Femtochemistry:
A theoretical overview
II – Transient spectra and excited states
Mario Barbatti
mario.barbatti@univie.ac.at
This lecture can be downloaded at
http://homepage.univie.ac.at/mario.barbatti/femtochem.html
lecture2.ppt
Photoinduced chemistry and physics
Energy (eV)
10
VR
Singlet
Triplet
Ph
Fl
PA
0
PA – photoabsorption
conical intersection
avoided crossing
1 fs
10-102 fs
102-104 fs
VR – vibrational relaxation 102-105 fs
intersystem crossing 105-107 fs
Fl – fluorescence
106-108 fs
Ph – phosforescence
1012-1017 fs
Nuclear coordinates
Femtosecond phenomena
time-resolved experiments
4
Conventional UV absorption spectrum
ade
absorption
gua
0
cyt
thy
Static spectrum: information is integrated over time
5
Ultra-short laser pulses
Transient spectrum: information is time resolved
Fluorescence spectrum
Time resolved spectra
1.0
static
0.8
transient
0.6
0.4
0.2
0.0
450
500
550
600
650
700
 (nm)
7
Transient (time-dependent) spectra: pump-probe
Mestdagh et al. J. Chem. Phys. 113, 240 (2000)
pump
t
w
Dt
Dt
+
and probe
td ~2000 fs
td < 200 fs
td < 200 fs
Pump
t = 60 fs
 = 618 nm
Probe
t = 6 fs
probe wavelength
 = 560 - 710 nm
Mathies et al. Science 240, 777 (1988)
absorption
excited state absorption (ionization)
0
transmission
0
transmission
1
1
spontaneous emission (fluorescence)
stimulated emission
2
1
1
Transmission due to
ground state depletion
0
Ground state absorption
Stimulated
emission
2
Excited state
absorption
0
14
Bacteriorhodopsin
15
geometry optimization
16
Topography of the potential energy surface
17
Topography of the excited-state potential energy surface
We want determine:
• minima
• saddle points
• minimum energy paths
• conical intersections
18
Newton-Raphson
A bit of basic mathematics: The Newton-Raphson’s Method
f(x)
Prove it!
f  xn 
xn1  xn 
f '  xn 
0
x
xR
x3
x2
x1
Numerical way to get the root of a function
19
Newton-Raphson
To find the extreme of a function, apply Newton-Raphson’s Method
to the first derivative
df/dx
f(x)
0
x
0
x
xe x3
xe
x2
x1
f '  xn 
xn1  xn 
f ' '  xn 
20
Geometry optimization
Taylor expansion:
E x
k 1
  E x   gx  x
Gradient vector:
Hessian matrix:
k
k T
k 1
1 k 1 k T
 x   x  x  Hx k x k 1  x k 
2
k
 r1 
x    , ri   xi , yi , zi 
 
rN 
 E / r1 
g x     


E / rN 
  2 E / r12
 2 E / r1r2   2 E / r1rN 
 2

2
2

E
/

r

r

E
/

r

2 1
2

H x   




 2
2
2 

E
/

r

r


E
/

r

N
1
N 
Szabo and Ostlund, Modern Quantum Chemistry, Appendix C
21
Geometry optimization
At xe,
g(xe) = 0
xe
xk
xe  x k  H 1 x k gx k 
Prove it!
If H-1 is exact: Newton-Raphson Method
If 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.
22
Problem 1:
• Get the gradient g
Numerical
Expensive, unreliable, however available for any method for which
excited-state energies can be computed
E  x1  E  x1  Dx   E  x1  Dx 

x1
2Dx
1 gradient = 2 x 3N energy calculations!
Analytical
Fast, reliable, but not generally available
Two ways to get the derivative of x2
dx 2
 2x
dx
dx 2  x  Dx    x  Dx 

dx
2Dx
2
2
23
Present situation of quantum chemistry methods
Methods allowing for excited-state calculations:
Method
MR-CISD
EOM-CC
SAC-CI
CC2 / ADC
CASPT2
MRPT2
CISD/QCISD
MCSCF
DFT/MRCI
OM2
TD-DFT
TD-DFTB
FOMO/AM1
Single/Multi
Reference
MR
SR
SR
SR
MR
MR
SR
MR
MR
MR
SR
SR
MR
Analytical
gradients













Coupling
vectors













Computational
effort
Typical implementation
Columbus
Aces2
Gaussian
Turbomole
Molpro
Gamess
Molpro / Gaussian
Columbus / Molpro
S. Grimme (Münster)
W. Thiel (Mülheim)
Turbomole
M. Elstner (Braunschweig)
Mopac (Pisa)
24
Problem 2:
• Get the Hessian H (or H-1)
Hessian has NxN = N2 elements
Normally second derivatives are computed numerically
Hessian 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)
Example: update in the BFGS method:
k
k 1
k
k 1 T 




