (1F (117 / (Ii-rd“ ezq2.(r2—TI)GIQ3.(T3-r1)
( 33. ) 2f 16 1 2 .3 ___—(2mg 12 (Is

267

 

I3 "7' '2
(1.2 (13 01777.7r c3 _, _,
>< . —'—++2—qz-qa
(61-3 + "1%)(73 +7712?) f: f3

 

X 35,132 90(7-‘22 F1)P0(F3-, 772)!)0071: F3)

— 6,22 [7002, F1)p1(F32 F2)p1(F1~ Fa)

+ 9 po(Fz. Fi)b€(Fs2 F2)33(F1« 71’3)

_ gfuBVEW p003, 7‘} 223073, @2301, F2)
— 3po(F2,F1)29?(F32F2)5i(7712F3)

+ €pl3l/6w‘7-y poll-".27 F1)8’1‘(F3, F2)s‘f(F1, F3)
— 35737 700071, F3)Sg(F22 7-"l)*9il(7?3v F2)

+ 35237 po(F3, 7"*2)-""g(7?22 7‘31)330‘9 773)

— 2p1(F1, F2)SB(F22 F1)8l(F3» F2)

+ 6237 p1(F1, 508372, Fll3il'F3w 7:72)

- 5m P1('F3, F2)33(F22F1)3i(fivfii)

+ 357% P0073: F2)p1 (F22 F1)p1 (F1? F3)

- ,32 pow—"12F3)P1(F22F1)P1(F3aF2)

+ 3p1('F2,F1)Sg(F32 F2)3i(7712 7:23)

— 36"5"6“’7" p1(F2, F1)86‘(F3,772)3‘f(7712f3)

-701(‘F22F1)~9?(F32F2)83(F1,F3)

+ 6 P1071,773)3113(‘F2a771)37(7_‘32F2)
— 35,32 p1(F],fi,)s’13(F2,F1)83(F3,F2)

—o —‘ ,3 -o -o ' -o -o
— 5,232 [)1(7‘3-. 7‘2)31 (7'22 T1)83(7'12 7'3)

268

 

"L L.

 

+ 2:3 ( 1m 12323202, 77088302 F2)
_ % S[11101371090202 F1)5’1‘3(1‘{,f72)
+.2;‘1(a,>-:,>.:‘2(r2 22233222»
$an 23> "232 ml- <>>

X [_ffl3fllu463p4l/E‘,;12u + 6t3I‘3H46l‘2M4l/67p1u

_6511373467147111262'73231 __ €7'u371467t1714vfx331213] . (9.370)

The contribution from the R-part of the double-exchange of the C-term is composed

of

(‘éfllvF,CR,2X> : <%fLF,CR1.2X> + (%}kF,CR2,2X) + (M’IIJVF.CR3,2X)+ (v3HF “CR4 2X) (9.371)

VHF.CR1,‘2X> <‘/HF,CR‘2.2x), (VlllflCR3,2x>, and (VHF,CR4.2x

where ( 3N 3N M M ). Denoting

CR1 E (v3lIIVF,CI{l,2X> ,

CR2 E (VHF CR2 2x) ,
CR3 E (V3 VHF CR3 2X) ,
CR4 E <%ILF,CR4.2X> ,
these are given by
922527 1
CR1 = (2—fn)fi It: 8 fd‘FldFQd’FEs fwddlzd‘zs 82(12‘0'2—7'1)e"’3'("3-r1)

'72
Q21 Q22 (131 Q3
((1% + "7%)(613 + mi)

3 7
x l6 1 “‘1 (”5132726712313 + 6’321‘2672l‘3 + 652%672’9)

 

269

, we have

CR2

x [ 67317172 (6

It _.I‘..—.II _. H ..II 31174..
(911(12.I‘1)sl‘2(7‘3 72)S‘03(Il.7'3) — 811(7‘,2 r1)902(13 7'2)31‘7(r1,7'3))
1 A,‘ ‘ I —o —o —o --o —o —o —o —o —o —o
+ 677 ‘1”1 011232 3111(7‘217‘1)(/)1(7‘3-. 7'2)/)0(7'197‘3) ‘ P0031 7'2),01(7‘1>7'3))

' 1' ‘1' .‘c‘W t It -' -0 H
4.115311771632272 (311(7'2-7‘1)512(71 7‘3)100(7'3 7‘2)

I‘ —o —o H I1 —o —0 fl —0 —' -o -'
‘311(7'2e7‘1)- 902(7711 7'3)P1(7"3> 7"2) + 511(T2>Tl)'912(r32r2)/70(T1=r13)

— 371710217‘1)902(7_'3>7‘72)p1(77’1>773)> ] - (57-372)

2 .
C 2 -' -o d 1 1 r r 1 r r
(g;—:) f—;8 fdrldrgdr3/(——2W)6dq2dq3602°(72‘ ~1)€q3'(3- 1)

B 7 ‘7
(12B1q22qglq.2

((12 +777 720013 +7772 7r)

fl —0 —o I1 -o -o [1‘ —~o -o
6I72fl3 _ 6132;136’72172) 511(7‘39T2)502(T21771)313(T11T3)

 

13272
13
'7' 6 171711 ('6'3272ép2“3 + 6’32”2672”3 _ 652“3672"2)
X8710] 7'3)90 2(7'217'1)913(773 T2)
3
+ 6 171711 (—6332.326“2“3 + 6fi2p2672113 + 632I136‘72772)
X31103, 7‘2)312(7'2> 717303011 T3)
fl
+ 6 171711 (5132726112113 - 6732772572773 + 6527‘36727‘2)
X 90101,T3)912(72171)313(T31T2)

- fl -. -. I1
— 63171“10132u26'12#3 301(7"3> 7‘2)312(7"2> 7"1)31 (7711773)
+if‘131'711/EB2’Y2V p1('I-::33 F2) p1(F2-> Fl )/)0('Fla E3) — p0(F21F1)/)1(F11 773))
5,7 l [J] _, _, _. .. ... ._.
+5 171 1033 2.72 (31 (7‘1, r3)p0('r2,1‘1)p1(7‘39 7‘2)
#1 a -0 -0 4 "’ "'
—81 (7'317'2)P0(7'2>7“1)Pl(7'1>7'3)

+37171077317:72)p1(‘7?2>771)00(7—‘71>7‘73) — 901(7'1 7“3)Pl(’"2 r1)p1(r3,r2))

+i€.{31’71171(63212796311131 + 6152736322,, _ 62327363211 — 6172771652,?)

270

CR3

11 _, _, V _. _. _. —.
x 311(7‘3, r2)812(7'12 7'3)P0("‘22 7'1)
_ . 63171316323232 2’51 (F2, F2)st,‘3(F1. 71:32.32. F1)
— l fi'31771u63'32;11u (— 831(F3F2)8:2(F1, 'FEg)p1(F'2,f'1)
+ 30 (’1, 72031 (7‘3, 7'2)/)1 (T21 7'1)

[1 '7 l1 4 —o '7 —o —o —o —o
‘311(7 7'3 72) 02(72 7‘1)PI(7‘1~7‘3) + 311(7‘227‘1)812(7'127100003272)

—sll(r",2 r1).s(,2(r1, -r )p1(r3.r2) + 811(7‘3.7“2)812(7‘227‘1)po(7‘127"3))

— z' 65171"672”1U (32121 (7‘3. 772%? (771» 7:3)701 (772» F1)

52

u .. .. _. .. _. ..
+311(7"127'3)30 (T2,r1)p1(r3,7‘2)

#1 _. _. _. _. _. _. u 13
+Sol(7‘1,72)~9f2(73272)pl(71273) — 501(71273)912(7°227‘1)101(7"327‘2)

l. 3 _, _, __, _,
—3:10—22F1)912(7‘32T2)Po(7‘127‘3))J- (9.373)

