Rotational spectroscopy of isotopic cyclopropenone, c-H_2C_3O, and determination of its equilibrium structure
Holger S. P. Müller, Ananya Brahmi M., Jean-Claude Guillemin, Frank Lewen, Stephan Schlemmer
AAstronomy & Astrophysics manuscript no. c-H2C3O_06 © ESO 2021February 9, 2021
Rotational spectroscopy of isotopic cyclopropenone, c -H C O, anddetermination of its equilibrium structure (cid:63)
Holger S. P. Müller , Ananya Brahmi M. , Jean-Claude Guillemin , Frank Lewen , and Stephan Schlemmer I. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germanye-mail: [email protected] Univ Rennes, Ecole Nationale Supérieure de Chimie de Rennes, CNRS, ISCR − UMR 6226, 35000 Rennes, FranceReceived 08 Dec 2020 / Accepted 21 Jan 2021
ABSTRACT
Context.
Cyclopropenone was first detected in the cold and less dense envelope of the giant molecular cloud Sagittarius B2(N). It wasfound later in several cold dark clouds and it may be possible to detect its minor isotopic species in these environments. In addition,the main species may well be identified in warmer environments.
Aims.
We aim to extend existing line lists of isotopologs of c -H C O from the microwave to the millimeter region and create one forthe singly deuterated isotopolog to facilitate their detections in space. Furthermore, we aim to extend the line list of the main isotopicspecies to the submillimeter region and to evaluate an equilibrium structure of the molecule.
Methods.
We employed a cyclopropenone sample in natural isotopic composition to investigate the rotational spectra of the mainand O-containing isotopologs as well as the two isotopomers containing one C atom. Spectral recordings of the singly and doublydeuterated isotopic species were obtained using a cyclopropenone sample highly enriched in deuterium. We recorded rotationaltransitions in the 70 −
126 GHz and 160 −
245 GHz regions for all isotopologs and also in the 342 −
505 GHz range for the main species.Quantum-chemical calculations were carried out to evaluate initial spectroscopic parameters and the di ff erences between ground-stateand equilibrium rotational parameters in order to derive semi-empirical equilibrium structural parameters. Results.
We determined new or improved spectroscopic parameters for six isotopologs and structural parameters according to di ff erentstructure models. Conclusions.
The spectroscopic parameters are accurate enough to identify minor isotopic species at centimeter and millimeterwavelengths while those of the main species are deemed to be reliable up to 1 THz. Our structural parameters di ff er from earlier ones.The deviations are attributed to misassignments in the earlier spectrum of one isotopic species. Key words.
Molecular data – Methods: laboratory: molecular – Techniques: spectroscopic – Radio lines: ISM – ISM: molecules –Astrochemistry
1. Introduction
Cyclopropenone is one of the smallest aromatic molecules(Anonymous 1983; Wang et al. 2011), after cyclopropenylidene( c -C H ), and the first cyclic molecule with a ketone functionalgroup that has been detected in space. It was first found in thecold and less dense envelope of the giant star-forming regionSagittarius B2(N) (Hollis et al. 2006). More recently, c -H C Owas detected in four dark clouds or prestellar cores, TMC-1,B1-b, L483, and Lupus-1A (Loison et al. 2016), and in the low-mass star-forming region L1527 (Araki et al. 2017).Benson et al. (1973) presented the first account on the rota-tional spectrum of cyclopropenone. These latter authors reportedbetween 6 and 14 rotational transitions for five isotopic speciesrecorded between 9 GHz and 40 GHz. The main isotopolog andthe two isotopomers containing one C atom were studied in asample in natural isotopic composition. Benson et al., employedisotopic enriched samples to investigate isotopologs containing (cid:63)
Transition frequencies from this work as well as related data fromearlier work are given for each isotopic species as supplementary ma-terial. We also provide quantum numbers, uncertainties, and residualsbetween measured frequencies and those calculated from the final setsof spectroscopic parameters. The data are available at CDS via anony-mous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http: // cdsweb.u-strasbg.fr / cgi-bin / qcat?J / A + A / O or two D. Additionally, they derived structural parametersof c -H C O and determined its dipole moment and rotational g values through Stark and Zeeman measurements, respectively.Later, Guillemin et al. (1990) extended the data set of the mainisotopic species to 247 GHz. These data are su ffi cient to identify c -H C O in cold astronomical sources up to the lower submil-limeter region, possibly up to ∼
500 GHz, but are not enough forsearches in warmer astronomical environments. Although cyclo-propenone has so far only been identified in colder sources, itmay well be detected in the warmer and denser parts of star-forming regions in the future. Propanal (Lykke et al. 2017),propene, and propenal (Manigand et al. 2021) are recent ex-amples of molecules initially found only in cold astronomicalsources later found in the hot corino, the warmer and denser partsof a low-mass star-forming region—of IRAS 16293 − c -HDC O, which isthe most promising isotopolog to be found in cold dark molec-ular clouds. Therefore, we investigated the rotational spectra offive isotopologs of cyclopropenone in the millimeter region to
Article number, page 1 of 11 a r X i v : . [ a s t r o - ph . I M ] F e b & A proofs: manuscript no. c-H2C3O_06 facilitate their detection in space and extended measurements ofthe main isotopic species to the lower submillimeter region toenable more secure searches in warmer astronomical sources.
