Non-empirical description of nuclear collective motion
with optimized basis for multi-reference DFT
Tohoku University

Moemi Matsumoto
Collaborators: Yusuke Tanimura and Kouichi Hagino
Phys. Rev. C 108, L051302 (2023).
DFT2024@Kobe 2024/2/21

Nuclear collective motion“Collective” aspects in nuclei by microscopic theory: essential problem
Especially, large amplitude collective motion is very important
186Pb

Nuclear fission

Shape coexistence

Non-empirical description of collective motion based on DFT
J. Dobaczewski, J. Phys.: Conf. Ser. 312 092002 (2011). / A. N. Andreyev. et al. Nature 405, 430 (2000).

Many-body states for collective motions
Generator coordinate method
Basis: a priori selected (non-orthogonal)

X
X
X X X
fa4 |
| a i +fi =| i =| ffai1 |=faa|i |+f| a i |+ffia3 =
2a =
a ifi
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Weight factors

ai + · · ·Relying on empirical / phenomenological theory !

<latexit sha1_base64="980HV1H4Um25xOXbFgoQ+2cmXL4=">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</latexit>Many-body states for collective motions
Generator coordinate method
Basis: a priori selected (non-orthogonal)

X
X
X X X
fa4 |
| a i +fi =| i =| ffai1 |=faa|i |+f| a i |+ffia3 =
2a =
a ifi
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Weight factors are variational parameteraai + · · ·<latexit sha1_base64="980HV1H4Um25xOXbFgoQ+2cmXL4=">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</latexit>

Relying on empirical / phenomenological theory !

Our method

Optimize both weight and basis at same time

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Non-empirical description of collective motion

Optimized-GCM
Trial function:

Slater determinant Single-particle states (orthonormal set)

：Hamiltonian kernel：Norm kernel

Total energy to be minimized:<latexit sha1_base64="BF+eE9f+8LHjDe7lCdVWeVB/3TI=">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</latexit>

h
(ab)
⇢ ↵ =

| bi
：Transition DMh a| bi†a | a↵ a：EDF：HF Hamiltonian

Optimized-GCM
Calculate gradient of Energy
・Single particle states：Hamiltonian kernel

・Weight function：Norm kernel<latexit sha1_base64="BF+eE9f+8LHjDe7lCdVWeVB/3TI=">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</latexit>

h
(ab)
⇢ ↵ =

| bi
：Transition DMh a| bi†a | a↵ a：EDF

Minimum search with CG method：HF HamiltonianSet up
16O

Ground state
Axial / reflection / time-reversal symmetrybasis expansion with axial harmonic oscillator• Skyrme (w/o time-odd components)
SIII

M. Beiner et al., Nucl. Phys. A 238, 29 (1975).

• No coulomb interaction

Initial states: WS potentials with different deformation

Results 16O
Hartree-Fock<latexit sha1_base64="0fTyWOQdkgjnCnpmmjD/osyF4AM=">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</latexit>

GCM−142.02 MeV| i=| i<latexit sha1_base64="UxLzuwEQ/Sh6+6wOs9Bozmr0JL0=">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</latexit>| i=-0.4 MeV

X

fa |

X

fa |

a

<latexit sha1_base64="UAKg7YihbdTGT9UVSgQhYm4p8bY=">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</latexit>(a)(Q2 )i −142.40 MeV
ai

Bases are local ground statesobtained by constrained HF (next slide)<latexit sha1_base64="uGsyHH3zuHPndqxKFHtYjgDje+0=">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</latexit>

Optimized-GCM

| i=

a

Not enough in conventional method !

-0.7 MeVai−143.10 MeV

Results 16O“Usual” selection of the bases in GCM:
The ground states that minimize the energy
for given values of deformation-110h<latexit sha1_base64="sNQFB22IPwcrjXUQkC9i9u1s+gY=">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</latexit>

Energy (MeV)

-115
-120

a |H|

ai

-125

Optimal bases are NOT
the ground statesof the each deformations

-130
-135
-140
-145
-8 -6 -4 -2

Blue: Constrained HF
Red: Optimized-GCM

0 2 4
Q2 (fm2)

6

8 10

Results 16O“Usual” selection of the bases in GCM:
The ground states that minimize the energy
for given values of deformation-110h<latexit sha1_base64="sNQFB22IPwcrjXUQkC9i9u1s+gY=">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</latexit>

Energy (MeV)

-115
-120

a |H|

ai

Blue: Constrained HF
Red: Optimized-GCM-125

One needs to take into accountexcitations of nuclei
in determining a collective coordinate

