Improved semi-experimental equilibrium structure and high-level theoretical structures of ketene

Ketenes (R₂C=C=O) represent an important functional group in organic chemistry.^1,2 The reactivity of ketenes, albeit modulated by the nature of the substituents, is generally very high with respect to cycloaddition reactions and the addition of nucleophiles. As such, ketenes are precursors to the family of carboxylic acid derivatives, including anhydrides, carboxylic acids, esters, and amides (Scheme ). It is the central relationship to these and other functional groups that places ketenes as important intermediates in diverse areas of science, including synthetic organic chemistry,^1–3 petroleum refining,⁴ photolithography,⁵ atmospheric chemistry,⁶ combustion,⁷ and astrochemistry.⁸ The parent molecule of the family, ketene (H₂C=C=O), is a paradigm in terms of structure, bonding, and reactivity. It is a cornerstone of structural organic chemistry and has attracted substantial interest from both experimental and theoretical communities. In the current study, we describe the determination of a highly precise, gas-phase molecular structure for ketene using state-of-the-art methods for both experiment and theory.

Ketene (H₂C=C=O, C_2v, ethenone) has been identified as an interstellar molecule, with its initial detection in Sgr B2 in 1977.⁹ Its detection in that source was later confirmed,¹⁰ and various subsequent studies detected ketene in other galactic^11–15 and extra-galactic¹⁶ sources. Ketene has been generated from the photolysis of interstellar ice analogs comprising a variety of chemical compositions (O₂ + C₂H₂, CO₂ + C₂H₄, CO + CH₄).⁸ The central, electrophilic carbon atom of ketene reacts with common interstellar molecules H₂O, NH₃, CH₃OH, and HCN to form acetic acid, acetamide, methyl acetate, and pyruvonitrile, respectively.^17–19 Some of these reaction products have potential prebiotic importance in the interstellar medium. Ketene can also be generated from the decomposition of two known interstellar molecules: acetic acid²⁰ and acetone.²¹ The acylium cation (H₃C–C≡O⁺, C_3v),²² which has recently been detected in the ISM,²³ is a protonated form of ketene. Due to its prevalence and possible role in the chemical reactions of extraterrestrial environments, the observation of rotational transitions of ketene in its ground and vibrationally excited states is important to radioastronomers. Previously, the rotational spectra of various ketene isotopologues have been observed and assigned up to a frequency of 350 GHz. In this work, we extended the frequency range to 750 GHz for 15 ketene isotopologues, including all singly substituted heavy-atom isotopologues of ketene, ketene-d₁, and ketene-d₂. The doubling of the spectral range for ketene isotopologues, along with new isotopologue analyses, expands the capabilities for radioastronomers to search for ketene spectral lines in different extra-terrestrial environments.

The pure rotational spectra of ketene and its isotopologues have been extensively studied for over seventy years. The rotational spectrum of ketene and its two deuteriated isotopologues, [2-²H]-ketene (HDC=C=O) and [2,2-²H]-ketene (D₂C=C=O), were measured in the early 1950s.^24,25 Subsequently, heavy-atom substituted isotopologues [2-¹³C]-ketene and [¹⁸O]-ketene were measured in 1959 and 1963,^26,27 but the final single-substitution isotopologue, [1-¹³C]-ketene, was not reported until 1990.²⁸ In 1966, the proton spin–rotation and deuterium nuclear quadrupole constants were determined for ketene, [2-²H]-ketene, and [2,2-²H]-ketene.²⁹ In 1976, the frequency range was extended into the millimeter-wave region (up to 220 GHz) for [2-²H]-ketene and [2,2-²H]-ketene, permitting analysis of their centrifugal distortion.³⁰ More recent studies expanded the measured frequency range to 350 GHz for various isotopologues, refined the least-squares fits, and measured the spectra of new isotopologues [¹⁷O]-ketene,³¹ [1,2-¹³C]-ketene,³² [2,2-²H, 1-¹³C]-ketene,³² [2,2-²H, 2-¹³C]-ketene,³² and [2,2-²H, ¹⁸O]-ketene.³² For ketene, itself, Nemes et al. reported the measurement of 82 a-type, ΔK_a = 0 transitions up to 800 GHz.³³  Figure 1 shows the 11 isotopologues for which rotational spectra have previously been reported (black) and the five isotopologues newly measured in this work (blue).