2. Experimental details
The synthesis of normal cyclopropenone was the same as inGuillemin et al. (1990) and followed the procedure described byBreslow et al. (1977). The synthesis of the cyclopropenone sam-ple highly enriched in deuterium followed largely Breslow &Oda (1972) with n -Bu SnH replaced by n -Bu SnD. Please notethat cyclopropenone decomposes at room temperature. It is verystable in a sealed container at temperatures below ∼
240 K. Wekept the sample in dry ice or liquid nitrogen for our experiments.
The investigation of the rotational spectra of cyclopropenone iso-topologs was carried out with two di ff erent spectrometers. Weemployed two 7 m coupled glass cells, each with an inner diam-eter of 10 cm, in a double path arrangement for measurements inthe 70 −
126 GHz region, yielding an optical path length of 28 m.We used a 5 m double path cell with a 10 cm inner diameterfor the 160 −
245 GHz and the 342 −
505 GHz ranges. Both spec-trometers use Virginia Diode, Inc. (VDI), frequency multipliersdriven by Rohde & Schwarz SMF 100A synthesizers as sources,and Schottky diode detectors. Frequency modulation was em-ployed to reduce baseline e ff ects with demodulation at twice themodulation frequency. This causes absorption lines to appear ap-proximately as second derivatives of a Gaussian. Additional in-formation on the spectrometers is available in Ordu et al. (2012)and Martin-Drumel et al. (2015), respectively.We recorded individual transition frequencies in all three fre-quency windows covering 10 MHz for well-predicted lines up to100 MHz in the search for first lines of the singly deuterated iso-topolog. The pressure was around 1.0 Pa in the 3 mm region astest measurements showed that the peak intensity was best be-tween ∼ ∼ −
20 kHz, and larger uncertainties up to ∼
100 kHz wereused for example for weaker lines and lines close to other lines.Initial measurements using the highly deuterated cyclo-propenone sample yielded no clear signs of rotational transi-tions of c -D C O, but instead strong signals of c -H C O. Wesuspected rapid D-to-H exchange on the cell walls, even thoughthis may appear unusual for a molecule with substantial aro-matic character. On the other hand, at least some unsaturatedmolecules are known to exchange D and H quite readily; forexample, the reaction between HC N and D O was used to gen-erate DC N (Spahn et al. 2008). We conditioned the cell wallswith ∼
200 Pa D O for two hours and observed strong signals of c -D C O and very weak ones of c -H C O afterwards. The sig-nals of c -HDC O, identified later, were in between, roughly afactor of four weaker than those of c -D C O. Fig. 1.
Model of the cyclopropenone molecule. Carbon atoms are sym-bolized by gray spheres which are numbered. Hydrogen atoms are indi-cated by small, light gray spheres and the oxygen atom by a red sphere.
3. Quantum-chemical calculations
We carried out quantum-chemical calculations at the RegionalesRechenzentrum der Universität zu Köln (RRZK) using the com-mercially available program Gaussian 09 (Frisch et al. 2013).We performed B3LYP hybrid density functional (Becke 1993;Lee et al. 1988), Møller-Plesset second- (MP2) and third-orderperturbation theory (MP3) calculations (Møller & Plesset 1934),along with coupled cluster calculations with single and doubleexcitations augmented by a perturbative correction for triple ex-citations, CCSD(T) (Raghavachari et al. 1989). We employedcorrelation consistent basis sets which were augmented with dif-fuse basis functions aug-cc-pVXZ (X = T, Q, 5; Dunning, Jr.1989), which we abbreviate here to 3a, 4a, and 5a, respectively.These basis sets were further augmented with core-correlatingbasis functions in some cases, yielding the aug-cc-pwCVXZ ba-sis sets (Peterson & Dunning, Jr. 2002), which we denote 3aC,4aC, and 5aC, respectively.Equilibrium geometries were determined by analytic gradi-ent techniques, harmonic force fields by analytic second deriva-tives, and anharmonic force fields by numerical di ff erentiation ofthe analytically evaluated second derivatives of the energy. Themain goals of these anharmonic force field calculations were toevaluate initial spectroscopic parameters for the minor isotopicspecies of cyclopropenone and first-order vibration-rotation pa-rameters (Mills 1972); see also Sect. 6. Core electrons were keptfrozen in MP2, MP3, and CCSD(T) calculations unless “ae” in-dicates that all electrons were correlated.
4. Spectroscopic properties of cyclopropenone
Cyclopropenone is a planar asymmetric rotor with κ = (2 B − A − C ) / ( A − C ) = − . −
1. Its dipole moment of 4.39 D (Bensonet al. 1973) is along the a -inertial axis which is aligned withthe CO bond; see Fig. 1. The strongest transitions are R -branchtransitions ( ∆ J = +
1) with ∆ K a = ∆ K c = +
1. The Boltz-mann peak at 300 K is near 480 GHz. Q -branch transitions with ∆ K a = ∆ K c = − ∆ K a = ± Article number, page 2 of 11olger S. P. Müller et al.: Rotational spectroscopy of isotopic c -H C O Fig. 2.
