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A.J. Merer, A.H. Steeves, H.A. Bechtel and R.W.Field, unpublished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

A1g+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1

A2u+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1

A1u 1 1 1 1 1 1 1 1 1 1 1 1 1 1

A2g 1 1 1 1 1 1 1 1 1 1 1 1 1 1

B1g+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1

B2u+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1

B1u 1 1 1 1 1 1 1 1 1 1 1 1 1 1

B2g 1 1 1 1 1 1 1 1 1 1 1 1 1 1

E+ 2 2 2 2 2 2 0 0 0 0 0 0 0 0 E 2 2 2 2 2 2 0 0 0 0 0 0 0 0 E1 2 2 2 2 0 0 2 2 0 0 0 0 0 0 E2 2 2 2 2 0 0 2 2 0 0 0 0 0 0 Eg 2 2 2 2 0 0 0 0 2 2 0 0 0 0 Eu 2 2 2 2 0 0 0 0 2 2 0 0 0 0

This octuple group G4(8) = G32 is

isomorphic with the double group G16

(2) = G32 used for H2C=CH2 by Merer and Watson in 1973.

Why???

We can approximately visualize the average form of our cis/trans bent acetylene as

(0.5H)2C=C(0.5H)2

How do the tunneling splittings in G4(8)

get to be K-staggerings in G4?

In the multiple-valued coordinate system there are 8 identical minima and therefore 8 localized vibrational basis functions

Bending (H12) and torsional (H13) tunneling motions give the following bending-torsional (bt) splittings

E(btAlg+) = E0 + 2H12 + 4H13

E(btBlg+) = E0 + 2H12 4H13

E(btEg) = E0 2H12 E(btE1) = E0 + 2H12 E(btE+) = E0 2H12,

Species of rotational levels JKaKc

Species of final bending-torsional-rotational (btr) wavefunctions must belong to one of the four single valued representations:

btrAlg+ or btrA2g

or btrB2u+ or btrB1u

E(btAlg+) = E0 + 2H12 + 4H13 (only J4n,even)

E(btBlg+) = E0 + 2H12 4H13 (only J4n+2,even)

E(btEg) = E0 2H12 (only Joe, Joo) E(btE1) = E0 + 2H12 (doesn’t exist)E(btE+) = E0 2H12 (doesn’t exist)

K-level staggerings

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 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)

PIa Genb , , , 1, 2

________________________________________________

E e , , , 1, 2 (ab)(12) a2b , , + , 2, 1 (ab)(12)* a2bcd , , , 2, 1 E* cd , , + , 1, 2 E a2 +, , , 1, 2 (ab)(12) b , , + , 2, 1 (ab)(12)* bcd +, , , 2, 1

E* a2cd , , + , 1, 2 E d +, , +, 1, 2

(ab)(12) a2bd , , + +, 2, 1 (ab)(12)* a2bc +, , , 2, 1 E* c , , + , 1, 2

E a2d , , +, 1, 2

(ab)(12) bd , , + +, 2, 1 (ab)(12)* bc , , , 2, 1 E* a2c , , + , 1, 2

E abc /2+, , /2+, 1, 2

(ab)(12) ac 3/2 , , + /2+, 2, 1 (ab)(12)* a /2+, , /2 , 2, 1 E* ab 3/2 , , + /2 , 1, 2

E a3bc /2+, , /2+, 1, 2

(ab)(12) a3c /2 , , + /2+, 2, 1 (ab)(12)* a3 /2+, , /2 , 2, 1 E* a3b /2 , , + /2 , 1, 2

E abcd /2+, , /2+, 1, 2 (ab)(12) acd /2 , , + /2+, 2, 1 (ab)(12)* ad /2+, , /2 , 2, 1 E* abd /2 , , + /2 , 1, 2 E a3bcd /2+, , /2+, 1, 2 (ab)(12) a3cd 3/2 , , + /2+, 2, 1 (ab)(12)* a3d /2+, , /2 , 2, 1 E* a3bd 3/2 , , + /2 , 1, 2

(a) G4 = C2h for trans bent acetylene (b) G4 = C2v for cis bent acetylene

PGa E C2(y) i (xz) PGa E C2(x) (xy) (xz)

PIb E (ab)(12) (ab)(12)* E* PIb E (ab)(12) (ab)(12)* E*

Ag 1 1 1 1 A1 1 1 1 1

Au 1 1 1 1 A2 1 1 1 1

Bg 1 1 1 1 B1 1 1 1 1

Bu 1 1 1 1 B2 1 1 1 1

(c) G4 = C2v for vinylidene (d) G4 = C2v for cis bent acetylene

PGa E C2(z) (yz) (xz) PGa E C2(y) (xy) (yz)

PIb E (12) (12)* E* PIb E (ab)(12) (ab)(12)* E*

A1 1 1 1 1 A1 1 1 1 1

A2 1 1 1 1 A2 1 1 1 1

B1 1 1 1 1 B1 1 1 1 1

B2 1 1 1 1 B2 1 1 1 1

cis

trans cis

trans

linear saddlesaddle

saddle

saddle

0

60o

60o

60o

60o 0

Q3

Q6

Q3 (trans bend)

Q6 (in-planecis bend)

H

H

C Cag

bu

H

H

C C

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