chemical reactions as rare events: transition state theory ...luca/course_tu/10_tst_tis.pdf · with...

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

Extending the scale

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

{Ri}

E

Potential Energy Surface: {Ri}

(3N+1)dimensional

109

106

103

1

1015 109 103 1

Length(m)

Time (s)

Microscopicregime

Mesoscopicregime

Macroscopicregime

few processes

few atoms

many atoms

many processes

continuum

average overall processes

more deta

ils

more proce

sses

Thermodynamics:p, T, V, N

Chemical energy conversion: catalysis

Fre

e en

ergy

Non-catalytic free-energy barrier

Adsorption

Reaction

Desorption

Reaction coordinate

Product(s)

Reactant(s)

Issues:● Reaction rate: proportional to exp ( F / kT) Δ● Selectivity: eliminate or at least reduce the undesired products

ΔFnon-cat

ΔFcat

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

• 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 .

Study of rare events

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

Basic quantity of Markov processes:


Jump probability

(Normalization)

Master equation:

Initial condition:

Conserved quantity:

Solution:

Stationary probabilities:


Suppose, W is not known, but we want to measure it, through statistical sampling.

Ensemble average or, via ergodicity, time average:

mean residence time

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

Number of A B during→

Total time spent in A

transition probability per unit time

mean first passage time


Rate constant

The inverse of the matrix element has a simple meaning:

Road map






TST:

Not “=”, due to existence of (small) buffer region

Transition state theory: vocabulary

Eql. (Gibbs) distribution


Definition:

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



Heaviside step function

Road map






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:

Transition state theory: rate constant

Road map






Transition state theory: BennetChandler approach

Key quantity (constant): reactive flux

Translational invariance:

Correlation function

For :

In TST:

Now, , for small :


Algorithm:1) Choice of reaction coordinate

Intuition or methods previous lecture2) 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






Harmonic Transition State Theory:

normalized local tangent at i

Elastic band:

Climbing image:

Nudged Elastic Band

Road map






Transition Path Sampling

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

Assuming it is a Markov process:

The statistical weight of a path

Depends on initial distribution and specific dynamics

point in phase space

initial conditions

Definition of transition path ensemble:

sum over all pathways

path starts in A ends in B

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

Transition Path Sampling

Sampling the path ensemble

Task: generating trajectories with frequency proportional to their weightold 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)

stationarydistribution

Shifting move

Time reversal

Sampling the path ensemble: moves

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

Partitioning of the space:

Traditional umbrella sampling:

For path probability:

Computing averages via umbrella sampling

Path that starts in A and visit B at least once

Path ensemble: rate constant

Connection with reactiveflux formalism

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

Path ensemble: rate constant

Road map






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

Given and the free energy:

Good So so(one not enough)

Bad(diffusive barrier)

Testing the reaction coordinate

Compute:

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

Transition Path Sampling: weak points

Road map







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

overall state

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

Overall states:

Transition Interface Sampling: rate constant

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: rate constant

overall state

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 τtrans < t < τstable like in BC theory.

Transition Interface Sampling: effective positive flux

(operational definition for MD)

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

Transition Interface Sampling: conditional crossing probability

Introducing the weighted average:

For i<j<k the direct flux from i through k is given by the direct flux from i through j < k times the conditional probability of reaching k before i after crossing j whilecoming directly from i

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 ijφ of trajectoriescrossing i and coming directly from j,

is the probability of reaching l before m

Transition Interface Sampling: flux and probability theorems

For i<j<k

For i<j<k<l

These are exact (no Markovian assumption)

: special cases

Transition Interface Sampling: rate constants

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

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

{Ri}

E

Potential Energy Surface: {Ri}

(3N+1)dimensional

109

106

103

1

1015 109 103 1

Length(m)

Time (s)

Microscopicregime

Mesoscopicregime

Macroscopicregime

few processes

few atoms

many atoms

many processes

continuum

average overall processes

more deta

ils

more proce

sses

Thermodynamics:p, T, V, N

+ –

Homogeneous crystal nucleation

Count number of connected particles

Cristal nucleation of LennardJonesium

Homogeneous crystal nucleation of LennardJonesium

Committor analysis

: Transition State Ensemble



Free energy isolevelsSpacing: 1 kT

50%10% 90%

fcc corebcc surf

~ bcc corebcc surf

Homogeneous crystal nucleation of diamond

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

chemical reactions as rare events: transition state theory ...luca/course_tu/10_tst_tis.pdf · with...

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