Lecture 6: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.

1. MIT
10.637
Lecture 6
Transition state searches
and PES sampling
hjkulik@mit.edu
September 23, 2014

2. MIT
10.637
Lecture 6
N > 3, General PES features
from H. B. Schlegel, Wayne State U.
We focus on a reaction coordinate defined by critical DOF, not full 3N-6!

3. MIT
10.637
Lecture 6
Energy minimizations
Ñf =
¶f
¶x1
, ,
¶f
¶xn
æ
è
ç
ö
ø
÷
Ñf = 0
Force = -Ñ(Energy)Minimize:
Gradient on our PES in
terms of all coordinates
(internal, cartesian):
For :
Any stationary point
(a) local minimum,
(b) global minimum,
(c) saddle point.
What we usually
want!
What we want
today!

4. MIT
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Lecture 6
Finding saddle points
• Saddle points are more difficult to find than
minima.
• Interpolation methods: assume reactant and
product minima are known and that a TS is
located “between” these two end-points. May only
find a geometry very close to the TS, rather than
the true TS.
• Local methods: propagate geometry using
information about the function, first and second
derivatives – do not need to know reactant/product
but usually need to have a good estimate of the
TS. Once TS is found, can get reaction path by
following steepest descent to R and P from TS.

5. MIT
10.637
Lecture 6
Saddle points in small systems
For extremely small systems: PES is 3N-6 variables, so it may be possible to
map out the complete PES in limited cases.
For simple small/intermediate systems: “coordinate driving” or constrained
optimizations.
For cases where there are a few internal degrees of freedom that describe the
difference between reactant and product: e.g. torsion angle, bond distances
for breaking/forming.
Choose fixed values for selected coordinates and remaining variables are
optimized, adiabatically mapping energy as f(reaction variables). TS is
geometry where residual gradients for fixed variables are “sufficiently small”.

6. MIT
10.637
Lecture 6
Saddle points in small systems
Coordinate driving:
Reaction variables are good if they have large coefficients in the Hessian
eigenvector with negative eigenvalue at the TS.
BUT we only know the Hessian of the TS after it has been found, making the
reaction coordinate selection very use-rbiased.
If only one or two variables change significantly between
Reactants and products, this method works well,
e.g. Rotation of a methyl group – torsion angle
HNC to HCN – HCN angle
SN2 reactions X +CH3Y  XCH3 + Y – XC and CY distances (pictured)
More than two reaction coordinates becomes difficult and rarely is successful.

7. MIT
10.637
Lecture 6
Synchronous transit
Linear synchronous transit: can consider this a
coordinate driving method.
All degrees of freedom – Cartesian or internal – are varied linearly between
reactant and product.
No optimization is performed.
Assumptions:
1) all variables change at the same rate along the reaction path.
2) transition state is the highest energy structure along the interpolation line.
However, assuming all variables changing the same amount all at the same time
is a bad approximation. This method rarely leads to a good estimate of the TS.

8. MIT
10.637
Lecture 6
Synchronous transit
R
TS
P
Quadratic synchronous
transit (QST): Approximate
reaction path by a parabola
instead of a straight line.
Find the maximum on the LST,
generate QST by minimizing
energy perpendicular to the
LST path.
Search for maximum on QST
path.

9. MIT
10.637
Lecture 6
The Hessian/Force matrix
(3N-6)x(3N-6) matrix
with elements:
Hij ( f ) =
¶2
f
¶xi¶xj
H( f ) =
¶2
f
¶x1
2
¶2
f
¶x1¶x2
¶2
f
¶x1¶xn
¶2
f
¶x2¶x1
¶2
f
¶x2
2
¶2
f
¶x2¶xn
¶2
f
¶xn¶x1
¶2
f
¶xn¶x2
¶2
f
¶xn
2
é
ë
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
(3N-6)
(3N-6)
Approximate PES around stationary point by harmonic potentials.

