Efficient approaches to multidimensional quantum dynamics: Dynamical pruning in phase, position and configuration space Henrik R. Larsson April 20, 2018 Group Prof. Hartke / Christiana Albertina University of Kiel, Germany Group Prof. Tannor / Weizmann Institute of Science, Rehovot, Israel
How to do molecular quantum dynamics simulations? ? 1 / 13
How to do molecular quantum dynamics simulations? ? � I A I ( t ) | I � direct-product basis wave function I ≡ { i 1 , i 2 , . . . , i D } , i κ ∈ [1 , N κ ] | I � ≡ � D κ =1 | χ κ j κ � | Ψ( t ) � HUGE tensor A , size � D κ =1 N κ i ∂ t | Ψ( t ) � H × A H IJ = � I | ˆ H | J � • TD-FCI: Standard approach in mol. quantum dynamics • Problem: Curse of dimensionality (exponential scaling) 1 / 13
How to do molecular quantum dynamics simulations? ? � I A I ( t ) | I � direct-product basis wave function I ≡ { i 1 , i 2 , . . . , i D } , i κ ∈ [1 , N κ ] | I � ≡ � D κ =1 | χ κ j κ � | Ψ( t ) � HUGE tensor A , size � D κ =1 N κ i ∂ t | Ψ( t ) � H × A H IJ = � I | ˆ H | J � • TD-FCI: Standard approach in mol. quantum dynamics • Problem: Curse of dimensionality (exponential scaling) • Possible loophole: Employ bases that lead to sparse tensors A � Dynamical Pruning (DP) 1 / 13
Dynamical Pruning (DP) PvB phase space bases pW TD-FCI MCTDH DVR FGH primitive basis SPF ... (SPF repre- ( A tensor) sentation) Gauß-Grid
Dynamical Pruning (DP) PvB phase space bases pW TD-FCI MCTDH DVR FGH primitive basis SPF ... (SPF repre- ( A tensor) sentation) Gauß-Grid
DVR/Coordinate-space-localised functions • Exploit locality of | Ψ � in position space: ⇒ • Add/remove neighbors if | A i | > θ / | A i | < θ • Used by Hartke 1 , Wyatt 2 and others. • Easiest to use: DVR/pseudospectral functions • Bonus: Potential is diagonal V ij = δ ij V ( x i ) 1 B. Hartke, Phys. Chem. Chem. Phys. , 2006, 8 , 3627, J. Sielk et al., Phys. Chem. Chem. Phys. , 2009, 11 , 463–475. 2 L. R. Pettey and R. E. Wyatt, Chem. Phys. Lett. , 2006, 424 , 443 –448, L. R. Pettey and R. E. Wyatt, Int. J. Quantum Chem. , 2007, 107 , 1566–1573. 2 / 13
Dynamical Pruning (DP) PvB phase space bases pW TD-FCI MCTDH DVR FGH primitive basis SPF ... (SPF repre- ( A tensor) sentation) Gauß-Grid
Phase-space-localised von Neumann basis � � 1 � , � − α ( x − x n ) 2 + i · p l · ( x − x n ) 4 exp 2 α α = σ p � x | ˜ g n , l � = 2 σ x π • Basis is localised at ( x n , p l ). • Problem: Poor convergence. p x 3 / 13
Phase-space-localised von Neumann basis � � 1 � , � − α ( x − x n ) 2 + i · p l · ( x − x n ) 4 exp 2 α α = σ p � x | ˜ g n , l � = 2 σ x π √ √ FGH (N points) PvN ( N × N points) • Basis is localised at ( x n , p l ). ( x 0 , + P ) • Problem: Poor convergence. · • Solution 1: 3 g n , l vN p 0 ⇐ ⇒ Projected von Neumann (PvN/PvB): | g i � = � δ x δ p j | χ j �� χ j | ˜ g i � ; { χ i } : DVR ∆ x x x Non-Orthogonal! (PvB: biorthogonal basis) 3 A. Shimshovitz and D. J. Tannor, Phys. Rev. Lett. , 2012, 109 , 070402, D. J. Tannor et al., Adv. Chem. Phys. , 2018, 163 , in press 3 / 13
Phase-space-localised von Neumann basis � � 1 � , � − α ( x − x n ) 2 + i · p l · ( x − x n ) 4 exp 2 α α = σ p � x | ˜ g n , l � = 2 σ x π • Basis is localised at ( x n , p l ). • Problem: Poor convergence. • + p p • Solution 1: 3 0 Projected von Neumann (PvN/PvB): | g i � = � j | χ j �� χ j | ˜ g i � ; { χ i } : DVR − p • Non-Orthogonal! (PvB: biorthogonal basis) • Solution 2: 4 x Projected Weylets (pW): � � 1 � � � �� � − α ( x − x n ) 2 � sin 4 exp � x | � 8 α φ nl � = p l x − x n − π 8 α π Orthogonal! Less sparse than PvB! 3 A. Shimshovitz and D. J. Tannor, Phys. Rev. Lett. , 2012, 109 , 070402, D. J. Tannor et al., Adv. Chem. Phys. , 2018, 163 , in press 4 B. Poirier and A. Salam, J. Chem. Phys. , 2004, 121 , 1690, H. R. Larsson et al., J. Chem. Phys. , 2016, 145 , 3 / 13 204108
Example of a PvB propagation 4 / 13
Multidimensions: Hamiltonian times state: H · A Unpruned case • Assume a SoP Hamilton-Tensor: H = h (1) ⊗ h (2) + . . . • D : dimension, n : 1D basis size, n 2 D : size of H ; n D : size of A • Scaling of H · A : O ( n D +1 ) by sequential summation (as done in electronic integral transformations) 3 D. J. Tannor et al., Adv. Chem. Phys. , 2018, 163 , in press, H. R. Larsson et al., J. Chem. Phys. , 2016, 145 , 204108. 5 / 13
