Transition States and

Reaction Paths

Computational Chemistry lab

2013

Potential Energy Surface (PES)A 3N-6 -dimensional hypersurface in, where N is the

number of atoms.

Minima– equilibrium

structures of one or

several molecules

Saddle points

Saddle point - Transition state

Minimal maximum –

maximum in one direction, minimum in other directions

Transition State

/A bE k Tk Ae

−=

k - rate constant

EA – activation energy

T – temperature

kb – Boltzmann constant

Arrhenius equation

Multidimensional Optimization

- gradient

∇ f=0 - Stationary Point (minimum, maximum, or saddle point)

Force: F=-∇ E (Potential energy)

The coordinates can be

Cartesian (x,y,z..),

or internal, such as

bondlength and angle

displacements)

Hessian matrix

ji xx

f

ij fH∂∂

∂=

2

)(

Quadratic approximation

around a stationary point

Approximate the complex energy landscape by harmonic potentials

A matrix of second-order derivatives

of the energy with respect to atomic

coordinates (e.g., Cartesian or internal

coordinates)

Sometimes called force matrix –

matrix size of (3N-6)x(3N-6)

),...( )()(

1

)( st

n

ststxxx =

r]0)([ )(

=∇st

xEr

∑∑= =

−−+−∇+=n

i

n

j

st

jj

st

ii

st

ij

ststst xxxxxHxxxExExE1 1

)()()(

21)()()( ))()(())(()()(

rrrrrr

∑=

=n

i

k

iij

k

jk lHl1

)()(εHessian matrix diagonalization (eigenproblem)

Eigenvalues2

kk mωε = kω - vibrational frequencies

Eigenvectors )(k

jl give normal coordinates ∑=

−=n

i

ii

k

ik xxlq1

)0()( )(

∑=

+=n

k

kk

st

n

st

n qxxExxE1

2

21)()(

11 ),...(),...( ε

∑∑= =

−−+=n

i

n

j

st

jj

st

ii

st

ij

st

n

st

n xxxxxHxxExxE1 1

)()()(

21)()(

11 ))()((),...(),...(r

Approximate the complex energy landscape by harmonic potentials

around a stationary point

Quadratic approximation

),...( )()(

1

)( st

n

ststxxx =

r]0)([ )(

=∇st

xEr

Eigenvalues2

kk mωε = kω - vibrational frequencies

Eigenvectors)(k

jl give normal coordinates ∑=

−=n

i

ii

k

ik xxlq1

)0()( )(

)real(0 All −> kk ωε

minimum

)imaginary(0 All −< kk ωε

maximum

)realimaginary/(0 −<> kk ωε

saddle point

Transition state – ε1<0, εk>0 (k>1)

Eigenvector l1- the path directionq

1

q2

∑=

+=n

k

kk

st

n

st

n qxxExxE1

2

21)()(

11 ),...(),...( ε

Algorithms for Finding Transition

StatesNo general methods which are guarantied to find!

Global methods – interpolation between reactant and product

Linear Synchronous Transit (LST)

Quadratic Synchronous Transit (QST)

Local methods – augmented Newton-Raphson

Eigenvector following

Berny algorithmSynchronous Transit-Guided Quasi-Newton (STQN)

method – QST+quasi-Newton

Force-field parameters in molecular mechanics are defined for equilibrium structures and can be inapplicable to transition structures.

Quantum calculations only

(recent semi-empirical potentials are applicable too)

Reactants

Saddle point

Products

Reactants

LST

Linear synchronous transit (LST)-search for a maximum along a linear path between reactants and products

Reactants

Saddle point

Products

Reactants

LST

QST

Linear synchronous transit (LST)- search for a maximum along a linear path between reactants and products

Quadratic synchronous transit (QST) - search for a maximum along an arc connecting reactants and products, and for a minimum in all directions perpendicular to the arc

Best case - the transition state is found.

General case – search is finished in a wrong saddle point or in a point with wrong number of negative eigenvalues (>1).

Newton(-Raphson) method:

One-step search for

quadratic functions kkk Fq ε/=

For arbitrary functions

Finds the closest stationary point (either

minimum, maximum, or saddle point).

Transition state – ε1(R)<0, εk(R)>0 (k>1)

Quasi-Newton – no explicit computation

of the Hessian matrix.

qk- normal coordinates, ,1

0 ∑=

=−n

k

kkq lRR ∑=

−=∇n

k

kkFE1

0 )( lR

Quadratic approximation : )()()( 2

21

1

0 kk

n

k

kk qqFEE ε∑=

−−= RR

εk and lk - eigenvalues and eigenvectors of the Hessian matrix at R0

)(/)()(1

1 ik

n

k

ikikii F RRRlRR ε∑=

+ +=

R 0

Rts

),...( 1 nxx=R - (3N-6) -dimensional vectors

Rational function optimization

(RFO)

∑=

+−

+=n

k kik

ikikii

F

1

1)(

)()(

λε R

RRlRR

ε1=min(εk) < λk≡λ<ε2/2

Berny algorithm (Gaussian, Opt=TS)

Eigenvector Following Method (Gaussian, Opt=EF; Hyperchem)

2/)4( 2

1

2

111 F++= εελ

∑>

−=≡>1

2 )/(:1i

iik Fk ελλλ

Clamber “up valley”

Clamber “up valley”

Synchronous Transit-Guided Quasi-Newton (STQN)

approach to the quadratic region -quadratic synchronous transit

complete the optimization-eigenvector-following algorithm

Reactants

Saddle point

Products

Eigenvector following

QST

TS estimateGaussian, Opt=QST2;

Hyperchem: input reactants and products only, automatic estimate of transition state

Gaussian, Opt=QST3: input reactants, products, and estimate of transition state

Is the correct TS found?

Look at the transition state geometry to make

sure it’s the right one.

Use several algorithms.

Try several estimates of transition state.

Follow reaction path to be sure that the

transition state connects the correct

reactants and products.

Reaction PathsSteepest descent path from transition state to reactants

and products

Gaussian - IRC

New point xk+1 –minimization on the hyperspheresurface

IRC-internal reaction coordinate