Ultrafast processes in molecules
31
Ultrafast processes in molecules Mario Barbatti [email protected] – Non-crossing rule and conical intersection
description
Ultrafast processes in molecules. IV – Non-crossing rule and conical intersections. Mario Barbatti [email protected]. The non-crossing rule. - PowerPoint PPT Presentation
Transcript of Ultrafast processes in molecules
Ultrafast processes in molecules
Mario [email protected]
IV – Non-crossing rule and conical intersections
2
The non-crossing rule
von Neumann and Wigner, Z. Phyzik 30, 467 (1929)Teller, JCP 41, 109 (1937)
“For diatomics, the potential energy curves of the electronic states of the same symmetry species cannot cross as the
internuclear distance is varied.”
3
Simple argument
Suppose a two-level molecule whose electronic Hamiltonian is H(R), where R is the internuclear coordinate.
Given a basis of unknown orthogonal functions f1 and f2, we want to solve the Schrödinger equation
Check it!
2/1212
2221122111 4
21
21 HHHHHE
2/1212
2221122112 4
21
21 HHHHHE
0 iEH 2,12211 icc iii ff
0det2221
1211
EHH
HEH *jijiij HRHH ff
The energies are given by
4
2/1212
2221122111 4
21
21 HHHHHE
2/1212
2221122112 4
21
21 HHHHHE
Simple argument
5
2/1212
2221122111 4
21
21 HHHHHE
2/1212
2221122112 4
21
21 HHHHHE
21 EE
00
12
2211
HHH (i)
(ii)
It is unlikely (but not impossible) that by varying the unique parameter R conditions (i) and (ii) will
be simultaneously satisfied.
Simple argument
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Rigorous proof
Naqvi and Brown, IJQC 6, 271 (1972)
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Conical intersections
• In diatomics the unique parameter R is not enough to satisfy the two conditions for crossing.
• In polyatomics there are 3N-6 internal coordinates!
What does happen if the molecule has more than one degree of freedom?
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Conical intersections
Suppose a two-level molecule whose electronic Hamiltonian is H(R), where R are the nuclear coordinates.
Given a basis of unknown orthogonal functions f1 and f2, we want to solve the Schrödinger equation
2/1212
2221122111 4
21
21 HHHHHE
2/1212
2221122112 4
21
21 HHHHHE
0 ie EH 2,12211 icc iii ff
0det2221
1211
EHH
HEH *jijeiij HHH ff R
The energies are given by
9
Conical intersections
In a more compact way:
2/1212
22,1 HE
where
221121 HH 22112
1 HH and
A degeneracy at Rx will happen if
0 XR
012 XH R
In general, two independent coordinates are necessary to tune these conditions.
10
Conical intersections
In a more compact way:
2/1212
22,1 HE
where
221121 HH 22112
1 HH and
Expansion in first order around Rx for :
Rs
RGRG
RRRR
X
XX
XX HH
12
21
2211
2121
21
XiiXi H RG
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Conical intersections
In a more compact way:
2/1212
22,1 HE
where
221121 HH 22112
1 HH and
In first order around Rx each of these terms are:
RsRGG XXX1221 Xii
Xi H RG
RgRGG XXX1221
RfR XXHH 121212
And the energies in a point RX + R are in first order:
2/1212
212122,1 RfRgRs XXXE
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Conical intersections
2/1212
212122,1 RfRgRs XXXE
Writting
then
sincossincos
RsRsfyfRgxgR
yx
RsRfRg
yxsyfxg ˆˆ;ˆ;ˆ srfg
2/122222,1 yfxgysxsE yx
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Conical intersections
2/1212
212122,1 RfRgRs XXXE
2/122222,1 yfxgysxsE yx
Atchity, Xantheas, and Ruedenberg, J. Chem. Phys. 95, 1862 (1991)
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Conical intersections
2/1212
212122,1 RfRgRs XXXE
What does happen if the molecule is distorted along a direction that is perpendicular to g and f?
