Holding Computations of Conical Intersections
to a Gold Standard
The Laser Spectroscopy Facility
Department of Chemistry
Conical Intersections in Chemistry
M. A. Robb and co-workers.
JAHN-TELLER THEOREM
For any non-linear molecule
in a degenerate electronic state,
there exists a displacement of the
nuclei along at least one
non-totally symmetric normal
coordinate, that gives rise to a
distortion of the molecular
geometry with a concomitant
lowering of the energy.
CONICAL INTERSECTIONS
Connecting the Jahn-Teller Parameters and the PES
 e ,i 
Di 
1 2

i
2 c
1
k i2
2
i  3 2
g ii
Ki 

a e
Linear J-T coupling constant of mode i
Quadratic J-T coupling constant of mode i
i
Spin-orbit coupling parameter (cm-1)
with
 i  D i   e ,i
J-T stabilization energy due to mode i
1
2
 total    i   iB
i
Equilibrium vibrational frequency (cm-1) of mode i
 iB  2 Di e,i Ki
Total J-T stabilization energy
Barrier to pseudorotation
Jahn-Teller Active Molecules
•Linear Jahn – Teller coupling - h(ωe)+ l (D)
•Quadratic Jahn – Teller coupling - q (K)
•Spin-orbit coupling - SO (aξe)
•Quantizing the nuclear moton - T
•Multi-mode effects – Di, Ki, ωei
Jahn-Teller Active Radicals
Cyclopentadienyl Radicals
Benzene Cations
C5X5
C6X3Y3
X = H or D
Cycloheptatrienyl Radical
X = H or F, Cl, Br
Y = H or F, Cl, Br
~ 2
X E1g (D6h) or
~ 2 "
X E (D3h)
Silver Trimer
C7X7
Ag3
~2 "
X E (D5h)
X=H
~2 "
X E 2 (D7h)
~
X 2E' (D3h)
Brief History of Studies of Jahn-Teller
Effect in Cyclopentadienyl
Experimental
Spectroscopy
Ab Initio Theory
(stabilization energy in cm-1)
Thrush, 1956
Liehr, 1956 (560)
Liebling & McConnell, 1965
Snyder, 1960 (728)
Carrington, et al., 1965
Hobey & McLachlan, 1960 (495)
Porter & Ward, 1968
Borden & Davidson, 1979 (2484)
Englman & Ramsey, 1970
Meyer, et al., 1979 (5072)
Purins & Feeley, 1973
Bearpark, Robb, & Yamamoto, 1999 (2147)
Engelking & Lineberger, 1977
Cunha & Canuto, 1999 (1614)
Nelson, et al., 1983
Kiefer, et al., 2001 (1655)
Yu, et al., 1988, 1993
Zilberg & Hass, 2002 (2554)
Bernstein, et al., 1995, 1999
Molecular Orbitals involved in
Jahn-Teller Distortion of C5H5
C2v
Diene
Allyl
Distortion
Distortion
D5h
C2v
Pseudorotation around the
C5H5 PES
E
Ra
Rb
Ra
E
Rb
Cyclopentadienyl Excitation Spectrum
C5H5
C5D5
C5H5 Emission from Origin
Experimental
82
Ab initio fundamentals
and overtones
131141
142
21
31 42
3
4
132
22
Fit fundamentals
and overtones
j=1/2
nj 1
2
5
6
7
8
9
10
11 12
Ab initio Jahn-Teller
Fit Jahn-Teller
Simulation
Experimental
0
500
1000
1500
-1
E (cm )
2000
2500
C5H5 Emission from 111
Experimental
21
Ab initio fundamentals
and overtones
Fit fundamentals
and overtones
j=1/2
nj 1
j=3/2
nj 1
82
71
2
2
3
4
5
3
22
6
7
4
8
5
9
6
10
7
11 12
8 9 10 11
Ab initio
Jahn-Teller
Fit Jahn-Teller
