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FACULTY OF MATHEMATICS AND NATURAL SCIENCES
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Local modes in
vibrational (and
rotational)
spectroscopy
(Picture courtesy
of Python M)
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FACULTY OF MATHEMATICS AND NATURAL SCIENCES
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Where is
Wuppertal?
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The Wuppertal monorail
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FACULTY OF MATHEMATICS AND NATURAL SCIENCES
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Vibrational energies of small molecules
Density o
f sta
tes
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cm-1
0 4000 8000 cm-1
Spectrum?
n
I Simple and structured! Extremely complicated,
wall-to-wall lines?
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FACULTY OF MATHEMATICS AND NATURAL SCIENCES
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H-X stretching vibrations modelled by
Morse oscillators:
Hamiltonian:
Eigenvalues:
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HCAO model for H2X molecule stretch (Harmonically Coupled Anharmonic Oscillators)
Hamiltonian:
re
Diagonalized in basis set of products of
Morse-oscillator eigenfunctions
[1] Child MS, Lawton RT. Faraday Discuss Chem Soc 1981, 71:273–285.
[2] Mortensen OS, Henry BR, Mohammadi MA. J Chem Phys 1981, 75:4800–4808
[3] Child MS, Halonen L. Adv Chem Phys 1984, 57:1–58
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FACULTY OF MATHEMATICS AND NATURAL SCIENCES
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Vibrational
energies
patterns:
Normal mode,
Local mode…
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FACULTY OF MATHEMATICS AND NATURAL SCIENCES
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Vibrational intensities Empirical fact: Stretching part of
dipole moment function modelled
well by „Molecular Bond“ function
Favours vibrational transitions to states with all N
excitation quanta localized in one bond.
[1] HALONEN, L., 1998, Adv. chem. Phys., 104.
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Summary:
Excited stretching states form
„polyads“ with the structure
predicted by HCAO
Observed transitions from the vibrational ground
state end in lowest levels of polyad.
Results transferable to molecules with n > 2
equivalent H-X stretching vibrations.
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Example experiment: ICLAS (Intracavity Laser Absorption Spectroscopy)
[1] Campargue A: http://www.chem.uni-wuppertal.de/sphers/han-sur-
lesse/campargue/camparguenew.html
[2] Bertseva E, Kachanov AA, Campargue A. Chem Phys Lett 2002, 351:18–26.
• Extremely sensitive
• Absoption path lengths of order-of-magnitude 100 km
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Example experiment: ICLAS for H2S (Intracavity Laser Absorption Spectroscopy)
[1] Ding Y, Naumenko O, Hu S-M, Bertseva E, Campargue A.. J Mol Spectrosc 2003, 217:222–238.
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[1] Ding Y, Naumenko O, Hu S-M, Bertseva E, Campargue A.. J Mol Spectrosc 2003, 217:222–238.
Example experiment: ICLAS for H2S (Intracavity Laser Absorption Spectroscopy)
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1972 Dorney and Watson CH4 8-fold and 6-fold clusters
1978 Zhilinskii and Pavlichenkov H2O 4-fold clusters (Erb)
1978 Harter and Patterson Rotational energy surfaces and clusters
1991 Lehmann Local mode theory and clusters
1992 Kozin et al H2Se 4-fold clusters observed
1993 Kozin and Jensen H2Se 4-fold cluster theory (Erbs)
1994 Jensen and Bunker H2X 4-fold cluster symmetry
1996 Kozin et al H2Te 4-fold clusters (exp and theory)
1997 Jensen et al Review paper on 4-fold clusters
2000 Jensen Review paper on LMT and clusters
2012 Jensen Another review paper on LMT and clusters
Rotational Energy Level Clusters
[Mol. Phys. 98, 1253-1285 (2000)]
[WIREs Comput. Mol. Sci. (Wiley Interdisciplinary
Reviews) 2, 494–512 (2012)]
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JJ,0
JJ,1
JJ-1,1
JJ-1,2
JJ-2,2
JJ-2,3
JJ-3,3
JJ-3,4
JJ-4,4
JJ,0
JJ,1
JJ-1,1
JJ-1,2
J
Ae=6.26, Be=6.11, Ce=3.09 cm-1
H2Te Rigid Rotor Energy Levels
[E(J
KaK
c)-
E(J
J0)]
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J
JJ,0
JJ,1
JJ-1,1
JJ-1,2
JJ,0
JJ,1
JJ-1,1
JJ-1,2
JJ-2,2
JJ-2,3
JJ-3,3
JJ-3,4
JJ-4,4
Actual H2Te Energy Levels
[E(J
KaK
c)-
E(J
J0)]
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1R 1L
2L 2R
1
2
1
1 1
2 2
2
JJ,0
JJ,1
JJ-1,1
JJ-1,2
in C2v(M)
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[3] S.N. Yurchenko, W. Thiel, and P. Jensen, J. Mol. Spectrosc. 245, 126 (2007)
Programs: XY3 and TROVE
(Theoretical Rotation-Vibration Energies)
[1] H. Lin et al., J. Chem. Phys. 117, 11265 (2002)
[2] S.N. Yurchenko et al., Mol. Phys. 103, 359 (2005) and references given there.
