A. J. Merer Institute of Atomic and Molecular Sciences, Taipei, Taiwan

Least squares fitting of perturbed vibrational polyadsnear the isomerization barrier in the S1 state of C2H2

J. H. BarabanP. B. Changala

R. W. Field Massachusetts Institute of Technology, Cambridge, MA

B = bending[31B1 = 3141 plus 3161]

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 stretch34B2

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 (au) = 764.90; 6 (bu) = 768.26 cm1

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 Ja Ga = 2 A Ja(Qtr a P)

acting between harmonic levels |v4> and |v6>, has imaginary matrix elements :

To get rid of the i s, multiply all |v6> functions by (i)v6. Everything then becomes real.

<v4+1 v6; K | H | v4 v6+1; K> = 2 i A a46 K √(v4+1)(v6+1)

<v4 v6+1; K | H | v4+1 v6; K> = 2 i A a46 K √(v4+1)(v6+1)

where = ½ [√4/6 + √6/4 ]

Successive transformations of the HamiltonianWith 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!

C2H2, A1Au: Rotational constants for the B3 polyad~

Vibrational origins, relative to T00 at 42197.57 cm1

T0 (43) T0 (63)

2295.10 (10)2314.79 (9)

T0 (4261)T0 (4162)

2321.59 (7)2279.47 (9)

Coriolis2Aa 18.363 (9) Bb 0.802 (3)2Aa, DK 0.023 (2)

Darling-Dennisonk4466 51.019 (9) k4466, DK 0.224 (8)

RotationA (63) 13.00 (5) A (43) 13.12 (5)

BC (63) 0.1406 (72) BC (43) 0.0798 (102)1.0870 (28) 1.0685 (30)B (43)B (63)

Parameters for the other two levels are interpolated, except A for 4261 and 4162, which are corrected by 0.41 (5) cm1.+

r.m.s.error = 0.028 cm1

cm1

Comparison of bending polyad fits (cm1)B2 31B2 51B232B2

k4466 51.68 (2) 60.10 (17) 66.50 (12) 51.60 (40)

2Aa 18.45 (1) 20.625 (17) 23.56 (11) 18.03 (10)

Bb 0.798 (2) 0.784 (5) 0.808 (14) 0.751 (15)

x46 11.39 (8) 28.40 (4) 37.97 (2) 13.2 (8)

r.m.s 0.012 0.032 0.024 0.025

B3 31B3 32B3

k4466 51.02 (1) 57.87 (12)

Bb

2Aa 18.36 (1) 20.60 (5)

0.802 (3) 0.779 (8)

r.m.s. 0.028 0.045* 0.036

Broken polyad

* Combined fit with the interacting 2131B1 polyad

Final least squares fitto the interacting 31B3

and 2131B1 polyads

Dots are observed termvalues and lines are calculated. Some of thehigher-order rotational constants are not very realistic, but theyreproduce the J-structure!

= 0.045 cm1

Darling-Dennison resonance

3163

314162

3143

314261

213141

213161

k266 = 8.66 ± 0.16 cm1

k244 = 7.3 ± 1.1 cm1

3163 lies far belowthe rest of the polyad;x36 is very large!

43700 43750 43800

44700 44750 44800

45650 45700 45750 45800

E / cm 1

46650 46700 46750 46800

33B2

32B2

31B2

B2 4262 4161

Excitation of 3 unravels the bending polyads

K=0 K=2

K=0 au & bu, B5K=2, B5

K=4K=3, 3151

IR-UV double resonance via X, 3+4 Q branch

E / cm 1

46175.4 cm 1 46227.1 cm 1

46160 46180 46200 46220 46240

One photon excitation from X, v=0

0 1 2 3 4 5 623456P R

46192.2 cm 1

K=1

8

3 4 5

7 6

4

5 345 2 3 4 567

5P RQ

P

R

Q

P

R

QRQ

6 12

1

5 34 5

4 5234563 2 3

2 3 4 51

23456 2 3 4 5 61 21 3 4 535 74P R P RQ6 27 4358

123456 Q

P Q R3 4 5 6 70 1 2345 123456 27

213142

42 3R31B4

45 3 267

345

6 45 321

5 4 3 2

Q

Q

PP

K' =1K' =2

31B4

~

~

A.J. Merer, A.H. Steeves, J.H. Baraban, H.A. Bechtel, R.W. Field, J. Chem. Phys. 134, 244310 (2011).

