On the Reaction Rate Calculation Methods using Path Sampling

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1 On the Reaction Rate Calculation Methods using Path Sampling Paris, France France 28 November, 2006 Titus S. van Erp Centre for Surface Chemistry and Catalysis, KULeuven Leuven, Belgium

2 Introduction Straightforward MD is incapable of studying rare events. Since 1934: The Reactive Flux (RF) method (Eyring, Wigner, Kecks, Bennett, Chandler,..) has been a standard method to tackle the timescale problem. 1999: the Transition Path Sampling (TPS) (Dellago, Bolhuis, Chandler,..) approach was developed as an alternative to the RF methods. Very recent, new path sampling approaches have been introduced such as TIS, PPTIS, FFS, Milestoning

3 Questions I want to address: What are the fundamental differences between Reactive Flux methods and Path Sampling methods and the differences between the different path sampling methods? What are their advantages and disadvantages?

4 For example, an acclaimed advantage of TPS is that is does not require a Reaction Coordinate. Is this statement really justified??? What is actually meant with this statement?? What is the Reaction Coordinate?? What about the new generation path sampling methods such as (PP)TIS, FFS, Milestoning??

5 Reactive flux approach Calculation of Free Energy barrier + transmission coefficient ΔF Reaction coordinate Constrained Dynamics can be used to evaluate the free energy profile Then, calculate the transmission coefficient by releasing MD trajectories from the top of the barriere

6 Reactive flux approach The free energy profile is sufficient to obtain the Transition State Theory (TST) approximation for the reaction rate. Both, the free energy barrier and the transmission coefficient dependent on the choice of RC The final result, however, is independent to the RC. Still, if the chosen RC is not somewhat close to a good RC this results in hysteresis, a low transmission coefficient and, hence, a very low efficiency

7 Transition Path Sampling a MC approach using dynamical (MD) pathways An effective way to sample Reactive trajectories An algorithm to calculate Reaction Rates order parameter Product State: B Reactant State: A

8 Transition Path Sampling a MC approach using dynamical (MD) pathways An effective way to sample Reactive trajectories An algorithm to calculate Reaction Rates order parameter SHOOTING

9 Transition Path Sampling a MC approach using dynamical (MD) pathways An effective way to sample Reactive trajectories An algorithm to calculate Reaction Rates order parameter SHIFTING

10 Transition Path Sampling a MC approach using dynamical (MD) pathways An effective way to sample Reactive trajectories An algorithm to calculate Reaction Rates order parameter No Reaction Coordinate required! The order parameter does not have to say anything about the actual shape of the barrier

11 Transition Path Sampling a MC approach using dynamical (MD) pathways An effective way to sample Reactive trajectories An algorithm to calculate Reaction Rates order parameter RC

12 Transition Path Sampling a MC approach using dynamical (MD) pathways An effective way to sample Reactive trajectories An algorithm to calculate Reaction Rates order parameter RC

13 Transition Path Sampling a MC approach using dynamical (MD) pathways An effective way to sample Reactive trajectories An algorithm to calculate Reaction Rates order parameter

14 Transition Path Sampling a MC approach using dynamical (MD) pathways An effective way to sample Reactive trajectories An algorithm to calculate Reaction Rates order parameter

15 Transition Path Sampling a MC approach using dynamical (MD) pathways An effective way to sample Reactive trajectories An algorithm to calculate Reaction Rates order parameter

16 Transition Path Sampling a MC approach using dynamical (MD) pathways An effective way to sample Reactive trajectories An algorithm to calculate Reaction Rates order parameter

17 Transition Path Sampling a MC approach using dynamical (MD) pathways An effective way to sample Reactive trajectories An algorithm to calculate Reaction Rates order parameter Final result is independent of the chosen order parameter

18 Transition Path Sampling a MC approach using dynamical (MD) pathways An effective way to sample Reactive trajectories An algorithm to calculate Reaction Rates order parameter However, can one still say that the order parameter is really different from a reaction coordinate as used in thermodynamic integration and umbrella sampling??

