Natural orbital methods for ab initio nuclear structure Patrick - - PowerPoint PPT Presentation

Natural orbital methods for ab initio nuclear structure

Patrick Fasano June 20, 2018

Department of Physics University of Notre Dame

Natural orbitals for Nuclear Structure – Outline

• 1. No-Core Configuration Interaction (NCCI) Overview
• 2. Natural Orbital Definition
• 3. Description of He Nuclei with Natural Orbitals

Natural orbitals for nuclear structure 1

Natural orbitals for Nuclear Structure – Outline

• 1. No-Core Configuration Interaction (NCCI) Overview
• 2. Natural Orbital Definition
• 3. Description of He Nuclei with Natural Orbitals

Natural orbitals for nuclear structure 1

Basics of NCCI

Begin with single-particle Hilbert space spanned by orthonormal single-particle basis {|α⟩}: ˆ h |nljm⟩ = ϵnljm |nljm⟩ This space has an (countably) infinite dimension; computationally, we must truncate to a finite number of single-particle states. Construct a many-body basis of Slater determinants with good M: {|Ψα⟩} = { |πα1πα2 · · · παZνα1να2 · · · ναN⟩

• ∑

i

mi = M }

Natural orbitals for nuclear structure 2

Basics of NCCI – The Curse of Dimensionality

4He 6He 6

L i

7

Li

8Be 10B 1 2C 16O

100 104 108 1012 1016 1020 1024

Dimension

2 4 6

Nshell 10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

8

10

9

basis size

• 35
• 30
• 25
• 20
• 15
• 10

ground state energy (MeV)

4He, FCI trunc. 4He, Nmax trunc. 6Li, FCI trunc. 6Li, Nmax trunc.

NCFC

JISP16 (No Coulomb)

Basis grows too fast keeping all possible Slater determinants, i.e. Full Configuration Interaction (FCI). →Can we eliminate some Slater determinants we don’t need?

Natural orbitals for nuclear structure 3

Nmax Truncation

All Slaters with a total number of oscillator quanta N =

A

∑

α=1

Nα ≤ N0 + Nmax are included in the basis, where Nα is the oscillator quantum number of the α − th particle, and N0 is the number of oscillator quanta in the lowest configuration. Nmax-truncation has been preferred traditionally because it allows exact center-of-mass factorization, and can lead to faster convergence with respect to basis size than FCI-truncation.

Natural orbitals for nuclear structure 4

NCCI Basis Size

100 102 104 106 108 1010 1012

Dimension

2 4 6 8 10 12 14 16

Nmax

Natural orbitals for nuclear structure 5

Many-body truncation

• 1. Assign each single-particle state a

weight wα (e.g. harmonic oscillator quanta N = 2n + ℓ) and sort orbitals by that weight.

• 2. Assign a weight to the Slater

determinants by Wα = ∑ wαi.

• 3. Truncate based on weight of Slater

determinant Wα ≤ Wmax.

Natural orbitals for nuclear structure 6

Many-body truncation

• 1. Assign each single-particle state a

weight wα (e.g. harmonic oscillator quanta N = 2n + ℓ) and sort orbitals by that weight.

• 2. Assign a weight to the Slater

determinants by Wα = ∑ wαi.

• 3. Truncate based on weight of Slater

determinant Wα ≤ Wmax.

Natural orbitals for nuclear structure 6

Many-body truncation

• 1. Assign each single-particle state a

weight wα (e.g. harmonic oscillator quanta N = 2n + ℓ) and sort orbitals by that weight.

• 2. Assign a weight to the Slater

determinants by Wα = ∑ wαi.

• 3. Truncate based on weight of Slater

determinant Wα ≤ Wmax.

Natural orbitals for nuclear structure 6

Convergence of NCCI Calculations

By completeness, a calculation in the infinite space → independence from parameters in the single-particle basis (i.e. ℏω). Convergence is signalled by independence of the calculated value from Nmax and b = (ℏc)/ √ (mNc2)(ℏω).

1 2 3 1 4 2

Harmonic

• scillator

1 2 3 4

N

Natural orbitals for nuclear structure 7

Convergence of NCCI Calculations

• /

+

• ()
• ℏω ()
• P. Fasano et al., in preparation

Natural orbitals for nuclear structure 8

Convergence of NCCI Calculations

• /

+

• ()
• ()
• ℏω ()
• P. Fasano et al., in preparation

Natural orbitals for nuclear structure 9

Natural orbitals for Nuclear Structure – Outline

• 1. No-Core Configuration Interaction (NCCI) Overview
• 2. Natural Orbital Definition
• 3. Description of He Nuclei with Natural Orbitals

Natural orbitals for nuclear structure 10

Natural Orbitals for Nuclear Physics

• Attempt to formulate a “natural”

basis for performing NCCI calculations.

