theoretical approaches to chemical reactivities - minh tho nguyen

8/20/2019 Theoretical Approaches to Chemical Reactivities - Minh Tho NGUYEN

http://slidepdf.com/reader/full/theoretical-approaches-to-chemical-reactivities-minh-tho-nguyen 1/44

Theoretical Approaches

to Chemical Reactivities

• I. Potential Energy Surfaces (PES)

• II. Use of PES in the Study of Chemical

Phenomena

• III. High Accuracy (Theoretical)

Thermochemistry

----------------------------------------------------------

Master in Physical and Theoretical Chemistry

Quy Nhon University, 25 – 31 March 2013

by Minh Tho NGUYEN, K.U.Leuven, Belgium



Born-Oppenheimer Approximation

-----------------------------------------------------------------

• H Ψ = E Ψ (1)

• H(r,R) = TN(R) + Te(r) + VNe(r,R) + Vee(r) + VNN (R) (2)

WIth fixed nuclei, the electronic motion can be described by:

• [Te(r) + Vee(r) + VNe(r,R) + VNN(R)]. φ(r,R) = εn(R). φn(r,R) (3)

• Total molecular WF:

Adiabatic Representation is based on the fact that a

nuclear mass is >1800 times as large as the electron

mass; nuclei move much slower than electrons.

( , ) ( ) ( , )n n

n

r R R r R χ Ψ = Φ∑



• One term WF: Ψ (r,R) = χn(R) φn(r,R)

• Define an Adiabatic Halmiltonian:

• The nuclear wavefunction:

- The validity of the adiabatic approximation resides in the assumption

that the nuclear kinetic energy is substantially smaller than the energy

gap between adiabatic electronic states.

The non-adiabatic coupling arising from the nuclear kinetic energy could

be considered as a perturbation of the total electronic energy and as a

consequence, the nuclei should contain a small amount of kinetic

energy for a slow motion.- Electronic eigenfunction is real, the coupling terms are small and can

be removed Born-Oppenheimer Approximation:

[TN(R) + Vn(R) ] χn(R) = En χn(R)

Hn(adia) = TN(R) + εn(R) + Λnn(R)

Hn (adia) χn (R) = E χn (R)



• The electronic energy Vn(R) plays the role of potential for the

nuclear motions.

• This motion occurs on a potential energy surface: a molecular

system undergoes a chemical reaction when its nuclei move

smoothly on a PES which is nothing else than its electronic

energies.

• The nuclear motion from one to another region induces a

reorganisation of its electronic density to adapt to the novel

nuclear configuration, but not the transition to other electronicstates.

• although both terms are often used to describe the separation

of electronic motion from that of the nuclear motion, there is a

difference between the adiabatic approximation and BOapproximation.

• while the BO PES is independent of the nuclear masses, the

diagonal coupling terms in the Adiabatic approximation depend

on the latter.



• To achieve high accuracy, the coupling term must be retained.

In view of the practical difficulties in evaluating the non-

adiabatic coupling terms, non-Born-Oppenheimer calculations

are presently achievable for atoms and small diatomic

molecules.

• In a perturbation treatment of the BOA, the first-order correction

to the BO electronic energy due to the nuclear motion is the

Diagonal BO Corrections (DBOC).

• in the DBOC, only the diagonal terms are computed from

EDBOC = <Ψ (r,R) / TN / Ψ (r,R) >

which thus depends on the nuclear mass.

Small at the chemical energy scale but very relevant for a

spectroscopic energy scale ( ~100 cm-1

). In some cases,DBOC needs to be added to obtain accurate heats of formation.



Features of Potential Energy Surfaces

The total energy E of a molecular system is calculated as a function of

internal coordinates qk:

E = E(qk) k from 1 to 3N-6

Search for stationary points where energy gradients (first derivatives of

energy) vanish

gk = δE / δqk k = 1. …. 3N-6

With energy Hessians (second derivatives), local minima (equilibrium

structures) and first-order saddle points (transition structures) can be

located and characterized

Hkl = δ2E / δqk δql k, l = 1 ……. 3N-6

All Eigenvalues positive: λk > 0 minimum

One negative eigenvalue: λk < 0 first-order saddle-point



• Statement 1:

An equilibrium structure corresponds to an

energy minimum.

