Gas-phase formation and spectroscopic characterization of the disubstituted cyclopropenylidenes c -C 3 (C 2 H) 2 , c -C 3 (CN) 2 , and c -C 3 (C 2 H)(CN) ⋆

Aims. The detection of c -C 3 HC 2 H and possible future detection of c -C 3 HCN provide new molecules for reaction chemistry in the dense interstellar medium (ISM) where R-C 2 H and R-CN species are prevalent. Determination of chemically viable c -C 3 HC 2 H and c -C 3 HCN derivatives and their prominent spectral features can accelerate potential astrophysical detection of this chemical family. This work characterizes three such derivatives: c -C 3 (C 2 H) 2 , c -C 3 (CN) 2 , and c -C 3 (C 2 H)(CN). Methods. Interstellar reaction pathways of small carbonaceous species are well replicated through quantum chemical means. Highly accurate cc-pV X Z-F12/CCSD(T)-F12 ( X = D,T) calculations generate the energetics of chemical formation pathways as well as the basis for quartic force field and second-order vibrational perturbation theory rovibrational analysis of the vibrational frequencies and rotational constants of the molecules under study. Results. The formation of c -C 3 (C 2 H) 2 is as thermodynamically and, likely, as stepwise favorable as the formation of c -C 3 HC 2 H, rendering its detectability to be mostly dependent on the concentrations of the reactants. Both c -C 3 (C 2 H) 2 and c -C 3 (C 2 H)(CN) will be detectable through radioastronomical observation with large dipole moments of 2.84 D and 4.26 D, respectively, while c -C 3 (CN) 2 has an exceedingly small and likely unobservable dipole moment of 0.08 D. The most intense frequency for c -C 3 (C 2 H) 2 is ν 2 at 3316.9 cm − 1 (3.01 µ m), with an intensity of 140 km mol − 1 . The mixed-substituent molecule c -C 3 (C 2 H)(CN) has one frequency with a large intensity, ν 1 , at 3321.0 cm − 1 (3.01 µ m), with an intensity of 82 km mol − 1 . The molecule c -C 3 (CN) 2 lacks intense vibrational frequencies within the range that current instrumentation can readily observe. Conclusions. Both c -C 3 (C 2 H) 2 and c -C 3 (C 2 H)(CN) are viable candidates for astrophysical observation, with favorable reaction profiles and spectral data produced herein, but c -C 3 (CN) 2 will not be directly observable through any currently available remote sensing means, even if it forms in large abundances.


Introduction
The detection of ethynyl cyclopropenylidene (c-C 3 HC 2 H), the first substituent-functionalized cyclopropenylidene derivative to be observed in space (Cernicharo et al. 2021), has sparked renewed interest in its parent molecule, c-C 3 H 2 . A proposed mechanism for the formation of c-C 3 HC 2 H from the abundant c-C 3 H 2 molecule and ethynyl radical (·C 2 H) suggests that c-C 3 HC 2 H can be produced through rapid substituent addition followed by hydrogen dissociation (Fortenberry 2021). This reaction represents one of the few known cosmic chemical pathways utilizing c-C 3 H 2 . The existence of c-C 3 HC 2 H questions the idea that c-C 3 H 2 is a carbonaceous "dead end" (Lin et al. 2016) in space. In TMC-1, the column density of c-C 3 HC 2 H is calculated to be 3.1(8) × 10 11 cm −2 , giving a relative abundance to H 2 of 3.1 × 10 −11 . In comparison to c-C 3 H 2 , c-C 3 HC 2 H has an abundance of 5.3 × 10 −3 (Cernicharo et al. 2021). While present in relatively low densities, the overall available concentrations may make the detection of other c-C 3 H 2 derivatives, as well as derivatives of c-C 3 HC 2 H itself, possible.
