Ultrafast processes in molecules Mario Barbatti [email protected] – Transient spectra and excited states
Feb 23, 2016
SingletTriplet
avoided crossing 102-104 fsconical intersection 10-102 fs
PA – photoabsorption 1 fs
VR – vibrational relaxation 102-105 fs
Energy (eV)
0
10
Nuclear coordinates
PhFl
PA
VR
Fl – fluorescence 106-108 fsintersystem crossing 105-107 fs
Ph – phosforescence 1012-1017 fs
Femtosecond phenomena
4
time-resolved experiments
5
Static spectrum: information is integrated over time
0
absorptionade
gua
thy
cyt
Ultra-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
Fluo
resc
ence
spe
ctru
m
(nm)
Time resolved spectra
static
transient
Transient (time-dependent) spectra: pump-probe
Mestdagh et al. J. Chem. Phys. 113, 240 (2000)
w
t
Dt
+
Dt
pump
and probe
td ~2000 fs
td < 200 fs
td < 200 fs
Mathies et al. Science 240, 777 (1988)
probe wavelength
= 618 nm
t = 60 fs
= 560 - 710 nm
t = 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
Bacteriorhodopsin
16
geometry optimization
17
Topography of the potential energy surface
18
Topography of the excited-state potential energy surface
We want determine:• minima• saddle points• minimum energy paths• conical intersections
19
Newton-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-Raphson
21
kkkTkkkkTkkk EE xxxHxxxxxgxx 1111
21
Taylor expansion:
221
2
22
212
21
221
221
2
//
/////
NN
N
EE
EEEEE
rrr
rrrrrrrr
xH
Hessian matrix:
NE
E
r
rxg
/
/ 1
Gradient vector:
iiii
N
zyx ,,,1
r
r
rx
Geometry optimization
Szabo and Ostlund, Modern Quantum Chemistry, Appendix C
22
Geometry 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:
• Get the gradient g
Numerical
Expensive, unreliable, however available for any method for which excited-state energies can be computed
x
xxExxExxE
DDD
211
1
1
1 gradient = 2 x 3N energy calculations!
Analytical
Fast, reliable, but not generally available
xdxdx 2
2
x
xxxxdxdx
DDD
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 methods
Methods allowing for excited-state calculations:
25
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
p p*
The electronic configuration changes quickly after the photoexcitation
28
Minima 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 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
Ene
rgy
(eV
)
LIICMin S1
MXS 3
V.Exc.
S0
S1
S2
30
Minima in the excited states
NH
O
NHCH
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
Tota
l num
ber o
f 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)
Frac
tion
of tr
ajec
torie
s
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
Tota
l num
ber o
f 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)
Frac
tion
of tr
ajec
torie
s
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
Tota
l num
ber o
f 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)
Frac
tion
of tr
ajec
torie
s
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 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 characters
33
Minima may have different diabatic characters
E
X
np*
pp*
Change of diabatic character
Adiabatic surface
np
p*
np
p*
34
Initial relaxation may involve several states
E
35
Relaxation keeping the diabatic character
Merchán et al. J. Phys. Chem. B 110, 26471 (2006)
36
Relaxation 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
0 1 2 3 4 5 60
1
2
3
4
5
6
7
0 1 2 3 4 5 60
1
2
3
4
5
6
7
0 1 2 3 4 5 60
1
2
3
4
5
6
7
0 1 2 3 4 5 60
1
2
3
4
5
6
7
0 1 2 3 4 5 60
1
2
3
4
5
6
7
0 1 2 3 4 5 60
1
2
3
4
5
6
7
0 1 2 3 4 5 60
1
2
3
4
5
6
7
0 1 2 3 4 5 60
1
2
3
4
5
6
7
dMW
(amu1/2Å)
6S1
np*
dMW
(amu1/2Å)
E8
p*
4H3
p*
dMW
(amu1/2Å)
Ene
rgy
(eV
)
2E
pp*
B3,6
np*
Ene
rgy
(eV
)
2H3
pp*E
nerg
y (e
V)
E3
pp*
4S3
np*
In general, multiple paths are available
38
Common reaction paths: efficiency
pp*/csnp*
X C
R1
R2 R3
R4
np*/cs
Ener
gy
np*
Reaction path
C O
R1
R2
pp*/cs
pp*X C
R1
R2 R3
R4
p-1sp*p-3s*
n-1s
N H
R1
R2
39
0 90 180 270 3600
90
180
(°)
(°
)
0 90 180 270 3600
90
180
(°)
(°
)
0 fs120 fs
170 fs200 fs
The trapping effect9H-adenine
Ener
gy
Reaction path
Ener
gy
Reaction path
0 90 180 270 3600
90
180
(°)
(°)
2-pyridone
Ener
gy
Reaction path
Ener
gy
Reaction path
40
4
6
8
4
6
0 5 10
4
6
3T1
pp*/cspp*
np*
Ene
rgy
(eV
)
6E
pp*/cspp*
np*out-of-plane O
np*/cspp*
np*
dMW
(Å.amu1/2)
E5
pp*/cspp*
np*
6,3B
np*/cspp*
np*
Radiationless decay: thymine
Zechmann and Barbatti, J. Phys. Chem. A 112, 8273 (2008)
41
Radiationless decay: lifetime
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
adenineN
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
adenineN
N
NH2
NH
N
adenineN
N
NH2
NH
N NH O
pp*/cs
pp*
pp*/cs
pp*
np*/csnp* np*/csnp*p-1s
p*p-3s*n-1s
p-1sp*p-3s*
n-1s
42
excited-state intramolecular proton transferESIPT
43
Proton Transfer in 2-(2'-Hydroxyphenyl)benzothiazole (HBT)
325 350 400 450 500 550 600 650 700
Elsaesser and Kaiser, Chem. Phys. Lett. 128, 231 (1986)
44
ESIPT reaction schemes
pump
ketoform
NOH
S1
S0
emission
DtN
OHNOH
NOH
reaction path
electronicconfigurationchange
several modes contribute
45
0.000
0.005
DT/T0
0.015
0 1 2 3 4 ps 6
0.000
0.005
0.010
0.015
gas phase
solution
DT/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
0.000
0.005
DT/T0
0.015
0 1 2 3 4 ps 6
0.000
0.005
0.010
0.015
gas phase
solution
probe = 570 nmResolution: 30 fs
• Barbatti, Aquino, Lischka, Schriever, Lochbrunner, Riedle, PCCP 11, 1406 (2009)
ESIPT: environment effects
ESIPT: QM/MM simulations
• Ruckenbauer, Barbatti, Lischka, unpublished
50
Next lecture
• Adiabatic approximation• Non-adiabatic corrections