1. Combined Dunham Analysis of Rotational and Electronic Transitions of CH+
Shanshan Yu, Brian J. Drouin, John C. Pearson,
and Takayoshi Amano
Jet Propulsion Laboratory, California Institute of Technology,
Pasadena, CA 91109
June 20, 2016
Copyright All rights reserved.
ISMS, Illinois

2. CH+ as an Interstellar Molecule
A couple of unidentified near-UV lines detected by Dunham (1937).
Douglas and Herzberg identified them based on their laboratory observations to be lowest-J A1-X1+ electronic transitions of CH+ (1941).
This is one of the first interstellar molecules identified.
The electronic spectra, in particular the A1Π – X1Σ+ band, have been
investigated extensively since.
On the other hand, the pure rotational transitions were not known until 2010.
Identification of five lines, J=2-1 to 6-5 toward NGC 7027 with ISO (1997). These observations were , however, of low-resolution of uncertainties of ~ 600 MHz.
First definite identification with J=1-0 toward DR 21 with the Herschel Space observatory (2010).
CH+ is one of the three interstellar molecules detected in the near UV region before microwave detection of interstellar molecules emerged. Since the first laboratory observation of the A-X band by Douglas and Herzberg, the electronic spectroscopy was pursued extensively.
On the other hand, the observation of the pure rotational lines were not made until 2010.
Due to lack of precise rotational transition frequencies, astronomical identification of CH+ by the rotational transitions were not successful until quite recently, apart from an ISO observation in 1997, although it was low resolution observations..
----- Meeting Notes (6/17/16 03:46) -----
June 20, 2016
ISMS, Illinois

3. Rotational Spectroscopy of CH+
First laboratory observation of the J=1-0 transition.
Ap. J. 716, L1 (2010); JCP. 133, (2010)
THz spectroscopy at JPL
Extension up to J=3-2 ( 12CH+, 13CH+ )
up to J=4-3 ( 12CD+ )
Still not extensive enough for reliable determination of the spectroscopic constants
Following the sub-mm observations, at JPL we extended the measurements up to 2 THz, and observed those lines of 3-2 for 12CH+ and 13CH+ and 4-3 for CD+. Still number of lines is not extensive enough for good determination of the molecular constants for individual fit for each isotopologue.
In this work, a combined Dunham analysis has been done with available electronic data.
Dunham analysis with all the available high-resolution data
June 20, 2016
ISMS, Illinois

4. Observation of THz lines of CH+
For production of CH+, an extended negative glow discharge in a gas mixture of CH4 (~0.5 mTorr) diluted in He ( 60 mTorr) was used.
The optimum discharge current was about 15 mA and the axial magnetic field of 160 Gauss was applied.
The discharge cell was cooled down to liquid nitrogen temperature.
In this investigation, the J=2-1 and 3-2 lines were observed for 12CH+ and 13CH+.
For CD+, J=2-1 to J=4-3 were measured.
JPL THz radiation sources: Multiplier chains
BWO
Briefly describe the experimental procedure.
Used an extended negative glow discharge in small amount of CH4 in He cooled to LN2 temperature. Used a THz spectrometer at JPL, frequency range was extended up to 2 THz. The details of the system is described in these papers, so I do not repeat the details.
B. J. Drouin et at, RSI. 76, (2005)
J. C. Pearson et al, RSI. 82, (2011)
June 20, 2016
ISMS, Illinois

5. Dunham Analysis
Dunham (1932) formulated the vibration-rotation energy for diatomic molecules
Ykl ≈ μC-(k+2l)/2
Watson (1980) worked out the mass dependence in detail.
where μC is the charge corrected reduced mass;
Vibration-rotation energies for a diatomic molecule can be expressed by this Dunham formula. Here the Dunham coefficients, Y_{kl} depend on the reduced mass. Watson investigated more detailed mass dependence by including the mass dependence corrections due to the Born-Oppenheimer breakdown. Here U_{kl} are the mass independent Dunham coefficients. \mu_C is the charge corrected reduced mass, and for a singly charged ion, C is unity. These mass independent Dunham coefficients were determined from least-squares fit.
At the time of our detection of the sub-mm wave rotational lines, Muller carried out a Dunham fit with the electronic bands data and the sub-mm data.
Müller (2010) carried out a Dunham analysis of the A1Π-X1Σ+ electronic transitions with, in addition, then newly observed sub-mm lines. 　
June 20, 2016
ISMS, Illinois

6. -doubling in 1 electronic states
Each rovibronic level of 1 states splits into two components due to the
-doubling.
The levels that have the parity (-1)J are classified as e-levels, while the
levels of the parity -(-1)J f-levels.
The rovibronic states of 1+ states are all e-levels, and those of 1- states
all f-levels. Therefore the e parity states of 1 state interact with 1+,
while the f parity states of 1 state interact with 1- .
In most cases, the -doubling
is taken to be
±(1/2)qJ(J+1).
Λ-doubling parameters are defined as
In most analyses, the lambda-doubling is expressed as ………
However, if you look into the symmetry of the states, we realize that e-parity levels of 1\Pi state interact with 1\Sigma^+, and f-parity levels with 1\Sigma^- states. So the magnitude of the shifts for e- and f-parity levels are not equal.
If we use this expression, it would affect the rotational constant.
As shown here, the splitting is not symmetric, as a results, the rotational constant and the lambda doubling constant are not determined independently.
In principle, B, q, and q’
cannot be determined independently in 1Π electronic states.

