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NEW LINE LISTS WITH INTENSITIES FOR THE
ROVIBRATIONAL TRANSITIONS WITHIN THE NH
X 3 Σ− AND OH X2Π GROUND STATES
James S.A. Brookea*, Peter F. Bernathb, Colin M. Westernc, Iouli E. Gordond, Gang
Lid, Marc C. van Hermerte & Gerrit C. Groenenboomf
a: Department of Chemistry, University of York,York, UK. b: Department of Chemistry & Biochemistry, Old
Dominion University, Norfolk,VA, USA. c: School of Chemistry, University of Bristol, Bristol, UK. d: HarvardSmithsonian Center for Astrophysics, Cambridge, MA, USA. e: Department of Chemistry, Gorlaeus Laboratories,
Leiden University,The Netherlands. f: Theoretical Chemistry, Institute for Molecules and Materials (IMM), Radboud
University Nijmegen, Nijmegen, The Netherlands.
Funded by:
Abundances
Spectrum of
the Sun
from the
ACE satelliteNH
Line lists
• Quantum number assignments, line positions
and intensities
• Positions
– Recorded directly from laboratory spectra
• Intensities
– Obtained with a combination of experimental and
theoretical methods
– Require potential energy curve and dipole
moment function
Potential Energy Curves – Experimental
Spectrum
• Spectrum is obtained from lab observations
Line
assignment
and fit
• Lines are assigned and a fit of line positions
provides molecular constants for each
vibrational level.
Molecular constants
fit
• Equilibrium constants are obtained from fitting
to 𝐺(𝑣) and 𝐵𝑣 molecular constants:
•
𝜔𝑒 𝑧𝑒 𝑣 +
Equilibrium constants
RKR1
Potential energy curve
𝐺 𝑣 = 𝜔𝑒 𝑣 +
•
1 4
2
1
2
𝐵𝑣 = 𝐵𝑒 − 𝛼𝑒 𝑣 +
− 𝜔𝑒 𝑥𝑒 𝑣 +
1 2
2
1
2
1 2
2
+ 𝛾𝑒 𝑣 +
+ 𝜔𝑒 𝑦𝑒 𝑣 +
+ 𝛿𝑒 𝑣 +
1 3
2
1 3
2
• RKR procedure generates potential energy
curve – Bob Le Roy’s RKR1 program
+
Dipole Moment Function (DMF) Theoretical
• Line intensities cannot be accurately obtained
from the experimental spectra that we use.
• Obtained from ab initio methods
Ab initio
Electronic (transition) dipole
moment function
LEVEL and PGOPHER to Final Line List
Case (b) vibrational wavefunctions and matrix elements
LEVEL
Potential energy curve
+
Dipole moment function
Hund’s case (b) to (a)
•
•
•
•
LEVEL does not include electron spin.
MEs are v′Λ′𝑁′ μ(𝑟) vΛ𝑁 - (Hund’s case (b))
PGOPHER needs case (a) MEs - v′Ω′𝐽′ μ(𝑟) vΩ𝐽 .
Transformation equation was derived:
v ′ Ω′ 𝐽′ 𝑇𝑞𝑘 vΩ𝐽 = −1
×
−1
𝐽′−Ω′
𝑁−𝑁′ +𝑆 ′ +𝐽+𝑘+Λ
𝐽′
𝑘
−Ω′ 𝑞
𝐽
Ω
−1
2𝑁 + 1 2𝑁′ + 1
𝑁′,𝑁′′
×
𝑁′ 𝐽′
𝐽 𝑁
𝑆
𝑘
𝑁′ 𝑘
−Λ′ 𝑞
𝑁
Λ
𝐽′
𝑆
Ω′ −Σ
𝑁′
−Λ′
v′Λ′𝑁′ 𝑇𝑞𝑘 vΛ𝑁
– Derivation published in CN paper in ApJS [1]
[1] Brooke JSA, Ram RS, Western CM, Li G, Schwenke DW, and Bernath PF. ApJS 210, 15 (2014)
𝐽
𝑆
Ω −Σ
𝑁
−Λ
The Herman-Wallis Effect
• Rotation - centrifugal force causes change in bond
length
• Results in a change in the vibrational
wavefunctions
• MEs are changed due to this and the
DMF
• Effect is greater in NH than C2 due to
light H atom
The Herman-Wallis Effect - NH
• The noticeable effect of
the Herman-Wallis
effect for NH is an
increase in the strength
of the R branch and a
decrease in the strength
of the P branch
• To the right is the
calculated spectrum of
the NH X 3 Σ− (1,0) band.