The gas-phase infrared spectrum of ketene was recorded nearly 85 years ago.³⁴ The rotational structure in the infrared spectrum was first analyzed by Halverson and Williams,³⁵ followed by Harp and Rasmussen,³⁶ Drayton and Thompson,³⁷ Bak and Andersen,³⁸ and Butler et al.³⁹ The infrared spectra of [2-²H]-ketene and [2,2-²H]-ketene were reported in 1951,⁴⁰ and the first vibration-rotation bands of ketene and [2,2-²H]-ketene were observed by Arendale and Fletcher in 1956.^41,42 The first complete analysis of the rotationally resolved infrared spectra of the nine fundamental states present in ketene, [2-²H]-ketene, and [2,2-²H]-ketene was performed by Cox and Esbitt in 1963.⁴³ Nemes explored the Coriolis coupling present in ν₅, ν₆, ν₈, and ν₉ of ketene in two separate studies in 1974 and 1978.^44,45 A similar study was done on [2,2-²H]-ketene by Winther et al., where ν₅, ν₆, ν₈, and ν₉ were examined.⁴⁶ A high-resolution infrared analysis of the four A₁ vibrational states for ketene, [2-²H]-ketene, and [2,2-²H]-ketene was performed by Duncan et al.,⁴⁷ followed by the high-resolution infrared study of ν₅, ν₆, ν₇, and ν₈ fundamental states and two overtone states for ketene by Duncan and Ferguson.⁴⁸ The ν_9, ν₆, and ν₅ bands in [2,2-²H]-ketene were analyzed with high-resolution infrared spectroscopy by Hegelund et al.⁴⁹ Escribano et al.⁵⁰ examined the ν₁ band of ketene in 1994, which is coupled to other vibrational modes. The fundamental states, ν₅ and ν₆, were re-examined by Campiña et al.⁵¹ in 1998 along with the observation of ν₆ + ν₉ in 1999 by Gruebele et al.⁵² Johns et al.⁵³ were able to update the ground-state spectroscopic constants derived from millimeter-wave data along with the infrared data provided by Campiña et al.⁵¹ and Escribano et al.⁵⁰ Nemes et al.⁵⁴ revisited the non-linear least-squares fitting of ν₅, ν₆, and ν₉ and were able to remove most of the Coriolis perturbation contributions to the A_ν rotational constants and derive new experimental Coriolis ζ constants. The vibrational energy manifold up to 3400 cm⁻¹ is shown in Fig. 2, using the experimental frequencies where available and supplementing with computed values where experimental values are not available.

The observation and assignment of various ketene isotopologues have been used for several structural determinations presented in Table I.^28,55–58 The first zero-point average structure (r_z), accounting for harmonic vibrational corrections, centrifugal distortion, and electronic corrections, was determined in 1976.⁵⁵ A second r_z structure was calculated in 1987,⁵⁶ after several studies examining the vibrational spectra of ketene isotopologues facilitated a more physically realistic general harmonic force field to be applied to the structure calculation.^47,48,59 The additional measurement of [1-¹³C]-ketene by Brown et al.²⁸ enabled the first complete substitution structure (r_s) determination, where every atom was isotopically substituted at least once in the structure determination. The first semi-experimental equilibrium structure (r_e^SE), using vibration–rotation interaction corrections calculated from an anharmonic force field, was calculated in 1995 with the rotational constants of the six isotopologues available at that time.⁵⁷ The six isotopologues in this r_e^SE structure provide 12 independent moments of inertia, which is more than sufficient to determine the four structural parameters of ketene. Guarnieri et al.^31,32,60 measured and assigned rotational transitions for five new isotopologues and determined an updated r_e^SE from the rotational constants of 11 isotopologues (22 independent moments of inertia) using vibration–rotation corrections calculated at the MP2/cc-pVTZ level in 2010.⁵⁸ The two r_e^SE structures are generally in good agreement but disagree somewhat with respect to the r_C–H value.

The foundation for semi-experimental equilibrium (r_e^SE) structure determination was pioneered by Pulay, Meyer, and Boggs,⁶³ and the accuracy of the structures obtained using this approach has been exemplified in various studies.^64–67 The accuracy of semi-experimental structures using different computational methods was investigated in the 1990s and 2000s^68–74 and was comprehensively reviewed by Puzzarini⁷⁵ and Puzzarini and Stanton.⁷⁶ The coupled-cluster method for both geometry optimizations and anharmonic force-field calculations, along with sufficiently large basis sets for the molecule of interest, was shown to provide the most accurate structural parameters.^73,77 For larger molecules, where coupled-cluster methods are not feasible, structural parameters derived from density functional methods, e.g., B3LYP/SNSD, display reasonable accuracy.⁷⁸ A number of our recent studies have shown remarkable agreement of CCSD(T)/cc-pCV5Z or CCSD(T)/cc-pCV6Z r_e structures with r_e^SE structures determined using CCSD(T)/cc-pCVTZ vibration–rotation corrections.^79–86 The small number of atoms in ketene allows the coupled-cluster approach to be utilized in this work, affording an r_e^SE structure of similar precision and accuracy to those recent studies.

