The sampling bottleneck TransitionFast: oscillations around each minimum. StateS l o w: the jump over the barrier fromone minimum to the other. Energy Evan’t Hoff-Arrhenius relationship: a -Ea kBT t jump ~ t vibe kbT Reactants ProductsExample: Reaction coordinate kcal Ea » 17 ; T = 300 K mol For a thermally activated process, t vib » 10-8 s timescale to observe event in real time is so slow. It would take 1015 1 fs MD t jump » 1s steps to observe the event directly!
How to sample rare events• Map out the PES• Constrained minimization• (Guided) synchronous transit• Nudged elastic band/string method• Dimer method• Monte Carlo• Umbrella sampling• Metadynamics
Mapping a complete PESFor very smallsystems: we canmap out a fully abinitio potentialenergy surface.e.g. CH4+Cl, CH5+,etc. see work ofBowman group. But this PES mapping is impractical for anything but very small systems.
Constrained minimizationsWorks well for simplereaction coordinates: Fails for complexinterpolate between reaction pathways.reactant and products,using some geometricconstraints to definePES
Synchronous transitQuadratic synchronoustransit (QST): Search for amaximum along an arc Rbetween R and P, minimum onperpendicular direction.Guided QST:Start with QST and then follow TSeigenvector to the saddle point.Works well only for simple Preaction coordinates and smallmolecule systems.
Nudged elastic band Chain-of-states method: string of images/geometries is used to describe the minimum energy pathway.A chain gang initial state final state guesses Springs keep interpolated images separated: Our chain of states are propagated on the potential energy surface until we find a minimum energy path.
Nudged elastic band Reactants, intermediates, products Saddle point Mueller potential
Nudged elastic band Reactants, intermediates, products Saddle point minimum energy path ÑE ( Ri ) ^ = 0 Mueller potential
Nudged elastic band Reactants, intermediates, products Saddle point minimum energy path NEB initial guess from interpolation Mueller potential
Nudged elastic band Reactants, intermediates, Fi products ^ i ti ˆ F i Saddle point S F i FNEB minimum i energy path NEB initial guess from interpolation NEB image Mueller potential
Nudged elastic bandNEB image force: ^ SF i NEB = F +F i iFi^ = - ( ÑE ( Ri ) - ÑE ( Ri ) × t it i ) ˆˆ FiTrue forces: ignore component thatminimizes energy parallel to path,only minimize perpendicularly. ^ i ti ˆ F iF = k ( Ri+1 - Ri - Ri - Ri-1 ) t i S i ˆ S F iSpring forces: only want F i NEBcomponent of this force that keepsimages separated (along path).
Nudged elastic bandClimbing image method: improved resolution of the saddle point F = -ÑE ( R j ) + 2ÑE ( R j ) × t jt j CI j ˆ ˆTrue forces: componentparallel to band is inverted,image moves up the band. with CISpring forces: CI imagefeels no spring forces. no CI
Nudged elastic bandVariable springs method: improved resolution of the saddle point ì æ Emax - Ei ö ï kmax - Dk ç ï ÷ki = í è Emax - Eref ø if Ei > Eref ï ï î kmin if Ei < Eref variable springs ErefSpring forces: stiffersprings for high energy fixedpoints to ensure resolution springsof the saddle point.
Nudged elastic bandImproved tangent method: for improved stability When parallel forces are large and LEPS potential perpendicular are small, path can get unstable, kinking. Improved tangent: ì Ri+1 - Ri ï if Ei+1 > Ei > Ei-1 NEB ï Ri+1 - Ri ti = í ï Ri - Ri-1 if Ei+1 < Ei < Ei-1 ï Ri - Ri-1 MEP î Resulting NEB path looks like MEP!
Nudged elastic bandPractical challenges:1) Stability of the calculation depends on the number of images. If images are too close together, we may be unstable.2) Convergence to minimize forces may be slow and depends on the minimizer used.3) Initial estimates of MEP based on cartesian interpolation may be poor (vs. internal coordinates). Initial estimate needs to be good to speed convergence.4) Rotations, translations may enter erroneously into path.5) Our idea of what the MEP should be biases the solution that we can find.
