1. Lanczos Representation Methods in Application to Rovibrational Spectroscopy Calculations Hong Zhang and Sean Smith Quantum & Molecular Dynamics Group Center for Computational Molecular Science The University of Queensland, Australia

2. 1. Lanczos representation methods 1. Lanczos representation filter diagonalisation in application to the calculations of ro-vibrational spectroscopy and unimolecular dissociation resonances, which includes recent J > 0 calculations and parallel computing implementation. 2. TI Lanczos subspace wavepacket method; real Lanczos artificial boundary inhomogeneity (LABI); real single Lanczos propagation ABI in bimolecular reactions. 3. Comparison with real Chebyshev wavepacket method.

3. 2. Methodologies 1. Transform the primary representation (DVR or FBR) of the fundamental 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 state energies, resonance energies and widths, S matrix elements in reactive scattering.

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

5. Solving subspace eigen-problem or linear system

7. 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

8. Real Chebyshev wavepacket propagation (1) [Zhang H, Smith SC, JCP, 117: 5174, 2002]

9. Real Chebyshev wavepacket propagation (2)

10. Lanczos Subspace Filter Diagonalization Generalized eigenvalue equation: WB=SB  S jj’ =(  (E j )|  (E j’ )) W jj’ =(  (E j )|T M |  (E j’ ))

11. Low Storage Filter Diagonalization Time evolution operator

12. J > 0 Hamiltonian: expression

13. J > 0 Hamiltonian: expansion Multiplying the Hamiltonian on the left side by and utilizing some integral and differential formula, one can obtain the coupled equations of motion in internal coordinates

14. J > 0 Hamiltonian: integral formula

15. J > 0 Hamiltonian: differential formula

16. J > 0 Hamiltonian: coupled equations m = 0 for m = 1 for

17. J > 0 Hamiltonian: angular DVR 1) Diagonalizing the coordinate operator to obtain the DVR points and transformation matrix: 2) Gauss-Jacobi quadrature

18. J > 0 Hamiltonian: DVR

19. 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

20. Matrix-vector Multiplication

21. Processor assignment a) Master processor (ID = 0) Perform the main propagation; write auto- correlation functions or ,  elements; all other related works except the matrix- vector multiplications. b) Working processors Perform the matrix-vector multiplications for each  component.

22. 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. Thus some processors might need to wait for others. For the highest or the lowest  components, only one Coriolis coupling required.

23. 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.

24. 3. Recent Results 3. Recent Results HO 2 (J = 0-50): using both Lanczos and Chebyshev method from parallel computing. HOCl/DOCl (J = 0- 30): using parallel Lanczos method.

25. Table 1 Selected HO 2 bound state energies in eV for J = 1 (even symmetry) for comparison. (Ref: X. T. Wu and E. F. Hayes, JCP, 1997,107, 2705, LFD: Zhang H, Smith SC, JCP, 2003, 118, 10042) nLFDCFDRef 1.000271.000270 10.295249.295272 20.447875.447876.447903 30.574994.575033

26. Fig. 1 J = 6 (even symmetry) bound state energies for HO 2 using both Lanczos FD and Chebyshev FD method. Circles from Lanczos FD method while squares from Chebyshev FD method.

27. Table 2 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/LSFDLHFDBunkerARKaKa 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)

28. Table 3 Calculated HOCl ro-vibrational state energies in cm -1 with spectroscopic assignments for J = 20 (Zhang, Smith, Nanbu & Nakamura, JPCA, 2006, 110: 5468). 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

29. Table 4 Comparison of experiments and calculations for selected HOCl far infrared transitions in cm -1 (Zhang, Smith, Nanbu & Nakamura, JPCA, 2006,, 110: 5468). 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

30. Fig. 2 Parallel computation of resonances for J = 20, 30, 40 and 50 cases in HO 2 dissociation. The quantum average rates (green solid line) are compared with statistical/classical results of Troe. et. al.

31. 4. Conclusions 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.

32. Acknowledgements Prof. Nanbu, Prof. Nukamura and CC members; Prof. Sean Smith and CCMS members; Prof. R. Kosloff and Prof. V. A. Mandelshtam; JSPS Invitation Fellowship; Australian Research Council; Supercomputer time from APAC, UQ, CC, IMS.