Issue 
A&A
Volume 559, November 2013



Article Number  A125  
Number of page(s)  9  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/201322375  
Published online  26 November 2013 
The rotational spectrum of ^{12}C_{2}HD in the ground and excited bending states: an improved rovibrational global analysis
^{1} Dipartimento di Chimica “Giacomo Ciamician”Università di Bologna, via F. Selmi 2, 40126 Bologna, Italy
email: claudio.degliesposti@unibo.it; luca.dore@unibo.it
^{2} Dipartimento di Chimica Industriale “Toso Montanari”, Università di Bologna, viale del Risorgimento 4, 40136 Bologna, Italy
email: luciano.fusina@unibo.it; filippo.tamassia@unibo.it
Received: 26 July 2013
Accepted: 11 October 2013
Rotational transitions of ^{12}C_{2}HD were recorded in the range 100–700 GHz for the vibrational ground state and for the bending states v_{4} = 1 (Π), v_{5} = 1 (Π), v_{4} = 2 (Σ^{+} and Δ), v_{5} = 2 (Σ^{+} and Δ), v_{4} = v_{5} = 1 (Σ^{−}, Σ^{+} and Δ), v_{4} = 3 (Π and Φ) and v_{5} = 3 (Π and Φ). The transition frequencies measured in this work were fitted together with all the infrared rovibrational transitions involving the same bending states available in the literature. The global fit allowed a very accurate determination of the vibrational, rotational, and ℓtype interaction parameters for the bending states up to v_{4} + v_{5} = 3 of this molecule. The results reported in this paper provide a set of information very useful for undertaking astronomical searches in both the mm−wave and the infrared spectral regions. The parameters from the global fit can be used to calculate accurate rest frequencies for rotational transitions in the ground state or in excited vibrational states involving the bending modes. Pure rotational transition frequencies up to 1 THz are listed.
Key words: molecular data / methods: laboratory: molecular / techniques: spectroscopic / catalogs
© ESO, 2013
1. Introduction
Acetylene can be found in several astronomical environments: in molecular clouds (Lacy et al. 1989), in massive young stellar objects and planet forming zones (Lahuis & van Dishoeck 2000; Bast et al. 2013), in circumstellar envelopes of AGB stars (Ridgway et al. 1976; Matsuura et al. 2006; Fonfría et al. 2008), and it has been identified in cometary comae (Mumma et al. 2003) as well. This unsaturated hydrocarbon may react with radicals – atomic C, CN, and CH – to form complex molecules in starless cores (Herbst 2005), it plays a key role in the formation of circumstellar carbon chain molecules (Cherchneff & Glassgold 1993) and it is even a possible precursor of benzene in a carbonrich PPN (Woods et al. 2002).
However, ^{12}C_{2}H_{2} does not have a permanent dipole moment and cannot be detected by (sub)millimetre telescopes, therefore interstellar acetylene has been detected by observing its vibrationrotation bands. The only detectable submillimetre features could be those due to some Pbranch highJ transitions of the ν_{5} ← ν_{4} difference band in the THz region (Yu et al. 2009). On the other hand, noncentrosymmetric isotopologues of acetylene, such as the subject of this paper ^{12}C_{2}HD, do have a small permanent electric dipole moment. The first astronomical detection of ^{12}C_{2}HD is recent, namely it has been observed in Titan’s atmosphere through the Composite InfraRed Spectrometer (CIRS) mounted on the Cassini spacecraft (Coustenis et al. 2008). From these observations it was possible to derive the D/H ratio on Titan, which was previously determined only through the transitions of the CH_{4}/CH_{3}D pair.
Molecules containing less abundant isotopes are very relevant from an astrophysical point of view. Several species containing D, ^{13}C, ^{15}N, ^{18}O, among the most important, provide a tool to assess isotopic ratios in several astronomical environments (see for instance Herbst 2003; Caselli & Ceccarelli 2012; Bézard 2009). The D/H isotopic ratio is of particular interest for several reasons. It is an important experimental constraint on the Big Bang models, as deuterium was formed in abundance only in this event. It can also provide key information on the chemical processes that lead to the formation of complex organic molecules (Sandford et al. 2001). Although the cosmic D/H ratio is of the order of magnitude of 10^{5}, abundances of a few percent with respect to their parent species can be produced in the interstellar medium through isotopic fractionation mechanisms (Herbst 2003). It is known that in cold dense interstellar clouds Denrichment proceeds through gas phase ionmolecule exothermic reactions, but also through gasgrain chemistry (Sandford et al. 2001). Alternative routes for achieving DH fractionation in more energetic environments and of interest for complex molecules are: a) gas phase unimolecolar photodissociation; and b) ultraviolet photolysis in Denriched ice mantles (Sandford et al. 2001).
