Chemical Reactions As Rare Events: Transition State Theory and Beyond

Chemical reactions as rare events: transition state theory and beyond.

Extending the scale

Thermodynamics: p, T, V, N Length continuum (m) ils Macroscopic 1 ta de regime e or average over 3 m 10 all processes Potential Energy many atoms es Surface: {R } 6 Mesoscopic s i 10 es (3N+1)dimensional regime oc pr few atoms many processes e 109 or E Microscopic m regime few processes

1015 109 103 1 Time (s)

{R } nd i Essentials of computational chemistry: theories and models. 2 edition. C. J. Cramer, JohnWiley and Sons Ltd (West Sussex, 2004). Ab initio atomistic thermodynamics and statistical mechanics of surface properties and functions

K. Reuter, C. Stampfl, and M. Scheffler, in: Handbook of Materials Modeling Vol. 1, (Ed.) S. Yip, Springer (Berlin, 2005). http://www.fhi-berlin.mpg.de/th/paper.html Chemical energy conversion: catalysis

Non-catalytic free-energy barrier Reactant(s)

ΔFnon-cat y g r e n e

e e r

F Reaction Product(s)

ΔFcat

Adsorption Desorption

Reaction coordinate Issues: ● Reaction rate: proportional to exp ( Δ F / kT)

● Selectivity: eliminate or at least reduce the undesired products Further reading on rare events techniques:

“Efficient sampling of Rare Events Pathways” Daniele Moroni, PhD thesis. http://wwwtheor.ch.cam.ac.uk/people/moroni/thesis.html

Study of rare events

• the mechanism: understanding the relevant features of the process, and the identification of a (set of) coordinates, called the reaction coordinate, that explains how the reaction proceeds. • the transition states: what are the dividing passages, what is the relevant change that the system must undergo to switch state • the rate constants: the transition probabilities per unit time. For the process A → B we call it . It can be considered as the frequency of the event, so that is the lifetime of state A. Corresponding concepts hold for the reversed process and . Road map

Setting the stage: The random telegraph

Transition state theory: the vocabulary TST: rigorous definition of the rate constant ... … and how to calculate it (BennetChandler approach)

the Nudged Elastic Band approach

the Transition Path Ensemble and its sampling testing the reaction coordinate: the committor analysis

Transition Interface Sampling

Road map

Setting the stage: The random telegraph

Transition state theory: the vocabulary TST: rigorous definition of the rate constant ... … and how to calculate it (BennetChandler approach)

the Nudged Elastic Band approach

the Transition Path Ensemble and its sampling testing the reaction coordinate: the committor analysis

Transition Interface Sampling

Setting the stage: The random telegraph

Jump probability

(Normalization)

Basic quantity of Markov processes:

Setting the stage: The random telegraph

Master equation:

Initial condition:

Conserved quantity:

Solution:

Stationary probabilities:

Setting the stage: The random telegraph Suppose, W is not known, but we want to measure it, through statistical sampling.

Ensemble average or, via ergodicity, time average: Number of A → B during

Total time spent in A transition probability per unit time The inverse of the matrix element has a simple meaning:

mean first passage time mean residence time

Rate constant Equality holds only if transition is instantaneous (not valid for “real” systems) Road map

Setting the stage: The random telegraph

Transition state theory: the vocabulary TST: rigorous definition of the rate constant ... … and how to calculate it (BennetChandler approach)

the Nudged Elastic Band approach

the Transition Path Ensemble and its sampling testing the reaction coordinate: the committor analysis

Transition Interface Sampling

Transition state theory: vocabulary

Not “=”, due to existence of (small) buffer region Eql. (Gibbs) distribution TST:

Transition state theory: vocabulary

Definition:

Transition state theory: vocabulary

Velocities? Assume dynamic evolution, e.g., NVTMD. Invoking ergodicity:

Transition state theory: vocabulary

Heaviside step

function Road map

Setting the stage: The random telegraph

Transition state theory: the vocabulary TST: rigorous definition of the rate constant ... … and how to calculate it (BennetChandler approach)

the Nudged Elastic Band approach

the Transition Path Ensemble and its sampling testing the reaction coordinate: the committor analysis

Transition Interface Sampling

Transition state theory: rate constant

We introduce a freeenergy term:

Transition state theory: rate constant

For a double well: approximate the integral with Gaussian around the minimum

Dynamical problem (rate constant) turned into static (freeenergy difference). Note the prefactor!

