Graeme HenkelmanUniversity of Texas at Austin

κ-dynamics Solid-state nudged elastic band

Methods for calculating rare event dynamics and pathways of solid-solid phase transitions

4.06Å

4.06Å

4.38Å

4.38Å

rocksalt

wurtzite

Classes of Dynamical SystemsRoughSmoothEnergy Landscape:

Final State:U

nknown

Know

n

Initial

Final

Initial

Initial Final

?

Transition state theoryA statistical theory for calculating the rate of slow thermal processes -- rare event dynamicsRequires an N-1 dimensional dividing surface that is a bottleneck for the transition:

Harmonic transition state theory

Find saddle points on the energy surfaceRate of escape through each saddle point region:

kHTST =

QNi=1 �iQN�1j=1 �†j

exp

✓��E

kBT

◆

�E

R

Px = x

†

kTST =1

2

⌦�(x� x†)|v�|

↵R

KMC: the Good, the Bad, and the UglyThe GoodFor rare event systems where transition rates are defined by first-order rate constants, KMC is an exact stochastic solution to the kinetic master equation.

The BadIt is very hard to determine all possible kinetic events available to the simulation.It is very hard to calculate an exact rate for a process, let alone all of them.Typically limited to transition state theory (e.g. harmonic TST within aKMC).

The UglyIt’s a lot of work calculating all possible events and rates to find one trajectory.

where

where μ is random on (0,1]

Dynamical corrections to TSTTST assumes that all trajectories that cross the TS are reactive trajectories.

: the ratio of successful trajectories to number of crossing points

ratio between theTST and true rate:

both total escape rate and by product:

A successful trajectory:

1) trajectory must go directly to products without recrossing the TS

2) trajectory must start in initial state

Example:

violation of TST

Dynamical correction factor

The κ-dynamics algorithm1. Choose a reaction coordinate and sample a transition state (TS) surface

2. Launch short time trajectories until one goes directly to a product and starts in the initial state; record the number, N

3. If no successful trajectory within Nmax, push TS up in free energy and go to 2

4. Calculate kTST for successful TS surface (parallel tempering, WHAM)

5. Update the simulation clock by

where μn are random numbers on (0,1]

6. Repeat procedure in product state

1

2

3

4 5

6

7

C.-Y. Lu, D. E. Makarov, and G. Henkelman, J. Chem. Phys. 133, 201101 (2010).

Sketch of proof that κ-dynamics is exact(1) Branching ratioThe probability ofreaching state j is:

A. Voter and J. D. Doll JCP 82, 1 (1985)

where

sampled(2) Reaction time

where is the true rateWhat we want:

What we have in κ-dynamics: where

(1 success)(N-1 failures)

Connection with TST rate: Erlang N-distribution

Combining:

the correct distribution based upon the true rate

Reaction coordinate test

TST+κhTST

Al/Al(100), hop on frozen surface

kTST

Exchange on relaxed surface

κ-dyn κ-dynamics rates are correct and independent of reaction coordinate

κ-dynamics

Bond-boost: specifies the stretch in the most-stretched bond:

R. A. Miron and K. A. Fichthorn, JCP 119, 6210 (2003)

Bond stretchingparameter

Branching ratio testComparison of reaction products for classical dynamics and κ-dynamics

Al/Al(100),relaxed surface

Reaction mechanism

Bran

chin

g ra

tio direct MDκ-dynamicsharmonic TST

0.0

0.5

1.0

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

����

�������

�

��������200K 400K

κ-dynamics trajectories

Al/Al(100) island ripening

Al/Al(100) pyramidcollapse

Solid-solid phase transitions4.06Å

4.06Å

4.38Å

4.38Å

rock salt wurtzite

Some transitions involve both change in atomic and cell degrees of freedom

E.g. CdSe:

wurtzite

Potential parameters: E. Rabani, J. Chem. Phys. 116, 258 (2002).

Nudged Elastic Band Methodwurtzite

Pratt, Elber, Karplus, ... and others Initial

Final

F!F

FSFNEB

MEPNEB

i

i i

i

ii

i+1

i!1

ˆ

Pioneering work

Nudged elastic band

[1] H. Jónsson, G. Mills, and K.W. Jacobsen, in Classical and Quantum Dynamics in Condensed Phase Simulations, 385 (1998).[2] G. Henkelman and H. Jónsson, J. Chem. Phys. 113, 9978 (2000).[3] G. Henkelman, B. P. Uberuaga, and H. Jónsson, J. Chem. Phys. 113, 9901 (2000).