Laboratory rotational spectra have been observed in the past for several monodeuterated species of acetylene, including also ^{13}C containing isotopologues (Wlodarczak et al. 1989; Matsumura et al. 1980; Cazzoli et al. 2008; Degli Esposti et al. 2013). Rotational transitions of ^{12}C_{2}HD were measured up to 418 GHz for the ground vibrational state (GS) and for the v_{4} = 1 and v_{5} = 1 excited states, ν_{4} and ν_{5} being the trans and cis bending modes, respectively (Wlodarczak et al. 1989). Pure rotational transitions in the GS up to J′′ = 10, around 650 GHz, were observed for ^{12}C_{2}HD, H^{12}C^{13}CD and H^{13}C^{12}CD (Cazzoli et al. 2008). In the same study, ab initio calculations were performed at various levels in order to predict the electric dipole moment for these species and the equilibrium structure of acetylene. The calculated dipole moment does not show sizable variations upon isotopic substitution of one carbon atom, and is approximately 0.01 D for all monodeuterated isotopologues in the GS. The ν_{5} ← ν_{4} difference band and associated hot bands for ^{12}C_{2}HD have been recorded recently in the farinfrared (FIR) region, between 60 and 360 cm^{1}, using the synchrotron radiation at the Canadian Light Source (PredoiCross et al. 2011). The same band for the H^{12}C^{13}CD and H^{13}C^{12}CD isotopologues was also detected and analysed.
As far as the infrared (IR) region is concerned, several papers have been published on ^{12}C_{2}HD. The most recent are the investigations of the bending states up to v_{4} + v_{5} = 3 (Fusina et al. 2005a) and of the stretchingbending bands in the 1800–4700 cm^{1} spectral region (Fusina et al. 2005b). In both cases the spectra were recorded by Fourier transform infrared spectroscopy (FTIR). A study of the integrated band intensities in the 25–2.5 μm window was also published (Jolly et al. 2008).
In the present paper we report on the observation of the pure rotational transitions of ^{12}C_{2}HD up to 657 GHz, that is in bands 3, 6, 7, 8 and 9 of the Atacama Large Millimeter Array (ALMA). A total of 168 transitions were assigned in the GS and in various excited vibrational bending states. The rotational lines detected in this work were fitted together with the FIR (PredoiCross et al. 2011) and IR lines (Fusina et al. 2005a). The spectroscopic parameters obtained from the final global fit are determined with an excellent accuracy.
The high accuracy of the millimetre and submillimetrewave data presented in this paper, together with the increasing sensitivity of new observation systems, such as ALMA, will favour the observation of transitions belonging to this species.
It should also be stressed that the dipole moment of ^{12}C_{2}HD is strongly enhanced by the bending vibrations. Therefore, considering that some chemically rich regions, e.g. IRC+10216 (Cernicharo et al. 2011), show a high degree of vibrational excitation, this will facilitate the detection of emission lines in the bending states ν_{4} or ν_{5} of ^{12}C_{2}HD. In Sect. 3.1, the dipole moment variation in excited bending states is discussed.
2. Experimental details
The sample of ^{12}C_{2}HD was purchased from CDN Isotopes (98.9% purity). The rotational spectra of ^{12}C_{2}HD were observed in selected frequency regions between 100 and 700 GHz, using a sourcemodulation mmwave spectrometer which employs Gunn oscillators (RPG Radiometer Physics GmbH and J. E. Carlstrom Co.) as main radiation sources covering the fundamental frequency range 75–124 GHz. Higher frequencies were generated using three different frequency multipliers (VDI – Virginia Diodes, Inc. and RPG). The Gunn oscillators were phaselocked to the suitable harmonic of the frequency emitted by a computercontrolled frequency synthesizer (Schomandl), referenced to an external rubidium frequency standard (SRS Stanford Research System). This guaranteed an absolute accuracy of ca. 20 Hz to the frequency scale. A liquidheliumcooled InSb detector (QMC Instr. Ltd.) was employed. The Gunn oscillators were frequency modulated at 6 kHz, and the detected signals were demodulated by a lockin amplifier tuned at twice the modulation frequency, so that the second derivative of the actual spectrum profile was detected by the computercontrolled acquisition system.
Transition frequencies were recovered from a lineshape analysis of the spectral profile (Dore 2003); their accuracy, estimated by repeated measurements, was in the range 5–30 kHz depending on the signaltonoise ratio of the recorded lines.
The absorption cell was a 3.5 m long, 10 cm in diameter glass tube equipped with polyethylene windows. A double pass arrangement based on a wire grid polarizer and a roof mirror (Ziurys et al. 1994; Dore et al. 1999) was employed to increase the absorption path. Sample pressures of a few tens of mTorr were employed during the measurements.
3. Results and discussion
3.1. Rotational analysis
Fig. 1 The J = 5 ← 4 transition of ^{12}C_{2}HD in the v_{5} = 1, ground, and v_{4} = 1 vibrational states. The lowfrequency components of each ℓdoublet are displayed. Fourteen scans are coadded, the total integration time is 850 s. 