LEVEL and PGOPHER to Final Line List
Case (a) matrix elements
case (b)
to (a)
PGOPHER
Einstein As
Line list with positions and intensities
Case (b) vibrational wavefunctions and matrix elements
LEVEL
Potential energy curve
+
Dipole moment function
Line Strength Equations
• PGOPHER diagonalises the Hamiltonian matrix, and
combines the resulting eigenvectors with the case (a)
MEs and line positions to calculate Einstein A values:
𝐴𝐽′ →𝐽′′
′ Ω′ 𝐽′ 𝑇 𝑘 vΩ𝐽 |2
|
v
𝑝
= 3.136 189 32 × 10−7 𝜈 3
(2𝐽′ + 1)
• where v ′ Ω′ 𝐽′ 𝑇𝑝𝑘 vΩ𝐽 is the ME.
• The Einstein A values are also converted to oscillator
strengths (f-values):
𝑓𝐽′ ←𝐽′′
1 (2𝐽′ + 1)
= 1.499 193 78 27 2
𝐴𝐽′ →𝐽′′
′′
ν (2𝐽 + 1)
NH X 3 Σ− state rovibrational and pure
rotational transitions
• NH is present in cool stars, comets, diffuse interstellar clouds,
the Sun, and probably the upper atmospheres of extrasolar
planets.
• Magnetic trapping, combustion
• Vibrational transitions have been used to calculate the actual
nitrogen abundance in cool stars and the Sun.
• Boudjaadar et al. 1986 - The Δv=1 sequence up to v′=5 [2]
• Ram et al. 1999 - more transitions in the same sequence in
1999 [3]
• Ram and Bernath 2010 - (6,5) band [4]
• Robinson et al. 2007 – rotational transitions in in v=1 and 2 [5]
[2] Boudjaadar D, Brion J, Chollet P, Guelachvili G, and Vervloet M. JMS 119, 352 (1986)
[3] Ram RS, Bernath PF, and Hinkle KH. JCP 110, 5557 (1999)
[4] Ram RS and Bernath PF. JMS 260, 115 (2010)
[5] Robinson A, Brown J, Flores-Mijangos J, Zink L, and Jackson M. Mol Phys 105, 639 (2007)
NH X 3 Σ − state calculations
• No new line
position fit used Ram and
Bernath 2010
[4]
• Dipole
moment
function
calculated by
Gerrit
Groenenboom
– not
published [6]
[6] Campbell WC, Groenenboom GC, Lu HI, Tsikata E, and Doyle JM. PRL, 100, 083003 (2008)
NH X 3 Σ − spectra
Comparison to Observed Spectra
Einstein Av′v′′ and Lifetimes
Einstein Av′v′′ (s-1)
Band
Ours
D [7]
R&W [8]
M&R [9]
1-0
27.19
51.7
34.9
31.9
2-1
57.91
92.3
69.18
3-2
90.14
144.4
108.12
4-3
121.40
144.49
5-4
148.70
173.94
6-5
168.92
191.70
v=1 Lifetimes (ms)
Observed []
Calculated []
Ours
37.0 ± 0.5stat +2.0
−0.8 syst
36.99
36.77
[7] Dodd JA, Lipson SJ, Flanagan DJ, Blumberg WAM, Person JC and Green BD. JCP 94, 4301 (1991)
[8] Rosmus P and Werner H-J. J. Mol. Struct 60, 405 (1980)
[9] Meyer W and Rosmus P. JCP 63, 2356 (1975)
NH Line list
• All possible bands calculated between observed v
levels (up to v=6), up to J between 25 and 44,
depending on the band.
• Rotational transitions included
v'
0
0
0
0
0
0
0
0
v'' J'
0 2.0
0 3.0
0 4.0
0 5.0
0 6.0
0 7.0
0 8.0
0 9.0
J''
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
F'
1
1
1
1
1
1
1
1
F''
1
1
1
1
1
1
1
1
p' p''
e e
e e
e e
e e
e e
e e
e e
e e
N' N'' Observed
1 0 32.50499
2 1 65.21272
3 2 97.79301
4 3 130.23878
5 4 162.51736
6 5
...
7 6
...
8 7
...
Calculated
32.504992
65.212716
97.793016
130.238779
162.517360
194.591348
226.422257
257.971567
Residual
E''
A
f
-0.00000
-0.0077 8.361500E-3 1.977382E-5
0.00000 32.4973 8.082840E-2 3.989194E-5
-0.00001 97.7100 2.917113E-1 5.879492E-5
0.00000 195.5030 7.137733E-1 7.710589E-5
0.00000 325.7418 1.416599E+0 9.502893E-5
...