The rotational spectra of synthesized samples of ketene and deuteriated ketene, described later, were continuously collected in the segments 130–230, 235–360, 350–500, and 500–750 GHz. The instrument covering the 130–360 GHz range has previously been described.^87–89 The 350–500 and 500–750 GHz segments were obtained with a newly acquired amplification and multiplication chain. VDI Mini SGX (SGX-M) signal generator, with external multipliers WR4.3X2 and WR2.2X2, generates 350-500 GHz and, with external multipliers WR4.3X2 and WR1.5X3, generates 500-750 GHz. These spectral segments were detected by using VDI zero-bias detectors WR2.2ZBD and WR1.5ZBD, respectively. The complete spectrum from 130 to 750 GHz was obtained using automated data collection software over ∼12 days with these experimental parameters: 0.6 MHz/s sweep rate, 10 ms time constant, and 50 kHz AM and 500 kHz FM modulation in a tone-burst design.⁹⁰ The frequency spectra were combined into a single spectral file using Assignment and Analysis of Broadband Spectra (AABS) software.^91,92 Pickett’s SPFIT/SPCAT programs⁹³ were used for least-squares fits and spectral predictions, along with Kisiel’s PIFORM, PLANM, and AC programs for analysis.^94,95 Additional short-frequency ranges of the spectrum were collected with an increased number of scans for low-abundance isotopologues. In our least-squares fits, we assume a uniform 50 kHz frequency measurement uncertainty for our measured transitions, 50 kHz for literature values that did not specify an uncertainty, and 25 kHz for transitions reported by Guarnieri et al.^31,32,60 Least-squares fitting output files are provided in the supplementary material.

In this study, we measured and assigned the rotational spectrum from 130 to 750 GHz for the primary (Fig. 3) and deuterium-substituted ketene isotopologues, including their heavy-atom isotopologues, ¹³C and ¹⁸O. All ¹⁷O-substituted isotopologues, including the new detection of [2-²H, ¹⁷O]-ketene and [2,2-²H, ¹⁷O]-ketene, were measured from 230 to 500 GHz. The reduced frequency coverage is due to lower signal-to-noise ratios (S/N) for the hardware configurations outside that range. Transitions of [1,2-¹³C]-ketene could not be measured or assigned due to the proximity of its transitions to those of [2-¹³C]-ketene and the inherently lower S/N for an isotopologue ∼0.0121% the intensity of the main isotopologue. Thus, we were unable to improve upon the spectroscopic constants presented by Guarnieri.³² Due to the planarity condition, each isotopologue provides two independent moments of inertia. With 16 isotopologues used for the new structure determination, an r_e^SE structure for ketene was obtained by using 32 independent moments of inertia.

Calculations were carried out using a development version of CFOUR.⁹⁶ The ketene structure was first optimized at the CCSD(T)/cc-pCVTZ level of theory. The optimized geometry and the same level of theory were subsequently used for an anharmonic, second-order vibrational perturbation theory (VPT2) calculation, wherein cubic force constants are evaluated using analytical second derivatives at displaced points.^97–99 Magnetic property calculations were performed for each isotopologue to obtain the electron-mass corrections to the corresponding rotational constants. The “best theoretical estimate” (BTE), as described previously, is based on a CCSD(T)/cc-pCV6Z optimized structure with four additional corrections^79–86 that address the following:

1. Residual basis set effects beyond cc-pCV6Z.

2. Residual electron correlation effects beyond the CCSD(T) treatment.

3. Effects of scalar (mass–velocity and Darwin) relativistic effects.

4. The fixed-nucleus approximation via the diagonal Born–Oppenheimer correction.

The equations used to calculate these corrections and the values of each of these corrections for ketene are provided in the supplementary material (S7–S13 and Table S-V). One of the most important factors of the algorithm used to determine the BTE is the estimation of residual basis set effects, specifically estimated as the difference between the (directly computed) cc-pCV6Z geometry at the CCSD(T) level of theory and the estimate of the CCSD(T) geometry at the basis set limit. Following others,¹⁰⁰ the latter is estimated by assuming an exponential convergence pattern with respect to the highest angular momentum basis functions present in the basis. A complete explanation of the BTE calculation is provided in the supplementary material.

