1. N. Moazzen-Ahmadi, J. Norooz Oliaee
A Hamiltonian to obtain a global frequency analysis of all the vibrational bands of ethane
N. Moazzen-Ahmadi, J. Norooz Oliaee
Institute for Quantum Science and Technology
Department of Physics and Astronomy
University of Calgary
Before I start, I would like to thank Jon Hougen for countless hours of discussion on group theory as pertains to ethane and for teaching me the finer aspects of group theory.
Canadian space agency

2. Motivation
Ethane is used as a tracer in the atmospheres of Jovian planets to understand the methane cycle.
To provide high quality line parameters for HITRAN and GEISA databases.
To study torsion mediated vibrational interactions.
On this slide I have listed some motivation for why we are doing this work. Ethane is one of the important hydrocarbon tracers in planetary atmospheres and their moons. It is used to understand the methane cycle in these atmospheres.
The Q-branch of the torsional fundamental around 30 micron and all other bands systems of ethane at higher frequency have been observed in planetary spectra. Therefore, there is a need for high quality laboratory line parameters to properly interpret these spectra.
Of course, ethane is intrinsically interesting because of its torsion mediated vibrational interactions, and so this talk is on our efforts to develop a Hamiltonian which cis suitable for a global frequency analysis of all the vibrational fundamentals of ethane.

3. Symmetry classification of the Torsion-rotation operators in the group
Let me start with the first step which is the torsion-rotation Hamiltonian, appropriate for analysis of the torsional bands.
We use the symmetry of relevant operators to construct the terms in the Hamiltonian subject to the requirements that these terms be time reversal, Hermitian, and totally symmetric. The symmetry of rotational operators are given on the left column and those of the torsional operators are given on the right.

4. Torsion-rotation Hamiltonian
The torsion-rotation Hamiltonian up to quartic terms is given on this slide. We use a two step procedure to calculate the torsion-rotation energy. First, we solve for the torsional energy and the torsional eigenvectors. These are then used to obtain the matrix elements of torsional operators required for the torsion-rotation Hamiltonian matrix. This matrix which includes the lowest 9 torsional states is diagonalized to obtain the total energy. This Hamiltonian works very well. We can fit the torsional spectra to experimental accuracy.

5. Torsional bands Resolution 0.0025 cm-1 Path Length 172 m
Q-branch fundamental
P-branch
Resolution cm-1
Path Length m
Pressure Torr
Temperature K
Three segments of the torsional spectra at high resolution are shown here. The Q-branch of the fundamental, its simulation, a segment in the P-branch, its simulation and finally a segment of the Q-branch of the hot band and its simulation. Because these bands are forbidden in lowest order the torsional spectrum must be recorded under high optical density conditions.
Q-branch 24-4
N. Moazzen-Ahmadi et al., JMS (2001).

6. 1-state analysis
This is what I call the 1-state analysis, 16 parameters were used. The characteristics of the fit for the torsional bands are shown in this table.

7. Symmetry classification of the vibrational operators
Choices of phase leading to real matrix algebra:
The matrix representation of vibrational coordinates with symmetry are identical to the matrix representation of the translation operators
The matrix representation of vibrational coordinates with symmetry are identical to the matrix representation of , except for except for classes where they differ by a minus sign.
Phase convention of Papusek and Aliev is used for the matrix elements of the vibrational coordinates (real) and momenta (purely imaginary).
The symmetry classification for the doubly degenerate modes were done following two papers by Jon Hougen. Here a denotes the atoms in the frame and b the atoms in the top. This linear combination with + sign transforms according E1d and the combination with the minus sign transform according to E2d symmetry. We do our calculation using real matrix algebra and to do so we have to make certain choices of phase. These are:…….
Other choices of phase may lead to complex matrix algebra.
J.T. Hougen, Can. J. Phys., 42, 1920 (1964).
J.T. Hougen, Can. J. Phys., 43, 935 (1965).

8. 2-state analysis: 9 bandV 9
= 1
34
24 4 9
Methyl rock
The extension to 2-state analysis was done by including the lowest frequency doubly degenerate fundamental, the nu9 band. This slide shows the bands which were added in the 2-state analysis. The 3nu4 band is observable because of the strong Coriolis mixing with the ground torsional state of nu9=1. 4 gs

9. 9 band
Coriolis-type
l-doubling
Selection rules:
The Coriolis interaction is shown at the top, a 27 by 27 matrix is used to calculate the overall energy. This Hamiltonian also works well. This table shows the quality of the fit, which is extremely good. We used 26 parameters to fit over 2200 frequencies.
N. Moazzen-Ahmadi et al., JCP (2001).