2 . ‘
C I 1 ,4 .. _. ._. _. ~
(29—__fA)B f128 fdfldfédfii f_(27r)6d(72(1625€"12‘(r2—r1)6103-(7‘3—1'1)
1r

’ *1 “r
(15 (12 (131712

(92 + m12rXQ3 '7‘ mgr)

x [2313131 (652.726

 

#2733 _ 5772732532733 — 632773532772)

1 p _. _. _. _.
x (331(72,r1)812(r3,T2)S‘;3(7‘1~7‘3)

[‘1 -o H _I‘ -o 1 -o l7 -o -0
—3so (12.232332 :2) 3(F1. F2.s>+2-1(r2.r1>’3<u.r2>sg3< 2, >)

+ 63171“ 5.3222 351522771) (3P0(F327::2)PO(F12 F33) " 91(fi33'F2)pl(F13F3))
_ 63131771 53272 s’:1(f'2,7"'1)2p1(7:§5,72:2)po(7_"1272:3)

+ 2‘ 5731717‘15‘32727‘2 ( — 3. ”1 (12 r1)sg2(71,F3)p1)(fi32F2)

+ 301(1‘2 77'31):2(F1,fi3)p1(173,f'2)

[12(

+3801(7‘2,T1)«S‘02(7‘3272)p0(7"1,7‘3) — 80102270817327‘2)Pl(7‘1 7‘3)

271

CR4

2 .
9A 047 _._._, l _‘dw «_q ‘.- ?_~
dr dr (17‘ —d d- 33232 317633-33 ’17
(an) f2 8/ l 2 3/(27T)6 (I2 (138
B '1 '7
(12 fi1‘12243193 2

((12 + m%)(q3 + mi)

3
[ 6 171"1 (—652725,12,13 + 5132172672113 + 5132173672‘9)

 

X (3 3:1(F3, F2)sg2 (F2, F1).933(F1, F3)

—3T1(F3,F2)532(F 72 F1)513(F12F3) _ 2311(F3,F2)s;‘2(F2,F1).933(F1,F;))
+ 533171“ (532225112113 — 53217257233 + 57723363272)

x (3 331(771, F3).9$2(F2, F1)33‘3(F’3. F2)
—-sin(Fl,F3)sg2(F2,F1)81l3(F32F2) — 2331(F12F3)3u2(F22F1)313(F32F2))
+ 66171”155272(- 3 831(F12 F3)PO(F22 F1)P0(F32 F2)

+ 3 3310732 F2)PO(F22 F1).00(F1: F3)

+311(F1,F3)p0(F2,F1)p1(F3, F2) - 911(F3,F2)p0(F2,F1)p1(F1,F3)

— 2 3:10:32 F2)P1(F22 F1)PO(F12 F3) + 2 331(F12F3)P1(F22 F1)p1(F3, FD)
+ z’ 61317112652721» (@0032,F1)p0(F1,F3)po(F3,F2)

— 2 p(1(F12F3)/11(F32F2)P1(F2a F1)

—Po(F2~F1)P1(F32F2)/71(F1~'33))

+ 22313131632327? (31‘103,F2)s-’1"2(F1,fis)po(v‘2.fi)

— 3331073, F2)Sg2(F1,F3)po(F22F1)

272

I" —o —o I”! —I —0 -' -*
+ 2311(7‘3, r2)302(7‘1.7‘3)101(7°2a 7‘1) )

_ If 5731311’5327‘1" ( — 5.7171(771, 7‘35)S:2(F32F2)p0(F22F1)
+ 3 831 (FL @532 (F3: 772)P0(F2v F1)

— 2 531071, F3)s:2(7_‘§;, F2)P1(F22 F1)

— 3 331(1‘3, Fs)832(F22F1)p0(F3aF2)
+3‘1‘1(-F1,733;)332(F2,F1)p1(F3,F2)

+ 3 331 (F3, F2).932(F2, F1)P0(F12 F3)

— 371073, F2)832(F22 F1)/)1(F1, F3)

—287102,F2)si’3(F2.F1)po<F1.F3))

 

_ 2' 6731313672771" (_ 5’1‘1 (7‘3, 172).??? (F1, F3)po(Fz. F1)
+ 3 .931 (F3, F2)sg2(F1, F3)P0(F22 F1)

— 2 5311073, F2)832(F12F3)01(F22 F1)

— 3 351072, F1)832(F12 F3)p0(F3, F2)

+ 5310-32,F1)s‘f2(F‘1,F'2)p1(F2.F2)

— 3 3:1 (F2, 130332 (F3, F2)100(F12 F3)

+ 531(F2,F1)sf2(F3,F2)P1(F1, F3)

+ 2 3‘1‘1(F2,F1)sg2(F1,F3)p1(F32F2)

+ 2 331(F1,F3)sf2(F2,F1)p1(F32F2)

+ 28‘1”(F2,F1)sf2(f?s,7"2)po(f’1,fia))] . (9375)

For symmetric IN M, this reduces to

'2
. :' ‘ 7 1 3 _. —O _' 1 ... _.
(W12¥'("R'ZX’INM> = _ (5%) $55; [(171617‘2d'r3 / qumqa

273

3
‘1 7’2 31 32
x(2772477241)(3373-53-31) ‘72 ‘12 Q3 713

(Q2 + "1222)(Q3 + mgr)

x [#3171" 63272” 00(F12 F3)/)0(F22 F1 WOO—‘32 772)] 2

 

 

(9.376)
while for unpolarized PNM, one has
(LSHFWCRhPNM) = .Q_A 204 3 fair dr (1F [31/1 —dq dq
2_f2 f28 123 69:36 23
x 8122.(F2—F1),2q3.(F3—F1) Q21Q22Q31Q32
‘ (q; +m3><q§ +m3.)
X [73 63171126322212 Pn(F1,F3)Pn(F22F1)Pn(F32F2)] -
(9.377)

For time-reversal invariant systems, there is no appreciable reduction in the size

of the expressions. Thus, we avoid repeating the expressions.

C-term contribution for specific systems

Combining the expressions for the direct, single and double-exchanges for the case of
symmetric INM, the contribution of the C-term to the HF energy of symmetric INM

becomes

2f,r 16

(12517131 clm2 or;
x —4 F + 2411—2 (73)
((1% + 777?.)(713 + 7733.) i( f3 f2?

X (— 6 (53121 p0(F1)/)0(F3e F2)/90(F2~ F3)

2
1
(VHFCJNM) = (94‘) — fdfingdF3/(——27r1)6dq"2(1q3e<12-(F2—F1) iq3(F3— r1)

 

 

+ 3 6,3121[70(F12F3)P0(F2~F1)P0(F32F2))

C. ' V ‘ 18 ‘ _, _, _, _, _, _.
— (if—é (”171 6 3272" Q22in2 P(1(7‘1~.7‘3)/)0(7‘227‘1)Po("“3»7‘2) ] - (9.378)

71'

274

The contribution of the C-term to the HF energy of unpolarized PN M becomes

 

2
. r - 1 1 .2 7 ~ -~ ~ .7
(VHF,C,PIVM = 9A _ dFld'f'gdfij '—d(i'2(l(f3 81(12.(12-—r1) ezq3.(r3—I1)
3” 2f 16 (27:)6

le Q31 Ci 777-2 C3
x . , —4 , 7' + 2—' 7’ ~65)
((1% +m£)(q§ +7723.) ll f: 13‘” ’

X (— 6 (55171Pn(F1)Prz(F3a71:2)0710727773)

 

 

+ 3 6,3171 1011013, F3)pn(F2$ F1)pn(F39 772))

(:4 r . 7 -o —o -9 —o --0 —¢
“* 6‘f—2 661711163272” Q262Q32 Pn(7‘1,7‘3)Pn(7‘2, T1)Pn(7"3, 7'2) J - (9-379)
7|"
For time-reversal invariant systems, the HF energy contribution from the C-term
is given by the sum of the direct, single and double—exchanges. Since there is no
appreciable reduction in the complexity of the expressions even after the assumption

of time-reversal invariance, the expressions are not reproduced here.