Section of the submillimeter spectrum of c -H C O. The K c = J − J . Also demonstrated is the 1:3 para to ortho ratio for transitions with K a being even and odd, respectively; seeSect. 4 for further details. Carbon has two stable isotopes with mass numbers 12 and13 and with terrestrial abundances of 98.89% and 1.11%, respec-tively (Berglund & Wieser 2011). The respective abundances are99.76%, 0.04%, and 0.20% for O, O, and O, and 99.98%and ∼ O or C in the ketogroup (C1) have C symmetry. Spin-statistics caused by the twoequivalent H lead to ortho and para states with a 3:1 intensity ra-tio; see Fig. 2. The ortho states are described by K a being odd.The doubly deuterated isotopolog also has C symmetry, butthe ortho to para ratio is 2:1, and the ortho states are describedby K a being even. The e ff ect of substituting C at C2 or C3 isthe same because the two carbon atoms are structurally equiv-alent. The corresponding species is therefore referred to as the C2 isotopolog. We note that we do not consider isotopologswith more than one C. The abundance of the C2 isotopologis ∼ C S , as is that of the HD isotopolog. No nontrivial spinstatistics exist in these two isotopologs. The HD isotopolog hasa very small b -dipole moment component of ∼ C2 isotopolog is ∼
5. Spectroscopic results
Pickett’s programs SPCAT and SPFIT (Pickett 1991) were usedto calculate and fit the rotational spectra of the cyclopropenoneisotopologs. The results of Guillemin et al. (1990) were used forthe main species. Too few transition frequencies have been pub-lished for the minor isotopic species to reliably determine quar-tic centrifugal distortion parameters. We evaluated ground-staterotational and equilibrium quartic centrifugal distortion parame-ters for six isotopic species of cyclopropenone at the B3LYP / X isoexp of isotopic spectroscopic parame-ters by scaling the calculated value X isocalc with the ratio of theexperimental and calculated value X mainexp / X maincalc of the main iso-topolog. These values were used as starting parameters to repro-duce the limited sets of transition frequencies of the minor iso- topic species. One important aspect in this procedure, and in allof our fitting, was to determine or to float as few parameters aspossible to improve the weighted rms (wrms) of the fit as a usefulmeasure of the quality of the fit. This implies that we searched ineach fitting round for the spectroscopic parameter that led to thegreatest reduction of the wrms, and that was useful in the contextof parameters already determined or floated in the fit. It was suf-ficient to float B and C in the cases of the C1, C2, and O iso-topologs of cyclopropenone. One transition of the O species,4 , − , , and two transitions of the C2 species, 10 , − , and 15 , − , , had large residuals between the reported ex-perimental frequency and the calculated one that we decreasedthe weights of these transitions initially; eventually, these wereomitted from the final fits. The reported frequencies were largerthan in our final calculations by 0.46 MHz for the line attributedto the O species and by 43.51 and 92.56 MHz, respectively,for the two lines assigned to the C2 isotopolog. We also had tofloat A in order to satisfactorily reproduce the transition frequen-cies of the doubly deuterated isotopolog. The adjusted rotationalparameters of c -D C O di ff ered somewhat more from the initialparameters than the adjusted B and C values of the other minorisotopic species. We assumed the deviations in the case of c -HDC O would be roughly half as large as those of c -D C O. Wecorrected the c -HDC O accordingly to improve the calculationsof this isotopolog for the first searches of rotational transitions.
Our investigations started with measurements of the main iso-topic species in the 342 −
505 GHz region. We targeted thestrongest R -branch transitions with K a ≤ K a =
29 in subsequent steps. Transitions with K a ≤ K a and exceeded 10 MHz for thehighest K a transitions. We then searched for Q -branch transitionswith ∆ K a = ∆ K a = ∆ J =
0. Later, very limited measurements werecarried out in the 160 −
245 GHz region and then again more ex-tensive ones in the 70 −
126 GHz region. The ∆ K a = ∆ J = K a =
8, and the ∆ K a = K a = ↔
7. Whereas the rotational spectrum of cy-clopropenone is quite sparse on the level of the strongest lines,it is much richer on the level of the weakest transitions recorded,which means that the desired line has an increased chance ofbeing blended with or being close to a usually unassigned line.In the end, we recorded 398 di ff erent, novel transitions,which corresponds to 311 di ff erent frequencies because of 87 un-resolved asymmetry doublets. The majority of these doublets areprolate paired transitions, which have the same J and K a for eachpair, as may be expected for an asymmetric rotor somewhat closeto the prolate symmetric limit; seven are oblate paired transitionswith K c = J in each case. In addition, we remeasured three tran-sitions already reported by Guillemin et al. (1990). These datawere combined with 14 transition frequencies from Benson et al.(1973) and 45 from Guillemin et al. (1990), which correspondedto 51 transitions. Uncertainties of 30 kHz were used for theselines as stated by Guillemin et al. (1990) for their data and com-mensurate for the data of Benson et al. (1973).We subjected the transition frequencies initially to fits em-ploying Watson’s A reduction, as done previously (Guilleminet al. 1990). We also tried Watson’s S reduction because cy-clopropenone is quite close to the prolate symmetric limit, inwhich case the S reduction is usually preferable. As can be seenin Table 1, we required an almost complete parameter set up Article number, page 3 of 11 & A proofs: manuscript no. c-H2C3O_06
Table 1.
Present and previous experimental spectroscopic parameters (MHz) of the main isotopolog of cyclopropenone and details of the fits incomparison to values calculated at the B3LYP /
3a level.
S reduction A reductionParameter Present B3LYP / a B3LYP / a Present Previous b Parameter A AB BC CD K × ∆ K × D JK × ∆ JK × D J × ∆ J × d × − − δ K × d × − − δ J × H K × − − Φ K × H KJ × Φ KJ × H JK × − − Φ JK × H J × − Φ J × h × φ K × h × φ JK × h × φ J × J max
48 48 27 K a , max
29 29 12No. transitions 463 463No. frequencies 370 370 59rms 0.0248 0.0243 0.028rms (new) c c c d c c c Notes.