7. Least-squares Fit
Least-squares fit was performed with sub-mm and THz rotational lines and
the available high-resolution A-X electronic transition frequencies .
12CH+ (Carringon and Ramsay, 1982):
0-0, 0-1,
1-0, 1-1, 1-2, 1-3,
2-1, 3-1
(Hakalla et al, 2006):
0-0, 0-1, 2-1
13CH+ (Bembenek, 1997):
0-0, 0-1, 1-0, 1-1,
2-0, 2-1
12CD+ (Bembenek et al, 1987):
0-0, 0-1, 1-0, 1-2, 1-3,
2-0, 2-1, 3-1
13CD+ (Bembenek, 1997):
0-0, 1-0
Selection rules:
R- and P-branches: e - e,( f – f )
Q-branch: e - f
Since 1Σ- states are not known to exist and
they should lie very high in energy if any,
q’ must be very small and may be negligible.
The least-squares fit was carried out by using only the Q-branch lines of the A-X system, and the Λ–doubling was neglected.
Number of total weighted data; 296
Number of variable parameters; 47
Corrected some misassingments
June 20, 2016
ISMS, Illinois

8. Mass independent Dunham coefficients determined from the fit.
Excerpt of the fitting results ( in cm-1 )
12CH+ (0 – 0) band CH+ (2 – 0) band
Q(1) a (2) b (-3)
Q(2) (0) (-4)
Q(3) (-4) (-1)
Q(4) (0) (4)
Q(5) (6) (-4)
Q(6) (8) (-21)
Q(7) (5) (1)
Q(8) (5) (2)
Q(9) (-5) (-10)
Q(10) (-5) (-8)
Q(11) (2)
This shows the fitted U parameters. These values are MHz based values, that means the input data were all of MHz units and the reduced mass was in atomic mass units. This part shows examples of the fitted residuals.
The fit was quite reasonable considering the accuracy of the measurements.
a Carringon and Ramsay
b Hakalla

9. Ykl coefficient converted from Ukl
The physical meaning of mass independent Dunham coefficients Ukl is difficult to grasp. We converted Ukl back to Ykl that are correlated to conventional spectroscopic constants.
12CH+
Hakalla, Eur. Phys. D 38, 481(2006)
This work
A1Π X1Σ+
(22) (22) MHz
　 (93)
(24)
(54) (36)
-687.(36) (108)
(150) (108)
(22) (22) cm-1
(140) (150)
2.6301(24) (26)
(140)
  A1Π X1Σ+
Y01 ≈ Be (15) (23)
Y02 ≈ -De (20) (39)
Y03 ≈ He (50) (63)
Y11 ≈ -αe (94) (97)
Y21 ≈ γe (56) (76)
Y12 ≈ -βe (20) (146)
Y10 ≈ ωe (29) (38)
Y20 ≈ -ωexe (27) (33)
Y30 ≈ ωeye (147) (184)
Y00 ≈ Te (40)
The physical meaning of the mass independent Dunham coefficients is difficult to grasp. So we converted back the U coefficients to Y coefficients for each isotopologue.
June 20, 2016
ISMS, Illinois

10. Molecular Constants Constants derived from the Dunham fit (in MHz).
Constants obtained from the conventional fit of individual isotopologue.
Here we list the spectroscopic parameters derived the Y coefficients. The can be compared with the parameters obtained from the conventional fits for each isotopologue. Because the number of rotational lines are quite limited, only three lines for CH+ and 13CH+, and four lines for CD+, we also included the 0-0 band of the A-X electronic band in the fit.
We can see the good agreements between the two sets of the constants. The accuracy of the constants from the conventional fits appears to be better. However, in the fits the THz and sub-mm lines were heavily weighted, so the fit is more an exact fit. So the accuracy may be unrealistically small.
Fits were done with THz lines and the (0-0) band of the A-X system.
June 20, 2016
ISMS, Illinois

11. Discussion and conclusion
The mass dependence of qkl defined below is different from that of Ykl.
Müller stated that the mass dependence of qkl should be μC-2-(k+2l)/2. However, in this investigation, we did not incorporate the Λ-doubling and only the f–levels in the excited A1Π state were included in the fit.
EΛ(v,J)=(1/2)J(J+1)Σ qkl(v+1/2)k[J(J+1)]l
Good fits of the Dunham fit of only f-levels in the excited A state indeed vindicates the assumption that the interaction with 1Σ– state(s) is negligibly small.
The Dunham coefficients determined here would be good
for prediction of the vibration-rotation transition frequencies.
June 20, 2016
ISMS, Illinois

12. Derived Vibration-rotation Transitions (cm-1)
12CH CH CD CD+
P(10)
P( 9)
P( 8)
P( 7)
P( 6)
P( 5)
P( 4)
P( 3)
P( 2)
P( 1)
R( 0)
R( 1)
R( 2)
R( 3)
R( 4)
R( 5)
R( 6)
R( 7)
R( 8)
R( 9)
June 20, 2016
ISMS, Illinois

13. Acknowledgements
A part of this research was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
A part of research was supported by Natural Science and Engineering Research Council of Canada ( NSERC )
June 20, 2016
ISMS, Illinois