If (one of the Cartesian coordinates), then: Thus:

Road map

Setting the stage: The random telegraph

Transition state theory: the vocabulary TST: rigorous definition of the rate constant ... … and how to calculate it (BennetChandler approach)

the Nudged Elastic Band approach

the Transition Path Ensemble and its sampling testing the reaction coordinate: the committor analysis

Transition Interface Sampling

Transition state theory: BennetChandler approach Correlation function

For :

Key quantity (constant): reactive flux

In TST:

Translational invariance:

Transition state theory: BennetChandler approach

Now, , for small :

Transition state theory: BennetChandler approach

Algorithm: 1) Choice of reaction coordinate Intuition or methods previous lecture 2) Free energy calculation Via umbrella sampling, metadynamics, ... 3) Evaluation of the transmission coefficient

BennetChandler approach: transmission coefficient

Flux through the surface

Only reactive trajectories Road map

Setting the stage: The random telegraph

Transition state theory: the vocabulary TST: rigorous definition of the rate constant ... … and how to calculate it (BennetChandler approach)

the Nudged Elastic Band approach

the Transition Path Ensemble and its sampling testing the reaction coordinate: the committor analysis

Transition Interface Sampling

Nudged Elastic Band

Harmonic Transition State Theory:

Elastic band:

normalized local tangent at i

Climbing image:

Road map

Setting the stage: The random telegraph

Transition state theory: the vocabulary TST: rigorous definition of the rate constant ... … and how to calculate it (BennetChandler approach)

the Nudged Elastic Band approach

the Transition Path Ensemble and its sampling testing the reaction coordinate: the committor analysis

Transition Interface Sampling

Transition Path Sampling

Discretized (sequence of states) of a trajectory of length (a path):

point in phase space The statistical weight of a path Depends on initial distribution and specific dynamics

Assuming it is a Markov process:

initial conditions

Transition Path Sampling

Definition of transition path ensemble: path starts in A ends in B

sum over all pathways

In case of deterministic dynamics: time propagator (e.g., velocityverlet)

Sampling the path ensemble

Task: generating trajectories with frequency proportional to their weight old path new path

Use detailed balance for overall conditional probability

Since:

Fulfilled by Metropolis rule:

Sampling the path ensemble: moves

Shooting move Select time slice at random in the “old” path perturb the state (easiest: change momenta) new path generated by evolving backward and forward the modified state. accept via

(in particular, reject if does not go from A to B)

stationary distribution

Sampling the path ensemble: moves Shifting move

Time reversal

Sampling the path ensemble: algorithm

Sampling the path ensemble: computing averages

Set B defined by order parameter

Probability that a trajectory that starts in A reaches λ at time t Computing averages via umbrella sampling Traditional umbrella sampling:

Partitioning of the space:

For path probability:

Path ensemble: rate constant

Path that starts in A and visit B at least once

Connection with reactiveflux formalism

Path ensemble: rate constant

1. Calculate the average hB(t) AB in the path ensemble, i.e. paths that start in A and visit B at least once

2. If the time derivative d hB(t) AB ∗ displays a plateau go to next step, otherwise repeat step 1 with a longer time t

3. Calculate the correlation function C(t') for fixed t ∈ [0, t] using umbrella sampling

4. Determine C(t) = C(t ) hB (t) AB / hB (t ) AB in the entire interval [0, t].

5. Compute the derivative C(t). The rate constant kAB is the value of the plateau

Road map

Setting the stage: The random telegraph

Transition state theory: the vocabulary TST: rigorous definition of the rate constant ... … and how to calculate it (BennetChandler approach)

the Nudged Elastic Band approach

the Transition Path Ensemble and its sampling testing the reaction coordinate: the committor analysis

Transition Interface Sampling

Testing the reaction coordinate

Committor: probability that a trajectory started from configuration r ends in state B. It indicates the commitment of r to the basin of attraction of B.

