reaction rate theory reactors, crystallizers, etc. The Peters Lab where the future is a random variable Peters et al. JACS (2008) Peters, Chem. Eng. Sci. (2012) catalysis on amorphous materials solute precipitate nucleation Duff et al. J. Chem. Phys. (2013) Goldsmith et al., J. Chem. Phys. (2013)
F (q) continuum q phase field CG-MD f = Ma iħ ∂ t Y = H Y
F (q) continuum q phase field Rare events philosophy: CG-MD Don’t wait to cross the barrier. Don’t alter the Hamiltonian. Compute F (q) and use non-eq. stat. mech. to get rate. f = Ma Rate law becomes the “generation” term in macroscopic balance equations. iħ ∂ t Y = H Y effort ~ D F ‡ /k B T
reaction coordinate = summary of the mechanism + … - dimensionality reduction - preserve thermo & dynamics - barriers, rates, and kinetic trends - formulate simple theories - test assumptions in theories of reaction dynamics 4
Sometimes the reaction coordinate is obvious When it is not, we use the committor x p B ( x ) = 0 for reactants p B ( x ) = 1 for products p B ( x ) ≈ ½ for ‡ ’s
method 1: trial and error committor analysis histogram of (crude) p B -estimates Bolhuis et al. Ann. Rev. Phys. Chem. 2002 Peters, J. Chem. Phys. 2006, Ann. Rev. Phys. Chem. 2016
• spectral theory: no A, B Markov state models Diffusion map • transition path theory: First step analyses Backward Kolmogorov from dynamics in 3N dimensions to scalar • variational principles: measure of progress Mincut, TSD Berezhkovskii-Szabo • statistical inference: Neural network Likelihood maximization Inertial LMax
committor: universal reaction coordinate? backward equation: L † p B ( x ) = 0 p B ≈ 0 for reactants p B ≈ 1 for products other definitions are closely related: p B ( x ) = ∫d v r( v ) p B ( x, v ) velocity- avg’d forward committor p B ( x ) ≈ a ( y 1 L ( x ) + b ) shifted + scaled 1 st left eigenfn concensus ? , y 1 p B ( x ) , L ( x ) are abstract mathematical objects with no simple physical interpretation
common features of extraordinary rate theories CNT: Gibbs et al. 1) Kinetic trends: rates at many conditions from one calculation 2) Activation parameters and prefactors with physical interpretation 3) Extract molecular TST: Eyring et al. level insight from data ETT: Marcus Peters, J. Phys. Chem. B . (2015)
common features of extraordinary rate theories 4) corollary theories CNT: Gibbs et al. Poon lnJ/J 0 eq. [additive] Bronsted eq. TST: Eyring et al. ln[k/k 0 ] (ionic strength) 1/2 ETT: Marcus Dutton’s rule linker distance Peters, J. Phys. Chem. B. (2015)
common features of extraordinary rate theories CNT: Gibbs et al. why are they TST: Eyring et al. so amazing? all three began from an accurate q with simple physical interpretation ETT: Marcus Peters, J. Phys. Chem. B . (2015)
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: B A 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 Peters et al. JACS (2008) dynamically accurate Fokker-Planck eqn.) Does the committor itself satisfy all three? 1. purely configurational the committor depends on conditions, e.g. p B = p B ( x, T) so theories based directly on p B cannot predict trends Peters, Ann. Rev. Phys. Chem . (2016, in press)
• spectral theory: no A, B Markov state models Diffusion map • transition path theory: First step analyses Backward Kolmogorov • variational principles: Mincut, TSD Berezhkovskii-Szabo • statistical inference: Neural network High throughput mechanistic Likelihood maximization hypothesis testing: only LMax goes directly for scalar q Inertial LMax
Aimless Shooting Likelihood maximization mechanistic hypothesis: q ( x ) = reaction coordinate if correct, q is sufficient… test via trial function likelihood of observing the data ● forward trajectory → B ● forward trajectory → A Peters and Trout, J. Chem. Phys. (2006) Peters, Beckham, Trout, J.Chem.Phys. (2007) correct hypothesis, i.e. the correct form of p B model motivated by KLBS theory variable length & permutation versions: reaction coordinate, maximizes L Mullen et al. J. Chem. Theory and Comp. (2015)