• Observables should converge faster

in “natural” basis.

• Define “natural” → maximize
• ccupation of lowest orbitals
• Minimizing depletion of Fermi sea,

not minimizing energy!

• Built from many-body calculation, so

maybe “aware” of correlations.

rowe2010:collective-motion

Natural orbitals for nuclear structure 11

Natural Orbitals for Nuclear Physics

• Attempt to formulate a “natural”

basis for performing NCCI calculations.

• Observables should converge faster

in “natural” basis.

• Define “natural” → maximize
• ccupation of lowest orbitals
• Minimizing depletion of Fermi sea,

not minimizing energy!

• Built from many-body calculation, so

maybe “aware” of correlations.

rowe2010:collective-motion

Natural orbitals for nuclear structure 11

Natural Orbitals for Nuclear Physics

• Attempt to formulate a “natural”

basis for performing NCCI calculations.

• Observables should converge faster

in “natural” basis.

• Define “natural” → maximize
• ccupation of lowest orbitals
• Minimizing depletion of Fermi sea,

not minimizing energy!

• Built from many-body calculation, so

maybe “aware” of correlations.

rowe2010:collective-motion

Natural orbitals for nuclear structure 11

Natural Orbitals for Nuclear Physics

Natural orbitals are the eigenvectors of the one-body RDM ραβ = ⟨α| ˆ ρ |β⟩ One-Body Reduced Density Matrix (RDM) ˆ ρ = ∑

αβ

|α⟩ ⟨Ψ| a†

αaβ |Ψ⟩ ⟨β|

ρ(x, x′) = A ∫ Ψ(x, x2, . . . , xA)Ψ∗(x′, x2, . . . , xA)dx2 · · · dxA

• Hermitian operator on the single-particle space;
• Depends on some reference many-body state |Ψ⟩;
• Contains all single-particle behavior in |Ψ⟩;
• Number operator expectation values on diagonal

ραα = ⟨α|Ψ|α⟩ = ⟨Ψ|Nα|Ψ⟩

Natural orbitals for nuclear structure 12

Natural Orbitals for Nuclear Physics

A change of basis on the single- particle space:

• does not change the

single-particle space;

• does not change the FCI

many-body space;

• does change a truncated

many-body space. We must sort our new natural or- bitals by occupation.

Natural orbitals for nuclear structure 13

Natural Orbitals – Two examples

Four-state, two-orbital system: 0s1/

2, 1s1/ 2

Eigenvector in initial basis:

1 2 0s 0s

N

0s 1s 0s 1s

N 2

1s 1s

N 4

Density matrix:

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

Eigenvectors of :

0s1 2 1 2 0s1 2 1 2 1s1 2 1s1 2 1 2 0s1 2 1 2 1s1 2

Eigenvector in natural orbital basis:

0s 0s

0s1/2 1s1/2

Natural orbitals for nuclear structure 14

Natural Orbitals – Two examples

Four-state, two-orbital system: 0s1/

2, 1s1/ 2

Eigenvector in initial basis:

|Ψ⟩ = 1 2 (

• (0s↑)(0s↓)

⟩

• N=0

+

• (0s↑)(1s↓)

⟩ −

• (0s↓)(1s↑)

⟩

• N=2

+

• (1s↑)(1s↓)

⟩

• N=4

)

Density matrix:

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

Eigenvectors of :

0s1 2 1 2 0s1 2 1 2 1s1 2 1s1 2 1 2 0s1 2 1 2 1s1 2

Eigenvector in natural orbital basis:

0s 0s

0s1/2 1s1/2

Natural orbitals for nuclear structure 14

Natural Orbitals – Two examples

Four-state, two-orbital system: 0s1/

2, 1s1/ 2

Eigenvector in initial basis:

|Ψ⟩ = 1 2 (

• (0s↑)(0s↓)

⟩

• N=0

+

• (0s↑)(1s↓)

⟩ −

• (0s↓)(1s↑)

⟩

• N=2

+

• (1s↑)(1s↓)

⟩

• N=4

)

Density matrix:

ρ =     

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

    

Eigenvectors of :

0s1 2 1 2 0s1 2 1 2 1s1 2 1s1 2 1 2 0s1 2 1 2 1s1 2

Eigenvector in natural orbital basis:

0s 0s

0s1/2 1s1/2

Natural orbitals for nuclear structure 14

Natural Orbitals – Two examples

Four-state, two-orbital system: 0s1/

2, 1s1/ 2

Eigenvector in initial basis:

|Ψ⟩ = 1 2 (

• (0s↑)(0s↓)

