High Dimensional Dynamical Systems: Theory and Computational Realisation for Molecular Dynamics

1. H. Waalkens, A. Burbanks, R. Schubert, S. WigginsSchool of MathematicsUniversity of Bristol High Dimensional Dynamical Systems: Theory and Computational Realisation for Molecular Dynamics (“or, phase spacemechanismsunderlying the dynamics”) Funded by the Office of Naval Research: Grant No. N00014-01-1-0769. Dr. Reza Malek-Madani

2. What do you want to know about these systems? • No ergodicity need to understand the mechanisms in phase space • governing…. • Dynamics of reaction, e.g. rates, reaction paths. Generally, how does the • reaction proceed? • Phase space geometry of reaction, e.g. what parts of phase space • participate in reaction? (important for “sampling strategies,” importance • sampling) • ???Control??? (beyond a “black box” approach)

3. Can’t you answer these questions with existing methods? What motivates “new methods?” • Many methods require assumptions on the dynamics to “get an answer,” e.g. RRKM Theory, umbrella sampling,… When are such assumptions valid? • Some sampling methods involve “modification of the dynamics” in order to overcome the “rare event problem.” What are the mechanisms in phase space • underlying “rare events,”“multiple time scales, ” “memory?” • Motivation: new experimental techniques, advances in laser spectroscopy, single molecule methods, yield real time dynamical information (complex systems have complex dynamics)

4. Growing Realization of the Ubiquity of Non-Ergodicity in Complex Systems….. • B. K. Carpenter [2005] Nonstatistical Dynamics in Thermal Reactions of Polyatomic Molecules.Annual Review of Physical Chemistry, 56, 57-89. • R. T. Skodje, X. M. Yang [2004] The Observation of Quantum Bottleneck States. International Reviews in Physical Chemistry, 23(2), 253-287. • A. Bach, J. M. Hostettler, P. Chen [2005] Quasiperiodic Trajectories in the Unimolecular Dissociation of Ethyl Radicals by Time Frequency Analysis. J. Chem. Phys. 123, 021101.

5. Only 78% of trajectories dissociate • Remaining trajectories have • lifetimes >>2 ps Analysis Tools: C. Chandre, S. Wiggins, T. Uzer [2003] Time-Frequency Analysis of Chaotic Systems. Physica D, 181, 171-196. L.-V. Arevalo, S. Wiggins [2001] Time-Frequency Analysis of Classical Trajectories Of Polyatomic Molecules. International Journal of Bifurcation and Chaos, 11, 1359-1380.

6. What can dynamical systems theory do for you? • Provides the framework for answering these questions— dynamics: phase space:mechanism (cannot deduce dynamics from the topology of the potential energy landscape) • Classify trajectories in terms of “qualitatively different behaviour,” e.g. reactive vs. non-reactive, fast slow time-scales, with invariant manifold techniques • Provide new, and more efficient, computational methods (based on exact dynamics) for computing reaction rates, reaction paths, understanding “rare events,” and incorporating and quantifying quantum mechanical effects

7. Recent Progress: Phase Space Transition State Theory (Original ideas--Wigner, Eyring, Polanyi) S. Wiggins, L. Wiesenfeld, C. Jaffe, T. Uzer [2001] Impenetrable Barriers in Phase Space. Physical Review Letters, 86(24), 5478-5481. T. Uzer, J. Palacian, P. Yanguas, C. Jaffe, S. Wiggins [2002] The Geometry of Reaction Dynamics. Nonlinearity, 15(4), 957-992. • Construct “dividing surfaces” with no (local) re-crossing and minimal flux. • These dividing surfaces “locally separate” the energy surface • These dividing surfaces are hemispheres of a (2n-2)d sphere (on an energy surface), whose “equator” is a (2n-3)d sphere that is a NHIM • Transport between components of the energy surface can only occur through the stable and unstable manifolds of the NHIM, which have the geometrical structure of “spherical cylinders,” • All of these geometrical structures can be realized through computationally efficient algorithms

8. HCN/CNH Isomerization: Benchmark Problem H n=3 degrees of freedom (Jacobi coordinates) 3D configuration space, 6D phase space, 5D energy surface C N J. Gong, A. Ma, S. A. Rice [2005] Isomerization and dissociation dynamics of HCN In a picosecond infrared laser field: A full-dimensional classical study. J. Chem. Phys. 122(14), 144311. J. Gong, A. Ma, S. A. Rice [2005] Controlled subnanosecond isomerization of HCN To CNH in solutions. J. Chem. Phys. 122(20), 204505.

9. Decoupling of the motion in terms of the normal form coordinates H. Waalkens, A. Burbanks, S. Wiggins [2004] Phase Space Conduits for reaction in Multidimensional systems: HCN Isomerization in three dimensions J. Chem. Phys. 121(13), 6207-6225.

10. Rigorous definition of a ``dynamical reaction path’’

11. Projections of Phase Space Structures into Configuration Space dividing surface DS NHIM (equator of DS) stable and unstable manifolds - manifolds can be realized through Poincare-Birkhoff normal form NF - explicit formulae for the manifolds in terms of NF coordinates- local pieces of stable and unstable manifolds can be ``globalized’’ by integrating trajectories

12. ``Reactive volume’’ -dynamics in the potential well is not ergodic-configuration space perspective is highly misleading (9% of initial conditions in HCN well can react)

13. H. Waalkens, A. Burbanks, S. Wiggins [2005] Efficient Procedure to Compute the Microcanonical Volume of Initial Conditions that Lead to Escape from a Multidimensional Potential Well. Physical Review Letters, 95, 084301. “Reactive Volume” flux form mean passage time flux through the dividing surface Flux is obtained “for free” from the normal form: H. Waalkens, S. Wiggins [2004] Direct construction of a dividing surface of minimal flux for multi-degree-of-freedom Systems that cannot be recrossed. J. Phys A: Math. Gen.37, L435-L445.

14. Application to HCN (by symmetry) (from normal form)

15. Brute-force Monte Carlo Calculations Survival probability:Uniformly sample initial conditions in the HCN component with respect to the measure and integrate them until they leave saturation value As a by-product of thecomputation (without integrating trajectories), we obtain the energy surface volume of the HCN component

16. Comparing Computational Efforts: Brute-Force Monte Carlo vs. Our Method Brute-Force Monte Carlo 10 M points, integrated for 500 ps Our method: M points, integrated (on average) for 0.174 ps => Efficiency 1: 30 000

17. “Rare Events”???? Muller-Brown Potential (2 DOF for simplicity) Deep well at “top” Shallow well at “bottom”

18. Iso-residence times for trajectories entering a well on the dividing surface Trajectories entering the shallow well Trajectories entering the deep well

19. Distribution of residence times alongfor trajectories entering the top well

20. Summary • Advances in theory, and the implementation of algorithms, enables the treatment of high dimensional problems • Dynamical systems theory provides a “dynamically exact” reaction rate theory (“transition state theory”) • From the dynamical systems framework we obtain a formula for the reactive volume which is more computationally efficient than classical Monte Carlo approaches • New notion of “dynamical reaction path” that respects the exact dynamics • Heteroclinic and homoclinic orbits as the skeleton of “rare events,” “routes to transition,” “dynamical memory” • The geometrical structures in phase space provide the framework for the quantum description Papers Available from http://lacms.maths.bris.ac.uk