Femtochemistry: A theoretical overviewFemtochemistry: A theoretical overview

Mario Barbattimario.barbatti@univie.ac.at

II – Transient spectra and excited states

This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture2.ppt

SingletTriplet

Photoinduced chemistry and physicsPhotoinduced chemistry and physics

avoided crossing 102-104 fs

conical intersection 10-102 fsPA – photoabsorption 1 fs

VR – vibrational relaxation 102-105 fs

Energy (eV)

0

10

Nuclear coordinates

PhFl

PA

VR

Fl – fluorescence 106-108 fs

intersystem crossing 105-107 fs

Ph – phosforescence 1012-1017 fs

Femtosecond phenomenaFemtosecond phenomena

4

time-resolved experiments

5Static spectrum: information is integrated over time

Conventional UV absorption spectrumConventional UV absorption spectrum

0

absorptionade

gua

thy

cyt

Ultra-short laser pulsesUltra-short laser pulses

Transient spectrum: information is time resolved

7

450 500 550 600 650 700

0.0

0.2

0.4

0.6

0.8

1.0

Flu

ore

sce

nce

sp

ect

rum

(nm)

Time resolved spectraTime resolved spectra

static

transient

Transient (time-dependent) spectra: pump-probeTransient (time-dependent) spectra: pump-probe

Mestdagh et al. J. Chem. Phys. 113, 240 (2000)

t

+

t

pump

and probe

d ~2000 fs

d < 200 fs

d < 200 fs

Mathies et al. Science 240, 777 (1988)

probe wavelength

= 618 nm

= 60 fs

= 560 - 710 nm

= 6 fs

Pump

Probe

0

absorption

1

transmission

2

stimulated emission

0

excited state absorption (ionization)

1

transmission

1

spontaneous emission (fluorescence)

Transmission due to ground state depletion

11

Excited stateabsorption

00

22

Stimulated emission

00

Ground state absorption

14

15

BacteriorhodopsinBacteriorhodopsin

16

geometry optimization

17

Topography of the potential energy surfaceTopography of the potential energy surface

18

Topography of the excited-state potential energy surfaceTopography of the excited-state potential energy surface

We want determine:• minima• saddle points• minimum energy paths• conical intersections

19

Newton-RaphsonNewton-Raphson

A bit of basic mathematics: The Newton-Raphson’s Method

0xR

x

f(x)

x1x2x3

n

nnn xf

xfxx

'1

Numerical way to get the root of a function

Prove it!

20

To find the extreme of a function, apply Newton-Raphson’s Method to the first derivative

0xe

f(x)

0 x

df/dx

xxe

x1x2x3

n

nnn xf

xfxx

''

'1

Newton-RaphsonNewton-Raphson

21

kkkTkkkkTkkk EE xxxHxxxxxgxx 1111

21

Taylor expansion:

221

2

22

212

21

221

221

2

//

//

///

NN

N

EE

EE

EEE

rrr

rrr

rrrrr

xH

Hessian matrix:

NE

E

r

r

xg

/

/ 1

Gradient vector:

iiii

N

zyx ,,,1

r

r

r

x

Geometry optimizationGeometry optimization

Szabo and Ostlund, Modern Quantum Chemistry, Appendix C

22

Geometry optimizationGeometry optimization

At xe, g(xe) = 0

kkke xgxHxx 1 Prove it!

xe xk

If H-1 is exact: Newton-Raphson MethodIf H-1 is approximated: quasi-Newton Method

When g = 0, an extreme is reached regardless of the accuracy of H-1, provided it is reasonable.

23

Problem 1:Problem 1:

• Get the gradient g

Numerical

Expensive, unreliable, however available for any method for which excited-state energies can be computed

x

xxExxExxE

211

1

1

1 gradient = 2 x 3N energy calculations!