The xrefit module of CFOUR calculates the moments of inertia and the semi-experimental equilibrium structure using the experimental, S-reduced rotational constants and computational electron-mass distribution and vibration–rotation corrections. The xrefiteration program was used to reveal insight into the contributions of additional isotopologues in refining the structure.⁸² The routine begins by determining a structure using a single isotopic substitution at each position and then sequentially adding the most uncertainty-minimizing isotopologue to the structural least-squares fit until all available isotopologues are incorporated. The routine is also useful in assessing the quality of the fit for each isotopologue, since a problematic fit may be readily apparent as a deviation in the xrefiteration plot.

Computational output files are provided in the supplementary material.

Ketene was synthesized by pyrolysis of acetone (HPLC grade, Sigma-Aldrich) at atmospheric pressure using a lamp described by Williams and Hurd¹⁰¹ and collected in a −130 °C cold trap [Scheme ]. After collection, the cold trap was isolated from the pyrolysis apparatus and placed under vacuum to remove volatile impurities; ketene was then transferred to a stainless-steel cold trap for spectroscopic investigation. A mixed deuterio-/protio-solution of acetone-d_x was produced by a procedure modified from Paulsen and Cooke,¹⁰² using acetone, D₂O, and lithium deuteroxide (LiOD). The reaction mixture was distilled, yielding ∼50% deuterium-enriched acetone. Pyrolysis and purification of this mixture produced ketene, [2-²H]-ketene, and [2,2-²H]-ketene [Scheme ]. An independent sample of [2,2-²H]-ketene [Scheme ] with high deuterium incorporation was generated by pyrolysis of acetone-d₆ (99.5%, Oakwood Chemical).

Table II provides the experimental spectroscopic constants in the S-reduction and I^r representation for all 16 ketene isotopologues used in the semi-experimental equilibrium structure determination. The [1,2-¹³C]-ketene spectroscopic constants provided in Table II are those reported by Guarnieri et al.³² due to the inability to measure this isotopologue in this work. Table II includes, for the normal isotopologue, the previously determined spectroscopic constants by Guarnieri and Huckauf⁶⁰ and the computed constants [CCSD(T)/cc-pCV6Z] for comparison to the experimental values determined in this work. The CCSD(T) values for all other isotopologues can be found in the supplementary material. Experimental rotational constants B₀ and C₀ determined by Guarnieri and Huckauf⁶⁰ are in exceptional agreement with this work, but A₀ is not in such good agreement. The computed rotational constants are in good agreement with the experimentally determined ones previously reported, with the largest discrepancy being in B₀ (0.58%). Centrifugal distortion constants determined by Guarnieri and Huckauf⁶⁰ are also in excellent agreement with those determined here, with the largest difference in d₂ (0.81%). Neither work was able to determine D_K, but Guarnieri et al. used the previously determined value from Johns et al.,⁵³ while the present work used the CCSD(T) value. The CCSD(T) value was utilized to maintain a consistent treatment with the sextic centrifugal distortion constants, which were also held fixed at the CCSD(T) values since experimental values are not available. The CCSD(T) values for the centrifugal distortion constants display the expected level of agreement with the experimental values, with the largest discrepancy in d₂ (22%). There are only two sextic distortion constants that were determined both in this work and by Guarnieri and Huckauf,⁶⁰  H_JK and H_KJ, and both are in reasonable agreement (28%). H_J was determined in this work, while it was held fixed in Guarnieri and Huckauf,⁶⁰ similar to D_K. H_K is held fixed at two different values in the two works for the same reason as D_K. This work held the off-diagonal sextic centrifugal distortion constants, h₁, h₂, and h₃, fixed at their respective CCSD(T) values, while they were held at zero in the previous studies. This difference seemed to negate the need for the two octic centrifugal distortion constants utilized in the previous studies for the frequency range and K_a range measured in this work. The largest relative discrepancy in the CCSD(T) sextic centrifugal distortion constants that were determined is in H_J (63%), which is not unexpected due to it being orders of magnitude smaller than the other constants.

The band structure for all ketene isotopologues is typical for a highly prolate molecule, with each band corresponding to a singular J″+1 value with different K_a values. The transitions of the band structure lose K_a degeneracy for K_a = 3 for all isotopologues, with the protio-isotopologues losing degeneracy in bands at higher frequencies. The current work expanded the range of quantum numbers of the transitions assigned for ketene to include J″+1 = 7 to 41 and K_a = 0 to 5. The breadth of the transitions assigned in this work is shown in Fig. 5, where all transitions newly measured are in black and previously measured transitions are in various colors indicating their source. All ketene isotopologue least-squares fits were limited to transitions with K_a ≤ 5, even though transitions with higher K_a were observed. This is because a single-state, distorted-rotor Hamiltonian was unable to provide a least-squares fit below the assumed measurement uncertainty of 50 kHz when incorporating transitions above K_a = 5. This failure of the single-state least-squares fit to model all of the observable transitions is due to coupling between the vibrational ground and excited states that has been extensively studied in the infrared spectra of ketene and summarized in the Introduction.^{44–49,51,52,54} An analysis of the vibrational coupling is beyond the scope of the current work. The cutoff of K_a ≤ 5 was implemented for all isotopologues to maintain a consistent amount of spectroscopic information. A similar procedure was applied to previous studies with HN_3, which also has a ground state coupled to low-energy fundamental states,¹¹⁰ and was shown to provide an r_e^SE structure with complete consistency with the BTE.^83,87