XY3: Rotation-vibration energies for
pyramidal, ammonia-type
molecule in isolated electronic
state, calculated variationally.
TROVE: Rotation-vibration energies for
any molecule in isolated electronic
state, calculated variationally.
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cc-pwCVTZ [1]
refined by fitting to experimental
vibrational term values [2]
[1] D. Wang,Q. Shi, and Q.-S. Zhu, J. Chem. Phys., 112, 9624 (2000)
[2] S.N. Yurchenko et al., Chem. Phys. 290, 59 (2003)
Variational rotation-vibration
calculations for PH3
Vibrational basis set:
6)(2 321 bendinv Vvvvv
J ≤ 80
Potential energy surface:
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Six-fold energy clusters
-6
-5
-4
-3
-2
-1
0
0 10 20 30 40 50 60 70 80
A1/A2
A2/A1
E
E
Cluster = A1 A2 2E in C3v(M)
J
[E(J
K)-
E(J
J)]
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3 PCS
= (132)1 PCS
4 PCS
= (23)*1 PCS
6 PCS
= (13)*1 PCS
Cluster = A1 A2 2E in C3v(M)
2 PCS
= (123)1 PCS
5 PCS
= (12)*1 PCS
1 PCS
= E 1 PCS
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FACULTY OF MATHEMATICS AND NATURAL SCIENCES
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L. Fusina, G. Di Lonardo, J. Mol. Struct. 517-518, 67 (2000)
EJK
- E
Jmax
/
cm-1
Watson-type Hamiltonian for PH3
J=
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• (,,) define orientation of molecule (xyz) relative to
laboratory (XYZ).
• (,) define orientation of Z axis relative to molecule (xyz).
Rotational coordinates
xyz is molecule-fixed; XYZ is space-fixed
x
y
z
x
y
z
X
Y
Z
Z
N
N
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For states with m = J (and J large),
is the probability distribution
for the orientation of the angular
momentum relative to the molecule
),(,mJF
i
Integration over all vibronic coordinates and
For a rovibronic eigenstate
is the probability distribution for the orientation
of the Z axis relative to the molecule. Z
J ),(, mJFJ
dVF iimJ sin),(*
,
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z z z
1
1
A
1
1
A
2
6
A
1
2
E
2
3
E
1
4
E 2
5
E
x y
z z z
y
y y
y y x
x x
x x
),(, mJF“Top cluster states” for J = m = 40,
vibrational ground state of PH3
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FACULTY OF MATHEMATICS AND NATURAL SCIENCES
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Then invert the unitary transformation to obtain
1
1
A =
Primitive cluster states j PCS
First symmetrize, e.g.
= 2
6
A
with similar expressions for the E functions....
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1 PCS
-6
-5
-4
-3
-2
-1
0
0 10 20 30 40 50 60 70 80J=20
= 94º
z
y x
dVF iiJmJ sin),(*
,
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1 PCS
-6
-5
-4
-3
-2
-1
0
0 10 20 30 40 50 60 70 80J=80
= 113º
z
y x
eq = 123º
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FACULTY OF MATHEMATICS AND NATURAL SCIENCES
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PH3 intensity calculations
Ab initio (CCSD(T)/aug-cc-pVTZ)
dipole moment surfaces
Rotation-vibration
wavefunctions from
the variational
calculation
Experimental data
from L. R. Brown,
R. L. Sams, I. Kleiner,
C. Cottaz, and L. Sagui,
J. Mol. Spectrosc. 215,
178-203 (2002)
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FACULTY OF MATHEMATICS AND NATURAL SCIENCES
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PH3 cluster transitions
Large line strengths
at high J
Can the lower states
be populated somehow?
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Similar effects for
XHD2 or XH2D molecules?
?
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XH2D: Cluster-free molecules
PH2D
BiH2D SbH2D
Reduced rotational term values
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XHD2: Molecules with rotational energy cluster formation
PHD2
BiHD2 SbHD2
Reduced rotational term values
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FACULTY OF MATHEMATICS AND NATURAL SCIENCES
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Local mode phenomena manifest themselves as
• clustering of vibrational energy levels at high
vibrational excitation
• clustering of rotation–vibration energy levels at high
rotational excitation.
Thus local mode behaviour is
induced by both vibrational and
rotational excitation.
Conclusions
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Thanks & Acknowledgments
The principal doer:
Sergei N. Yurchenko, Wuppertal, Ottawa,
Mülheim, Dresden
Other doers:
Miguel Carvajal, Huelva
Hai Lin, Denver
Serguei Patchkovskii, Ottawa
A fellow ponderer: Walter Thiel, Mülheim
Thanks for support from the European Commission,
the German Research Council (DFG), and the Foundation
of the German Chemical Industry (Fonds der Chemie).
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