9.10 Symbolic derivation of EDF-NNN-DME for

time-reversal invariance

In this section, we discuss some of the ingredients of the symbolic derivation of EDF-
NNN-DME, i.e. the EDF that we obtain after the application of the DME to the
HF energy from chiral EFT N NN interaction at N2LO. The derivation is performed
for time-reversal invariant systems. This should be kept in mind in subsequent dis-
cussions. In the final step which involves angular integrations, we assume spherical
symmetry, which also implies time-reversal invariance. The symbolic steps required
to relax this assumption and treat deformed time-reversal invariant systems is dis-
cussed in appendix 9.10.5. For the complete detail of the symbolic derivation, refer
to Ref. [161].

275

9.10.1 Generic DME ansatz

In section 7.2.4, we discussed the merits of the symbolic derivation of EDF-NNN—DME
starting from a generic DME ansatz. Here, we develop the generic ansatz for the basic
nonlocal densities (”1071,73 + :33), (”2(17'1 + Jig, F1) and (”3 (F1 + 553, F1 + :32). These
non—local densities can be scalar/ vector, isoscalar/isovector. The basic objective of
the DME ansatz is to approximate each of these non-local densities in terms of local

densities, dependent on F1, and 7r—functions which can depend on 52 and/or :33.

In developing the DME ansatz for these nonlocal densities, we follow the same
scheme as in section 5.3.6 Where we relied on short-range / Taylor series expansion of
the nonlocal densities. However, the formal expansion that is given in appendix 9.5.7
in relation to the application of the DME to the HF energy from NN interactions
cannot be used. With the aim of implementing the symbolic machinery in the most
general way, the DME ansatz we need at this point should be general and complex
enough so that it can be adopted to different special cases. For instance, for the
analytical coupling calculation discussed in section 9.11, we adopted the generalized
PSA-DME. This will have the benefit of minimizing the effort required to implement
a different approximation of the nonlocal densities and obtain a new EDF (perhaps

both in form and couplings).

Imagine one wants to build an expansion scheme for §“1(F1,f'1 + :33). In this case,
we have a single nonlocality coordinate, £3. The simplest route to a possible ansatz

is one that applies Taylor series to the density about 1"]

(“1(F1,7"'1+f3) ~ alum +F.s-v'<"1<F1.F')

 

 

276

 

 

where we truncated the expansion at second order, and hence we generalize it to

(“W—"1.771 +373) 1’ “<1’O(Q)C"1(Fi) + II“1‘1(Q)2‘7'3 -F’<"1(F1,F')

 

2
+ $119262) (f3 - 6') (.411 (f1)

 

Needless to say, the n—functions and Q in Eqs. (9.381) are to be fixed by the
choosen analytical / parameterized DME scheme. But, this route does not lead to the

most general form of the ansatz. Rather, we follow a circuitous way which leads to a

 

general ansatz with which Eq. (9.381), for example, can be recovered as a special case.
Note that if we write ("1(F1,F1+ 553) as g"1 (Ff+ 172, if — Fri/2), where if = F+1/2a':'3
and 7" = —:i"3, we can apply the DME derived in section 5.3. In order to arrive at a
final expression which is separable in F1 and :33, we perform a short-range expansion
(Taylor series) and truncate beyond second-order terms. Performing these set of steps

yields

_, _, _, ‘ ,0 , q, 1 ,0 _, _. _,
§”1(T‘1, 7'1 + (133) 2’ Hgl(k173) [77811 (k133)§‘ 1 (7'1) + 5 Wfl (k1133).’133 ' ng‘l (7'1)

1 #11)

2
+-8- 7r2 (kzr3)(:i:'3 - V) g“1(f'1):|
, U l .1 ,u _,
— 2Hi1(k:r3)$31[7r31 (max? 1m)
1 ,1 _‘ -' 1 ,u _,
5 will (le3).l'3 -V<i1 1(r1)
1 p1,] _, " 2 [11.1/1 _,
+g7l’2 (kI3) 13V (1 (7‘1)

1
+ ~22 H31 (A7173)3?§<;1 (F1) 1 (9382)

and the same form holds for (“1(1-“'1. F1 + :32) after replacing £3 with :32. While

(“2(F1+f2.771) = (’12‘(T1.7_"1+372) :

<"2(F1 + 4173;771) = (“2*(FLF1 +53) , (9.383)

by time-reversal invariance. Hence, the corresponding ansatz is the same as the one
in Eq.(9.382) as all quantities are. real for the case of time-reversal invariant systems.
("3071 + :33, F1 + 552) has two nonlocality coordinates. Still, the set of steps applied to

arrive at Eq. (9.382) can be applied to obtain

—0 —-o —o —o l' —o —o l —o 1 —o —o a —o
§”3(T1 + $3,T1+ 1’2) ’1 Hair“, 1132,13) [C‘3(7‘1) + 5(132 + 1133) 'VCu3(7‘1)
1 -. 2
+ g ((552 + 53) ' V) (”3(F1)]
- l‘ —+ —0 U V
_ 211131-(k7 $211.3) ($21 _ $31)

—0

v” w 1 —o -0 3V -4
X [C113 1(7‘1) + ‘2‘ (1132 +333) ’ V§i13 1(7'1)
1 _, _, -* 2 “3,111 _,
+§ ($2 + 5173) ' V <1 (7'1)

It -0 -0 —o —-o p a
H23,.(k,a:2,1:3) (.173 + 11:3 — 21:2 - 2:3) (23(7'1).

(9.384)

1
24

Unlike the 7r—functions discussed so far, 11:13,. are not manifestly separable in fig
and 533 and in fact depend on the relative orientation of the two vectors. This has
an important implication for the analytical calculation of the couplings. Refer to

section 9.11.

Key points on the DME ansatz

One notes that in the generic DME expansions given in Eqs.(9.382)-(9.384), each non-
local density can be either pan or 5'0/1. To recap the notations used in the Eq.(9.382)-
(9.384), for example, qfl’ul refers to the local cartesian tensor spin—current density
that results at first order gradient with respect to the relative gradient operator. To

elaborate on this point, consider the ansatz as applied to p0(F1,F1+;rT;.;). The resulting

278

 

Iqt

mp— ...

 

expansion reads

_. 1 s ~ ~
Half-"33) [7T8(k.173)p0(7‘1) + — “flit-Falls ' VPOU‘I)

P()(F12F1 + ’33) "-1“ 2
1 _. 2
+3 war-rm (a - v) pow]
V "-‘ _V _. 1 7 .., _’ '1! —¢
_ .: II’f(k:c3) x31 [flammabilw + 5 ni<kar3>x3 - V] 1th)
1 . ~ 2 u
1 p 2 “1 -' 9 385
+ i] H2(k1‘3)$3 (2 (T1)- ( ' )

Hence, in the case where g“1 2 po /1, then (ill'ul = y'all, i.e. the current density. Since
we are dealing with time-reversal invariant systems, 3.0/1 : 6. However, we did not set
it to zero in Eqs. (9.382)-(9.383) as we wrote the equation to hold for both scalar and
vector densities, and when ("I = Fig/1, then cf 1~”1 = J0/1~I‘1"1’ i.e. cartesian tensor
spin-current density. As to €51 in Eq (9.385) and in general, it refers to a second
order correction term in the expansion. It is analogous3 to iAp — T + 3/ 5 16% p of the

DME discussed in section 5.3.3. Obviously, €51 = O in the case where c” 1 = 5'0/1. The

same notation applies to (”2 and (”3.

