Rare event techniques are used to sample rare thermal activation processes that cannot be directly observed in simulation or experiment. Common methods include constrained minimization, nudged elastic band, dimer method, and Monte Carlo. Nudged elastic band constructs a chain of images describing the minimum energy pathway between reactants and products on a potential energy surface. Monte Carlo uses random sampling to infer properties from many probabilistic experiments weighted by the Boltzmann factor. These methods help characterize transition states that are challenging to describe with a single reaction coordinate.

1. Rare event techniques
Heather J Kulik
hkulik@stanford.edu
03/11/13

2. The sampling bottleneck
Transition
Fast: oscillations around each minimum. State
S l o w: the jump over the barrier from
one minimum to the other.
Energy
E
van’t Hoff-Arrhenius relationship: a
-Ea kBT
t jump ~ t vibe kbT
Reactants
Products
Example:
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!

3. 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

4. Mapping a complete PES
For very small
systems: we can
map out a fully ab
initio potential
energy surface.
e.g. CH4+Cl, CH5+,
etc. see work of
Bowman group.
But this PES mapping is impractical
for anything but very small systems.

5. Constrained minimizations
Works well for simple
reaction coordinates: Fails for complex
interpolate between
reaction pathways.
reactant and products,
using some geometric
constraints to define
PES

6. Synchronous transit
Quadratic synchronous
transit (QST): Search for a
maximum along an arc
R
between R and P, minimum on
perpendicular direction.
Guided QST:
Start with QST and then follow
TS
eigenvector to the saddle point.
Works well only for simple P
reaction coordinates and small
molecule systems.

7. 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.

8. Nudged elastic band
Mueller
potential

9. Nudged elastic band
Reactants,
intermediates,
products
Saddle point
Mueller
potential

10. Nudged elastic band
Reactants,
intermediates,
products
Saddle point
minimum
energy path
ÑE ( Ri ) ^ = 0
Mueller
potential

11. Nudged elastic band
Reactants,
intermediates,
products
Saddle point
minimum
energy path
NEB initial
guess from
interpolation
Mueller
potential

12. 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

13. Nudged elastic band
NEB image force:
^ S
F i
NEB
= F +F
i i
Fi^ = - ( ÑE ( Ri ) - ÑE ( Ri ) × t it i )
ˆˆ
Fi
True forces: ignore component that
minimizes energy parallel to path,
only minimize perpendicularly. ^
i ti
ˆ
F
i
F = k ( Ri+1 - Ri - Ri - Ri-1 ) t i
S
i
ˆ S
F
i
Spring forces: only want F i
NEB
component of this force that keeps
images separated (along path).

14. Nudged elastic band
Climbing image method: improved resolution of the saddle point
F = -ÑE ( R j ) + 2ÑE ( R j ) × t jt j
CI
j
ˆ ˆ
True forces: component
parallel to band is inverted,
image moves up the band. with CI
Spring forces: CI image
feels no spring forces. no CI

15. Nudged elastic band
Variable 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
Eref
Spring forces: stiffer
springs for high energy
fixed
points to ensure resolution
springs
of the saddle point.

16. Nudged elastic band
Improved 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!

17. Nudged elastic band
Practical 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.

18. Dimer method
Two separated states, R1
and R2. R1
States are pushed up the
PES, inverting the force
along the lowest
frequency mode.
Also rotate the dimer to R2
sample PES. Dimer
force
Method scales well with
increasing complexity! Real
force

19. 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.

20. Monte Carlo
If we are interested in macroscopic averages, these are thermodynamical
quantities 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 of
many probabilistic, random experiments.
Today:
(1) scientific simulations
(2) used to simulate real events, e.g.
Stock market, etc.

21. Basic principle of Monte Carlo
p
1
Acircle = Asquare =1
4
y
For random selections, we are either a “hit”
inside the circle or a “miss” outside the circle.
nhits Acircle p
= =
0
nhits + nmiss Asquare 4 0 x 1
hit miss
We can estimate p in this way, but it will take thousands
of attempts to get a reasonable estimate!

22. Monte Carlo
Can 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 W0ij
Accept j with some probability Pij

23. A Monte Carlo algorithm
Compute
Initial Perturb the
energy for
configuration system
perturbation
Accept
Is DE<0?
Accept with
probability
P α e-ΔE/kT

24. Monte Carlo timescales
Monte Carlo timescale has no true meaning:
not a dynamical timescale but a measure of how much phase space has
been sampled.
We get an
property
average of our
property at long
MC timescales.
MC timescale
We can bias our dynamics: way perturbations are sampled can be
determined by the kind of phenomena we’re trying to describe: exchange
across long distances, nearest neighbor exchanges, etc.

25. Practical challenges for TSs
Characterization of the TS is only as good as the energetic
model 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.

26. Beyond MEPs
Conical intersections: beyond the Born-oppenheimer approximation, coupling of
states and transfer between states govern key phenomena.
Also: electron transfer (Marcus theory), proton transfer (need quantum nuclear
effects), allosteric transitions that are difficult to describe by a single coordinate…

27. 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).