Rigorous analysis of ab initio calculations for parabolic quantum dots
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(2) Outline. 1. Background and formalism Overview and motivation Quantum dots as artificial nuclei The Harmonic oscillator and model spaces. 2. The full configuration interaction method Formulation How to analyze FCI Numerical results. 3. Coupled cluster methods (CC) Brief outline of method “Imagined” convergence analysis. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 2 / 44.
(3) Background and formalism. Outline. 1. Background and formalism Overview and motivation Quantum dots as artificial nuclei The Harmonic oscillator and model spaces. 2. The full configuration interaction method Formulation How to analyze FCI Numerical results. 3. Coupled cluster methods (CC) Brief outline of method “Imagined” convergence analysis. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 3 / 44.
(4) Background and formalism. Overview and motivation. Overview of this talk. The ultimate goal of this work is to understand coupled cluster methods (CC) applied to nuclei. One may consider (parabolic) quantum dots as minimal model for nuclei, in a sense “artificial nuclei” We will analyze the full configuration interaction (FCI) method for quantum dots We will also see some illuminating numerical results Finally, we will discuss CC methods and discuss how these may be analyzed rigorously for quantum dots Very little physics, only method talk: an outline of rigorous mathematical analysis. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 4 / 44.
(5) Background and formalism. Overview and motivation. Overview of this talk. The ultimate goal of this work is to understand coupled cluster methods (CC) applied to nuclei. One may consider (parabolic) quantum dots as minimal model for nuclei, in a sense “artificial nuclei” We will analyze the full configuration interaction (FCI) method for quantum dots We will also see some illuminating numerical results Finally, we will discuss CC methods and discuss how these may be analyzed rigorously for quantum dots Very little physics, only method talk: an outline of rigorous mathematical analysis. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 4 / 44.
(6) Background and formalism. Overview and motivation. Overview of this talk. The ultimate goal of this work is to understand coupled cluster methods (CC) applied to nuclei. One may consider (parabolic) quantum dots as minimal model for nuclei, in a sense “artificial nuclei” We will analyze the full configuration interaction (FCI) method for quantum dots We will also see some illuminating numerical results Finally, we will discuss CC methods and discuss how these may be analyzed rigorously for quantum dots Very little physics, only method talk: an outline of rigorous mathematical analysis. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 4 / 44.
(7) Background and formalism. Overview and motivation. Overview of this talk. The ultimate goal of this work is to understand coupled cluster methods (CC) applied to nuclei. One may consider (parabolic) quantum dots as minimal model for nuclei, in a sense “artificial nuclei” We will analyze the full configuration interaction (FCI) method for quantum dots We will also see some illuminating numerical results Finally, we will discuss CC methods and discuss how these may be analyzed rigorously for quantum dots Very little physics, only method talk: an outline of rigorous mathematical analysis. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 4 / 44.
(8) Background and formalism. Overview and motivation. Overview of this talk. The ultimate goal of this work is to understand coupled cluster methods (CC) applied to nuclei. One may consider (parabolic) quantum dots as minimal model for nuclei, in a sense “artificial nuclei” We will analyze the full configuration interaction (FCI) method for quantum dots We will also see some illuminating numerical results Finally, we will discuss CC methods and discuss how these may be analyzed rigorously for quantum dots Very little physics, only method talk: an outline of rigorous mathematical analysis. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 4 / 44.
(9) Background and formalism. Overview and motivation. Motivation. Why do all this rigorous analysis? People disagree on published results. Curse of dimensionality ⇒ computational constraints: dim. of Hilbert space. ∼. exp(A),. A = no. particles. If we don’t understand FCI/CC for quantum dots, then what with nuclei? (See next slides.) Understanding might also lead to new or better methods, or make them easier to implement. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 5 / 44.
(10) Background and formalism. Quantum dots as artificial nuclei. Parabolic quantum dots A harmonic oscillator (HO) trap. We place A electrons in the trap They interact via Coulomb repulsion This gives us the Hamiltonian (h̄ = m = 1 etc) A. H = H0 + V = ∑ h(i) + i=1. with. 1 1 h(i) = − ∇2i + r2i 2 2. 1 u(i, j) 2∑ i6=j. u(i, j) =. λ kri − rj k. (1). λ = O(1) to O(10). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 6 / 44.
(11) Background and formalism. Quantum dots as artificial nuclei. Parabolic quantum dots A harmonic oscillator (HO) trap. We place A electrons in the trap They interact via Coulomb repulsion This gives us the Hamiltonian (h̄ = m = 1 etc) A. H = H0 + V = ∑ h(i) + i=1. with. 1 1 h(i) = − ∇2i + r2i 2 2. 1 u(i, j) 2∑ i6=j. u(i, j) =. λ kri − rj k. (1). λ = O(1) to O(10). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 6 / 44.
(12) Background and formalism. Quantum dots as artificial nuclei. Parabolic quantum dots A harmonic oscillator (HO) trap. We place A electrons in the trap They interact via Coulomb repulsion This gives us the Hamiltonian (h̄ = m = 1 etc) A. H = H0 + V = ∑ h(i) + i=1. with. 1 1 h(i) = − ∇2i + r2i 2 2. 1 u(i, j) 2∑ i6=j. u(i, j) =. λ kri − rj k. (1). λ = O(1) to O(10). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 6 / 44.
(13) Background and formalism. Quantum dots as artificial nuclei. Parabolic quantum dots A harmonic oscillator (HO) trap. We place A electrons in the trap They interact via Coulomb repulsion This gives us the Hamiltonian (h̄ = m = 1 etc) A. H = H0 + V = ∑ h(i) + i=1. with. 1 1 h(i) = − ∇2i + r2i 2 2. 1 u(i, j) 2∑ i6=j. u(i, j) =. λ kri − rj k. (1). λ = O(1) to O(10). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 6 / 44.
(14) Background and formalism. Quantum dots as artificial nuclei. Parabolic quantum dots A harmonic oscillator (HO) trap. We place A electrons in the trap They interact via Coulomb repulsion This gives us the Hamiltonian (h̄ = m = 1 etc) A. H = H0 + V = ∑ h(i) + i=1. with. 1 1 h(i) = − ∇2i + r2i 2 2. 1 u(i, j) 2∑ i6=j. u(i, j) =. λ kri − rj k. (1). λ = O(1) to O(10). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 6 / 44.
(15) Background and formalism. Quantum dots as artificial nuclei. Quantum dots vs. nuclei. There are strong similarities between no-core shell model approach to nuclei and parabolic quantum dots: Quantum dots:. Nuclei:. 1 Rd , d = 1, 2, 3, spin2 HO confinement, h̄ω fixed Singluar two-body interaction λ/krij k Purely discrete spectrum. 1 R3 , spin- , isospin 2 HO pseudo-confinement, h̄ω variational parameter Highly singular NN(N)-interactions; unknown Complicated spectrum, continua. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 7 / 44.
(16) Background and formalism. Quantum dots as artificial nuclei. Quantum dots vs. nuclei. There are strong similarities between no-core shell model approach to nuclei and parabolic quantum dots: Quantum dots:. Nuclei:. 1 Rd , d = 1, 2, 3, spin2 HO confinement, h̄ω fixed Singluar two-body interaction λ/krij k Purely discrete spectrum. 1 R3 , spin- , isospin 2 HO pseudo-confinement, h̄ω variational parameter Highly singular NN(N)-interactions; unknown Complicated spectrum, continua. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 7 / 44.
(17) Background and formalism. Quantum dots as artificial nuclei. Quantum dots vs. nuclei. There are strong similarities between no-core shell model approach to nuclei and parabolic quantum dots: Quantum dots:. Nuclei:. 1 Rd , d = 1, 2, 3, spin2 HO confinement, h̄ω fixed Singluar two-body interaction λ/krij k Purely discrete spectrum. 1 R3 , spin- , isospin 2 HO pseudo-confinement, h̄ω variational parameter Highly singular NN(N)-interactions; unknown Complicated spectrum, continua. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 7 / 44.