Dipole moments, vibrational term values, populations and intensity calculations for several excited vibrational bending states of ^{12}C_{2}HD.
The very small dipole moment of ^{12}C_{2}HD was first determined by Matsumura et al. (1980), who performed Starkeffect measurements on the J = 2 ← 1 rotational transition of ^{12}C_{2}HD in the v_{4} = 1,3 and v_{5} = 1,3 states. The obtained dipole moment values are: − 0.02359(5) D for v_{4} = 1, 0.05601(9) D for v_{5} = 1, − 0.09077(26) D for v_{4} = 3 and 0.1472(21) D for v_{5} = 3. These experimental values led to extrapolate a GS moment of 0.01001(15) D, since the dipole moments of the v_{4} = 1 and v_{4} = 3 states have opposite sign with respect to that of the GS. The increase of the dipole moment values due to vibrational excitation causes a considerable intensity enhancement of the excited state rotational lines, so that their detection is much easier than expected from an evaluation of the population factors. At room temperature (kT = 207 cm^{1}), the population factors are N_{4}/N_{0} = exp ( − 518/kT) = 0.0819 and N_{5}/N_{0} = exp ( − 677/kT) = 0.0378, which should reduce the intensity of the rotational lines by a factor of 12 for v_{4} = 1 and of 26 for v_{5} = 1. It should be noticed that the bending states are doubly degenerate. However, the factor 2, which doubles the population of v_{4} = 1 and v_{5} = 1, has no effect on the intensity of the rotational lines. In fact, the ℓtype doubling removes the degeneracy of the levels and rotational transitions are allowed only within each set. On the other hand, from the experimental μ values one can calculate the following ratios: (μ_{4}/μ_{0})^{2} ≃ 5.6 and (μ_{5}/μ_{0})^{2} ≃ 31, which partly compensate the unfavourable population factors of the excited states. The intensity ratio between rotational lines in the excited bending states and in the GS can be calculated as $\begin{array}{ccc}\frac{{\mathit{I}}_{\mathit{v}}}{{\mathit{I}}_{\mathrm{0}}}\mathrm{=}{\left(\frac{{\mathit{\mu}}_{\mathit{v}}}{{\mathit{\mu}}_{\mathrm{0}}}\right)}^{\mathrm{2}}{\mathrm{e}}^{\mathrm{}{\mathit{E}}_{\mathit{v}}\mathit{/}\mathit{kT}}\mathrm{\xb7}& & \end{array}$(1)The rotational transitions in v_{4} = 1 are expected to be less intense than those of the GS by only a factor 12/5.6 ≃ 2, whereas the lines in v_{5} = 1 should be even stronger than those in the GS by a factor 31/26 ≃ 1.2. Figure 1 shows a 250 MHz frequency scan in which the J = 5 ← 4 transitions of v_{5} = 1, ground, and v_{4} = 1 states are simultaneously present. The experimental intensity ratios nicely agree with the predicted ones. Anyway, for any combination of bending vibrational quanta, the dipole moment can be calculated by applying the usual expression for the vibrational dependence $\begin{array}{ccc}{\mathit{\mu}}_{\mathit{v}}\mathrm{=}{\mathit{\mu}}_{\mathrm{0}}\mathrm{+}\sum _{\mathit{i}}\mathit{\delta}{\mathit{\mu}}_{\mathit{i}}{\mathit{v}}_{\mathit{i}}& & \end{array}$(2)as reported in Eq. (11) of Matsumura et al. (1980). The values of the parameters δμ_{i} are: δμ_{4} = − 0.03360(13) D and δμ_{5} = 0.04600(17) D, as given in Table 3 of the same reference. For mixed excitations of ν_{4} and ν_{5} there is a less significant enhancement on the dipole moment, and therefore on the intensity, since δμ_{4} and δμ_{5} have opposite sign and comparable magnitude. Indeed, only transitions in the combination v_{4} + v_{5} could be detected, whereas in higherorder mixed vibrational excitations they were too weak to be observed.
A summary of the intensity predictions at 300 K and 500 K for rotational lines of some vibrationally excited bending states is reported in Table 1. It is evident a large temperature effect, therefore at 500 K the transitions in vibrational states up to the doubly excited ν_{4} and ν_{5} are stronger than the groundstate ones.
