Free Access
Issue
A&A
Volume 656, December 2021
Article Number A148
Number of page(s) 21
Section Atomic, molecular, and nuclear data
DOI https://doi.org/10.1051/0004-6361/202141616
Published online 15 December 2021

© ESO 2021

1 Introduction

After the early discovery of sugar formation by base-catalyzed aldol condensations of formaldehyde (H2CO) in the presence of calcium (Butlerow 1861), collectively known as formose reactions (Loew 1889), H2CO has been considered as a plausible primordial source of sugars on prebiotic Earth (Delidovich et al. 2014). Additionally, the syntheses of glycine (C2H5NO2) and glycolic acid (C2H4O3) by Strecker synthesis require H2CO, as well as hydrogen cyanide (HCN), ammonia (NH3), or other amines and aldehydes for the formation of amino acids (Choughuley et al. 1975). The Strecker-type syntheses and the related Bucherer-Bergs synthesis from alpha-aminonitriles (Ruiz-Mirazo et al. 2014) could explain the presence of amino acids in meteorites and model prebiotic reactions including the classic Miller-Urey experiment (Miller 1953; Oró et al. 1959; Choughuley et al. 1975). The formose and Strecker reactions highlight the relevance of formaldehyde as a key precursor in prebiotic chemical evolution (Menor-Salván 2018), but its role during bio-genesis on Earth is still a subject of much debate (Cleaves II 2008). Recently, the formation of sugar mixtures compatible with formose reactions has been observed after ultraviolet irradiation of interstellar ices (Meinert et al. 2016) as well, and this might further result in formaldehyde oligomerization (Furukawa et al. 2019). These findings highlight the possible relevance of formaldehyde in the interstellar medium (ISM) and its reactions in current prebiotic chemical evolution models. The origin of formaldehyde itself is less clear.

The presence of H2CO in the ISM was first reported in 1969 by Snyder et al. (1969) from 15 radio sources with galactic and extragalactic origins, becoming the first organic polyatomic molecule (OPM) identified in the ISM. Since then, it has been detected through absorption and emission bands as continually being present in vast and varied astrophysical objects, such as molecular clouds, envelopes of low- and high-mass protostars, commentary media, and asteroids (see the main detection of formaldehyde and references therein in Appendix A). Consequently, H2CO is recognized as one of the most abundant OPMs known in space (Mangum et al. 2008). Due to its ubiquity, H2CO and some of its isotopologs (e.g., H213CO; H213C16O; H212C17O or H2C18O), Zuckerman et al. (1969); Maret (2003); Blair et al. (2008) are commonly selected for tracing and modeling the astrophysical properties of the gases (e.g., kinetic energy, kinetic temperature, and density) in objects such as galaxies (Shuter 1982; Muhle et al. 2007), cold and dense molecular clouds (Tang et al. 2017), and in star-forming regions (Wiebe et al. 2008). Formaldehyde is known to be a very labile molecule under cold ISM conditions (Cody et al. 2011) and is considered a crucial chemical intermediary in the formation pathways leading to interstellar complex organic molecules (iCOMs) at low temperatures (Vasyunin & Herbst 2013; Barone et al. 2015).

Chemical reactions above 20 K and up to ~250 K are largely accepted to be governed by gas-phase chemistry in molecular processes (Eschenmoser 1994; Lazcano & Miller 1996). Formaldehyde gas-phase reactions are frequently linked to para-H2CO formation, while the chemistry in grain-iced mantles normally refers to the formation of its nuclear isomer, ortho-H2CO (Roueff et al. 2006; Ceccarelli et al. 2001). In this way, defining an ortho-to-para ratio (OTP) for H2CO can help to distinguish whether its formation has been produced in either the gas-phase or on icy-grain surfaces (Eschenmoser 1994; Guzmán et al. 2011, 2013). Below 20 K (the CO snow line), H2CO has been suggested to form on surfaces of icy interstellar dust grains by a double hydrogenation process of CO (CO + ⋅H → HCO + ⋅H → H2CO, (Tielens & Whittet 1997; Woon 2002). After a migration of the dust to warmer regions, a sublimation process occurs desorbing formaldehyde from the grains and entering in the gas-phase. H2CO desorption energy has experimentally been defined to be 7.21 ± 0.21 kcal mol−1 if grain sublayers are composed of iced water and to be 6.48 ± 0.12 kcal mol−1 if grains are based on olivine. In both cases a zeroth-order kinetic velocity of ω0 = 1028 mol cm−2 s−1 is obtained (Noble et al. 2012). In addition, CO hydrogenation seems to be 250 times more efficient in H2 O-rich ices than in CO-rich ice mantles, as estimated by Ceccarelli et al. (2001). This might justify why a portion of the H2CO abundances detected in some regions of ISM cannot be explained solely through gas-phase formation mechanisms with the abundances detected in excess typically attributed to dust grain formation (Hasegawa & Herbst 1993; Tielens & Whittet 1997; Woon 2002; Roueff et al. 2006; Persson et al. 2018). Despite the frequent H2CO detection, neither gas- nor grain-phase formations match the abundances that most modern astrochemical models predict. This is usually attributed to the fact that gas-phase reactions of formaldehyde and their associated molecular formation mechanisms in space are poorly understood (Dickens & Irvine 1999; Schöier et al. 2002; Ceccarelli et al. 2002; Roueff et al. 2006). The main aim of this paper therefore is quantum chemically to pinpoint viable molecular formation mechanisms of neutral H2CO in the gas-phase in cold regions of the ISM under local thermal equilibrium (LTE) conditions where this molecule has been detected.

