1. Parallel Computations in Quantum Lanczos Representation Methods Hong Zhang and Sean C. Smith Quantum & Molecular Dynamics Group Center for Computational Molecular Science The University of Queensland, Australia

2. Outline 1.Introduction 2. Methodologies 3. Results 4. Conclusions & Future Work

3. 1. Introduction Lanczos (1950) 1. Wyatt: Recursive residue generation method 2. Iung and Leforestier; Yu and Nyman; Carrington: Spectrally transformed Lanczos algorithm 3. Manthe, Seideman, and Miller; Karlsson: GMRES/QMR DVR-ABC 4. Guo; Carrington: Symmetry adapted Lanczos method 5. Zhang and Smith: Lanczos representation methods.

4. 1.1 Lanczos representation methods 1. Lanczos representation filter diagonalisation in unimolecular dissociation and rovibrational spectroscopy: recent non-zero total angular momentum (J > 0) calculations using parallel computing implementation. 2. TI Lanczos subspace wavepacket method; real Lanczos artificial boundary inhomogeneity (LABI); real single Lanczos propagation ABI in bimolecular reactions.

5. 3. Systems: HOCl and HO 2 Importance in atmospheric chemistry and in combustion chemistry, e.g., balance of stratospheric ozone, etc.

6. [Zhang & Smith, PhysChemComm, 6: 12, 2003]

7. 2. Methodologies 1. Transform the primary representation (DVR or FBR) of the fundamental scattering equations (e.g., TI Schrödinger equation or TI wavepacket- Lippmann-Schwinger equation) into the tridiagonal Lanczos representation. 2. Solve the eigen-problem or linear system within the subspace to obtain either eigen-pairs or subspace wavefunctions. 3. Extract physical information, e.g., bound states, resonances, and scattering (S) matrix elements.

8. Representation transformation [Kouri, Arnold & Hoffman, CPL, 203: 166, 1993; Zhang H, Smith SC, JCP, 116: 2354, 2002]

9. Solving subspace eigen-problem or linear system

10. Algorithm (a) Choose M th element of  (E j ) to be arbitrary (but non- zero) and calculate (b) For k=M-1, M-2, …, 2, update scalar  k-1 : (c) Determine constant (  true = c  ) by normalization or by

11. Comparison with other eigen-problem/linear system solvers QMR/MINRESQL/QR Forward recursion Efficiency very important, as linear system solver must be repeated for many different filter energies throughout spectrum. [Yu & Smith, BBPC,101: 400, 1997; Yu & Smith, CPL, 283: 69, 1998; Zhang & Smith, JCP, 115: 5751, 2001; Zhang & Smith, PCCP, 3: 2282, 2001]

12. Representation Parallel computing Conceptually difficult Computationally difficult: cpu time and memory In propagation, the most time consuming part is the matrix-vector multiplication. We use Message-passing interface (MPI) to perform parallel computation. Propagation Final analysis

13. J > 0 DVR Hamiltonian

14. Matrix-vector multiplication

15. Processor assignment a) Master processor (ID = 0) Perform the main propagation; write ,  elements in bound and resonance calculations; all other related works except the matrix-vector multiplications. b) Working/slave processors (ID = 1, 2, …) Perform the matrix- vector multiplications for each  component.

16. Communications According to the Coriolis coupling rules, only two nearest neighbouring  components need to communicate. Load balancing Load balancing j min is different for each  component, but j max is the same, i.e, the DVR size for  angle is changeable. For the highest or the lowest  components, only one Coriolis coupling required.

17. Timing Due to the communications and loading balance issues, the model doesn’t scale ideally with (J+1) for even spectroscopic symmetry or J for odd spectroscopic symmetry. However, one can achieve wall clock times (e.g., for even symmetry J = 6 HO 2 case) that are within about a factor of 2 of J = 0 calculations. For non parallel computing, the wall clock times will approximately be a factor of 7 of J = 0 calculations.

18. 3. Recent Results 3. Recent Results HO 2 : J = 0-50 bound state energies and resonance energies and widths using both Lanczos and Chebyshev method from parallel computing. HOCl/DOCl: J = 0-30 ro-vibrational spectroscopy.