x

x
x

x
1
1
T


H k  ΛH k 1Λ 
T
k
k 1
k
k 1 





x

x
g

g

 x k  x k 1 g k  g k 1 T 

Λk  1 
T
 x k  x k 1  g k  g k 1 


25
excited state relaxation
26
p
p*
The electronic configuration changes quickly after the photoexcitation
27
Minima in the excited states
E
“Spectroscopic”
minimum
Global
minimum
X
• “Spectroscopic” minima are close to the FC region
• Global minima often are counter-intuitive geometries
28
Minima in the excited states
7.0
6.5
V.Exc.
6.0
5.5
Energy (eV)
5.0
4.5
S2
S1
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
S0
0
Min S1
2
4
6
LIIC
8
10
MXS 3
29
Minima in the excited states
O
O
HN
Ground state minimum
HN
HC
S1 “spectroscopic” minimum
30
Relaxation in the excited states
O
NH
O
HC
(2)
0.6
NH
0.4
S2
1.45
1.0
0.8
1.40
0.6
0.4
S1-S2 Gap
0.2
1.35
0.2
0.0
0.0
0
50
100
150
1.60
Bond length (Å)
1.2
Energy (eV)
0.8
1.50
R(C6-N)
1.4
200
0
(d)
20
40
60
1.50
1.45
1.40
1.35
R(C2-C3)
R(C4-C5)
R(C2-O)
1.30
1.25
1.20
0
50
100
Time (fs)
150
200
80
100
S2  S1
S1  S2
12
1.55
Total number of hoppings
Fraction of trajectories
1.0
(c)
(b)
(1)
Bond length (Å)
(a)
10
8
6
4
2
0
0
20
40
60
80
100
Time (fs)
Barbatti et al., in Radiation Induced Molecular Phenomena in Nucleic Acid ( 2008)
31
Surface can have different diabatic characters
Merchan and Serrano-Andres, JACS 125, 8108 (2003)
32
Minima may have different diabatic characters
E
Change of diabatic character
np*
Adiabatic surface
p*
p
n
pp*
p*
p
n
X
33
Initial relaxation may involve several states
E
34
Relaxation keeping the diabatic character
Merchán et al. J. Phys. Chem. B 110, 26471 (2006)
35
Relaxation changing the diabatic character
1.732
[1.772]
Barbatti et al. J.Chem.Phys. 125, 164323 (2006)
36
In general, multiple paths are available
7
7
Energy (eV)
6
6
5
5
4
4
4
3
3
3
5
2
2
2
1
H3
1
2
3
4
5
1
6
S3
6
1
2
3
4
5
0
6
np*
4
4
4
3
3
3
2
2
2
B3,6
1
E3
0
1
2
3
4
5
0
6
7
7
6
6
pp*
5
1
2
3
4
5
4
3
3
2
4
5
6
p*
E8
6
0
0
1
2
3
4
5
6
1/2
dMW (amu Å)
np*
5
4
3
1
0
0
2
6
5
1
1
7
5
5
H3
0
0
6
pp*
4
1
7
7
p*
2
4
0
0
Energy (eV)
np*
6
pp*
0
Energy (eV)
7
2
2
1
E
6
1
0
S1
0
0
1
2
3
4
1/2
dMW (amu Å)
5
6
0
1
2
3
4
1/2
dMW (amu Å)
5
6
37
Common reaction paths: efficiency
R1
R1
X
C O
R2
Energy
R2
np*/cs
np*
C R
4
R3
pp*/cs
np*
Reaction path
R1
X
pp*
R2
R1
N
C R
4
R3
H
R2
n-1s
p-3s*
pp*/cs
p*
p-1s
38
The trapping effect
9H-adenine
170 fs
90
200 fs
Reaction path
0 fs
120 fs
2-pyridone
0
90
180
270
360
180
 (°)
 (°)
0
Energy
 (°)
Energy
180
90
0
Reaction path
0
90
180
 (°)
270
39
360
Radiationless decay: thymine
8
6
Energy (eV)
4
pp*
3
6
4
np*
np*/cs
6,3
T1
pp*
pp*/cs
np*
6
pp*
np*
pp*
6
4
pp*/cs
B
np*/cs
np*
out-of-plane O
E
pp*
pp*/cs
np*
E5
0
5
10
1/2
dMW (Å.amu )
Zechmann and Barbatti, J. Phys. Chem. A 112, 8273 (2008)
40
Radiationless decay: lifetime
NH2
H
N
N
N
pyrrole
Occupation
1.00
adenine
S3
S4
O
H
N
N
N
H
pyridone
S2
S1
S0
0.75
S1
S1
0.50
0.25
S2
S2
S0
S3
0.00
0
50
100
0
50
S0
100
0
50
100
150
Time (fs)
pp*
n-1s
p-3s*
p*
p-1s
pp*/cs
np*
np*/cs
41
excited-state intramolecular proton transfer
ESIPT
42
Proton Transfer in 2-(2'-Hydroxyphenyl)benzothiazole (HBT)
H O
N
proton transfer
S
enol
absorption
 (nm) 325
350
O
H
N
8000 cm
400
-1
450
S
keto
emission
500
550
600 650 700
und Einstellungen\schrieve\Desktop\goldegg\hbt-cw_Graph1
43
Elsaesser and Kaiser, Chem. C:\Dokumente
Phys. Lett.
128, 231 (1986)
ESIPT reaction schemes
H O
N
H
Dt
S1
pump
O
N
H O
N
emission
H O
N
S0
electronic
configuration
change
keto
form
reaction path
several modes contribute
44
DT/T0
0.005 signal at the keto wave number appears after only 30 fs
Emission
0.000
DT/T
solution
0.015
0.010
0.005
0.000
0
1
2
3
4
ps
Lochbrunner, Wurzer, Riedle, J. Phys. Chem. A 107 10580 (2003)
6
45
46
Internal conversion should play a role
47
ESIPT
gas phase
0.015
probe = 570 nm
Resolution: 30 fs
DT/T0
0.005
0.000
solution
0.015
0.010
0.005
0.000
0
1
2
3
4
ps
6
Schriever et al., Chem. Phys. 347, 446 (2008)
Barbatti et al., PCCP 11, 1406 (2009)
48
Next lecture
• Adiabatic approximation
• Non-adiabatic corrections
Contact
mario.barbatti@univie.ac.at
This lecture can be downloaded at
http://homepage.univie.ac.at/mario.barbatti/femtochem.html
lecture2.ppt
49