A very limited number of excitedstate transitions of ^{12}C_{2}HD have previously been observed only for v_{4} = 1,3 (Π) and v_{5} = 1,3 (Π) (Wlodarczak et al. 1989; Matsumura et al. 1980). We have considerably enlarged the previous dataset by measuring more than 150 new line frequencies, spanning J values from 1 to 10, corresponding to transitions in the ground state, in the first excited vibrational bending states v_{4} = 1 and v_{5} = 1 (Π), in the doubly excited v_{4} = 2 and v_{5} = 2 states (Σ^{+} and Δ), in the combination state v_{4} = v_{5} = 1 (Σ^{−}, Σ^{+} and Δ) and in the triply excited v_{4} = 3 and v_{5} = 3 states (Π and Φ). The search of the rotational lines was guided by predictions based on the spectroscopic constants determined in a previous analysis of 4888 IR and FIR data and 21 rotational transitions available at that time (PredoiCross et al. 2011). On average, the newly observed lines were found some hundreds of kHz away from the initial predictions. The measured transition frequencies are listed in Table 2, along with predictions of unobserved lowJ transitions and of the ones occurring at higher frequency up to 1 THz.
The term values of the observed rotational levels in the ground and excited vibrational states are in the range 2 − 2140 cm^{1}. With the exception of the ground state, multiplets of rotational lines were always observed for each J + 1 ← J transition, because of ℓtype resonance effects. Before performing the final rovibrational global analysis (see Sect. 3.2), a series of statebystate leastsquares fits were done to check the consistency of the MW measurements. The pure rotational spectra have been analysed using the formalism originally developed by Yamada et al. (1985) and already employed to fit the excitedstate rotational spectra of a large number of linear carbon chains (see for example Bizzocchi & Degli Esposti 2008, and references therein). The model is slightly different from the one used to perform the global rovibrational analysis (see Sect. 3.2 for a detailed description), because the vibrational dependence of the various parameters is neglected, so that effective constants for each vibrational state were determined by these preliminary fits. Briefly, rotational and vibrational ℓtype resonance effects have been treated by diagonalization of rovibrational matrices with offdiagonal elements which include q_{t}, ${\mathit{q}}_{\mathit{t}}^{\mathit{J}}$, r_{tt′} and ${\mathit{r}}_{\mathit{t}{\mathit{t}}^{\mathrm{\prime}}}^{\mathit{J}}$ spectroscopic parameters (t and t′ being equal to 4 or 5). The vibrational energy differences between the interacting ℓ sublevels (Δℓ_{t} = ± 2) of each doubly or triply excited vibrational state have been expressed through the effective values of the constants g_{44}, g_{55} and g_{45}, which produce ℓdependent energy contributions. In addition, the ℓdependence of rotational and quartic centrifugal distortion constants have been taken into accounts trough the γ^{tt′} and δ^{tt′} parameters, respectively. Generally, not all of the required constants can be statistically determined from the rotational transitions of a single vibrational state, and some assumptions had to be necessarily made. The ℓtype doubling parameters q_{t} and ${\mathit{q}}_{\mathit{t}}^{\mathit{J}}$ were fitted for the v_{t} = 1 (t = 4 or 5) and v_{t} = 3 states (where degenerate ℓ = ± 1 levels do exist), but constrained to interpolated values for the v_{t} = 2 states, in order to avoid high correlations with the g_{44} and g_{55} constants. The latter constant has a rather large value (ca. 5.2 cm^{1}), so that rotational ℓtype resonance effects are weak in the v_{5} = 3 state, where no splitting of the  ℓ  = 3 lines could be observed. As far as the v_{4} = v_{5} = 1 combination state is concerned, where rotational and vibrational ℓtype resonance effects are simultaneously present, q_{t} and ${\mathit{q}}_{\mathit{t}}^{\mathit{J}}$ constants were held fixed at the values determined for the singly excited bending states, while the parameters involved in the vibrational ℓtype resonance, namely g_{45}, r_{45} and ${\mathit{r}}_{\mathrm{45}}^{\mathit{J}}$ were refined. The various statebystate fits were repeated iteratively in order to achieve full consistency of the obtained parameters. The results of the eight leastsquares fits performed are collected in Tables 3 and 4.
Measured and predicted^{a} transition frequencies (MHz) of ^{12}C_{2}HD in the ground and excited bending states^{b}.