Linear approximation fails
Crossing seam
XE 122,1 sE
RperpendRx
E1
E2
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Conical intersections
What does happen if the molecule is distorted along a direction that is parallel to g or f?
Linear approximation fails
Crossing seam
E
RparallelRx
E1
E2
2/1212
212122,1 RfRgRs XXXE
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Branching space
• Starting at the conical intersection, geometrical displacement in the „branching space“ lifts the degeneracy linearly.
• The branching space is the plane defined by the vectors g and f.
• Geometrical displacements along the other 3N-8 internal coordinates keep the degeneracy (in first order). These coordinate space is called „seam“ or „intersection“ space.
Note that XXX EEH 12211212 hf
2112 h Non-adiabatic coupling vector
For this reason the branching space is also referred as g-h space.
See the proof, e.g., in Hu at al. J. Chem. Phys. 127, 064103 2007 (Eqs. 2 and 3)
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• The coupling vectors define one of the directions of the branching space around the conical intersections, which is important for the localization of these points of degeneracy.
Why are non-adiabatic coupling vectors important?
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Conical intersections are not rare“When one encounters a local minimum (along a
path) of the gap between two potential energy surfaces, almost always it is the shoulder of a
conical intersection. Conical intersections are not rare; true avoided intersections are much less
likely.”
Truhlar and Mead, Phys. Rev. A 68, 032501 (2003)
R
E
e << 1 a.u.
R
E
0~
2/432
N
tot
tot
CIVminV e ~ O(1) is the density of zeros in the
Hel matrix.
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Conical intersections are connected
Crossing seam
Minimum on the crossing seam (MXS)
Energy
R
4
5
6
7
8
-100
-50
0
50
100
4060
80100
120
Ene
rgy
(eV
)
(degrees)
degrees)
Ethylidene
Pyramidalized
H-migration
Barbatti, Paier and Lischka, J. Chem. Phys. 121, 11614 (2004)
C3V
Crossing seam in ethylene
20 40 60 80 100 120 1400
10
20
30
0
10
20
30
0
10
20
30
Hydrogen Migration (degrees)
Twisted
planarP
yram
idal
izat
ion
(deg
rees
) Ethylid. MXS
Pyramid. MXS
H-mig. c.i.
Example of dynamics results
Torsion + Pyramid.
H-migration
Pyram. MXS
Ethylidene MXS
~7.6 eV
t ~ 100-140 fs
60%
11%23%
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Conical intersections are distorted
2/12222
2,1 221 yxyxyxdE gh
yxgh
It can be rewritten as a general cone equation (Yarkony, JCP 114, 2601 (2001)):
2
22
ghgh d
hg asymmetry parameter
2/122 hg ghd pitch parameter
ggs
ghx d
hhs
ghy d
tilt parameters
2/1212
212122,1 RfRgRs XXXE
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h01 g01
Energy
Example: pyrrole
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h01 g01
Energy
Photoproduct depends on the direction that the molecule leaves the intersection
26Burghardt, Cederbaum, and Hynes, Faraday. Discuss. 127, 395 (2004)
Example: protonated Schiff Base
In gas phase In water
NH2+
r
Q
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0 20 40 60 80 100 120 140 160 180 200-202468
10
0 20 40 60 80 100 120 140 160 180 200-202468
10ener
gy (e
V)
S0 S1 S2
time (fs)
gashpase
aqua
Ruckenbauer, Barbatti, Niller, and Lischka, JPCA 2009
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29
0 50 100 150 200 250 300 350 400 450 5000
30
60
90
120
150
180
210
240
270
300
330
360
cent
ral t
orsi
on (°
)
time (fs)0 50 100 150 200 250 300 350 400 450 500
0
30
60
90
120
150
180
210
240
270
300
330
360
cent
ral t
orsi
on (°
)
time (fs)
gasphase water
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Intersections don’t need to be conical!
Bersuker, The Jahn Teller effect, 2006
Yarkony, Rev. Mod. Phys. 68, 985 (1996)