Even though all notations and conventions have been explicitly given in Ta-

ble 1.2, we recap the ones used for the 7r-functions. Taking (”1 as an example,

.0 .0 ,0 . .
#51 mill ,wgl refer to the 7r—functlons of the local part of the non—local densrty

,1 .1 ,1 .
(“1 and n31 ,7r’1‘1 ,ngl refer to the zeroth, first and second 7r—functlons for the

local density that appears at first order with respect to the relative gradient opera-

” Q g

tor. Hence, these 7r—functions are equal to 7T6), 71f, n5, #3, mi, 71'; or 7r3, 7rf, n3, 7rd, 7r1J , 71";

when §“1 = 100/1 or §“2 = .§'0/1 respectively. The example given in Eq.(9.385) il-

 

3It does not mean they are the same.

279

 

lustrates this statement explicitly. Since we are dealing with time-reversal invariant

~ — ~ - r - . u- 1r
systems, 713,7ri',7r§,7r3.71{.7r§ are Irrelevant. HO’ and I‘ll2 can be seen as some common

. ll.“ . . .
prefactors while H2’ is the 7r—function of the second order correction.

Comments on the DME ansatz

The following comments are at play concerning the DME ansatz: (i) The ansatz
is designed to be a general template on which all the known (and perhaps future)
analytical/parameterized DMEs can be mapped. This can be done by setting the
various n—functions to the values dictated by the analytical/parameterized DME at
hand. This allows for a minimal effort to adopt the symbolic machinery to specific
cases. (ii) As mentioned at the beginning of this section, any discrepancy between the
EDF that results after the application of the DME, and the exact NNN HF energy
is solely due to the DME ansatz of the nonlocal densities and the n—functions. This
is a trivial statement in the case of two-body interactions. However, it is not so for
three-body interactions as unless one makes a convenient coordinate choice, it is not
trivial to treat even the non-DME part exactly. Thus, by improving the n—functions,

one can hope to get better and better accuracy.

9.10.2 The G-tensors and their analytical forms

In sections 721- 7.2.1, we identified the three generic forms (Eqs. (9.406)-(7.2.1)) of
the terms that occur in the HF energy from the chiral EFT NNN interaction at N2LO.
We refer to the interaction form factors that enter these equations as G-tensors. They

are of the form

51 11 B2 72

Glfll'yli3272($2,$3,q2’(137w) E [(19 d9 eid’2i2eirf3f3 Q2 Q3 (12 (I3 ,
(’2 Q3 (<13 + ""%)(q§ + mi)

(9.386)

 

280

 

 

13 7'
'* r '1 ”- (1293
03703,.173,q-2,(13.w) E [(19, (I9. e'q2'126'q3'r3 . ,
* ’2 “3 ((15 + "18%)(93 +7713.)
(9.387)
qflql
an,» 1.. _ , i“? .1 3 3
C11 (1'3,(13) : [(lflq3 (3 ((31.13 m. (9.388)
L 71’

where (19,12 and qu3 refer to the differential solid angles of the two vectors, and
w is the angle between :32 and 173. In the following, we derive the analytical forms
of these tensors. Indices 51,71, 132,72 can take values {1,2,3} corresponding to the
cartesian labels {.1:, y, 2}. To obtain the analytical form for 03171527261», 11:3, q2, q3, w),

we define

m -: 2'31}: (1697
F (:L,q) dflqe -———— (9.389)

(12+m73'

In the case where the vector 55 is along the 2? direction, we denote the F ”(ii, q) tensor

as F ffli’, q), which equals

,Q‘ 5 3
Ff‘r(;i:',q) = 5),, /(que'q"'-q—q—q—. (9.390)

2 + 771.3,

Denoting F,11 E Ff‘,Ff2 E Fzyy,Ff3 E Ff“, we obtain

 

 

 

2 .
q 31(qrr)
. = —4 , 9.391
F. (liq) 7rq2+m72r (gm) ( )
2 .
V (I 310151?)
FW .. = —4 9.392
2 .
. q . J1(Q~’L‘)
1W~ . = —— 2 .. —4 . 9.393
z ($.61) (12 +771; ( Mom) 7r (qr) ) ( )

Next, we need to define a convenient coordinate system. If one could define both 5132
and 5:3 to be along the 2 direction of the coordinate system, the G-tensor could be
calculated very easily. But in the actual case, one cannot, in general, define both 932

and 5:3 to be along the :2 direction. Let :32 be such that it is along the 2 direction and

281

.73 be in the .1: — 2 plane. with angle (12 between .82 and 1'3. Next, write

Gdlll'd2l2(332e$3a(12103-93) = F1‘31*~’2(f2,qe)F7172(f3,q3)

= 651132 F 5131 (4132, (I2) Fi’17'2(1’39113-,w) » (9394)
where due to the specific coordinate system chosen, the G-tensor will be nonzero
only when {31 = 32. The same cannot be said about 71 and 72. In the next step,
we concentrate on F 71 72(1‘3, q3,w). One can calculate this quantity by performing a
few steps involving rotation of the coordinate system. Hence, rotate the coordinate
system with respect to the y-axis such that 5:3 aligns with the new 2’ axis. The

rotation matrix for this operation reads

cos(w) 0 —sin(w)
R(w) = 0 1 0

sin(w) 0 cos(w)

The transformation of the various terms of F7172 is (obviously only the vectors get

affected)

(1 => RACE,
(9.395)
where we have left out other trivial terms. Using these intermediate results
' . , _ _ —1 —1 , fl “ ,
F7172 (1'37 (13' W) — 6Itll‘2 12111171 (w) RI12'72 (“2) F21 2(1‘3, (13) ' (9‘396)

282

Plugging this result into Eq. (9.394), one obtains the analytical form

131/31

0‘3111‘:’2"2(:r2.:173,(12.93.92) = 5131132R_l (WlR—l (to) F2 (£732,02)F§‘"(-’1332(13)-

ml Mg

(9.397)

To calculate the related but different G tensor, C(37(1*2,173,q2,q3,w), we define

We represent the case where the vector i" is along the z axis by F f (f, q)

4.. 3
F501", 4) = 33,. f 49.6“ q

7
q2 + 771?.

where 63,3 E 623 and representing F,1 E F}, F,2 E F3, F23 E Ff, we obtain

F:(.1:,q) = 0,
Fzy(;l7,q) = 0,
. q .
Fz ;, = 4 ————,—' , .
.(r q) I flq2+m§ 11(93‘)

Plugging this result
Gfi”(:r2,$3,qz,qa.w) = 638 63,.R;,1(w)Ff(:v2,q2) Ff(rvs,q3)—
Finally, we have a trivial tensor G defined in Eq.(9.388) is given by

(137(i’, q) E Ffll(.ii, q).

