Femtochemistry: A theoretical overview
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Femtochemistry: A theoretical overviewFemtochemistry: A theoretical overview
Mario [email protected]
IV – Non-crossing rule and conical intersections
This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture4.ppt
2
The non-crossing ruleThe 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 argumentSimple 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 1 and 2, we want to solve the Schrödinger equation
Check it!
2/12
122
221122111 421
21
HHHHHE
2/12
122
221122112 421
21
HHHHHE
0 iEH 2,12211 icc iii
0det2221
1211
EHH
HEH *jijiij HRHH
The energies are given by
5
2/12
122
221122111 421
21
HHHHHE
2/12
122
221122112 421
21
HHHHHE
21 EE
0
0
12
2211
H
HH (i)
(ii)
It is unlikely (but not impossible) that by varying the unique parameter R conditions (i) and (ii) will
be simultaneously satisfied.
Simple argumentSimple argument
6
Rigorous proofRigorous proof
Naqvi and Brown, IJQC 6, 271 (1972)
Suppose that the degeneracy occurs at R0:
0201 RERE
R0
E
R
R+R
E1 = E0 + E1
E2 = E0 + E2
21 EE
To have a “crossing” not only the degeneracy condition is necessary. It is also need:
7
Rigorous proofRigorous proof
011 EH e
Schrödinger equation for the first state:
Small displacement from R0 to R0 + R:
1001
10101
00
EERRE
RR
HHRRH e
Neglecting terms in 2:
0011100 EHEH
Multiply by to the left and integrate over the nuclear coordinates: 0*2 R
01
021
01
02 EH Prove it!
8
Rigorous proofRigorous proof
After repeating the same steps for the second state:
01
021
01
02 EH
01
022
01
02 EH
Because of the second condition
01
022
01
021 EE
21 EE
Therefore 001
02 H
Expanding in Taylor to the first order
RRH
H 00
102 RH
9
Rigorous proofRigorous proof
Having a crossing between two states requires two conditions:
001
02 RH
0201 RERE (i) degeneracy
(ii) crossing
It is unlikely (but not impossible) that by varying the unique parameter R conditions (i) and (ii) will
be simultaneously satisfied.(a) If only (i) is satisfied it is not a crossing, but a complete state degeneracy
for any R.
012 (for any R)
001
021
01
02 EH
(b) If the states have different symmetries, (ii) is trivially satisfied because:
10
Conical intersectionsConical 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?
11
Conical intersectionsConical 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 1 and 2, we want to solve the Schrödinger equation
2/12
122
221122111 421
21
HHHHHE
2/12
122
221122112 421
21
HHHHHE
0 ie EH 2,12211 icc iii
0det2221
1211
EHH
HEH *jijeiij HHH R
The energies are given by
12
Conical intersectionsConical intersections
In a more compact way:
2/1212
22,1 HE
where
221121
HH 221121
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.
13
Conical intersectionsConical intersections
In a more compact way:
2/1212
22,1 HE
where
221121
HH 221121
HH and
Expansion in first order around Rx for :
Rs
RGRG
RRRR
X
XX
XX HH
12
21
2211
2121
21
XiiXi H RG
14
Conical intersectionsConical intersections
In a more compact way:
2/1212
22,1 HE
where
221121
HH 221121
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:
2/12
12
2
12122,1 RfRgRs XXXE
15
Conical intersectionsConical intersections
2/12
12
2
12122,1 RfRgRs XXXE
Writting
then
sincos
sin
cos
RsRs
fyfR
gxgR
yx
Rs
Rf
Rg
yxsyfxg ˆˆ;ˆ;ˆ srfg
2/122222,1 yfxgysxsE yx
16
Conical intersectionsConical intersections
2/12
12
2
12122,1 RfRgRs XXXE
2/122222,1 yfxgysxsE yx
Atchity, Xantheas, and Ruedenberg, J. Chem. Phys. 95, 1862 (1991)
17
Conical intersectionsConical intersections
2/12
12
2
12122,1 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
18
Conical intersectionsConical 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
2/12
12
2
12122,1 RfRgRs XXXE
19
Branching spaceBranching 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 keeps 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)
20
• 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?Why are non-adiabatic coupling vectors important?
21
Conical intersections are not rareConical 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
<< 1 a.u.
R
E
0~
2/432
N
tot
tot
CIV
minV ~ O(1) is the density of zeros in the Hel matrix.
22
Conical intersections are connectedConical 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
)
(degre
es)
degrees)
EthylideneEthylidene
PyramidalizedPyramidalized
H-migrationH-migration
Barbatti, Paier and Lischka, J. Chem. Phys. 121, 11614 (2004)
CC3V3V
Crossing seam in ethylene
Torsion Torsion + Pyramid.+ Pyramid.
H-migrationH-migration
Pyram. MXSPyram. MXS
Ethylidene MXSEthylidene MXS
~7.6 eV~7.6 eV
~ 100-140 fs~ 100-140 fs
60%60%
11%11%23%23%
26
Conical intersections are distortedConical intersections are distorted
2/1
22222,1 22
1yxyxyxdE gh
yxgh
It can be rewritten as a general cone equation (Yarkony, JCP 114, 2601 (2001)):
2
22
ghgh d
hg asymmetry parameter
2/122hg ghd pitch parameter
ggs
ghx d
hhs
ghy d
tilt parameters
2/12
12
2
12122,1 RfRgRs XXXE
29Burghardt, Cederbaum, and Hynes, Faraday. Discuss. 127, 395 (2004)
Example: protonated Schiff Base
In gas phase In water
NH2+
r
30
0 20 40 60 80 100 120 140 160 180 200-2
0
2
4
6
8
10
0 20 40 60 80 100 120 140 160 180 200-2
0
2
4
6
8
10
energ
y (
eV
)
S0 S1 S2
time (fs)
gashpase
aqua
Ruckenbauer, Barbatti, Niller, and Lischka, JPCA 2009
32
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)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
33
Intersections don’t need to be conical!Intersections don’t need to be conical!
Bersuker, The Jahn Teller effect, 2006
Yarkony, Rev. Mod. Phys. 68, 985 (1996)
34
Next lectureNext lecture
• Finding conical intersections
This lecture can be downloaded athttp://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture4.ppt