Rovibronic Analysis of the State of the NO3 Radical

Henry Tran, Terrance J. Codd, Dmitry Melnik, Mourad Roudjane, and Terry A. Miller

Laser Spectroscopy Facility

The Ohio State University

Columbus, Ohio 43210

Introduction• NO3 has 4 vibrational modes:

1 ( - symmetric stretch) 2 ( - out of plane bend) 3 ( - degenerate stretch) 4 ( - degenerate bend)bend)

• From the vibronic analysis, we have assigned the 3 , 4 and 3 + 4 fundamental bands assuming overall strong Jahn-Teller (JT) coupling in this state.

• To confirm this assumption and obtain more information about this state, we have analyzed the rotational structure of these bands in the state. The parallel bands have vibronic symmetry The perpendicular bands have vibronic

symmetry

Moderate resolution spectrum of the . state of NO3 with assignments.

U

v=0

v=1

e

e

a2

a1

No JT JT1+JT2

e

a2

a1

e

Strong JT2

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

Problem Formulation• There are two approaches to resolving the rotational spectra

1 1

2 2

1 2

1 2

1 2

1 1 1

2 2 2

( ) 0

0 ( )

( )

( )

A E A Ed od od

A E A Ed od od

E A E A E Eod od d odE A E A E Eod od od d

A A E E

A H A E H H

A H A E H H

E H H H E H

E H H H H E

1 1

2 2

1 2

1 2

1 2

1 1 1

2 2 2

( ) 0

0 ( )

( )

( )

A E A Ed od od

A E A Ed od od

E A E A E Eod od d odE A E A E Eod od od d

A A E E

A H A E H H

A H A E H H

E H H H E H

E H H H H E

0

0 0

00

0

0

0

Includes Jahn-Teller effects and coupling between vibronic levels.

Hd 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 310 and 31

0 410 Bands of the A2E″ State of NO3 Radical, ISMS, 2013.

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 310 and 31

0 410 Bands of the A2E″ State of NO3 Radical, ISMS, 2013.

[2] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

1 1

2 2

1 2

1 2

1 2

1 1 1

2 2 2

( ) 0

0 ( )

( )

( )

A E A Ed od od

A E A Ed od od

E A E A E Eod od d odE A E A E Eod od od d

A A E E

A H A E H H

A H A E H H

E H H H E H

E H H H H E

0

0 0

00

0

0

0

Simulation of [1]

[1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

Simulation of [1]

[1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

Simulation of [1]

[1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

Simulation of [1]

[1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

Simulation of [1]

[1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

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.

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 NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

Simulation of [1]

[1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

Simulation of [1]

[1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

Simulation of [1]

[1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

Simulation of • Split lines.

Simulation of [1]

[1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

Simulation of [1]

[1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

Simulation of [1]

[1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

Simulation of [1]

[1] Kentarou Kawaguchi, Ryuji. Fujimori, Jian Tang, Takashi Ishiwata. FTIR Spectroscopy of NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

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 NO3: Perturbation Analysis of the ν3+ν4 State, J. Phys. Chem. A, 117 (50), pp 13732–13742 (2013).

[1]

Split Line Analysis• Certain experimental lines “split” from the simulated lines.

R-Branch ofP-Branch of

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 NO3. Dissertation, The Ohio State University (2014).

Split Line AnalysisAssignments 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?

Perpendicular Band Analysis• The perpendicular bands are unsatisfactorily fit by the symmetric top model.

Possible that the degenerate vibronic levels allow JT terms.

1 1

2 2

1 2

1 2

1 2

1 1 1

2 2 2

( ) 0

0 ( )

( )

( )

A E A Ed od od

A E A Ed od od

E A E A E Eod od d odE A E A E Eod od od d

A A E E

A H A E H H

A H A E H H

E H H H E H

E H H H H E

0

0 0

00

0

0

0

Perpendicular Band Analysis• The perpendicular bands are unsatisfactorily fit by the symmetric top model.

Possible that the degenerate vibronic levels allow JT terms.

• Need to consider a model with JT distortion terms and interstate coupling.