⋆ Additional data obtained from spectroscopic analysis alongside structural information for all molecules can be found at https:// egrove.olemiss.edu/chem_facpubs/33 Given that reaction between c-C 3 H 2 and ·C 2 H is shown to be energetically allowed at the low temperatures of the interstellar medium (ISM; Fortenberry 2021), the detection and abundances of c-C 3 HC 2 H are far from surprising, as its precursors are some of the most abundant carbonaceous species in space. Both c-C 3 H 2 and its 2 D and 13 C isotopologues are found in a myriad of cosmic environments (Thaddeus et al. 1981Matthews & Irvine 1985;Seaquist & Bell 1986;Cox et al. 1987;Vrtilek et al. 1987;Madden et al. 1989;Lucas & Liszt 2000;Oike et al. 2004;Teyssier et al. 2004;Qi et al. 2013;Nixon et al. 2020;Bell et al. 1986;Majumdar et al. 2017;Gomez Gonzalez et al. 1986;Madden et al. 1986;Spezzano et al. 2013). Its abundances rival those of CH 3 OH in cold regions, such as the Magellanic Clouds (Heikkilä et al. 1999), and have been used as a standard for the calculation of hydrogen abundances in Sagittarius B2 (Corby et al. 2018). Likewise, ·C 2 H is ubiquitous in space (Tucker & Kutner 1978;Tucker et al. 1974;Huggins et al. 1984;Henkel et al. 1988), having been one of the first interstellar molecules detected, along with its isotopologues (Combes et al. 1985;Vrtilek et al. 1985;Saleck et al. 1994).
Similarly, the large presence of the ·CN radical in various cosmic environments (Adams 1941;Jefferts et al. 1970;Henkel et al. 1988;McKellar 1940;Turner & Gammon 1975; A&A 671, A95 (2023) Fig. 1. Molecular structure of c-C 3 (C 2 H) 2 . Paron et al. 2021) makes a c-C 3 H 2 nitrile derivative possible as well. While the search for c-C 3 HCN is still ongoing (Cernicharo et al. 2021), its energetically viable formation mechanism most strongly resembles the formation mechanism of c-C 3 HC 2 H, both in reaction coordinate structures and in exothermicity (Flint & Fortenberry 2022). This mechanism, combined with high reactant concentrations in several molecular clouds, suggests its detection to be the next most probable out of any currently proposed substituted cyclopropenylidene. The likelihood that c-C 3 HCN also exists in the ISM means that a potential derivative of it, too, must be further considered.
The common occurrence of the cyano and ethynyl substituents within astrophysical discussions is not only a result of radical abundances. Cyano-functionalized hydrocarbons comprise a large fraction of the astrochemical detections within the last five years (Lee et al. 2021b,a;McGuire et al. 2018McGuire et al. , 2020McGuire et al. , 2021Cernicharo et al. 2020;Loomis et al. 2021;Xue et al. 2020) as a result of the enhanced dipole moment of such molecules compared to the pure hydrocarbon parent molecule. Likewise, ethynyl groups and other C≡C linkages are common in both observed chemical species as well as in the reaction pathways that produce them. Hydrocarbons are among the most frequently detected molecules in interstellar environments, and the vast majority of them have at least one degree of unsaturation (McGuire et al. 2021). Such unsaturated molecules appear to be the driving force of interstellar chemistry, and they may play a key role in the synthesis of polycyclic aromatic hydrocarbons (PAHs), species thought to be responsible for the unidentified infrared bands (Tielens 2008). As such, the R-CN and R-C 2 H functional groups are key to the continued study of interstellar astrochemistry as it currently is understood.
Following its laboratory detection by McCarthy et al. (1997), c-C 3 HC 4 H, a monosubstituted c-C 3 H 2 derivative with an elongated substituent chain, compared to that of c-C 3 HC 2 H, has been hypothesized in the literature as a possible candidate for interstellar detection (Chandra et al. 2005). However, its disubstituted isomer c-C 3 (C 2 H) 2 (shown in Fig. 1) has eluded even laboratory detection, despite being one of the most stable isomers of C 7 H 2 , as determined quantum chemically (Thaddeus et al. 1998;Dua et al. 2000). Though c-C 3 (C 2 H) 2 has been considered previously using ab initio methods, the results are limited to optimized geometries, harmonic frequencies, and IR intensities (Thimmakondu & Karton 2017). Moreover, these results identify c-C 3 (C 2 H) 2 as having a large dipole moment of 3.77 D, bringing it into the realm of detectability through radioastronomical observation. Additionally, the C-H antisymmetric stretch (ω = 3450, λ = 2.899 µm) of c-C 3 (C 2 H) 2 was shown to have an intensity of 122 km mol −1 , making it ripe for infrared detection as well. Similar ring-containing C 5 N 2 structures have received even less attention, with only one mention of c-C 3 (CN) 2 , shown in Fig. 2, in the literature to date (Jiang et al. 2004). The linear  and bent isomers of C 5 N 2 , as well as its anionic form, have been discussed in prior studies (Singh & Chaturvedi 1989;Belbruno et al. 2001;Yang et al. 2011). The general lack of information regarding this molecule is possibly linked to the lower intrinsic stability of linear C 2n+1 N 2 compounds (Belbruno et al. 2001;Jiang et al. 2004). Laboratory studies have been successful in forming the linear isomer of C 5 N 2 (Smith et al. 1994), as verified by Tittle et al. (1999). However, c-C 3 (CN) 2 , like c-C 3 (C 2 H) 2 , has seemingly also eluded laboratory observation.
