# Potential Energy Surfaces, Energy Minimization Methods, and Transition State Modeling - PowerPoint PPT Presentation

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## Potential Energy Surfaces, Energy Minimization Methods, and Transition State Modeling

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### Stationary Points - points on the PES where the gradient (or force) is zero; ... related system or chemical intuition' and a preconceived notion of the mechanism. ... – PowerPoint PPT presentation

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Title: Potential Energy Surfaces, Energy Minimization Methods, and Transition State Modeling

1
Potential Energy Surfaces, Energy Minimization
Methods, and Transition State Modeling

2
Outline
• Potential Energy Surface (PES)
• Energy Minimization Methods (Algorithms)
• Global Minimum Structure
• Transition State Modeling
• Reaction Pathway Modeling

3
Potential Energy Surface
4
Potential Energy Surface Terms
• Gradient - the first derivative of the energy
with respect to geometry (X, Y Z) also termed
the Force (strictly speaking, the negative of the
• Stationary Points - points on the PES where the
gradient (or force) is zero this includes
Maxima, Minima, Transition States (which are
first order saddle points), and higher order

5
PES Terms...
• In order to distinguish among the latter, one
must examine the second derivatives of the PES
with respect to geometry the matrix of these is
termed a Hessian (or force) matrix.
• Diagonalization of this matrix yields
Eigenvectors which are normal modes of vibration
the Eigenvalues are proportional to the square of
the vibrational frequency. (IR spectra can be
derived from these)

6
Sign of 2nd Derivatives
• The sign of the second derivative can be used to
distinguish between Maxima and Minima on the PES
• Minima on the PES have only positive eigenvalues
(vibrational frequencies)
• Maxima or Saddle Points (maximum in one direction
but minimum in other directions) have one or more
negative (imaginary) frequencies.
• A frequency calculation must be performed to
determine the sign of the vibrational frequencies.

7
Potential Energy Surface
8
Energy Minimization Algorithms
9
Energy Only (Univariate) Method
• Simplest to implement
• Proceeds one direction until energy increases,
then turns 90ยบ, etc.
• Least efficient
• many steps
• steps are not guided
• Not used very much.

10
Steepest Descent Method
• Simplest method in use
• Follows most negative gradient (max. force)
• Fastest method from a poor starting geometry
• Converges slowly near the energy minimum
• Can skip back and forth across a minimum.

11
• Adds history to simplicity of steepest descent
method to implicitly gather 2nd derivative
information to guide the search.
• Variations on this procedure are the
Fletcher-Reeves, the Davidon- Fletcher-Powell and
the Polak-Ribiere methods.

12
Second Derivative Methods
• The 2nd derivative of the
energy with respect to
X,Y,Z the Hessian

determines the pathway.
• Computationally more
involved, but generally
fast and reliable, esp.
near
the minimum.
• Quasi-Newton, Newton-Raphson,
block diagonal Newton-Raphson

13
Approaches to Locating the Global Minimum Energy
Structure
• Systematic Dihedral driving (manual or automatic)
• Randomization-minimization (Monte Carlo)
• Molecular dynamics (Newtons laws of motion)
• Simulated Annealing (reduce T during MD run)
conformations modify slightly retain lowest
energy ones, repeat)
• Trial error (poor)
• Methods are tedious, but absolutely necessary
if the result is to be
meaningful!

14
• What does the global minimum energy structure
really mean?
• Does reaction/interaction of interest necessarily
occur via the lowest energy conformation?
• What other low energy conformations are
available? (Boltzmann distribution/ensemble of
conformations and probability/entropy
considerations may be important).

15
Transition State Modeling
16
Transition State Modeling
• A Transition State is a stationary point for
which the second derivative of the energy with
respect to the reaction coordinate is negative,
but second derivatives in all other directions
are positive.
• The T.S. is the highest point along the lowest
energy pathway between reactants and products.
• A frequency calculation on a transition state
structure yields one and only one negative
(imaginary) frequency.

17
Transition States Why Difficult?
• Reactants and products are well defined molecular
entities Transition States are not.
• It is thought that T.S. exhibit elongated bonds,
partial bonding, and may have some aspects of
electronically excited states associated with
them.
• T.S. cannot be observed experimentally therefore
no parameters can be devised for modeling them.