10. MIT
10.637
Lecture 6
The Hessian/Force matrix
Diagonalize Hessian
(eigenvalue problem)
From E. Eliav,
Tel Aviv U
minimum maximum saddle point
eklj
(k)
= Hijli
(k)
i=1
n
å
ek = mwk
2
Eigenvalues: Eigenvectors:
normal
coordinates
Harmonic
frequencies
εk>0 εk<0 εk>0, except one εj<0
ωk all real ωk all imaginary one imaginary ωj on RC

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Lecture 6
Eigenvector following
• In the vicinity of transition state, can use
information from second derivative (Hessian)
matrix.
• Can be expensive to compute all second
derivatives for large systems (3N-6x3N-6) so
some method to compute only elements needed
or to approximate can speed up approach.
• Need good initial guess for the transition state.
• Works best if Hessian already has only one
negative eigenvalue – (e.g. augmented-Hessian
NR) and need actual Hessian and not just initial
approximation! Expensive!

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Lecture 6
Synchronous transit
Improving on LST and QST:
Another class of methods starts with QST solution and then follows eigenvector
to the saddle point – “guided QST”
Introduced by Schlegel and coworkers, Synchronous Transit-guided Quasi-
Newton (STQN) methods:
-Can use parabola or circle arc for interpolation
-Then use tangent to guide search towards the TS.
-Once TS region is located, optimization is switched to quasi-Newton-Raphson
-Can require either just reactants and products or reactants, transition, state and
products.
Note: these methods generally all work for simple systems and coordinates: the
interpolation may be nonsensical in Cartesians but meaningful in internals, etc.

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Lecture 6
Rational function optimization
Rational function optimization:
Recall from geometry optimizations lecture – RFO expands the function in terms
of a rational approximation and then have to solve for a shift parameter that
makes eigenvalues positive. The quadratic approximation – requires step length to
be equal to the trust radius also.
Partial rational function optimization (P-RFO) for TS searches:
Now use two shift parameters l and lTS:
Choose lTS such that search will maximize the energy in this direction, but step
can be outside trust radius. QA or “TRIM” uses l=-lTS :

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Lecture 6
Problems with eigenvector
following
• Need one eigenvector weakly coupled to others to
follow (i.e. small 3rd order derivatives).
• NR may fail to converge if the eigenvectors are too
coupled and there’s not just one eigenvalue to
follow.
• Need a good geometry to start from.
• Hessian is expensive. NR TS search requires
explicit Hessian from the start.

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Lecture 6
The dimer method
Eliminates the need to calculate the
Hessian:
Dimer is oriented towards lowest curvature
mode by rotation and translation.
Curvature is estimated by finite difference
from only the forces on the two images,
avoiding the need for a Hessian
calculation.
Once the dimer is aligned on the lowest
curvature mode, force parallel to the dimer
is inverted from the total force, making the
dimer climb up to the saddle point:

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Lecture 6
Nudged elastic band
Chain-of-states method: string of interpolated
images/geometries is propagated. This is a modification of the
elastic band (Elber & Karplus) method.
A chain gang initial state final state guesses
(reactants) (products)
Springs keep interpolated
images separated:
Our chain of states are propagated on the
potential energy surface until we find a
minimum energy path.
Typically 4-20 images

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Lecture 6
Nudged elastic band
Nudged elastic band is a method that propagates a series of interpolated images
towards the minimum energy path.
Initial interpolations of the images are usually linear – and if done in Cartesian
coordinates (common), one has to be careful as there can be clashing/overlap of
atoms.
Spring forces keep the images spaced equally. Otherwise they’d fall downhill to the
reactant and product states. We apply the spring force along the tangent (i.e. along
the path):
i-1 i i+1
Spring
constant
Distance
i+1 to i
Distance
i to i-1
Tangent
unit vector

18. MIT
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Lecture 6
Nudged elastic band
i-1 i i+1
The energy is minimized perpendicular to the path – i.e. forces along the path
are projected out and replaced with the spring force.
The perpendicular force is the true force minus the component that is parallel to
the path:
The total force in NEB is the sum of the spring force along the tangent and the true
force perpendicular to the tangent:

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10.637
Lecture 6
Nudged elastic band
Mueller
potential

20. MIT
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Lecture 6
Nudged elastic band
Mueller
potential
Reactants,
intermediates,
products
Saddle point

21. MIT
10.637
Lecture 6
Nudged elastic band
Mueller
potential
Reactants,
intermediates,
products
Saddle point
minimum
energy path
ÑE Ri( )^
= 0