• Statement 2:

A transition structure for a chemical processfor a reaction must be a first-order saddle point;

a saddle-point that is a maximum in only one

direction and a minimum in all other

perpendicular directions.



Reaction mechanism via TS structures

Products

Reactants

Transition

Structure (TS)

Reaction Coordinate

Potential

Energy

(Eelec+ Vnn)

S 3



• Statement 3:

A Reaction Coordinate should be considered as a combinationof internal coordinates.

In practice, only portions of a potential energy surface can be mapped out, andrelevant energy profiles established

A minimum energy reaction pathway is determined by the intrinsic reactioncoordinate (IRC).

The path of steepest descent from a saddle-point is not unique, because the shape ofa PES depends on the particular choice of coordinates that describes the geometry.More than one combination of bond lengths, bond angles and dihedral angles can beemployed to represent the same structure.

Expl: - NH3 inversion

- a 1,2-H-shift

- cycloaddition



Vibrations Energy Surfaces

• Harmonic oscillator approximation of motions on a PES:

E(q) = E0 + ½ Σij Fij (qi – qi0) (q j – q j

0) Fij = δ2E / δqi δq j = force constants

The harmonic vibrational frequencies for a molecule can be computed from

the force constant matrix, the geometry and the atomic masses by solution

of the Wilson-GF method:

det (GF – εI) = 0

where G-1 is the kinetic energy matrix (geometry + masses) and eigenvalues

ε are related to the vibrational frequencies:

ω = 1/2π √ ε

The normal coordinates of vibration are those linear combinations of the

coordinates for which the GF matrix equation can be solved. In mass-

weighted coordinates (G being constant times the unit matrix), the normal

coordinates are just the eigenvectors of F.



• The eigenvectors associated with the negative eigenvalue (the normal mode

associated with the imaginary frequency) is the single direction in which the

surface is a maximum, and is termed the Transition Vector

• Statement 4.

Symmetry of Stationary Points:

- A transition vector cannot belong to a degenerate representation

(a second-order saddle point, ex: linear water);

- A transition vector must be symmetric for all symmetry operations of

the TS that also leave the reactants and products unchanged(ex. NH3 inversion; 1,3-H-shift in propene (C2), X

- + CH3X (σv)…)

- the transition vector must be antisymmetrical for any of the symmetry

operations of the TS that interconvert the reactants and products.



1

ˆ (r,R) (r,R) M M

A B

nuc elec nuc tot

A B A AB

Z Z H T E T E

R= >

= + + = +∑ ∑

Topological features on a potential energy surface

Potential Energy Surface



• Approaches to Chemical Reactivities

Statistical MechanicsClassical

Quantum Mechanics

ψ(r,R,t)

V(R): PESψ(R)

ψ(r,R)

Chemical

Dynamics

Chemical

Kinetics

- Molecular Structure

- Spectroscopic Properties

- Thermochemical Parameters

Rate

Constants

k(E,t,T,P)

Quantum Chemical Methods:

- Molecular Orbital Theory

-Density Functional Theory

Reaction

Mechanisms

Time Dependent

Direct Statistical Methods

Born-Oppenheimer

Approximation

f (E,T,P)

f (E,t)



• A PES describe only the static situation of a molecular

system. Real molecules have more than infinitesimal

kinetic energy, and is not likely to follow the intrinsic

reaction path. A reaction is dynamic by nature.Nevertheless, the intrinsic reaction coordinate

provides a convenient description of the progress of a

reaction, and plays also as the starting point in the

calculations of reaction rates by statistical ratetheories (kinetics) or chemical dynamics (trajectory

calculations).

