Download presentation

Presentation is loading. Please wait.

Published byCali Jaycox Modified over 2 years ago

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A.J. Merer, A.H. Steeves, H.A. Bechtel and R.W.Field, unpublished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 ) 22 11 x CbCb CaCa z x CaCa 11 22 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 11 22 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

Similar presentations

OK

Outline 1. Introduction 2. User community and accuracy needs 3. Which large-amplitude motions 4. Which tools = which sym. operations 5. Example (in progress)

Outline 1. Introduction 2. User community and accuracy needs 3. Which large-amplitude motions 4. Which tools = which sym. operations 5. Example (in progress)

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on needle stick injury images Ppt on pi in maths draw Download ppt on surface area and volume for class 9th Ppt on regular expression generator Ppt on latest technology in communication Ppt on features of ms word Ppt on landforms for 4th grade Ppt on power grid failure in india Ppt on principles of peace building institute of africa Ppt on edgeworth box