22. MIT
10.637
Lecture 6
Nudged elastic band
Mueller
potential
Reactants,
intermediates,
products
Saddle point
minimum
energy path
NEB initial
guess from
interpolation
a la LST

23. MIT
10.637
Lecture 6
Nudged elastic band
Mueller
potential
Reactants,
intermediates,
products
Saddle point
minimum
energy path
NEB initial
guess from
interpolation
NEB image
i
Fi
NEB
Fi
S
Fi
^
Fi
ˆti

24. MIT
10.637
Lecture 6
Nudged elastic band
i
Fi
NEB
Fi
S
Fi
^
Fi
ˆti
Fi
NEB
= Fi
^
+Fi
S
Fi
^
= - ÑE Ri( )-ÑE Ri( )× ˆti( )
Fi
S
= k Ri+1 -Ri - Ri -Ri-1( ) ˆti
Spring forces: only want
component of this force that keeps
images separated (along path).
True forces: ignore component that
minimizes energy parallel to path,
only minimize perpendicularly.
NEB image force:

25. MIT
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Lecture 6
Nudged elastic band
Mueller
potential
Reactants,
intermediates,
products
Converged
NEB path
NEB image

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Lecture 6
Nudged elastic band
The climbing image extension to nudged elastic band ensures that an image
converges to the saddle point. While NEB guarantees images to be equally spaced,
no guarantee that there’s an image at the top of the barrier – in fact, it’s very
unlikely.
Select the highest energy image and invert the force:
This maximum energy image is not affected by the spring forces.
Qualitatively: this image moves up the PES along the band and down in energy
perpendicular to the band. Need to have enough images close to the CI to get a
good estimate of the path.
Total force Force parallel
to the tangent

27. MIT
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Lecture 6
Nudged elastic band
Variable spring constants have been introduced to ensure convergence of the
most important points along the minimum energy path – around the saddle point.
This is extra important when climbing image is used because you need to have a
good estimate of the tangent to the path near the saddle point.
Choose strong forces near the saddle point and weak forces further away.
The force constants are linearly scaled based on the energy of the image:
Higher energy endpoint

28. MIT
10.637
Lecture 6
Nudged elastic band
Example of climbing image results
vs. standard NEB results
Example of variable springs results
vs. standard NEB results
Reaction coordinate
RelativeEnergy
Variable springs
Fixed springs
Reaction coordinate
RelativeEnergy
Climbing
image
Standard

29. MIT
10.637
Lecture 6
Nudged elastic band
NEB paths can be difficult to converge: when
the force parallel to the MEP is large
compared to the force perpendicular, when
there are inflection points. Kinks can form on
the band and oscillate.
Improved tangent method revises the
definition of the tangent from:
A slight improvement ensures equispaced
images even in areas of large curvature:
LEPS potential + harmonic
oscillator with NEB path in
dashed line showing kinks

30. MIT
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Lecture 6
Nudged elastic band
Adding in perpendicular spring force when the angle between the vectors between
neighboring images (Ri-Ri-1 vs Ri+1-Ri) deviated from zero helps.
However, this leads to corner cutting on the MEP in curved regions. If the saddle
point is in a curved region, then the saddle point energy will be overestimated.
Improved tangent method:
Use only the tangent to the image that’s higher in energy:
Special case where image I is at maximum or minimum:

31. MIT
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Lecture 6
Nudged elastic band
Modified tangent NEB
converges on the LEPS+H.O.
potential
Modified tangent smooths out convergence.
Can also show that there’s a stability criterion
for kinking in the path. The following must be
satisfied:
F < 2CR
Where F is the perpendicular force to the path
and C is the curvature around the MEP.
Path will always become unstable for enough
images because R goes down for large #
images.

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Lecture 6
String method
The string method finds a minimum energy path (f*) connecting points A+B. The
MEP satisfies the constraint:
The force perpendicular to the path is zero.
The string method is similar to the NEB method – it propagates a path of images,
e.g.
where a is the normalized arc length between A and B.
There’s a constraint that the parameterization is preserved when the string
deforms or the local arc length is constant along the string:
or
This means that the elastic energy in the string is distributed evenly.