Watson’s S and A reductions were used in the representation I r . Numbers in parentheses are one standard deviation in units of the leastsignificant figures. Empty fields indicate parameters not used in the fit. ( a ) Ground-state rotational and equilibrium centrifugal distortion parameters. ( b ) Guillemin et al. (1990). ( c ) The labels (new), (B), and (G) refer to our new data, those from Benson et al. (1973), and from Guillemin et al.(1990). ( d ) Weighted rms, unit-less. to sixth order in the A reduction; only Φ K was not used in thefit. Two fewer parameters were used in the S reduction, albeitwith a slightly larger wrms of 0.972 compared to 0.921. Nev-ertheless, we decided to view the S reduced fit as the preferredone. Trial fits with Φ K or H K , H J , and h from B3LYP /
3a cal-culations added to the fit as fixed parameters did not improvethe quality of the fits and resulted in changes of the remainingspectroscopic parameters roughly corresponding to their uncer-tainties. Moreover, we were uncertain as to the reliability of thequantum-chemically calculated sextic distortion parameters; seealso Sect. 7.1. Therefore, these parameters were omitted fromthe final fits. The quantum-chemically calculated spectroscopicparameters are given in Table 1 for comparison, along with theparameters from Guillemin et al. (1990). We also provide rmsvalues for the whole data set and for the individual sources forcompleteness. The rms is meaningful if only one uncertainty wasused in all instances or if the uncertainties di ff er only by a factorof a few. In the present case, the uncertainties di ff er by around afactor of ten. In this and similar cases, the rms is dominated bythe lines with large residuals. C1, C2, and O isotopologs
We began our investigations of the minor isotopologs of cyclo-propenone containing one C or O in the 160 −
245 GHz regionsearching for the stronger R -branch transitions first with K a ≤ K a =
5. Additional measurements were carriedout in the 70 −
126 GHz region, in which we recorded R -branchtransitions up to K a = ∆ K a = ∆ J = C. The extent of measuredlines was more limited for the O because of its lower abun-dance.The final line lists of the C1, C2, and O isotopologsconsist of 104, 95, and 48 di ff erent and new transition frequen-cies, respectively, in addition to the small number of previouslyreported lines (Benson et al. 1973) for which uncertainties of70 kHz were assigned. The resulting sets of spectroscopic pa-rameters, determined as described at the beginning of this sec-tion, had not only B and C floated for C1 and C2, but also A and two respectively three quartic distortion parameters. It wassu ffi cient to float only B and C in the case of the O isotopolog.The parameter values, their uncertainties, and additional detailsof the fits are presented in Table 2 as minimum parameter sets
Article number, page 4 of 11olger S. P. Müller et al.: Rotational spectroscopy of isotopic c -H C O Table 2.
Spectroscopic parameters (MHz) from a minimum parameterset, a maximum parameter set, and initial parameters along with detailsof the fits of the C1, C2, and O isotopologs of cyclopropenone.
Parameter Minimum Maximum Initial C1 isotopolog A B C D K × D JK × D J × d × − − − d × − − − J max
21 (15) 21 (15) K a , max − c
115 (6, − a No. freqs. 110 (6, − c
110 (6, − a rms 0.0271 0.0186wrms b C2 isotopolog A B C D K × D JK × D J × d × − − − d × − − − J max
30 (6) 30 (6) K a , max − c
106 (5, − a No. freqs. 100 (5, − c
100 (5, − a rms 0.0184 0.0180wrms b O isotopolog A B C D K × D JK × D J × d × − − − d × − − − J max
19 (14) 19 (14) K a , max − c
61 (10, − a No. freqs. 58 (10, − c
58 (10, − a rms 0.0545 0.0525wrms b Notes.
Watson’s S reduction was used in the representation I r . Numbersin parentheses of parameters are one standard deviation in units of theleast significant figures. Numbers in parentheses associated with num-bers of quantum numbers, transitions, and lines, respectively, refer todata from the previous study (Benson et al. 1973). Parameters withoutuncertainties were estimated and kept fixed in the analyses; see Sect. 5. ( a ) With omitted lines after the comma; see Sect. 5. ( b ) Weighted rms,unitless.
Fig. 3.
Section of the millimeter spectrum of cyclopropenone highly en-riched in deuterium. Transitions with small but rapidly increasing asym-metry splitting are shown for K a = c -HDC O. We assigned thesetransitions first because of their easily recognizable patterns. which are the preferred parameter sets. We also tested the num-ber of parameters that can be determined with su ffi cient signif-icance. These were all quartic centrifugal distortion parametersfor the isotopomers with one C and all but ∆ D K for the Oisotopolog. The resulting values are given as maximum parame-ter sets in the same table along with the initial parameters whosederivation was also described at the beginning of this section.