Similar to the Orbital, PES is a mathematical object,

born from the BO approximation. However, its

becomes a powerful concept in chemistry



Construction of a PES: location of energy

minima and first-order saddle points.

A full geometry optimization which minimizes ormaximizes the electronic energy with respect to

all nuclear coordinates allows the stationary

points to be located.

In practice, portions of a PES is schematically

represented using an arbitrary reactioncoordinate, that are related to the reactions

under consideration.



Locating MinimaThere are numerous methods and strategies for the location of minima:

- Energy-only- Gradients / Hessians

- One coordinate / Multi-coordinate

The main feature of an optimization algorithm is to find a minimum on a

surface by construction of a series of points that explore a portion of the

surface and proceed progressively closer to a local minimum Surface is unknown, the search begins adopting a simple mathematical

form for the surface and adjusts this as more data are generated. Most

commonly, the surface is modeled by a quadratic polynominal:

E(q) = E(qo) + Σi Ai (qi – qio) + ½ Σij Bij (qi – qi

o) (q j – q jo)



Criteria:

- speed of convergence to minimum

- stability and reliability- overall cost:

Analytical gradient requires the same

amount of computer time as the energy Numerical gradient for N dimensions

requires N times energy

Numerical Hessian takes 5 to 10 to N timeslonger than gradient

method using derivatives requires more timein a step, but faster in terms of the total

number of steps.



Energy Only (Univariate) Method

• Simplest to implement, widest range of applicability

• Least efficient

– many steps for N dimensions

– steps are not guided

– slow convergence

• used when no gradient

is available

One coordinate is changed

at a time and the calculation cycle

over all of the coordinates: known as

“AXIAL iteration method” or

“sequential univariate search”

• Energy only but numerical derivatives:



• Energy-only but numerical derivatives:

1. Calculate the energy at the starting geometry and at positive andnegative displacement of each of the coordinate; obtaining derivatives

2. Find the minimum on the model surface; if the predicted change in thecoordinate is sufficiently small, the optimizations is stopped

3. Step to the minimum, calculate its energy; step the same distancebeyond, and recalculate the energy

4. Fit a parabola of the three new poins, find a minimum, calculate energy.

5. Recalculate the numerical first derivative by stepping along eachcoordinate, and readjust the model surface to fit the available data.

Steps 2-5 repeated until convergence is achieved. The number of stepsrequired is still proportional to N2



Gradient Algorithm: Steepest Descent Method

• Simplest method but more efficient ifanalytical gradient available

• Estimate of the 2d derivative, andimprove the estimate as theoptimization proceeds

• Methods differ by the updates of 2dderivatives

• Follows most negative gradient (max.force)

• Fastest method from a poor starting

geometry

N + 2 steps• Converges slowly near the energyminimum

• Can skip back and forth across aminimum (in a cyclic system)

• An order of magnitude increase inefficiency w.r.t. energy-only algorithm

Quasi-Newton, Newton-Raphson,

block diagonal Newton-Raphson



Gradient Algorithm: Steepest Descent Method

1. obtain an estimate of the Hessian Hij (unity matrix,

empirical estimate...)2. calculate the energy, E, and the gradient vector g i

3.update Hessian

4. carry out an accurate minimization along the lineconnecting the current point and the previous point (to

save, fitting a cubic or quartic curve to estimate the

position of the minimum from the fitted curve rather

than from recalculation of energy)5. calculate displacement using gi and Hij :

Δq = - H-1 g

S d D i ti



Second Derivatives

• The best method: on a quadratic surface, theminimum can be found in one step.

• Drawback is the cost for Hij (5 to 10 time aslong as gradient calculations)

• the most efficient optimization is the gradient

method with a good guess for Hij• It is imperative to have a good guess for

Hessian matrix for an optimization at a high

level (don’t start without Hij). It can be obtainedfrom a lower-lever

• for difficult cases (shallow wells, stronglycoupled coordinates (ring)), analytical Hij are

needed.



