Einstein A coefficients for vibrational-rotational

transitions of NO

Mauricio Gutiérrez1 and John Ogilvie2

1Georgia Institute of Technology, Atlanta GA2Universidad de Costa Rica, San José, Costa Rica

68th International Symposium on Molecular Spectroscopy

June 20th, 2013

Why Einstein A coefficients of NO?

• Einstein A coefficient: proportionality factor between the intensity of spectral lines and the relative populations.

• There is considerable uncertainty in vibrational distributions of NO products in several reactions.

• Our method has been applied to other molecules, but not to NO.

P. Houston et al, J. Phys. Chem. A 114, 11292 (2010)

Einstein A coefficients

J. F. Ogilvie, The vibrational and rotational spectrometry of diatomic molecules (Academic Press, 1998)G. Herzberg, Spectra of diatomic molecules (D. Van Nostrand Company, 1950)

Einstein A coefficients

J. F. Ogilvie, The vibrational and rotational spectrometry of diatomic molecules (Academic Press, 1998)G. Herzberg, Spectra of diatomic molecules (D. Van Nostrand Company, 1950)

Einstein A coefficients

Radial dipole moment

D. M. Dennison, Phys. Rev. 28, 318 (1926)G. Herzberg, Spectra of diatomic molecules (D. Van Nostrand Company, 1950)

Einstein A coefficients

Radial dipole moment Angular dipole moment

D. M. Dennison, Phys. Rev. 28, 318 (1926)G. Herzberg, Spectra of diatomic molecules (D. Van Nostrand Company, 1950)

Einstein A coefficients

Radial dipole moment Angular dipole moment

D. M. Dennison, Phys. Rev. 28, 318 (1926)G. Herzberg, Spectra of diatomic molecules (D. Van Nostrand Company, 1950)

TDM II: radial contribution

R. Herman and R. F. Wallis, J. Chem. Phys. 23, 4 (1955) R. H. Tipping and J. F. Ogilvie, J. Mol. Spec. 96, 442 (1982)

Herman-Wallis approach:

• Apply perturbation theory.• Re-express the matrix elements in terms of a

purely vibrational part and a vibrational-rotational interaction.

TDM II: radial contribution

R. Herman and R. F. Wallis, J. Chem. Phys. 23, 4 (1955) R. H. Tipping and J. F. Ogilvie, J. Mol. Spec. 96, 442 (1982)

Herman-Wallis approach:

• Apply perturbation theory.• Re-express the matrix elements in terms of a

purely vibrational part and a vibrational-rotational interaction.

TDM III: vibrational part

Y.-P. Lee et al, Infrared Physics and Technology 47, 227 (2006).

TDM III: vibrational part

Y.-P. Lee et al, Infrared Physics and Technology 47, 227 (2006).

TDM IV: vibrational matrix elements

Dunham’s potential

Dunham’s method:

• Use the harmonic oscillator eigenfunctions as a basis and apply perturbation theory with Dunham’s potential.

• Obtain symbolic expressions for the matrix elements.

J. F. Ogilvie, The vibrational and rotational spectrometry of diatomic molecules (Academic Press, 1998)

Results I: testing our method (HCl)

Fundamental bandP branch

E. Arunan et al, J. Chem. Phys. 97, 3 (1992)

35

30

25

20

15

Ein

stein

coeffi

cien

t /

s-1

0 2 4 6 8 10J

40

Results I: testing our method (HCl)

E. Arunan et al, J. Chem. Phys. 97, 3 (1992)

40

35

30

25

20

15

Ein

stein

coeffi

cien

t /

s-1

0 2 4 6 8 10J

Fundamental bandP branch

Results I: testing our method (HCl)

E. Arunan et al, J. Chem. Phys. 97, 3 (1992)

Fundamental bandR branch

Ein

stein

coeffi

cien

t /

s-1

0 2 4 6 8 10J

12

14

16

18

20

Results I: testing our method (HCl)

E. Arunan et al, J. Chem. Phys. 97, 3 (1992)

0 2 4 6 8 10J

Fundamental bandR branch

Ein

stein

coeffi

cien

t /

s-1

12

14

16

18

20

Results II: Einstein A coefficients for NO

M. Gutiérrez and J. F. Ogilvie, unpublished

Fundamental band (Ω = ½)P branch

5 10 15 200J

Ein

stein

coeffi

cien

t /

s-1

9

8

7

6

10

Results II: Einstein A coefficients for NO

M. Gutiérrez and J. F. Ogilvie, unpublished

Ein

stein

coeffi

cien

t /

s-1

Fundamental band (Ω = ½)Q branch

5 10 15 200J

5

4

3

2

1

Results II: Einstein A coefficients for NO

M. Gutiérrez and J. F. Ogilvie, unpublished

J

Ein

stein

coeffi

cien

t /

s-1

Fundamental band (Ω = ½)R branch

5

6

7

8

5 10 15 200

Results II: Einstein A coefficients for NO

M. Gutiérrez and J. F. Ogilvie, unpublished

J

Ein

stein

coeffi

cien

t /

s-1

Fundamental band (Ω = ½)R branch

5

6

7

8

5 10 15 200

• 2∏1/2, 2∏3/2

• Δv = 1, 2

• v = 10

• J = 20.5

Conclusions

• We have calculated the spontaneous emission coefficients for vibration-rotational transitions with Δv = 1, 2 up to v = 10 for NO in its electronic ground state.

• Using the same method, we calculated coefficients for HCl and they agree with previous results.

• Future work: comparison with results from ab initio methods.

Acknowledgements

• John Ogilvie

• Ken Brown’s group