3.2. Global rovibrational analysis
Rovibrational transitions involving the vibrational states presently studied, except the v_{4} = 3 state (Φ), were already observed in the FIR and IR regions (PredoiCross et al. 2011; Fusina et al. 2005b). They have been fitted together with the rotational transitions presently measured. The model Hamiltonian adopted for the global analysis represents an extension up to three quanta of the bending excitation, i.e. v_{4} + v_{5} = 3, of the Hamiltonian for a molecule with two bending vibrations which has been described in detail by Herman et al. (1991). It was already used for ^{13}C_{2}HD (Degli Esposti et al. 2013) and ^{12}C_{2}HD (Fusina et al. 2005a) itself. The term values of the rovibrational levels of the transitions were obtained by diagonalizing the appropriate energy matrix containing the following vibrational (G^{0}) and rotational (F) diagonal contributions: $\begin{array}{ccc}{\mathit{G}}^{\mathrm{0}}\mathrm{\left(}{\mathit{v}}_{\mathrm{4}}\mathit{,}{\mathit{\ell}}_{\mathrm{4}}\mathit{,}{\mathit{v}}_{\mathrm{5}}\mathit{,}{\mathit{\ell}}_{\mathrm{5}}\mathrm{\right)}& \mathrm{=}& {\mathit{\omega}}_{\mathrm{4}}^{\mathrm{0}}{\mathit{v}}_{\mathrm{4}}\mathrm{+}{\mathit{\omega}}_{\mathrm{5}}^{\mathrm{0}}{\mathit{v}}_{\mathrm{5}}\mathrm{+}{\mathit{x}}_{\mathrm{44}}^{\mathrm{0}}{\mathit{v}}_{\mathrm{4}}^{\mathrm{2}}\mathrm{+}{\mathit{x}}_{\mathrm{45}}^{\mathrm{0}}{\mathit{v}}_{\mathrm{4}}{\mathit{v}}_{\mathrm{5}}\mathrm{+}{\mathit{x}}_{\mathrm{55}}^{\mathrm{0}}{\mathit{v}}_{\mathrm{5}}^{\mathrm{2}}\\ & & \mathrm{+}{\mathit{g}}_{\mathrm{44}}^{\mathrm{0}}{\mathit{\ell}}_{\mathrm{4}}^{\mathrm{2}}\mathrm{+}{\mathit{g}}_{\mathrm{45}}^{\mathrm{0}}{\mathit{\ell}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{5}}\mathrm{+}{\mathit{g}}_{\mathrm{55}}^{\mathrm{0}}{\mathit{\ell}}_{\mathrm{5}}^{\mathrm{2}}\\ & & \mathrm{+}{\mathit{y}}_{\mathrm{444}}{\mathit{v}}_{\mathrm{4}}^{\mathrm{3}}\mathrm{+}{\mathit{y}}_{\mathrm{445}}{\mathit{v}}_{\mathrm{4}}^{\mathrm{2}}{\mathit{v}}_{\mathrm{5}}\mathrm{+}{\mathit{y}}_{\mathrm{455}}{\mathit{v}}_{\mathrm{4}}{\mathit{v}}_{\mathrm{5}}^{\mathrm{2}}\mathrm{+}{\mathit{y}}_{\mathrm{555}}{\mathit{v}}_{\mathrm{5}}^{\mathrm{3}}\\ & & \mathrm{+}{\mathit{y}}_{\mathrm{4}}^{\mathrm{44}}{\mathit{v}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{4}}^{\mathrm{2}}\mathrm{+}{\mathit{y}}_{\mathrm{4}}^{\mathrm{45}}{\mathit{v}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{5}}\mathrm{+}{\mathit{y}}_{\mathrm{4}}^{\mathrm{55}}{\mathit{v}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{5}}^{\mathrm{2}}\\ & & \mathrm{+}{\mathit{y}}_{\mathrm{5}}^{\mathrm{44}}{\mathit{v}}_{\mathrm{5}}{\mathit{\ell}}_{\mathrm{4}}^{\mathrm{2}}\mathrm{+}{\mathit{y}}_{\mathrm{5}}^{\mathrm{45}}{\mathit{v}}_{\mathrm{5}}{\mathit{\ell}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{5}}\mathrm{+}{\mathit{y}}_{\mathrm{5}}^{\mathrm{55}}{\mathit{v}}_{\mathrm{5}}{\mathit{\ell}}_{\mathrm{5}}^{\mathrm{2}}\end{array}$$\begin{array}{ccc}\mathit{F}\mathrm{\left(}{\mathit{v}}_{\mathrm{4}}\mathit{,}{\mathit{\ell}}_{\mathrm{4}}\mathit{,}{\mathit{v}}_{\mathrm{5}}\mathit{,}{\mathit{\ell}}_{\mathrm{5}}\mathrm{\right)}& \mathrm{=}& [{\mathit{B}}_{\mathrm{0}}\mathrm{}{\mathit{\alpha}}_{\mathrm{4}}{\mathit{v}}_{\mathrm{4}}\mathrm{}{\mathit{\alpha}}_{\mathrm{5}}{\mathit{v}}_{\mathrm{5}}\mathrm{+}{\mathit{\gamma}}_{\mathrm{44}}{\mathit{v}}_{\mathrm{4}}^{\mathrm{2}}\mathrm{+}{\mathit{\gamma}}_{\mathrm{45}}{\mathit{v}}_{\mathrm{4}}{\mathit{v}}_{\mathrm{5}}\mathrm{+}{\mathit{\gamma}}_{\mathrm{55}}{\mathit{v}}_{\mathrm{5}}^{\mathrm{2}}\\ & & \mathrm{+}{\mathit{\gamma}}^{\mathrm{44}}{\mathit{\ell}}_{\mathrm{4}}^{\mathrm{2}}\mathrm{+}{\mathit{\gamma}}^{\mathrm{45}}{\mathit{\ell}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{5}}\mathrm{+}{\mathit{\gamma}}^{\mathrm{55}}{\mathit{\ell}}_{\mathrm{5}}^{\mathrm{2}}\\ & & \mathrm{+}{\mathit{\gamma}}_{\mathrm{444}}{\mathit{v}}_{\mathrm{4}}^{\mathrm{3}}\mathrm{+}{\mathit{\gamma}}_{\mathrm{445}}{\mathit{v}}_{\mathrm{4}}^{\mathrm{2}}{\mathit{v}}_{\mathrm{5}}\mathrm{+}{\mathit{\gamma}}_{\mathrm{455}}{\mathit{v}}_{\mathrm{4}}{\mathit{v}}_{\mathrm{5}}^{\mathrm{2}}\mathrm{+}{\mathit{\gamma}}_{\mathrm{555}}{\mathit{v}}_{\mathrm{5}}^{\mathrm{3}}\\ & & \mathrm{+}{\mathit{\gamma}}_{\mathrm{4}}^{\mathrm{44}}{\mathit{v}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{4}}^{\mathrm{2}}\mathrm{+}{\mathit{\gamma}}_{\mathrm{4}}^{\mathrm{45}}{\mathit{v}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{5}}\mathrm{+}{\mathit{\gamma}}_{\mathrm{4}}^{\mathrm{55}}{\mathit{v}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{5}}^{\mathrm{2}}\\ & & \mathrm{+}{\mathit{\gamma}}_{\mathrm{5}}^{\mathrm{44}}{\mathit{v}}_{\mathrm{5}}{\mathit{\ell}}_{\mathrm{4}}^{\mathrm{2}}\mathrm{+}{\mathit{\gamma}}_{\mathrm{5}}^{\mathrm{45}}{\mathit{v}}_{\mathrm{5}}{\mathit{\ell}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{5}}\mathrm{+}{\mathit{\gamma}}_{\mathrm{5}}^{\mathrm{55}}{\mathit{v}}_{\mathrm{5}}{\mathit{\ell}}_{\mathrm{5}}^{\mathrm{2}}][\mathit{M}\mathrm{}{\mathit{k}}^{\mathrm{2}}]\\ & & \mathrm{}[{\mathit{D}}_{\mathrm{0}}\mathrm{+}{\mathit{\beta}}_{\mathrm{4}}{\mathit{v}}_{\mathrm{4}}\mathrm{+}{\mathit{\beta}}_{\mathrm{5}}{\mathit{v}}_{\mathrm{5}}\mathrm{+}{\mathit{\delta}}_{\mathrm{44}}{\mathit{v}}_{\mathrm{4}}^{\mathrm{2}}\mathrm{+}{\mathit{\delta}}_{\mathrm{45}}{\mathit{v}}_{\mathrm{4}}{\mathit{v}}_{\mathrm{5}}\mathrm{+}{\mathit{\delta}}_{\mathrm{55}}{\mathit{v}}_{\mathrm{5}}^{\mathrm{2}}\\ & & \mathrm{+}{\mathit{\delta}}^{\mathrm{44}}{\mathit{\ell}}_{\mathrm{4}}^{\mathrm{2}}\mathrm{+}{\mathit{\delta}}^{\mathrm{45}}{\mathit{\ell}}_{\mathrm{4}}{\mathit{\ell}}_{\mathrm{5}}\mathrm{+}{\mathit{\delta}}^{\mathrm{55}}{\mathit{\ell}}_{\mathrm{5}}^{\mathrm{2}}][\mathit{M}\mathrm{}{\mathit{k}}^{\mathrm{2}}{]}^{\mathrm{2}}\\ & & \mathrm{+}[{\mathit{H}}_{\mathrm{0}}\mathrm{+}{\mathit{h}}_{\mathrm{4}}{\mathit{v}}_{\mathrm{4}}\mathrm{+}{\mathit{h}}_{\mathrm{5}}{\mathit{v}}_{\mathrm{5}}][\mathit{M}\mathrm{}{\mathit{k}}^{\mathrm{2}}{]}^{\mathrm{3}}\end{array}$with M = J(J + 1) and k = ℓ_{4} + ℓ_{5}.
Effective spectroscopic constants determined from statebystate fits of the rotational transitions measured for the ground and v_{4} = 1, v_{5} = 1, v_{4} = v_{5} = 1 states of ^{12}C_{2}HD^{a}.
Effective spectroscopic constants determined from statebystate fits of the rotational transitions measured for the v_{4} = 2, v_{4} = 3, v_{5} = 2 and v_{5} = 3 states of ^{12}C_{2}HD^{a}.
