continuum q phase field CG-MD f = Ma i t Y = H Y F (q) - - PowerPoint PPT Presentation

reaction rate theory

Peters et al. JACS (2008) Goldsmith et al., J. Chem. Phys. (2013)

reactors, crystallizers, etc.

solute precipitate nucleation

catalysis on amorphous materials

Peters, Chem. Eng. Sci. (2012) Duff et al. J. Chem. Phys. (2013)

The Peters Lab

where the future is a random variable

iħ∂tY =HY f = Ma CG-MD phase field continuum

q F(q)

iħ∂tY =HY f = Ma CG-MD phase field continuum

q F(q)

Rare events philosophy: Don’t wait to cross the barrier. Don’t alter the Hamiltonian. Compute F(q) and use non-eq. stat. mech. to get rate. Rate law becomes the “generation” term in macroscopic balance equations.

effort ~ DF‡/kBT

4

• dimensionality reduction
• preserve thermo & dynamics
• barriers, rates, and kinetic trends
• formulate simple theories
• test assumptions in theories of reaction dynamics

reaction coordinate = summary of the mechanism + …

Sometimes the reaction coordinate is obvious When it is not, we use the committor pB(x) = 0 for reactants pB(x) = 1 for products pB(x) ≈ ½ for ‡’s x

histogram of (crude) pB-estimates

Bolhuis et al. Ann. Rev. Phys. Chem. 2002 Peters, J. Chem. Phys. 2006, Ann. Rev. Phys. Chem. 2016

method 1: trial and error committor analysis

Neural network Likelihood maximization Inertial LMax

• spectral theory: no A, B
• transition path theory:
• variational principles:
• statistical inference:

Markov state models Diffusion map First step analyses Backward Kolmogorov Mincut, TSD Berezhkovskii-Szabo

from dynamics in 3N dimensions to scalar measure of progress

L†pB(x) = 0 pB ≈ 0 for reactants pB ≈ 1 for products

committor: universal reaction coordinate? pB(x) ≈ a ( y1

L(x) + b)

• ther definitions

are closely related:

shifted + scaled 1st left eigenfn velocity-avg’d forward committor

pB(x) = ∫dv r(v) pB(x, v)

backward equation:

pB(x), , y1

L(x) are abstract

mathematical objects with no simple physical interpretation concensus ?

common features of extraordinary rate theories

CNT: Gibbs et al. TST: Eyring et al. ETT: Marcus

Peters, J. Phys. Chem. B. (2015)

1) Kinetic trends: rates at many conditions from one calculation 2) Activation parameters and prefactors with physical interpretation 3) Extract molecular level insight from data

Peters, J. Phys. Chem. B. (2015)

ln[k/k0] (ionic strength)1/2 [additive] lnJ/J0 linker distance

4) corollary theories Poon eq. Bronsted eq. Dutton’s rule

common features of extraordinary rate theories

CNT: Gibbs et al. TST: Eyring et al. ETT: Marcus

common features of extraordinary rate theories

CNT: Gibbs et al. TST: Eyring et al. ETT: Marcus

Peters, J. Phys. Chem. B. (2015)

why are they so amazing? all three began from an accurate q with simple physical interpretation

So what is the ideal reaction coordinate?

• 1. Purely configurational: q = q(x)
• 2. Foliation: isosurfaces of q(x) should open

around A and close on B as q increases.

• 3. Dynamically consistent projections:

q(x) should be sufficient to predict the committor

if sufficient to predict the committor, then sufficient to predict the rate (even if not sufficient to give dynamically accurate Fokker-Planck eqn.)

Does the committor itself satisfy all three?

• 1. purely configurational

the committor depends on conditions, e.g. pB = pB(x, T) so theories based directly on pB cannot predict trends

A B

Peters et al. JACS (2008)

Peters, Ann. Rev. Phys. Chem. (2016, in press)

Neural network Likelihood maximization Inertial LMax

• spectral theory: no A, B
• transition path theory:
• variational principles:
• statistical inference:

Markov state models Diffusion map First step analyses Backward Kolmogorov Mincut, TSD Berezhkovskii-Szabo

High throughput mechanistic hypothesis testing: only LMax goes directly for scalar q

Aimless Shooting

• forward trajectory → B
• forward trajectory → A

Peters and Trout, J. Chem. Phys. (2006) Peters, Beckham, Trout, J.Chem.Phys. (2007) form of pB model motivated by KLBS theory variable length & permutation versions: Mullen et al. J. Chem. Theory and Comp. (2015)

mechanistic hypothesis: q(x) = reaction coordinate if correct, q is sufficient… correct hypothesis, i.e. the correct reaction coordinate, maximizes L

