1. 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

2. H2H2 H1H1 CbCb CaCa z x (a) trans H2H2 H1H1 CbCb CaCa z x (b) cis H2H2 H1H1 CbCb CaCa z (c) vinylidene x

3. Results for no bond breaking = trans and cis acetylene 1. For rovibronic symmetry species & nuclear spin statistics use permutation-inversion group G 4 = C 2h = C 2v 2. For symmetry species of electronic, vibrational, & rotational parts of basis functions use group G 4 (8) = G 32 3. For selection rules for perturbations between levels of cis-bent acetylene and trans-bent acetylene use G 4 4. There are energy level splittings caused by LAM tunnelings in G 4 (8) vibrational levels 5. But there are only rotational K-stack staggerings in G 4 rovibrational levels: K a = 4n, K a = 4n+2, and K a = odd

4. A.J. Merer, A.H. Steeves, H.A. Bechtel and R.W.Field, unpublished

5. 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

6. H1H1 H2H2 (b) The trans acetylene configuration a i (  1,  2 ) 22 11 x CbCb CaCa z x CaCa 11 22 H2H2 H1H1 z CbCb (a) Trans and cis acetylene - no bond breaking CaCa H1H1 H2H2 CbCb z (c) The trans configuration a i (  1,  2 )  1  2 x LAM CCH bends Motion on two circles centered on the C atoms -2  /3 <  1,  2 < + 2  /3 LAM H migration motion Motion on one ellipse centered at center of mass  1,  2 are unrestricted

7. 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 theory and to “extended permutation-inversion groups”

8. CaCa 11 22 H2H2 H1H1 x z CbCb CaCa -1-1 -2-2 H2H2 H1H1 x z CbCb  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 G 4, but not for wavefunction There are 8 identical trans minima.

10. This octuple group G 4 (8) = G 32 is isomorphic with the double group G 16 (2) = G 32 used for H 2 C=CH 2 by Merer and Watson in 1973. Why??? We can approximately visualize the average form of our cis/trans bent acetylene as (0.5H) 2 C=C(0.5H) 2

11. How do the tunneling splittings in G 4 (8) get to be K-staggerings in G 4 ? In the multiple-valued coordinate system there are 8 identical minima and therefore 8 localized vibrational basis functions Bending (H 12 ) and torsional (H 13 ) tunneling motions give the following bending-torsional (bt) splittings E( bt A lg + ) = E 0 + 2H 12 + 4H 13 E( bt B lg + ) = E 0 + 2H 12  4H 13 E( bt E g ) = E 0  2H 12 E( bt E 1 ) = E 0 + 2H 12 E( bt E + ) = E 0  2H 12,

12. Species of rotational levels J KaKc Species of final bending-torsional-rotational (btr) wavefunctions must belong to one of the four single valued representations: btr A lg + or btr A 2g  or btr B 2u + or btr B 1u 

13. E( bt A lg + ) = E 0 + 2H 12 + 4H 13 (only J 4n,even ) E( bt B lg + ) = E 0 + 2H 12  4H 13 (only J 4n+2,even ) E( bt E g ) = E 0  2H 12 (only J oe, J oo ) E( bt E 1 ) = E 0 + 2H 12 (doesn’t exist) E( bt E + ) = E 0  2H 12 (doesn’t exist) K-level staggerings

14. Future work 1. Try to find more examples of applications of this group theory and this K-staggering formalism in cis-bent and trans-bent S 1 acetylene spectra (A. Merer & Bob Field’s group) 2. Look for applications in H. Kanamori’s old (~ 2004 unpublished) T 1 acetylene spectra.

16. Next 2 slides show structure of coordinate transformations E One copy of Permutation-Inversion (ab)(12) group G 4 (ab)(12)* E* E Another copy of Permutation-Inversion (ab)(12) group G 4 (ab)(12)* E* E Another copy of Permutation-Inversion (ab)(12) group G 4 (ab)(12)* E* 8 copies in all 

20. 1. Tunneling path is nearly circular in  1,  2 space 2. Note very high barrier to linear configuration. Consider only bent forms of HCCH This allows us to avoid quasi-linear molecule complications