1. DMITRY G. MELNIK AND TERRY A. MILLER The Ohio State University, Dept. of Chemistry, Laser Spectroscopy Facility, 120 W. 18th Avenue, Columbus, Ohio 43210 ANALYSIS OF THE ROTATIONALLY-RESOLVED SPECTRA OF THE VIBRONICALLY ACTIVE MOLECULES

2. Background Rotational structure of the isolated vibronic states of the Jahn-Teller active molecules has been well studied. a,b,c In many cases complications arise: i. Physical significance of the parameters is not transparent c ii. Simple models fail to predict spectra to experimental accuracy iii. Fundamentally fail in case of overlapping spectra involving closely spaced interacting bands. d,e Models are needed that permit for the fitting of experimental data of such spectra which present transparent interpretation of the determined parameters. We aim at building a physically simple model that can harness growing computational powers to analyze complex rotational structure, e.g., within vibronically excited states. a J. K. G. Watson, J. Mol. Spectrosc, 103, 125 (1984) c J. M. Brown, Mol. Phys., 20, 817, (1971) b J. T. Hougen, J. Mol. Spectrosc, 81, 73 (1980) d D. G. Melnik, T. A Mille and J. Liu, TI15, 67 th MSS, 2012 e M. Roudjane, T. J. Codd and T. A. Miller, TI03, 67 th MSS, 2012

3. Traditional approach to the analysis of rotational spectra. 1. Full Hamiltonian 2. Isolation of the vibronic problem 3. Constructing the effective rotational Hamiltonian for an isolated state a. zeroth order (expectation values) b. contact transformation to account for vibronic interactions with other states

4. Isolated state vs. Coupled state Hamiltonian. Select an isolated group of closely spaced interacting levels of interest Perform contact transformation (1) to isolate the group from other levels Perform contact transformation (2) to reduce full Hamiltonian to the ERH This can be rather cumbersome!

5. Nature and symmetry of rovibronic coupling terms. 1.We separate vibronic and rotational problems We assume that the vibronic Hamiltonian is solved and evaluate H ROT + H SO in the basis set of its eigenfunctions. 2.We recognize that generally H ROT and H SO contain vibronic operators, where 3.We choose the target MS group (in the present case, D 3h ) 4.We evaluate vibronic matrix elements of H mR = H mROT + H SO in the basis of eigenfunctions of the chosen group of vibronic levels and characterize the resulting spin and rotational operators according to the total spin-vibronic symmetry.

6. Interaction map between vibronic levels of a D 3h molecule. Spin-rovibronic Hamiltonian,  (  ) – notation inherits the phase of the multiplying spin and rotational operators H 1 Hamiltonian is antisymmetric to time-reversal, vanishes within nondegenerate states

7. Vibronic bands of NO 3 radical 4. Experimental spectrum: 550+ transitions (with S/N > 2). Many transitions are split Simulation: 176 resolved transitions Exp. Sim. a a M.-W. Chen, Ph.D. Thesis, OSU, 2011 1.A large number of transitions appears to be “split” 2.Split components vary in relative intensity across the spectrum 3.Quantum number dependence of the splitting magnitude is not fully understood. R 0 (N) + R 0 (N) - R 3 (N) + R 3 (N) - N=3 N=4 N=5

8. Possible origins of the line doubling 1.Removal of K-degeneracy by dynamic JT-distortion. Only levels of total and symmetry are allowed by the nuclear spin statistics. However, hence at least one of these levels is nuclear spin-forbidden 2. transitions gain intensity through a spin-dependent interaction F 2 (J’=N’-1/2) F 1 (J’=N’+1/2) F 2 (J”=N”-1/2) F 1 (J”=N”+1/2) Interacting levels are separated by ~ 2BN, therefore substantial intensity borrowing via this mechanism is highly unlikely. 3. Interaction with nearby “dark levels” ? F 1 (J’=N’-1/2) F 2 (J’=N’+1/2)