Approaches to Global Minimum

Large number of possible isomers, conformers...

• Dihedral driving (systematic)

• Randomization-minimization (Monte Carlo)

• Molecular dynamics (Newton’s laws of motion)

• Simulated Annealing (reduce T during MD run)

• Genetic Algorithms (start with a population ofconformations; modify slightly; retain lowest

energy ones, repeat)• Trial & error (poor)

All 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 energyconformation?

• What other low energy conformations are

available? (Boltzmann distribution and

probability/entropy considerations may be

important).



Locating Transition Structure (TS)

• Reactants and products are well defined

molecular entities; TSs are not.• TSs are observed experimentally yet; therefore

no parameters can be devised for modeling

them. The only information for guessing themis result from literature on similar / close

reactions

• It is thought that TSs exhibit elongated bonds,partial bonding, and may have some aspects

of excited states associated with them



TS Locating Difficulties...

• We ‘know’ relatively little about thegeometry of TS’s; most of what we

‘know’ is based on calculation.Guessing TS geometries is moredifficult than guessing the geometry ofa stable structure.

chemical intuition is important inguessing a TS



Locating TS

• Mathematically, one has to go uphill to locate a TS, the

algorithms available are less efficient

• the PES in the vicinity of the TS is ‘flatter’ than the surface near

a minimum, therefore it may be more difficult to predict the

shape of a TS.

• The reaction coordinate is not known in advance and must be

determined as part of the optimization.

• There may not exist a unique T.S. There are different TSs

corresponding to different processes. One often obtains the

unwanted TS!

(methyl rotation...)



“ Guessing” a TS Geometry

• Base the guess on a previously calculated,

related system, or ‘chemical intuition’ and apreconceived notion of the mechanism.

• Use an ‘average’ of the reactant and productgeometries: Linear Synchronous Transit

method).• a Quadratic Synchronous Transit method, in

which minima perpendicular to the LST areconnected.

• Don’t be disappointed if optimizations fail:several attempts are needed!

Locating TS



Locating TS

• Perform a low level calculation (HF, B3LYP/3-21G or /6-31G*)....

• Because TS’s involve partial bonding, lower levels of theoryare not always useful

• If found, use that result as starting point for higher levelcalculation, with analytical frequencies at the first point

• Verify with a frequency calculation at the same high level oftheory and basis set

• When failed: change the geometry, basis set, method …

• In shallow surface, or coupled coordinates, analytical

Hessians at every point may be needed to have the rightcurvature



Confirming a Possible TS

• Must be a first order saddle point on PES smoothly

connecting reactant to product. – Verify that the Hessian yields one and only onenegative (imaginary) frequency.

– Animate the normal coordinate corresponding to theimaginary frequency; it should connect reactants and

products (have vibrations consistent with expected bondbreaking and bond forming).

– It is imperative to run the IRC pathway when not sureabout the identity of the minima.

• Confirm the TS by different levels of theory (HF, MP2, DFTbasis sets ...)



• Linear Synchronous Transit (LST):



Linear Synchronous Transit (LST):

1. Assume that the reaction path is a straight line

connecting reactants and products.2. the LST TS is always higher than the true TS.

From this, one can minimize the energy w.r.t. allcoordinates perpendicular to the linear path.

3. The resulting point is lower in energy than the trueTS.

4. The reaction path can now be described by aparabola or quadratic curve that connect the

minima quadratic synchronous transit (QST)The new maximum is a better estimate.., and can berepeated until the TS is found.



LST and QST Approaches

• Advantage: only energy no gradient and less



• Advantage: only energy, no gradient, and less

costly than a surface fitting.

• Linear interpolation of distance matrices

gives a reasonable pathway

• Problems when the path is strongly curve,

and the reactants and products are very

different.

• In any case, when using a low-level method,

this give a quick scan of the surface to get

indication on the shape.

• The estimate TS can certainly be valubale for

optimization using other methods.



