Einstein A coefficients for vibrational-rotational transitions of NO

Mauricio Gutiérrez 1 and John Ogilvie 2 1 Georgia Institute of Technology, Atlanta GA 2 Universidad de Costa Rica, San José, Costa Rica 68 th International Symposium on Molecular Spectroscopy June 20 th , 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 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)

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)

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) 40 35 30 25 20 15 0 Fundamental band P branch 2 4 J 6 E. Arunan et al, J. Chem. Phys. 97, 3 (1992) 8 10

Results I: testing our method (HCl) 40 35 30 25 20 15 0 Fundamental band P branch 2 4 J 6 E. Arunan et al, J. Chem. Phys. 97, 3 (1992) 8 10

Results I: testing our method (HCl) 20 18 16 14 12 0 Fundamental band R branch 2 4 J 6 E. Arunan et al, J. Chem. Phys. 97, 3 (1992) 8 10

Results I: testing our method (HCl) 20 18 16 14 12 0 Fundamental band R branch 2 4 J 6 E. Arunan et al, J. Chem. Phys. 97, 3 (1992) 8 10

Results II: Einstein A coefficients for NO 10 Fundamental band (Ω = ½) P branch 9 8 7 6 0 5 M. Gutiérrez and J. F. Ogilvie, unpublished 10 J 15 20

Results II: Einstein A coefficients for NO 5 Fundamental band (Ω = ½) Q branch 4 3 2 1 0 5 M. Gutiérrez and J. F. Ogilvie, unpublished 10 J 15 20

Results II: Einstein A coefficients for NO 8 Fundamental band (Ω = ½) R branch 7 6 5 0 5 M. Gutiérrez and J. F. Ogilvie, unpublished 10 J 15 20

Results II: Einstein A coefficients for NO 8 Fundamental band (Ω = ½) R branch • 7 2 ∏ 1/2 , 2 ∏ 3/2 • Δv = 1, 2 6 • v = 10 5 • J = 20.5

0 5 M. Gutiérrez and J. F. Ogilvie, unpublished 10 J 15 20

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