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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

Potential Energy Surface

(a first order saddle point)

(a first order saddle point)

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.

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)

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.

Potential Energy Surface

Energy Minimization Algorithms

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.

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.

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.

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

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!

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).

Transition State Modeling

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.

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.

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!

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.

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)

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!

LST and QST Approaches

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).

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.)

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

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.

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.

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.

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)

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