1. Einstein A coefficients for vibrational-rotational transitions of NO
Mauricio Gutiérrez1 and John Ogilvie2
1Georgia Institute of Technology, Atlanta GA
2Universidad de Costa Rica, San José, Costa Rica
68th International Symposium on Molecular Spectroscopy
June 20th, 2013

2. 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.
Considerable uncertainty concerning vibrational distribution of the NO products from equation (2).
Our method has been applied to other molecules, but not to NO.
P. Houston et al, J. Phys. Chem. A 114, (2010)

3. 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)

4. 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)

5. 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)

6. 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)

7. 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)

8. TDM II: radial contribution
Herman-Wallis approach:
Apply perturbation theory.
Re-express the matrix elements in terms of a purely vibrational part and a vibrational-rotational interaction.
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)

9. TDM II: radial contribution
Herman-Wallis approach:
Apply perturbation theory.
Re-express the matrix elements in terms of a purely vibrational part and a vibrational-rotational interaction.
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)

10. TDM III: vibrational part
Y.-P. Lee et al, Infrared Physics and Technology 47, 227 (2006).

11. TDM III: vibrational part
Y.-P. Lee et al, Infrared Physics and Technology 47, 227 (2006).

12. 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.
Instead of solving explicitly the wavefunctions for the vibrational states, you just obtain symbolic expressions for the matrix elements of the powers of x.
Maple function which gives you the symbolic expression for the matrix element for an arbitrary perturbation order.
J. F. Ogilvie, The vibrational and rotational spectrometry of diatomic molecules (Academic Press, 1998)

13. Results I: testing our method (HCl)40 35 30
Einstein coefficient / s-1 25 20
Instead of solving explicitly the wavefunctions for the vibrational states, you just obtain symbolic expressions for the matrix elements of the powers of x.
Maple function which gives you the symbolic expression for the matrix element for an arbitrary perturbation order. 15
Fundamental band
P branch 2 4 6 8 10 J
E. Arunan et al, J. Chem. Phys. 97, 3 (1992)

14. Results I: testing our method (HCl)40 35 30
Einstein coefficient / s-1 25 20
Instead of solving explicitly the wavefunctions for the vibrational states, you just obtain symbolic expressions for the matrix elements of the powers of x.
Maple function which gives you the symbolic expression for the matrix element for an arbitrary perturbation order. 15
Fundamental band
P branch 2 4 6 8 10 J
E. Arunan et al, J. Chem. Phys. 97, 3 (1992)

15. Results I: testing our method (HCl)20 18 16
Einstein coefficient / s-1 14
Instead of solving explicitly the wavefunctions for the vibrational states, you just obtain symbolic expressions for the matrix elements of the powers of x.
Maple function which gives you the symbolic expression for the matrix element for an arbitrary perturbation order. 12
Fundamental band
R branch 2 4 6 8 10 J
E. Arunan et al, J. Chem. Phys. 97, 3 (1992)

16. Results I: testing our method (HCl)20 18 16
Einstein coefficient / s-1 14
Instead of solving explicitly the wavefunctions for the vibrational states, you just obtain symbolic expressions for the matrix elements of the powers of x.
Maple function which gives you the symbolic expression for the matrix element for an arbitrary perturbation order. 12
Fundamental band
R branch 2 4 6 8 10 J
E. Arunan et al, J. Chem. Phys. 97, 3 (1992)

17. Results II: Einstein A coefficients for NO10
Fundamental band (Ω = ½)
P branch 9 8
Einstein coefficient / s-1 7
Instead of solving explicitly the wavefunctions for the vibrational states, you just obtain symbolic expressions for the matrix elements of the powers of x.
Maple function which gives you the symbolic expression for the matrix element for an arbitrary perturbation order. 6 5 10 15 20 J
M. Gutiérrez and J. F. Ogilvie, unpublished

18. Results II: Einstein A coefficients for NO5
Fundamental band (Ω = ½)
Q branch 4 3
Einstein coefficient / s-1 2 1 5 10 15 20 J
M. Gutiérrez and J. F. Ogilvie, unpublished

19. Results II: Einstein A coefficients for NO8
Fundamental band (Ω = ½)
R branch 7
Einstein coefficient / s-1 6 5 5 10 15 20 J
M. Gutiérrez and J. F. Ogilvie, unpublished

20. Results II: Einstein A coefficients for NO8
Fundamental band (Ω = ½)
R branch
2∏1/2, 2∏3/2
Δv = 1, 2
v = 10
J = 20.5 7
Einstein coefficient / s-1 6 5 5 10 15 20 J
M. Gutiérrez and J. F. Ogilvie, unpublished

21. 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.

22. Acknowledgements
John Ogilvie
Ken Brown’s group