Analytical

Fast, reliable, but not generally available

xdxdx

22

x

xxxxdxdx

2

222

Two ways to get the derivative of x2

24

Method Single/Multi Reference

Analytical gradients

Coupling vectors

Computational effort

Typical implementation

MR-CISD MR Columbus EOM-CC SR Aces2 SAC-CI SR Gaussian CC2 / ADC SR Turbomole CASPT2 MR Molpro MRPT2 MR Gamess CISD/QCISD SR Molpro / Gaussian MCSCF MR Columbus / Molpro DFT/MRCI MR S. Grimme (Münster) OM2 MR W. Thiel (Mülheim) TD-DFT SR Turbomole TD-DFTB SR M. Elstner (Braunschweig) FOMO/AM1 MR Mopac (Pisa)

Present situation of quantum chemistry methodsPresent situation of quantum chemistry methods

Methods allowing for excited-state calculations:

25

Problem 2:Problem 2:

• Get the Hessian H (or H-1)

Hessian has NxN = N2 elementsNormally second derivatives are computed numericallyHessian matrix is too expensive!

Use approximate Hessian:1. Compute H in inexpensive method (3-21G basis, e.g.)2. Do not compute. Use guess-and-update schemes (MS, BFGS)

11

111

11

kkTkk

TkkkkT

kkggxx

xxxxΛΛHH

11

11

kkTkk

Tkkkk

kggxx

ggxx1Λ

Example: update in the BFGS method:

26excited state relaxation

27

The electronic configuration changes quickly after the photoexcitation

28

Minima in the excited statesMinima in the excited states

E

X

“Spectroscopic” minimum

Globalminimum

• “Spectroscopic” minima are close to the FC region• Global minima often are counter-intuitive geometries

29

Minima in the excited statesMinima in the excited states

0 2 4 6 8 10

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

En

erg

y (e

V)

LIICMin S1

MXS 3

V.Exc.

S0

S1

S2

30

Minima in the excited statesMinima in the excited states

NH

O

NH

CH

O

Ground state minimum S1 “spectroscopic” minimum

31

0 20 40 60 80 1000

2

4

6

8

10

12

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.35

1.40

1.45

1.50

0 50 100 150 2001.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

1.600 50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

Tot

al n

umbe

r of

hop

ping

s

Time (fs)

S2 S1 S1 S2

Ene

rgy

(eV

)

S1-S2 Gap

R(C6-N)

Bon

d le

ngth

(Å

)

R(C2-C3) R(C4-C5) R(C2-O)

Bon

d le

ngth

(Å

)

Time (fs)

Fra

ctio

n of

traj

ecto

ries

S2

NH

CHO

NH

O(a)

(c)

(b)

(d)

(1)

(2)

0 20 40 60 80 1000

2

4

6

8

10

12

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.35

1.40

1.45

1.50

0 50 100 150 2001.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

1.600 50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

Tot

al n

umbe

r of

hop

ping

s

Time (fs)

S2 S1 S1 S2

Ene

rgy

(eV

)

S1-S2 Gap

R(C6-N)

Bon

d le

ngth

(Å

)

R(C2-C3) R(C4-C5) R(C2-O)

Bon

d le

ngth

(Å

)

Time (fs)

Fra

ctio

n of

traj

ecto

ries

S2

NH

CHO

NH

O(a)

(c)

(b)

(d)

(1)

(2)

0 20 40 60 80 1000

2

4

6

8

10

12

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.35

1.40

1.45

1.50

0 50 100 150 2001.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

1.600 50 100 150 200

0.0

0.2

0.4

0.6

0.8

1.0

Tot

al n

umbe

r of

hop

ping

s

Time (fs)

S2 S1 S1 S2

Ene

rgy

(eV

)

S1-S2 Gap

R(C6-N)

Bon

d le

ngth

(Å

)

R(C2-C3) R(C4-C5) R(C2-O)

Bon

d le

ngth

(Å

)

Time (fs)

Fra

ctio

n of

traj

ecto

ries

S2

NH

CHO

NH

O(a)

(c)

(b)

(d)

(1)

(2) NH

CHO

NH

O(a)

(c)

(b)

(d)

(1)

(2)

