The Vibrational Jahn-Teller Effect in
Non-degenerate Electronic State
Bishnu P Thapaliya, The University of Akron
ISMS,U of Illinois, Champaign/Urbana, June 22-26, 2015
Ram S Bhatta
Mahesh B Dawadi
David S Perry
Introduction: Jahn-Teller Effect
Introduced by Herman Jahn and Edward Teller (1936)
JT theorem – “A configuration of a polyatomic molecule for
an electronic state having orbital degeneracy cannot be stable
with respect to all displacements of the nuclei unless in the
original configuration the nuclei all lie on a straight line.”1
γ
Born-Oppenheimer approximation:
- Adiabatic separation of electronic and nuclear motion
Example: E⊗e problem
I. B. Bersuker, Chem.Rev. 2001, 101, 1067-1114
1. H. A. Jahn and E. Teller, Proc. R. Soc. Lond. A 161, 220 (1937)
Vibrational Jahn-Teller Distortion
Cr(CO)3(C6H6)
Application of the Jahn-Teller theorem to the
vibrational case:
- based on adiabatic separation of high
frequency vibrations from low frequency
vibration
Degenerate excited states of high-frequency
vibrations are geometrically distorted along
low-frequency degenerate coordinates.
• 1A1 state - 18 e- filled shell
• E-type CO stretch vibrational state:
Applies most molecules of C3v and higher
symmetry
- Spontaneous J-T distortion along the e-type
Cr-C stretch: ~ 1 x 10-3 Å.
- Stabilization energy: 0.08 cm-1
B3LYP// 6-31+G(2d,p)
M.E. Kellman, Chem. Phys. Lett. 87, 171 (1982); Jan Makarewicz, VIIth Int. Conf. on High-Res. Vib. Spectrosc., Liblice, Czechoslovakia (1982).
3
TheVibrational Jahn-Teller Effect:
Spontaneous distortion is very small.
Vibrational JTE is weak compared
to electronic JTE.
Cr(CO)3(C6H6)
CO stretches
Restoring potential remains strong.
Wavenumber / cm-1
Surface splitting due to vibrational
JTE is substantial.
2010
2000
1990
1980
ρ0
0
0.1
B3LYP//6-31+G(2d,p)
0.2 ρCr-C / Å
4
Vibrational CIs at C3v:
The Vibrational Jahn-Teller Effect (E⊗e)
CH3CN
Cr(CO)3(C6H6)
Asymmetric E-type CH stretches
Asymmetric E-type CO stretches
Wavenumber / cm-1
Wavenumber / cm-1
2010
3206
2000
3204
1990
0˚
5˚
MP2/6-311+G(3df,2p)
10˚
3202
ρ
15˚ CCN bend
1980
0
0.1
B3LYP//6-31+G(2d,p)
0.2 ΡCr-C/ Å
5
Methanol Asymmetric CH-Stretch Frequencies
Conical intersections (CIs)
Ab initio: CCSD(T)/aug-cc-pVTZ)
ρ = 0°
Global Minima
γ = 60˚, 180˚, 300˚
C3v geometry
ρ = 62.4°
Staggered
Torsional Saddles
γ = 0˚, 120˚, 240˚
Eclipsed-CI
ρ =91.9°
Eclipsed
Jahn-Teller distortion coordinate
Eclipsed-CI
6
Methanol Asymmetric
CH-Stretches:
Frequencies and Force
constants
12-D
normal
mode
calc.
CCSD(T)/aug-cc-pTVZ
Torsional minimum energy path
3-D
normal
mode
calc.
local CH
harmonic
force
constants
L.-H. Xu, J.T. Hougen, R.M. Lees, J. Mol. Spectrosc. 293-294, 38 (2013).
7
Extended High-order Jahn-Teller Model
A. Viel and W. Eisfeld, J. Chem. Phys. 120, 4603 (2004).
❖ Express the 2-level problem in the real (Cartesian) basis
æ U +V
0 ö æ W
H =ç
÷+ç
U +V ø è Z
è 0
with
Z ö
÷
-W ø
U =U 0g +U 3g cos 3g +U 6g cos6g +...
V = V 0g +V 3g cos 3g +V 6g cos6g +...
W = W 1g cosg +W 2g cos2g +W 4g cos 4g +W 5g cos5g +...
Z = W 1g sin g -W 2g sin2g +W 4g sin 4g -W 5g sin5g +...
❖ Each of the Fourier coefficients is expanded in a power series in ρ.
W 1g = l1(1) r + 3!1 l1(3) r 3 + 5!1 l2(5)r 5 +...
W 2g = 2!1 l1(2) r 2 + 4!1 l2(4) r 4 + 6!1 l2(6)r 6 +...
W 4g = 4!1 l1(4) r 4 + 6!1 l1(6)r 6 +...