With such complex ring-chain molecules brought into the realm of possibility for astrophysical observation by the detection and formation mechanisms of their monosubstituted relatives, the gap between the potential for these species and the lack of information in the literature concerning them must be bridged. This work presents the results of highly accurate quantum chemical calculations for the purpose of spectroscopically characterizing c-C 3 (C 2 H) 2 and c-C 3 (CN) 2 as well as the mixed-substituent derivative c-C 3 (C 2 H)(CN), shown in Fig. 3. The rovibrational data reported could lend support to future experimental work as well as NASA projects, such as the James Webb Space Telescope (JWST). Thermodynamics of reaction were calculated for the formation of c-C 3 (C 2 H) 2 , c-C 3 (CN) 2 , and c-C 3 (C 2 H)(CN) to create a basis for their existence in astrophysical environments. Reaction pathways for c-C 3 (C 2 H) 2 and c-C 3 (CN) 2 are proposed in order to provide additional astrochemical motivation for this spectroscopic search. Such formation mechanisms build upon existing cyclopropenylidene chemistry, bringing the field of astrochemistry closer to making the fullest use of this ubiquitous family of molecules. These cyclopropenylidene derivatives in particular may be necessary for explaining the formation and growth of larger unsaturated hydrocarbon systems and soot grains.

Reactions in the ISM
The regions of space where these substitution reactions are expected to take place lack a significant photon flux.
A95, page 2 of 11 A. R. Flint et al.: A&A proofs, Consequently, processes that occur in the cold ISM cannot benefit from photon collisions and the subsequent influx of energy to propel their chemistry. The only energy available to the reaction pathway is the energy that is inherent to it via the reactants (Puzzarini 2022). This places constraints on the chemical mechanisms such that any energetically disallowed reaction pathwaysones that contain thermodynamic or kinetic barriers -are filtered out. The combined energy of the reacting species must represent a maximum on the reaction coordinate. The association of the species for bimolecular reactions, like the ones proposed within this work, must be initially barrierless to fulfill this requirement (Puzzarini 2022;Grosselin & Fortenberry 2022). The net reaction must also be exothermic.
Bimolecular associations that occur within the ISM, as a consequence of being exothermic in nature, often result in the creation of a submerged intermediate structure that contains an excess of energy with respect to the reactants. This excess energy must be dissipated, which is assumed to occur via exit of a leaving group that carries away the energy in the form of kinetic energy. The dissipation of energy through radiative means is assumed to be infeasible for the submerged intermediates on the c-C 3 (C 2 H) 2 and c-C 3 (CN) 2 reaction coordinates without destruction of the complex (Puzzarini 2022). Radiative association becomes slowed compared to other processes, such as the removal of a leaving group, as the relative velocity of the reactants decreases. In the coldest regions of the ISM, such as TMC-1 and other areas where neutral-neutral reactions are plentiful, radiative association would therefore be hindered (Herbst 2021). As a result, the rate of hydrogen dissociation is assumed to control the final step(s) of the reaction pathway.
Exit from the pathway from a more shallow potential well (i.e., a decrease in E Products -E Intermediate ) for a barrierless exit is assumed to be faster than the same process for a deeper potential well (Tielens 2005). Additionally, despite the assumption that radiative dissipation will not occur for any submerged intermediates, exit from shallower wells on the reaction coordinate would decrease the opportunity for radiative dissipation of energy to compete with dissociation of a leaving group (Herbst 2001(Herbst , 2021.