Relaxation in the excited statesRelaxation in the excited states

Barbatti et al., in Radiation Induced Molecular Phenomena in Nucleic Acid ( 2008)

32Merchan and Serrano-Andres, JACS 125, 8108 (2003)

Surface can have different diabatic charactersSurface can have different diabatic characters

33

Minima may have different diabatic charactersMinima may have different diabatic characters

E

X

n

Change of diabatic character

Adiabatic surface

n

n

34

Initial relaxation may involve several statesInitial relaxation may involve several states

E

35

Relaxation keeping the diabatic characterRelaxation keeping the diabatic character

Merchán et al. J. Phys. Chem. B 110, 26471 (2006)

36

Relaxation changing the diabatic characterRelaxation changing the diabatic character

Barbatti et al. J.Chem.Phys. 125, 164323 (2006)

[1 .7 7 2 ]

1 .7 3 2

[1 .7 7 2 ]

1 .7 3 2

[1 .7 7 2 ]

1 .7 3 2

37

In general, multiple paths are available In general, multiple paths are available

38

Common reaction paths: Common reaction paths: efficiencyefficiency

*/csn

X C

R1

R2R3

R4

n*/cs

Ene

rgy

n

Reaction path

C O

R1

R2

*/cs

X C

R1

R2R3

R4

-1s-3s

n-1s

N H

R1

R2

39

0 90 180 270 3600

90

180

(°)

(°

)

0 90 180 270 3600

90

180

(°)

(°

)

0 fs

120 fs

170 fs

200 fs

The trapping effectThe trapping effect9H-adenine

Ene

rgy

Reaction path

Ene

rgy

Reaction path

0 90 180 270 3600

90

180

(°)

(°)

2-pyridone

Ene

rgy

Reaction path

Ene

rgy

Reaction path

40

4

6

8

4

6

0 5 10

4

6

3T1

*/cs*

n*

Ene

rgy

(eV

)

6E

*/cs*

n*out-of-plane O

n*/cs*

n*

dMW

(Å.amu1/2)

E5

*/cs*

n*

6,3B

n*/cs*

n*

Radiationless decay:Radiationless decay: thyminethymine

Zechmann and Barbatti, J. Phys. Chem. A 112, 8273 (2008)

41

Radiationless decay:Radiationless decay: lifetimelifetime

0 50 100

0.00

0.25

0.50

0.75

1.00

0 50 100 0 50 100 150

S3

S2

S1

S0

S4

Occ

upat

ion

S2

Time (fs)

S3 S

1

S0

S2

S1

S0

pyridonepyrrole

NH

adenine

N

N

NH2

NH

N NH O

0 50 100

0.00

0.25

0.50

0.75

1.00

0 50 100 0 50 100 150

S3

S2

S1

S0

S4

Occ

upat

ion

S2

Time (fs)

S3 S

1

S0

S2

S1

S0

pyridonepyrrole

NH

adenine

N

N

NH2

NH

N

adenine

N

N

NH2

NH

N NH O

*/cs

*/cs

n*/csn n*/csn-1s

-3sn-1s

-1s-3s

n-1s

42

excited-state intramolecular proton transferESIPT

43

Proton Transfer in 2-(2'-Hydroxyphenyl)benzothiazole (HBT)

Elsaesser and Kaiser, Chem. Phys. Lett. 128, 231 (1986)

44

ESIPT reaction schemes

pump

ketoform

NOH

S1

S0

emission

tN

OHNOH

N

OH

reaction path

electronicconfigurationchange

several modes contribute

45

T/T

Lochbrunner, Wurzer, Riedle, J. Phys. Chem. A 107 10580 (2003)

Emission signal at the keto wave number appears after only 30 fs

46

47

Internal conversion should play a role

48

ESIPTESIPT

probe = 570 nmResolution: 30 fs

Schriever et al., Chem. Phys. 347, 446 (2008)Barbatti et al., PCCP 11, 1406 (2009)

49

Next lecture

• Adiabatic approximation• Non-adiabatic corrections

Contactmario.barbatti@univie.ac.at

This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture2.ppt