Vibrational and rotational ℓtype resonances are expressed by offdiagonal matrix elements (Herman et al. 1991) containing the following parameters: $\begin{array}{ccc}{\mathit{r}}_{\mathrm{45}}& \mathrm{=}& {\mathit{r}}_{\mathrm{45}}^{\mathrm{0}}\mathrm{+}{\mathit{r}}_{\mathrm{445}}\mathrm{(}{\mathit{v}}_{\mathrm{4}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{+}{\mathit{r}}_{\mathrm{455}}\mathrm{(}{\mathit{v}}_{\mathrm{5}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{+}{\mathit{r}}_{\mathrm{45}}^{\mathit{J}}\mathit{M}\\ {\mathit{q}}_{\mathit{t}}& \mathrm{=}& {\mathit{q}}_{\mathit{t}}^{\mathrm{0}}\mathrm{+}{\mathit{q}}_{\mathit{tt}}{\mathit{v}}_{\mathit{t}}\mathrm{+}{\mathit{q}}_{\mathit{t}{\mathit{t}}^{\mathrm{\prime}}}{\mathit{v}}_{{\mathit{t}}^{\mathrm{\prime}}}\mathrm{+}{\mathit{q}}_{\mathit{t}}^{\mathit{J}}\mathit{M}\mathrm{+}{\mathit{q}}_{\mathit{t}}^{\mathit{JJ}}{\mathit{M}}^{\mathrm{2}}\mathrm{+}{\mathit{q}}_{\mathit{t}}^{\mathit{k}}\mathrm{(}\mathit{k}\mathrm{\pm}\mathrm{1}{\mathrm{)}}^{\mathrm{2}}\\ {\mathit{\rho}}_{\mathit{t}}& \mathrm{=}& {\mathit{\rho}}_{\mathit{t}}^{\mathrm{0}}\mathrm{+}{\mathit{\rho}}_{\mathit{tt}}{\mathit{v}}_{\mathit{t}}\mathrm{+}{\mathit{\rho}}_{\mathit{t}{\mathit{t}}^{\mathrm{\prime}}}{\mathit{v}}_{{\mathit{t}}^{\mathrm{\prime}}}\mathrm{+}{\mathit{\rho}}_{\mathit{t}}^{\mathit{J}}\mathit{M}\mathrm{+}{\mathit{\rho}}_{\mathit{t}}^{\mathit{JJ}}{\mathit{M}}^{\mathrm{2}}\\ \mathrm{and}& & \\ {\mathit{\rho}}_{\mathrm{45}}^{\mathrm{0}}& \mathrm{+}& {\mathit{\rho}}_{\mathrm{45}}^{\mathit{J}}\mathit{M}\mathit{.}\end{array}$The global fit included 5317 IR and 168 MW transitions. Overlapping lines were given zero weight. The uncertainty for the FIR and IR data was in the range 5.0 × 10^{5} − 2.0 × 10^{4} cm^{1}, and 1.0 × 10^{7} cm^{1} for the MW data. The MW blended lines were given an uncertainty of $\sqrt{\mathrm{2}}\hspace{0.17em}\mathrm{\times}\mathrm{1.0}\hspace{0.17em}\mathrm{\times}\hspace{0.17em}{\mathrm{10}}^{7}\hspace{0.17em}{\mathrm{cm}}^{1}$. Finally, 489 IR transitions, 9.2%, were excluded in the final fit because they were overlapping (271) or their observedcalculated values (218) were larger than 5 times their estimated experimental uncertainty.
For the 3ν_{4}(Φ) state, the J′′ = 3 e/f components were not resolved. For the 3ν_{5}(Φ) state, all the e/f components of the rotational lines were not resolved. The couples of overlapping lines were identified by the same frequency (see Table 2) and their observedcalculated values were derived from the comparison of the experimental frequency with the average of the frequencies calculated for the two components. Sixty nine statistically well determined parameters, which are collected in Table 5, were refined with a final rms value of 3.06 × 10^{4} cm^{1} for the IR data and 19 kHz for the MW data. A few parameters in the model not reported in Table 5 were nevertheless allowed to vary during the fitting procedure but they resulted statistically undetermined and were constrained to zero. Some of the parameters are highly correlated, i.e. ${\mathit{\omega}}_{\mathrm{4}}^{\mathrm{0}}$, ${\mathit{x}}_{\mathrm{44}}^{\mathrm{0}}$ and y_{444}, ${\mathit{g}}_{\mathrm{44}}^{\mathrm{0}}$ and ${\mathit{y}}_{\mathrm{4}}^{\mathrm{44}}$, γ^{44} and ${\mathit{\gamma}}_{\mathrm{4}}^{\mathrm{44}}$. The results in Table 5 can be compared with those obtained from the previous analysis, see Table 2 of PredoiCross et al. (2011). Sixty two parameters are common to both sets. The inclusion of the MW data allowed the determination of 7 additional constants. Three of these are related to the ν_{4} bending states, partly because experimental data for the v_{4} = 3(Φ) state have been obtained for the first time. Values and signs of all the common parameters are consistent in both sets: the differences between new and old values being in the range 0–1% (31 parameters), 1–10% (18 parameters), 10–50% (7 parameters), 50–100% (5 parameters) and 1 parameter, h_{4}, differs by 306%. The inclusion of the MW data also yields a significant improvement of the precision of the main constants which contribute to the rotational energy. For example, the values of B_{0}, α_{t}, and ${\mathit{q}}_{\mathit{t}}^{\mathrm{0}}$ constants are ca. 5 times more precise than those determined previously (PredoiCross et al. 2011). It should be pointed out that in the present global analysis 60 additional IR transitions were discarded in the final fit, compared with the previous fit (PredoiCross et al. 2011), adopting the same rejection limits. The discarded lines are scattered over most of the bands. Considering the high accuracy of the assigned MW transitions, this result is particularly pleasing since it confirms the validity of the calibration of the IR data, which span a wide wavenumber range, from about 90 to 2100 cm^{1}.