Our procedure for studying the rotational spectrum of the D iso-topolog was very similar to that for the minor isotopic speciesdescribed in the previous section. The use of a sample highly en-riched in deuterium made it possible to record considerably moretransitions extending to higher quantum numbers. The spectralrecordings were richer in lines than those of the sample in natu-ral isotopic composition because of the presence of lines causedby the HD species and even some of the main species. In addi-tion, we identified lines of traces of CH Cl which was used as asolvent in the course of the preparation of the deuterated sampleand was di ffi cult to remove completely.The R -branch transitions with ∆ K a = K a =
9, transitions with ∆ K a = ∆ J = ∆ K a = K a = ← ff erent strategy to search for transitionsof the HD isotopolog because no previous data exist. A pat-tern of two or more transitions is obviously more decisive thana single line. The best candidates are nearly prolate or nearly Article number, page 5 of 11 & A proofs: manuscript no. c-H2C3O_06 oblate paired R -branch transitions because the small asymme-try splitting changes rapidly with J . The asymmetry splittingof the K c = J nearly oblate paired transitions was deemed tobe too large in the 160 −
245 GHz region. As can be seen inFig. 3, the K a = J ” =
12 to 14 compared very well with the calculated split-tings. We searched for R -branch transitions with equal or lower K a with an improved calculation of the rotational spectrum. Sub-sequently, we sought Q -branch transitions with ∆ K a = K a = K a = ← b -type transitions of c -HDC O as thesewere calculated to be even weaker than many of the ∆ K a = b -type transitions were covered by accident;even much longer integration times may not have helped be-cause the number of lines with similar or larger intensities wasso large that it would be di ffi cult to identify unblended lines un-ambiguously. In addition, it is important to note that ∆ K a = K -dependent pa-rameters than transitions with ∆ K a = K a .The minimum parameter sets for both deuterated isotopologswere determined as described above. Interestingly, we obtaineda satisfactory fit for 172 di ff erent frequencies of c -D C O afterfloating only two of the five quartic centrifugal distortion param-eters, whereas we required four for 126 di ff erent frequencies of c -HDC O; this is probably a consequence of the choice of thecovered transitions. In the fit of c -D C O with the full parame-ter set, ∆ D J and ∆ D JK improved the wrms by similar amounts.The parameter values, their uncertainties, and additional detailson the fits are presented in Table 3.
6. Structural parameters of cyclopropenone
Several methods exist to derive structural parameters of amolecule from its moments of inertia, which are inversely pro-portional to the rotational parameters. The results depend on themodel to a varying degree, and some methods do not even yieldthe same result for di ff erent data sets in theory. Data for di ff er-ent isotopologs are required for molecules with three or moresymmetry-inequivalent atoms; it is usually desirable to substi-tute each symmetry-inequivalent atom once to be able to obtainreliable structural parameters.The ground-state e ff ective (or r ) structure is the moststraightforward model in which the structural parameters are fitto the ground-state moments of inertia. The ground-state valuesare usually the ones determined first, and are often the only ones.The ground-state moments of inertia contain vibrational contri-butions from the zero-point vibrations, causing the r structure tobe one of the less meaningful structure models, with sometimesrelatively large changes between isotopic data sets.The di ff erences in the moments of inertia caused by thesubstitution of one or more symmetry-inequivalent atoms canbe used to determine the Cartesian coordinates of these atoms(Kraitchman 1953; Costain 1958). The resulting substitution (or r s ) structure reduces the e ff ects of vibrational contributions tothe ground-state moments of inertia, albeit to a varying degree.The r s structure is equivalent to r ∆ I , the structure that is obtainedby fitting structural parameters to the di ff erences of the momentsof inertia (Rudolph 1991). This structure is, in turn, equivalentto the r I ,(cid:15) structure, in which vibrational contributions (cid:15) i to theground-state moments of inertia, that is I ii , = I ii , e + (cid:15) i , with i = a , b , c , are assumed to be equal for di ff erent isotopologs of Table 3.
Spectroscopic parameters (MHz) from a minimum parameterset, a full parameter set, and initial parameters along with details on thefits of the mono and doubly deuterated isotopologs of cyclopropenone.
Parameter Minimum Full InitialHD isotopolog A B C D K × D JK × D J × d × − − − d × − − − J max
20 (0) 20 (0) K a , max a isotopolog A B C D K × D JK × D J × d × − − − d × − − − J max
24 (10) 24 (10) K a , max a Notes.
Watson’s S reduction was used in the representation I r . Numbersin parentheses of parameters are one standard deviation in units of theleast significant figures. Numbers in parentheses associated with num-bers of quantum numbers, transitions, and lines, respectively, refer todata from the previous study (Benson et al. 1973). Parameters withoutuncertainties were estimated and kept fixed in the analyses, see Sect. 5. ( a ) Weighted rms, unitless. a given molecule. The advantage of taking (cid:15) i into account ex-plicitly is that rotational parameters of isotopic species to bestudied can be estimated much more accurately, in particularfor atoms with more than two isotopes or for multiply substi-tuted isotopologs (Epple & Rudolph 1992; Müller & Gerry 1994;Müller et al. 2019), if residuals of known isotopologs can betransferred to the values of the desired isotopolog. In the case ofH CS, Müller et al. (2019) extrapolated the residuals of H C S,H C S, and H C S to H C S, and those of H C S, H C S,and H
CS to H C S.The equilibrium (or r e ) structure of a molecule in the po-tential minimum is the most meaningful structure. However, itis also the most elusive structure because data of more thanone isotopolog are needed for molecules with three or more Article number, page 6 of 11olger S. P. Müller et al.: Rotational spectroscopy of isotopic c -H C O Table 4.
