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 2016. All rights reserved. ISMS, Illinois

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

Rotational Spectroscopy of CH+ First laboratory observation of the J=1-0 transition. Ap. J. 716, L1 (2010); JCP. 133, 244305(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

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, 093131 (2005) J. C. Pearson et al, RSI. 82, 093105 (2011) June 20, 2016 ISMS, Illinois

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

-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.

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

Mass independent Dunham coefficients determined from the fit. Excerpt of the fitting results ( in cm-1 ) 12CH+ (0 – 0) band 13CH+ (2 – 0) band Q(1) 23591.845a (2) 26658.843b (-3) Q(2) 23581.732 (0) 26641.301 (-4) Q(3) 23566.526 (-4) 26614.941 (-1) Q(4) 23546.194 (0) 26579.714 (4) Q(5) 23520.671 (6) 26535.535 (-4) Q(6) 23489.878 (8) 26482.317 (-21) Q(7) 23453.723 (5) 26419.999 (1) Q(8) 23412.108 (5) 26348.392 (2) Q(9) 23364.897 (-5) 26267.350 (-10) Q(10) 23311.967 (-5) 26176.725 (-8) Q(11) 23253.153 (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

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+ 356357.(22) 425029.6(22) MHz 　 -57.8481(93) -41.645 ------ 0.00361(24) -27470.(54) -14831.9 (36) -687.(36) 74.7(108) -3.2177(150) 0.7974(108)   1864.402(22) 2857.561(22) cm-1 -115.8317(140) -59.3179(150) 2.6301(24) 0.2253(26) 24118.7262(140)   A1Π X1Σ+ Y01 ≈ Be 355746(15) 425060.9(23) Y02 ≈ -De -57.50(20) -41.969(39) Y03 ≈ He -0.002101(50) 0.002834(63) Y11 ≈ -αe -27375.0(94) -14865.8(97) Y21 ≈ γe -714.0(56) 96.0(76) Y12 ≈ -βe -3.75(20) 1.184(146)   Y10 ≈ ωe 1857.693(29) 2857.714(38) Y20 ≈ -ωexe -109.254(27) -59.468(33) Y30 ≈ ωeye 0.1784(147) 0.2828(184) Y00 ≈ Te 24120.748(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

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

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

Derived Vibration-rotation Transitions (cm-1) 12CH+ 13CH+ 12CD+ 13CD+ P(10) 2422.1555 2416.2960 1868.1741 1859.1342 P( 9) 2457.5352 2451.4454 1886.3791 1877.1241 P( 8) 2492.1250 2485.8127 1904.2755 1894.8102 P( 7) 2525.9441 2519.4164 1921.8576 1912.1869 P( 6) 2558.9902 2552.2539 1939.1175 1929.2465 P( 5) 2591.2467 2584.3083 1956.0467 1945.9804 P( 4) 2622.6880 2615.5545 1972.6354 1962.3794 P( 3) 2653.2834 2645.9619 1988.8739 1978.4336 P( 2) 2682.9999 2675.4980 2004.7521 1994.1334 P( 1) 2711.8036 2704.1291 2020.2601 2009.4689 R( 0) 2766.5383 2758.5443 2050.1260 2039.0089 R( 1) 2792.4038 2784.2635 2064.4646 2053.1944 R( 2) 2817.2256 2808.9484 2078.3944 2066.9779 R( 3) 2840.9731 2832.5690 2091.9062 2080.3505 R( 4) 2863.6180 2855.0970 2104.9910 2093.3034 R( 5) 2885.1351 2876.5077 2117.6405 2105.8284 R( 6) 2905.5055 2896.7822 2129.8466 2117.9176 R( 7) 2924.7199 2915.9110 2141.6023 2129.5641 R( 8) 2942.7840 2933.8994 2152.9022 2140.7625 R( 9) 2959.7252 2950.7738 2163.7423 2151.5089   June 20, 2016 ISMS, Illinois

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