Simulation
Experimental
0
500
1000
1500
-1
E (cm )
2000
2500
C5H5 Emission from 121
Experimental
Ab initio fundamentals
and overtones
132
82
142
21
Fit fundamentals and overtones
j=1/2
nj 1
j=3/2
nj 1
2
3
2
4
5
3
6
7
4
8
5
9
6
10
7
11 12
8 9 10 11
Ab initio
Jahn-Teller
Fit Jahn-Teller
Simulation
Experimental
0
500
1000
1500
-1
E (cm )
2000
2500
C5D5 Emission From Origin
Experimental
Ab initio fundamentals
and overtones
82
2142
42
142
21
44
132
2182
84
22
Fit fundamentals
and overtones
j=1/2
nj 1
2
3
4
5
6
Ab initio Jahn-Teller
Fit Jahn-Teller
Simulation
Experimental
0
500
1000
1500
-1
E (cm )
2000
2500
Jahn-Teller Parameters
C5H5
calculated
experimental
GRHF
CASSCF (minimum)
CASSCF (intersection)
mode
ωi
Di
ρimin
εi
ωi
Di
ρimin
εi
Di
ρimin
εi
9
-
-
-
-
3030
<0.01
<0.01
<1
<0.01
<0.01
<1
10
1320
0.36
0.14
477
1411
0.68
0.18
959
0.98
0.26
1387
11
1041
0.57
0.19
594
1058
0.34
0.15
360
0.48
0.18
509
12
872
0.19
0.12
166
815
0.19
0.13
155
0.30
0.16
245
εtotal
1237
1474
2147
C5D5
experimental
calculated
GRHF
CASSCF (minimum)
CASSCF (intersection)
mode
ωi
Di
ρimin
εi
ωi
Di
ρimin
εi
Di
ρimin
εi
9
-
-
-
-
2237
<0.01
<0.01
<1
<0.01
<0.01
<1
10
1353
0.63
0.18
852
1378
0.87
0.21
1199
1.24
0.25
1719
11
861
0.39
0.17
336
836
0.36
0.17
301
0.51
0.20
431
12
-
-
-
-
716
<0.01
<0.01
<1
<0.01
<0.01
4
εtotal
1188
1500
2147
Cyclopentadienyl Geometric Distortion
k=
0
-1
1
-2
S
k
 S
0
2
 4k

 
 5

 S cos
ΔS
(Å/rad.)
Symmetry
coordinate
C5H5
exp
C5D5
C5(H/D)5
calc
C-C-C bend
0.012
0.0093
0.0080
C-C stretch
0.059
0.059
0.066
C-C-H bend
0.022
0.013
0.020
C-H stretch
~
<0.001
~<0.001
<0.001
~
Benzene Cation Experimental Results
C6F6+, C6H3F3+ LIF jet-cooled excitation and emission spectra T. A. Miller and
V. E. Bondybey, in Molecular Ions: Spectroscopy, Structure, and Chemistry (North-Holland,
1983), The Jahn-Teller Effect in Benzenoid Cations: Theory and Experiment, pp. 201-229.
ZEKE and MATI Spectroscopy
C. H. Kwon and M. S. Kim, J. Chem. Phys. 120, 11578 (2004).
C6H6+, C6D6+
ZEKE and MATI Spectroscopy
R. Linder, K. Müller-Dethlefs, E. Wedum, K. Haber, and E. R. Grant, Science 271, 1698 (1996).
R. Linder, Dissertation, TU Müchen, 1996.
C. H. Kwon, H. L. Kim and M. S. Kim, J. Chem. Phys. 119, 4305 (2003).
A. B. Burrill, Y. K. Chung, H. A. Mann, and P. M. Johnson, J. Chem. Phys. 120, 8587 (2004).
IR Spectroscopy of Ar·C6(H/D)6+
R. G. Satink, H. Piest, G. von Helden, and G. Meijer, J. Chem. Phys. 111, 10750 (1999); J.
Bakker, R. G. Satink, G. von Helden, and G. Miejer, Phys. Chem. Chem. Phys. 4, 24
(2002); J. Bakker, L. Mac Aleese, R. G. Satink, G. von Helden, and G. Meijer,
unpublished results.
Computation
J. Eiding, R. Schneider, W. Domcke, H. Koppel, and W. von Neissen, Chem. Phys. Lett. 177,
345 (1991).
