H. Waalkens, A. Burbanks, R. Schubert, S. Wiggins

School of Mathematics

University of Bristol

High Dimensional Dynamical Systems: Theory and Computational Realisation for Molecular Dynamics (“or, phase space mechanisms underlying 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 What do you want to know about these systems?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 Can’t you answer these questions with existing methods? What with existing methods? What motivates “new methods?”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 spaceWhat are the mechanisms in phase space

underlying underlying “rare events,”“rare events,” “multiple time scales, ” “memory?”“multiple time scales, ” “memory?”

•Motivation: 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-Growing Realization of the Ubiquity of Non-Ergodicity in Complex Systems…..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

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.

Analysis Tools:Analysis Tools:

6 What can dynamical systems theory do for What can dynamical systems theory do for you?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 Recent Progress: Phase Space Transition State Theory State Theory (Original ideas--Wigner, Eyring, Polanyi)(Original ideas--Wigner, Eyring, Polanyi)

• 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

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.

RS n 32

8HCN/CNH Isomerization: Benchmark ProblemBenchmark Problem

n=3 degrees of freedom (Jacobi coordinates)

3D configuration space, 6D phase space,

5D energy surface

H

R

rC

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.

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

11 Projections of Phase Space Structures into Configuration Space

NHIMstable and unstable

manifoldsdividing surface DS

- 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

4S 3SRS3 (equator of DS)

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.

i

i

i

tt

,enter

,enter

,enter

DS

DS

DS

i

i,enterDS

1)!1(

1 nn

)dd1

k

n

kk qp

i

ii

tN ,enterDSreact

mean passage time

flux through the dividing surface

“Reactive Volume”

flux form

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

tN 2react;HCN

ps174.0t

2h0085.0 (from normal form)

(by symmetry)

eV

h0717.0

3

react;HCN N

15 Brute-force Monte Carlo Calculations

HCN

reactHCN;

HCN

react-nonHCN; 1)(N

N

N

NPs

eV

h795.0

3

HCN N

dpdpdpdrdRdHE Rr)(

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 the computation (without

integrating trajectories), we obtain the energy

surface volume of the HCN component

91.01)(HCN

reactHCN; N

NPs

16 Comparing Computational Efforts: Brute-Force Monte Comparing Computational Efforts: Brute-Force Monte Carlo vs. Our MethodCarlo vs. Our Method

Our method:

M points, integrated (on average) for 0.174 ps

Brute-Force Monte Carlo

10 M points, integrated for 500 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

02 p

20 SummarySummary• 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