(18) Background and formalism. Quantum dots as artificial nuclei. Quantum dots vs. nuclei. There are strong similarities between no-core shell model approach to nuclei and parabolic quantum dots: Quantum dots:. Nuclei:. 1 Rd , d = 1, 2, 3, spin2 HO confinement, h̄ω fixed Singluar two-body interaction λ/krij k Purely discrete spectrum. 1 R3 , spin- , isospin 2 HO pseudo-confinement, h̄ω variational parameter Highly singular NN(N)-interactions; unknown Complicated spectrum, continua. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 7 / 44.
(19) Background and formalism. Quantum dots as artificial nuclei. Quantum dots vs. nuclei. There are strong similarities between no-core shell model approach to nuclei and parabolic quantum dots: Quantum dots:. Nuclei:. 1 Rd , d = 1, 2, 3, spin2 HO confinement, h̄ω fixed Singluar two-body interaction λ/krij k Purely discrete spectrum. 1 R3 , spin- , isospin 2 HO pseudo-confinement, h̄ω variational parameter Highly singular NN(N)-interactions; unknown Complicated spectrum, continua. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 7 / 44.
(20) Background and formalism. The Harmonic oscillator and model spaces. Multi-indices. We need the concept of a multi-index to ease notation. Definition (Multi-index) A tuple of d integers nj : n = (n1 , n2 , · · · , nd ),. nj ≥ 0.. We think of it as a vector of integers. The “length” of n: |n| = n1 + n2 + · · · + nd . We will use the multi-index to specify q.n.’s in each spatial direction x, y, z, . . .. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 8 / 44.
(21) Background and formalism. The Harmonic oscillator and model spaces. The harmonic oscillator Hamiltonian I suppose we all know the HO and it’s eigenfunctions: A. H0 = ∑ h(i) = ∑ α c†α cα i=1. α. Here, α ≡ (n, σ) = (space q.n., spin q.n.) Single-particle functions φα (x): φα (x) = φn (r) χσ (s) , | {z } | {z } space w.f. spin w.f.. d α = |n| + |{z} 2 shell. Separation of variables: φn (r) ≡ φn1 (r1 ) · · · φnd (rd ). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 9 / 44.
(22) Background and formalism. The Harmonic oscillator and model spaces. The harmonic oscillator Hamiltonian I suppose we all know the HO and it’s eigenfunctions: A. H0 = ∑ h(i) = ∑ α c†α cα i=1. α. Here, α ≡ (n, σ) = (space q.n., spin q.n.) Single-particle functions φα (x): φα (x) = φn (r) χσ (s) , | {z } | {z } space w.f. spin w.f.. d α = |n| + |{z} 2 shell. Separation of variables: φn (r) ≡ φn1 (r1 ) · · · φnd (rd ). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 9 / 44.
(23) Background and formalism. The Harmonic oscillator and model spaces. The harmonic oscillator Hamiltonian I suppose we all know the HO and it’s eigenfunctions: A. H0 = ∑ h(i) = ∑ α c†α cα i=1. α. Here, α ≡ (n, σ) = (space q.n., spin q.n.) Single-particle functions φα (x): φα (x) = φn (r) χσ (s) , | {z } | {z } space w.f. spin w.f.. d α = |n| + |{z} 2 shell. Separation of variables: φn (r) ≡ φn1 (r1 ) · · · φnd (rd ). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 9 / 44.
(24) Background and formalism. The Harmonic oscillator and model spaces. The harmonic oscillator Hamiltonian I suppose we all know the HO and it’s eigenfunctions: A. H0 = ∑ h(i) = ∑ α c†α cα i=1. α. Here, α ≡ (n, σ) = (space q.n., spin q.n.) Single-particle functions φα (x): φα (x) = φn (r) χσ (s) , | {z } | {z } space w.f. spin w.f.. d α = |n| + |{z} 2 shell. Separation of variables: φn (r) ≡ φn1 (r1 ) · · · φnd (rd ). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 9 / 44.
(25) Background and formalism. The Harmonic oscillator and model spaces. Examples of quantum numbers and shells n. |n|. |n| = nx + ny shell |n| = 4. one dim. general case. two dim. Finally, c†α creates particle in orbital α c†α1 c†α2 · · · c†αA |−i ≡ |α1 α2 · · · αA i | {z }. Slater determinant. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 10 / 44.
(26) Background and formalism. The Harmonic oscillator and model spaces. Examples of quantum numbers and shells n. |n|. |n| = nx + ny shell |n| = 4. one dim. general case. two dim. Finally, c†α creates particle in orbital α c†α1 c†α2 · · · c†αA |−i ≡ |α1 α2 · · · αA i {z } |. Slater determinant. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 10 / 44.
(27) Background and formalism. The Harmonic oscillator and model spaces. Single-particle function expansions Arbitrary single-particle functions expanded in HO functions: |ψi =. ∑ |αi hα|ψi = ∑ cα |αi α. α. ψ(x) = hx|ψi = ∑ cα φα (x) α. Projects onto. Expansion in eigenspaces:. space with HO. ∞. |ψi =. energy N + d/2. ∑ PN |ψi N=0. We define shell-probability p(N): p(N) ≡ hψ|PN |ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 11 / 44.
(28) Background and formalism. The Harmonic oscillator and model spaces. Single-particle function expansions Arbitrary single-particle functions expanded in HO functions: |ψi =. ∑ |αi hα|ψi = ∑ cα |αi α. α. ψ(x) = hx|ψi = ∑ cα φα (x) α. Projects onto. Expansion in eigenspaces:. space with HO. ∞. |ψi =. energy N + d/2. ∑ PN |ψi N=0. We define shell-probability p(N): p(N) ≡ hψ|PN |ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 11 / 44.
(29) Background and formalism. The Harmonic oscillator and model spaces. Single-particle function expansions Arbitrary single-particle functions expanded in HO functions: |ψi =. ∑ |αi hα|ψi = ∑ cα |αi α. α. ψ(x) = hx|ψi = ∑ cα φα (x) α. Projects onto. Expansion in eigenspaces:. space with HO. ∞. |ψi =. energy N + d/2. ∑ PN |ψi N=0. We define shell-probability p(N): p(N) ≡ hψ|PN |ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 11 / 44.
(30) Background and formalism. The Harmonic oscillator and model spaces. The shell-probability Recall the shell-probability p(N): p(N) ≡ hψ|PN |ψi ,. PN projects onto N’th shell. We have: p(N) =. ∑. |cn |2. |n|=N. n2. n2 N. N n. N n1. n1 n3. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 12 / 44.
(31) Background and formalism. The Harmonic oscillator and model spaces. Many-body harmonic oscillator It is important to keep in mind that: Mathematically, we may treat the many-body |Ψi as a higher-dimensional one-body function! Trivial separation property of HO gives: A 1 1 H0 = ∑ h(i) = − ∇2R + R2 , 2 2 i=1. R = (r1 , · · · , rA ) ∈ RAd. The Slater determinants are eigenfunctions: H0 |α1 · · · αA i = (α1 + · · · + αA ) |α1 · · · αA i “Shell number” for this interpretation: N = |N| = |n1 | + |n2 | + · · · + |nA |. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 13 / 44.
(32) Background and formalism. The Harmonic oscillator and model spaces. Many-body harmonic oscillator It is important to keep in mind that: Mathematically, we may treat the many-body |Ψi as a higher-dimensional one-body function! Trivial separation property of HO gives: A 1 1 H0 = ∑ h(i) = − ∇2R + R2 , 2 2 i=1. R = (r1 , · · · , rA ) ∈ RAd. The Slater determinants are eigenfunctions: H0 |α1 · · · αA i = (α1 + · · · + αA ) |α1 · · · αA i “Shell number” for this interpretation: N = |N| = |n1 | + |n2 | + · · · + |nA |. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 13 / 44.
