1. 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

2. B = bending [3 1 B 1 = 3 1 4 1 plus 3 1 6 1 ] 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

3. A-axis Coriolis Darling- Dennison

4. 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 ) = 764.90; 6 (b u ) = 768.26 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

5. 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 ]

6. 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!

7. C 2 H 2, A 1 A u : Rotational constants for the B 3 polyad ~ Vibrational origins, relative to T 00 at 42197.57 cm  1 T 0 (4 3 ) T 0 (6 3 ) 2295.10 (10) 2314.79 (9) T 0 (4 2 6 1 ) T 0 (4 1 6 2 ) 2321.59 (7) 2279.47 (9) Coriolis 2A  a 18.363 (9) BbBb 0.802 (3) 2A  a, D K  0.023 (2) Darling-Dennison k 4466  51.019 (9) k 4466, D K 0.224 (8) Rotation A (6 3 ) 13.00 (5) A (4 3 ) 13.12 (5) B  C (6 3 ) 0.1406 (72) B  C (4 3 ) 0.0798 (102) 1.0870 (28)1.0685 (30) B (4 3 ) B (6 3 ) Parameters for the other two levels are interpolated, except A for 4 2 6 1 and 4 1 6 2, which are corrected by 0.41 (5) cm  1.  + r.m.s.error = 0.028 cm  1 cm  1

8. Comparison of bending polyad fits (cm  1 ) B2B2 31B231B2 51B251B2 32B232B2 k 4466  51.68 (2)  60.10 (17)  66.50 (12)  51.60 (40) 2A  a 18.45 (1)20.625 (17)23.56 (11)18.03 (10) BbBb 0.798 (2)0.784 (5)0.808 (14)0.751 (15) x 46 11.39 (8)28.40 (4)37.97 (2)13.2 (8) r.m.s0.0120.0320.0240.025 B3B3 31B331B3 32B332B3 k 4466  51.02 (1)  57.87 (12) BbBb 2A  a 18.36 (1) 20.60 (5) 0.802 (3) 0.779 (8) r.m.s.0.0280.045*0.036 Broken polyad * Combined fit with the interacting 2 1 3 1 B 1 polyad

9. Final least squares fit to the interacting 3 1 B 3 and 2 1 3 1 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!  = 0.045 cm  1 Darling-Dennison resonance 31633163 314162314162 31433143 314261314261 213141213141 213161213161 k 266 = 8.66 ± 0.16 cm  1 k 244 =  7.3 ± 1.1 cm  1 3 1 6 3 lies far below the rest of the polyad; x 36 is very large!

10. Excitation of 3 unravels the bending polyads

11. C 2 H 2 : the cis-3 1 6 1 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, 244310 (2011) K-staggering (Tunnelling splitting)

12. Rotational constants from fitting of cis-3 1 6 1 (cm  1 ) T0T0 ± 0.02 46165.36 A ± 0.01 14.02 B ± 0.0011 1.1258C ± 0.00131.0370 10 3  JK ± 0.30 0.74 S 3.906 ± 0.020 † † 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.

13. 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

14. 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: 47237.19 259.96 303.04 391.19 65.85 22.77 131.23 KT 0 / cm  1 01230123 Conclusion: there is a K-staggering of +6.31 cm  1 in trans-5 3

16. 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 0.036 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.

17. Rotational constants for the trans-3 2 B 3 polyad (cm  1 ) Vibration Coriolis D-D T 0 (3 2 4 3 ) 46412.97 (9)T 0 (3 2 6 3 )46291.90 (7) T 0 (3 2 4 1 6 2 )46516.49 (80)T 0 (3 2 4 2 6 1 )46504.60 (67) 2A  a 2A  a, D K 22.40 (3)  0.027 (5) B  b (6 3 /4 1 6 2 ) 0.832 (78) B  b (4 1 6 2 /4 2 6 1 ) 0.661 (16) B  b (4 2 6 1 /4 3 ) 0.436 (43) k 4466  45.73 (19) k 4466, D K  0.339 (61) K-stagger S (3 2 4 3 )S (3 2 6 3 )4.21 (11)  4.63 (10) S (3 2 4 1 6 2 )S (3 2 4 2 6 1 )1.62 (152)  3.68 (147) S (3 2 6 3 ), D J S (3 2 4 3 ), D J 0.025 (fixed)0.034 (6) Rotation A (3 2 4 3 )16.682 (22)A (3 2 6 3 )14.563 (20) B  C (3 2 4 3 )B  C (3 2 6 3 ) 0.0713 (85) 0.0552 (179) 101 data points; r.m.s. error = 0.036 Rotational constants for 3 2 4 1 6 2 and 3 2 4 2 6 1 interpolated B (3 2 4 3 )B (3 2 6 3 )1.0817 (33)1.0779 (36) _ _

18. 29 cm  1 K-staggering