⟩

• N=0

+

• (0s↑)(1s↓)

⟩ −

• (0s↓)(1s↑)

⟩

• N=2

+

• (1s↑)(1s↓)

⟩

• N=4

)

Density matrix:

ρ =     

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

    

Eigenvectors of ρ:

• 0s′

1/ 2

⟩ = 1 √ 2

• 0s1/

2

⟩ − 1 √ 2

• 1s1/

2

⟩

• 1s′

1/ 2

⟩ = 1 √ 2

• 0s1/

2

⟩ + 1 √ 2

• 1s1/

2

⟩

Eigenvector in natural orbital basis:

0s 0s

0s1/2 1s1/2

Natural orbitals for nuclear structure 14

Natural Orbitals – Two examples

Four-state, two-orbital system: 0s1/

2, 1s1/ 2

Eigenvector in initial basis:

|Ψ⟩ = 1 2 (

• (0s↑)(0s↓)

⟩

• N=0

+

• (0s↑)(1s↓)

⟩ −

• (0s↓)(1s↑)

⟩

• N=2

+

• (1s↑)(1s↓)

⟩

• N=4

)

Density matrix:

ρ =     

1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2

    

Eigenvectors of ρ:

• 0s′

1/ 2

⟩ = 1 √ 2

• 0s1/

2

⟩ − 1 √ 2

• 1s1/

2

⟩

• 1s′

1/ 2

⟩ = 1 √ 2

• 0s1/

2

⟩ + 1 √ 2

• 1s1/

2

⟩

Eigenvector in natural orbital basis:

|Ψ⟩ =

• (0s′

↑)(0s′ ↓)

⟩

0s1/2 1s1/2

Natural orbitals for nuclear structure 14

Natural Orbitals – Two examples

Eigenvector in initial basis:

|Ψ⟩ = 1 + √ 3 4 |(0s↑)(0s↓)⟩

• N=0

+ 1 − √ 3 4 |(0s↑)(1s↓)⟩ − 1 − √ 3 4 |(0s↓)(1s↑)⟩

• N=2

+ 1 + √ 3 4 |(1s↑)(1s↓)⟩

• N=4

Density matrix:

1 2 1 4 1 2 1 4 1 4 1 2 1 4 1 2

Eigenvectors of :

0s1 2 1 2 0s1 2 1 2 1s1 2 1s1 2 1 2 0s1 2 1 2 1s1 2

Eigenvector in natural orbital basis:

3 4 0s 0s 1 4 1s 1s

0s1/2 1s1/2

Natural orbitals for nuclear structure 15

Natural Orbitals – Two examples

Eigenvector in initial basis:

|Ψ⟩ = 1 + √ 3 4 |(0s↑)(0s↓)⟩

• N=0

+ 1 − √ 3 4 |(0s↑)(1s↓)⟩ − 1 − √ 3 4 |(0s↓)(1s↑)⟩

• N=2

+ 1 + √ 3 4 |(1s↑)(1s↓)⟩

• N=4

Density matrix:

ρ =     

1/ 2

−1/

4 1/ 2

−1/

4

−1/

4 1/ 2

−1/

4 1/ 2

    

Eigenvectors of :

0s1 2 1 2 0s1 2 1 2 1s1 2 1s1 2 1 2 0s1 2 1 2 1s1 2

Eigenvector in natural orbital basis:

3 4 0s 0s 1 4 1s 1s

0s1/2 1s1/2

Natural orbitals for nuclear structure 15

Natural Orbitals – Two examples

Eigenvector in initial basis:

|Ψ⟩ = 1 + √ 3 4 |(0s↑)(0s↓)⟩

• N=0

+ 1 − √ 3 4 |(0s↑)(1s↓)⟩ − 1 − √ 3 4 |(0s↓)(1s↑)⟩

• N=2

+ 1 + √ 3 4 |(1s↑)(1s↓)⟩

• N=4

Density matrix:

ρ =     

1/ 2

−1/

4 1/ 2

−1/

4

−1/

4 1/ 2

−1/

4 1/ 2

    

Eigenvectors of ρ:

• 0s′

1/ 2

⟩ = 1 √ 2

• 0s1/

2

⟩ − 1 √ 2

• 1s1/

2

⟩

• 1s′

1/ 2

⟩ = 1 √ 2

• 0s1/

2

⟩ + 1 √ 2

• 1s1/

2

⟩

Eigenvector in natural orbital basis:

