Extended permutation-inversion groups for simultaneous treatment of the rovibronic states of trans-acetylene, cis-acetylene, and vinylidene Jon T. Hougen a and Anthony J. Merer b a National Institute of Standards and Technology, Gaithersburg, MD 20899, USA b Institute of Atomic and Molecular Sciences, Academia Sinica, PO Box 23-166, Taipei, Taiwan 10617, and Department of Chemistry, University of British Columbia, Vancouver, BC, Canada V6T 1Z1
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Extended permutation-inversion groups for simultaneous treatment of the rovibronic states of trans-acetylene, cis-acetylene, and vinylidene Jon T. Hougen.
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Extended permutation-inversion groups for
simultaneous treatment of the rovibronic states
of trans-acetylene, cis-acetylene, and vinylidene
Jon T. Hougena and Anthony J. Mererb
aNational Institute of Standards and Technology, Gaithersburg, MD 20899, USAbInstitute of Atomic and Molecular Sciences, Academia Sinica, PO Box 23-166, Taipei, Taiwan 10617, and Department of Chemistry, University of British Columbia, Vancouver, BC, Canada V6T 1Z1
H2
H1
CbCaz
x
(a) trans
H2H1
CbCaz
x
(b) cis
H2
H1
CbCaz
(c) vinylidene
x
Results for no bond breaking = trans and cis acetylene 1. For rovibronic symmetry species & nuclear spin statistics use permutation-inversion group G4 = C2h = C2v
2. For symmetry species of electronic, vibrational, & rotational parts of basis functions use group G4
(8) = G32
3. For selection rules for perturbations between levels of cis-bent acetylene and trans-bent acetylene use G4
4. There are energy level splittings caused by LAM tunnelings in G4
(8) vibrational levels
5. But there are only rotational K-stack staggerings in G4
rovibrational levels: Ka= 4n, Ka= 4n+2, and Ka= odd
A.J. Merer, A.H. Steeves, H.A. Bechtel and R.W.Field, unpublished
What theoretical tools are necessary to understand cis-bent acetylene, trans-bent acetylene (& vinylidene) =bent acetylene without (with) bond breaking?
1. Laboratory-fixed coordinate system Molecule-fixed coordinate systems Multiple-valued molecule-fixed coordinate systems Coordinate transformations under group operations
2. Point groups & Permutation-inversion groups & Extended permutation-inversion groups Limited identities in the group theory
3. Large amplitude motions Tunneling between equivalent minima High-barrier tunneling Hamiltonian
H1
H2
(b) The trans acetylene configuration ai(1,2)
2
1
x
Cb
Caz
x
Ca
1
2
H2
H1
zCb
(a) Trans and cis acetylene - no bond breaking
Ca
H1
H2
Cb z
(c) The trans configuration ai(1, 2)
1
2
x
LAM CCH bendsMotion on two circlescentered on the C atoms-2/3 < 1, 2 < + 2/3
LAM H migration motionMotion on one ellipsecentered at center of mass 1, 2 are unrestricted
LAMs lead to a multiple valued coordinate system
The coordinates {, , 1, 2} = {K-rotation, HCCH torsion, HCC bend, CCH bend} for a given configuration in space are not unique.
A multiple valued coordinate system leads to “limited identities” in the group theoryand to “extended permutation-inversion groups”
Ca
1
2
H2
H1
x
zCb
Ca
-1
-2
H2
H1
x
zCb
1, 2 -1, -2
1. Apply also + “Limited Identity”
2. Apply also + “Limited Identity”
There is 1 real identity and 7 limited identities = identity in PI group G4, but not for wavefunctionThere are 8 identical trans minima.
Character table for the group G4(8), for trans and cis HCCH. Limited identities in red.
a ac ab abc e d a2 a2d c a2c b bd bc a2bc a3 a3c a3b a3bc cd a2cd a2b a2bd a2bcd bcd ad acd abd abcd Species a3d a3cd a3bd a3bcd
1. Try to find more examples of applications of this group theory and this K-staggering formalism in cis-bent and trans-bent S1 acetylene spectra (A. Merer & Bob Field’s group)
2. Look for applications in H. Kanamori’s old (~ 2004 unpublished) T1 acetylene spectra.
Next 2 slides show structure of coordinate transformations
E One copy of Permutation-Inversion(ab)(12) group G4
(ab)(12)*E*
E Another copy of Permutation-Inversion(ab)(12) group G4
(ab)(12)*E*
E Another copy of Permutation-Inversion(ab)(12) group G4
(ab)(12)*E* 8 copies in all
Transformation properties of the coordinates , , , , 1, 2
under symmetry operations of the eight-fold extended group G4(8)