The heavy-atom isotopologues of ketene were observed at natural abundance from the synthesized protio-sample of ketene, and only ^aR_0,1 transitions could be observed and measured. The [2-²H]-ketene and [2,2-²H]-ketene isotopologues were the only other isotopologues where ^aQ_0,−1 transitions could be measured. The [2-²H]-ketene isotopologue has a small predicted b-dipole moment, μ_b = 0.048 D, but no b-type transitions were sufficiently intense to be observed in our spectrum. All of the heavy-atom isotopologues for [2-²H]-ketene and [2,2-²H]-ketene were observable at natural abundance in their deuterium-enriched samples. For the previously observed isotopologues, the published spectroscopic constants were used as predictions for the spectral region measured in this work. Once rotational constants were obtained for several known isotopologues, a preliminary semi-experimental equilibrium structure was obtained, which provided very accurate equilibrium rotational constants for new isotopologues. Along with CCSD(T) vibrational and electronic corrections and centrifugal distortion constants, the predicted rotational constants were used to assign transitions for the previously unidentified isotopologues: [2-²H, 1-¹³C]-ketene, [2-²H, 2-¹³C]-ketene, [2-²H, ¹⁸O]-ketene, [2-²H, ¹⁷O]-ketene, and [2,2-²H, ¹⁷O]-ketene. This technique greatly expedited the search for these transitions, as the CCSD(T)/cc-pCVTZ rotational constants predictions were not accurate enough to easily identify and assign the transitions. All heavy-atom isotopologue transitions measured from 500 to 750 GHz and all three ¹⁷O-isotopologues measured from 230 to 500 GHz required averaging 20 scans due to low S/N. These low S/N transitions were collected by acquiring 10 MHz windows around each predicted transition. Despite the various isotopologue substitutions, the spectral pattern (Fig. 6) of ketene was relatively consistent due to the highly prolate nature of the molecule.

The rotational spectra of three separate ketene samples are shown in Fig. 6, where (a) corresponds to the protio-isotopologue from 420 to 434 GHz, (b) corresponds to the [2-²H]-ketene isotopologue from 420 to 434 GHz, and (c) corresponds to the [2,2-²H]-ketene isotopologue from 411 to 425 GHz along with their respective heavy-atom isotopologues. The rotational spectrum of ketene is sparse with the bands of R-branch transitions with constant J values, separated by ∼16 GHz or ∼2C, allowing for the assignment of multiple isotopologues within one sample spectrum. Thus, there was little issue with transitions overlapping, which would cause a poor determination of the transition frequencies. Each spectrum contains unassigned transitions belonging to excited vibrational states of ketene isotopologues, as shown in Fig. 6. Data distribution plots for the isotopologues, showing the breadth of quantum numbers observed, are provided in the supplementary material.

In contrast to several of our recent r_e^SE structure determinations,^79–86 the spectroscopic constants determined in the S-reduction (⁠$B0β(S)$⁠) were converted to equilibrium constants (⁠$Beβ$⁠) and used directly in the least-squares fitting of the semi-experimental equilibrium structure without conversion to the determinable constants. In those previous studies, with the exception of [1,3-²H]-1H-1,2,4-triazole, the A-reduction and S-reduction rotational constants for each isotopologue were converted to the determinable constants from which the centrifugal distortion has been removed. These determinable constants are then averaged before converting to the equilibrium constants (⁠$Beβ$⁠) for r_e^SE structure determination. As noted previously, ketene is an extreme prolate top (κ = −0.997) with only a-type transitions; thus, the A-reduction spectroscopic constants cannot be determined as accurately as the S-reduction constants. As a result, the S-reduction specific vibration–rotation interaction corrections determined in the CFOUR anharmonic frequency calculation were used to obtain the equilibrium rotational constants. Thus, the centrifugal distortion corrections used to determine the equilibrium rotational constants are the computed ones rather than the experimental ones. Ideally, the spectroscopic constants would be determined in both the A- and S-reductions, and each rotational constant would be converted to a determinable constant. If the determinable constants were similar, it would give confidence that the rotational and quartic centrifugal distortion constants were determined with sufficient accuracy to be included in the structure determination, regardless of the size of the transition dataset. A similar problem might have arisen in the HN₃ semi-experimental equilibrium structure determinations,^83,87 but the HN₃ dataset included b-type transitions, along with a-type transitions, which gave enough spectroscopic information to successfully determine the A-reduction spectroscopic constants.

