1. Rovibronic Analysis of the State of the NO 3 Radical Henry Tran, Terrance J. Codd, Dmitry Melnik, Mourad Roudjane, and Terry A. Miller Laser Spectroscopy Facility The Ohio State University Columbus, Ohio 43210

2. Introduction Moderate resolution spectrum of the. state of NO 3 with assignments.

3. U v=0 v=1 e e a2a2 a1a1 No JT JT 1 +JT 2 e a2a2 a1a1 e Strong JT 2 Physically, this should correspond to localization in one of three minima, corresponding to lowered symmetry molecular structure. Degeneracy is ro-vibronic Near triple degeneracy Influence of JT Coupling on Rotational Structure

4. Problem Formulation There are two approaches to resolving the rotational spectra 0 0 0 0 0 0 0 0 Includes Jahn-Teller effects and coupling between vibronic levels. H d is an oblate symmetric top including centrifugal distortion and spin orbit where vibronic levels are isolated. [1] [1] Mourad Roudjane, Terrance J. Codd, and Terry A. Miller. High Resolution Cavity Ring Down Spectroscopy of the 3 1 0 and 3 1 0 4 1 0 Bands of the A 2 E″ State of NO 3 Radical, ISMS, 2013.

5. Parallel Band Analysis We have analyzed parallel bands.  Spectra collected by high resolution, jet-cooled, cavity ring down spectroscopy. [1] The parallel bands are fit with an oblate symmetric top Hamiltonian including centrifugal distortion and spin-rotation. [1]  We use the ground state constants recorded by Kawaguchi et al. [2]  Transitions were assigned iteratively and a least squares regression of free parameters was used to fit the simulation. [1] Mourad Roudjane, Terrance J. Codd, and Terry A. Miller. High Resolution Cavity Ring Down Spectroscopy of the 3 1 0 and 3 1 0 4 1 0 Bands of the A 2 E″ State of NO 3 Radical, ISMS, 2013. [2] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013). 0 0 0 0 0 0 0 0

6. Simulation of [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

7. Simulation of [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

8. Simulation of [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

9. Simulation of [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

10. Simulation of [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

11. Simulation of Intensity of high J lines (25/2- 33/2) at 8343 cm -1 are not well simulated. Weaker lines on blue end of spectrum missing from simulation.

12. Simulation of Lower rotational temperature. Lines are less dense and spectrum is well simulated.  Very good experimental spectrum. [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

13. Simulation of [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

14. Simulation of [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

15. Simulation of [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

16. Simulation of Split lines.

17. Simulation of [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

18. Simulation of [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

19. Simulation of [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

20. Simulation of [1] [1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

21. Comparison of Simulations [2] [1] E. Hirota, T. Ishiwata, K. Kawaguchi, M. Fujitake, N. Ohashi, and I. Tanaka, J. Chem. Phys, 107, 2829 (1997). [2] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO 3 : Perturbation Analysis of the ν 3 +ν 4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013). [1]

22. Split Line Analysis Certain experimental lines “split” from the simulated lines. R-Branch of P-Branch of

23. Split Line Analysis We assume the split occurs from an accidental degeneracy of a bright level and a dark level which causes the two levels to be mixed allowing the dark level to borrow intensity from the bright state. Define where I is intensity and B and R refer to the blue and red end of the doublet respectively. where is the frequency in cm-1 and B and R are as defined above. Then we may derive [4] [4] Codd, Terrance. Spectroscopic Studies of the State of NO 3. Dissertation, The Ohio State University (2014).

24. Split Line Analysis Assignments of split lines in (RMS = 148 Mhz) We have calculated the estimated unperturbed frequency of each split line. One of each pair of unperturbed values should match predicted frequency given by the model. Difference is calculated and most differences are within experimental error. Where does the dark level come from?

25. Perpendicular Band Analysis

26. 0 0 0 0 0 0 0 0 Includes Jahn-Teller effects and coupling between vibronic levels. H d is an oblate symmetric top where vibronic levels are isolated.

27. Perpendicular Band Analysis U v=0 v=1 e e a2a2 a1a1 e a2a2 a1a1 e Includes Jahn-Teller effects and coupling between vibronic levels. H d is an oblate symmetric top where vibronic levels are isolated. 0 0 0 0 0 0 0 0

28. Projection of the full ro-vibronic operator to vibronic 3x3 basis Bra-ket operator form: Matrix form: The Hamiltonian An A state becomes isolated

29. Projection of the full ro-vibronic operator to vibronic 3x3 basis Bra-ket operator form: Matrix form: The Hamiltonian We use “extended” projection operator including time- reversal operator and Hermitian conjugation operations to build the full Hamiltonian.

