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

i Basics of NCCI Begin with single-particle Hilbert space spanned by orthonormal single-particle basis {| α ⟩} : ˆ h | nljm ⟩ = ϵ nljm | 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 : � { } � ∑ {| Ψ α ⟩} = | π α 1 π α 2 · · · π α Z ν α 1 ν α 2 · · · ν α N ⟩ m i = M � � � Natural orbitals for nuclear structure 2

Basics of NCCI – The Curse of Dimensionality 10 24 16 O 10 20 2 C 1 -10 10 16 4 He, FCI trunc. 10 B 4 He, N max trunc. Dimension ground state energy (MeV) -15 6 Li, FCI trunc. 8 Be 10 12 6 Li, N max trunc. Li 7 -20 i NCFC 6 L 6 He JISP16 10 8 (No Coulomb) 4 He -25 10 4 -30 10 0 -35 1 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10 2 4 6 basis size N shell 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

A N max Truncation All Slaters with a total number of oscillator quanta ∑ N = N α ≤ N 0 + N max α = 1 are included in the basis, where N α is the oscillator quantum number of the α − th particle, and N 0 is the number of oscillator quanta in the lowest configuration. N max -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 10 12 10 10 10 8 Dimension 10 6 10 4 10 2 10 0 0 2 4 6 8 10 12 14 16 N max Natural orbitals for nuclear structure 5

Many-body truncation 1. Assign each single-particle state a weight w α (e.g. harmonic oscillator quanta N = 2 n + ℓ ) and sort orbitals by that weight. 2. Assign a weight to the Slater determinants by W α = ∑ w α i . 3. Truncate based on weight of Slater determinant W α ≤ W max . Natural orbitals for nuclear structure 6

Many-body truncation 1. Assign each single-particle state a weight w α (e.g. harmonic oscillator quanta N = 2 n + ℓ ) and sort orbitals by that weight. 2. Assign a weight to the Slater determinants by W α = ∑ w α i . 3. Truncate based on weight of Slater determinant W α ≤ W max . Natural orbitals for nuclear structure 6

Many-body truncation 1. Assign each single-particle state a weight w α (e.g. harmonic oscillator quanta N = 2 n + ℓ ) and sort orbitals by that weight. 2. Assign a weight to the Slater determinants by W α = ∑ w α i . 3. Truncate based on weight of Slater determinant W α ≤ W max . 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. ℏ ω ). Convergence is signalled by independence of the calculated value from N max and b = ( ℏ c ) / ( m N c 2 )( ℏ ω ) . √ 4 2 0 4 3 1 3 N 2 0 2 1 1 Harmonic oscillator 0 0 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

rowe2010:collective-motion 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 occupation of lowest orbitals • Minimizing depletion of Fermi sea, not minimizing energy! • Built from many-body calculation, so maybe “aware” of correlations. Natural orbitals for nuclear structure 11

rowe2010:collective-motion 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 occupation of lowest orbitals • Minimizing depletion of Fermi sea, not minimizing energy! • Built from many-body calculation, so maybe “aware” of correlations. Natural orbitals for nuclear structure 11

rowe2010:collective-motion 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 occupation of lowest orbitals • Minimizing depletion of Fermi sea, not minimizing energy! • Built from many-body calculation, so maybe “aware” of correlations. Natural orbitals for nuclear structure 11

Natural Orbitals for Nuclear Physics One-Body Reduced Density Matrix (RDM) Natural orbitals are the eigenvectors of the one-body RDM ρ αβ = ⟨ α | ˆ ρ | β ⟩ ∑ | α ⟩ ⟨ Ψ | a † ρ = ˆ α a β | Ψ ⟩ ⟨ β | αβ ∫ ρ ( x , x ′ ) = A Ψ( x , x 2 , . . . , x A )Ψ ∗ ( x ′ , x 2 , . . . , x A ) d x 2 · · · d x A • 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 ρ αα = ⟨ α | Ψ | α ⟩ = ⟨ Ψ | N α | 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

Eigenvector in initial basis: 1 0 s 0 s 0 s 1 s 0 s 1 s 1 s 1 s 2 0 2 4 Density matrix: 0 0 1 2 1 2 0 0 1 2 1 2 0 0 1 2 1 2 0 0 1 2 1 2 Eigenvectors of : 1 1 0 s 1 2 0 s 1 2 1 s 1 2 2 2 1 1 1 s 1 2 0 s 1 2 1 s 1 2 2 2 Eigenvector in natural orbital basis: 0 s 0 s Natural Orbitals – Two examples N N N Four-state, two-orbital system: 0 s 1 / 2 , 1 s 1 / 2 1s 1 / 2 0s 1 / 2 Natural orbitals for nuclear structure 14

Density matrix: 0 0 1 2 1 2 0 0 1 2 1 2 0 0 1 2 1 2 0 0 1 2 1 2 Eigenvectors of : 1 1 0 s 1 2 0 s 1 2 1 s 1 2 2 2 1 1 1 s 1 2 0 s 1 2 1 s 1 2 2 2 Eigenvector in natural orbital basis: 0 s 0 s Natural Orbitals – Two examples Four-state, two-orbital system: 0 s 1 / 2 , 1 s 1 / 2 Eigenvector in initial basis: | Ψ ⟩ = 1 � � � � ⟩ ⟩ ⟩ ⟩ � ( 0 s ↑ )( 0 s ↓ ) � ( 0 s ↑ )( 1 s ↓ ) � ( 0 s ↓ )( 1 s ↑ ) � ( 1 s ↑ )( 1 s ↓ ) 2 ( + − + ) � �� � � �� � � �� � N = 0 N = 2 N = 4 1s 1 / 2 0s 1 / 2 Natural orbitals for nuclear structure 14

Eigenvectors of : 1 1 0 s 1 2 0 s 1 2 1 s 1 2 2 2 1 1 1 s 1 2 0 s 1 2 1 s 1 2 2 2 Eigenvector in natural orbital basis: 0 s 0 s Natural Orbitals – Two examples Four-state, two-orbital system: 0 s 1 / 2 , 1 s 1 / 2 Eigenvector in initial basis: | Ψ ⟩ = 1 � � � � ⟩ ⟩ ⟩ ⟩ � ( 0 s ↑ )( 0 s ↓ ) � ( 0 s ↑ )( 1 s ↓ ) � ( 0 s ↓ )( 1 s ↑ ) � ( 1 s ↑ )( 1 s ↓ ) 2 ( + − + ) � �� � � �� � � �� � N = 0 N = 2 N = 4 Density matrix: 1s 1 / 2   0 0 1 / 1 / 2 2 0 0  1 / 1 /  2 2   ρ =   0 0 1 / 1 / 2 2   0 0 1 / 1 / 2 2 0s 1 / 2 Natural orbitals for nuclear structure 14