9. Perturbation by a “dark level” We can deduce the separation of the unperturbed (basis) levels and, as well is the magnitude of the coupling between them Approximation: two interacting levels, only one of which is “bright”

10. Vibronic structure of NO 3 NO 3 belongs to D 3h MS group and exhibits strong Jahn-Teller effect No vibronic Linear Linear and Quadratic Interaction Jahn-Teller Jahn-Teller Single vibrational mode, v 4 =1 a < 4 cm -1 14.3 cm -1 2.6 cm -1 Multiple vibrational modes E” level outside of the manifold Likely candidate: an E” level close to A” 1 level! a T. Codd, M. Roudjane, M.-W. Chen and T. A. Miller, WJ05, 68 th MSS, 2013

11. Interaction between a” 1 and e” levels Symmetry-supported interaction between a” 1 and e” vibronic levels: Selection rules: Interaction strength: Energy gap: Expected line doubling pattern (from 2x2 interaction scheme): Rapidly increases with N Rapidly increases with K

12. Interaction strength Energy gap, deperturbed F 1 component F 2 component Deperturbed energy gaps and interaction strength 1.Weak N-dependence for both (E  -E  ) and h. for N>4 2. Similar doublings for K=6 are also observed abnormally strong perturbation, likely and accidental resonance with a “third party” level

13. Possible interaction schemes No dramatic and systematic dependence of the (E  -E  ) on quantum numbers  the interaction is likely to be diagonal in K. Hamiltonian perturbing level Remark vibronic level of the electronic state within ~ 1GHz of the bright level No such level is predicted by vibronic calculations a – extremely unlikely. a T. Codd, M. Roudjane, M.-W. Chen and T. A. Miller, WJ05, 68 th MSS, 2013 highly excited vibronic level of the electronic state within ~ 1GHz of the bright level Requires the presence of the E” electronic character in the vibronic eigenfunction of the dark level – extremely unlikely. highly excited vibronic level of the electronic state highly excited vibronic level of the electronic state within ~ 1GHz of the bright level Plausible

14. Strength and chance of interaction z- component of the Coriolis interaction: Estimate: Contain derivatives of the rotational constants with respect to non-totally symmetric coordinates Chance of finding a perturbing level by calculating vibronic level density at the energy of the level a. a R. A. Marcus, J. Chem. Phys, 20, 359 (1952) For E=7600 cm -1, the density of the vibronic levels in the ground state is ~ 3.6 levels per cm -1. (a crude approximation, not accounting for anharmonic and vibronic coupling effects).

15. Preliminary results and the next steps 1.We have proposed an approach for analyzing the spectra of the strongly coupled closely lying levels based exclusively on the symmetry arguments. 2.Symmetry argument allows for qualitative characterizing the problem and defining the effective rovibronic Hamiltonian suitable for quantitative treatment. 3.It has been shown that it is quite unlikely that the doubling of rotational levels of the 4 1 state is accountable for the interaction within the levels of the A electronic state, but quite possible due to the interaction with the highly excited levels of the ground state. 4.Parameters of the vibronic levels such as 4 1 levels of the excited state can be obtained by the analysis of the deperturbed level structure. 5. Development of the computational multifold models for quantitative analysis of perturbed spectra are underway.

16. Acknowledgements Colleagues: Dr. Mourad Roudjane Dr. Rabi Chhantyal Pun Terrance Codd, Neal Kline OSU NSF UoL

17. Par. GHz value A9.338(3) B8.064(3) C4.893(3)  aa +4.26(2)  bb -4.59(2)  cc +1.72(1)  bc 1.96(3) Expected to be negative in the second order PT Expected to vanish in the second order PT Example: isopropoxy radical spin-orbit interaction PJT interaction In principle, the exact value of the ERH parameter, e.g.,  aa, can be calculated as a sum of m th order perturbation contributions, rigorous calculations GHz