Likelihood maximization

test via trial function likelihood of observing the data

Peters, Chem. Phys. Lett., 554, 248 (2012)

narrow committors & high transmission coefficients auto-reverts to original LMax ( b=0 ) for overdamped dynamics

inertial Likelihood Maximization (iLMax)

rxn probability pRX(q, vq) instead of pB(q)

• prob. to reach B in Kramers-Grote-Hynes:

likelihood that trial q explains shooting data

1 2

( , ) {( ) }

RX

p q q erfc a q q bq         

‡

( ) ( ) ( ') ( ')

( ) ( ) ( ') ( ') , ,

( , ) (1 ( , ))

k k k k

B A k k k k RX RX

L p q q p q q

 

 

 

x x x x

results from 12D spin-boson model

Beckham, Peters, JPCLett, (2011)

P*=5.68, T*=0.888, Rcutoff=2.5Å

‡

liquid fcc

Aimless Shooting

long (ns) diffusive transition paths fcc crystal nucleation from supercooled LJ fluid



likelihood max = high throughput hypothesis testing

• Size:

– nDellago(2008) – nFrenkel=nYLM – det(I1· I2 · I3) – [det(I1· I2 · I3)]1/3 – [det(I1· I2 · I3)]1/6

• Shape:

– Imax / Imin – [det(I1· I2 · I3)]1/3 / Imin – nsurface / (ncluster)2/3

• Size · Structure:

– q6

cl · n

– q6

cl · n

– <css>∙n

• Size/(elongatedness):

– n / (Imax / Imin)

• Structure

– q2

box, q4 box, q6 box, q8 box

– w2

box, w4 box, w6 box, w8 box

– q2

cl, q4 cl , q6 cl , q8 cl

– w2

cl, w4 cl, w6 cl, w8 cl

– q2

box, q4 box, q6 box, q8 box

– q2

cl, q4 cl, q6 cl, q8 cl

– Q2

glob, Q4 glob, Q6 glob, Q8 glob

– W2

glob, W4 glob, W6 glob, W8 glob

– <Css> = average coordination of solid particles [Parrinello et al.] – Lechner-Dellago avg. local ql’s

Steinhardt, van Duijnveldt and Frenkel, ten Wolde and Frenkel, Ruiz-Montero, Donadio and Parrinello, Degranges and Delhommelle, Torquato, Debenedetti, Moroni, Bolhuis, Glotzer, Lechner and Dellago,… PRL, JCP, JACS, JPCB, PRE,…. from 1981 to 2011

ten Wolde, Ruiz-Montero, Frenkel, JCP, 1996

about 60 cords used for nxn in supercooled LJ

Beckham, Peters, JPCLett, (2011)

Beckham, Peters, JPCLett, (2011)

q = Q6

cl·n

40,000 dof to 1 !

i.e. Frenkel x Bolhuis P*=5.68, T*=0.888, Rcutoff=2.5Å

‡

liquid fcc

Likelihood maximization tests 60 mechanistic hypotheses in minutes

±

all consequential dynamics preserved 

 

19

Knott et al. J. Am. Chem. Soc. (2013) Hydrolysis of Cellulose by Cellulase bond making/breaking reactions 

Juraszek and Bolhuis, Biophysical J. (2008) Rate Constant and Reaction Coordinate of Trp-Cage Folding in Explicit Water

One coordinate to replace pFOLD(x) ? probably not… A reaction coordinate for the key transitions? Yes

bottleneck transitions in protein folding 

Mullen et al. J. Chem. Theory and Comp. 2014 Geissler, et al.

• J. Phys. Chem. B. (1999)

ion-pair (NaCl) dissociation

?

• which solvent coordinate ?
• if known, would k → 1.0 ?
• GHT = VTST: yes or no ?

Grote-Hynes theory: non-Markovian friction model Multidimensional harmonic bath: exact solution by transition state theory

two decades of convincing arguments and counterarguments…

Chandler, Faraday Disc. (1988) “Grote-Hynes theory is a version of multidimensional TST.” Pollak, J. Chem. Phys. (1991) “Once [the variational] transition state has been identified there are no further corrections, dynamical or otherwise.” Garrett and Truhlar, JPCB. (2000) “The success of GH theory means …

• ptimization of the dividing surface

does remove the recrossing.” Dakhnovskii and Ovchinnikov,

• Phys. Lett. (1985).“[BCHO]
• scillators should be considered

nonphysical… we can only determine their combination which is equal to the memory function.” Hynes, Faraday Disc. (1988) “GH theory … is more general than multidimensional TST, and reduces to it for the special case of a [BCHO]”.

vs.

