nano266 - lecture 14 - transition state modeling

Modeling Transition States

Shyue Ping Ong

What is a transition state?

Particular configuration along a reaction coordinate corresponding to the highest potential energy along the reaction coordinate.

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Applications

Reaction rates •  Activated states •  Barriers to reactions •  Lowest energy pathway from reactant to products

Diffusion

•  Activation energies for migration •  E.g., migration of defects (radiation damage), Li migration for

batteries

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The potential energy surface

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The rate equation

Atomic vibrations are on the order of 0.1-1ps, which implies ν ~ 1012-1013. This is a frequently used approximation to avoid having to undertake a potentially complex calculation of the frequency directly.

~60 meV change in ΔEa leads to an order of magnitude change in the rate.

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r =νe−ΔEakbT

The Nudged Elastic Band Method

Method for finding the minimum energy path (MEP) when initial and final endpoints are known.

Initial guesses for MEP typically given by linear interpolations between the start and end points. The interpolated points are known as images.

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Source: http://theory.cm.utexas.edu/henkelman/research/saddle/

{rx} = x{ri}+ (1− x){rf}

Forces in NEB

Conceptually, this is like adding springs between images.

Images are then moved in accordance to the force using an optimization algorithm.

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Fi = Fis

||−∇E(Ri ) ⊥

∇E(Ri ) ⊥ =∇E(Ri )−∇E(Ri ) ⋅ t̂ iFi

s

||= k Ri+1 −Ri − Ri −Ri-1( ) ⋅ t̂ i

Henkelman, G.; Uberuaga, B. P.; Jónsson, H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths, J. Chem. Phys., 2000, 113, 9901, doi:10.1063/1.1329672.

Climbing Image NEB

Slight modification of NEB

Effect: •  Moves the potential energy surface along the elastic band and

down the potential surface perpendicular to the band. •  Other images in the band define the one degree of freedom for

which a maximization of the energy is carried out. •  Good approximation of saddle point. •  No additional cost.

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Fimax = −∇E(Rimax)+ 2∇E(Rimax

)||

= −∇E(Rimax)+ 2∇E(Rimax

).t̂ imax t̂ imax


CI vs Regular NEB

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Variable springs

Use stronger springs near saddle point

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ki! =kmax −Δk

Emax −Ei

Emax −Eref

$

%&&

'

()) if Ei > Eref

kmax −Δk if Ei < Eref

*

+,,

-,,


General NEB calculation procedure

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Full relaxation of start and end points (good force convergence is critical!)

Choose # of images, and generate guesses

Perform NEB calculation

Challenges in NEB calculations

Stability of calculation depends on number of images. Too few images -> lack of resolution. Too many -> unstable convergence. Linear interpolation between start and end points may lead to very bad guesses for MEP Convergence towards MEP may be extremely slow. Force convergence criteria typically much more stringent. Guess for the MEP may bias the final solution.

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Oxygen vacancy diffusion in YSZ

NANO266 13 Kushima, A.; Yildiz, B. Oxygen ion diffusivity in strained yttria stabilized zirconia: where is the fastest strain?, J. Mater. Chem., 2010, 20, 4809, doi:

10.1039/c000259c.

Hydrogen diffusion on TM surfaces

NANO266 14 Kristinsdóttir, L.; Skúlason, E. A systematic DFT study of hydrogen diffusion on transition metal surfaces, Surf. Sci., 2012, 606, 1400–

1404, doi:10.1016/j.susc.2012.04.028.

Methanol oxidation on Cu surfaces

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Sakong, S.; Groß, A. Density functional theory study of the partial oxidation of methanol on copper surfaces, J. Catal., 2005, 231, 420–429, doi:10.1016/j.jcat.2005.02.009.

Spinel compounds for multivalent electrodes

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Liu, M.; Rong, Z.; Malik, R.; Canepa, P.; Jain, A.; Ceder, G.; Persson, K. a. Spinel compounds as multivalent battery cathodes: a systematic evaluation based on ab initio calculations, Energy Environ. Sci., 2014, 00, 1–11, doi:10.1039/C4EE03389B.

NEB calculations -> KMC

Kinetic Monte Carlo – MC simulation method for the time evolution of natural processes.

•  For studies of diffusivity or reaction rates, requires as input environment-dependent activation barriers(usually from NEB)

Basic algorithm:

•  Start with an equilibrium atomic arrangement •  Determine migration probabilities Γ based on the paths available. Γ is calculated

based on Arrhenius form. •  Sample a random number in (0,1], and the migration event k is chosen such that:

•  Time is updated by Δt given by:

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Concentration Dependent Diffusion in LiCoO2

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Van Der Ven, A.; Ceder, G. Lithium Diffusion in Layered Li[sub x]CoO[sub 2], Electrochem. Solid-State Lett., 2000, 3, 301–304.

NEB + Percolation Theory

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Lee, J.; Urban, A.; Li, X.; Su, D.; Hautier, G.; Ceder, G. Unlocking the potential of cation-disordered oxides for rechargeable lithium batteries., Science, 2014, 343, 519–22, doi:10.1126/science.1246432.

Growing string method

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Peters, B.; Heyden, A.; Bell, A. T.; Chakraborty, A. A growing string method for determining transition states: comparison to the nudged elastic band and string methods., J. Chem. Phys., 2004, 120, 7877–86, doi:10.1063/1.1691018.

Begins as two string fragments, one associated with the reactants and the other with the products. Each string fragment is grown separately until the fragments converge. Once the two fragments join, the full string moves toward the MEP. Typically finds saddle points much more quickly when linearly interpolated guess is very far from MEP.

Dimer Method

Use to finding saddle points when final state is not known. Involves two replicas of the system, a ‘dimer’, which is used to transform the force in such a way that optimization leads to convergence to a saddle point rather than a minimum.

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nano266 - lecture 14 - transition state modeling

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