(33) Background and formalism. The Harmonic oscillator and model spaces. Many-body harmonic oscillator It is important to keep in mind that: Mathematically, we may treat the many-body |Ψi as a higher-dimensional one-body function! Trivial separation property of HO gives: A 1 1 H0 = ∑ h(i) = − ∇2R + R2 , 2 2 i=1. R = (r1 , · · · , rA ) ∈ RAd. The Slater determinants are eigenfunctions: H0 |α1 · · · αA i = (α1 + · · · + αA ) |α1 · · · αA i “Shell number” for this interpretation: N = |N| = |n1 | + |n2 | + · · · + |nA |. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 13 / 44.
(34) Background and formalism. The Harmonic oscillator and model spaces. Many-body harmonic oscillator It is important to keep in mind that: Mathematically, we may treat the many-body |Ψi as a higher-dimensional one-body function! Trivial separation property of HO gives: A 1 1 H0 = ∑ h(i) = − ∇2R + R2 , 2 2 i=1. R = (r1 , · · · , rA ) ∈ RAd. The Slater determinants are eigenfunctions: H0 |α1 · · · αA i = (α1 + · · · + αA ) |α1 · · · αA i “Shell number” for this interpretation: N = |N| = |n1 | + |n2 | + · · · + |nA |. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 13 / 44.
(35) Background and formalism. The Harmonic oscillator and model spaces. Many-particle function expansions. Arbitrary many-body functions expanded in HO Slater det.’s: |Ψi =. ∑. α1 ···αA. cα1 ···αA |α1 · · · αA i Projects onto space with. Expansion in eigenspaces:. total HO energy N + Ad/2. ∞. |Ψi =. ∑ PN |Ψi N=0. Again, we define “shell”-probability p(N): p(N) ≡ hΨ|PN |Ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 14 / 44.
(36) Background and formalism. The Harmonic oscillator and model spaces. Many-particle function expansions. Arbitrary many-body functions expanded in HO Slater det.’s: |Ψi =. ∑. α1 ···αA. cα1 ···αA |α1 · · · αA i Projects onto space with. Expansion in eigenspaces:. total HO energy N + Ad/2. ∞. |Ψi =. ∑ PN |Ψi N=0. Again, we define “shell”-probability p(N): p(N) ≡ hΨ|PN |Ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 14 / 44.
(37) Background and formalism. The Harmonic oscillator and model spaces. Many-particle function expansions. Arbitrary many-body functions expanded in HO Slater det.’s: |Ψi =. ∑. α1 ···αA. cα1 ···αA |α1 · · · αA i Projects onto space with. Expansion in eigenspaces:. total HO energy N + Ad/2. ∞. |Ψi =. ∑ PN |Ψi N=0. Again, we define “shell”-probability p(N): p(N) ≡ hΨ|PN |Ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 14 / 44.
(38) Background and formalism. The Harmonic oscillator and model spaces. Model spaces: cutting down ∞ dimensions. Direct product space: Allow only |n| ≤ Nmax in single-particle space. VDP = Span {|α1 · · · αA i | max |ni | ≤ Nmax } ⊂ HA. α2. ← (complete A-body Hilbert space). Energy cut space: Restrict total HO energy instead: ( VEC = Span |α1 · · · αA i | α1. Simen Kvaal (University of Oslo). ). ∑ |ni | = N ≤ Nmax i. = (P0 + P1 + · · · + PNmax )HA. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 15 / 44.
(39) Background and formalism. The Harmonic oscillator and model spaces. Model spaces: cutting down ∞ dimensions. Direct product space: Allow only |n| ≤ Nmax in single-particle space. VDP = Span {|α1 · · · αA i | max |ni | ≤ Nmax } ⊂ HA. α2. ← (complete A-body Hilbert space). Energy cut space: Restrict total HO energy instead: ( VEC = Span |α1 · · · αA i | α1. Simen Kvaal (University of Oslo). ). ∑ |ni | = N ≤ Nmax i. = (P0 + P1 + · · · + PNmax )HA. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 15 / 44.
(40) The full configuration interaction method. Outline. 1. Background and formalism Overview and motivation Quantum dots as artificial nuclei The Harmonic oscillator and model spaces. 2. The full configuration interaction method Formulation How to analyze FCI Numerical results. 3. Coupled cluster methods (CC) Brief outline of method “Imagined” convergence analysis. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 16 / 44.
(41) The full configuration interaction method. Formulation. Configuration Interaction (CI) Variational formulation of eigenvalue problem: Find the |Ψi ∈ H that minimizes the energy: E = min. |Ψi∈H. hΨ|H|Ψi hΨ|Ψi. Rayleigh-Ritz: Restrict to model space V ⊂ H : Find the |Ψh i ∈ V that minimizes the energy: Eh = min. |Ψh i∈V. hΨh |H|Ψh i hΨh |Ψh i. This is CI with respect to V , using VDP or VEC gives FCI. Let |Φi i be basis for V . We obtain the matrix formulation Huh = Eh uh ,. Simen Kvaal (University of Oslo). Hij = hΦi |H|Φj i. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 17 / 44.
(42) The full configuration interaction method. Formulation. Configuration Interaction (CI) Variational formulation of eigenvalue problem: Find the |Ψi ∈ H that minimizes the energy: E = min. |Ψi∈H. hΨ|H|Ψi hΨ|Ψi. Rayleigh-Ritz: Restrict to model space V ⊂ H : Find the |Ψh i ∈ V that minimizes the energy: Eh = min. |Ψh i∈V. hΨh |H|Ψh i hΨh |Ψh i. This is CI with respect to V , using VDP or VEC gives FCI. Let |Φi i be basis for V . We obtain the matrix formulation Huh = Eh uh ,. Simen Kvaal (University of Oslo). Hij = hΦi |H|Φj i. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 17 / 44.
(43) The full configuration interaction method. Formulation. Configuration Interaction (CI) Variational formulation of eigenvalue problem: Find the |Ψi ∈ H that minimizes the energy: E = min. |Ψi∈H. hΨ|H|Ψi hΨ|Ψi. Rayleigh-Ritz: Restrict to model space V ⊂ H : Find the |Ψh i ∈ V that minimizes the energy: Eh = min. |Ψh i∈V. hΨh |H|Ψh i hΨh |Ψh i. This is CI with respect to V , using VDP or VEC gives FCI. Let |Φi i be basis for V . We obtain the matrix formulation Huh = Eh uh ,. Simen Kvaal (University of Oslo). Hij = hΦi |H|Φj i. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 17 / 44.
(44) The full configuration interaction method. Formulation. Configuration Interaction (CI) Variational formulation of eigenvalue problem: Find the |Ψi ∈ H that minimizes the energy: E = min. |Ψi∈H. hΨ|H|Ψi hΨ|Ψi. Rayleigh-Ritz: Restrict to model space V ⊂ H : Find the |Ψh i ∈ V that minimizes the energy: Eh = min. |Ψh i∈V. hΨh |H|Ψh i hΨh |Ψh i. This is CI with respect to V , using VDP or VEC gives FCI. Let |Φi i be basis for V . We obtain the matrix formulation Huh = Eh uh ,. Simen Kvaal (University of Oslo). Hij = hΦi |H|Φj i. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 17 / 44.
(45) The full configuration interaction method. How to analyze FCI. Questions of accuracy and convergence When studying convergence, accuray, etc., norms are useful: = hΨ|Ψi. ← standard L2 norm. kΨk21. = hΨ|H0 |Ψi. ← “energy norm”. i |Ψ. V. We study errors of the approximations: |δΨi. =. |Ψh i − |Ψi. δE. =. Eh − E. Q |Ψi. =. (1 − P) |Ψi. Simen Kvaal (University of Oslo). H. |δΨi. kΨk2. |Ψ i h. ← error in numerical solution. ← error in energy ← error in projection onto V. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 18 / 44.