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Lecture 6
String method
Using the constraint developed for the string along with the definition of the MEP,
we can rewrite the expression for the MEP:
Where t is the tangent vector and l is a Langrange multiplier that imposes the
constraint.
In order to propagate a string towards the MEP is to carry out steepest descent
dynamics in the string space:
The expression for NEB written in the same terminology is:

34. MIT
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Lecture 6
String method
Can propagate the string with damped dynamics approach instead to ensure
rapid convergence:
where gs is a friction coefficient. Can propagate these with Verlet algorithm:
Where we have a modified friction factor and force:
The values of the Lagrange multipliers can be solved iteratively, based on the fact
that the distance between images is constant and the expression for propagation.

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Lecture 6
String method
• Main difference between NEB and string is
that the images get repositioned at every
iteration in the string method.
• Extensions to the string method include:
– Growing string method: start two strings – one
from reactants and one from products. Grow
them until they join and then propagate towards
MEP.
– Frozen string method: Same as growing string
but once images are added, they are frozen.
More efficient than growing string.
– Finite temperature string: incorporates
temperature effects.

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Lecture 6
Further considerations
• Good coordinate choices are very important for
transition state optimization
• Good coordinate choice can enlarge the convergence
region, reducing requirement of a good starting
geometry. Poor coordinates, decrease convergence
region.
• Eigenvector following methods work for stiff systems –
small eigenvalues in Hessians will be a problem.
• Small eigenvalues/soft modes in the Hessian are
better suited to path-based methods.
• Transition state may be a minimum for a certain
symmetry (vs. the symmetry of the reactants and
products).

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Lecture 6
Dynamic free energy methods
• For large systems, it may be difficult to
optimize transition states via the methods
we’ve discussed so far.
• Instead, may want to carry out simulation
dynamically.
• There’s also the consideration of free
energy (G), which has been excluded so
far – only have looked at enthalpy (H).

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Lecture 6
Umbrella sampling
Direct dynamics will not sample high-energy regions near the TS if there’s
an activation energy.
Need to bias the sampling, use a penalty to the potential to force sampling of high
energy regions:
The biasing potential can be made to be sufficiently steep, so the energy far from
r0 is very high in energy. Then will only sample near r0.
This method is known as umbrella sampling.
The ensemble is not Boltzmann but it can be deconvoluted.

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Lecture 6
Umbrella sampling
Reaction coordinate
Probability
Umbrella sampling locally
enhances probability in low-
probability regions by altering
the potential.
Perform many biased simulations at different
positions along reaction path, calculate free
energy – called Potential of Mean Force:
this is done by unweighting and stitching
together the underlying free energy function.
Weighted histogram-analysis method (WHAM) is a common approach to
stitching together the free energy: deconvolute contribution of force constant from
curvature of underlying free energy surface.

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Lecture 6
Steered molecular dynamics
Steered molecular dynamics is a technique where constant force or velocity is
applied to select atoms to move them from one configuration to another.
A set of collective variables is chosen to define progress along a reaction
coordinate, e.g. proton transfer in malonaldehyde:
Collective variable:
dist(1)-dist(2).
Once reaction coordinate is propagated, can get PMF:
Guiding potential:
Work: PMF:
(1)
(2)
May need to run
several (hundreds?)
of SMD runs to get
accurate PMF.

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Lecture 6
Metadynamics
Introduced in 2002 by Laio and Parrinello, metadynamics is a way to speed up
dynamics and sample free energy landscapes of complex systems.
Collective variables (up to 3, typically 2) are selected to define the free energy
landscape, e.g. coordination number of relevant species participating in a
reaction, bond distances or angles, etc. The most challenging aspect of
metadynamics is choosing the right collective variables.
A biasing Gaussian potential is added to the real potential:
New Gaussian is added at every tG in space of collective variables, S are the
coordinates, s is the collective variable, w is the height of the Gaussian, ds is the
width of the Gaussian.

42. MIT
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Lecture 6
Metadynamics
The potential wells are gradually filled
up minima until the full landscape is
traversed.
The free energy surface is obtained by
taking the negative image of all of the
Gaussians.
If Gaussians are too large or added too
frequently, free energy surface will be
inaccurate, Gaussians too small or
infrequently added – will take too long for
simulation to finish.