Mechanisms
Geometry optimization and harmonic frequency calculations of the starting materials and products of the c-C 3 (C 2 H) 2 and c-C 3 (CN) 2 formation pathways are based on coupled cluster theory at the singles, doubles, and perturbative triples level [CCSD(T)] (Raghavachari et al. 1989;Hampel et al. 1992;Knowles et al. 1993;Shavitt & Bartlett 2009;Crawford & Schaefer III 2000) in conjunction with the explicitly correlated F12b formalism [CCSDT-F12b] (Adler et al. 2007) and the cc-pVXZ-F12 basis set Yousaf & Peterson 2008;Knizia et al. 2009). This combination is referred to as F12-TZ. Optimization and harmonic frequency calculations for all intermediates on the pathways were performed with the B3LYP functional (Yang et al. 1986;Becke 1993;Lee et al. 1988;Stephens et al. 1994;Vosko et al. 1980) and the aug-cc-pVTZ basis set (Dunning 1989;Kendall et al. 1992;Woon & Dunning 1993). The single-point energies of the optimized intermediate geometries were computed with F12-TZ (Ramal-Olmedo et al. 2021), which were corrected with the B3LYP/aug-cc-pVTZ zero-point vibrational energies (ZPEs). All calculations of reaction coordinate minima made use of the MOLPRO2022.2 (Werner et al. 2012(Werner et al. , 2022(Werner et al. , 2020 quantum chemical software package. Locatable transition states on the reaction pathway were optimized at the B3LYP/aug-cc-pVTZ level of theory with GAUSSIAN16 (Frisch et al. 2016). The F12-TZ single-point energy at this geometry was computed and corrected with the B3LYP/augcc-pVTZ ZPE, and the same was done for the intermediates. Gabedit (Allouche 2010(Allouche , 2017 was used to visualize the motion corresponding to the vector of the imaginary frequency of the geometry in order to confirm the connection between two minima. When necessary for mechanistic analysis, the molecular orbitals (MOs) of a geometry were computed in GAUSSIAN16 with ROHF/aug-cc-pVTZ, as this is the reference for the F12-TZ energies, (Roothaan 1951;Brinkley et al. 1974;Watts et al. 1993) and visualized with Gabedit. Natural bond orbitals (NBOs; Foster & Weinhold 1980;Reed & Weinhold 1983;Carpenter 1987;Reed et al. 1988) were computed with UHF/cc-pVTZ for specified minima in GAUSSIAN16 in order to extract additional information about the MOs, including which orbitals are participating in bondingto-antibonding delocalization. Analyses of these orbital calculations provide a guide for the electronic basis on which the reaction coordinate proceeds.
The radical character of the reaction pathways within this work required that single-reference methods such as CCSD(T)-F12 be used carefully, as such species are commonly prone to influence from multireference effects during computation. For calculations performed via F12-TZ, Lee & Taylor (1989) define the T1 diagnostic that provides a measure of how greatly the system is affected by static correlation. For T1 diagnostic values below 0.02, the system is generally accepted as well behaved under a single-reference frame. The T1 diagnostics for all molecules on both reaction pathways were evaluated to ensure that static correlation would not significantly contaminate the calculated results.
Previous studies on c-C 3 H 2 substitution (Fortenberry 2021;Flint & Fortenberry 2022) show that in every case where a substitution reaction of this type is permitted thermodynamically, no barrier to association to form the initial submerged well is present. Removal of the leaving group upon exit of a submerged well, in every case, either crosses no exit barrier or crosses a barrier that is kinetically accessible to the reaction. When subjected to a second substitution, c-C 3 HC 2 H and c-C 3 HCN are expected to also lack hindering entrance and exit barriers, and thus barrierless associations were assumed for all reaction pathways calculated in this work. This follows experimental data (Smith 2006(Smith , 2011 that show a decrease in reaction rate between a radical and a neutral unsaturated species in the gas phase as the temperature decreases. Despite the assumption that no barriers to hydrogen dissociation are present, the dissociation itself can be impacted by other factors, such as the nature of the intermediates and dissociation complexes. Where deemed necessary for analyzing how these dissociations occur, relaxed potential energy scans (PESs) generated the single-point energies of a series of geometries along a specified reaction coordinate at step sizes of 0.01 Å. The reaction coordinate variable was defined as an internuclear distance, and at fixed variable values, the remainder of the geometry was optimized. The PES calculations were performed at the F12-TZ//B3LYP/aug-cc-pVTZ combined level of theory in order to remain consistent with the reaction pathway.