-130
-135
-140
-145
-8 -6 -4 -2

0 2 4
Q2 (fm2)

6

8 10

Results 16O
100
80
60
40
20
0
-20
-40
-60
-80
-8 -6 -4 -2

800
600
400
Q6 (fm6)

Q4 (fm4)

Q4 / Q6 of optimal bases (the next higher moment)
Q4<0

Q4>0
CHF
Optimized

200
0
-200
-400

0 2 4
Q2 (fm2)

6

8 10

-600
-8 -6 -4 -2

0 2 4
Q2 (fm2)

6

8 10

There are large fluctuations in higher moments→ Q2 alone does not make a good collective coordinate
at least both Q2 and Q4 are needed cf.) C. V. N. Kumar and L. M. Robledo, PRC 108, 034312 (2023).
N. Hizawa, and K. Hagino, PRC 109, 014312 (2024).

Results 16O

0.04
0.02

44
2
0
-2
-4

0.06
0.04
0.02

-4

44
2
0
-2
-4

0.06
0.04
0.02

-4

44
2
0
-2
-4

0.06
0.04
0.02

-2
-4

44
2
0
-2
-4

0.06
0.04
0.02

4
2
0
-2
-4
44
2
0
-2
-4

0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0

0.08
0.06
0.04
0.02

4 0.082 0 2 4r (fm)
0.06

0.04
0.02

0
0
0
0
0
0
2 0 2 4 4 2 0 2 4 4 2 0 2 4 4 2 0 2 4 4 2 0 2 4 4 2 0 2 4
r (fm)r (fm)r (fm)r (fm)r (fm)r (fm)

0
0.08

0.06
0.04
0.02
0

ρ(r,z) (fm-3)

0

0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
4 0.0820
2
r (fm)

ρ(r,z) (fm-3)-2

2

z (fm)4) 6
Q4Q(fm6 (fm )04

z (fm)

-2

2

0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
4 0.0820
2
r (fm)

ρ(r,z) (fm-3)

0.06

-4

0

4

68 10
8 10

ρ(r,z) (fm-3)z (fm)

02 0 2 4
40.08
r (fm)
2
0.06
0
0.04
-2
0.02
-4
0
4 2 0 2 4 4 2 0 2 4 4r (fm)r (fm)
4

-2

2

0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
4 0.0820
2
r (fm)

ρ(r,z) (fm-3)z (fm)-4

0

4

ρ(r,z) (fm-3)

-2

2

0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
4 0.0820
2
r (fm)

ρ(r,z) (fm-3)z (fm)

0.02

0

4

10
88 10

ρ(r,z) (fm-3)

0.04

2

0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
4 0.0820
2
r (fm)66z (fm)0.06

4

z (fm)

2
2
0
0
-2
-2
-4
-4
(e) 4 OptGCM
4 2 0 2 44
r (fm) 2
2
0
0
-2
-2
-4
-4

z (fm)0.08

z (fm)

4

ρ(r,z) (fm-3)

CHF

ρ(r,z) (fm-3)z (fm)

(d) 4

80
60
200
40
20
0
0
-200
-20
-40
-400
-60
-600
-80 -8 -6 -4 -2 0 2 4-8 -6 -4 -2 0 2 4 2 6
2) )
2 (fm
Q2Q
(fm

400

ρ(r,z) (fm-3)z (fm)-145 10-10-8 -61 -42 -2 3 0 4 2 54 66
2)
Q2 (fmNumberρ(r,z) (fm-3)

CHF
OptGCMz (fm)-140 10-8

ρ(r,z) (fm-3)z (fm)-6

ρ(r,z) (fm-3)-135 10

16O

CHF
Optimized

600(c) 100

Energy (MeV)

10-4

Q4 (fm4)

-130

10-2

ρ(r,z) (fm-3)z (fm)

Eigenvalue

-125

100

ρ(r,z) (fm-3)-120z (fm)

Energy (MeV)

-115
(a)

800

100
80
-110
(b)60
-115
40
-120
20
-125
0
-130
-20
-135
-40
-60
-140
-80
-145
87 108-8 -8-6 -6-4-4 -2-2 00 22 44
2
QQ2 2(fm(fm2))z (fm)-110

Summaryü Generator coordinate method with variational basis optimization→ This method can describe collective motion
without pre-fixed collective degree of freedomü

16O: bases with high excitation energy were obtainedü One needs to take into account excitations of nuclei
in determining a collective coordinate

Future Work＊ Analysis of collective motions in excited states(angular momentum projection)＊ Connection to existing collective model

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