Femtochemistry: A theoretical overviewFemtochemistry: A theoretical overview
Mario [email protected]
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
This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture2.ppt
