A. J. Merer Institute of Atomic and Molecular Sciences, Taipei, Taiwan Least squares fitting of perturbed vibrational polyads near the isomerization barrier in the S 1 state of C 2 H 2 J. H. Baraban P. B. Changala R. W. Field Massachusetts Institute of Technology, Cambridge, MA

B = bending [3 1 B 1 = plus ] A.H. Steeves, H.A. Bechtel, A.J. Merer, N. Yamakita, S. Tsuchiya and R.W. Field, J. Mol. Spectrosc. 256, 256 (2009). Trans bend C=C stretch 34B234B2

A-axis Coriolis Darling- Dennison

Vibrational angular momentum The two low-lying bending fundamentals, 4 (torsion) and 6 (cis-bend) are almost degenerate: [Utz et al, 1993] 4 (a u ) = ; 6 (b u ) = cm  1 They correlate with the 5 (  u ) vibration of the linear molecule, so that they possess a vibrational angular momentum. This has two effects: A- and B-axis Coriolis coupling, for all vibrational levels Darling-Dennison resonance, for their overtones and combinations

A phase complication The A-axis Coriolis operator, H =  2 A J a G a =  2 A J a (Q tr  a P) acting between harmonic levels |v 4 > and |v 6 >, has imaginary matrix elements : To get rid of the i s, multiply all |v 6 > functions by (i) v 6. Everything then becomes real. = 2 i A  a 46 K  √(v 4 +1)(v 6 +1) =  2 i A  a 46 K  √(v 4 +1)(v 6 +1) where  = ½ [√ 4 / 6 + √ 6 / 4 ]

Successive transformations of the Hamiltonian With a diagonalization routine that attempts to preserve the energy order of the basis states, Step 1: Transform away the large  K=0 off-diagonal elements of the D-D resonance and A-axis Coriolis coupling. The resulting functions still have well-defined K. Step 2: Transform away the  K= ±2 asymmetry elements. The resulting functions still have well-defined even-K or odd-K character. Step 3: Transform away the  K= ±1 elements of the B-axis Coriolis coupling. These elements are the smallest, and do not scramble the K values unduly. This can still break down at the local avoided crossings!

C 2 H 2, A 1 A u : Rotational constants for the B 3 polyad ~ Vibrational origins, relative to T 00 at cm  1 T 0 (4 3 ) T 0 (6 3 ) (10) (9) T 0 ( ) T 0 ( ) (7) (9) Coriolis 2A  a (9) BbBb (3) 2A  a, D K  (2) Darling-Dennison k 4466  (9) k 4466, D K (8) Rotation A (6 3 ) (5) A (4 3 ) (5) B  C (6 3 ) (72) B  C (4 3 ) (102) (28) (30) B (4 3 ) B (6 3 ) Parameters for the other two levels are interpolated, except A for and , which are corrected by 0.41 (5) cm  1.  + r.m.s.error = cm  1 cm  1

Comparison of bending polyad fits (cm  1 ) B2B2 31B231B2 51B251B2 32B232B2 k 4466  (2)  (17)  (12)  (40) 2A  a (1) (17)23.56 (11)18.03 (10) BbBb (2)0.784 (5)0.808 (14)0.751 (15) x (8)28.40 (4)37.97 (2)13.2 (8) r.m.s B3B3 31B331B3 32B332B3 k 4466  (1)  (12) BbBb 2A  a (1) (5) (3) (8) r.m.s *0.036 Broken polyad * Combined fit with the interacting B 1 polyad

Final least squares fit to the interacting 3 1 B 3 and B 1 polyads Dots are observed term values and lines are calculated. Some of the higher-order rotational constants are not very realistic, but they reproduce the J-structure!  = cm  1 Darling-Dennison resonance k 266 = 8.66 ± 0.16 cm  1 k 244 =  7.3 ± 1.1 cm  lies far below the rest of the polyad; x 36 is very large!

Excitation of 3 unravels the bending polyads

C 2 H 2 : the cis band group (46200 cm  1 ) 3.9 cm  A.J. Merer, A.H. Steeves, J.H. Baraban, H.A. Bechtel and R.W. Field, J. Chem. Phys. 134, (2011) K-staggering (Tunnelling splitting)

Rotational constants from fitting of cis (cm  1 ) T0T0 ± A ± B ± C ±  JK ± S ± † † S is the shift of the K=1 levels above the position predicted from the K=0 and 2 levels (K-staggering parameter). r.m.s. error =0.019 Data from K= 0 – 2 only.

Cis-C 2 H 2 does not show bending polyad structure, since  4  6 = 250 cm  1, compared to 3 cm  1 for trans-C 2 H 2. K-staggering is easy to model for cis-C 2 H 2, and for trans levels that are not part of polyads. For trans- bending polyads it is a serious extra complication. K-staggering

K-staggering in trans-5 3 The ratio of the K=3  1 and 2  0 intervals is 1.993:1, close to the expected 2:1. The K=1  0 interval should be one quarter of the K=2  0 interval (16.46 cm  1 ), but is 6.31 cm  1 greater. The trans level 5 3 lies about 60 cm  1 above the calculated isomerization barrier. Watson (JMS 98, 133 (1982)) has given the energies of its K=0  3 states: KT 0 / cm  Conclusion: there is a K-staggering of cm  1 in trans-5 3

Steps in the fitting of the trans-3 2 B 3 polyad Full data setCoriolis + D-D0.989 r.m.s./ cm  1 What?How? K=0 and 2 onlyCoriolis + D-D0.036 Full data setCoriolis + D-D + K-staggering0.111 Full data setCoriolis + D-D + K-staggering and its J-dependence The J-dependence of the K-staggering is the same as allowing the two tunnelling components of a vibrational level to have different B rotational constants.

Rotational constants for the trans-3 2 B 3 polyad (cm  1 ) Vibration Coriolis D-D T 0 ( ) (9)T 0 ( ) (7) T 0 ( ) (80)T 0 ( ) (67) 2A  a 2A  a, D K (3)  (5) B  b (6 3 / ) (78) B  b ( / ) (16) B  b ( /4 3 ) (43) k 4466  (19) k 4466, D K  (61) K-stagger S ( )S ( )4.21 (11)  4.63 (10) S ( )S ( )1.62 (152)  3.68 (147) S ( ), D J S ( ), D J (fixed)0.034 (6) Rotation A ( ) (22)A ( ) (20) B  C ( )B  C ( ) (85) (179) 101 data points; r.m.s. error = Rotational constants for and interpolated B ( )B ( ) (33) (36) _ _

29 cm  1 K-staggering