10. 9 band
This slide shows four small portions of the spectrum. On the higher frequency side, torsional doubling is essentially what we expect from the difference in the barrier heights in the two states. As we move to lower frequency, there is a gradual increase in the torsional splitting, indicating larger and larger coupling between the gs and nu9. Near level crossing, the last segment, the splitting is very large. Now the transitions in the 3nu4 overtone are strong enough to be observed.
0.4 Torr, 2 m, -140 C, cm-1

11. gs 3-state analysis 1 V = 1 1 V = 1 1 2 3 41 V 9
= 1 1 V 3
= 1 1 2 3 4 gs
The 3 band of ethane is not infrared active. However it is Raman allowed. The ground torsional state of v3=1 stack interacts strongly with nu4 =4 of gs stack through Fermi interaction.

12. 3 band
Coriolis-type
Fermi-type
l-doubling
The Fermi interaction is shown at the top, The Hamiltonian matrix has now four 9*9 diagonal blocks. The characteristics of the fits between the 2-state and 3-state fits are compared in the table which shows that the standard deviation and chisqured improve in the 3-state fit. 37 parameters were varied in the 3-state fit.
37 parameters were varied.

13. CD3CD3: 3 band
I don’t have a simulation for normal ethane. However, to give you some idea of how good the model is here is a simulation for fully deuterated ethane. Trace (a) is the simulation and (b) is the experimental trace.
(a) Theory; (b) Experiment
Selection rules:

14. 4-state analysis: 12  9 and 9 + 4  4
The next torsional stack is that of v12=1, the symmetric methyl rock. This vibrational fundamental is not infrared active. Furthermore, nu12 fundamental is a very weak Raman band. To obtain information on this stack, we recorded the difference band nu12-nu9, which is infrared active but weak due to the population of the initial state. The nu9+nu4-nu4 hot band whose upper state interacts with ground torsional level of v12=1 is also included in the 4-state analysis.
(a) Torsional fundamental 4; (b) 24 4; (c) 9 fundamental ; (d) 9 +4 4 hot band;
(e) 3 fundamental; (f) the forbidden 34 band, made bright by mixing with 9; and (g) the infrared-active difference band 12 9.
L. Borvayeh et al., JMS 250, 51 (2008).

15. 12  9 difference band
Resolution cm-1
Path Length m
Pressure Torr
Temperature K
Selection rules:
The nu12-nu9 was recorded under high optical density conditions. Two bands are expected, a perpendicular band, with selection rules given here, which was observed, and a parallel band which must be too weak because we did not observe it.
Perpendicular band (observed):
Parallel band (not observed):
L. Borvayeh et al., JMS 250, 51 (2008).

16. 4-state analysis: 12  9 and 9 + 4  4 bands
Coriolis-type
Fermi-type
l-doubling
M9,12
The interaction between the two torsional stacks is shown here. We had to increase the size of the Hamiltonian matrix by two more 9 by 9 blocks.

17. 4-state analysis: 12  9 and 9 + 4  4
Again, the Hamiltonian works well and chi-square decreases substantially. The Hamiltonian had 67-parameters, 65 of which were varied.
67-parameter model, 65 of which were varied.

18. 5-state analysis: 6 band
Now on to the third band system of ethane around 7 micron. To the lower side is the nu6 fundamental. To do the analysis we had to also include the torsional stack for v8=1. The experimental conditions for this spectrum are shown here. The upper state of this band interacts with v4=0 of v8=1 and v4=1 of v12=1.
Resolution cm-1
Path Length m
Pressure mTorr
Temperature K

19. 5-state analysis: 6 band
XXX
The Hamiltonian matrix has now 11 diagonal blocks and includes six torsional stacks. XXX represent the torsion mediated vibrational couplings.

20. 5-state analysis: 6 band
82-parameter Hamiltonian, 77 of which were varied.
Here are some of the new interactions which we had to include. Again the Hamiltonian works well. This was a 82 parameters Hamiltonian, 77 of which varied.
JQSRT180, 7–13(2016).

21. The standard deviation for nu6 is 37 times 10^-5 cm-1 and other components fit the same as before.

22. 6-state analysis: 8, 12 4 4, 9 24 24 bands
(In progress)
Currently, we are working on a 6-state analysis which includes frequencies from three additional bands listed at the top. We have made excellent progress using several additional intervibrational couplings and extending the Hamiltonian matrix to 121 by 121. Although we can fit the nu9+2nu4-2nu4 to experimental accuracy, our standard deviation for the other two bands is about 30 MHz. We would like to improve this to about 10 MHz. We think we can do this in the next couple months. If so, we can start thinking about the band system of ethane around 3 micron. Our Hamiltonian model has now 100 parameters with 8000 lines in the data set.
In summary, I would say the Hamiltonian for ethane is in a good shape and there is hope that we may be able to tackle the 3-micron band system relatively soon.
100-parameter Hamiltonian, ~8000 lines