• Locating a TS: Coordinate Driving:

In many reaction, the change of one coordinatedominates the region of the surface,the TS can belocated by following this coordinate.

Problem encountered when the path should becurved so that a second coordinate begins todominate (e.g. 1,2-H-shift): the method fails giving adiscontinuity on the surface, or to lead to a dead-endvalley.

at best, a supperposition of many single coordinatecurves could help identify the saddle region.

• Walking up Valleys:



• Walking up Valleys:

A simple approach to improve the coordinate driving method is to take a

step of a specific length and then determine the direction that corresponds

to the shallowest path up the valley.

Eigenvector Following Method (EF): In following the appropriate vector of

the Hessian at each step.

Hij needs to be recalculated frequently high cost.

If a guess starts with a negative eigenvalue of the Hessian and the

transition vector is corresponding, and it remains negative, even the

magnitude and the direction could be changed, the EF method usually

leads to a proper TS after a limited number of steps.

If the Hessians are available (from calculations at a lower level), and a

guess with a clear direction, the EF method represents a reliable method

for finding TSs.

In constrat to the minima, there is no guaranty that the process converges

to a TS.



Symmetry and Stationary Points

• In favorable circumstances, symmetry can be

used to turn a TS optimization into a minimumlocation.

• Constraint could lead to faster convergence

avoiding the involvement of lower frequencymodes.

• If a higher symmetry structure is suspected, it

is often more efficient to optimize the highersymmetry structure directly and test it w.r.t.

distortion to lower symmetry (HP=NH)



Choice of coordinates

• Faster convergence if the coordinate system is chosen

with some care. Strong coupling between coordinatesinvariably causes difficulties for optimizations (stiff =stretch + bend; loose = internal rotation + inversion;cycles, linear = inherent strong coupling and also

redundant coordinates).• Coupling of the transition vector with the low

frequency mode leads to problem for TSs.

• Loosely bound clusters: flat surafce with large

geometrical changes may occur with small energychanges

• Use of internal coordinates with dummy atoms (5-rings, cycloadditions)

Summary for TS searching



Summary for TS searching

• Coordinate Driving, or potential energy surface scan

(1D) or small grid (2D) locate the transition region.

• LST – QST

• Once a reasonable guess is obtained, and the

gradient and the hessians can be coumputed /

estimated, the reaction coordinate may be frozen in aminimization. Then the eigenvector following method

with g and H represents the best strategy.

• The problem is to update the Hessians. In somedifficult cases, recalculation of the full H at each

iteration may be necessary even it is the most

expensive one.

Wh t t d h ti i ti f il



What to do when optimization fails

1. Forces too large: error in the input: poor geometry, orbad coordinate system.

reconstruct the Z-matrix too avoid strong coupling2. Negative eigenvalue during an optimization: structure

is not a minimum or numerical problem with Hessianupdates follow the negative eigenvector; or improve

the Hij.If this persists, just impose the symmetry, or freeze therelevant coordinate. When the structure is optimized,release the frozen coordinate and reoptimize.

3. Too many negative eigenvalues:

reoptimizefollowing the lower frequency vector, or update forbetter Hessians.

4. No negative eigenvalue during a TS optimization: badgeometry guess or bad Hessian

5 Eigenvalues too large error in input or bad



5. Eigenvalues too large error in input, or bad

Hessian

6. Eigenvalues too small shallow minimum, or

redundancy (more than 3N-6 coordinates), orHessian are not good updated.. tighten the

convergence criteria (RMS gradient)

7. Number of steps exceeded: difficult case, reoptimize

rather than increase the number. Don’t go beyond 50

iterations, it won’t converge.

8. Change in point group detected during an

optimization: z-matrix does not reflect the fullsymmetry, or optimization inadvertently distorts the

molecule use the Cartesian coordinates.

.........



Danger of Computational Science

Black BoxGarbage.in Garbage.out

Users are fully and solely responsible

for the accuracy of the data and their

interpretations.

theoretical approaches to chemical reactivities - minh tho nguyen

Documents