1 1

2 2

1 2

1 2

1 2

1 1 1

2 2 2

( ) 0

0 ( )

( )

( )

A E A Ed od od

A E A Ed od od

E A E A E Eod od d odE A E A E Eod od od d

A A E E

A H A E H H

A H A E H H

E H H H E H

E H H H H E

Includes Jahn-Teller effects and coupling between vibronic levels.

Hd is an oblate symmetric top where vibronic levels are isolated.

Perpendicular Band Analysis• The perpendicular bands are unsatisfactorily fit by the symmetric top model.

Possible that the degenerate vibronic levels allow JT terms.

• Need to consider a model with JT distortion terms and interstate coupling. Moreover, we would like to be able to continuously transition between limit of small and large

Jahn-Teller coupling.

U

v=0

v=1

e

e

a2

a1

e

a2

a1

e

1 1

2 2

1 2

1 2

1 2

1 1 1

2 2 2

( ) 0

0 ( )

( )

( )

A E A Ed od od

A E A Ed od od

E A E A E Eod od d odE A E A E Eod od od d

A A E E

A H A E H H

A H A E H H

E H H H E H

E H H H H E

Includes Jahn-Teller effects and coupling between vibronic levels.

Hd is an oblate symmetric top where vibronic levels are isolated.

1 1

2 2

1 2

1 2

1 2

1 1 1

2 2 2

( ) 0

0 ( )

( )

( )

A E A Ed od od

A E A Ed od od

E A E A E Eod od d odE A E A E Eod od od d

A A E E

A H A E H H

A H A E H H

E H H H E H

E H H H H E

0

0 0

00

0

0

0

Projection of the full ro-vibronic operator to vibronic 3x3 basis

Bra-ket operator form:

Matrix form:

i ijd od d od

i j i

H H H i i H i j H

1 1

1

1

1

1 1 1( )

( )

( )

A E A Ed od od

E A E Eod d odE A E Eod od d

A E E

A H A E H H

E H H E H

E H H H E

The Hamiltonian

1 1

2 2

1 2

1 2

1 2

1 1 1

2 2 2

( ) 0

0 ( )

( )

( )

A E A Ed od od

A E A Ed od od

E A E A E Eod od d odE A E A E Eod od od d

A A E E

A H A E H H

A H A E H H

E H H H E H

E H H H H E

An A state becomes isolated

Projection of the full ro-vibronic operator to vibronic 3x3 basis

Bra-ket operator form:

Matrix form:

i ijd od d od

i j i

H H H i i H i j H

1 1

1

1

1

1 1 1( )

( )

( )

A E A Ed od od

E A E Eod d odE A E Eod od d

A E E

A H A E H H

E H H E H

E H H H E

The Hamiltonian

We use “extended” projection operator including time-reversal operator and Hermitian conjugation operations to build the full Hamiltonian.

3

3

3

1

1

3

1

1

ˆ

2exp

3

ˆ ˆ

2ex

ˆ

p3

v

v

j j

v v

Е Е

iC Е E

C

A A

Е

A A

N

Е

N

iC N N C

N N

A A

1 1

1

1

12 2

1 1 1 1 12 2

1 12 2

1 1

( )

( )

( )

A E A Ed

A E EEd

A E EEd

A E E

A H A E h N h N

E h N H E h N

E h N h N H E

2 ,d zH C N B N N

In which, diagonal part of Hamiltonian:

Coupling parameters:

are vibrational coordinate dependent components of rotational tensor.

1

1

1 1

12

41

24

EExx yy xy

A Exx yy xy

h E B B iB E

h A B B iB E

B

•To illustrate the theoretical approach, we will not explicitly consider spin effects.

Vibronic basis set is delocalized.

To treat cases with strong JT2 interaction, we need to develop rovibronic Hamiltonian in basis set of vibronic functions localized at the wells of PES

resulting from JT2 interaction.

The Hamiltonian

A11

3

1

3

1

3

Ex2

6

1

6 1

6

Ey

1

2

1

2

S1 S2 S3

• We first perform {A1, E+, E-} {A1, Ex, Ey} basis transformation.