Ground-state rotational parameters B i , of cyclopropenone isotopic species, vibrational corrections ∆ B i , v a , resulting semi-empirical equi-librium rotational parameters B SE i , e , and inertia defects ∆ . b B3LYP MP2 ae-MP2Species B i /∆ B i , ∆ B i , v B SE i , e ∆ B i , v B SE i , e ∆ B i , v B SE i , e main A B C ∆ − − − C1 A B C ∆ − − − C2 A B C ∆ − − − O A B C ∆ − − − A B C ∆ − − − A B C ∆ − − − Notes. ( a ) ∆ B i , v = (cid:80) j α B i j / ff erent quantum-chemical means as detailed in Sect. 3. ( b ) All numbers in units of MHz, except ∆ inunits of amu Å . symmetry-inequivalent atoms, and for each isotopolog we re-quire the knowledge of several vibration-rotation parameters ac-cording to B e = B + (cid:88) j α Bj − (cid:88) j ≤ k γ Bjk − ... (1)in order to evaluate the equilibrium parameter B e from theground-state value B . Here, the α Bj are first-order vibrationalcorrections, the γ Bjk are second-order vibrational corrections, andso on. Equivalent formulations hold for A e and C e . An attrac-tive and lately very common approach is to calculate (cid:80) j α Bj / B i , e from the experimental ground-state values (Stanton et al. 1998). Second and higher order vibra-tional contributions are neglected. Numerous quantum-chemicalprograms are available to carry out such calculations; examplesare mentioned in Sect. 3.We used B3LYP, MP2, and ae-MP2 calculations with a triplezeta basis set to evaluate the first-order vibrational correctionsfor isotopologs of cyclopropenone which are summarized in Ta-ble 4 together with ground-state values and the resulting semi-empirical equilibrium values. The inertia defect ∆ = I cc − I bb − I aa is also given for each set of rotational parameters. We employed the RU111J program (Rudolph 1995) to de-rive semi-empirical equilibrium structural parameters r SE e foreach of the three quantum-chemical calculations. The results aregiven in Table 5 together with structural parameters purely fromquantum-chemical calculations and with the r s parameters fromBenson et al. (1973). The di ff erences between the r s parametersand our r SE e parameters turned out to be larger than expected andare discussed in detail in Sect. 7.2. We calculated subsequently r I ,(cid:15) structures, which are equivalent to r s structures as mentionedabove, in order to evaluate the dependence of the resulting pa-rameters on diverse input data. The first set of input data wereour present rotational data from six isotopic species as summa-rized in Table 4. The second set employed our starting values asdescribed in Sect. 5. The third set were the rotational parametersfrom Benson et al. (1973), but without A ( C2), as it deviated by61 MHz from the value of our preferred fit. The results of thesestructure fits are also given (in reverse order) in Table 5.
7. Discussion
As can be seen in Table 1, the J and K a quantum number rangesand also the number of di ff erent transition frequencies of themain isotopolog of cyclopropenone have been greatly increasedin the course of the present investigation with respect to the pre- Article number, page 7 of 11 & A proofs: manuscript no. c-H2C3O_06
Table 5.
Quantum-chemical and experimental bond lengths (pm) and bond angles (deg) of cyclopropenone.
Method a r (CO) r (C − C) r (C = C) b r (CH) ∠ (OCC) ∠ (C3C1C2) b ∠ (C1C2H)B3LYP /
3a 120.157 142.672 133.964 108.054 151.999 56.001 153.841B3LYP /
4a 120.000 142.626 133.894 108.005 152.005 55.989 153.852B3LYP / /
3a 120.681 143.399 135.270 107.970 151.858 56.284 154.166MP2 /
4a 120.388 143.046 134.907 107.850 151.865 56.270 154.150MP2 /
5a 120.313 142.962 134.815 107.817 151.868 56.264 154.142ae-MP2 / / / /
3a 119.628 142.810 134.383 107.698 151.934 56.133 153.478ae-MP3 / /
3a 120.684 143.735 135.308 108.155 151.921 56.158 153.739CCSD(T) /
4a 120.346 143.347 134.912 108.030 151.928 56.144 153.731ae-CCSD(T) / r sc r I ,(cid:15) (old) d r I ,(cid:15) (old) e r I ,(cid:15) (new) f r SE e (B3LYP) g r SE e (MP2) g r SE e (ae-MP2) g Notes.
All values from this work unless indicated otherwise. Numbers in parentheses are one standard deviation in units of the least significantfigures. ( a ) Quantum-chemical calculations as detailed in Sect. 3; structural parameter determinations as described in Sect. 6. ( b ) Derived parameterin present calculations. ( c ) Benson et al. (1973); structural parameters derived from substitution coordinates; ∠ (C1C2H) calculated from ∠ (C2C3H)and ∠ (C3C1C2). ( d ) This work; rotational parameters from Benson et al. (1973), but A ( C2) omitted; see also Sects. 6. ( e ) This work; rotationalparameter derived as described in Sect. 5 using data from Benson et al. (1973) and Guillemin et al. (1990). ( f ) This work; rotational parametersfrom this work. ( g ) Quantum-chemical method in parentheses as detailed in Sect. 6; see also Table 4. vious study (Guillemin et al. 1990). The previous spectroscopicparameters agree well with ours in the A reduction taking intoaccount the uncertainties in that work. The B3LYP /
3a quantum-chemically calculated ground-state rotational and equilibriumquartic centrifugal distortion parameters agree quite well withour experimental values in both reductions. The comparison ofthe sextic centrifugal distortion parameters is more mixed as faras experimental values are available; trial fits showed that theexperimental values are only slightly a ff ected by the absence ofthree and one sextic parameter in the S and A reduction, respec-tively. This shows that the deviations seen in particular in H KJ and the corresponding Φ KJ are not mainly caused by the omis-sion of the three or one parameter. Moreover, a trial fit with h in the fit yielded a value of 19 ± µ Hz, not compatible with thecalculated 96 µ Hz.Our final fit in the S reduction has two parameters less thanthe one in the A reduction at the expense of a slightly largerwrms. We consider this improvement to be su ffi cient to favor theS reduction. Additional reasons for preferring the S reduction