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• One term WF: Ψ r,R = χnR φnr,R• Define an Adiabatic Halmiltonian: • The nuclear wavefunction: - The validity of the adiabatic approximation resides in the assumption that the nuclear k

Theoretical Approaches

to Chemical Reactivities

• I Potential Energy Surfaces (PES)

• II Use of PES in the Study of Chemical

( , ) n( ) n( , )

n

• 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 electronic states.

• 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 BO

approximation.

• 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:

Hkl = δ 2 E / δqk δql k, l = 1 …… 3N-6

 All Eigenvalues positive: λ k > 0  minimum

One negative eigenvalue: λ k < 0  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 (E elec + V nn )

• Statement 3:

A Reaction Coordinate should be considered as a combination

of internal coordinates

 In practice, only portions of a potential energy surface can be mapped out, and

relevant energy profiles established

 A minimum energy reaction pathway is determined by the intrinsic reaction

coordinate (IRC).

The path of steepest descent from a saddle-point is not unique, because the shape of

a 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 be employed 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 Σ ij Fij (qi – qi0 ) (qj – qj 0 ) Fij = δ 2 E / δq i 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:

• 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 ( σ v )…)

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

operations of the TS that interconvert the reactants and products.

Topological features on a potential energy surface

Potential Energy Surface

• Approaches to Chemical Reactivities

Statistical Mechanics Classical

- 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 rate

theories (kinetics) or chemical dynamics (trajectory

Construction of a PES: location of energy

minima and first-order saddle points.

A full geometry optimization which minimizes or maximizes the electronic energy with respect to all nuclear coordinates allows the stationary

Locating Minima

There 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(q o ) + Σi Ai (qi – qio ) + ½ Σij Bij (qi – qio ) (qj – qjo )

 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 times longer than gradient

 method using derivatives requires more time

in 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 :

 1 Calculate the energy at the starting geometry and at positive and

negative displacement of each of the coordinate; obtaining derivatives

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

 3 Step to the minimum, calculate its energy; step the same distance

beyond, 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 each

coordinate, and readjust the model surface to fit the available data.

 Steps 2-5 repeated until convergence is achieved The number of steps required is still proportional to N 2

Gradient Algorithm: Steepest Descent Method

• Simplest method but more efficient if

analytical gradient available

• Estimate of the 2d derivative, and

improve the estimate as the

• Can skip back and forth across a

minimum (in a cyclic system)

• An order of magnitude increase in

efficiency 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 H ij (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 line

connecting 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

Second Derivatives

• The best method: on a quadratic surface, the

minimum can be found in one step.

• Drawback is the cost for Hij (5 to 10 time as

long 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 obtained from a lower-lever

•  for difficult cases (shallow wells, strongly

coupled coordinates (ring)), analytical Hij are

needed.

Approaches to Global Minimum

• 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 of

conformations; 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 energy

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 them

is result from literature on similar / close

TS Locating Difficulties

• We ‘know’ relatively little about the

geometry of TS’s; most of what we

‘know’ is based on calculation

Guessing TS geometries is more

difficult than guessing the geometry of

a stable structure.

 chemical intuition is important in

guessing 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 a

preconceived notion of the mechanism

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

method)

which minima perpendicular to the LST are

connected

• Don’t be disappointed if optimizations fail:

several attempts are needed!

• If found, use that result as starting point for higher level

calculation, with analytical frequencies at the first point

• Verify with a frequency calculation at the same high level of theory 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 right curvature

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 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).

– It is imperative to run the IRC pathway when not sure

about the identity of the minima.

• Confirm the TS by different levels of theory (HF, MP2, DFT basis sets )

Locating a TS:

• Surface Fitting: the simplest way is to fit an

analytical expression to computed energies

Problems:

1.Acceptable functional form must be foundfor dimension surface

multi-2 A large number a energy should be computed

3 The fitted surface may not be sufficiently accurate

in the region of the barrier to provide an acceptable estimate of the TS geometry

Analytical form is needed in chemical dynamics

and trajectory calculations, in conformation

analysis, or the crossing of ground and excited

states

• 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 all

coordinates perpendicular to the linear path

3 The resulting point is lower in energy than the true

TS

4 The reaction path can now be described by a

parabola or quadratic curve that connect the

minima quadratic synchronous transit (QST)

The new maximum is a better estimate , and can be repeated until the TS is found

LST and QST Approaches

• Advantage : only energy, no gradient, and less costly than a surface fitting.

• Linear interpolation of distance matrices 

gives a reasonable pathway

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 coordinate

dominates the region of the surface,the TS can be located by following this coordinate

Problem encountered when the path should be

curved so that a second coordinate begins to

dominate (e.g 1,2-H-shift): the method fails giving a discontinuity on the surface, or to lead to a dead-end valley

 at best, a supperposition of many single coordinate curves could help identify the saddle region

• 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.