methods for calculating rare event dynamics and …bemod12/slides/henkelman_bemod12.pdfmethods for...
TRANSCRIPT
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