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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
(a first order saddle point)
(a first order saddle point)
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
    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.

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
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.

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)
  • 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!

14
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).

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)
  • To get the best energy value, do single point
    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
  • combination of radicals
  • CH3. CH3. CH3CH3
  • addition of radicals to alkenes
  • 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.
  • 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.

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
  • 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.)

31
LST and QST Approaches
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