488.2592 2.465530E+0 1.126338E-4
...
682.8505 3.920697E+0 1.299393E-4
...
909.2728 5.836083E+0 1.469401E-4
Desc Source
rR1(1)
K
rR1(2)
FL
rR1(3)
FL
rR1(4)
FL
rR1(5)
FL
rR1(6) None
rR1(7) None
rR1(8) None
• NH paper has been submitted to JCP
[email protected]
OH X2Π Ground State “Meinel” Transitions
• Important in atmospheric chemistry
– Oxidation of hydrocarbons
– Airglow
• Found in extra-terrestrial environments:
– Stars, comets, interstellar medium, extraterrestrial atmospheres
• Rotational and rovibrational transitions from
– Bernath and Colin 2009 [10]
– Martin-Drumel et al. 2011 [11]
[10] Bernath PF, & Colin R. JMS 257, 20 (2009)
[11] Martin-Drumel MA, Pirali O, Balcon D, Brechignac Ph, Roy P, & Vervloet M. Rev. Sci. Instr 82, 113106 (2011)
Dipole Moments (μv), Calculated Using
Various DMFs
Our Einstein As compared to HITRAN
Observed
Not
Observed
OH Spectrum – (2,0) band
Conclusions
• New line list with positions and intensities
produced for X 3 Σ− state vibration-rotation
transitions, up to v=6.
• Line lists also produced for OH up to v=13.
• Transformation equation doesn’t work as
expected for OH, but intensities are an
improvement compared to what’s currently
available in HITRAN.
• The combination of experimental and theoretical
results is very effective and will be used for more
molecules in the future. Thanks for listening!
Transition Dipole Moment Matrix Elements
• It then overlaps them and multiplies the result by the electronic
(transition) dipole moment function, which is then integrated to
give the transition dipole moment matrix element (TDMME).
R/P OH Intensities (2,0)
with
transformation
Our DMF
N only
Our DMF
with
transformation
HITRAN DMF
N only
HITRAN DMF
v=0 Vibrationally Averaged Dipole
Moments, μ0
Year
1974
1974
1975
1987
1992
2014
Authors
Scarl and Dalby []
Das et al. []
Meyer and Rosmus []
Goldfield and Kirby []
Cantarella et al. []
This work
aCalculated
μ0
1.389±0.075
1.5155a
1.5546a
1.480
1.4827
1.5246
by Cantarella et al. using the reported data of Das et al. and
Meyer and Rosmus
[] Scarl EA and Dalby FW. Can. J. Phys. 52, 1429 (1974)
[] Das G, Wahl AC, and Stevens WJ. JCP 61, 433 (1974)
[] Goldfield EM and Kirby KP. JCP 87, 3986 (1987)
[] Cantarella E, Colot F, and Liévin J. Phys. Scripta. 46, 489 (1992)
Full Calculation Method
Start
Einstein As and f-values – line list with
positions and intensities
Spectrum
PGOPHER
Line
assignment
and fit
Case (a) matrix elements
Hund’s case (b) to (a)
Molecular constants
fit
Equilibrium constants
Case (b) vibrational wavefunctions and transition
dipole moment matrix elements
LEVEL
Start
Ab initio
RKR1
Potential energy curve
End
+
Electronic (transition)
dipole moment function
OH X2Π Ground State Transitions
NH X 3 Σ− state rovibrational and pure
rotational transitions
• NH is present in cool stars, comets, diffuse interstellar clouds,
the Sun, and probably the upper atmospheres of extrasolar
planets.
• Magnetic trapping, combustion
• Vibrational transitions have been used to calculate the actual
nitrogen abundance in cool stars and the Sun.
• Boudjaadar et al. 1986 - The Δv=1 sequence up to v′=5 []
• Ram et al. 1999 - more transitions in the same sequence in
1999 []
• Ram and Bernath 2010 - (6,5) band []
• Robinson et al. 2007 – rotational transitions in in v=1 and 2 []
[] Boudjaadar D, Brion J, Chollet P, Guelachvili G, and Vervloet M. JMS 119, 352 (1986)
[] Ram RS, Bernath PF, and Hinkle KH. JCP 110, 5557 (1999)
[] Ram RS and Bernath PF. JMS 260, 115 (2010)
[] Robinson A, Brown J, Flores-Mijangos J, Zink L, and Jackson M. Mol Phys 105, 639 (2007)