(46) The full configuration interaction method. How to analyze FCI. Questions of accuracy and convergence When studying convergence, accuray, etc., norms are useful: = hΨ|Ψi. ← standard L2 norm. kΨk21. = hΨ|H0 |Ψi. ← “energy norm”. =. |Ψh i − |Ψi. δE. =. Eh − E. Q |Ψi. =. (1 − P) |Ψi. Simen Kvaal (University of Oslo). i |Ψ. V. We study errors of the approximations: |δΨi. H. |δΨi. kΨk2. |Ψ i h. ← error in numerical solution. ← error in energy ← error in projection onto V. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 18 / 44.
(47) The full configuration interaction method. How to analyze FCI. Questions of accuracy and convergence When studying convergence, accuray, etc., norms are useful: ← standard L2 norm. kΨk21. = hΨ|H0 |Ψi. H. ← “energy norm”. i |Ψ. P |Ψi V. We study errors of the approximations: |δΨi. =. |Ψh i − |Ψi. δE. =. Eh − E. Q |Ψi. =. (1 − P) |Ψi. Simen Kvaal (University of Oslo). Q |Ψi. = hΨ|Ψi. |δΨi. kΨk2. |Ψ i h. ← error in numerical solution. ← error in energy ← error in projection onto V. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 18 / 44.
(48) The full configuration interaction method. How to analyze FCI. Questions of accuracy and convergence When studying convergence, accuray, etc., norms are useful: ← standard L2 norm. kΨk21. = hΨ|H0 |Ψi. H. ← “energy norm”. i |Ψ. P |Ψi V. We study errors of the approximations: |δΨi. =. |Ψh i − |Ψi. δE. =. Eh − E. Q |Ψi. =. (1 − P) |Ψi. Simen Kvaal (University of Oslo). Q |Ψi. = hΨ|Ψi. |δΨi. kΨk2. |Ψ i h. ← error in numerical solution. ← error in energy ← error in projection onto V. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 18 / 44.
(49) The full configuration interaction method. How to analyze FCI. Questions of accuracy and convergence II. Theorem (A priori error estimate (see Babuska and Osborn)) There exists a constant C1 , dependent on u(i, j) only, such that the error |δΨi is bounded by kδΨk ≤ C1 kQΨk1 = C1 hQΨ|H0 |QΨi1/2 . There exists a constant C2 such that the energy error is bounded by δE ≤ C2 kδΨk2 ≤ C2 C1 kQΨk21 . We need to understand . . . Approximating properties of basis function expansions in |Φi i. “What does P |Ψi capture?” Behaviour of exact A-body wave function |Ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 19 / 44.
(50) The full configuration interaction method. How to analyze FCI. Questions of accuracy and convergence II. Theorem (A priori error estimate (see Babuska and Osborn)) There exists a constant C1 , dependent on u(i, j) only, such that the error |δΨi is bounded by kδΨk ≤ C1 kQΨk1 = C1 hQΨ|H0 |QΨi1/2 . There exists a constant C2 such that the energy error is bounded by δE ≤ C2 kδΨk2 ≤ C2 C1 kQΨk21 . We need to understand . . . Approximating properties of basis function expansions in |Φi i. “What does P |Ψi capture?” Behaviour of exact A-body wave function |Ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 19 / 44.
(51) The full configuration interaction method. How to analyze FCI. Questions of accuracy and convergence II. Theorem (A priori error estimate (see Babuska and Osborn)) There exists a constant C1 , dependent on u(i, j) only, such that the error |δΨi is bounded by kδΨk ≤ C1 kQΨk1 = C1 hQΨ|H0 |QΨi1/2 . There exists a constant C2 such that the energy error is bounded by δE ≤ C2 kδΨk2 ≤ C2 C1 kQΨk21 . We need to understand . . . Approximating properties of basis function expansions in |Φi i. “What does P |Ψi capture?” Behaviour of exact A-body wave function |Ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 19 / 44.
(52) The full configuration interaction method. How to analyze FCI. Properties of basis functions. Already seen: |Φi i are harmonic oscillator eigenfunctions in Ad dimensions. Let P project onto V = VEC , and Q = 1 − P: P = P0 + P1 + · · · + PNmax Consider expansion of some wave function |Ψi: D. |Ψi = (P + Q) ∑ ci |Φi i = ∑ ci |Φi i + i. i=1. ∞. ∑. ci |Φi i. i=D+1. P |Ψi is the best approximation in V , in both k · k and k · k1 norms. If the ci , i > D are small, it is also good Behaviour of ci will be related to the analytic properties of |Ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 20 / 44.
(53) The full configuration interaction method. How to analyze FCI. Properties of basis functions. Already seen: |Φi i are harmonic oscillator eigenfunctions in Ad dimensions. Let P project onto V = VEC , and Q = 1 − P: P = P0 + P1 + · · · + PNmax Consider expansion of some wave function |Ψi: D. |Ψi = (P + Q) ∑ ci |Φi i = ∑ ci |Φi i + i. i=1. ∞. ∑. ci |Φi i. i=D+1. P |Ψi is the best approximation in V , in both k · k and k · k1 norms. If the ci , i > D are small, it is also good Behaviour of ci will be related to the analytic properties of |Ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 20 / 44.
(54) The full configuration interaction method. How to analyze FCI. Properties of basis functions. Already seen: |Φi i are harmonic oscillator eigenfunctions in Ad dimensions. Let P project onto V = VEC , and Q = 1 − P: P = P0 + P1 + · · · + PNmax Consider expansion of some wave function |Ψi: D. |Ψi = (P + Q) ∑ ci |Φi i = ∑ ci |Φi i + i. i=1. ∞. ∑. ci |Φi i. i=D+1. P |Ψi is the best approximation in V , in both k · k and k · k1 norms. If the ci , i > D are small, it is also good Behaviour of ci will be related to the analytic properties of |Ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 20 / 44.
(55) The full configuration interaction method. How to analyze FCI. Properties of basis functions. Already seen: |Φi i are harmonic oscillator eigenfunctions in Ad dimensions. Let P project onto V = VEC , and Q = 1 − P: P = P0 + P1 + · · · + PNmax Consider expansion of some wave function |Ψi: D. |Ψi = (P + Q) ∑ ci |Φi i = ∑ ci |Φi i + i. i=1. ∞. ∑. ci |Φi i. i=D+1. P |Ψi is the best approximation in V , in both k · k and k · k1 norms. If the ci , i > D are small, it is also good Behaviour of ci will be related to the analytic properties of |Ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 20 / 44.
(56) The full configuration interaction method. How to analyze FCI. Properties of basis functions. Already seen: |Φi i are harmonic oscillator eigenfunctions in Ad dimensions. Let P project onto V = VEC , and Q = 1 − P: P = P0 + P1 + · · · + PNmax Consider expansion of some wave function |Ψi: D. |Ψi = (P + Q) ∑ ci |Φi i = ∑ ci |Φi i + i. i=1. ∞. ∑. ci |Φi i. i=D+1. P |Ψi is the best approximation in V , in both k · k and k · k1 norms. If the ci , i > D are small, it is also good Behaviour of ci will be related to the analytic properties of |Ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 20 / 44.
(57) The full configuration interaction method. How to analyze FCI. Properties of basis functions. Already seen: |Φi i are harmonic oscillator eigenfunctions in Ad dimensions. Let P project onto V = VEC , and Q = 1 − P: P = P0 + P1 + · · · + PNmax Consider expansion of some wave function |Ψi: D. |Ψi = (P + Q) ∑ ci |Φi i = ∑ ci |Φi i + i. i=1. ∞. ∑. ci |Φi i. i=D+1. P |Ψi is the best approximation in V , in both k · k and k · k1 norms. If the ci , i > D are small, it is also good Behaviour of ci will be related to the analytic properties of |Ψi. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 20 / 44.