arethat it is more versatile than the A reduction as the latter can ex-hibit convergence problems for molecules close to the prolate oroblate symmetric limit. Moreover, Watson (1977) recommendedto use the S reduction because such fits yield smaller correla-tion coe ffi cients. This conclusion was recently reemphasized inan extensive study of the rotational spectra of SO O and S O in their lowest two and three vibrational states, respectively, withdetailed analyses of the fits (Margulès et al. 2020). Our spectroscopic parameters for the C1, C2, and Oisotopologs in Table 2 agree very well with our initial values bothfor the minimum and maximum parameter sets. The agreementis good in the cases of the deuterated isotopologs, as can be seenin Table 3. The parameter D K from the full parameter sets of bothdeuterated isotopologs agrees very well with the initial values,which may indicate that the small changes in D K seen for theisotopomers containing one C may nevertheless be too large.The initial spectroscopic parameters of all minor isotopicspecies were derived from quantum-chemically calculated onesscaled with the respective ratios between experimental and cal-culated values of the main species as described in slightly moredetail at the beginning of Sect. 5. Such scaling was used, for ex-ample, to evaluate some sextic centrifugal distortion parametersof two minor isotopologs of TiO (Kania et al. 2011). Cazzoliet al. (2014) demonstrated in a study of the rotational spectrumof H S that such scaling works very well even for heavy-atomsubstitutions of relatively light molecules such as H S. Higherlevel quantum-chemical calculations, as employed by Cazzoliet al. (2014), may lead to better starting values than the lowerlevel calculations used here. However, higher level calculationsare computationally more demanding, in particular for somewhatlarger molecules. Moreover, Morgan et al. (2018) showed thatconsiderable computational e ff ort is necessary to obtain good-quality, systematically converged results even for the fairly smallmolecule formaldehyde, H CO. The quality of the scaling ofcentrifugal distortion parameters may be limited in addition bythe lack of vibrational corrections. Quartic and sextic equilib-
Article number, page 8 of 11olger S. P. Müller et al.: Rotational spectroscopy of isotopic c -H C O rium centrifugal distortion parameters can be calculated withseveral quantum-chemical programs, but the derivation of vibra-tional corrections is, to the best of our knowledge, not possiblewith any publicly available program.Scaling of quantum-chemically derived distortion parame-ters should be better than scaling by appropriate powers of theratios of 2 A − B − C , B + C , and B − C , as done, for example, inthe recent case of isotopic H CS (Müller et al. 2019). This typeof scaling usually works quite well for heavy-atom substitutions,but less so for H to D substitutions, in particular for moleculeswith few atoms. Fixing parameters of minor isotopologs, whichcannot be determined su ffi ciently well, to values from the mainisotopic species usually works less well, but is in most cases bet-ter than fixing such parameters to zero.We may ask ourselves what the advantages and disadvan-tages of our minimum parameter fits are compared to our maxi-mum parameter fits. Increasing the parameter set beyond the op-timum increases the uncertainties, usually the correlation amongthe parameters, and frequently a ff ects the parameter values out-side the initial uncertainties. Therefore, transition frequenciescalculated with fewer parameters are often better in cases of in-terpolation or modest extrapolation compared to values from alarger parameter set. On the other hand, calculated uncertaintiesquickly become too optimistic upon extrapolation for a smallerparameter set. Extrapolation in J from our present data should bereasonable to two times the upper frequency limit for the strong R -branch transition which is ∼ K a is much more limited.The ground-state inertia defects in Table 4 are relativelysmall and positive, as is common for planar molecules with smallout-of-plane vibrational amplitudes. The semi-empirical equilib-rium inertia defects obtained from the ground-state rotationalconstants after subtracting the first-order vibrational correctionsare much smaller in magnitude and negative, which is also verycommon. Our semi-empirical equilibrium structural parameters in Table 5are largely very similar among the three di ff erent sets. The CObond length has fairly large uncertainties, but displays almostno scatter; the C = C bond shows, in contrast, more pronounced,though still modest variations, which are only a few times thecombined uncertainties for the parameters employing B3LYPand ae-MP2 corrections. The CH bond varies to an approxi-mately equal degree, albeit with larger uncertainties than theC = C bond.The previously published r s structure (Benson et al. 1973)agrees, at best, reasonably well with our r SE e structures; the C = Cbond length di ff erence is, with more than 4 pm, particularlylarge. We calculated r I ,(cid:15) structures that are equivalent to the r s structure (Rudolph 1991). Using our ground-state rotational dataof six isotopologs, as summarized in Table 4, we obtained struc-tural parameters close to our r SE e values; the largest di ff erenceof 0.4 pm occurs in the CO bond length and is even within thecombined uncertainties. The initial parameters of five isotopicspecies, as derived in Sect. 5, yielded essentially the same pa-rameters, the main di ff erences are larger uncertainties in someparameters, most likely caused by the absence of data of the HDisotopolog; even the fit with values from Benson et al. (1973)yields results quite similar to the other two r I ,(cid:15) fits. The devia-tion of the r s structure parameters therefore appears to be causedby the di ff erence of 61 MHz in A ( C2) which is mainly caused by two lines falsely attributed to the C2 isotopolog; see alsoSect. 5.The quantum-chemical calculations in Table 5 show verysmall changes between basis sets and also rather small changesbetween the di ff erent methods. The bond lengths display theusual shortening upon increasing basis set size and upon correla-tion of all electrons as far as applicable. The bond lengths di ff ersubstantially among the di ff erent methods. The B3LYP param-eters show little change with basis set size and agree quite wellwith our r SE e values. The largest deviation is seen for the C = Cbond length, which is too short by about 0.5 pm. The C − C bondlength is also too short, but by smaller amounts, whereas the COand CH bonds were calculated slightly too long. The MP2 / / r SE e structure, the MP2 /