B. E. Applegate and T. A. Miller, J. Chem. Phys. 117, 10654 (2002).
A. Avoird and V. F. Lotrich, J. Chem. Phys. 120, 10069 (2004).
BENZENE CATION PES
Benzene Cation Pseudorotation
1
1
1
1
1
1
C6H6+ ZEKE Spectrum
00
Ab initio fundamentals
and overtones
81
21
41
61
71
82
Fit fundamentals
and overtones
Ab initio quadratic
Jahn-Teller
201
(e1g, e1u, e2u)
111
191
202
Fit and split
quadratic
Jahn-Teller
nj 1
j=1/2
2
nj
1 2
j=3/2
3
4
3
Ab initio linear JahnTeller (e2g)
4
5 6
5
6
7
7
8
x30
x30
Fit linear Jahn-Teller
Simulation
Experimental
0
500
cm-1
1000
1500
9
C6D6+ ZEKE Spectrum
00
Ab initio fundamentals
81
and overtones
41
71
82
101
21
61
4181
Fit fundamentals
and overtones
Ab initio quadratic
Jahn-Teller
201
(e1g, e1u, e2u)
202
191
111
Fit and split
quadratic
Jahn-Teller
nj 1
j=1/2
nj
1
j=3/2
2
2
3
Ab initio linear JahnTeller (e2g)
3
4
4
5
6 7
5
6
8
7 8
9
9 10
10
11 12
x4
x4
Fit linear Jahn-Teller
Simulation
Experimental
0
500
cm-1
1000
1500
400
600
800
1000
cm-1
1400
1200
1400
191+|1/2, 2>
1200
141+|1/2, 2>
4181
1600
3
1600
82141
6171
2141
131
81141
91
41+|3/2, 3>
41+|3/2, 4>
81191
141+|3/2, 1>
141+|3/2, 2>
201+|3/2, 2>*
201+|3/2, 1>*
2
21191
21101
2
41+|3/2, 3>
41+|3/2, 4>
1000
101
141
191*
|1/2, 2>*
41*
1
6171
1
41+|1/2, 3>
800
141+|3/2, 1>
141+|3/2, 2>
Ar-C6D6+
191*
4181*
101*
600
141
400
201+|3/2, 1>*, 81201
201+|3/2, 2>*, 81201
41 *
Benzene
Cation
IR
Spectra
81201
Ar-C6H6+
3
4
1800
4
1800
Benzene Cation Jahn-Teller Energy
Stabilization and Geometric Distortion
S
Exp
εT
εB
Minimum
Intersection
εT
εB
εT
εB
726
1
757
-9
1542
62
+
821
3
1019
5
2094
86
1237
0
1474
0
2147
0
C5H5
0
 2k
 3
 S cos




ΔS (Å/rad)
Exp
Calc
C6H6+
C6D6+
C6(H/D)6+
C-C-C bend
0.032
0.033
0.029
C-C stretch
0.038
0.036
0.037
C-C-H bend
0.022
0.014
0.011
C-H stretch
0.0005
0.008
0.0009
Calc
C6H6+
C6F6
 S
Symmetry
coordinate
Stabilization Energy (cm-1)
in e2g modes
Molecule
k
Jahn-Teller Parameters
C6H6+
C6D6+
Constant
Ab initio calc.
Exp. fit
Ab initio calc.