3 4 0s 0s 1 4 1s 1s

0s1/2 1s1/2

Natural orbitals for nuclear structure 15

Natural Orbitals – Two examples

Eigenvector in initial basis:

|Ψ⟩ = 1 + √ 3 4 |(0s↑)(0s↓)⟩

• N=0

+ 1 − √ 3 4 |(0s↑)(1s↓)⟩ − 1 − √ 3 4 |(0s↓)(1s↑)⟩

• N=2

+ 1 + √ 3 4 |(1s↑)(1s↓)⟩

• N=4

Density matrix:

ρ =     

1/ 2

−1/

4 1/ 2

−1/

4

−1/

4 1/ 2

−1/

4 1/ 2

    

Eigenvectors of ρ:

• 0s′

1/ 2

⟩ = 1 √ 2

• 0s1/

2

⟩ − 1 √ 2

• 1s1/

2

⟩

• 1s′

1/ 2

⟩ = 1 √ 2

• 0s1/

2

⟩ + 1 √ 2

• 1s1/

2

⟩

Eigenvector in natural orbital basis:

|Ψ⟩ = √ 3 4

• (0s′

↑)(0s′ ↓)

⟩ + √ 1 4

• (1s′

↑)(1s′ ↓)

⟩

0s1/2 1s1/2

Natural orbitals for nuclear structure 15

Natural Orbitals for NCCI

How we use natural orbitals to accelerate convergence:

• 1. Perform an initial many-body NCCI calculation in an oscillator

basis.

• 2. Compute an approximate one-body reduced density matrix from
• ne of the many-body states.
• 3. Diagonalize the one-body reduced density matrix to obtain a

new basis.

• 4. Transform all input Hamiltonian matrix elements.
• 5. Diagonalize many-body Hamiltonian in new many-body basis.

Natural orbitals for nuclear structure 16

Natural orbitals for Nuclear Structure – Outline

• 1. No-Core Configuration Interaction (NCCI) Overview
• 2. Natural Orbital Definition
• 3. Description of He Nuclei with Natural Orbitals

Natural orbitals for nuclear structure 17

Results with Natural Orbitals

• scillator

20

• 7.75
• 7.70
• 7.65
• 7.60
• 7.55
• 7.50

E (MeV)

5 10 15 20 25 30 35 40

ℏω (MeV) Daejeon16

3He 1/21 +

natural orbitals 12 14 20

5 10 15 20 25 30 35 40

ℏω (MeV)

• P. Fasano et al., in preparation

Natural orbitals for nuclear structure 18

Results with Natural Orbitals

• scillator

8 10 12 14 20

1.5 1.6 1.7 1.8 1.9 2.0

rp (fm)

5 10 15 20 25 30 35 40

ℏω (MeV) Daejeon16

3He 1/21 +

natural orbitals 6 8 10 12 14 20

5 10 15 20 25 30 35 40

ℏω (MeV)

• P. Fasano et al., in preparation

Natural orbitals for nuclear structure 19

Results with Natural Orbitals

• scillator

12 14

• 31
• 30
• 29
• 28
• 27
• 26
• 25
• 24

E (MeV)

5 10 15 20 25 30 35 40

ℏω (MeV) Daejeon16

8He 01 +

natural orbitals 8 10 12

5 10 15 20 25 30 35 40

ℏω (MeV)

• Ch. Constantinou et al., in preparation

Natural orbitals for nuclear structure 20

Results with Natural Orbitals

• scillator

2 4 6 8 10 12 14

1.25 1.45 1.65 1.85 2.05 2.25

rp (fm)

5 10 15 20 25 30 35 40

ℏω (MeV) Daejeon16

8He 01 +

natural orbitals 2 4 6 8 10 12

5 10 15 20 25 30 35 40

ℏω (MeV)

• Ch. Constantinou et al., in preparation

Natural orbitals for nuclear structure 21

Natural Orbitals – Decompositions

Natural orbitals decomposed into harmonic oscillator functions:

• Ch. Constantinou et al., in preparation

Natural orbitals for nuclear structure 22

Natural Orbitals – Decompositions

Natural orbitals decomposed into harmonic oscillator functions:

• Ch. Constantinou et al., in preparation

Natural orbitals for nuclear structure 23

Collaborators

• Valentino Constantinou (U. Notre Dame, Monmouth College)
• Mark Caprio (U. Notre Dame)
• Pieter Maris (Iowa State U.)
• James Vary (Iowa State U.)

Natural orbitals for nuclear structure 24

Summary

• Goal: Try to solve the many-body problem starting with a

realistic NN (and 3N) interaction.

• Convergence assessed based on independence from

single-particle basis and many-body truncation.

• Picking better basis functions leads to better convergence!

Natural orbitals for nuclear structure 25