grote-hynes, pollak, and dynamics of the committor in ion...

Solvent Coordinates Basic Coordinates Coordination number, N i • number of waters within a given radius of cation, anion, ion pair, or both ions Interionic water density, ρ ii • density of water between the ions Grote-Hynes Regimes Relevant Timescales • Solute – inverse barrier frequency ω B -1 , time for solute coordinate to traverse barrier without friction • Solvent – bath memory time τ mem , time for solvent bath to respond to changes in the solute coordinate Likelihood Maximization Test many coordinates with one set of aimless shooting trajectories Incorporate coordinate velocity to select for high transmission coefficient Inertial Likelihood Maximization Solute Coordinate: ion-pair distance, r ion Seems like a simple problem, but… Na + Cl – in TIP3P Water Original Likelihood Maximization q LMax = q(r ion , N B , r opt ) with ΔlnL = +185 Inertial Likelihood Maximization q iLMax = q(r ion , ρ ii , N B ) with ΔlnL = +255 It is possible to obtain a high transmission coefficient from a coordinate that imperfectly describes the committor. This finding is fortunate because Ballard and Dellago showed that detailed non-local information from the first three solvation shells is needed to accruately predict the committor. Reaction Mechanism All waters near ion-pair midpoint (gray) increase the interionic density ρ ii , but only properly oriented waters (red) coordinate to both ions, N B . ΔlnL[q(r ion , ρ ii )] = +233 while ΔlnL[q(r ion , N B )] = +218, showing that ρ ii is the dominant solvent parameter. Free Energy Surfaces No simple rotation of the reaction coordinate will remove the metastable intermediates in the free energy landscape or the nonlinear forces from anharmonicity in the potential energy surface. We conclude these features make recrossing an intrinsic part of ion-pair dissociation. Background Methods Transition Path Sampling Monte Carlo in trajectory space • dynamics are not biased along q Basin definitions • large: capture fluctuations • too large: encroach on barrier Aimless Shooting Moves Independent realizations of p B Variable length trajectories allow for narrow basin definitions Transition State Theory (TST) and Solvent-Induced Friction TST: One-directional, equilibrium flux from reactants to products through a dividing surface q(x) = q ‡ Friction: Interactions between bath coordinates and reaction coordinate • induces recrossing of dividing surface • causes k TST to overestimate k Models of Reactions in Solution Grote-Hynes: Generalized Langevin Equation (GLE) • incorporated friction into dynamics • derived dynamic correction to TST: k = κ GH k TST,1D Pollak: Bath of harmonic oscillators • included bath modes in Hamiltonian • variationally optimized the dividing surface, including both solute and solvent degrees of freedom • equivalent to GLE • obtained exact rate from TST: k = k TST,(3N)D Applications to Atomistic Solvents Use the solute coordinate • account for correlated recrossing using the transmission coefficient, κ • obscures the reaction mechanism, as κ does not differentiate between coordinate error and natural friction in the dynamics Find a better reaction coordinate • committor, p B is probability of relaxing to products • ideal reaction coordinate cuts configuration space along isocommittors • contains no mechanistic information about common physical characteristics of transition states Goal Test whether friction-induced recrossing can be eliminated for reactions in atomistic solvents For ion-pair dissociation in water, identify an improved solvent reaction coordinate Grote-Hynes, Pollak, and Dynamics of the Committor in Ion Pair Dissociation Ryan Gotchy Mullen 1 , Joan-Emma Shea 2 , Baron Peters 1,2 (1) Department of Chemical Engineering, (2) Department of Chemistry & Biochemistry, University of California, Santa Barbara, CA, 93106-5080 Summary Pollak eliminated recrossing in harmonic oscillator model by locating the dividing surface at the p B = ½ isosurface. For ion-pair dissocation, κ[p B ] < 1, dynamic recrossing is an inescapable consequence of dimensionality reduction to a single coordinate. Inertial likelihood maximization improves upon the ion-pair distance by identifying a solvent coordinate with a high transmission coefficient. Free