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