Spectroscopic methods
In order to provide a means of detecting the products of the examined reaction mechanisms, the theoretical spectroscopic A95, page 3 of 11 A&A 671, A95 (2023) data in this work were computed by quartic force fields (QFFs), which are fourth-order Taylor series expansions of the potential energy portion of the internuclear Watson Hamiltonian (Fortenberry & Lee 2019). The QFFs utilized in this work are also based on coupled cluster theory at the singles, doubles, and perturbative triples level [CCSD(T)] (Raghavachari et al. 1989;Shavitt & Bartlett 2009;Crawford & Schaefer III 2000) in conjunction with the explicitly correlated F12b formalism [CCSD(T)-F12b] (Adler et al. 2007).
In this work, QFFs were computed for c-C 3 (C 2 H) 2 , c-C 3 (CN) 2 , and c-C 3 (C 2 H)(CN) with three different levels of theory, which all utilize the CCSD(T)-F12b method. Two QFFs employed the cc-pVXZ-F12 basis set Yousaf & Peterson 2008;Knizia et al. 2009), where X is D or Z. These combinations are referred to hereafter as F12-DZ and F12-TZ, respectively. The other QFF employed the cc-pCVDZ-F12 basis set (Hill & Peterson 2010) with additional corrections for scalar relativity utilizing canonical CCSD(T) with the cc-pVTZ-DK basis set (Douglas & Kroll 1974;Jansen & Hess 1989). This combination is referred to hereafter as F12-DZ-cCR (Watrous et al. 2021). The F12-DZ combination is the only level of theory at which QFFs are computed for c-C 3 (C 2 H) 2 and c-C 3 (C 2 H)(CN) due to the increase in computational cost of the other two levels of theory utilized in this work.
The QFFs for c-C 3 (CN) 2 have a symmetry-internal coordinate (SIC) system of 10325 points with atom labels corresponding to Fig. 2 defined below: The QFFs for c-C 3 (C 2 H) 2 and c-C 3 (C 2 H)(CN) were run directly in Cartesian coordinates and are composed of 115 746 points and 145 824 points, respectively. Table 1. Thermodynamics of reaction for formation of c-C 3 (C 2 H) 2 , c-C 3 (CN) 2 , and c-C 3 (C 2 H)(CN).
Notes. Elements in the ·X column denote the radical used as the starting material to achieve the final result. All energies reported in kcal mol −1 .
The SIC QFFs begin with the optimization of the geometry at each level of theory. The geometry was then displaced by 0.005 Å or 0.005 radians to form the QFF. Single-point energies were computed at each displacement. These single-point energies were then refit by a least-squares procedure to yield the equilibrium geometry. A refit to the function minimum zeroed the gradients and yielded the force constants and the new equilibrium geometry. The force constants were converted from SICs to Cartesian coordinates utilizing the INTDER program (Allen 2005). For the Cartesian QFFs, each geometry was first optimized at the F12-DZ level of theory. The optimized geometry was then displaced along the x, y, and z axes by 0.005 Å to form the QFF .
For all of the QFFs, the rovibrational spectroscopic data was computed utilizing the Cartesian force constants in second order vibrational perturbation theory (Watson 1977;Papousek & Aliev 1982;Franke et al. 2021) in the SPECTRO software package (Gaw et al. 1996). To further increase the accuracy of the data, type-1 and type-2 Fermi resonances, Fermi resonance polyads (Martin & Taylor 1997), Coriolis resonances, and Darling-Dennison resonances were taken into account (Martin & Taylor 1997;Martin et al. 1995).
