1. Thomas Halverson and Bill Poirier Tom.Halverson@ttu.edu Texas Tech University Department of Physics 6-9-13

2. 2 I. I.What we mean by exact II. II.Exponential scaling and basis truncation III. III.Theoretical Overview IV. IV.Code specific V. V.Results VI. VI.Future work

3. 3

4. 4 Exact Quantum Dynamics Nuclear dynamics Born-Oppenheimer Approximation Bound states Rovibrational SpectroscopyTISE: Exact to numerical convergence In general, all quantities of interest can be calculated from the eigenfunctions and eigenenergies

5. 5 Choice of potential Problem specific Ab initio methods/ DFT Basis Choice Direct product vs. not-direct product Symmetry Point group Permutation/ Inversion Exponential Scaling Diagonalization Method Exact Quantum Dynamics Choice of potential Basis Choice Diagonalization Method Symmetrized Gaussians Direct Diagonalization (in 1-D)

6. 6

7. 7 Exponential Scaling and Basis Truncation direct product basis For a direct product basis, the size of the Hamiltonian matrix grows exponentially with the degrees of freedom (D) At large dimensionality problems become intractable

8. 8 Exponential Scaling and Basis Truncation A method for decreasing basis size, without decreasing accuracy basisefficiency: Increases basis efficiency: Types Polyad truncation Energy truncation Phase space truncation

9. 9

10. 10 Theoretical Overview Density operator for lowest K states of a given H Weyl symbol: Wigner PDF Phase space region R Quasi-classical approximation* *B. Poirier and J. C. Light, J. Chem. Phys. 111, 4869 (1999); B. Poirier and J. C. Light, J. Chem. Phys. 113, 211 (2000).

11. 11 Theoretical Overview Density operator in some basis Phase space Region R’

12. 12 R R’R’ Theoretical Overview

13. 13 Theoretical Overview Each square is denoted by a two indices (m, n) double density The are of each square has area  2 =  (double density) The red square is (m=5/2, n=3/2) and is at coordinate (x = 5/2, p = 3/2 * B. Poirier and A. Salam, J. Chem. Phys. 121, 1690 (2004); B. Poirier and A. Salam, J. Chem. Phys. 121, 1704 (2004).

14. 14 Theoretical Overview Collective well-localized Very small overlap Can apply phase space truncated Linearly Independent Real valued * B. Poirier and A. Salam, J. Chem. Phys. 121, 1690 (2004); B. Poirier and A. Salam, J. Chem. Phys. 121, 1704 (2004).

15. 15 Theoretical Overview Phase Space Truncation Weyl-Heisenberg Lattice

16. 16 Theoretical Overview Start with DPB: F mn (x) = f m1n1 (x 1 )×f m2n2 (x 2 )×...Retain: H(m 1 Δ, m 2 Δ,..., n 1 Δ, n 2 Δ) < E max Defeats Exponential Scaling

17. 17

18. 18 Code Specifics Stand alone, single code Dimension independent Massively Parallel   Utilizes standard dense matrix routines (Scalapack Library)   effective scaling tested up to 4104 cores Designed for usability

19. 19

20. 20 Results Coupled anharmonic oscillator A=C=0, B=1/2, D=α Multidimensional Hamiltonian is a sum of one-dimensional Hamiltonians + coupling term

21. 21 Results Basis size Converged states Harmonic Oscillator BasisSymmetrized Gaussians 3%2%0.2%0.06%3%2%0.2%0.06% ~10008206933362411015(1018)954(997)178(312)31(115) ~3000162513887035472893(2931)2755(2812)456(786)75(162) ~1000038853361185715269231(9378)8268(8697)1683(2184)428(872) ~30000876979844543387626466(26918)23778(24695)6546(8625)1775(4351) ~1000002061418355119701069885501(89227)75691(81625)20525(24364)4105(10761) ~175000-- 150276(150276)136595(136595)48416(48416)25176(25176) T. Halverson and B. Poirier, J. Chem. Phys. 137, 224101 (2012).

22. 22 Results Basis size Converged states Harmonic Oscillator BasisSymmetrized Gaussians 3%2%0.2%0.06%3%2%0.2%0.06% ~10008206933362411015(1018)954(997)178(312)31(115) ~3000162513887035472893(2931)2755(2812)456(786)75(162) ~1000038853361185715269231(9378)8268(8697)1683(2184)428(872) ~30000876979844543387626466(26918)23778(24695)6546(8625)1775(4351) ~1000002061418355119701069885501(89227)75691(81625)20525(24364)4105(10761) ~175000-- 150276(150276)136595(136595)48416(48416)25176(25176) T. Halverson and B. Poirier, J. Chem. Phys. 137, 224101 (2012).