(58) The full configuration interaction method. How to analyze FCI. Properties of basis functions for A = d = 1. For A = d = 1, we have hx|Φi (x)i −→ φn (x): √ 2 φn (x) = (2n n! π)−1/2 Hn (x)e−x /2 Exponential fall-off, smooth, increasing number of oscillations. x. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 21 / 44.
(59) The full configuration interaction method. How to analyze FCI. Properties of basis functions for A = d = 1. For A = d = 1, we have hx|Φi (x)i −→ φn (x): √ 2 φn (x) = (2n n! π)−1/2 Hn (x)e−x /2 Exponential fall-off, smooth, increasing number of oscillations. x. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 21 / 44.
(60) The full configuration interaction method. How to analyze FCI. Properties of basis functions for A = d = 1. For A = d = 1, we have hx|Φi (x)i −→ φn (x): √ 2 φn (x) = (2n n! π)−1/2 Hn (x)e−x /2 Exponential fall-off, smooth, increasing number of oscillations. x φ0 (x). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 21 / 44.
(61) The full configuration interaction method. How to analyze FCI. Properties of basis functions for A = d = 1. For A = d = 1, we have hx|Φi (x)i −→ φn (x): √ 2 φn (x) = (2n n! π)−1/2 Hn (x)e−x /2 Exponential fall-off, smooth, increasing number of oscillations. x φ1 (x). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 21 / 44.
(62) The full configuration interaction method. How to analyze FCI. Properties of basis functions for A = d = 1. For A = d = 1, we have hx|Φi (x)i −→ φn (x): √ 2 φn (x) = (2n n! π)−1/2 Hn (x)e−x /2 Exponential fall-off, smooth, increasing number of oscillations. x φ2 (x). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 21 / 44.
(63) The full configuration interaction method. How to analyze FCI. Properties of basis functions for A = d = 1. For A = d = 1, we have hx|Φi (x)i −→ φn (x): √ 2 φn (x) = (2n n! π)−1/2 Hn (x)e−x /2 Exponential fall-off, smooth, increasing number of oscillations. x φ3 (x). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 21 / 44.
(64) The full configuration interaction method. How to analyze FCI. Properties of basis functions for A = d = 1. For A = d = 1, we have hx|Φi (x)i −→ φn (x): √ 2 φn (x) = (2n n! π)−1/2 Hn (x)e−x /2 Exponential fall-off, smooth, increasing number of oscillations. x φ4 (x). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 21 / 44.
(65) The full configuration interaction method. How to analyze FCI. Properties of basis functions for A = d = 1. For A = d = 1, we have hx|Φi (x)i −→ φn (x): √ 2 φn (x) = (2n n! π)−1/2 Hn (x)e−x /2 Exponential fall-off, smooth, increasing number of oscillations. x φ5 (x). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 21 / 44.
(66) The full configuration interaction method. How to analyze FCI. Properties of basis functions for A = d = 1. For A = d = 1, we have hx|Φi (x)i −→ φn (x): √ 2 φn (x) = (2n n! π)−1/2 Hn (x)e−x /2 Exponential fall-off, smooth, increasing number of oscillations. x φ6 (x). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 21 / 44.
(67) The full configuration interaction method. How to analyze FCI. Properties of basis functions for A = d = 1. For A = d = 1, we have hx|Φi (x)i −→ φn (x): √ 2 φn (x) = (2n n! π)−1/2 Hn (x)e−x /2 Exponential fall-off, smooth, increasing number of oscillations. x φ10 (x). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 21 / 44.
(68) The full configuration interaction method. How to analyze FCI. Properties of basis functions for A = d = 1. For A = d = 1, we have hx|Φi (x)i −→ φn (x): √ 2 φn (x) = (2n n! π)−1/2 Hn (x)e−x /2 Exponential fall-off, smooth, increasing number of oscillations. x φ20 (x). Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 21 / 44.
(69) The full configuration interaction method. How to analyze FCI. Hermite function approximation: 1D result Consider the expansion ψ(x) =. ∞. ∑ cn φn (x). n=0. Theorem (Approximation by Hermite functions (See S.K. ’09)) Assume ψ(x) falls off exponentially. Then ψ(x) ∈ H k (R) if and only if ∞. ∑ |cn |2 nk. <. +∞. n=0. That is, p(n) = |cn |2 ∼ n−(k+1) . What is H k (R)? All (weak) partial derivatives up to order k exist and are L2 -integrable Rapid fall-off of p(n) (and cn ) ⇔ ψ(x) is smooth Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 22 / 44.
(70) The full configuration interaction method. How to analyze FCI. Hermite function approximation: 1D result Consider the expansion ψ(x) =. ∞. ∑ cn φn (x). n=0. Theorem (Approximation by Hermite functions (See S.K. ’09)) Assume ψ(x) falls off exponentially. Then ψ(x) ∈ H k (R) if and only if ∞. ∑ |cn |2 nk. <. +∞. n=0. That is, p(n) = |cn |2 ∼ n−(k+1) . What is H k (R)? All (weak) partial derivatives up to order k exist and are L2 -integrable Rapid fall-off of p(n) (and cn ) ⇔ ψ(x) is smooth Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 22 / 44.
(71) The full configuration interaction method. How to analyze FCI. Hermite function approximation: 1D result Consider the expansion ψ(x) =. ∞. ∑ cn φn (x). n=0. Theorem (Approximation by Hermite functions (See S.K. ’09)) Assume ψ(x) falls off exponentially. Then ψ(x) ∈ H k (R) if and only if ∞. ∑ |cn |2 nk. <. +∞. n=0. That is, p(n) = |cn |2 ∼ n−(k+1) . What is H k (R)? All (weak) partial derivatives up to order k exist and are L2 -integrable Rapid fall-off of p(n) (and cn ) ⇔ ψ(x) is smooth Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 22 / 44.
(72) The full configuration interaction method. How to analyze FCI. Hermite function approximation: 1D result Consider the expansion ψ(x) =. ∞. ∑ cn φn (x). n=0. Theorem (Approximation by Hermite functions (See S.K. ’09)) Assume ψ(x) falls off exponentially. Then ψ(x) ∈ H k (R) if and only if ∞. ∑ |cn |2 nk. <. +∞. n=0. That is, p(n) = |cn |2 ∼ n−(k+1) . What is H k (R)? All (weak) partial derivatives up to order k exist and are L2 -integrable Rapid fall-off of p(n) (and cn ) ⇔ ψ(x) is smooth Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 22 / 44.
(73) The full configuration interaction method. How to analyze FCI. Hermite function approximation: 1D result Consider the expansion ψ(x) =. ∞. ∑ cn φn (x). n=0. Theorem (Approximation by Hermite functions (See S.K. ’09)) Assume ψ(x) falls off exponentially. Then ψ(x) ∈ H k (R) if and only if ∞. ∑ |cn |2 nk. <. +∞. n=0. That is, p(n) = |cn |2 ∼ n−(k+1) . What is H k (R)? All (weak) partial derivatives up to order k exist and are L2 -integrable Rapid fall-off of p(n) (and cn ) ⇔ ψ(x) is smooth Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 22 / 44.
(74) The full configuration interaction method. How to analyze FCI. Numerical calculation. 102. We consider f (x) = (1 + 2|x|)e−x. log(|cn |). g(x) = f 0 (x) We have:. 100. 10−2. |cn | ∼ n−1.28 |cn | ∼ n−0.74 Notice: k not necessarily an integer!. Simen Kvaal (University of Oslo). 2 /2. 10−4 0 10. 101. 102. 103. log(n + 1). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 23 / 44.