43. MIT
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Lecture 6
Summary
• For small systems with simple reaction coordinates:
coordinate driving, linear synchronous transit,
quadratic synchronous transit, or synchronous transit-
guided-Quasi-Newton can be good approaches.
• For intermediate systems: string method or nudged
elastic band
• For large systems with many DOF or to compute free
energies: steered MD, metadynamics, umbrella
sampling
• Coordinate selection and initial guess for a
reaction pathway are almost always challenging.

44. MIT
10.637
Lecture 6
Survey!
bit.ly/lec-6

45. MIT
10.637
Lecture 6
Transition state theory
Reaction coordinate
Energy
E
a
kBT
Reactants
Products
Transition State
Activation energy is one piece of the
puzzle – full exploration of TST allows
us to calculate absolute rates, through
prefactor calculations.
A potential energy surface connects
the reactants and products: this is a QM
concept!
The activated complex or transition
state is at the maximum potential
energy. Once configuration is achieved,
fast decay to products.
Concentration of the transition state is
dependent upon equilibrium with
reactants, vibration along the path.
45

46. MIT
10.637
Lecture 6
The Eyring equation
Entropy of
activation
term
Enthalpy of
activation
term
Macroscopic Eyring equation:
Microscopic Eyring equation:
These are the molecular partition
functions of A and B and the
activated complex, AB.
The Eyring equation is a rigorous method of obtaining kinetics from free energy of
activation and may be derived from statistical mechanics (not here):

47. MIT
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Lecture 6
The Eyring equation and TST
In the derivation of the Eyring equation (not
shown), one vibrational frequency is
stripped out of the molecular partition
function for the activated complex.
This is assumed to be the mode that
carries the complex over the barrier to the
products.
Also: note that TST assumes once the
activated complex is formed, it readily
proceeds to products and does not cross
back, which can be inaccurate.

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Lecture 6
Interpretation of
48
The potential energy surface is
represented by changes in the internal
coordinates of the molecule(s).
These correspond to vibrational
modes that alter internal
coordinates or translation of the
reactants with respect to each
other.
At the TS, we have one “molecule”
and motion is characterized by the
vibrational modes.
One imaginary vibrational frequency
characterizes the transition state and
correlates to the attempt frequency
in our derivation of the Eyring
equation.
Imaginary frequency of the
transition state for HCNHNC
isomerization

49. MIT
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Lecture 6
Partition function review
#
Maxwell-Boltzmann
E
Fraction
molecules
in state i.
b= 1/kBT
energy of
the state
Boltzmann distribution
Molecular partition function
sum over
j states:
sum over
i levels:
1 2 3 4 5 6 1 2 3 4
5
6g=5
g=1

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Lecture 6
Molecular partition function
contributions are derived from solving eigenvalue
problems in quantum mechanics.

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Lecture 6
Molecular degrees of freedom
Linear Nonlinear
Translational 3 3
Rotational 2 3
Vibrational 3N-5 3N-6
For zero field,
potential energy
derived only
from vibrational
coordinates.
e.g. H2O: 3 normal modes
symmetric stretch asymmetric stretchbend

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Lecture 6
Vibrational zero-point energy
n=0
n=1
n=2
1
2
~!
3
2
~!
5
2
~!
From Q.M., molecules
have zero point energy:
ZPE
Reaction coordinate
+ B
P
AB‡
A
E0
Ea
Energy
Enthalpy of activation:

53. MIT
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Lecture 6
Contributions to prefactors
DOF form f(TX)
if
if
Order
½
½
0
1
1
10
108 cm-1
53

54. MIT
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Lecture 6
Prefactors from quantum
chemistry
• Must have structures that are at minima or
saddle points – vibrationally characterize
them to get zero point energy and vibrational
partition function.
• ZPE corrects the enthalpy of activation.
• Rotational partition function may also be
calculated and its contribution is added to the
partition functions.
• Electronic terms are neglected.
• Translational terms only depend on
temperature.

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Lecture 6
Summary
• It is possible to use a saddle point structure
and minimum energy structures to obtain
absolute rates of reaction by calculating
prefactors from statistical mechanics.
• However, relative rates, where the prefactors
are assumed to be the same for different
reactions are typically preferred unless the
method is extremely accurate.