Femtochemistry: A theoretical overview
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Femtochemistry: A theoretical overviewFemtochemistry: A theoretical overview
Mario [email protected]
V – Finding conical intersections
This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture5.ppt
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2Antol et al. JCP 127, 234303 (2007)
pyridonepyridoneformamideformamide
Where are the conical intersections?
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Conical intersection Structure Examples
Twisted 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 intersectionsPrimitive conical intersections
X C
R1
R2
R3
R4
X C
R1
R2R3
R4
X C
R1
R2 R3
R4
C
R1R2
R3
H
C O
R1
R2
X Y
R1
R2
X
R1 R2
H
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(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: Conical intersections: Twisted-Twisted-pyramidalizedpyramidalized
Barbatti et al. PCCP 10, 482 (2008)
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(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: Conical intersections in rings: Stretched-Stretched-bipyramidalizedbipyramidalized
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The biradical character
Aminopyrimidine MXS CH2NH2+ MXS
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The biradical character
2 1*
S0 ~ (2)2
S1 ~ (2)1(1*)1
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One step back: single -bonds
Barbatti et al. PCCP 10, 482 (2008)
0 30 60 900
10
Rigid torsion (degrees)
2
2
CHCH22SiHSiH22
0 30 60 900
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Rigid torsion (degrees)
2
2
CHCH22CHCH22
2
0 30 60 900
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Rigid torsion (degrees)
CHCH22NHNH22++
0 30 60 900
10
Rigid torsion (degrees)
2
2
CHCH22CHFCHF
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One step back: single -bonds
0 30 60 900
10
Rigid torsion (degrees)
2
2
CC22HH44
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One step back: single -bonds
Michl and Bonačić-Koutecký, Electronic Aspects of Organic Photochem. 1990
The energy gap at 90° depends on the electronegativity difference () along
the bond.
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One step back: single -bonds
depends on:• substituents• solvation• other nuclear coordinates
For a large molecule is always possible to find an adequate geometric configuration that sets to the intersection value.
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Urocanic acid
• Major UVB absorber in skin• Photoaging • UV-induced immunosuppression
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Finding conical intersectionsFinding 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
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Penalty functionPenalty function
2
2
221 1ln
2 c
EEcc
EEf 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
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Gradient projection methodGradient 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
g
gg 2
Gradient E2
JTIJIJ
TIJp E
IJ hhggIg
Projection of gradient of EJ
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Gradient projection methodGradient 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
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Lagrange-Newton MethodLagrange-Newton Method
A simple example:
Optimization of f(x)Subject to (x) = k
Lagrangian function:
kxxfxL )()()(
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
xL
xxL
xL
xxlxxL
0
, 020
2000
xL
x
LLxlxxL
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Lagrange-Newton MethodLagrange-Newton Method
0
, 020
2000
xL
xxL
xL
xxlxxL
0
, 020
2000
xL
x
LLxlxxL
k0 0
x 0
kx
Lx
x
xx
L
0
0
0
020
2
0
xL
xL
xxL
00
20
2
kxx
00
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Lagrange-Newton MethodLagrange-Newton Method
kx
Lx
x
xx
L
0
0
0
020
2
0
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.
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Lagrange-Newton MethodLagrange-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
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Lagrange-Newton MethodLagrange-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
000k
0h
0g
khg
000
00
2
1
†
†
†JI
IJ
IJ
IJ
IJIJIJ
EE
LL
kx
Lx
x
xx
L
0
0
0
020
2
0
Compare with the simple one-dimensional example:
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Lagrange-Newton MethodLagrange-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
000k
0h
0g
khg
000
00
2
1
†
†
†JI
IJ
IJ
IJ
IJIJIJ
EE
LL
λq ,,, 21Solve these equations for
Update λq ,,, 21
Repeat until converge.
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Comparison of methodsComparison 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)
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Crossing of states with different multiplicitiesCrossing of states with different multiplicitiesExample: thymineExample: thymine
Serrano-Pérez et al., J. Phys. Chem. B 111, 11880 (2007)
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Crossing of states with different multiplicitiesCrossing of states with different multiplicities
Lagrangian function to be optimized:
M
iiiJIIIJ KEEEL
11
Now the equations are:
JI
IJ
IJ
IJ
IJIJ
EE
LL
λ
q
0k
g
kg
1†
†
0
00
0IJH
Different from intersections between states with the same multiplicity, when different
multiplicities are involved the branching space is one
dimensional.
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Three-states conical intersectionsExample: cytosine
Kistler and Matsika, J. Chem. Phys. 128, 215102 (2008)
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Conical intersections between three statesConical intersections between three states
Lagrangian function to be optimized:
M
iiikJIkIJJkJIIIJK KHHHEEEEEL
132121
This leads to the following set of equations to be solved:
K
0
λ
ξ
ξ
q
000k
000h
000g
khg
E
LL IJIJ
†
†
†
Matsika and Yarkony, J. Chem. Phys. 117, 6907 (2002)
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29Devine et al. J. Chem. Phys. 125, 184302 (2006)
Example of application: photochemistry of imidazoleExample of application: photochemistry of imidazoleFast H elimination
Slow H elimination
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30Devine et al. J. Chem. Phys. 125, 184302 (2006)
Example of application: photochemistry of imidazoleExample of application: photochemistry of imidazoleFast H elimination
Slow H elimination
Fast H elimination: * dissociative state
Slow H elimination: dissociation of the hot ground state formed by internal conversion
How are the conical intersectionsin imidazole?
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Predicting conical intersections: ImidazolePredicting conical intersections: Imidazole
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32Barbatti et al., J. Chem. Phys. 130, 034305 (2009)
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2.5 3.0 3.5 4.0 4.5 5.0 5.5
3.0
3.5
4.0
4.5
5.0E
ne
rgy
(eV
)
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!
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Pathways to the intersections
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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)
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
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Next lectureNext lecture
• Transition probabilities
This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture5.ppt