19 Transition Interface Sampling an improvement upon the TPS rate algorithm van Erp, Moroni, and Bolhuis, JCP 118, 7762 (2003)

20 TIS theory Product State Reactant state

21 TIS theory : by straightforward MD :very small number, can be calculated using following exact factorization:

22 TIS theory Condition: i) You re at the point to cross the surface in one single dt timestep ii) while was crossed more recently than is the chance that will be crossed before under these two conditions

23 TIS theory : All possible pathways that start at and end at, divided by all possible pathways that start at and end at either or with at least one crossing with

24 TIS Algorithm: MD Tom Caremans

25 TIS Algorithm:1st path simulation

26 TIS Algorithm: 2nd path simulation

27 Bad Good TPS Fixed path length that must be rather long and lies partly within the stable states. Slow convergence for recrossing events: cancellation between positive and negative terms Shooting and shifting (Time-reversal) moves US-type simulation can be performed with shorter paths using a correctionfactor, but the statistical error in the correction prohibits much advancement. US-type simulation provides not so much information about the mechanism A bit more flexibility in the state definitions A and B Flexible path length that is much shorter on average and lies solely on the barrier region Fast convergence: binary type of MC sampling. No shifting TIS Pathways are automaticly shorter for the path ensembles with an interface that has to be crossed closer to state A TIS path ensembles provides a natural data set to analyze success and failure at each step in the reaction A and B definitions must be really stable: trajectories entering A or B should commit to these states.

28 Partial Path TIS Lowering of the history dependence to reduce the average path length

29 Partial Path TIS Lowering of the history dependence to reduce the average path length

30 Partial Path TIS Lowering of the history dependence to reduce the average path length

31 Partial Path TIS Lowering of the history dependence to reduce the average path length

32 Partial Path TIS

33 Milestoning Very similar to PPTIS, but stronger Markovian assumption: the system remains at equilibrium at each crossing interface Provides additional dynamical information by taking track of the time-distribution of the hopping probabilities

34 Forward Flux Sampling (FFS) Based on the same rate equations as TIS The way pathways are generated is very different from TIS

35 Forward Flux Sampling (FFS) Based on the same rate equations as TIS The way pathways are generated is very different from TIS TIS: keep one of the succesfull paths and start shooting

36 Forward Flux Sampling (FFS) Based on the same rate equations as TIS The way pathways are generated is very different from TIS FFS: keep all the succesfull paths and start stochastic release

37 Forward Flux Sampling (FFS) Based on the same rate equations as TIS The way pathways are generated is very different from TIS FFS: keep all the succesfull paths and start stochastic release

38 Efficiency scaling as function of the barrier height and quality of the RC. TvE, JCP 125, (2006)

39 Efficiency scaling as function of the barrier height and quality of the RC : TIS shows no hysteresis! TvE, JCP 125, (2006)

40 Efficiency scaling as function of the barrier height and quality of the RC : TIS shows no hysteresis! TvE, JCP 125, (2006)

41 Other path sampling methods: Milestoning and PPTIS with small interface separation give systematic errors TvE, JCP 125, (2006)

42 Other path sampling methods: FFS, a large number of pathways in the low populated tails of the distributions at are need to end up with the correct uniform distribution at TvE, JCP 125, (2006)

43 Do path sampling methods require a reaction coordinate? The final result doesnot depend on it, but same is true for RF methods. Still it seems that the efficiency is less sensitive to what reaction coordinate/order parameter you chose compared to RF methods at least for TIS/TPS. But is a truly RC free method possible?

44 Truly order parameter-free sampling: Use the time as transition-parameter

45 Bad Good TIS FFS PPTIS Milestoning Exact Exact Memory loss approximation Assumption of full equilibrium at each crossing interface Average path length can be long ~50 % reduction compared to TIS Much shorter than TIS 50 % reduction compared to PPTIS Works for all kind of dynamics Only stochastic dynamics See TIS See TIS Moderate correlation between pathways. No correlation between different path ensembles Strong correlations between pathways and path-ensembles Similar to TIS Few correlations. No correlation between different Path ensembles Equilibrium systems or nonequilibrium systems for which there exist a known steady state distribution. Equilibrium/Non- Equilibrium. No information about distribution is needed See TIS See TIS Systems should obey separation of timescales See TIS See TIS Time dependent crossing probabilities Efficiency is relatively insensitive to the chosen RC Efficiency depends strongly on the RC Systematic error or loss efficiency Systematic error that depends on the RC

46 The End Thank you for your attention! (this presentation and additional information will be downloadable from or search transition interface sampling in google)

Dynamical Rare event simulation techniques for equilibrium and non-equilibrium systems arxiv: v2 [cond-mat.stat-mech] 31 Mar 2011

Dynamical Rare event simulation techniques for equilibrium and non-equilibrium systems arxiv:1101.0927v2 [cond-mat.stat-mech] 31 Mar 2011 Contents Titus S. van Erp Centrum voor Oppervlaktechemie en Katalyse,