283

(9.398)

(9.399)

(9.400)
(9.401)
(9.402)

(9.403)

(9.404)

(9.405)

 

9.10.3 Sample DME simplification

After the specification of the DME ansatz and the analytical calculation of the G-
tensors, the next logical step is to plug them into the respective exact HF energy
terms to obtain the EDF. We discuss the DME simpification by considering a term

that has the form of Generic-Form-l. Consider

<V3N 0 0 0) = [drldr2d7'3 €102 (193 (12 93G"1715272(:r2,x3,q2,q3,w)

X 56171 5.6272 90(F11773)PO(F2~F1)PO(F3sF2)1 (9406)

where we recap that w is the angle between :32 and 53. According to the description
given in section 7.2.1, this means, Eq. (9.406) corresponds to a case of Eq. (7.2.1)

where 01 = 0, (:2 = 0 03 = 1 and

C“1(F1,F3) = .00(F12F3)a
(““7732 771) = 90(‘F31F1),
(”30:9, 771) = 100072171) 1
#1151252112 = 6317163272, (9.407)

where due to there being only scalar densities, the tensor #113532 is independent
of {#1, pg, 113}. Refer to section 7.2.1 for the explanation of this notation. Plugging
the DME ansatz discussed in the previous section, and noting that L071) = 0 for
time—reversal invariant systems, one obtains

H F.Cl .p0p0p0

<f/3N ) 2 / dFld-fgdf3 dq2 (1(13 ngg Git-31718272 ($21 $31 (123 (13‘. Cd) 6 6

(3111 3212

_, 1 - .. _,
X (H8091?) [7r5(k1‘2)p0(r1) + § ”flk4172l-l'3 ° V/)0(7‘1)
1 _. 2
+g 7r'§’(l€1‘2)(552 - V) 91071)]

284

 

I t -*
+2—4 HQ] (4"1'2l-1’g‘130‘10

3 1 3 ~ ..
X (H3043) [”g(k173)P0(7‘1) + § 77f(’\’41-’3)-173 ' V100(7'1)
1 ~ 2
+§ 75(k.113)(.i’3-V) [Mill]

+— H: (km) 3301))

(9.408)

where <30?) is the second-order correction density in the DME of the scalar density
with the given coordinates. For instance, the generalized PSA—DME discussed in

appendix 9.5.3 sets this correction density as
2 "" 1 —0 -o 3 2 —o
C0(7'1) = 5 A/)0(7‘1) — 7"001) + g kpp0(7‘1). (9.409)

The expressions given in appendix 9.9 show that we already have a very large number
of terms in the exact HF energy, even for time-reversal invariant systems. The appli-
cation of the DME increases the number of terms by at least an order of magnitude,
further ruling out any hope for manual simplification. The next step in the symbolic
simplification involves angular integrations with respect to the orientation of :32 and

a

1'3 .

Angular integrations for spherical systems

From our sample DME simplification given in Eq. (9.408), it can be seen that we

can perform the angular integrations with respect to the orientations of :32 and 53

285

if we solve one remaining problem. I.e. obtain a separable expression for the 11er
7r—functions that couple :3; and $3. This holds for all 11:3,, 7r—functions used in
the DME ansatz of (”11(771 + 132,51 + 572), given in Eq. (9.384). Within the symbolic

approach, we solve this problem by expanding these problematic 7r—functions as

:I,,.(k.:i-'2,f3) z 2 Z f;~,:'.(ka:2>g:;.i.(kx3) ems)", (9.410)

n max [mnza.c(n)
11:0 m=0

where ,j',f,(k;r2) and gf,‘,¥,,(k;1:3) are unspecified scalar functions and the number of
terms in the inner summation depends on the value of n, expressed as (mmax(n)).
Note that the angular integrations (with respect to 552 and 553) do not require the
values of ,j‘,:,(k:r2) and gfi'f,,(ka:3) to be specified. In addition, the special symbolic
technique that we developed, Ref. [161], helped us avoid specifying mmax(n) which is
not known anyways. As to mum we found that nma,r = 5 suffices in the practical
implementation [161]. In section 9.11.3, we discuss how these scalar functions are
obtained for Ilifr derived from the generalized PSA-DME. In addition, we show that
at nmam Z 5, Eq. (9.410) becomes practically exact for these Tr—functions. Actually,
one can increase rim,t for any Hip if there is a need. The only problem with increasing
um“;c to a much higher value is the rapid increase in the time-complexity of the
symbolic computation. Similar to the application of the DME (ansatz) to the exact

HF ener , the ex ansion of II? introduces et another increase in the number of
gy p z,fr y

terms to simplify.

With this expansion at hand, we are able to perform the angular integrations
with respect to :32 and :33. At this point, due to the complexity of the problem,
we introduce the assumption of spherical symmetry. This implies that all the local
densities depend only on the magnitude of the radial vector. In other words, we now
have only three independent directions: f2, :33 and 7"}. Appendix 9.10.5 discusses how

we can relax this assumption. The generic form of the required angular integrations

286

for the case of spherical symmetry read

/(IQI2dS2I3 (i3 . 6)1(;1‘:2 . i3)1nli‘2 8' i73ln (i2 . 6)!) (5:2 ® fl)"1

"1

(.132 59 73):; (5:3 99 f1):g(.i:3 59 73):: , (9.411)

where 1,771.,72. and p are integers. The maximum power of gradient in the DME ansatz
for any density is fixed at two, hence, I e {0, 1,2} and also pc {0, 1,2}. Due to the
specific form of the ansatz, mc{0,1,2,3,4,5,6,7,8,9, 10} and n6 {0, l, 2}. The ex-
ponents 111, 712,113,714 e{0,1}. Even though the generic angular dependence given in

Eq. (9.411) is very complex, all these terms do not occur at the same time.

The origin of the various angular dependencies is (i) 3.6 and 332.6 are due to the
DME ansatz, (ii) $2.123 originates from the DME ansatz (Eqs. (9.382)-(9.384)), the ro-
tation matrix of the G tensors (Eq. (9.102)) and the expansion of 11:2}, (Eq. (9.410))
(iii) |:i‘:2 ® :i:3l comes from the rotation matrix in the G tensors and (iv) (:22 (8) 73)::
where is {1, 2, 3, 4} originate from the directional coupling in the DME of the vector
density, 32,“. Remember that in spherical symmetry, only the vector component of
the cartesian spin-current tensor density is nonzero. The exponents n1, n2, 71.3 and
714 can not be one at the same time i.e, at most only three of the exponents can be

one at the same time. This is due to the fact that only three local/non-local densities

(be it vector/ scalar) densities appear in all terms of the exact 3NF HF energy.

Due to the huge number of terms generated by the DME expansion, direct multi-
dimensional (four dimensions) angular integrations is both impossible and not re-
quired. Rather we developed a Mathematica rule-based technique to replace the
multi-dimensional integrals with four independent single dimensional integrals. The
merit of this technique is that one can calculate the single dimensional integrals once

and use their stored values in the whole computation. The angular integrations is

287

followed by several symbolic manipulation techniques to obtain the final EDF. The
details of these techniques can be found in Ref. [161]. We remark that the symbolic
automation enabled us to keep all higher-order terms in the final EDF (up to sixth

order) for whenever necessary.

9.10.4 Contributions to EDF-NNN-DME

In the following, we list the EDF, truncated at fourth-order, that results from the
application of the DME on specific contributions to the HF energy (of time-reversal
invariant systems) from chiral EFT NNN interaction at N2LO. Due to the assumption
of spherical symmetry that we imposed in the previous angular integration step, the
given expressions hold only for spherical systems. The actual values of the couplings
as a functional of the w—functions is found in the Mathematica files of Ref. [161].

First let us define the auxiliary quantity (3 /1(7“') as

«kl/10") = "I" ° firm/10*). (9.412)

Also note that Cfi/i (7") stands for the second-order correction density which in the case

of generalized PSA-DME reads
1 3
C0310:.) = 5400/15.) — 70/10:) + 3ka 100/10“). (9413)
Fourth order EDF from the D-term

D .. P3 '3 PO”? 2 ”8‘8 2 2
8 = / dr C. me + 6. pummm + 6. powwow
pi‘fi 2 2 ”OFF? 2
+6’1 101071900) + Cl 90(7‘11110'79‘10’)
J2 .. .. _. _.
+Cf0 0 Pom J09“) ' Jom + CfIJOJl 10107) J00?) -J1(r”)

288

J2 ._. .4 I. ’ d a ‘
+ch 1pm .1199 - .119?) + 6.wa pom [v - Jomli

+ci’0"°’“° pom J13<a - 4.7m + 63’1“") pm Jim - Mom
.000") W - Jim]2
+cfl"°““’1 pm 4399 - Aim + cf"’1“’1 pom J’m - 412(9) '

pIVJOVJl

+CI

pUVJlVJI

pmv Jo(7"')V J10") + C

(9.414)