5a values are slightly closer forthe C − C and CH bonds, whereas the ae-MP2 / /
3a level, and evenmore so at ae-MP3 / / ff ects of correlating all electrons com-pared to those with a frozen core are similar to those of MP2 cal-culations employing basis sets of triple zeta quality. If we assumethe relative di ff erences are the same for basis sets of quadru-ple zeta quality, we can estimate the following ae-CCSD(T) / r (CO) ≈ r (C − C) ≈ r (C = C) ≈ r (CH) ≈ r SE e values; this behavior is common for ae-CCSD(T) / ff ect is smallestfor the distant CH bond. Formaldehyde is probably the prototypi-cal molecule with a CO double bond. Its CO bond is only slightlylonger than that of c -H C O. The C = C bond lengths of c -H C Oand c -C H are essentially equal to the one in C H , and arethus typical CC double bonds. The C − C bonds of both cyclicmolecules are nevertheless much shorter than that of ethane,whose C − C bond can probably be viewed as a prototypical CCsingle bond. We note that these bonds are only slightly longerthan the aromatic CC bonds in benzene. Finally, the CH bondlengths in c -H C O and c -C H are substantially shorter thanin C H , but much longer than ∼
8. Conclusion and outlook
Our data for five minor isotopic species of cyclopropenone aresu ffi cient for astronomical searches in cold molecular cloudseven without any extrapolation, as the Boltzmann peak at 10 K isat ∼
95 GHz. Extrapolation of the data in J should be reasonableup to 450 or 500 GHz. Such extrapolations should be possible upto 1 THz for the main isotopic species, which is su ffi cient evenfor rotational temperatures around 300 K. However, we assumethat rotational temperatures of c -H C O in warmer sources willbe closer to 100 K based on findings for propanal, propenal, and
Article number, page 9 of 11 & A proofs: manuscript no. c-H2C3O_06
Table 6.
Equilibrium bond lengths (pm) and selected bond angles (deg) of cyclopropenone in comparison to related molecules.
Molecule r (CO) r (C − C) r (C = C) r (CH) ∠ (C3C1C2) ∠ (C1C2H) c -H C O a c -C H b CO c H d c -C H e H f Notes.
Numbers in parentheses are one standard deviation in units of the least significant figures. If no uncertainties are given, none were pub-lished. ( a ) This work, r SE e with ae-MP2 corrections. ( b ) Spezzano et al. (2012); r SE e , ae-CCSD(T) /
4C values; r (C = C) calculated from r (C − C) and ∠ (C3C1C2). ( c ) Lohilahti (2006); experimental r e values. ( d ) Harmony (1990); r ρ m values according to the second set of data; see Harmony & Taylor(1986) for the definition of this structure model. ( e ) Gauss & Stanton (2000); r SE e at the CCSD(T) / ( f ) Martin & Taylor (1996); r e values fromCCSD(T) calculations extrapolated to infinitely large basis set with ae corrections and empirical adjustments. propene in IRAS 16293 − c -HDC O is themost promising isotopolog to be found in cold molecular clouds.One of the most sensitive molecular line surveys of such sourcesis that by Cernicharo et al. (2020) between 31.4 and 50.3 GHzcarried out with the Yebes 40 m radio telescope with additionaldata from the 3 mm region taken earlier with the IRAM 30 mdish. Cernicharo et al. (2020) report the detection of HC O + andinvestigated the chemistry of oxygen-containing molecules inTMC-1. They derived a column density of (4 . ± . × cm − for c -H C O. The most constraining transition of c -HDC O inthat survey is the 3 − transition at 37833.721 MHz. The 3 σ upper limit to its intensity in brightness temperature is ∼ ffi ciency); the cor-responding intensity of the 3 − ortho transition of c -H C Ois 42 mK, yielding a 3 σ column density ratio of >
16 for c -H C O to c -HDC O (J. Cernicharo, private communication toHSPM, Nov. / Dec. 2020). This is less constraining than the corre-sponding ratios of, for example, ∼
34 and ∼
25 found for c -C H to c -C HD (Bell et al. 1986) and l -C H to l -C HD (Spezzanoet al. 2016), respectively, in TMC-1. However, there are sourceswith higher degrees of deuteration that may be considered if theircolumn densities of c -H C O are su ffi ciently large. A searchfor the cyclopropenone isotopomers with one C may be morepromising in the envelope of Sagittarius B2(N) because the Cto C ratio in the Galactic center is as low as 20 to 30 (Mülleret al. 2008, 2016; Halfen et al. 2017; Jacob et al. 2020).We determined structural parameters of cyclopropenonewhich compare very favorably with those of cyclopropenylideneas far as comparison is possible. The CO bond length agreesalmost exactly with that of formaldehyde. Relatively large devi-ations between our structural parameters and the earlier substi-tution structure were traced to misassignments of two transitionsto the C2 isotopolog which led to an overly large A rotationalparameter. Acknowledgements.
We are grateful to José Cernicharo for communication ofunpublished results of a search for c -HDC O in TMC-1. We acknowledge sup-port by the Deutsche Forschungsgemeinschaft via the collaborative researchcenter SFB 956 (project ID 184018867) project B3 as well as the Gerätezen-trum SCHL 341 / / INSU with INC / INP co-funded by CEA and CNES. Our research benefited fromNASA’s Astrophysics Data System (ADS).
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