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
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)
Einstein coefficient / s-1
40
35
30
25
20
15
Fundamental band
P branch
0
2
4
E. Arunan et al, J. Chem. Phys. 97, 3 (1992)
J
6
8
10
Results I: testing our method (HCl)
Einstein coefficient / s-1
40
35
30
25
20
15
Fundamental band
P branch
0
2
4
E. Arunan et al, J. Chem. Phys. 97, 3 (1992)
J
6
8
10
Results I: testing our method (HCl)
Einstein coefficient / s-1
20
18
16
14
12
Fundamental band
R branch
0
2
4
E. Arunan et al, J. Chem. Phys. 97, 3 (1992)
J
6
8
10
Results I: testing our method (HCl)
Einstein coefficient / s-1
20
18
16
14
12
Fundamental band
R branch
0
2
4
E. Arunan et al, J. Chem. Phys. 97, 3 (1992)
J
6
8
10
Results II: Einstein A coefficients for NO
Einstein coefficient / s-1
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
Einstein coefficient / s-1
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
Einstein coefficient / s-1
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
Einstein coefficient / s-1
8
Fundamental band (Ω = ½)
R branch
•
7
2∏ , 2∏
1/2
3/2
• Δv = 1, 2
6
• v = 10
• J = 20.5
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:
methods.
comparison with results from ab initio
Acknowledgements
• John Ogilvie
• Ken Brown’s group