… and VTST could not find optimal ‡-surface to resolve it.

Mullen et al. J. Chem. Theory and Comp. 2014 Mullen et al. J. Chem. Phys. (2014) Geissler, et al.

• J. Phys. Chem. B. (1999)

ion-pair (NaCl) dissociation

?

• which solvent coordinate ?
• if known, would k → 1.0 ?
• GHT = VTST: yes or no ?

k → 1.0 GHT = VTST testing theories

/ /

k saturated

while P(pB) still broad

P(pB) pB



different k ’s from dynamics

• r from free

energy barriers?

trends across series of similar reactions

from NaCl to all alkalai chlorides

Yonetani, J. Chem. Phys. (2015)



133 7

ion-desolvation in ionic crystal growth

Joswiak, Doherty, Peters spiral growth @ screw dislocation

Stack & Grantham,

• Cryst. Growth Des. 2010

BaSO4 growth

previous efforts

Stack et al. J. Am. Chem. Soc. 2012, 134, 11-14

wikipedia

metadynamics + reactive flux

a priori chosen solvent coordination number (metadynamics)

0 0.5 1

formate binding to TiO2 (rutile)

CN distance from surface

Mori et al. J. Chem. Theory Comput. 2013, 9, 5059-69

‡

no ‡’s at all ! kudos to Mori et al. for disclosure of limitations

*arbitrary shift in pmf

Cl- attachment/detachment in NaCl growth from water

NaCl (Halite)

poor rxn coordinate

hysteresis

aimless shooting + (inertial) likelihood maximization

detached attached trial coordinates: distances, coordination #s, energy gaps, local water density, etc.

How good are the new coordinates?

better

0 0.5 1

not great, but … …most improved?

baseline = solute only

good enough!

Why is k small? Would even better coordinates help?

31

0 - 2ps

p(q, t | ‡, 0) t =

32

2 - 4ps Why is k small? Would even better coordinates help?

p(q, t | ‡, 0) t =

33

4 - 6ps

Frozen near barrier

Why is k small? Would even better coordinates help?

p(q, t | ‡, 0) t =

pB vs. t : recrossing is innate. cannot eliminate with better q. so a model for F(q) will capture all kinetic trends

pulling ion out leaving unsolvated cavity solvation of kink site cavity

Marcus + Madelung ?

CAREER

www.engineering.ucsb.edu/~baronp/

• test mechanistic hypotheses
• dimensionality reduction with

clear mechanistic interpretation

• preserve thermo & dynamics
• barriers, rates, and trends
• test theories of reaction dynamics
• theories for series of related rxns

Ryan G. Mullen Mark Joswiak Joan-Emma Shea Peter Bolhuis Gregg Beckham Mike Doherty

谢谢

CTMC CHEM

David J. Gross

2004 Nobel, Physics

Alan J. Heeger

2000 Nobel, Chemistry

Herb Kroemer

2000 Nobel, Physics

Walter Kohn

1998 Nobel, Chemistry

Consider UCSB for your PhD or post-doc!

Shuji Nakamura

2014 Nobel, Chemistry

37

Chloride detachment at a corner site

Biasing CVs for metadynamics

• Height above terrace
• Cl coordination to Cl in top layer

Liu et al. Phys. Chem. Chem. Phys. 2011, 13, 13162-6

Aimless Shooting

• trajectories from 0 to t
• shoot from t/2±

t along the

• ld trajectory. (

t << t)

• choose momenta from exp[-

EK]

• propagate to 0 and t starting at

t/2± t on the new trajectory

• accept new trajectory if reactive

Original TPS: Dellago et al., J. Chem. Phys. 110 6617 (1999). Aimless shooting: Peters and Trout, J. Chem. Phys. (2006). * for NVE ensemble, select p from const. KE sphere

Aimless Shooting is TPS plus…

• shoots are independent

realizations of pB(x)

• shooting point distribution

~ p(TP|x)p(x|TP)

• easiest TPS to implement
• highly efficient even for

diffusive dynamics and long transition paths

Original TPS Aimless Shooting

An exact result for quadratic FES and constant diffusion tensor

• J. Langer, Ann. Phys. (1969)

Berezhkovskii and Szabo, JCP (2005) project onto 1D rxn coor in direction e direction e+ minimizes k(e) and gives kN if flux direction (i.e. the movie) reaction coordinate: e+ = ∂qpB direction (if q is a complete list)

exact but not practical … need many q, so 2nd derivatives of FES (A) at q‡ very expensive. …but “KLBS theory” is exact for the example problem: