The NEB and the CI-NEB methods
Jean-Sabin McEwen
Dept. of Chemical and Biomolecular Engineering
University of Notre Dame
Jason Bray
Rachel Getman
Dorrell McCallman
William F. Schneider
Elastic Band Methods
•
•
•
Class of methods deriving from the “chain-of-states” method that finds MEP between
two minima using the gradient of the PES but not the Hessian
– Does not require “good” estimate of MEP/TS geometries
Method:
– Locate minima on PES
– Draw a path between them
• Most common way to do this: linear interpolation
– Discretize the path into a finite # of points (called “images”)
• Each geometry is a linear interpolation of the two minima
• Optimization is performed at each point by analyzing the forces on the
geometry. Goal is to converge each image to the same MEP.
What is the MEP?
On the MEP, the forces along
the direction parallel to the reaction
pathway are finite, but they are 0 in
all other directions
• Complex surfaces contain many reaction pathways and
thus many MEPs. We want to make sure we converge
all images to the same MEP, so we connect them with a
theoretical spring:
• Spring force on each image is:
(
) (
F spring = ki+1 xi+1 - xi - ki xi - xi-1
)
The Plain Elastic Band Method (1/2)
The Plain Elastic Band Method (2/2)
k= 1.0
k= 0.1
Taken from: H. Jónsson, G. Mills, K. W. Jacobsen, Nudged Elastic Band Method for Finding Minimum Energy Paths of
Transitions, in Classical and Quantum Dynamics in Condensed Phase Simulations, Ed. B. J. Berne, G. Ciccotti and
D. F. Coker (World Scientific, 1998), page 385.
The Elastic Band Method (1/2)
• Since we want to converge to the MEP and not
necessarily to a stationary point, we need to project the
component of the total force perpendicular to the
reaction pathway out of the total force:
Fi EB = -ÑVi |^ +Fi spring
æ
xi+1 - xi-1 ö
÷ xi+1 - xi-1 + Fi spring
= -ÑVi - ç -ÑVi ×
ç
÷
||
x
x
||
è
i+1
i-1 ø
(
)
The Elastic Band Method (2/2)
– This way, the reaction path is updated @ each step (or every 10
steps, or …, depends on program, defaults, user inputs, etc.)
– Elastic Band method is converged when = 0 (or goes below
some specified convergence criterion.)
• If all spring constants are the same, the spring force is
minimized when images are equally spaced
• Perpendicular component of the molecular force zero-ed
when image is on the MEP
• Relaxation in the direction perpendicular to the reaction path
stretches images over the PES, much like pulling a rubber band
over a surface (hence “Elastic Band” Method)
Nudged Elastic band Method
– Removes perpendicular component of spring force:
Fi EB = -ÑVi |^ +Fi spring |
æ
æ
xi+1 - xi-1 ö
xi+1 - xi-1 ö
spring
÷ xi+1 - xi-1 + ç Fi
÷ xi+1 - xi-1
= -ÑVi - ç -ÑVi ×
×
ç
÷
ç
|| xi+1 - xi-1 || ø
|| xi+1 - xi-1 || ÷ø
è
è
(
)
(
)
– This way, spring forces do not interfere with relaxation to MEP
• Takes away unnatural forces perpendicular to the
reaction path
• This clean version of the EB force brings about an
optimization algorithm referred to as “nudging,”
hence “Nudged” Elastic Band
Climbing Image Nudged Elastic Band Method
• EB methods do not necessarily find the “TS”
– Sometimes require interpolation between the two highest energy
images
• Cure: Climbing Image Elastic Band Methods
– After a few geometry convergence steps in the EB method,
locates image with the highest E
• This image is now denoted MAX
Climbing Image Nudged Elastic Band Method
– Spring forces are removed from max, CI-NEB force becomes:
FMAX = -ÑVMAX
æ
xi+1 - xi-1 ö
÷ xi+1 - xi-1
+ 2 ç ÑVMAX ×
ç
÷
||
x
x
||
è
i+1
i-1 ø
(
)
• MAX moves down toward MEP and up to TS
• Upward movement denoted “climbing,” hence
“Climbing Image”-Nudged Elastic Band
NEB Methods
• Standard NEB (NEB)
– Pros
• Potential to converge faster b/c don’t have to wait
for TS image to “climb” to the top
– Cons
• Not always an image at the transition state
• Requires subsequent run with CINEB or dimer to
find TS
• Climbing-image NEB (CINEB)
– Pros
• Always an image at the transition state
– Cons
• Potential to converge at 2nd order saddle point
Examples
• NEB routine with Mathematica
• O2 dissociation on Pt(111)
• O2 dissociation on Pt(321)
• Formation of N2O on Pd(111)
General Procedures and Philosophies
1. Converge initial and final states with high
accuracy
–
not always) means TS is incorrect
–
–
–
EDIFFG = -0.03
2. Generate initial guess of pathway using
interpolation tools
–
–
Check for unrealistic bond lengths, etc.
4. Run NEB (standard or climbing-image)
until converges or 100 steps
–
–
–
–
–
–
If standard NEB, run CINEB , return to step
4
Converge with quasi-Newton-Raphson
(IBRION = 1, NSW = 10)
Calculate vibrational modes, verify only 1
imaginary mode exists, visualize imaginary
mode and verify expected behavior
Potential problems in verifying TS
• TS won’t converge: usually (but
Displace image in direction of unwanted
mode(s) and start new NEB
If unwanted mode has near-zero
frequency, then may adjust POTIM
6. If unconverged in 100 steps, resolve
potential problems
–
Relax images near possible local minima
• May require dividing pathway into
pieces and running separate
NEB’s
If hasn’t converged by 100 steps, generally
means intervention will be beneficial
5. If converged, verify transition state (TS)
–
• More than 1 imaginary mode
nebmake.pl, interpolate.pl
3. Verify feasibility of interpolated pathway
and adjust as needed
Continue trying to converge
Try to calculate vibrational modes anyway
Move on to step 6
–
–
–
Add images for better resolution (≤ 1 Å
between images)
Check that forces on each image decrease
If no problems, repeat step 4
Barrier on Pt(111) at 5/16 ML
1.38 Angstroms
O2(*) + *  2O(*)
White spheres: Pt
Blue Spheres: O
Red Spheres: O
Barrier on Pt(111) at 5/16 ML
1.41 Angstroms
O2(*) + *  2O(*)
White spheres: Pt
Blue Spheres: O
Red Spheres: O
Barrier on Pt(111) at 5/16 ML
1.80 Angstroms
O2(*) + *  2O(*)
White spheres: Pt
Blue Spheres: O
Red Spheres: O
Barrier on Pt(111) at 5/16 ML
2.27 Angstroms
O2(*) + *  2O(*)
White spheres: Pt
Blue Spheres: O
Red Spheres: O
Barrier on Pt(111) at 5/16 ML
2.56 Angstroms
O2(*) + *  2O(*)
White spheres: Pt
Blue Spheres: O
Red Spheres: O
Barrier on Pt(111) at 5/16 ML
2.79 Angstroms
O2(*) + *  2O(*)
White spheres: Pt
Blue Spheres: O
Red Spheres: O
Barrier on Pt(111) at 5/16 ML
3.14 Angstroms
O2(*) + *  2O(*)
White spheres: Pt
Blue Spheres: O
Red Spheres: O
Barrier on Pt(111) at 5/16 ML
3.00 Angstroms
O2(*) + *  2O(*)
White spheres: Pt
Blue Spheres: O
Red Spheres: O
Barrier on Pt(111) at 5/16 ML
2.87 Angstroms
O2(*) + *  2O(*)
White spheres: Pt
Blue Spheres: O
Red Spheres: O
Low Coverage O2 Dissociation: Pt(321)
O2(*) + *  2O(*)
• Four pathways
studied
• Lowest energy
pathway (ABCD)
connects most
stable O2 initial
state with most
stable O final
state
A
1.38 Å
White spheres: Pt
Red Spheres: O
Low Coverage O2 Dissociation: Pt(321)
O2(*) + *  2O(*)
• Four pathways
studied
• Lowest energy
pathway (ABCD)
connects most
stable O2 initial
state with most
stable O final
state
B
1.68 Å
White spheres: Pt
Red Spheres: O
Low Coverage O2 Dissociation: Pt(321)
O2(*) + *  2O(*)
• Four pathways
studied
• Lowest energy
pathway (ABCD)
connects most
stable O2 initial
state with most
stable O final
state
C
3.52 Å
White spheres: Pt
Red Spheres: O
Low Coverage O2 Dissociation: Pt(321)
O2(*) + *  2O(*)
• Four pathways
studied
• Lowest energy
pathway (ABCD)
connects most
stable O2 initial
state with most
stable O final
state
• Pathway FGH is
mirror image of
pathway ABC
White spheres: Pt
Red Spheres: O
D
4.88 Å
Formation of N2O on Pd(111)
Grey spheres: Pt
Red Spheres: O
Blue spheres: N
NO(*) + N(*)  N2O(*)
Formation of N2O on Pd(111)
Grey spheres: Pt
Red Spheres: O
Blue spheres: N
NO(*) + N(*)  N2O(*)
Formation of N2O on Pd(111)
Grey spheres: Pt
Red Spheres: O
Blue spheres: N
NO(*) + N(*)  N2O(*)
Formation of N2O on Pd(111)
Grey spheres: Pt
Red Spheres: O
Blue spheres: N
NO(*) + N(*)  N2O(*)
Formation of N2O on Pd(111)
Grey spheres: Pt
Red Spheres: O
Blue spheres: N
NO(*) + N(*)  N2O(*)
Formation of N2O on Pd(111)
Grey spheres: Pt
Red Spheres: O
Blue spheres: N
NO(*) + N(*)  N2O(*)
Formation of N2O on Pd(111)
Grey spheres: Pt
Red Spheres: O
Blue spheres: N
NO(*) + N(*)  N2O(*)
References
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•
•
•
•
•
•
Plain Elastic Band of Elber and Karplus: Chem. Phys. Lett., 139, 1987, 375.
– No projection of forces: PEB force is a sum of spring and total molecular forces.
Elastic Band Method of Ulitsky and Elber: J. Chem. Phys. Lett., 92, 1990, 15.
– Projects out parallel component of molecular force
Self-Penalizing Walk method of Czerminski and Elber: International Journal of Quantum Chemistry: Quantum
Chemistry Symposium 24, 1990, 167.
– Instead of a spring force uses a repulsive force and an attractive force couple to maintain connection but not
allow images to get too close. Projects out parallel component of molecular force.
Locally-Updated Plains method of Choi and Elber: J. Chem. Phys., 94, 1990, 751.
– No spring force, i.e. images are not connected. Parallel component of molecular force is projected out.
Plain Elastic Band method of Gillian and Wilson: J. Chem. Phys., 97, 1992, 3.
– For molecular dynamics.
Nudged Elastic Band Method of Jónsson, Mills, and Jacobsen:
– Projects out parallel component of molecular force and perpendicular component of spring force.
Improved Tangent Nudged Elastic Band Method of Henkelman and Jónsson: J. Chem. Phys., 113, 2000, 9978.
- Improves how the reaction path is calculated:
ti =
Ri - Ri-1
+
Ri+1 - Ri
|| Ri - Ri-1 || || Ri+1 - Ri ||
- Ensures equi-spacing even in regions of large curvature, gets rid of “kinkiness” of path
Climbing Image Nudged Elastic Band Method of Henkelman, Uberuaga, and Jónsson: J. Chem. Phys., 113, 2000, 9901.
Introduces climbing image: spring forces removed from NEB force of image with highest energy so that
it can move along the elastic band in search of the maximum. Searches to minimize total molecular force.