Estimator:

The real reaction coordinate!

: Transition State Ensemble Testing the reaction coordinate

Given and the free energy:

Compute:

Good So so Bad (one not enough) (diffusive barrier) Transition Path Sampling: weak points

1. Rates are computed using C(t). This correlation function converges to the correct result because of a cancellation of positive and negative fluxes. It can be improved using the effective positive flux.

2. Paths have a fixed length. As a result they might spend time in the stable states. This time is wasted as far as the rate constant is concerned, because only the first passage time counts.

3. An initial path must be generated before starting the path sampling

Road map

Setting the stage: The random telegraph

Transition state theory: the vocabulary TST: rigorous definition of the rate constant ... … and how to calculate it (BennetChandler approach)

the Nudged Elastic Band approach

the Transition Path Ensemble and its sampling testing the reaction coordinate: the committor analysis

Transition Interface Sampling

measure whether the backward overall state (forward) time evolution of x will reach interface i before j or not.

points of first crossing with interface i on a backward (forward) trajectory starting in x0

Overall states:

Transition Interface Sampling: rate constant

overall state

In principle, this formula is an operational way to compute the rate: start an infinite long trajectory and count the number of effective positive crossings, i.e. the crossings of ∂ B when coming directly from A. In practice, it one needs to enhance the transition probability (rare event!)

Transition Interface Sampling: overall states

TIS: rate constant, connection to TST

this function shows a linear regime for 0 < t < τstable , instead of only for τ trans < t < τstable like in BC theory. Transition Interface Sampling: effective positive flux

only one point (full circle) contributes to the flux across i, the first one coming directly (no recrossing of i) from j. The other two recrossings (open circles) cancel each other in the flux

(operational definition for MD)

Transition Interface Sampling: conditional crossing probability

Introducing the weighted average:

Probability for the system to reach interface l before m under the condition that it crosses at t = 0 interface i, while coming directly from interface j in the past.

Or, in the ensemble φij of trajectories crossing i and coming directly from j, is the probability of reaching l before m For i

probability of reaching k before i after crossing j while coming directly from i Transition Interface Sampling: flux and probability theorems

For i

These are exact (no Markovian assumption) Transition Interface Sampling: rate constants

: special cases

relating the flux through ∂ B to the flux through an interface closer to A

positive crossings through λ1

Transition Interface Sampling: algorithm

Applications of Transition Interface Sampling (and Forward Flux Sampling)

Extending the scale

1015 109 103 1 Time (s)

K. Reuter, C. Stampfl, and M. Scheffler, in: Handbook of Materials Modeling Vol. 1, (Ed.) S. Yip, Springer (Berlin, 2005). http://www.fhi-berlin.mpg.de/th/paper.html Homogeneous crystal nucleation

+ –

Homogeneous crystal nucleation of LennardJonesium

Cristal nucleation of LennardJonesium

Count number of connected particles

Homogeneous crystal nucleation of LennardJonesium

Committor analysis

: Transition State Ensemble

Homogeneous crystal nucleation of LennardJonesium

~ bcc core fcc core bcc surf bcc surf

10% 50% 90%

Free energy isolevels Spacing: 1 kT

Homogeneous crystal nucleation of diamond

(Ideal mixture) Ghiringhelli et al., PRL (2007). 2011: PSR J17191438 b (brown dwarf or planet)

2012: 55 Cancri e (carbon planet) Maybe also V886 Centauri (BPM 37093), known from the '60s