Potential Energy Surfaces,
Energy Minimization Methods,
and Transition State Modeling
Outline
Potential Energy Surface (PES)
 Energy Minimization Methods (Algorithms)
 Global Minimum Structure
 Transition State Modeling
 Reaction Pathway Modeling
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Potential Energy Surface
(a first order saddle point)
(a first order saddle point)
Potential Energy Surface Terms
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Gradient - the first derivative of the energy
with respect to geometry (X, Y & Z); also
termed the Force (strictly speaking, the
negative of the gradient is the force)
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
Saddle Points.
PES Terms...
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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)
Sign of 2nd Derivatives
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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.
Potential Energy Surface
Energy Minimization Algorithms
Energy Only (Univariate) Method
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Simplest to implement
Proceeds one direction
until energy increases,
then turns 90º, etc.
Least efficient
–
–
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many steps
steps are not guided
Not used very much.
Steepest Descent Method
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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.
Conjugate Gradient Method
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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.
Second Derivative Methods
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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
Approaches to Locating the Global
Minimum Energy Structure
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Systematic Dihedral driving (manual or automatic)
Randomization-minimization (Monte Carlo)
Molecular dynamics (Newton’s laws of motion)
Simulated Annealing (reduce T during MD run)
Genetic Algorithms (start with a population of
conformations; modify slightly; retain lowest energy
ones, repeat)
Trial & error (poor)
Methods are tedious, but absolutely necessary
if the result is to be meaningful!
Caveats about Minimum
Energy Structures
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What does the global minimum energy structure really
mean?
What other low energy conformations are accessible?
(Boltzmann distribution/ensemble of conformations
and probability/entropy considerations may be
important). Some properties, such as dipole moment,
are best computed as weighted averages of dipole
moments of various contributing conformations.
m = 0.48 m1 + 0.27 m2 + 0.14 m3 + 0.10 m4
For a molecule having four major conformations, contributing 48%,
27%, 14% and 10% respectively (Based on the Boltzmann distribution)
Caveats, cont’d.
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Does reaction/interaction of interest necessarily
occur via the lowest energy conformation?
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As an example, a study of the solvolysis of
some six-membered ring bromoamines shows
that reaction can best occur via a high energy
boat conformation:
Higher energy conformer in
solvolysis reaction
Solvolysis Rate (Water):
N CH3
(CH3)3C
N CH3
(CH3)3C
C(CH3)3
C(CH3)3
>>
H
Br
Br
H
(CH3)3C
Sole Product is:
N CH3
C(CH3)3
H
OH
(CH3)3C
If SN2,
expect:
N CH3
C(CH3)3
HO
H
Mechanism consistent with
observations
Br
H
(CH3)3C
H
Br
N CH3
Bu
C(CH3)3
N CH3
Bu
H O
C(CH3)3
H
N
CH3
C(CH3)3
H
Br
H
HO
-H
Bu
(CH3)3C
N CH3
N CH3
C(CH3)3
C(CH3)3
H
OH
Transition State Modeling
Transition State Modeling
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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.
Transition States; Why Difficult?
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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.
TS Modeling Difficulties...
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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!
More TS Modeling Difficulties...
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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.
Best Approach: Mixed Methods
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“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)
To get the “best” energy value, do single point
energy calculation with a method that includes
electron correlation (e.g., MP2)
“Guessing” a TS Geometry
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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!
LST and QST Approaches
Confirming a Possible TS
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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).
Frequency Calculation
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Keyword: Frequency, Freq or Frequencies
Output file has a table of frequencies listed in
order of increasing magnitude; imaginary
frequencies (negative; <0 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.)
Do all Reactions have a TS?
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No!!! There are numerous examples of barrierless reaction pathways:
– combination of radicals
CH3. + CH3.
CH3CH3
– addition of radicals to alkenes
CH3. + CH2=CH2
–
CH3CH2CH2.
gas phase addition of ions to neutral molecules
(CH3)3C +
CH2
CHCH3
(CH3)3CCH2CHCH3
Effect of Solvent on
Transition State Energy
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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.
Modeling a TS in Titan manually
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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.
Modeling a TS in Titan auto.
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Create a model of the reactant (or product);
minimize & save with a different filename.
Return to Build menu; select Reaction. Click
in 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.
Modeling a TS in Titan…(either)
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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)
Reaction Pathway Following
Follows closely from T.S. modeling
 Linear Synchronous Transit approach
 Quadratic Synchronous Transit approach
 Reaction Path Approach
– 10 or so minimum energy points equally spaced
between reactant and product
 Walking Up Valleys (least steep ascent; may not
lead to proper T.S.)
 Steepest Descent (from T.S.)
LST and QST Approaches