The dipole moment for c-C 3 (CN) 2 was computed at the F12-TZ level of theory, and the dipole moments for c-C 3 (C 2 H) 2 and c-C 3 (C 2 H)(CN) were computed at the F12-DZ level of theory. The intensities were computed at the B3LYP/aug-cc-pVTZ (Yang et al. 1986;Lee et al. 1988;Becke 1993;Dunning 1989) level of theory using GAUSSIAN16 (Frisch et al. 2016). The geometry optimizations, harmonic frequencies, singlepoint energies, and dipole moments were computed with the MOLPRO 2022.2 software package (Werner et al. 2022). Figure 4 displays the reaction pathway for the formation of c-C 3 (C 2 H) 2 from c-C 3 HC 2 H and ·C 2 H. Analogously, Fig. 5 shows the mechanism for the formation of c-C 3 (CN) 2 from c-C 3 HCN and ·CN. The relative ZPE-corrected electronic energies of all points on the pathways are summarized in Tables 1 and 2. Exothermicities for the single substitutions (Fortenberry 2021;Flint & Fortenberry 2022) show that a second substitution onto the cyclopropenylidene moiety to create c-C 3 (C 2 H) 2 is as thermodynamically favorable as the first. However, the formation of c-C 3 (CN) 2 was less exothermic than the formation of c-C 3 HCN by 2.94 kcal mol −1 . Notes. All energies reported in kcal mol −1 .  As shown previously for the monosubstituted derivatives (Fortenberry 2021;Flint & Fortenberry 2022), less of the potential energy gained by creation of the submerged well was required to be given up upon exit from the pathway by the diethynyl derivative at every point within the mechanism. Formation of c-C 3 (C 2 H) 2 is less likely to be impeded by a competing radiative stabilization process. Evaluation of the T1 diagnostics of all reactants, products, and intermediates showed negligible influence from static correlation effects. Several transition states gave a T1 diagnostic greater than 0.02, indicating possible multireference effects. Given the nature of these molecules and that the largest of the T1 diagnostic values for the transition states is 0.031, the impact of static correlation on these structures was neglected.

Reaction mechanisms
The disubstitution mechanisms are close analogues of the monosubstitution mechanisms for ·C 2 H and ·CN in most regards. However, some structures on the reaction coordinate differ slightly from previous iterations of this mechanism type. An I1 variant with C s symmetry is able to form due to chemically nonequivalent sites of radical addition to c-C 3 HC 2 H and c-C 3 HCN. This symmetric intermediate cannot lead to product formation on its own and must pass through a transition state to form the C 1 isomer of I1 if formed as a result of the initial collision. For the c-C 3 (C 2 H) 2 pathway, the TS I1 structure proved difficult to locate. An approximate upper-bound energy and structure were given for this transition state resulting from a coordinate scan, as the failure of the structure to converge to the correct molecular motion may be a consequence of the level of theory used and not the potential surface as a whole. An I2 structure with C 1 symmetry arising from displacement of a substituent group out of the plane was not able to form on these potential surfaces, unlike for the formation of c-C 3 HC 2 H and c-C 3 HCN. As a result, transition from the C s I1 structure to an I2 structure could not occur. A hydride shift from the C 1 I1 isomer to an I2 structure was also not locatable for these surfaces at this level of theory. Due to the nature of the available transition states, the reaction pathways were essentially stratified by the submerged well type.