Exp. fit
ωe,18
573
584
546
555
D18
0.42
0.51
0.38
0.46
K18
0.013
0.022
0.015
0.032
ε18π/3
240
293
206
245
ε180
246
306
213
262
ωe,17
1152
1161
844
856
D17
0.12
0.12
0.11
0.13
K17
-0.020
-0.008
-0.018
-0.008
ε17π/3
144
138
94
116
ε170
138
136
91
114
ωe,16
1571
1543
1518
1486
D16
0.23
0.18
0.29
0.24
K16
-0.012
-0.018
-0.013
-0.022
ε16 π/3
373
275
453
366
ε160
364
265
442
350
εT
757
707
753
727
εB
-9
1
-7
-1
Brief History of Studies of Jahn-Teller
Effect in Cycloheptatrienyl
Experimental Spectroscopy
Ab Initio Theory
McConnell, 1962
Longuet-Higgins & McEvens, 1957
Zwolenik, 1963
Lee & Wright, 1998
Silverstone, 1964
Elder & Parr, 1969
Koenig & Chang, 1978
Mach, 1989
Johnson, 1991
Lineberger, 1996
Maier, et al., 2002
Meijer, et al., 2003
Electronic Excitation Spectrum of Tropyl
REMPIa
1 / 2, n1
000
201
e1'
1 / 2, n2
e1'
1 / 2, n3
e1'
201 1 / 2, n2
201 1 / 2, n1
Peak F
LIF
Origin Peak A
* *
25700
25720
*
Peak C
Peak B
*
Peak E
*
*
Peak D
*
26000
27000
28000
29000
-1
Excitation Energy /cm
a
T. Pino, F. Güthe, H. Ding, J. P. Maier J. Phys. Chem. A 106, 10022 (2002)
Dispersed Fluorescence data via Peak A
125 m
375 m
Pumping frequency
0
1000
Artifacts of the discharge
Discharge Light
2000
3000
Red Shift/cm
4000
-1
5000
Dispersed Fluorescence data via Peak B
125 m
375 m
Pumping frequency
0
1000
Artifacts of the discharge
Discharge Light
2000
3000
Red Shift/cm
4000
-1
5000
Dispersed Fluorescence data via Peak C
125 m
375 m
Pumping frequency
0
1000
2000
3000
4000
Red Shift/cm
5000
-1
6000
7000
8000
Electronic Structure and Jahn-Teller distortion of the
~
~
X and A States of Tropyl Radical
~2
X E2 "
2
~2
A E3 "
2
A2
B1
Jahn-Teller active modes:
Linear:
e3
Quadratic:
2
A2
2
B1
Jahn-Teller active modes:
e1
Quadratic: e3 , e3
Linear:
e2 , e2
Jahn-Teller stabilization energy:
CASSCF(7,7)/6-31G* 2143 cm-1
EOMEA-CCSD/DZP
1424 cm-1
EOMEA-CCSD/TZ2P
1433 cm-1
Jahn-Teller stabilization energy:
CASSCF(7,7)/6-31G*
4120 cm-1
EOMEA-CCSD/DZP
1098 cm-1
~
2
Jahn-Teller Active Vibrational Modes in the X E 2 "State of Tropyl Radical
18 (e3') =898 cm-1
D18=0.11
16 (e3') =1569 cm-1
D16=0.56
17(e3') =1313 cm-1
D17=0.22
15 (e3') =3187 cm-1
D15=0.01
~2
A
E3 " State of Tropyl Radical
Jahn-Teller Active Vibrational Modes in the
7 (e1') =1015 cm-1
D7=0.72
6 (e1') =1519 cm-1
D6=0.14
5 (e1') =3205 cm-1
D5=0.01
Silver Trimer: Motivation
• Catalytic properties
• Models of surfaces and thin films of Ag
• A single-mode system with Jahn-Teller
effect subject to spin-orbit effects
• It exhibits interesting spectra
Laser-induced Fluorescence spectrum
Frequency/cm-1
Dispersed Fluorescence spectrum
Red Shift/cm-1
Vibronic coupling interactions
In the equilateral triangle configuration D3h
for the ground state: a1'2 e'1 → 2E'
for the excited state: a1'2 e"1 → 2E"
Two vibrational modes:
(i) A symmetric stretch of a1' symmetry
(ii) An asymmetric stretch of e' symmetry
• Jahn-Teller distortion of the 2E' ground and 2E"
electronic states along the e' coordinates
• The e' mode is linearly and quadratically JT
active!
Coordinate system and nuclear displacements
z
1
y
x
3
Conical Intersection
y
2
JT split electronic states
Ground state:
Excited state:
2
2
E"
B1
2
A2
1
( r12  r13  r23 )
3
1
Qex [b2 ] 
( r12  r13 )
2
Symmetric stretch R[a1 ] 
Asymmetric stretch
Bending mode
Qey [a1 ] 
1
(2r23  r12  r13 )
6
Computational approach
Ag in IB group,
5th
Symmetry
breaking
row in the periodic table
(1s)2(2s)2(2p)6(3s)2(3p)6(4s)2(3d)10
(4p)6(4d)10(5s)1
Electron Core Potential’s substitute for 28
electronsa
Construction of a new basis set for Ag:
cc-pVDZ
(5551)/[5431]
Emin
Emax
dmin
Xmin
dmax
X0
Xmax
State-averaging methodology between the JT split states
(symmetry breaking problems!)