Multidimensions: Hamiltonian times state: H · A Unpruned case • Assume a SoP Hamilton-Tensor: H = h (1) ⊗ h (2) + . . . • D : dimension, n : 1D basis size, n 2 D : size of H ; n D : size of A • Scaling of H · A : O ( n D +1 ) by sequential summation (as done in electronic integral transformations) Pruned case • Pruning: n D − n D → ˜ n D +1 ) scaling possible with new algorithm 3 • O (˜ • ONLY for orthogonal basis • Nonorthogonal basis: S − 1 PvB H PvB A • Pruned S − 1 not of SoP form: O (˜ n 2 D ) scaling 3 D. J. Tannor et al., Adv. Chem. Phys. , 2018, 163 , in press, H. R. Larsson et al., J. Chem. Phys. , 2016, 145 , 204108. 5 / 13
Application: 2D double well • Testing a pruned DVR (FGH), PvB and pW 6 / 13
Application: 2D double well • Testing a pruned DVR (FGH), PvB and pW • Accuracy versus basis size? 10 + 2 pW FGH 1 PvB Infidelity of the autocorrelation 10 − 2 10 − 4 10 − 6 10 − 8 10 − 10 10 − 12 0 10 20 30 40 50 60 70 Mean number of used basis functions / % 6 / 13
Application: 2D double well • Testing a pruned DVR (FGH), PvB and pW • Accuracy versus basis size? • Timing? 10 + 2 pW 1 × 10 6 FGH full FGH time 1 PvB pW 100000 FGH Infidelity of the autocorrelation PvB 10 − 2 10000 Needed time /s 10 − 4 1000 10 − 6 100 10 10 − 8 1 10 − 10 0 . 1 10 + 2 10 − 2 10 − 4 10 − 6 10 − 8 10 − 10 10 − 12 1 10 − 12 Infidelity of the autocorrelation 0 10 20 30 40 50 60 70 Mean number of used basis functions / % FGH/DVR: Potential diagonal, pW: Non-diagonal 6 / 13
Vibr. resonance dynamics of DCO 4 • DP-DVR with filter diagonalization + CAP • Controlled accuracy of pruning for energies and widths • Decay dynamics up to 200 ps with DP-DVR: Confirms polyad model • Comparison with velocity mapped images from Temps Group @ Kiel ∆ E / cm − 1 Γ / cm − 1 (a) ∆ E � 8902 cm − 1 : (2,2,2) 2 02 (b) ∆ E � 8942 cm − 1 : (0,5,0) 2 02 P label Expt. DP Expt. DP 3000 2000 1000 0 3000 2000 1000 0 -1000 v � 0 v � 0 P ( E D )/ arb. units 5 ((034)) 8778 8775 3.50 5.6 v � 1 v � 1 WKS 1.0 SAG 5 ((042)) 8821 8830 <2.00 1.1 0.8 Exp 0.6 5 ((222)) 8902 8895 1.06 1.2 0.4 5 (050) 8942 8950 1.79 0.13 0.2 5 (132) 9050 9029 0.34 0.28 0.0 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 5 (230) 9099 9096 0.20 0.32 (c) ∆ E � 9896 cm − 1 : (2,3,1) 2 02 (d) ∆ E � 10065 cm − 1 : (1,4,1) 2 02 5.5 027 — 9234 — 13 5 ((140)) 9272 9248 0.29 0.31 4000 3000 2000 1000 0 4000 3000 2000 1000 0 v � 0 v � 0 5.5 ((321)) — 9494 — 17 P ( E D )/ arb. units v � 1 v � 1 v � 2 v � 2 1.0 5.5 (043) 9614 9629 2.30 1.4 0.8 5.5 (223) 9686 9688 <5.00 5.5 0.6 0.4 5.5 ((051)) 9757 9762 0.83 0.64 0.2 5.5 ((133)) 9819 9805 <3.00 1.8 0.0 5.5 (231) 9896 9891 1.22 1.6 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 E D / cm − 1 E D / cm − 1 5.5 ((141)) 10065 10044 6.00 3.9 4 H. R. Larsson et al., arXiv:1802.07050; submitted to J. Chem. Phys, 2018. 7 / 13
Dynamical Pruning (DP) PvB phase space bases pW TD-FCI MCTDH DVR FGH primitive basis SPF ... (SPF repre- ( A tensor) sentation) Gauß-Grid
Multi-Configurational Time-Dependent Hartree (MCTDH) ∼ TD-CAS-SCF for nuclei • Single Particle Functions (SPF) | φ � : time-dependent, variationally optimised direct-product basis • Configurations | I � : Hartree-Product of SPFs mode combination shifts effort � I A I ( t ) | I ( t ) � single particle functions (SPF), wave function | I ( t ) � ≡ � D i κ ∈ [1 , n κ ] κ =1 | φ κ j κ ( t ) � | Ψ( t ) � tensor size � D i n i , n i ≤ N i 8 / 13
Multi-Configurational Time-Dependent Hartree (MCTDH) ∼ TD-CAS-SCF for nuclei • Single Particle Functions (SPF) | φ � : time-dependent, variationally optimised direct-product basis • Configurations | I � : Hartree-Product of SPFs mode combination shifts effort � I A I ( t ) | I ( t ) � single particle functions (SPF), wave function | I ( t ) � ≡ � D i κ ∈ [1 , n κ ] κ =1 | φ κ j κ ( t ) � | Ψ( t ) � tensor size � D i n i , n i ≤ N i • Mode combination: Combine strongly coupled modes to propagate multidimensional SPFs. • Shifts both computational effort and storage requirement 8 / 13
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