(75) The full configuration interaction method. How to analyze FCI. Numerical calculation. 102. We consider f (x) = (1 + 2|x|)e−x. log(|cn |). g(x) = f 0 (x) We have:. 100. 10−2. |cn | ∼ n−1.28 |cn | ∼ n−0.74 Notice: k not necessarily an integer!. Simen Kvaal (University of Oslo). 2 /2. 10−4 0 10. 101. 102. 103. log(n + 1). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 23 / 44.
(76) The full configuration interaction method. How to analyze FCI. Numerical calculation. 102. We consider f (x) = (1 + 2|x|)e−x. log(|cn |). g(x) = f 0 (x) We have:. 100. 10−2. |cn | ∼ n−1.28 |cn | ∼ n−0.74 Notice: k not necessarily an integer!. Simen Kvaal (University of Oslo). 2 /2. 10−4 0 10. 101. 102. 103. log(n + 1). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 23 / 44.
(77) The full configuration interaction method. How to analyze FCI. Weak differentiability, again Would it be sufficient to consider standard derivatives of ψ(x)? No! The concept of weak differentiability is essential to this result. Both the below functions are smooth everywhere except at x = 0. But the jump discontinuity distinguishes the two.. |cn | ∼ n−1. −1/2. |cn | ∼ n. Simen Kvaal (University of Oslo). f (x) ∈ H 1 (R) x. f 0 (x) ∈ H 0 (R) x. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 24 / 44.
(78) The full configuration interaction method. How to analyze FCI. Weak differentiability, again Would it be sufficient to consider standard derivatives of ψ(x)? No! The concept of weak differentiability is essential to this result. Both the below functions are smooth everywhere except at x = 0. But the jump discontinuity distinguishes the two.. |cn | ∼ n−1. −1/2. |cn | ∼ n. Simen Kvaal (University of Oslo). f (x) ∈ H 1 (R) x. f 0 (x) ∈ H 0 (R) x. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 24 / 44.
(79) The full configuration interaction method. How to analyze FCI. Weak differentiability, again Would it be sufficient to consider standard derivatives of ψ(x)? No! The concept of weak differentiability is essential to this result. Both the below functions are smooth everywhere except at x = 0. But the jump discontinuity distinguishes the two.. |cn | ∼ n−1. −1/2. |cn | ∼ n. Simen Kvaal (University of Oslo). f (x) ∈ H 1 (R) x. f 0 (x) ∈ H 0 (R) x. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 24 / 44.
(80) The full configuration interaction method. How to analyze FCI. Generalization to d dimensions Consider expansion of ψ(r), ψ(r) =. ∑ cn φn (r) n ∞. ∞. ∑ ··· ∑. =. n1 =0. cn1 ···nd φn1 (r1 ) · · · φnd (rd ). nd =0. How do we study the limit “large n” as we have a d-dimensional array of coefficients? n2. n. n2. n1. n1 n3. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 25 / 44.
(81) The full configuration interaction method. How to analyze FCI. Generalization to d dimensions II Solution: Study behaviour of shell probability p(N): p(N) ≡ hψ|PN |ψi ,. PN projects onto N’th shell. We have: p(N) =. ∑. |cn |2. |n|=N. n2. n2 N. N n. N n1. n1 n3. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 26 / 44.
(82) The full configuration interaction method. How to analyze FCI. Harmonic oscillator function approximation: general result Consider the expansion ψ(r) = ∑ cn φn (r). n. Theorem (Approximation by h.o. functions (See S.K. ’09)) Assume ψ(r) falls off exponentially. Then ψ(r) ∈ H k (Rd ) if and only if ∞. ∑ p(N)N k. <. +∞. N=0. That is, p(N) ∼ N −(k+1) . Rapid fall-off of p(N) ⇔ ψ(r) is smooth Notice: Valid for many-body wave-functions as well!. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 27 / 44.
(83) The full configuration interaction method. How to analyze FCI. Harmonic oscillator function approximation: general result Consider the expansion ψ(r) = ∑ cn φn (r). n. Theorem (Approximation by h.o. functions (See S.K. ’09)) Assume ψ(r) falls off exponentially. Then ψ(r) ∈ H k (Rd ) if and only if ∞. ∑ p(N)N k. <. +∞. N=0. That is, p(N) ∼ N −(k+1) . Rapid fall-off of p(N) ⇔ ψ(r) is smooth Notice: Valid for many-body wave-functions as well!. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 27 / 44.
(84) The full configuration interaction method. How to analyze FCI. Harmonic oscillator function approximation: general result Consider the expansion ψ(r) = ∑ cn φn (r). n. Theorem (Approximation by h.o. functions (See S.K. ’09)) Assume ψ(r) falls off exponentially. Then ψ(r) ∈ H k (Rd ) if and only if ∞. ∑ p(N)N k. <. +∞. N=0. That is, p(N) ∼ N −(k+1) . Rapid fall-off of p(N) ⇔ ψ(r) is smooth Notice: Valid for many-body wave-functions as well!. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 27 / 44.
(85) The full configuration interaction method. How to analyze FCI. Harmonic oscillator function approximation: general result Consider the expansion ψ(r) = ∑ cn φn (r). n. Theorem (Approximation by h.o. functions (See S.K. ’09)) Assume ψ(r) falls off exponentially. Then ψ(r) ∈ H k (Rd ) if and only if ∞. ∑ p(N)N k. <. +∞. N=0. That is, p(N) ∼ N −(k+1) . Rapid fall-off of p(N) ⇔ ψ(r) is smooth Notice: Valid for many-body wave-functions as well!. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 27 / 44.
(86) The full configuration interaction method. How to analyze FCI. Approximation in model space Let’s recall the “hyper-pyramid”/energy cut model space VEC : VA = Span {Slater det’s with h.o. energy ≤ Emax } As the Slater determinants are Ad-dimensional h.o. eigenfunctions, VA. = Span {Slater det’s with h.o. energy ≤ Nmax + Ad/2} = (P0 + P1 + · · · + PNmax ) H | {z } ≡P. We obtain for the error in the norm kQΨk2 =. ∞. ∑. −k p(N) ∼ Nmax. N=Nmax +1. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 28 / 44.
(87) The full configuration interaction method. How to analyze FCI. Approximation in model space Let’s recall the “hyper-pyramid”/energy cut model space VEC : VA = Span {Slater det’s with h.o. energy ≤ Emax } As the Slater determinants are Ad-dimensional h.o. eigenfunctions, VA. = Span {Slater det’s with h.o. energy ≤ Nmax + Ad/2} = (P0 + P1 + · · · + PNmax ) H | {z } ≡P. We obtain for the error in the norm kQΨk2 =. ∞. ∑. −k p(N) ∼ Nmax. N=Nmax +1. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 28 / 44.
(88) The full configuration interaction method. How to analyze FCI. Approximation in model space Let’s recall the “hyper-pyramid”/energy cut model space VEC : VA = Span {Slater det’s with h.o. energy ≤ Emax } As the Slater determinants are Ad-dimensional h.o. eigenfunctions, VA. = Span {Slater det’s with h.o. energy ≤ Nmax + Ad/2} = (P0 + P1 + · · · + PNmax ) H | {z } ≡P. We obtain for the error in the norm kQΨk2 =. ∞. ∑. −k p(N) ∼ Nmax. N=Nmax +1. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 28 / 44.
(89) The full configuration interaction method. How to analyze FCI. Approximation using Slater determinants. Using this information, we obtain the following: Theorem (Accuracy of FCI calculations) Suppose we solve the many-body problem with FCI using HO basis functions in an energy cut model space with parameter Emax = Nmax + Ad/2. Assume that the exact A solution |Ψi ∈ H k (RAd ) ⊗ Cq . Then: −(k−1)/2. kδΨk1 ≤ C1 Nmax and. −(k−1). δE ≤ C2 Nmax. The constants depends roughly linearly on the strength of the interactions.. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 29 / 44.
