Femtochemistry: A theoretical overviewFemtochemistry: A theoretical overview

Mario Barbattimario.barbatti@univie.ac.at

V – Finding conical intersections

This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture5.ppt

2Antol et al. JCP 127, 234303 (2007)

pyridonepyridoneformamideformamide

Where are the conical intersections?

3

Conical intersection Structure Examples

Twisted Polar substituted ethylenes (CH2NH2+)

PSB3, PSB4HBT

Twisted-pyramidalized Ethylene6-membered rings (aminopyrimidine)4MCFStilbene

Stretched-bipyramidalized

Polar substituted ethylenesFormamide5-membered rings (pyrrole, imidazole)

H-migration/carbene EthylideneCyclohexene

Out-of-plane O FormamideRings with carbonyl groups (pyridone,cytosine, thymine)

Bond breaking Heteroaromatic rings (pyrrole, adenine, thiophene, furan, imidazole)

Proton transfer Watson-Crick base pairs

Primitive conical intersectionsPrimitive conical intersections

X C

R1

R2

R3

R4

X C

R1

R2R3

R4

X C

R1

R2 R3

R4

C

R1R2

R3

H

C O

R1

R2

X Y

R1

R2

X

R1 R2

H

4

5

(b)

3 2

1

65

4(a)

(b)

3 2

1

65

4(a)

(b)(b)

3 2

1

65

4(a)

3 2

1

65

4(a)

Conical intersections: Conical intersections: Twisted-Twisted-pyramidalizedpyramidalized

Barbatti et al. PCCP 10, 482 (2008)

6

(a)

4

32

1

5

´

(b)

(a)

4

32

1

5

´

(b)

(a)

4

32

1

5

´

(a)

4

32

1

5

´

(b)(b)

Conical intersections in rings: Conical intersections in rings: Stretched-Stretched-bipyramidalizedbipyramidalized

7

The biradical character

Aminopyrimidine MXS CH2NH2+ MXS

8

The biradical character

2 1*

S0 ~ (2)2

S1 ~ (2)1(1*)1

9

One step back: single -bonds

Barbatti et al. PCCP 10, 482 (2008)

0 30 60 900

10

Rigid torsion (degrees)

2

2

CHCH22SiHSiH22

0 30 60 900

10

Rigid torsion (degrees)

2

2

CHCH22CHCH22

2

0 30 60 900

10

Rigid torsion (degrees)

CHCH22NHNH22++

0 30 60 900

10

Rigid torsion (degrees)

2

2

CHCH22CHFCHF

10

One step back: single -bonds

0 30 60 900

10

Rigid torsion (degrees)

2

2

CC22HH44

11

One step back: single -bonds

Michl and Bonačić-Koutecký, Electronic Aspects of Organic Photochem. 1990

The energy gap at 90° depends on the electronegativity difference () along

the bond.

12

One step back: single -bonds

depends on:• substituents• solvation• other nuclear coordinates

For a large molecule is always possible to find an adequate geometric configuration that sets to the intersection value.

13

Urocanic acid

• Major UVB absorber in skin• Photoaging • UV-induced immunosuppression

14

Finding conical intersectionsFinding conical intersections

Three basic algorithms:

• Penalty function (Ciminelli, Granucci, and Persico, 2004; MOPAC)• Gradient projection (Beapark, Robb, and Schlegel 1994; GAUSSIAN)• Lagrange-Newton (Manaa and Yarkony, 1993; COLUMBUS)

Conical intersection optimization:

• Minimize: f(R) = EJ

• Subject to: EJ – EI = 0HIJ =

0

Keal et al., Theor. Chem. Acc. 118, 837 (2007)

Conventional geometry optimization:

• Minimize: f(R) = EJ

15

Penalty functionPenalty function

2

2

221 1ln

2 c

EEcc

EEf JIJIR

Function to be optimized:

This term minimizes the energy average

Recommended values for the constants:

c1 = 5 (kcal.mol-1)-1

c2 = 5 kcal.mol-1

This term (penalty) minimizes the energy difference

)1ln( 2Ef p

16

Gradient projection methodGradient projection method

E

RperpendRx

E1

E2

E

RparallelRx

E1

E2

Minimize in the branching space:

Minimize in the intersection space:

EJ - EI

EJ

IJ

IJJIb EE

g

gg 2

Gradient E2

JTIJIJ

TIJp E

IJ hhggIg

Projection of gradient of EJ

17

Gradient projection methodGradient projection method

Gradient used in the optimization procedure:

pb ccc ggg 221 1

Constants:

c1 > 00 < c2 1

Minimize energy difference along the branching space

Minimize energy along theintersection space

18

Lagrange-Newton MethodLagrange-Newton Method

A simple example:

Optimization of f(x)Subject to (x) = k

Lagrangian function:

kxxfxL )()()(

Suppose that L was determined at x0 and 0. If L(x,) is quadratic, it will

have a minimum (or maximum) at [x1 = x0 + x, 1 = 0 + ], where

x and are given by:

0

, 020

2000

xL

xxL

xL

xxlxxL

0

, 020

2000

xL

x

LLxlxxL

19

Lagrange-Newton MethodLagrange-Newton Method

0

, 020

2000

xL

xxL

xL

xxlxxL

0

, 020

2000

xL

x

LLxlxxL

k0 0

x 0

kx

Lx

x

xx

L

0

0

0

020

2

0

xL

xL

xxL

00

20

2

kxx

00

20

Lagrange-Newton MethodLagrange-Newton Method

kx

Lx

x

xx

L

0

0

0

020

2

0

Solving this system of equations for x and will allow to find the extreme

of L at (x1,1). If L is not quadratic, repeat the procedure iteratively until

converge the result.