Fourth order EDF from the single-exchange piece of the D-like part of the

C-term

600’” = / czF{c3’3p3<a + C30 23209912199 + c3°’l‘l(ap1(a<.(a
+C3’OV”°‘V”° pm 6pm - 6pm
+c3’0W1'V’1 pom Wm - 6pm
+6?” p0 0121(5) Apom + 09091 Apl 900910109 A1010")
+3331» 3933(3) + 63"”1‘1p empmcfm
+c3’0‘O [239939) + c3031) om Mia-Jam

+c3’0" 2 mafia) Jim +C3’0‘”°""0 409991-34me

+c3°’°“° pom Jam 4sz + 63°V’1V’1pom [v- J3m1’
+C3O’1A’1pom Jim - Aim} . (9415)

Fourth order EDF from the double-exchange piece of the D-like part of
the C-term

CD2 -+ ”(3) 3 90”? 90911
s) J = [Me paw-9+0», W(flpim+c maplmcl‘m

2A
+6.30 ’0 p3(7")A/J0(F) +6.31’0p3m 4pm

289

+Cf’001 p1{)0(f)p1(7—)Ap1(:) + C3 p3 (”100—791(4)

+6322 p3<z7<33(17+6.36-”°’2’2 pomp3m<3m

p21

+63’22‘O p3m<3m + 632° p3m<3m
. 12 —o ._. _. _.
+C30 0 1000:.) J00?) ”10(7) + C91 “0 1P1('F)J0(F)°J1(F)
2 .3 .3 A J2 .. ..
+630’2p 3m J3m - J3m + 63 ’0 22 mm Jom - Jom
2 J2 12

+630 0 3m Jim Jim + 6322 2 3m Jim Jim
+ 63’2’2‘7’2 933m - Jim V7 - Jim
+ 632V’2V’2 pom [<7 - Jim1’
+632’2A’0p 3m Jim AJim + 632 2’ 2 32m Jim Jim
+6322J0’2Ap3(v7 Jim Jim + 632 “02 397 Jim Jim
+6322“0 6mm - Jim 6 - Jim
+632 p3mJim-AJim
333202? 433mm - Jim 3 3:32? 2m Jim - Jim
+630 22 397 Jim Jim + 6322’2V’2 W m Jimv Jim
““0"“ mm 6 3 Jim 6 - Jim
+ 6337222’2‘”1 mm . Jim 9 - Jim
pom [v Jim]’+ mm Jim - AJim
+ 63’2““2 pom Jim - AJim } . (9.416)

JlAJO

+C3

0VJ1VJ1

p JAJ
+C3 C310 1

Fourth order EDF from the Rl-double-exchange piece of the C-term

2

_. .3. J _. _.
62222 = / 672(632’2’2 p3m J3m - M) + 63’0 2 33m J3m - J3m
sz .3 _. _. _.
+632 ”J1(J7-J3mV-J3m
+6322“1 Jim - Jim 6 - Jim

290

+C;':"‘2J2 332m Jim - Jim + 33212021 mm Jim - Jim
+6 2 2 2 2m Jim- Jim

+6.3J”2J2"JJ0 \inm - Jim 6 3 Jim

+632J2‘“0 33m Jim - AJim + 63”“? Apim Jim - Jim
33:32? 33m Jim 3 Jim + 3:623 33m Jim - Jim
+63722J°VJ2 633m - Jim <7 - Jim

PIG—J) {7- 311.7(7‘76- 31-10:.)

+632°J2VJ2 633m 3 Jim 6 - Jim

p3m [37.333312

+ 63222“1 33m Jim - AJim

p VJ VJ

p VJ VJ

+ 6322’2‘“1 33m Jim - AJim } . (9.417)

Fourth order EDF from the R2-double-exchange piece of the C-term

01322:: _. P%‘3 2 2032121
3 ~ = fd+{63 pm3-3m+63 p3mp3m<3m

+632'J0J2 33m Jim - Jim + 63"”2 pm Jim - Jim

+63?“0 Jim - 33(3):“; - Jim + 63’22’2‘7’2 Jim - Jim 6 - Jim
+632 ‘2 2 33m Jim Jim + 6322222 Ap3m Jim Jim
+632 J2 J2 33m J3m J3m + 6322J2‘7J22Vpim J3mv J3(7
+632J2‘“0 p3m Jim - AJim + 63 ”OJ 2 mm Jim - Jim

<2J2 3 3 3132 3 3

+630 2 33m J3m - J3m + 630 233m Jim - J3m
+6322J0‘JJ2 633m - Jim 6 - Jim
+632‘JJ2‘JJ1 33m \‘7’ - Jim 37 - Jim
+63‘22J2‘7J2 633m - Jim 6 - Jim

291

[)UVJIV ll

+C5 J p007) [6 ° j1(’—'J)]2
+632'J0’2‘J2 33m Jim - AJim

d

+ 63‘2J2A'J2 33m Jim - AJim } . (9418)

Fourth order EDF from the R3-double-exchange piece of the C—term

. J2 .. .3
80333.21: : /dF{C§OJOpO(14) JO(7—’) J0(T_“) + CP] “0 1p1(7"')J0(7‘-‘) . J10?)
+2250 1 33333333331333 + 630 0Jim ~ J3mv 3 J3m

2 _. _. ..
+632VJ° J3m J3mv J3m + 632J2‘JJ2 Jim J3mv Jim
2J 2
+C6 MP0 0 A1000?) J00?) J00—") + 660 J0 C02(7?) J00?) J00.)
1.12 a -o -o -o —o —o
+63 2 J2 33m J3m - J3m + 63’2OJ2‘JJ2 v33m J3mv - J3m

33m [37 - 3730712 + 6322““ pom Jim - AJim
+C;1J0:§1 1(nJ0(J_J)1( J10?) + CgleOJl APIU‘) J0(7d) J10”)

p0VJO v.10

+C6

+66 J91“) J00?) "11(7?) + C: V7101“) ' J10?) V ' J0(7?)
2 2

.3 .3 3; J .3 .3
mm J3m - AJom + 630 2 33m Jim - J3m
+6322J2‘7J2 633m - Jim <7 - Jim

pJAJ

+632‘7J2‘7J2 pm <7 - Jim <7 - Jim
+CgOVJ1VJ1p0(’F) [6 . 371(T—j]2

+632J°AJ2p 3m Jim - AJim
+632J2AJ2 pim J3m - AJim } . (9419)

Fourth order EDF from the R4-double-exchange piece of the C-term
. ,3 M0 p2
8222222 = f d?{6 3 233m + 6 2 m p3m

292

+Cp0p1111/10(7)P1(7’7<1(77 + CPUA p0 1922(7) A000?)

+5101”)1 Apl PIJ(n/)1(flAP1(:) + C700 P32303900?)

+633 33333363 +cf’0"2133(333 m<3m
+6312121pi(J)<i(J) + C70 0 P007) Jam J3(J)

+0.71% 233m jib") - 3710‘”) + (370 1m(fl371(fl-373(J)
333327103333 . 33333-33333 3 3:?‘7203333 . 3333.333
+6¥°12W2 Jim 3333 J3m1 + 63013 Ap3m Jim - Jim
+6111]2 0 <§(J) Jam Jam + C701J3C3(fljom J0(4)

+ 63"“1‘2W0 633m Jim \‘7' - Jim

+ 63’0W22‘710 33m [6 - Jim]?

+c50"0‘1'10p0(33 J'i(33 Af3(33 + 6:2 “‘1 1 <10”) J3(J) J3m
+cf2’11011Apl(33J3(33 Jim + 6:2 ‘1 1 <30") J3m J3m
+63V”2’2‘”° 633m - Jim J - Jim

9107) 3m?) ' Ajifi’?)

3363’? 33m Jim - Jim + C7VP1J11W1§301(F) Jim 6 3 Jim
Pl('flv'%(fl§‘j1(fl

33m [VJ-Jim]?