Dimer methodTwo separated states, R1and R2. R1States are pushed up thePES, inverting the forcealong the lowestfrequency mode.Also rotate the dimer to R2sample PES. Dimer forceMethod scales well withincreasing complexity! Real force
String method• Similar to NEB in construction/cost.• Images propagated on path, following force perpendicular to path.• No springs, images are adjusted slightly following force propagation to ensure spacing.• Finite temperature extension.• “Growing string” method allows the path to change in time to allow for variations in MEP.
Monte CarloIf we are interested in macroscopic averages, these are thermodynamicalquantities that are difficult to extract from direct dynamics.Origin of Monte Carlo (1940s Los Alamos- Ulam/von Neumann):Rather than deterministic mathematical methods…Infer instead the answer from the outcome ofmany probabilistic, random experiments.Today:(1) scientific simulations(2) used to simulate real events, e.g.Stock market, etc.
Basic principle of Monte Carlo p 1Acircle = Asquare =1 4 yFor random selections, we are either a “hit”inside the circle or a “miss” outside the circle. nhits Acircle p = = 0nhits + nmiss Asquare 4 0 x 1 hit missWe can estimate p in this way, but it will take thousandsof attempts to get a reasonable estimate!
Monte CarloCan we pick where we sample such that their weight is proportional to e-bE? M e- b H v M A =å M Av ? A = å Av v=1 å e- b Hv v=1 v=1 Random sample probability-weighted sample Metropolis algorithm: “Walk” through phase space, with proper P for infinite time limit Random starting state Pick a trial state j from I with some rate W0ij Accept j with some probability Pij
A Monte Carlo algorithm Compute Initial Perturb the energy forconfiguration system perturbation Accept Is DE<0? Accept with probability P α e-ΔE/kT
Monte Carlo timescalesMonte Carlo timescale has no true meaning:not a dynamical timescale but a measure of how much phase space hasbeen sampled. We get an property average of our property at long MC timescales. MC timescaleWe can bias our dynamics: way perturbations are sampled can bedetermined by the kind of phenomena we’re trying to describe: exchangeacross long distances, nearest neighbor exchanges, etc.
Practical challenges for TSsCharacterization of the TS is only as good as the energeticmodel being used. Transition states often have open-shell, multi-reference character: CCSD(T), which is great for local minima, often fails for transition states as a result of triples amplitudes, multi- reference character. Density functionals that yield 1-2 kcal/mol error in minima often underestimate TSs by about 3 kcal/mol.
Beyond MEPsConical intersections: beyond the Born-oppenheimer approximation, coupling ofstates and transfer between states govern key phenomena.Also: electron transfer (Marcus theory), proton transfer (need quantum nucleareffects), allosteric transitions that are difficult to describe by a single coordinate…
Follow-up reading• Synchronous transit – T. A. Halgren and W. N. Lipscomb “The synchronous-transit method for determining reaction pathways and locating molecular transition states” Chem. Phys. Lett. (1977). – C. Peng and H. B. Schlegel “Combining synchronous transit and quasi- Newton methods to find transition states” Israel J. of Chem. (1993).• Chain of states techniques – D. Sheppard, R. Terrell, and G. Henkelman “Optimization methods for finding minimum energy paths” J. Chem. Phys. (2008). – W. E., W. Ren, and E. Vanden-Eijnden “String method for the study of rare events” Phys. Rev. B. (2002). – G. Henkelman and H. Jonsson “A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives”J. Chem. Phys. (1999).• Monte Carlo – D. P. Landau and K. Binder A Guide to Monte Carlo Simulations in Statistical Physics. Cambridge University Press 2nd Edition (2005). – R. H. Swendsen and J.-S. Wang “Nonuniversal critical dynamics in Monte Carlo simulations”. Phys. Rev. Lett. (1987).