Spectroscopic parameters (in cm^{1}) for the bending states of ^{12}C_{2}HD resulting from the simultaneous fit of all rovibrational and rotational transitions involving levels up to v_{4} + v_{5} = 3^{a}.
4. Conclusions
Rotational lines of ^{12}C_{2}HD were detected in the range 100–700 GHz. They were analysed together with the IR transitions reported in the literature in a global fit. Sixty nine parameters (rotational, vibrational, rovibrational, and ℓtype interaction) were determined with high precision. The analysis of the ℓtype resonances which affect the pure rotational spectra allows the determination of the small vibrational energy differences ΔG between different ℓsublevels of a given vibrational state, which can be directly calculated using the effective values of the fitted g_{tt′}, r_{45} and B_{v} constants in Tables 3 and 4.
Vibrational energy differences ( cm^{1}) between ℓsublevels of doubly and triply excited bending states of ^{12}C_{2}HD, as resulting from the rotational analysis and the global rovibrational analysis.
The reliability of these results can be checked by comparison with the corresponding values from the global analysis. Table 6 shows that the agreement between the two sets of ΔG values is very good, differences being mostly less than one percent, thus confirming that an accurate treatment of ℓtype resonances in the pure rotational spectra can provide precise information on the vibrational energy. The set of spectroscopic constants determined in this work is the most accurate and consistent available in the literature. From these constants it is possible to derive very accurate predictions for IR and MW spectra useful for astronomical searches. An extensive list of rotational frequencies up to 1 THz is reported in Table 2 for all the observed vibrational states, whose energies are reported in Table 1. The line strength of each transition can be calculated by the simple formula (Lafferty & Lovas 1978): $\begin{array}{ccc}\mathit{S}\mathrm{(}\mathit{J}\mathrm{+}\mathrm{1}\mathrm{\leftarrow}\mathit{J}\mathrm{)}\mathrm{=}\frac{\mathrm{(}\mathit{J}\mathrm{+}\mathrm{1}{\mathrm{)}}^{\mathrm{2}}\mathrm{}{\mathit{\ell}}^{\mathrm{2}}}{\mathrm{(}\mathit{J}\mathrm{+}\mathrm{1}\mathrm{)}}\mathrm{\xb7}& & \end{array}$(3)
Acknowledgments
The authors acknowledge the Università di Bologna and the Ministero della Ricerca e dell’Università for financial support under the grant PRIN09 “Highresolution Spectroscopy for Atmospherical and Astrochemical Research: Experiment, Theory and Applications”. The authors also thank Prof. G. Di Lonardo for helping the analysis of the infrared spectra.
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All Tables
Dipole moments, vibrational term values, populations and intensity calculations for several excited vibrational bending states of ^{12}C_{2}HD.
Measured and predicted^{a} transition frequencies (MHz) of ^{12}C_{2}HD in the ground and excited bending states^{b}.
Effective spectroscopic constants determined from statebystate fits of the rotational transitions measured for the ground and v_{4} = 1, v_{5} = 1, v_{4} = v_{5} = 1 states of ^{12}C_{2}HD^{a}.
Effective spectroscopic constants determined from statebystate fits of the rotational transitions measured for the v_{4} = 2, v_{4} = 3, v_{5} = 2 and v_{5} = 3 states of ^{12}C_{2}HD^{a}.
Spectroscopic parameters (in cm^{1}) for the bending states of ^{12}C_{2}HD resulting from the simultaneous fit of all rovibrational and rotational transitions involving levels up to v_{4} + v_{5} = 3^{a}.
Vibrational energy differences ( cm^{1}) between ℓsublevels of doubly and triply excited bending states of ^{12}C_{2}HD, as resulting from the rotational analysis and the global rovibrational analysis.
All Figures
Fig. 1 The J = 5 ← 4 transition of ^{12}C_{2}HD in the v_{5} = 1, ground, and v_{4} = 1 vibrational states. The lowfrequency components of each ℓdoublet are displayed. Fourteen scans are coadded, the total integration time is 850 s. 

In the text 
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