❖Eigenvalues, keeping Jahn-Teller interactions up to 4th-order
E± = (V 0g +U 0g ) + (V 3g +U 3g ) cos 3g + (V 6g +U 6g ) cos 6g
{
± (W
)
1g 2
+ (W
)
2g 2
+ (W
)
4g 2
+ 2W 1g (W 2g +W 4 g ) cos 3g + 2W 2gW 4g cos 6g
}
1
2
8
Model Parameters for
Methanol Asymmetric
CH-Stretches
CCSD(T)/aug-cc-pTVZ
❖ Points are combinations of ab initio
CH frequencies at γ = 0˚, 30˚, 60˚.
❖ Lines are appropriate power series
fits.
❖ Vibrational part of adiabatic
energies relative to the zero-point
level:
9
Methanol
CCSD(T)/aug-cc-pVTZ
CH Stretches, ν2 and ν9
Electronic Potential Energy
Wavenumber / cm-1
10,000
5,000
0
0˚
Fit RMS = 0.2 cm-1
40˚
80˚
ρ
Vibrational Jahn-Teller Effect
CCSD(T)/aug-cc-pVTZ
CH3OH
CH3SH
CH stretches
CH stretches
Wavenumber / cm-1
20
10
60˚
100˚
ρ
-10
-20
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Implications for spectroscopy
The vibrational Jahn-Teller effect accounts
for the splitting of A′ and A″ vibrations.
 10 - several 10’s cm-1
Test the limits of the adiabatic concept
 Limited frequency separation
 Intermediate frequency vibrations
Comparison to experiment
 Level patterns in torsionally excited states built
on CH stretches
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Implications for dynamics
Localized ultrafast energy transfer near CIs
 CIs accessible at low energies
 Competition with delocalized processes
A broadly applicable concept
 CIs are widespread
 Both unimolecular and collisional processes
 Applicability to solvent coordinates and Van der
waals modes
P. Hamm and G. Stock, Mol. Phys. 111, 2046 (2013). [Fig. 8]
13
Thank you
Questions or Comments?
14
13
Introduction: Vibrational Adiabaticity
Adiabatic separation of high- and low-frequency
vibrations.
1. Solve the motion “fast” motion at each point in the
large-amplitude coordinate space.
2. Solve the “slow” large-amplitude motion on the
potential formed by the fast states.
Becomes exact when the fast states are independent of
slow coordinates.
A crucial form of off-diagonal anharmonicity is
embodied in the dependence of the fast states on the slow
coordinates.
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Introduction: Vibrational Adiabaticity
Vibrationally adiabatic approximation in methanol
• 1 “slow” coordinate =torsion
B. Fehrensen, D. Luckhaus, M. Quack, M. Willeke, T. R. Rizzo, J. Chem. Phys. 119, 5534 (2003).
D. S. Perry, J. Mol. Spectrosc. 257, 1 (2009).
L.-H. Xu, J.T. Hougen, R.M. Lees, J. Mol. Spectrosc. 293-294, 38 (2013).
Vibrational conical intersections - Hamm and Stock
• At least 2 “slow” coordinates
• Ultrafast vibrational relaxation
• Malonaldehyde; (H2O)2
P. Hamm and G. Stock, Phys. Rev. Lett. 109, 173201 (2012).
P. Hamm and G. Stock, Mol. Phys. 111, 2046 (2013). [Fig. 8] at left
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Methanol CH Stretches, ν2 and ν9
Introduction: Types of Conical Intersections
C3
Criteria for a conical intersection:
- Zeroth-order splitting and coupling both zero
- Apply to both electronic and vibrational CIs
Symmetry-required conical intersections
- e.g., C3v: E⊗e
σ
Symmetry-allowed conical intersections
-e.g.,Cs: A′ & A″
Accidental conical intersections
-same symmetry
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Geometric Phase
Evolution of the high frequency vibrational
wavefunctions along a closed path
 Changes sign if 1 CI enclosed
 (-1)n if n CIs enclosed
Torsional minimum energy path

4 CIs enclosed ⇒ +1
States with mixed geometric phase
Torsional tunneling splittings are inverted
irrespective of geometric phase.


methanol
methylamine
D. C. Clary, Geometric Phase in Chemical Reactions. Science 309, 1195-1196 (2005).
20
Methanol Diabatization
3 diabatization schemes correspond to Xu, Hougen, Lees’ 3 limiting cases.
Jahn-Teller 1st-order
Jahn-Teller 2nd-order
Double-valued diabatic surface
Geometric phase = -1
Coupling is zero in Cs planes.
Geometric phase = +1
Coupling is zero in Cs planes.
Geometric phase = -1
Smallest coupling, zero in staggered.
L.-H. Xu, J.T. Hougen, R.M. Lees, J. Mol. Spectrosc. 293-294, 38 (2013).