Quantum Dynamics of Four-Atom Reactions within the Real Wave Packet Framework

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(Ht) 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 c(u) is still complex -- how to relate to q(u) = Re[c(u)] ? If c has no f(E) components for f(E) < 0 (or 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 + H2 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 s0as a function of effective time and then constructing PI afterwards

22. Cross Sections, Rate Constants Since we can compute PI(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, j1, j2, k1, v1, v2

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?