• For illustration we construct vibronic basis from localized symmetric wavefunctions using projection operators

Delocalized basis {A1, Ex, Ey}

Localized basis {S1, S2, S3}

Unitary transformation

1 1 2 3, , , ,x yA E E S S S

Localized vs. Delocalized Basis

1 1 1

1 1 1

1

1 2 3

1 1 1 1 1 11 1 11

1 1 1 1 1 11 1 12

1 1

/3 /3

/3

/3

0

2 /

13

3

2

3 3 3 3 3 3

2

3 3 3 3 3 3

3 3

A E A E A EEE EE EE

Sd d d

A E A E A EEE EE EE

Sd d d

A E EE

d

S S S

h h h h h hE E ES H H H

h h h h h hE E ES H H H

h hES H

b bb

bb b

b

1 11 1 1 11 1 2 /3

2

3 3 3 3

A E A EEE EE

Sd d

h h hb

hEb

EH H

1 1

1 1

2

2

2 2

1 12 2 ,

3 31 1

,3 3

ˆ

Sd A E z A E

A E z A E

i i

H C C N B B N N

H C C N B B N N

b e N e N

The Hamiltonian in Localized Basis

• To show parameter relationship, we express localized Hamiltonian in terms of “delocalized” parameters

1

1 1 1 12 2 212 2 2

,3 3 3 3

A E EEA E A E

z

C C C C h hEH N N N N N

2 2 2,2 4z

A B A BH CN N N N N

Diagonal term:

Asymmetric rigid rotor:

1 1 1

1 1 1

1

1 2 3

1 1 1 1 1 11 1 11

1 1 1 1 1 11 1 12

1 1

/3 /3

/3

/3

0

2 /

13

3

2

3 3 3 3 3 3

2

3 3 3 3 3 3

3 3

A E A E A EEE EE EE

Sd d d

A E A E A EEE EE EE

Sd d d

A E EE

d

S S S

h h h h h hE E ES H H H

h h h h h hE E ES H H H

h hES H

b bb

bb b

b

1 11 1 1 11 1 2 /3

2

3 3 3 3

A E A EEE EE

Sd d

h h hb

hEb

EH H

1 1

1 1

2

2

2 2

1 12 2 ,

3 31 1

,3 3

ˆ

Sd A E z A E

A E z A E

i i

H C C N B B N N

H C C N B B N N

b e N e N

The Hamiltonian in Localized Basis

• To show parameter relationship, we express localized Hamiltonian in terms of “delocalized” parameters

1

1 1 1 12 2 212 2 2

,3 3 3 3

A E EEA E A E

z

C C C C h hEH N N N N N

2 2 2,2 4z

A B A BH CN N N N N

Diagonal term:

Asymmetric rigid rotor:

• Transformation properties under axis rotation Rf (unitary):

• So all diagonal elements have the same eigenvalues.

*i iR H R H

1 1 1

1 1 1

1

1 2 3

1 1 1 1 1 11 1 11

1 1 1 1 1 11 1 12

1 1

/3 /3

/3

/3

0

2 /

13

3

2

3 3 3 3 3 3

2

3 3 3 3 3 3

3 3

A E A E A EEE EE EE

Sd d d

A E A E A EEE EE EE

Sd d d

A E EE

d

S S S

h h h h h hE E ES H H H

h h h h h hE E ES H H H

h hES H

b bb

bb b

b

1 11 1 1 11 1 2 /3

2

3 3 3 3

A E A EEE EE

Sd d

h h hb

hEb

EH H

The Hamiltonian in Localized Basis

• For the Hamiltonian in {S1, S2, S3} to have triply-degenerate eigenvalues, all off-diagonal terms must vanish. (Wells are isolated and vibronic levels are truly degenerate.)

1

1 1

1 1 21 1 10 ,

3 3 3 3

A E EE

od A E z A E

h hEH b C C N B B N N

• This is the other limit corresponding to a triply degenerate asymmetric rotor.

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.

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