(90) The full configuration interaction method. How to analyze FCI. Behaviour of exact wave function. Singular potential u(i, j) ⇒ well-known cusp conditions on wave functions across singularities (see Hoffmann-Ostenhof et al.) Ground state for two-electron dot with λ = 1: 2. 2. Ψ0 (r1 , r2 ) = (1 + cr12 )e−(r1 +r2 )/2 Pauli principle ⇒ smoothness varies for different wave functions Also some other interesting results are available: Work of Yserentant, Hackbush, Hoffmann-Ostenhof and others. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 30 / 44.
(91) The full configuration interaction method. How to analyze FCI. Behaviour of exact wave function. Singular potential u(i, j) ⇒ well-known cusp conditions on wave functions across singularities (see Hoffmann-Ostenhof et al.) Ground state for two-electron dot with λ = 1: 2. 2. Ψ0 (r1 , r2 ) = (1 + cr12 )e−(r1 +r2 )/2 Pauli principle ⇒ smoothness varies for different wave functions Also some other interesting results are available: Work of Yserentant, Hackbush, Hoffmann-Ostenhof and others. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 30 / 44.
(92) The full configuration interaction method. How to analyze FCI. Behaviour of exact wave function. Singular potential u(i, j) ⇒ well-known cusp conditions on wave functions across singularities (see Hoffmann-Ostenhof et al.) Ground state for two-electron dot with λ = 1: 2. 2. Ψ0 (r1 , r2 ) = (1 + cr12 )e−(r1 +r2 )/2 Pauli principle ⇒ smoothness varies for different wave functions Also some other interesting results are available: Work of Yserentant, Hackbush, Hoffmann-Ostenhof and others. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 30 / 44.
(93) The full configuration interaction method. How to analyze FCI. Behaviour of exact wave function. Singular potential u(i, j) ⇒ well-known cusp conditions on wave functions across singularities (see Hoffmann-Ostenhof et al.) Ground state for two-electron dot with λ = 1: 2. 2. Ψ0 (r1 , r2 ) = (1 + cr12 )e−(r1 +r2 )/2 Pauli principle ⇒ smoothness varies for different wave functions Also some other interesting results are available: Work of Yserentant, Hackbush, Hoffmann-Ostenhof and others. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 30 / 44.
(94) The full configuration interaction method. Numerical results. Convergence of parabolic dot FCI N is number of particles, R = Nmax , M is total angular momentum, S is total electron spin. Curves show δE/E. Relative error for N=3, λ=2. −1. 10. M=0, S=1, α = −1.2772 M=0, S=3, α = −2.1716 M=2, S=1, α = −1.3093 M=2, S=3, α = −2.5417. −2. 10. Relative error for N=5, λ = 0.2. −2. 10. M=0, S=1, α = −1.5150 M=0, S=5, α = −3.6563 M=3, S=1, α = −1.8159 M=3, S=5, α = −4.2117 −3. Relative error. Relative error. 10. −3. 10. −4. 10 −4. 10. −5. 10. 6. 8. 10. 12 R. Simen Kvaal (University of Oslo). 14. 16. 18. 20. 6. 8. 10. 12. 14. 16. 18. 20. R. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 31 / 44.
(95) The full configuration interaction method. Numerical results. Exponential (?) convergence in NCSM calculations. log(|E − Efadd |). 102. 100. 10−2. 0. 10. 20 30 40 N h̄ω = 24 MeV, |E − Efadd | ∼ Ce−0.15N. Simen Kvaal (University of Oslo). From Navratil & Barrett, PRC 57, p. 562 (1998). Convergence test of NCSM for 3 H, Nijmegen II effective interaction.. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 32 / 44.
(96) Coupled cluster methods (CC). Outline. 1. Background and formalism Overview and motivation Quantum dots as artificial nuclei The Harmonic oscillator and model spaces. 2. The full configuration interaction method Formulation How to analyze FCI Numerical results. 3. Coupled cluster methods (CC) Brief outline of method “Imagined” convergence analysis. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 33 / 44.
(97) Coupled cluster methods (CC). Brief outline of method. From CI to coupled cluster (CC). CI is a variational search within a linear space V . CC is a non-variational search within a non-linear space X (⊂ V ) X consists of functions on the form: |Ψi = eT |Φ0 i. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 34 / 44.
(98) Coupled cluster methods (CC). Brief outline of method. From CI to coupled cluster (CC). CI is a variational search within a linear space V . CC is a non-variational search within a non-linear space X (⊂ V ) X consists of functions on the form: |Ψi = eT |Φ0 i. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 34 / 44.
(99) Coupled cluster methods (CC). Brief outline of method. From CI to coupled cluster (CC). CI is a variational search within a linear space V . CC is a non-variational search within a non-linear space X (⊂ V ) X consists of functions on the form: |Ψi = eT |Φ0 i. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 34 / 44.
(100) Coupled cluster methods (CC). Brief outline of method. From CI to coupled cluster (CC). CI is a variational search within a linear space V . CC is a non-variational search within a non-linear space X (⊂ V ) X consists of functions on the form: |Ψi = eT |Φ0 i. Unperturbed HO ground state Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 34 / 44.
(101) Coupled cluster methods (CC). Brief outline of method. From CI to coupled cluster (CC). CI is a variational search within a linear space V . CC is a non-variational search within a non-linear space X (⊂ V ) X consists of functions on the form: |Ψi = eT |Φ0 i. Cluster operator; excitation operator Simen Kvaal (University of Oslo). Unperturbed HO ground state. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 34 / 44.
(102) Coupled cluster methods (CC). Brief outline of method. From CI to coupled cluster (CC). CI is a variational search within a linear space V . CC is a non-variational search within a non-linear space X (⊂ V ) X consists of functions on the form: |Ψi = eT |Φ0 i. True ground state; ansatz always valid Simen Kvaal (University of Oslo). Cluster operator; excitation operator. Unperturbed HO ground state. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 34 / 44.
(103) Coupled cluster methods (CC). Brief outline of method. Excitation operators. Suppose |Φ0 i is filled with states up to F : |Φ0 i = c†α1 · · · c†αA |−i. Emax. F. Let (ai ) be below and (bi ) above Fermi level. Define: Xab = c†b1 · · · c†bn can · · · ca1 Moves particles from below F to above F Notice: VDP .. Xab |Φ0 i. Simen Kvaal (University of Oslo). Xab11 ba22 ba33 |Φ0 i. generates basis for. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 35 / 44.
(104) Coupled cluster methods (CC). Brief outline of method. Excitation operators. Suppose |Φ0 i is filled with states up to F : |Φ0 i = c†α1 · · · c†αA |−i. Emax. F. Let (ai ) be below and (bi ) above Fermi level. Define: Xab = c†b1 · · · c†bn can · · · ca1 Moves particles from below F to above F Notice: VDP .. Xab |Φ0 i. Simen Kvaal (University of Oslo). Xab11 ba22 ba33 |Φ0 i. generates basis for. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 35 / 44.
(105) Coupled cluster methods (CC). Brief outline of method. Excitation operators. Suppose |Φ0 i is filled with states up to F : |Φ0 i = c†α1 · · · c†αA |−i. Emax. F. Let (ai ) be below and (bi ) above Fermi level. Define: Xab = c†b1 · · · c†bn can · · · ca1 Moves particles from below F to above F Notice: VDP .. Xab |Φ0 i. Simen Kvaal (University of Oslo). Xab11 ba22 ba33 |Φ0 i. generates basis for. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 35 / 44.
(106) Coupled cluster methods (CC). Brief outline of method. Excitation operators. Suppose |Φ0 i is filled with states up to F : |Φ0 i = c†α1 · · · c†αA |−i. Emax. F. Let (ai ) be below and (bi ) above Fermi level. Define: Xab = c†b1 · · · c†bn can · · · ca1 Moves particles from below F to above F Notice: VDP .. Xab |Φ0 i. Simen Kvaal (University of Oslo). Xab11 ba22 ba33 |Φ0 i. generates basis for. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 35 / 44.