30. In which, diagonal part of Hamiltonian: Coupling parameters: are vibrational coordinate dependent components of rotational tensor. To illustrate the theoretical approach, we will not explicitly consider spin effects. Vibronic basis set is delocalized. To treat cases with strong JT 2 interaction, we need to develop rovibronic Hamiltonian in basis set of vibronic functions localized at the wells of PES resulting from JT 2 interaction. The Hamiltonian

31. A1A1 ExEx EyEy S1S1 S2S2 S3S3 We first perform {A 1, E +, E - }  {A 1, E x, E y } basis transformation. For illustration we construct vibronic basis from localized symmetric wavefunctions using projection operators Delocalized basis {A 1, E x, E y } Localized basis {S 1, S 2, S 3 } Unitary transformation Localized vs. Delocalized Basis

32. The Hamiltonian in Localized Basis To show parameter relationship, we express localized Hamiltonian in terms of “delocalized” parameters Diagonal term: Asymmetric rigid rotor:

33. The Hamiltonian in Localized Basis To show parameter relationship, we express localized Hamiltonian in terms of “delocalized” parameters Diagonal term: Asymmetric rigid rotor: Transformation properties under axis rotation R  (unitary): So all diagonal elements have the same eigenvalues.

34. The Hamiltonian in Localized Basis For the Hamiltonian in {S 1, S 2, S 3 } to have triply-degenerate eigenvalues, all off-diagonal terms must vanish. (Wells are isolated and vibronic levels are truly degenerate.) This is the other limit corresponding to a triply degenerate asymmetric rotor.

35. Summary Rotational bands in the state were collected using high resolution, jet-cooled, cavity ring down spectroscopy. The parallel bands were fit using an oblate symmetric top model.  Despite the fact that the bands analyzed belonged to Jahn-Teller active modes, it is likely that the average vibrational structure is symmetric. Split lines were observed in the rotational bands and these are possibly a result of a degeneracy between a dark level and a bright level.  Our analysis lends credibility to this hypothesis. It is possible to continuously transform a Hamiltonian from a limit of weak JT effects to strong JT effects. We will fit the perpendicular bands using the more complete model taking into account coupling between vibronic levels and JT terms.

36. Acknowledgements The Miller Group  Dr. Terry A. Miller  Dr. Dmitry Melnik  Dr. Mourad Roudjane  Terrance J. Codd  Dr. Neal D. Kline  Meng Huang NSF OSU

37. Simulation We use the ground state constants presented by Fujimori et al. 1 Transitions were assigned iteratively and a least squares regression of free parameters was used to fit the simulation. The parallel bands were fit using only the part of the presented Hamiltonian corresponding to an oblate symmetric top with spin orbit and centrifugal distortion and satisfactory fits were obtained.  and are presented. [1] R. Fujimori, N. Shimizu, J. Tang, T. Ishiwata, and K. Kawaguchi. Fourier transform infrared spectroscopy of the ν 2 and ν 4 bands of NO 3. Journal of Molecular Spectroscopy, 283:10 - 17, 2012.

38. Question : Under what conditions are all eigenvalues (at least 3-fold) degenerate? Answer : All off-diagonal elements  H must vanish (i.e. wells are truly isolated – no tunneling interaction present) and ΔE 1 vanishes (the a and e levels are truly vibronically degenerate). The Hamiltonian in Localized Basis The transformation is a continuous transformation between high symmetry and the asymmetric top. To show parameter relationship, we express localized Hamiltonian in terms of “delocalized” parameters

39. Simulation of [2] R. Fujimori, N. Shimizu, J. Tang, T. Ishiwata, and K. Kawaguchi. Fourier transform infrared spectroscopy of the ν 2 and ν 4 bands of NO 3. Journal of Molecular Spectroscopy, 283:10 - 17, 2012. [3] E. Hirota, T. Ishiwata, K. Kawaguchi, M. Fujitake, N. Ohashi, and I. Tanaka, J. Chem. Phys, 107, 2829 (1997). [2] Hirota et al. Constants Our Constants [3]

40. Perpendicular Band Analysis

42. Transformation properties under axis rotation R  (unitary): Therefore,,, have the same set of eigenvalues Diagonal terms: Off-diagonal terms: Diagonal term H d (0): Asymmetric rigid rotor: The Hamiltonian in Localized Basis