Images connect initial and final states

Perpendicular component (potential):

NEB force on each image:

Parallel component (springs):

Danger of assuming a reaction coordinate

final

initial

ener

gy

reaction coordinate

drag

NEB

x 1-1-2 0

2

3

1

r AB saddle

initial

final

initial band

NEB

drag

drag direction

The x-coordinate separates initial from final state, but is not a suitable reaction coordinate:

Initial

Final

A drag calculation along x (black points) misses the saddle point where the reaction coordinate follows the y-direction

x

y reaction coordinate

The resulting barrier is under-estimated:

Cell variables vs atomic coordinates

x 1-1-2 0

2

3

1

r AB saddle

initial

final

initial band

NEB

drag

drag direction

a) atom dominated b) cell dominated

final(wurtzite)

4.38

Å4.

22 Å

4.06

Å

8.75 Å

9.36 Å

9.08 Å

10.98 Å

11.49 Å

initial(rock salt)

transitionstate

8.75 Å

4.20

Å4.

20Å

4.20

Å4.

06Å

4.06

Å4.

06Å

4.38

Å4.

38Å

4.38

Å

atom dominated cell dominated

Two pathways for the same solid-solid phase transition in CdSe

cell

NEB calculations in only atomic [1] or cell (RNM) [2] coordinates have the errors found in drag methods. Requires a unified approach

atoms

[1] D. R. Trinkle, R. G. Hennig, S. G. Srinivasan, et al., PRL 91, 025701 (2003).[2] K. J. Caspersen and E. A. Carter, PNAS 102, 6738 (2005).

Choice of cell representation

Cell coordinatesChoose a representation which does not allow for net rotation of the lattice

Changes in the cell are represented as strain

v3 = (h ,h ,h )3x 3y 3z

v1 = (h ,0,0)1x

= (h ,h ,0)2x 2yv2

periodic solid expanded periodic solid

ɛ = δ 0( )0 δ

Strain describes changes in the lattice which is invariant to the unit cell

Combining atomic and cell variables (G-SSNEB)

Single displacement vector:

Jacobian to combine different units

change in atom positionschange in cell shape

Requirement: reaction pathways should be independent of unit cell size and shape -- the path should be a property of the infinite solid

Unit of length, average distance between atoms:

Scaling of atomicdisplacements with size:

Choose J so that Jε has same units and scaling as ΔR:

Similar logic applies to the stress / force vector:

volume

Results

NEB

15 meV

ene

rgy /

ato

m (m

eV)

-60 -40

-20

20

0 13 meV

initial final

initial state

final state

saddle

G-SSNEBRNM

NEB

G-SSNEB

RNM

saddle

12 meV16 meV

ene

rgy /

ato

m (m

eV)

initial final

NEB

saddle

G-SSNEBRNM

initial state

final state

G-SSNEB

NEB

RNM

-60 -40

-20

20

0

saddle

Atom-dominated process: Cell-dominated process:RNM fails; NEB / G-SSNEB work NEB fails; RNM / G-SSNEB work

Scaling with system size

ene

rgy /

ato

m (m

eV)

-60

-40

-20

20

0

-1 1 0 -2 distance / !N (Å)

8 32 72

128

atoms / cell

128transition state

Minimum energy path is insensitive to unit cell size and shape:

Change of mechanismWith increasing system size:

ba

ene

rgy

(eV)

number of atoms in unit cell 10 10,000 1,000 100

0.1

1

10c

b

a

The mechanism changes from a concerted bulk process (cell dominated) to a local (atom dominated)

c

3d (bulkconcerted)

2d (line defect)

1d (point defect)

a

MoviesConcerted mechanism

Local mechanism

Local process in detail

-80

-60

-40

-20

0

ener

gy /

atom

(meV

)

0

1

2

initial final

a

cb

cb

a

Complex mechanism:

Research Group

Chun-Yaung (Albert) Lu

Daniel Sheppard

PenghaoXiao

Software tools

Research Group and CollaboratorsChun Yaung Lu (graduate student, now at LANL) Daniel Sheppard (graduate student, now at LANL) Penghao Xiao (graduate student)Rye Terrell (graduate student)Sam Chill (graduate student)Will Chemelewski (graduate student)

Dmitrii Makarov (U. Texas, Austin)Hannes Jónsson (U. Iceland)Andreas Pederson (U. Iceland)Duane Johnson (Iowa State/Ames Lab)

DOE - EFRC, SISGRNSF - CAREER, NIRTWelch FoundationAFOSR

Funding

Acknowledgments

http://theory.cm.utexas.edu/vtsttools/$ AKMC, Dimer, (SS)NEB, and dynamical matrix$ methods implemented in the VASP code

http://theory.cm.utexas.edu/bader/$ Bader charge density analysis

http://theochem.org/EON/$ The EON project for long time scale simulations

http://theory.cm.utexas.edu/code/tsase/$ Transition state simulation environment (for ASE)

Texas Advanced Computing CenterEMSL at PNNLCNM at ANL

Computer Time