The semi-experimental equilibrium structure parameters of ketene obtained from 16 isotopologues are shown in Fig. 7 and enumerated in Table IV. The 2σ statistical uncertainties of the bond distances are all <0.0007 Å, and the corresponding uncertainty in the bond angle is 0.014°. Overall, the precision and accuracy of the structural parameters are similar to those of HN₃⁸³ when comparing the r_e^SE calculated at the same level of theory. The 2σ statistical uncertainties of heavy-atom bond lengths are nearly identical for HN₃ and ketene (0.000 74 for N1–N2; 0.000 75 for N2–N3; and 0.000 69 for C1-C2; 0.000 66 for C1-O). The 2σ statistical uncertainty for the respective X–H bond is also quite similar (0.0003 for HN₃ and 0.0002 for ketene). More generally, the bond length accuracy of the r_e^SE of ketene is similar to our other works, including heterocyclic molecules,^79–82,111 and is of the same order of magnitude for the accuracy in the angles, with that in ketene being more accurately determined. This improvement in accuracy is largely due to having 8× more independent moments of inertia than structural parameters (three bond lengths and one angle) for ketene. Table IV presents the r_e^SE structural parameters determined in the complete analysis, as well as the recommended r_e^SE structural parameters, which take into account the limits of precision in their determination. The distinction between these sets of values is discussed in greater detail in the next section.

In accordance with previous structure determinations,^80–85 the effect of including the available isotopologues in the r_e^SE structure is examined using xrefiteration.^82, Figure 8 shows a plot of parameter uncertainty as a function of the number of incorporated isotopologues and reveals that the total uncertainty and the uncertainty of both the bond distances and bond angle have converged with the inclusion of the 11th isotopologue. Coincidentally, this is the same number of isotopologues used in the r_e^SE determination published by Guarnieri et al.⁵⁸ The composition of the set of 11 isotopologues, however, is different in each case. Guarnieri et al.⁵⁸ determined the r_e^SE with mainly protio- and [2,2-²H]-ketene isotopologues, while the first 11 isotopologues utilized by xrefiteration in the current work include a mix of protio-, [2-²H]-ketene, and [2,2-²H]-ketene isotopologues. The addition of the five other isotopologues beyond the 11th neither decreases nor increases the total uncertainty, which is similar to the situation observed with HN₃⁸³ but unlike the cases of thiophene⁸⁰ and 1H- and 2H-1,2,3-triazole,⁸⁵ where the statistical uncertainty increases with the inclusion of the final isotopologues. Figure 9 shows the structural parameter values and their uncertainties as a function of the number of ketene isotopologues, added in the same order as in Fig. 8. It is evident that the structural parameters are well-determined with the core set of isotopologues because they agree with the respective BTE values. The addition of further isotopologues, however, decreases the 2σ uncertainties for all parameters until the addition of the 11th isotopologue, similar to Fig. 8. The r_C–C and r_C–O bond lengths of the current r_e^SE are both smaller than the respective BTE values (by 0.0003 and 0.0004 Å, respectively), while there is quite close agreement between the r_C–H bond lengths (r_eC–H − r_e^SE_C–H = 0.000 04 Å). The value of the bond angle, θ_C–C–H, is slightly larger (0.009°) than the BTE value. Both heavy-atom bond distances of the r_e BTE structure are too large (Fig. 9) relative to their r_e^SE parameters, and the observed residuals are very similar to those we observed in HN₃,⁸³ 0.000 35 Å for the central r_N1–N2 bond and 0.000 41 Å for the terminal RN2-N3 bond. In the structural least-squares fitting from rotational constants, the most difficult atom to locate is the heavy atom nearest to the center of mass, but an error in its location would tend to make one heavy-atom distance too long and one too short, contrary to the observations in ketene and HN₃. This may suggest that these residual discrepancies, which are not present in the distances involving H atoms, are due to some systematic shift in the BTE distances related to the heavy atom backbone of these molecules.