+c”2 101112 3m Jim - AJim

p J A.)

+ C7p1VJ0VJ1

VJ VJ

+6522’2A‘12 33m Jim - AJim } . (9.420)

9. 10.5 Extension to deformed time-reversal invariant systems

The assumption of spherical symmetry was used to obtain the generic form of the

angular integrations in Eq. (9.411).

This can be relaxed to treat deformed time-

reversal invariant systems, though with a significant increase in the time—complexity

293

of the symbolic computation. In deformed systems, all local densities denoted here
by, ((771), depend on the orientation of 7"}. Hence, they come with their own direction,
increasing the number of available vector directions to a much higher value than
the three that we have in spherical systems (:32, f3 and F1). This entails changing
the angular integration technique discussed in appendix 9.10.3 with symbolic tensor
manipulation. To illustrate the difference between the two-techniques, consider a

schematic example, perhaps misleading due to its simplicity,

-o

Note that {7 is acting on some unspecified local density which depends on 7"}. In the
case of spherical symmetry, the action of 6 results in a gradient density that is along

7‘1. Hence, T is written as
T = [(1% dog |6|0082(92)sin(02), (9.422)

which is directly integrable. In deformed systems, one can no longer assume that the

direction of the local densities is along 1‘1. Hence, one spells out Eq. (9.421) as
T = Z 131,, v,- / d92i2,,-§:2,j, (9.423)
U

where z' and j label the cartesian coordinates of the labels. A symbolic algorithm,
a la the one developed for the angular integrations in appendix 9.10.3 needs to be
developed that not only does the integration but also groups the tensor components

so that a manifestly scalar EDF results. For details, refer to Ref. [162].

294

9.11 Analytical couplings for the EDF from chiral
EFT NNN interaction at N2LO

9.11.1 Functional form of the couplings

The main objective of this section is to motivate the analytical calculation of the

couplings of the EDFs given in appendix 9.10.4,Cf1C2 3 ,by isolating the forms and
types of integrals required. Note that at this stage, all required angular integra-

tions, viz, f d!) f (19 f dflxz and f de3 have been carried out. Of course,

(12’ Q31

f dflrl is left intact as F1 is the coordinate of the local EDF. To recap, we per-
formed f qu2 and f qu3 in appendix 9.10.2 where the G—tensors are discussed, while
f dflxz and f dQI3 are handled in appendix 9.10.3. Thus, the remaining integrals are
f dqg qg, f dq3 qg, f dccg 3:3 and f d333 93%. For the sake of notational simplicity, we use

H #2

J. fr as it is rather than their separable expansion discussed in appendix 9.10.3. In

the actual analytical calculation, the expansion is used. Refer to appendix 9.11.3 for

details. After dropping all pre-factors, the couplings take the following general forms.

Couplings from Generic-Form—l

All the couplings that originate from Generic-Form-l 7.2.1 take the form
1 2 3 .
Cg g C (k) 0‘ [51372 d4’33 dQ2 (193 $3 5’33 q; (1% 312,333 Ff($2,CI2)F2(3331(I3)
X H :71(k$3)7rs C1 0/1(k:v3)

Xmas.) Mamezm. how-3,10 , (9.424)
01‘

V1 2.3 "
0* ,. (1.) oc /d:c2d:v3d(12d(13$§$§(1§(1§ 231%Ff”(r2.q2)FZ”(xa»q:s)

295

 

 

 

(1.0/1

X 11:1 (A7133) 7T3 (1511.3)

xn§2(1:1:2)7r;2‘°/ (emu? (32,532) , (9.425)

where p, q, r, s, t, u, v, 13, '7 take only integer values, F311?” are given in Eqs.(9.391)-
(9.393) and Ff, F2 are as defined in Eqs.(9.400)-(9.402). In fact, 7‘, s, t, 11, vi? {0, 1, 2},
3,76 {1,2, 3}, while p,q6 {0, 1, 2, 3, 4}.

Couplings from Generic-Form-2

All the couplings that originate from Generic-Form-2 7.2.1 take the form

1 2.3 1
Cg ‘ ‘ (k) o< /d;1:2 (1:1:3 dqg dq3 2:3 1:3 q3 q3 [x3273 F:(.’L‘2,(I2)F:($3,Q3)

x 113.013sz k) $345233, 1)] , (9.426)

01'

1.2 3 . . ,1 ..
CC (‘ C (k) 0‘ /(l$2 d4103 ([92 (19313 I?) (1% (1% [372,173 Ff‘j($2,92)FzM($3a‘13)

x Harman113,492,299} , (9.427)

where all the specifications given at the end of Eq. (9.424) regarding the indices and

functions hold.

Couplings from Generic-Form-3

All the couplings that originate from Generic-For1n-3 7.2.1 take the form
(19293 .2 2 .P 13x? ..
C 0< (1333d031393 1'3 F; (13:03)
Xmas-3) 792”“ (M3) 11:391.) «PD/1929)} , (9.428)

296

where again all the specifications given at the end of Eq.(9.424) regarding the indices

hold and F ‘33 is defined in Eqs.(9.391)-(9.393).

9.11.2 Matching generalized PSA-DME against the DME-

ansatz

At this point, we have the expressions for the couplings expressed as a functional
of the 7r—functions of the DME ansatz discussed in appendix 9.10.1. In line with
the view that the DME ansatz is just a general symbolic template for an anlyti~
cal/parameterized DME, we fix the 7r—functions of the DME ansatz with the gener—
alized PSA-DME of appendix 9.5.3 (also discussed in sections 7.2.3) for the analytical

calculation of the couplings.

Adopting and matching the DME ansatz, Eq. (9.382), for pq(F1,F1 + 552) with its

generalized PSA-DME expansion, Eq. (9.211), the 7r—functions for this density read

' 1: cc )

0( [‘32) 3 [€sz ’

p _ j1(kF$2)

H2(kF$2) -- 3 "k'F' x2 ,
«3am» = 1, «((km) = 0, «((kpxg) = o, (9.429)

where we set k = kp. Similar a—functions hold for pq(F1 + 52, F1) due to the assump—
tion of time-reversal invariance, while for pq(F1, F1 + 553) and pq(fi + 553, F1) one simply
replaces 2:2 with 2:3. We leave the 7r—functions multiplying the time-odd densities

unspecified. As to pq(f'1 + :32, 771 + :33), following the same set of steps

j1(kplf2 — f3”

 

ngJ,,(kF, :32, 5,) = 3

 

kFl-‘f2—f3l ’
n” 1:37,“. 2 3”” .
2,fr( P 12 $3) (Cpl-52 _f3l 1 (9 430)

297

where again time-reversal invariance implies the same set of 7r—functions for pq(F1 +
f3..7_"1+ I2).

The 7r—functions for .§,(F1,f'1 + :32) read
j1(kp1.'2)

kF$2
nchprg) = 1, 7rlJ(kp2:2) = 0, 1r‘1’(kp:r-2) = 0. (9.431)

Hf (lap-1:2) = 3

7

Similar n—functions hold for 51,071 +272, F1), while for 31,071, F} +53) and §q(7"'1+:i:'3, F1),
one simply replaces 2:2 with 3:3. Again, we leave the 7r—functions multiplying the
time-odd densities unspecified. As to .§'q(f"1 + 52,13 + 53), following the same set of
steps

j1(kFl52 — 53')
kplih — 553|

 

Hf, 1.93.52, .33) = 3

7

(9.432)

where again time-reversal invariance implies the same 7r—functions for 51,071 + .53, F1 +

—o

22).