21

Lagrange-Newton MethodLagrange-Newton Method

In the case of conical intersections, Lagrangian function to be optimized:

M

iiiIJJIIIJ KHEEEL

121

minimizes energy of one state

restricts energy difference to 0

restricts non-diagonal Hamiltonian terms to 0

allows for geometric restrictions

22

Lagrange-Newton MethodLagrange-Newton Method

Lagrangian function to be optimized:

M

iiiIJJIIIJ KHEEEL

121

Expanding the Lagrangian to the second order, the following set of equations is obtained:

Kλ

q

000k

0h

0g

khg

000

00

2

1

†

†

†JI

IJ

IJ

IJ

IJIJIJ

EE

LL

kx

Lx

x

xx

L

0

0

0

020

2

0

Compare with the simple one-dimensional example:

23

Lagrange-Newton MethodLagrange-Newton Method

Lagrangian function to be optimized:

M

iiiIJJIIIJ KHEEEL

121

Expanding the Lagrangian to the second order, the following set of equations is obtained:

Kλ

q

000k

0h

0g

khg

000

00

2

1

†

†

†JI

IJ

IJ

IJ

IJIJIJ

EE

LL

λq ,,, 21Solve these equations for

Update λq ,,, 21

Repeat until converge.

24

Comparison of methodsComparison of methods

LN is the most efficient in terms of optimization procedure.

GP is also a good method. Robb’s group is developing higher-order optimization based on this method.

PF is still worth using when h is not available.

Keal et al., Theor. Chem. Acc. 118, 837 (2007)

25

Crossing of states with different multiplicitiesCrossing of states with different multiplicitiesExample: thymineExample: thymine

Serrano-Pérez et al., J. Phys. Chem. B 111, 11880 (2007)

26

Crossing of states with different multiplicitiesCrossing of states with different multiplicities

Lagrangian function to be optimized:

M

iiiJIIIJ KEEEL

11

Now the equations are:

JI

IJ

IJ

IJ

IJIJ

EE

LL

λ

q

0k

g

kg

1†

†

0

00

0IJH

Different from intersections between states with the same multiplicity, when different

multiplicities are involved the branching space is one

dimensional.

27

Three-states conical intersectionsExample: cytosine

Kistler and Matsika, J. Chem. Phys. 128, 215102 (2008)

28

Conical intersections between three statesConical intersections between three states

Lagrangian function to be optimized:

M

iiikJIkIJJkJIIIJK KHHHEEEEEL

132121

This leads to the following set of equations to be solved:

K

0

λ

ξ

ξ

q

000k

000h

000g

khg

E

LL IJIJ

†

†

†

Matsika and Yarkony, J. Chem. Phys. 117, 6907 (2002)

29Devine et al. J. Chem. Phys. 125, 184302 (2006)

Example of application: photochemistry of imidazoleExample of application: photochemistry of imidazoleFast H elimination

Slow H elimination

30Devine et al. J. Chem. Phys. 125, 184302 (2006)

Example of application: photochemistry of imidazoleExample of application: photochemistry of imidazoleFast H elimination

Slow H elimination

Fast H elimination: * dissociative state

Slow H elimination: dissociation of the hot ground state formed by internal conversion

How are the conical intersectionsin imidazole?

31

Predicting conical intersections: ImidazolePredicting conical intersections: Imidazole

32Barbatti et al., J. Chem. Phys. 130, 034305 (2009)

33

2.5 3.0 3.5 4.0 4.5 5.0 5.5

3.0

3.5

4.0

4.5

5.0E

ne

rgy

(eV

)

dMW

(Å.amu1/2)

Puckered NH EXS

Planar MXS

Geometry-restricted optimization (dihedral angles kept constant)

Crossing seam

It is not a minimum on the crossing seam, it is a maximum!

34

Pathways to the intersections

35

At a certain excitation energy:

1. Which reaction path is the most important for the excited-state

relaxation?

2. How long does this relaxation take?

3. Which products are formed?

36

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

S0

S1

S2

S3

S4

Ave

rage

adi

abat

ic p

opul

atio

n

Time (fs)

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

S0

S1

S2

S3

S4

Ave

rage

adi

abat

ic p

opul

atio

n

Time (fs)

Time evolution

37

38

Next lectureNext lecture

• Transition probabilities

Contactmario.barbatti@univie.ac.at

This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture5.ppt