Ultrafast processes in molecules

Mario Barbattibarbatti@kofo.mpg.de

V – Finding conical intersections

2

Antol et al. JCP 127, 234303 (2007)

pyridoneformamide

Where are the conical intersections?

3

Conical intersection Structure ExamplesTwisted 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 intersections

X C

R1

R2

R3

R4

X C

R1

R2 R3

R4

X C

R1

R2 R3

R4

C

R1 R2

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: Twisted-pyramidalized

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: Stretched-bipyramidalized

7

The biradical character

Aminopyrimidine MXS CH2NH2+ MXS

8

The biradical character

p2 p1*

S0 ~ (p2)2

S1 ~ (p2)1(p1*)1

9

One step back: single p-bonds

Barbatti et al. PCCP 10, 482 (2008)

0 30 60 900

10

Rigid torsion (degrees)

p2

pp*

p*2

CH2SiH2

0 30 60 900

10

Rigid torsion (degrees)

p2

pp*

p*2

CH2CH2

p2

pp*

sp*

0 30 60 900

10

Rigid torsion (degrees)

CH2NH2+

0 30 60 900

10

Rigid torsion (degrees)

p2

pp*

p*2

CH2CHF

10

One step back: single p-bonds

0 30 60 900

10

Rigid torsion (degrees)

p2

pp*

p*2

C2H4

b

11

One step back: single p-bonds

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

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

the bond.

12

One step back: single p-bonds

d depends on:• substituents• solvation• other nuclear coordinates

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

13

Urocanic acid

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

14

Finding 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 function

2

2

221 1ln

2 cEEccEEf 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 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

ggg 2

Gradient E2

JTIJIJ

TIJp E

IJ hhggIg

Projection of gradient of EJ

17

Gradient 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 Method

A simple example:Optimization of f(x)Subject to r(x) = k

Lagrangian function: kxxfxL )()()( r

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

xLx

xL

xL

xxlxxL

0, 020

2000

xLx

LLxlxxL

19

Lagrange-Newton Method

0, 020

2000

x

LxxL

xL

xxlxxL

0, 020

2000

xL

xLLxlxxL

k0r 0

x 0r

kxLx

x

xxL

0

0

0

020

2

0 rr

rxL

xLx

xL

00

20

2

kxx

00 rr

20

Lagrange-Newton Method

kxLx

x

xxL

0

0

0

020

2

0 rr

r

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 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 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

000k0h0gkhg

00000

2

1

†

†

†JI

IJ

IJ

IJ

IJIJIJ

EELL

dddd

kxLx

x

xxL

0

0

0

020

2

0 rr

rCompare with the simple one-dimensional example:

23

Lagrange-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

000k0h0gkhg

00000

2

1

†

†

†JI

IJ

IJ

IJ

IJIJIJ

EELL

dddd

λq dddd ,,, 21Solve these equations for

Update λq ,,, 21

Repeat until converge.

24

Comparison 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 multiplicitiesExample: thymine

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

26

Crossing of states with different multiplicities

Lagrangian function to be optimized:

M

iiiJIIIJ KEEEL

11

Now the equations are:

JI

IJ

IJ

IJ

IJIJ

EELL

λ

q

0kg

kg

ddd

1†

†

000

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 states

Lagrangian function to be optimized:

M

iiikJIkIJJkJIIIJK KHHHEEEEEL

132121

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

K0

λξξq

000k000h000gkhg

ELL IJIJ

dddd

†

†

†

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

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

Example of application: photochemistry of imidazoleFast H elimination

Slow H elimination

30

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

Example of application: photochemistry of imidazoleFast H elimination

Slow H elimination

Fast H elimination: ps* 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: 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

nerg

y (e

V)

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

0

2

4

6

8

10

0 1 2 3 4

0

2

4

6

8

0 1 2 3 4 5

S0 pp*

Ene

rgy

(eV

)

CS/ps*NH

CS/ps*CN

dMW

(Å.amu1/2)

CS/pp* CS/pp*

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?

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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

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Next lecture• Transition probabilities

Contactbarbatti@kofo.mpg.de