18
TS Modeling Difficulties...
• Mathematically, there is less attention paid to
saddle points than to minima, so there are fewer
algorithms available to locate them.
• It is generally thought that the PES in the
vicinity of the T.S. is flatter than the
surface near a minimum, therefore it may be more
difficult to predict the structure of a
transition state accurately. A single, unique
T.S. structure may not even exist!

19
More TS Modeling Difficulties...
• Because T.S. probably involve partial bonding,
lower levels of theory are not likely to be very
useful in modeling them accurately.
• We know relatively little about the geometry of
T.S. most of what we know is based on
calculation. Guessing T.S. geometries is more
difficult than guessing the geometry of a stable
structure.

20
Best Approach Mixed Methods
• Guess T.S. geometry (methods on next slide)
• Perform a low level (AM1 or PM3) semi-empirical
MO calculation as a transition structure.
• Use that result as starting point for higher
level (HF/3-21G or /6-31G) calculation.
• Verify with a frequency calculation at the same
level of theory and basis set as the geometry
optimization. (only one imaginary frequency, the
animation of which is consistent with the rxn.
step)
energy calculation with a method that includes
electron correlation (e.g., MP2)

21
Guessing a TS Geometry
• Base the guess on a previously calculated,
related system or chemical intuition and a
preconceived notion of the mechanism.
• Use an average of the reactant and product
geometries (Linear Synchronous Transit method in
Spartan or Gaussian).
• Some programs employ a Quadratic Synchronous
Transit method, in which minima perpendicular to
the LST are connected.
• Several attempts may be needed!

22
LST and QST Approaches
23
Confirming a Possible TS
• Must be a first order saddle point on PES
smoothly connecting reactant to product.
• Verify that the Hessian (matrix of 2nd
derivatives with respect to coordinates) yields
one and only one negative (imaginary) frequency.
• Animate the normal coordinate corresponding to
the imaginary frequency it should connect
reactants and products (have vibrations
consistent with expected bond breaking and bond
forming).

24
Frequency Calculation
• Keyword Frequency, Freq or Frequencies
• Output file has a table of frequencies listed in
order of increasing magnitude imaginary
frequencies (negative lt0 cm-1) are listed first.
• To animate a frequency in Titan, select Display,
Vibrations, then check the box next to the
frequency you wish to animate (you must first
perform a frequency calculation.)

25
Do all Reactions have a TS?
• No!!! There are numerous examples of
barrier-less reaction pathways
• CH3. CH3. CH3CH3
• CH3. CH2CH2 CH3CH2CH2.
• gas phase addition of ions to neutral molecules

26
Effect of Solvent on Transition State Energy
• Although there is insufficient experience to
provide a general answer, it is known that in one
important classes of reactions, SN2, the reaction
profile is very different in solvent than in the
gas phase.
• In the SN2 reaction, solvent actually creates an
energy barrier that is non-existent in the gas
phase.
• Obviously solvent can be very important and any
gas-phase calculations of reaction energies
should be used with caution when applied to
condensed (liquid) phase chemistry.

27
Modeling a TS in Titan manually
• Create model of product minimize (MMFF) save
it.
• Modify this model by constraining distances so
as to stretch the bonds that are forming or
breaking during the reaction to about 1.5 times
their normal length.
• Save under a new filename. Do a constrained
geometry optimization.
• Remove the constraints and perform a Transition
State Geometry calculation.

28
Modeling a TS in Titan auto.
• Create a model of the reactant (or product)
minimize save with a different filename.
turn on pairs of adjoining bonds that undergo
bond order changes in the reaction that you are
modeling. Curved arrows appear as if you had
written the reaction mechanism.
• When all bonds have been selected, click on the
reaction button ( ) at the bottom of the
screen. This builds a guess at the transition
state geometry from a library of optimized TSs.

29
Modeling a TS in Titan(either)
• Do a Transition State Geometry calculation at a
higher level of theory or with a larger basis set
as needed, requesting that frequencies be
computed.
• After optimization, examine the list of
frequencies animate the one imaginary frequency
to confirm that it follows the reaction
coordinate, i.e., the expected path between
reactant and product. (the presence of more than
one imaginary frequency indicates a higher order
saddle point if there are no imaginary
frequencies, a minimum has been found)

30
Reaction Pathway Following
• Follows closely from T.S. modeling
• Linear Synchronous Transit approach