A graphical representation of all the structural parameters for the current r_e^SE, the r_e^SE by East et al.,⁵⁷ the r_e^SE by Guarnieri et al.,⁵⁸ the BTE, and various coupled-cluster calculations with different basis sets is shown in Fig. 10. Upon cursory inspection, it seems there is excellent agreement among all of the structural parameters of the three r_e^SE structures (Table IV and Fig. 10), and all are quoted to similar precision. Because separate sets of discrepancies are involved with respect to the two previous r_e^SE structure determinations, we will discuss them separately. The heavy-atom distances from Guarnieri et al.⁵⁸ are essentially the same as our own, although with slightly larger 2σ uncertainties due to the smaller dataset compared to the present work, and BTE results for both parameters easily fall within the quoted 2σ limit. The agreement for the two parameters involving the hydrogen-atom position is not quite as good. The r_C-H bond distance from Guarnieri et al.⁵⁸ is in disagreement with our value by slightly more than the combined 2σ error estimates, and the BTE value falls well outside their 2σ error range. The angle, θ_C-C-H, is, indeed, in agreement with our value within the combined estimated 2σ error limits, but the BTE value of this parameter falls significantly outside their 2σ error range. We believe that the reason for these discrepancies is the impact of untreated coupling between vibrational states impacting the rotational constants. It is known that the ground state of ketene at high K_a values is affected by perturbations from low-lying vibrational states. We have chosen to employ only measurements for K_a = 0–5, which removed this problem. Guarnieri et al.,⁵⁸ however, used higher K_a transitions in their least-squares fits, which required the inclusion of higher-order centrifugal distortion terms, L_JK and L_JKK. These effective parameters distort the determined values of A₀ from the regression analysis, which may affect the structural parameters. We tested our conjecture by using our rotational constants for the set of isotopologues used by Guarnieri et al.⁵⁸ and found the resultant structure in essentially complete agreement with our own r_e^SE structure (Table IV), which is consistent with the analysis of the structure shown in Fig. 9. This indicates that the five additional isotopologues that we measured and included were not required to achieve this improved accuracy.

The situation with respect to the r_e^SE structure from East et al.⁵⁷ is more straightforward. The bond distances and angles reported by East et al.⁵⁷ are in complete agreement with the current r_e^SE values. This is somewhat surprising, given that the rotational constants are substantially less precise than the values determined in the present work and that the A₀ constant of [¹⁸O]-ketene used by East et al.⁵⁷ and determined experimentally by Brown et al.,²⁸ 287 350 (910) MHz, is clearly too large by about 5 GHz. The values of A₀ for all heavy-atom isotopologues should be nearly the same because they depend only on the distance of the hydrogen atoms from the a-axis. This is confirmed by the data in Table II. The remaining rotational constants used by East et al.⁵⁷ are all similar to those in the present work. We obtained an r_e^SE structure using the rotational constants and vibration–rotation interaction corrections presented in Table XIV of the East et al.⁵⁷ work and obtained a structure very closely resembling the one presented in that work for all parameters. This is also an interesting outcome, as the vibration–rotation interaction corrections used in that work are clearly inadequate, as evidenced by the residual inertial defects presented in their Table XIV⁵⁷ that vary in sign and order of magnitude across the six isotopologues. Despite the inadequacy of the vibration–rotation interaction corrections, the r_e^SE structure of East et al.⁵⁷ is in excellent agreement with the new r_e^SE but not quite in agreement with the θ_C–C–H value from the r_e BTE structure. These analyses are provided in the supplementary material and summarized in Tables S-V and S-VI.

The r_e^SE structure presented in this work, like the previously reported structures,^57,58 suffers from the impact of untreated Coriolis coupling between its ground state and its vibrationally excited states. Despite this limitation, the 2σ statistical uncertainties for the bond distances and bond angles are quite small (0.0002 to 0.0007 Å for the bond distances and 0.014° for the bond angle). For the bond distances, this statistical uncertainty is approaching the limit of the r_e^SE structure determination, which requires the assumption that there is one mass-independent equilibrium geometry. The mass independence of equilibrium structures is a tacitly accepted assumption of molecular structure determination by rotational spectroscopy that is no longer valid as the limits of accuracy and precision are extended, especially for parameters involving hydrogen atoms. As a simple test of this assumption, optimized geometries were obtained for ketene and [2,2-²H]-ketene with and without the diagonal Born–Oppenheimer correction (DBOC; SCF with the aug-cc-pCVTZ basis set). Of course, the r_e structure obtained from the normal optimization without the DBOC resulted in the same equilibrium geometry for both isotopologues. With the DBOC, however, the equilibrium C–D distance decreased relative to the C–H distance by 0.000 06 Å. This value, which is similar to that obtained for benzene,⁸⁶ suggests that the limit and trustworthiness of the r_e^SE structure for ketene and other C–H containing r_e^SE structures is on the order of 0.0001 Å, which is half of the 2σ statistical uncertainty of the r_e^SE C–H distance in this work. As a consequence of these relationships, our recommended values for the structural parameters of ketene are r_C–C = 1.3122 (7) Å, r_C–H = 1.0759 (2) Å, r_C–O = 1.1606 (7) Å, and θ_C–C–H = 119.086 (14)°, as shown in Fig. 7 and Table IV.