23. 23 Results Lowest 100000 computed energies for 3D coupled harmonic oscillator Solid line- “master list” energies (N = 302 352) Dotted line- energy truncated HO basis energies (N = 102 340) Dashed line- PST SG energies (N = 100 016) T. Halverson and B. Poirier, J. Chem. Phys. 137, 224101 (2012).

24. 24 Results Dimensionality Basis Size Cores2.0%0.2%0.06% 82392524240800 8 154323576883201280 821125111521607136521 82426271728203946235911 127975914466890120 1750871728168421219323128 161332136826500 1642473603202300 161179935761102642390 T. Halverson and B. Poirier, J. Chem. Phys. 137, 224101 (2012).

25. 25 Results Dimensionality Basis Size Cores2.0%0.2%0.06% 82392524240800 8 154323576883201280 821125111521607136521 82426271728203946235911 127975914466890120 1750871728168421219323128 161332136826500 1642473603202300 161179935761102642390 T. Halverson and B. Poirier, J. Chem. Phys. 137, 224101 (2012).

26. 26 Results Accuracy, (cm -1 ) Number of States, K Efficiency, K/N 10003518660.5% 100 1087718.7% 1022723.9% 11400.24% 0.1310.05% Quartic force field by Pouchan group* Accurately Computed States for N=58163 SG calculations converged using seven different basis sizes ranging from N=310 to N=102858 *C. Pouchan, M. Aouni, and D. Bégué, Chem. Phys. Lett. 334, 352 (2001).

27. 27 Results Accuracy, (cm -1 ) Number of States, K Efficiency, K/N 10003518660.5% 100 1087718.7% 1022723.9% 11400.24% 0.1310.05% Quartic force field by Pouchan group* Accurately Computed States for N=58163 SG calculations converged using seven different basis sizes ranging from N=310 to N=102858 *C. Pouchan, M. Aouni, and D. Bégué, Chem. Phys. Lett. 334, 352 (2001).

28. 28 Results Accuracy, (cm -1 ) Number of States, K Efficiency, K/N 10008545040.3% 100 34041.6% 103020.14% 1110.005% 0.130.001% Quartic force field by Pouchan group* Accurately Computed States for N=212197 SG calculations converged using six different basis sizes ranging from N=2042 to N=409582 *C. Pouchan, M. Aouni, and D. Bégué, Chem. Phys. Lett. 334, 352 (2001).

29. 29 Results Accuracy, (cm -1 ) Number of States, K Efficiency, K/N 10008545040.3% 100 34041.6% 103020.14% 1110.005% 0.130.001% Quartic force field by Pouchan group* Accurately Computed States for N=212197 SG calculations converged using six different basis sizes ranging from N=2042 to N=409582 *C. Pouchan, M. Aouni, and D. Bégué, Chem. Phys. Lett. 334, 352 (2001).

30. 30 Results Accuracy, (cm -1 ) Number of States, K Efficiency, K/N 10006029339.99% 100 81925.4% 1011000.73% 1730.048% 0.130.002% Quartic force field by Pouchan group* Accurately Computed States for N=150786 SG calculations converged using six different basis sizes ranging from N=1414 to N=340502 *C. Pouchan, M. Aouni, and D. Bégué, Chem. Phys. Lett. 334, 352 (2001).

31. 31 Results Accuracy, (cm -1 ) Number of States, K Efficiency, K/N 10006029339.99% 100 81925.4% 1011000.73% 1730.048% 0.130.002% Quartic force field by Pouchan group* Accurately Computed States for N=150786 SG calculations converged using six different basis sizes ranging from N=1414 to N=340502 *C. Pouchan, M. Aouni, and D. Bégué, Chem. Phys. Lett. 334, 352 (2001).

32. 32

33. 33 Future Work 1.Newer, Larger Clusters Kraken, Blue Waters 2.Newer, Larger Potentials Work with experimental and electronic structure groups 3.Improve accuracy Phase space region operators* * R. Lombardini and B. Poirier, Phys. Rev. E 74, 036705 (2006).

34. 34 Personnel Principle Investigator Dr. Bill Poirier Fellow Graduate Students Corey Petty Drew Brandon Support Robert A. Welch Foundation National Science Foundation Organizers Terry A. Miller Frank C. Delucia Anne B. McCoy Computer Resources Texas Tech University High Performance Computing Center University Of Texas-Austin Texas Advanced Computing Center