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

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

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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)
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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
  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.

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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)

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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.

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Potential Energy Surface
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Energy Minimization Algorithms
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Energy Only (Univariate) Method

• Simplest to implement
• Proceeds one direction until energy increases,
  then turns 90º, etc.
• Least efficient
• many steps
• steps are not guided
• Not used very much.

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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.

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Conjugate Gradient Method

• 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.

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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

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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)
• 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!

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Caveats about Minimum Energy Structures

• 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).

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Transition State Modeling
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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.

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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.

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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!

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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.

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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)
• To get the best energy value, do single point
  energy calculation with a method that includes
  electron correlation (e.g., MP2)

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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!

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LST and QST Approaches
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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).

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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.)

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Do all Reactions have a TS?

• No!!! There are numerous examples of
  barrier-less reaction pathways
• combination of radicals
• CH3. CH3. CH3CH3
• addition of radicals to alkenes
• CH3. CH2CH2 CH3CH2CH2.
• gas phase addition of ions to neutral molecules

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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.

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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.

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Modeling a TS in Titan auto.

• 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.

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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)

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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.)

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LST and QST Approaches