The pathways also present a new planar reaction intermediate with C 2v symmetry that is not seen within the monosubstitution mechanisms. Such planar structures, as is in the case of the c-C 3 H 3 radical (Hoffmann et al. 1984), usually experience Jahn-Teller distortions that force one group out of the plane of the molecule, giving rise to a pseudoplanar geometry for the remainder of the structure. Such planar intermediates were avoided in all cases for the monosubstituted pathways (Flint & Fortenberry 2022), which were optimized purely at the F12-TZ level of theory. To ensure that planar geometries for the disubstituted variants were not produced as artifacts of optimization at the B3LYP/aug-cc-pVTZ level of theory, planar structures were generated and optimized at B3LYP/aug-cc-pVTZ for c-C 3 HC 2 H and c-C 3 HCN. Harmonic frequency analysis showed an imaginary frequency that forces the substituent group out of the plane A95, page 5 of 11 A&A 671, A95 (2023) Fig. 6. Two highest-occupied and lowest-occupied MOs for C s isomer of I2. of the molecule. The planarity of one of the intermediates for the disubstituted pathways was thus deemed to be a consequence of the potential surface and not largely attributed to the level of theory at which the calculations were performed. Figures 6 and 7 show the frontier MOs of both I2 isomers for the two reaction pathways. Most interestingly, the lowestoccupied molecular orbital (LUMO) of the C 2v isomers of both pathways shows strong C-H antibonding character. The NBO analysis of these structures showed that the electronic occupation of the C-H σ * orbital for the C s isomer was reduced by a factor of over 2.5 compared to that of the C 2v isomer for both pathways. Additionally, the frontier MOs of the products (Fig. 8) replicated those of the C 2v I2 isomer. Upon hydrogen dissociation, the hydrogen atom harbored the new singly occupied molecular orbital (SOMO) of the system. The former I2 SOMO, having released its electron density to the leaving group, became the LUMO of the product, while the SOMO-1 became the highestoccupied molecular orbital (HOMO) of the product. We note that the C 2v I2 SOMO does not become the LUMO of c-C 3 (C 2 H) 2 -this orbital is instead displaced to the LUMO+2, as the antibonding orbitals along the ethynyl groups comprising the LUMO and LUMO+1 are slightly lower in energy. An overall MO analysis of these complexes set up the planar I2 isomer for more facile product formation from this intermediate. Despite this, the C s I2 isomer remained in control of product formation. Since an a 2 electronic excitation is required to fully populate both C 2v I2 LUMOs, this transition requires excess energy (Pople & Sidman 1957), rendering it unlikely to occur. A relaxed scan of the labile C-H coordinate for both I2 symmetries showed that the geometry and orbitals of the dissociating complex resemble that of the C 2v isomer until the approximate bond-breaking distance of C-H = 1.55 Å, where continued dissociation results in a large spike in the electronic energy. As the C-H interatomic distance begins to surpass the bond-breaking distance, the geometry of the complex must distort to the lower C s symmetry. At these C-H distances, the MOs of the C s isomer closely resemble those in Fig. 7 and the predicted shifts in orbital occupation upon dissociation discussed previously. The C 2v I2 isomer may provide a greater electronic incentive for product formation while the leaving group remains bound, but dissociation from either isomer is possible, as they merge toward the same complex as dissociation occurs in order to remain in the lowest energy orbital occupation. This phenomenon is expected to take place for the association to form either intermediate I2 for the c-C 3 (C 2 H) 2 and to form the c-C 3 (CN) 2 pathways as well.
Given that the exothermicities of reaction for the formation of c-C 3 HC 2 H and c-C 3 (C 2 H) 2 are nearly equal, the thermodynamic incentive for reaction is the same, provided that the primary formation pathway for both molecules involves the addition of ·C 2 H. The available densities of c-C 3 HC 2 H for reaction likely dictate the ability of this pathway to occur and therefore also control the expected concentrations of c-C 3 (C 2 H) 2 . Approximating the c-C 3 HC 2 H/c-C 3 H 2 density ratio to be equal to that of c-C 3 (C 2 H) 2 /c-C 3 HC 2 H, observations may find a column density for c-C 3 (C 2 H) 2 of roughly 1.6 × 10 9 or a relative density to H 2 of 1.6 × 10 −14 in TMC-1.
of 140 km mol −1 , and the second largest intensity is ν 14 at 592.8 cm −1 (16.9 µm), with an intensity of 71 km mol −1 . Both of these intensities are on the same magnitude or larger of the antisymmetric stretch of water, which is about 70 km mol −1 . The two intense frequencies for c-C 3 (C 2 H) 2 fall within the range of the Near Infrared Spectrograph (NIRSpec) instrument on JWST, which has high resolution and could aid in detection. These two frequencies should be intense enough to be observed with JWST if the abundances as estimated above are true. The harmonic vibrational frequencies and intensities have been previously computed at the CCSD(T)/cc-pVTZ level of theory (Thimmakondu & Karton 2017). Between the previous level of theory and the F12-DZ computed in this work, a majority of the harmonic frequencies agree within 5 cm −1 . The biggest differences come from the symmetric and antisymmetric stretches of carbon in the ring and the out-of-plane bends. While the two levels of theory for the harmonic intensities differ quantitatively from each other, they agree qualitatively in the relative intensities of the frequencies.