9.11.3 Application of Gegenbaur’s addition theorem

In appendix 9.10.3, we discussed the need for the separable expansion and its for-
mal solution of H§,f,.(k,5:'2,f3). With the generalized PSA-DME used to fix the
7r—functions of the DME ansatz, one can apply Gegenbaur’s addition theorem, which
is discussed in appendix 9.1.5 to the as of yet non-separable 7r—functions given in
the Eqs. (9.212), (9.213) and (9.230) : “3,”: ngr and Hifr. In this case, the non-
separability is due to the appearance of the following function in these 7r—functions

j1(kl4§3 — 52D
klfs -- f2|

f(k,:z‘7'2, :33) , (9.433)

 

298

whose Gegenbaur expansion is obtained by using Eqs. (9.59) and (9.50). It reads

_. _. 7r J3/2(kl5«:3 - 52')
A. .2‘ ..,,“ Z _ . . s
f( '1 1.2.13) \/; (kILEzg _ f2|)O:/2 .

/144 F(3/2) :01
7r #:2173112 “:0 g)J/:+1(k-T2)Jlt+1(kr3)C3/2(COS(6))’
(9.434)

 

 

where 6 is the angle between :52 and 553. A simple re-organization of the terms is needed
to recast Eq. (9.434) in to the form of Eq. (9.410). There are several interesting
characteristics of this expansion that make it highly applicable to this work: (i)
It converges very fast, (ii) In combination with the symbolic integration technique
mentioned in the next section (appendix 9.11.4), it makes the analytical computation
of the couplings possible (iii) the validity of truncating the expansion at some value of
p can readily be tested with Mathematica. Our numerical experiments show that p 2
5 suffices for a practically perfect reproduction of the exact quantity. Furthermore,
these numerical experiments show that the accuracy increases progressively as the
value at which 11 is truncated increases. This, seemingly trivial statement, is not
obvious just from the expansion formula given in Eq. (9.434). In the following we test
the accuracy of truncating the expansion at u = 5. For this, let us define f5(k, f2, :53)

as

 

 

_. - 144 r( 3 2) 5
43.42.43) = (/ k, / :04%)j,.+1((k:c2)jp+1(k$3)Cf/2(COS(9))
2331132 11: _0
(9.435)
and the ratio function R5(k,:1:2,1‘.3,0) as
R k _ (3f27f3) ,
( ante) — ( 52 1,3) (9.436)

299

We plot R5(k, 222,1:3,0) in Fig. 9.1, after absorbing the k dependence into X2 =

[9232 and X3 = km3. First, note that the region of interest for this work is set

 

   

Angle- — Angle = % Angle =%
1.5
6.0 g
n 1.0
N 4.0
2 0 0.5
0.0 0.0
Angle— - —- Angle- — % Angle= s—gr
1.5
1.0
0.5
0.0
= 3_1r

 

Angle

7
4
6.0
('0
N 4.0
2.0
0.0

0.0 2.0 4.0 6.0 0.0 2.0 4.0 6.0 0.0 2.0 4.0 6.0
X2 X2 X2

Angle = 1r

 

Figure 9.1: (Color online) R5(k, :02, x3, 0) for a set of angles.

by the range of nuclear interactions as :32 and 533 are relative coordinates and the
k parameter, viz, the DME length scale. Setting k = kp z 1.42fm close to the
saturation density of INM, and the a maximum range of about 5fm for the nuclear-

interaction range dependent pieces ,viz, $2 and :33, the region of interest becomes

300

X2 S 7.5 and X3 _<_ 7.5. As can be seen from the plot, Fig. 7.3, R5(k,;r2,;v3,0) is
close to one for most, if not all, of the region of interest. In fact, investigation of the
actual numerical values shows that R5(k,:1:2,:1:3,6) 2 0.995 for most of the physically
interesting region. Hence, for our calculation of the couplings, we truncated the
expansion at p = 5. Further validation of the accuracy of Gegenbaur’s expansion
truncated at p = 5 is discussed in section 7.3 where we compare results from Monte-
Carlo based calculation, which are essentially exact, with the ones based on the
truncated Gegenbaur’s expansion.

Finally, it should be mentioned that, the Gegenbaur expansion of expressions
containing higher order spherical bessel functions, such as j3(k|§:'3 — 52|)/(k|:i?3 — f2|)3,
have been found to be as accurate, though at this point their practical use is limited for
this work. In order to compare the dependence of the couplings on the 7r—functions,
we have generated another set of couplings by setting II’2’J7, = 105j3(k|i:'3—f2|)/(k|§:'3—

f2|)3. This is based on the 7r—functions extracted from the original DME of Ref. [170].

9.11.4 Analytical and symbolic integration

With the use of the 7r—functions specified in Eqs. 9.11.2 and the application of Gegen-
baur’s addition theorem, the final step in the analytical calculation of the couplings

is the four—dimensional integration with respect to f dqg dq3 dxg dIL'3 qg q§ :33 2:3.

@123
At this point, each term in the expression for any of the couplings is separable in
((12, 2:2) and (q3,:1:3). This can be seen from the form of the generic couplings given in
Eqs. (9.424)-(9.428) and the expansion discussed in appendix 9.11.3.

We first perform the analytical integrations with respect to q2 and q3. Note that

we have already absorbed all Q2 and q3 dependencies into the G-tensors. Hence, the

only integrals that we need to integrate out the (12 and q3 dependencies from the are

 

00 4 —mr
, , _. __ _ ___—- 9.4 7
f0 dq q2 772,2 10(qx) 7r 51:2 7rm 2:2: , ( 3)

 

 

 

ac «l f , 6 ;? J—mr ,,_—m;r
/ dq q J10”) = —7r —(—l) + 7r m P + 7r m C (9.438)
0

 

q2 + m'2 qx $3 2:152 2.173 l
00 (13 ' 6(1‘) e-mr e—mx
() c I

where q stands for either (12 or (13. These integrals can be obtained by following the
steps: (i) Rewrite the fractional prefactors, for instance, q4/(q2+m2) as q2 — 1712/ (q2+
m2) and (ii) Express the integral of the non—fractional piece in terms of 5(1‘) whereas

that of the other piece can be found by standard contour integration.

The subsequent integrations are with respect to 2:2 and 11:3. Even though each
relevant term is separable in 2:2 and $3 and thus the integrals calculable separately,

the required integrals are much more complex. In general, we need to obtain

/ e‘x/"a:"j,,(a:)j,,(x)da:, (9.440)
0

where u 2 0, p 2 0, whereas it > 0 is a real parameter and n is an integer. Conver-
gence of the integral requires that n+u+u Z 0. Note that in our problem, u = kp/mfl.
The application of Gegenbaur’s expansion introduces higher order spherical Bessel
functions. I.e. if the expansion is truncated at p = 5, up to sixth-order (35(1)) spher-
ical Bessel functions are introduced. However, we are not aware of any analytical
or symbolic integration technique that works for any values of the indices: {11, u, ,u}
which satisfy the convergence criterion. Furthermore, we need to perform calcula-
tion of hundreds these integrals. In Ref. [157], we discuss how we solve this problem
with the Mathematica package that we developed. In the package, we designed and
implemented a symbolic integration algorithm that can calculate these integrals for
any values of {71, V, p}, which satisfy the convergence criterion, and gives the exact
analytical expression.

In principle, we have analytical expressions for the couplings at this point. How-

ever, the actual Mathematica implementation of the DME starting from the exact

302

HF energy from the chiral EFT N NN interaction at N2LO has not been discussed.
In Ref. [161], we give a detailed presentation of the symbolic implementation. We
remark that the implementation is done in such a way that it can be adapted to
deformed time-reversal invariant systems as discussed in appendix 9.10.5 with only a
modest amount of change. The implementation for deformed time-reversal invariant
systems will be the subject of a subsequent publication [162]. A detailed analysis
and discussion of these couplings can be found in Ref. [160]. Finally, the lengths of
the coupling expressions prevent us from reproducing even a single coupling in here.

Consult the Mathematica files of Ref. [161].

303

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