View
22
Download
0
Category
Preview:
DESCRIPTION
Femtochemistry: A theoretical overview. V – Finding conical intersections. Mario Barbatti mario.barbatti@univie.ac.at. This lecture can be downloaded at http://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture5.ppt. Where are the conical intersections?. formamide. pyridone. - PowerPoint PPT Presentation
Citation preview
Femtochemistry: A theoretical overviewFemtochemistry: A theoretical overview
Mario Barbattimario.barbatti@univie.ac.at
V – Finding conical intersections
This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture5.ppt
2Antol et al. JCP 127, 234303 (2007)
pyridonepyridoneformamideformamide
Where are the conical intersections?
3
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
4
5
(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)
6
(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
7
The biradical character
Aminopyrimidine MXS CH2NH2+ MXS
8
The biradical character
2 1*
S0 ~ (2)2
S1 ~ (2)1(1*)1
9
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
10
Rigid torsion (degrees)
2
2
CHCH22CHCH22
2
0 30 60 900
10
Rigid torsion (degrees)
CHCH22NHNH22++
0 30 60 900
10
Rigid torsion (degrees)
2
2
CHCH22CHFCHF
10
One step back: single -bonds
0 30 60 900
10
Rigid torsion (degrees)
2
2
CC22HH44
11
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.
12
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.
13
Urocanic acid
• Major UVB absorber in skin• Photoaging • UV-induced immunosuppression
14
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
15
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
16
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
17
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
18
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
19
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
20
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.
21
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
22
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:
23
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.
24
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)
25
Crossing of states with different multiplicitiesCrossing of states with different multiplicitiesExample: thymineExample: thymine
Serrano-Pérez et al., J. Phys. Chem. B 111, 11880 (2007)
26
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.
27
Three-states conical intersectionsExample: cytosine
Kistler and Matsika, J. Chem. Phys. 128, 215102 (2008)
28
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)
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
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?
31
Predicting conical intersections: ImidazolePredicting conical intersections: Imidazole
32Barbatti et al., J. Chem. Phys. 130, 034305 (2009)
33
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!
34
Pathways to the intersections
35
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?
36
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
37
38
Next lectureNext lecture
• Transition probabilities
Contactmario.barbatti@univie.ac.at
This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture5.ppt
Recommended