C2H2: the cis-3161 band group (46200 cm1)

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)

Rotational constants from fitting of cis-3161 (cm1)

T0 ± 0.0246165.36

A ± 0.0114.02

B ± 0.00111.1258

C ± 0.00131.0370

103 JK ± 0.300.74

S 3.906 ± 0.020 †

† S is the shift of the K=1 levels above the position predictedfrom the K=0 and 2 levels (K-staggering parameter).

r.m.s. error = 0.019

Data from K= 0 – 2 only.

Cis-C2H2 does not show bending polyad structure, since 4 6 = 250 cm1, compared to 3 cm1 for trans-C2H2.

K-staggering is easy to model for cis-C2H2, and fortrans levels that are not part of polyads. For trans-bending polyads it is a serious extra complication.

K-staggering

K-staggering in trans-53

The ratio of the K=31 and 20 intervals is 1.993:1, close to the expected 2:1. The K=10 interval should be one quarter of the K=20 interval (16.46 cm1), but is 6.31 cm1 greater.

The trans level 53 lies about 60 cm1 above the calculated isomerization barrier. Watson (JMS 98, 133 (1982)) has given the energies of its K=03 states:

47237.19 259.96 303.04 391.19

65.85

22.77

131.23

K T0 / cm1

0123

Conclusion: there is a K-staggering of +6.31 cm1 in trans-53

46300 46400 46500 46600 46700

32B3

cis-

cis-32

2132B1

2151

B5

K' = 01

1

2

0

2

0 1 201 2

01 20 1 2

1 2

Bu Au AuBu

E / cm

Bu Au

BuK' =

K' =

K' =

K=1

4161

K=1

10 2

51B2

0 1 2K' = 01 2 1 20

cis-42

K=1

cis-63

K=0

cis-3162

K=1

cis-63

K=2

Bu BuAu

Stick diagram of the 32B3 polyad region IR-UV double resonance

Steps in the fitting of the trans-32B3 polyad

Full data set Coriolis + D-D 0.989

r.m.s./ cm1What? How?

K=0 and 2 only Coriolis + D-D 0.036

Full data set Coriolis + D-D + K-staggering 0.111

Full data set Coriolis + D-D + K-staggering and its J-dependence

0.036

The J-dependence of the K-staggering is the same as allowingthe two tunnelling components of a vibrational level to havedifferent B rotational constants.

Rotational constants for the trans-32B3 polyad (cm1)

Vibration

Coriolis

D-D

T0 (3243) 46412.97 (9) T0 (3263) 46291.90 (7)

T0 (324162) 46516.49 (80) T0 (324261) 46504.60 (67)

2Aa

2Aa, DK

22.40 (3)

0.027 (5)

Bb (63/4162) 0.832 (78)

Bb (4162/4261) 0.661 (16)

Bb (4261/43) 0.436 (43)

k4466 45.73 (19) k4466, DK 0.339 (61)

K-stagger S (3243) S (3263)4.21 (11) 4.63 (10)S (324162) S (324261)1.62 (152) 3.68 (147)

S (3263), DJS (3243), DJ 0.025 (fixed) 0.034 (6)

Rotation A (3243) 16.682 (22) A (3263) 14.563 (20)

BC (3243) BC (3263)0.0713 (85) 0.0552 (179)

101 data points; r.m.s. error = 0.036Rotational constants for 324162 and 324261 interpolated

B (3243) B (3263)1.0817 (33) 1.0779 (36)__

E / cm

1131B1

34B1

46750 46800 46850 46900 46950 47000

K=0, buK=0, au

K=1 K=2

Observed

Predicted

4675

0 K

=1

4675

5 K

=0

b u

4676

7 K

=1

4677

5 K

=0

b u

4678

1 K

=2 46

790

K=

1

4680

0 K

=2

1 1 22

11 2

4693

4 K

=2

4693

9 K

=1

4694

7 K

=2

4699

6 K

=0

a u

4701

6 K

=1

4697

8 ?

K=

0 a u

4696

8 K

=1

4689

3 K

=0

b u

4688

6 K

=2

4687

9 K

=1

4687

8 K

=2

4686

7 K

=1

4684

9 K

=2

4684

8 K

=0

a u

4684

2 K

=0

b u

4683

5 K

=1

4682

7 K

=0

b u46

822

K=

0 a u

4685

3 K

=1

4683

3 K

=1

4681

8 K

=0

a u

?

C2H2, K=0 2 ungerade levels, 46730-47020 cm

31B5

2131B3

1

1

2

21

1

2

2

1

1

1

2

2

00

00

0 0 0 0

0bu

0au

0 0

bubu

bu

au

au

au

au

aubu

bu

29 cm1

K-staggering