A new, highly precise, and accurate semi-experimental equilibrium (r_e^SE) structure for ketene (H₂C=C=O) has been determined from the rotational spectra of 16 isotopologues. The 2σ values for the r_e^SE structure of ketene, and also the discrepancies between the best theoretical estimate (BTE) and the r_e^SE structural parameters, are strikingly similar to those for the previous r_e^SE structure of hydrazoic acid (HNNN).⁸³ This outcome is noteworthy, although perhaps not surprising, given (i) the structural similarity between the two species and (ii) the highly over-determined datasets, which are a consequence of the large number of isotopologues relative to the number of structural parameters. In both cases, we found that extrapolation to the complete basis set limit provided slightly better agreement with the r_e^SE structure when the highest level calculation included in the extrapolation was CCSD(T)/cc-pCV6Z, as opposed to CCSD(T)/cc-pCV5Z. It is somewhat surprising that the high accuracy of the ketene structure did not require the full dataset from 16 isotopologues. The uncertainties in the structural parameters did not improve with the inclusion of the “last” five isotopologues in the xrefiteration analysis. This case stands in contrast to other molecules that we have studied, in which quite large numbers of isotopologues are required for convergence of the r_e^SE parameters.^82,84,85 Previous studies were steering us toward a generalization that “more is better” with respect to the number of isotopologues used in a structure determination, but the current case provides a counterexample. In the current case, the extra isotopologues do not degrade the quality of the structure, but they do not improve it. The current state of understanding of r_e^SE structure determination does not enable a prediction of the number of isotopologues that will be required for the statistical uncertainties of the structural parameters to converge.

The present work confirms the great utility of the BTE structure as a benchmark for the semi-experimental structures. Although both of the published r_e^SE structures for ketene, to which we have compared our own results, are generally in excellent agreement with the present work, the comparison to the BTE structure clearly establishes that the present approach of limiting the dataset to low K_a values (0–5) improved the results for a molecule in which perturbations of the ground state exist. There need to be further investigations of why the BTE structure predicts heavy-atom bond distances that are longer than the r_e^SE structure when all of the other structural parameters agree so well. The BTE values for the heavy-atom distances do not fall outside the 2σ statistical uncertainty of the r_e^SE values, but the small differences between BTE and r_e^SE values have now been observed in both ketene and HN₃. The general applicability to similar molecules and the origin of the effect merit additional study.

The current studies enhance the capability for radioastronomers to search for ketene in different extraterrestrial environments by extending the measured frequency range of ketene to 750 GHz as well as providing data for newly measured isotopologues, [2-²H, 1-¹³C]-ketene, [2-²H, 2-¹³C]-ketene, [2-²H, ¹⁸O]-ketene, [2-²H, ¹⁷O]-ketene, and [2,2-²H, ¹⁷O]-ketene. These new data will also be valuable for identifying these species in laboratory experiments, e.g., electric discharge, pyrolysis, and photolysis. Finally, these spectra contain a great many transitions from vibrationally excited states, which should prove valuable in analyzing and quantifying the numerous perturbations that exist between these low-lying vibrational states. We hope to pursue this topic in the future.

Computational output files, least-squares fitting for all isotopologues, data distribution plots for all non-standard isotopologues, xrefiteration outputs, equations used for calculating determinable constants and BTE corrections, and tables of S-reduction, A-reduction, and determinable constants, structural parameters, inertial defects, BTE corrections, and synthetic details are provided in the supplementary material.

We gratefully acknowledge the funding from the U.S. National Science Foundation for the support of this project (Grant No. CHE-1954270 to R.J.M and Grant No. CHE-1664325 to J.F.S.). We thank Maria Zdanovskaia for her assistance with generating figures. We thank Maria Zdanovskaia and Andrew Owen for their thoughtful commentary and review of the manuscript. We thank Tracy Drier for the construction of the ketene lamp used in this work.

The authors have no conflicts to disclose.

Houston H. Smith: Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Brian J. Esselman: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Samuel A. Wood: Investigation (equal); Methodology (equal); Writing – review & editing (equal). John F. Stanton: Formal analysis (equal); Investigation (equal); Software (equal); Writing – review & editing (equal). R. Claude Woods: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Supervision (equal); Writing – review & editing (equal). Robert J. McMahon: Funding acquisition (lead); Investigation (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.