(107) Coupled cluster methods (CC). Brief outline of method. Excitation operators. Suppose |Φ0 i is filled with states up to F : |Φ0 i = c†α1 · · · c†αA |−i. Emax. F. Let (ai ) be below and (bi ) above Fermi level. Define: Xab = c†b1 · · · c†bn can · · · ca1 Moves particles from below F to above F Notice: VDP .. Xab |Φ0 i. Simen Kvaal (University of Oslo). Xab11 ba22 ba33 |Φ0 i. generates basis for. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 35 / 44.
(108) Coupled cluster methods (CC). Brief outline of method. Excitation operators. Suppose |Φ0 i is filled with states up to F : |Φ0 i = c†α1 · · · c†αA |−i. Emax. F. Let (ai ) be below and (bi ) above Fermi level. Define: Xab = c†b1 · · · c†bn can · · · ca1 Moves particles from below F to above F Notice: VDP .. Xab |Φ0 i. Simen Kvaal (University of Oslo). Xab11 ba22 ba33 |Φ0 i. generates basis for. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 35 / 44.
(109) Coupled cluster methods (CC). Brief outline of method. The cluster operator T Recall the ansatz: |Ψi = eT |Φ0 i T is on the form T = T1 + T2 + · · · + TK where ···bn b1 b2 ···bn Tn = ∑ tab11 ab22 ···a X n a1 a2 ···an a,b. If K = 1, we get CCS (“singles”). K = 2 gives CCSD (“singles and doubles”), et.c. K = A is exact! We set t = (t(1) , · · · , t(K) ); a vector of all the amplitudes. T = T(t). Simen Kvaal (University of Oslo). ←− a linear function of the amplitudes t(n). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 36 / 44.
(110) Coupled cluster methods (CC). Brief outline of method. The cluster operator T Recall the ansatz: |Ψi = eT |Φ0 i T is on the form T = T1 + T2 + · · · + TK where ···bn b1 b2 ···bn Tn = ∑ tab11 ab22 ···a X n a1 a2 ···an a,b. If K = 1, we get CCS (“singles”). K = 2 gives CCSD (“singles and doubles”), et.c. K = A is exact! We set t = (t(1) , · · · , t(K) ); a vector of all the amplitudes. T = T(t). Simen Kvaal (University of Oslo). ←− a linear function of the amplitudes t(n). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 36 / 44.
(111) Coupled cluster methods (CC). Brief outline of method. The cluster operator T Recall the ansatz: |Ψi = eT |Φ0 i T is on the form T = T1 + T2 + · · · + TK where ···bn b1 b2 ···bn Tn = ∑ tab11 ab22 ···a X n a1 a2 ···an a,b. If K = 1, we get CCS (“singles”). K = 2 gives CCSD (“singles and doubles”), et.c. K = A is exact! We set t = (t(1) , · · · , t(K) ); a vector of all the amplitudes. T = T(t). Simen Kvaal (University of Oslo). ←− a linear function of the amplitudes t(n). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 36 / 44.
(112) Coupled cluster methods (CC). Brief outline of method. The cluster operator T Recall the ansatz: |Ψi = eT |Φ0 i T is on the form T = T1 + T2 + · · · + TK where ···bn b1 b2 ···bn Tn = ∑ tab11 ab22 ···a X n a1 a2 ···an a,b. If K = 1, we get CCS (“singles”). K = 2 gives CCSD (“singles and doubles”), et.c. K = A is exact! We set t = (t(1) , · · · , t(K) ); a vector of all the amplitudes. T = T(t). Simen Kvaal (University of Oslo). ←− a linear function of the amplitudes t(n). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 36 / 44.
(113) Coupled cluster methods (CC). Brief outline of method. CC equations Nonlinear search space X for CCSD· · · K: n o X = eT(t) |Φ0 i : t = (t(1) , · · · , t(K) ) Could attempt a variational search within X , but this is too complicated. Instead, non-variational amplitude equations are used: f(t) = 0,. fi (t) ≡ hΦi |e−T(t) HeT(t) |Φ0 i = 0. ∀i 6= 0. Energy expression: ECC = J(t) ≡ hΦ0 |e−T(t) HeT(t) |Φ0 i 1 = hΦ0 |H(1 + T2 + T12 )|Φ0 i 2. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 37 / 44.
(114) Coupled cluster methods (CC). Brief outline of method. CC equations Nonlinear search space X for CCSD· · · K: n o X = eT(t) |Φ0 i : t = (t(1) , · · · , t(K) ) Could attempt a variational search within X , but this is too complicated. Instead, non-variational amplitude equations are used: f(t) = 0,. fi (t) ≡ hΦi |e−T(t) HeT(t) |Φ0 i = 0. ∀i 6= 0. Energy expression: ECC = J(t) ≡ hΦ0 |e−T(t) HeT(t) |Φ0 i 1 = hΦ0 |H(1 + T2 + T12 )|Φ0 i 2. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 37 / 44.
(115) Coupled cluster methods (CC). Brief outline of method. CC equations Nonlinear search space X for CCSD· · · K: n o X = eT(t) |Φ0 i : t = (t(1) , · · · , t(K) ) Could attempt a variational search within X , but this is too complicated. Instead, non-variational amplitude equations are used: f(t) = 0,. fi (t) ≡ hΦi |e−T(t) HeT(t) |Φ0 i = 0. ∀i 6= 0. Energy expression: ECC = J(t) ≡ hΦ0 |e−T(t) HeT(t) |Φ0 i 1 = hΦ0 |H(1 + T2 + T12 )|Φ0 i 2. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 37 / 44.
(116) Coupled cluster methods (CC). Brief outline of method. CC equations Nonlinear search space X for CCSD· · · K: n o X = eT(t) |Φ0 i : t = (t(1) , · · · , t(K) ) Could attempt a variational search within X , but this is too complicated. Instead, non-variational amplitude equations are used: f(t) = 0,. fi (t) ≡ hΦi |e−T(t) HeT(t) |Φ0 i = 0. ∀i 6= 0. Energy expression: ECC = J(t) ≡ hΦ0 |e−T(t) HeT(t) |Φ0 i 1 = hΦ0 |H(1 + T2 + T12 )|Φ0 i 2. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 37 / 44.
(117) Coupled cluster methods (CC). Brief outline of method. Basics of the coupled cluster method VFCI VCISD VCIS |Φ0 i. Left: Illustration of CI spaces formed by K-fold excitations of |Φ0 i. Illustration of CCS, truncating T at T1 . Notice: eT = 1 + T + T 2 /2 + · · · contains higher order excitations CISD; covering more of VDP And so on . . . CC works extremely well because of the “exponentiating”. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 38 / 44.
(118) Coupled cluster methods (CC). Brief outline of method. Basics of the coupled cluster method VFCI Left: Illustration of CI spaces formed by K-fold excitations of |Φ0 i.. VCISD VCIS |Φ0 i. XS. Illustration of CCS, truncating T at T1 . Notice: eT = 1 + T + T 2 /2 + · · · contains higher order excitations CISD; covering more of VDP And so on . . . CC works extremely well because of the “exponentiating”. Simen Kvaal (University of Oslo). Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 38 / 44.
(119) Coupled cluster methods (CC). Brief outline of method. Basics of the coupled cluster method VFCI Left: Illustration of CI spaces formed by K-fold excitations of |Φ0 i.. VCISD VCIS |Φ0 i. XSD Simen Kvaal (University of Oslo). XS. Illustration of CCS, truncating T at T1 . Notice: eT = 1 + T + T 2 /2 + · · · contains higher order excitations CISD; covering more of VDP And so on . . . CC works extremely well because of the “exponentiating”. Rigorous analysis of ab initio calculations for parabolic quantum dots. ECT*2009. 38 / 44.
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