Single point CISD computationsb
Single point SO-CISD computationsb
aECP’s
taken from L.A. LaJohn, P.A. Christiansen, R.B. Ross, T. Atashroo and W.C.
Ermler, J. Chem. Phys. 87, 2812 (1987) (http://www.clarkson.edu/~pac/reps.html)
bThese were done with COLUMBUS (http://www.itc.univie.ac.at/~hans/Columbus)
SCF and CISD results along the C2v cut of the PES
of the ground electronic state of Ag3
2
SCF; A1
2000
2
SCF; B2
2
CISD; A1
2
-1
Potential Energy (cm )
CISD; B2
D3h
0
2B
2
2A
1
-0.6
ESCF(D3h)=-435.915590 a.u.
ECISD(D3h)=-436.821540 a.u.
0.0
Qey [a1] (Angstrom)
0.6
SO-CISD results along the C2v cut of the PES
of the ground electronic state of Ag3
-1
Potential Energy (cm )
5000
E1/2
E1/2
0
-0.5
0.0
Qey [a1] (Angstrom)
-1
ESOCI("0 cm ")=-436.842319 a.u.
("spin-orbit")=232 cm
-1
0.5
SCF and CISD results along the C2v cut of the PES
of the excited electronic state of Ag3
-1
Potential Energy (cm )
2000
2
SCF; A2
D3h
2
SCF; B1
2
CISD; A2
2
CISD; B1
2B
1
2A
2
0
-0.6
ESCF(D3h)=-435.804498 a.u.
ECISD(D3h)=-436.698733 a.u.
0.0
Qey [a1] (Angstrom)
0.6
SO-CISD results along the C2v cut of the PES
of the excited electronic state of Ag3
-1
Potential Energy (cm )
1400
E1/2
E1/2
700
0
-0.2
-1
0.0
ESOCI("0 cm ")=-436.719038 a.u. Qe [a ] (Angstrom)
y
1
-1
("spin-orbit")=19 cm
0.2
Ground State
Excited State
Theory
Experiment
Theory
Experiment
158
155
155
139
ece (e')/cm-1 -
-1.5
-
0.9
D
3.23
3.3
0.94
2
K
0.091
0.09
0.005
-0.17
aze /cm-1
232
232
19
30
n1 (a1')/cm-1
162
180
164
158.5
ad/(cm-1/Å2)
-
-
-
-10
n2 (e')/cm-1
aBilinear
Constant
Franck-Condon Factors
<u"|u'>
Fitted
aPredicted
<0|0>
0.7
0.7
<0|1>
0.6
0.6
Fitted
y
Excited State
V=3
<0|2>
0.5
0.3
<0|3>
0.1
0.1
<1|0>
0.59
0.56
<1|1>
0.1
0.2
<1|2>
0.6
0.6
<1|3>
0.3
x
0.017 Angstrom
y
V=1
Ground State
x
0.5
aPredicted
<2|0>
0.35
0.35
<2|1>
0.5
0.5
<2|2>
0.1
0.1
<2|3>
0.5
0.5
<3|0>
0.2
0.2
<3|1>
0.4
0.4
<3|2>
0.3
0.3
<3|3>
0.26
0.26
ahttp://www.chemistry.ohio-state.edu/~vstakhur/FC.php
These values were computed assuming an equilibrium geometry difference between
the ground and excited state of 0.017 Angst. and a one-dimensional case with regard
to the totally symmetric vibration.
Conclusions
•Jahn-Teller active molecules serve as excellent tests of our
understanding of conical intersections
•The spectra of Jahn-Teller active organic radicals can be
reproduced using analytical PESs, but require the inclusion of
other than traditional Jahn-Teller terms
•Modern computational chemistry codes can be utilized to
provide excellent initial estimates for Jahn-Teller parameters
• Best parameter estimates result from computations at the
global minimum rather than the conical intersection