inertial Likelihood Maximization (iLMax) rxn probability p RX ( q, v q ) instead of p B ( q ) prob. to reach B in Kramers-Grote-Hynes: ‡ ( , ) {( ) } 1 p q q erfc a q q bq 2 RX likelihood that trial q explains shooting data B A ( ) ( ) ( ') ( ') ( , ) (1 ( , )) k k k k L p q q p q q RX RX ( ) ( ) ( ') ( ') k k k k , , results from 12D x x x x spin-boson model narrow committors & high transmission coefficients auto-reverts to original LMax ( b=0 ) for overdamped dynamics Peters, Chem. Phys. Lett., 554, 248 ( 2012 )
fcc crystal nucleation from supercooled LJ fluid ‡ liquid Aimless Shooting long (ns) diffusive transition paths fcc P*=5.68, T*=0.888, R cutoff =2.5 Å Beckham, Peters, JPCLett , (2011)
likelihood max = high throughput hypothesis testing • Structure • Size : – q 2 box , q 4 box , q 6 box , q 8 box – n Dellago (2008) – w 2 box , w 4 box , w 6 box , w 8 box – n Frenkel =n YLM cl , q 6 cl , q 8 – det( I 1 · I 2 · I 3 ) – q 2 cl , q 4 cl – w 2 cl , w 4 cl , w 6 cl , w 8 – [det( I 1 · I 2 · I 3 )] 1/3 cl – [det( I 1 · I 2 · I 3 )] 1/6 – q 2 box , q 4 box , q 6 box , q 8 box ten Wolde, Ruiz-Montero, Frenkel, JCP , 1996 • Shape: – q 2 cl , q 4 cl , q 6 cl , q 8 cl – I max / I min about 60 cords used for nxn in – [det( I 1 · I 2 · I 3 )] 1/3 / I min – Q 2 glob , Q 4 glob , Q 6 glob , Q 8 glob supercooled LJ – W 2 glob , W 4 glob , W 6 glob , W 8 – n surface / ( n cluster ) 2/3 glob • Size · Structure: – < C ss > = average coordination of solid cl · n – q 6 particles [Parrinello et al.] cl · n – q 6 – Lechner-Dellago avg. local q l ’s – <c ss >∙n • Size/(elongatedness): Steinhardt, van Duijnveldt and Frenkel, ten Wolde and Frenkel, Ruiz-Montero, Donadio and Parrinello, Degranges – n / ( I max / I min ) and Delhommelle, Torquato, Debenedetti, Moroni, Bolhuis, Glotzer, Lechner and Dellago,… PRL, JCP, JACS, JPCB, PRE,…. from 1981 to 2011 Beckham, Peters, JPCLett , (2011)
‡ liquid Likelihood maximization tests 60 mechanistic hypotheses in minutes fcc P*=5.68, T*=0.888, R cutoff =2.5 Å 40,000 dof to 1 ! q = Q 6 cl · n all consequential dynamics preserved i.e. Frenkel x Bolhuis ± Beckham, Peters, JPCLett , (2011)
Knott et al. J. Am. Chem. Soc. (2013) Hydrolysis of Cellulose by Cellulase bond making/breaking reactions 19
One coordinate to replace p FOLD ( x ) ? probably not… A reaction coordinate for the key transitions? Yes Juraszek and Bolhuis, Biophysical J. (2008) Rate Constant and Reaction Coordinate of Trp-Cage Folding in Explicit Water bottleneck transitions in protein folding
ion-pair (NaCl) dissociation Geissler, et al. J. Phys. Chem. B. (1999) ? - 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 Mullen et al. J. Chem. Theory and Comp. 2014
two decades of convincing arguments and counterarguments… Chandler, Faraday Disc. (1988) Dakhnovskii and Ovchinnikov , “Grote -Hynes theory is a version of Phys. Lett. (1985). “[BCHO] multidimensional TST.” oscillators should be considered nonphysical… we can only Pollak , J. Chem. Phys. (1991) determine their combination which “Once [the variational] transition state is equal to the memory function.” has been identified there are no further vs. corrections, dynamical or otherwise.” Hynes , Faraday Disc. (1988) “GH theory … is more general than Garrett and Truhlar , JPCB. (2000) multidimensional TST, and reduces “The success of GH theory means … to it for the special case of a optimization of the dividing surface [BCHO]”. does remove the recrossing.” … and VTST could not find optimal ‡ - surface to resolve it.
ion-pair (NaCl) dissociation Geissler, et al. J. Phys. Chem. B. (1999) ? - which solvent coordinate ? - if known, would k → 1.0 ? - GHT = VTST: yes or no ? P(p B ) k → 1.0 / p B / GHT = VTST testing theories k saturated while P ( p B ) still broad Mullen et al. J. Chem. Theory and Comp. 2014 Mullen et al. J. Chem. Phys . (2014)
trends across series of similar reactions from NaCl to all alkalai chlorides 7 133 different k ’s from dynamics or from free energy barriers? Yonetani, J. Chem. Phys. (2015)
ion-desolvation in ionic crystal growth spiral growth @ screw dislocation Stack & Grantham, Cryst. Growth Des. 2010 Joswiak, Doherty, Peters
previous efforts BaSO 4 growth wikipedia metadynamics + reactive flux Stack et al. J. Am. Chem. Soc. 2012, 134 , 11-14
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