1. Quantum Dynamics of Four-Atom Reactions within the Real Wave Packet Framework Stephen K. Gray Chemistry Division Argonne National Laboratory Argonne, Illinois 60439 gray@tcg.anl.gov

2. Acknowledgements Gabriel Balint-Kurti: co-developer of the RWP method Evelyn Goldfield: co-developer of the four-atom implementation

3. Outline Introductory Remarks Real Wave Packet Framework: Cosine Iterative Equation Modified Schrödinger Equation Inferring Physical Observables Four-Atom Systems: Representation Dispersion Fitted Finite Differences Initial Conditions and Final State Analysis Cross Sections and Rate Constants Concluding Remarks

4. Introductory Remarks Real wave packet (RWP) method: An approach for obtaining accurate quantum dynamics involving the real part of a wave packet and Chebyshev iterations [Gray and Balint-Kurti] Can view it as a highly streamlined version of Tal- Ezer and Kosloff’s propagator Shares features with: Mandelshtam and Taylor’s Chebyshev expansion of the Green’s operator, Kouri and co-workers’ “time-independent” wave packets, Chen and Guo’s Chebyshev propagator

5. Cosine Iterative Equation

7. Cosine equation was successful, S. K. Gray, J. Chem. Phys. 96, 6543 (1992) However, cos(H  ) must still be evaluated in some way Can we do better?

8. Modified Schrödinger Equation Underlying time-independent Schrödinger equation has the same bound states (and scattering states) Solutions of the modified equation contain the same information as the more standard one

10. Inferring Physical Observables

11. But  (u) is still complex -- how to relate to q(u) = Re[  (u)] ? If  has no f(E) components for f(E) 0) Allows energy-resolved scattering and related quantities, e.g., S matrix elements and reaction probabilities, to be obtained from Fourier analysis of q.

13. Four-Atom Systems Diatom-diatom Jacobi coordinates, body-fixed z-axis is the R vector AB + CD  ABC + D

14. Representation J = total angular momentum quantum number p = parity K = projection of total angular momentum on a body-fixed axis (often an approximately good quantum number)

15. Gatti and co-workers; Goldfield; Chen and Guo Note: Most applications so far have assumed K to be good (centrifugal sudden approximation)

16. H and H q

17. Comments on H q : Three distance (or radial) kinetic energy contributions evaluated with either dispersion fitted finite differences (DFFD’s) or potential-optimized discrete-variable representations (PODVR’s) DFFD: Gray and Goldfied PODVR: Echave and Clary; Wei and Carrington

18. DFFD’s Can obtain signifcantly better Accuracy than standard FD approximation  Error in reaction probability for 3D D + H 2 reaction

19. V q Basis to grid, multiply By diagonal V, then Convert back to basis A key “trick” that allows large rotational bases to be treated Favorable, near linear scaling with problem size

20. Propagation and Analysis

21. Reaction Probabilities Write  I as FT of q (Meijer et al.) -- problem reduces to saving certain dq/ds and q at s 0 as a function of effective time and then constructing P I afterwards

22. Cross Sections, Rate Constants Since we can compute P I (E), I = some initial state, there is nothing special about constructing cross sections and rate constants The problem is the large number of I states that must be considered: I = J, p, K, j 1, j 2, k 1, v 1, v 2

23. A State-Resolved Cross Section:

24. Rate Constants

25. Approximation: J-Shifting Use result for a “reference” J to extrapolate to other J

26. Bowman has extensively discussed J-shifting The idea of using non-zero J values to base the J- shifting is not new -- previous work along related lines includes S. L. Mielke, G. C. Lynch, D. G. Truhlar, and D. W. Schwenke, Chem. Phys. Lett. 216, 441 (1993). H. Wang, W. H. Thompson, and W. H. Miller, J. Phys. Chem. A 102, 9372 (1998). J. M. Bowman and H. M. Shnider, J. Chem. Phys. 110, 4428 (1999). D. H. Zhang and J. Z. H. Zhang, J. Chem. Phys. 110, 7622 (1999).

27. Concluding Remarks For accurate quantum dynamics of three and four-atom systems, the RWP method is a good choice of methods -- less memory and more efficient than comparable complex wave packet calculations

28. However, to go beyond four-atoms requires (most likely) abandoning the detailed scattering theory approach involving complicated angular momentum bases and detailed state-resolved considerations Cumulative reaction probability and related approaches to direct evaluation of averaged quantities (Miller, Manthe) The use of parallel computers and Cartesian coordinates?