19. Table 1 Selected HO 2 bound state energies for J = 30 (even symmetry) for comparison. All energy units are in eV (Zhang & Smith, JPCA, 110: 3246, 2006 ). NDS/LSFDLHFDBunkerJ-shiftingKaKa KcKc ( 1, 2, 3 ) 1.124642.1244980.125357030(0,0,0) 2.125728.125729.1256170.127746130(0,0,0) 3.135318.135319.1350050.134914229(0,0,0) 4.146785.146786.1464580.146859328(0,0,0) 5.163396.1629590.163583427(0,0,0) 6.184702.1841270.185085526(0,0,0)

20. Fig. 1 Plot of the quantum logarithmic rates versus resonance energies. Thin dotted line - QM results; red line - Troe et al. SACM/CT calculations; green line – quantum average. (Zhang & Smith, JCP, 123: 014308, 2005 )

21. Fig. 2 Same as previous figure, except J = 30 (unpublished latest results).

22. Table 2 Selected low vibrational energies at J = 0 for HOCl for comparison. All energy units are in cm -1 (Zhang, Smith, Nanbu & Nakamura, JPCA, 2006, in press). n ( 1, 2, 3 ) This work 1 BowmanThis work 2 Experimental 10, 0, 00.000.0000.00 20, 0, 1650.58724.336724.98724.36 30, 1, 01261.971238.6171245.111238.62 40, 0, 21309.211444.1071442.311438.68 50, 1, 11926.921953.7481967.41 60, 0, 31963.222154.0282161.36 70, 2, 02522.282456.3632458.672461.21

23. Table 3 Calculated HOCl ro-vibrational state energies in cm -1 with spectroscopic assignments for J = 20 (Zhang, Smith, Nanbu & Nakamura, JPCA, 2006, in press). nquantumAR(J, K a, K c )(ν 1, ν 2, ν 3 )symmetry 1205.15209.0420, 0, 200, 0, 0even 2223.49229.0120, 1, 200, 0, 0even 3283.76288.9120, 2, 190, 0, 0even 4381.70388.7420, 3, 180, 0, 0even 21226.16229.0120, 1, 190, 0, 0odd 22283.71288.9120, 2, 180, 0, 0odd 23381.70388.7420, 3, 170, 0, 0odd

24. Table 4 Comparison of experiments and calculations for selected HOCl far infrared transitions in cm -1 (Zhang, Smith, Nanbu & Nakamura, JPCA, 2006, in press). n(J’, K a, K c )(J”, K a, K c ) ( 1, 2, 3 ) OBSCAL 111, 1, 1110, 0, 100, 0, 030.4726830.00 1311, 2, 910, 1, 100, 0, 071.1236670.03 2820, 5, 1520, 4, 160, 0, 0177.98236175.37 3310, 2, 811, 1, 110, 0, 149.3326048.93 4120, 3, 1820, 2, 190, 0, 199.3508098.43 5611, 5, 710, 4, 60, 0, 1188.69489187.03

25. Table 5 Comparison of experiments and calculations for selected HO 2 vibrational state energies in cm -1 from three latest PESs (Zhang & Smith, unpublished latest resuts). n ( 1, 2, 3 ) DMBE IVTroe et al.Xie at al.Experimental 10, 0, 00.00 0.0 20, 0, 11065.361270.891090.511097.6 30, 1, 01296.331610.651388.931391.8 40, 0, 22090.872436.942163.14 50, 1, 12359.352896.982462.26 60, 2, 02516.603134.912752.11 70, 0, 33080.663514.813216.67 81, 0, 03333.633588.553430.083436.2

26. Table 6 Comparison of experiments and calculations for selected DOCl vibrational state energies in cm -1 from the latest ab initio PES and LHFD calculations (Zhang & Smith, unpublished latest resuts). ncalexp ( 1, 2, 3 ) 10.000.0(0, 0, 0) 2722.588723.3(0, 0, 1) 3916.213909.6(0, 1, 0) 82358.013(0, 1, 2) 92532.055(0, 2, 1) 102669.2012665.6(1, 0, 0)

27. Table 7 Comparison of experiments and calculations for selected DOCl ro-vibrational energies in cm -1 from the latest ab initio PES (Zhang & Smith, latest results; Hu et al., J Mol Spec, 209, 105). nLHFDARexp(J, K a, K c ) ( 1, 2, 3 ) 03102.293096.8861(30, 0, 30)(1, 0, 0) 13112.573103.1051(30, 1, 30)(1, 0, 0) 23143.413138.8417(30, 2, 29)(1, 0, 0) 33194.803190.3545(30, 3, 28)(1, 0, 0) 43266.753261.8675(30, 4, 27)(1, 0, 0) 53359.253353.6688(30, 5, 26)(1, 0, 0) 63472.313465.6222(30, 6, 25)(1, 0, 0) 73605.933597.5640(30, 7, 24)(1, 0, 0)

28. 4. Conclusions and Future Work Development of Lanczos representation methods; Design of a parallel computing model; Combination of both has made rigorous quantum calculations possible for challenging J > 0 applications. Further comparative studies between QD/ST; Develop quantum statistical theories; Mixed QD/MD simulations in larger systems.

29. Acknowledgements Prof J. Troe (The University of Göttingen) Prof S. Nanbu (Kyushu University) Prof H. Nakamura (IMS) Dr Marlies Hankel CCMS/MD members Australian Research Council Supercomputer time from APAC & UQ