energy surfaces reveal multiple channels, stable intermediates and strong anharmonicity along important coordinates. Potential Applications Ion Pair Results Dynamics of the Committor We thank the NSF CAREER and Graduate Research Fellowship for funding. We thank Andrew Ballard and Cristoph Dellago for helpful discussions. Acknowledgments Ionic Crystal Growth Protein Self-Assembly & Fibril Formation k TST = 1 2 ! q e ! !"F ‡ Trajectory Analysis Trajectories that recross in both forward & time-reversed directions give conflicting information about coordinate accuracy. A dividing surface with no recrossing and κ = 1, like that of Pollak’s harmonic oscillator model, must be composed entirely of p B = ½ states. Ion Pair Results If recrossing can be eliminated, the committor itself should give κ[p B ] = 1. The top of F(p B ) is almost, but not completely, ﬂat as anMcipated from the backward Kolmogorov equaMon. Clearly, κ[p B ] < 1, so the dynamical fricMon associated with the water solvent molecules does not behave like the fricMon in the harmonic bath model. Recrossing persists even for the p B = ½ dividing surface. Gale & coworkers, JACS, 2006 Berne & coworkers, Nature, 2005 reaction coordinate Free Energy ✗ gas phase in polar solvent Reactants Products q ‡ ΔF ‡ q 1 m!! q = ! "V "q ! m d ! " (t ! ! ) ! q(t ) # + R(t ) direct force + friction + thermal fluctuations H = p 2 2 m + V ( q ) + 1 2 m i ! x i 2 + m i i ! ! i 2 x i + c i m i ! i 2 q " # $ % & ’ 2 solute modes + bath modes coupled to solute Non‐recrossing A B Recross in Both p B Mme q βF κ p(p B ) narrow narrow broad broad low low low high 0 1 0 1 time ps Friction on r ion 0 1 2 0 1 time ps Reactive Flux 2 3 4 5 6 7 8 0 2 4 6 8 ionion separation, r ion Å Free Energy kT Cl Na contact Cl Na solvent separated κ[r ion ] = 0.29 ± 0.02 0 1 0 1 2 committor, p B Probability Density 0 Κ 1 time Reactive Flux No recrossing Numerous recrossing 0 12 1 committor Probability Density Accurate coord Inaccurate coord ‡ A B contact ion pairs are only narrowly metastable, ΔF(r ion ) = 5.2 k B T Grote-Hynes theory works using solvent friction on r ion extracted from simulation correlated recrossing: dynamics or coordinate error? bimodal distribution suggests missing coordinate(s) Geissler, Dellago, Chandler, J Phys Chem B, 1999 which may be bilinearly coupled to r ion Truhlar, Garret, J Phys Chem B, 2000 0 1 2 0.0 0.5 1.0 time ps Reactive Flux 0 0.5 1 0 2 4 6 8 committor, p B Free Energy k B T κ[p B ] = 0.38 ± 0.09 ΔF(p B ) = 7.2 k B T F ‐∞ as p B 1 Rhee et al., J Phys Chem B, 2005 Rey, Guardia, J Phys Chem, 1992 Karim, McCammon, Chem Phys Letters, 1986 Fast Bath τ mem << ω B -1 Markovian ‡ GH solute coordinate bath coordinate A B ‡ Pollak solute coordinate bath A B solute coordinate bath A B Slow Bath ω B -1 << τ mem Nonadiabatic Dynamic Caging 0 1 2 0.0 0.5 1.0 time ps Reactive Flux r ion q LMax 0 0.5 1 0 1 2 committor, p B Probability Density 0 0.5 1 0 1 2 committor, p B Probability Density 0 1 2 0.0 0.5 1.0 time ps Reactive Flux r ion q iLMax κ[q LMax ] = 0.07 ± 0.15 is lower κ[q iLMax ] = 0.43 ± 0.02 improves N B = 0 ρ ii = 0.016 Å -3 N B = 1 ρ ii = 0.016 Å -3 N B = 1 ρ ii = 0.018 Å -3 r ion = 4.15 Å r ion = 3.85 Å r ion = 3.64 Å Cl – q iLMax =0 similar to harmonic valley, could be described by bilinearly coupling ρ ii to r ion . multiple channels and metastable intermediates This regime is identical to Kramers theory, which does not provide an accurate κ[r ion ]. Bath modes do not change during barrier crossing, exert constant force on solute coordinate Solute coordinate is restricted to a narrow range at barrier top. r opt Ciccotti et al., J Chem Phys, 1990 F = ma q( z ) = q T ( z ) + c A !! q T ( z ) p(p B | q LMax ) improves similar p(p B | q iLMax ) improvement Fennel et al., J Phys Chem B, 2009 ! B = 1 2 + 1 2 erf [ q( z ) + c V ! q( z )] coordinate + velocity coordinate + acceleration Ballard, Dellago, J Phys Chem B, 2012 Peters, Chem Phys Lett, 2012 Force-augmented coords Frozen-solvent coords τ mem = 160 fs is comparable to ω B -1 = 60 fs New Coordinates: 0 1 time Reactive Flux 0 1 time Reactive Flux solute coordinate bath A B ‡ GH 0 1 time Reactive Flux solute coordinate

grote-hynes, pollak, and dynamics of the committor in ion...

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