Shown in Table 4, c-C 3 (C 2 H)(CN) has one intense peak: ν 1 at 3321.0 cm −1 (3.01 µm), with an intensity of 82 km mol −1 . This intense frequency corresponds to the C-H stretch of the ethynyl group and is nearly the same as the ν 2 stretch in c-C 3 (C 2 H) 2 . This one intense peak for c-C 3 (C 2 H)(CN) falls within the range of the NIRSpec instrument on JWST. However, c-C 3 (CN) 2 does not have any intensities of around 70 km mol −1 A95, page 7 of 11 A&A 671, A95 (2023) Table 4. Harmonic and fundamental vibrational frequencies (in cm −1 ) and intensities (in km mol −1 ) of c-C 3 (C 2 H)(CN).
Though c-C 3 (CN) 2 has not been reported experimentally, the studies of Smith et al. (1994) and Tittle et al. (1999) show that while NC 4 N discharges in Ar matrices successfully produce the linear isomer of c-C 3 (CN) 2 , there are unidentified infrared features in the experimental spectrum of Smith et al. (1994) that were not computationally reproduced by Tittle et al. (1999). Upon examination of these "leftover" vibrational frequencies, c-C 3 (CN) 2 was found not to be responsible for the excess features. For this to be the case, the bright ν 4 (a 1 ) vibrational mode of c-C 3 (CN) 2 would be expected to be prominent due to the experimental methods of Smith et al. (1994) filtering out most vibrational features that do not belong to the totally symmetric, irreducible representation (Tittle et al. 1999). Therefore, c-C 3 (CN) 2 may not have yet been produced in a laboratory environment or may have only been produced in environments in which it immediately undergoes isomerization or some other chemical process. As such, additional and unknown chemistry of this molecule persists even though c-C 3 (CN) 2 appears to form easily. These aspects will be explored in future work.

Conclusions
This work proposes synthetic pathways consistent with the cold chemistry of the ISM for the formation of two disubstituted c-C 3 H 2 derivatives, c-C 3 (C 2 H) 2 and c-C 3 (CN) 2 , along with promising thermodynamics for c-C 3 (C 2 H)(CN). A second substitution of a radical species onto the ring moiety of c-C 3 HC 2 H or c-C 3 HCN is generally found to be as energetically favorable as the initial substitution mechanism. In particular, values for ∆E rxn are nearly equal for the formations of c-C 3 HC 2 H and c-C 3 (C 2 H) 2 . As a result, column densities of c-C 3 (C 2 H) 2 in TMC-1 are hypothesized to depend mostly on c-C 3 HC 2 H concentrations, which leads to a potential column density of c-C 3 (C 2 H) 2 within the range of current observational capabilities. In order to aid observation of these molecules that are predicted to form, QFF calculations provide chemically accurate spectroscopic data for the astrochemically viable c-C 3 (C 2 H) 2 and c-C 3 (CN) 2 as well as for a mixed-substituent molecule, c-C 3 (C 2 H)(CN). The intense ν 2 frequency of c-C 3 (C 2 H) 2 , corresponding to the C-H antisymmetric stretch, coupled with the large dipole moment of 3.84 D allow for its facile astrophysical detection. The strong agreement of these more advanced (and likely more accurate) QFF results with previous theory further validates the spectroscopic methodology used to compute them. Through all current means of astrophysical observation, c-C 3 (CN) 2 is likely undetectable due to its near-zero dipole moment as well as its low intensity and low frequency for most of its vibrational motions. In contrast, the considerable dipole moment of c-C 3 (C 2 H)(CN) when taken into account alongside the reasonably intense ν 1 C-H stretch and favorable thermodynamics of formation through either synthetic route may allow for interstellar detection of this molecule. Collectively, these data have the potential to add a new pure hydrocarbon, as well as a new nitrile, to the molecular ranks of the ISM. These molecules may be the key to unlocking hidden chemical pathways to more complex molecules, such as PAHs, as well as formation routes to soot grains and other essential astrophysical building blocks.