2 Formaldehyde and cold astrophysical regions

Diffuse clouds (DC) can be described as irregular spatial confinements of interstellar gas and dust (silicate grains and amorphous crystals) at low temperatures and densities from 10 to a few 100 particles × cm−3 (Ballesteros-Paredes et al. 1999; Ceccarelli et al. 2002; Yamamoto 2017; Velusamy et al. 2017). In this astrophysical environment, neutral atomic hydrogen (HI) dominates showing a column density of NHI < 5 × 1020 cm−2 (Millar et al. 1979; Liszt & Lucas 1995). In addition to enabling observations in direct lines of sight (LOS) from terrestrial and space telescopes (Wooden et al. 2004), the low particle density facilitates UV radiation transmission (which is the main source of energy, with a minor contribution from cosmic rays, CR) to penetrate the cloud, subsequently warming it. As a result, the kinetic temperature (Tk) of the gases ranges between 30 and 100 K (Ballesteros-Paredes et al. 1999) from external to the innermost regions with fluctuations in pressure from ~6100 to 7700 K cm−3 (Yamamoto 2017). Because of the abundance and related absorption produced by HI, UV radiation does not exceed of 13.60 eV (313.62 kcal mol−1) (Ballesteros-Paredes et al. 1999). Formaldehyde has been detected specifically in regions of this classification, along with other carbonaceous compounds, such as HCN, HCO, HNO, C2 H, and C3 H. The core of the cloud, known as a translucent cloud, is denser (5 × 104 cm−3) with a central dust visual extinction of ~2.5 magnitudes (van der Tak et al. 2000; Di Francesco et al. 2002). The dominant form of hydrogen in the core is neutral atomic with a column density (NHI) associated with 1.7 × 1021 cm−2 mag−1× AV (Liszt & Lucas 1995). Molecular hydrogen (H2) is formed in this region on dust grain surfaces reaching a fractional density (NHI/NH2) of 102 cm−3 at a temperatureof ~100 K (Liszt & Lucas 1995). The average abundance density detected for atomic nitrogen has been estimated to be 4.3 × 10−5 cm−3 and for atomic oxygen this is ~3.5 × 10−7 cm−3 (Yamamoto 2017), while H2CO shows a number density value of ~10−10 cm−3 (Millar et al. 1979; Liszt & Lucas 1995). In this way diffuse molecular clouds (DMCs) are prone to transforming interstellar atoms into molecules. As a case in point, formaldehyde forms at ~100 K under an average pressure of ~6.9 × 103 K cm−3.

Dark, cold, and dense molecular clouds (DCDMCs) are commonly associated with star-forming regions (Maciel 2013). The most common form of hydrogen is neutral molecular (H2) with a typical particle spatial density of 10 cm−3–10 cm−6 cm−3 (Spitzer 1998). The temperature inside these clouds is lower than in DMCs. It typically oscillates from 10 to 20 (Combes & Pineau des Forets 1999; Marmet 2000). The main source of energy is UV radiation (primary and secondary – UV pumping; Klessen & Glover 2014) generated by collisions between CRs and H2 that cause the molecular bonds to break. The excess energy per collision has been defined empirically to be 4.48 eV or, equivalently, 103.44 kcal mol−1 (Klessen & Glover 2014). A particular type of dark cloud, known as “Bok globules” (Bok & Reilly 1947), are small (0.2–1 pc; Launhardt et al. 2010), cold, and dense confinements of interstellar dust and gas in hydrostatic equilibrium at the edge of gravitational collapse where the prevailing hydrogen is also in the molecular form (Dickman 1977). Multiple molecular species have been detected inside Bok globules, such as formaldehyde (H2CO; Palmer et al. 1969; Rickard et al. 1977; Minn & Greenberg 1979), carbon monoxide (CO; Wang et al. 1995; Cecchi-Pestellini et al. 2001), or carbon monosulfide (CS; Wang et al. 1995; Turner 1995), as well as isotopologs of these species such as C18 O (Wang et al. 1995). Additionally, nitrogenated compounds as ammonia (NH3; Minn & Greenberg 1979; Turner 1995; Bourke et al. 1995), and some ionized species such as HCO+ (Turner 1995; Bourke et al. 1995), are also present in these astrophysical objects. The H2CO abundances relative to molecular hydrogen detected toward this type of clouds are a few orders of magnitude higher than DMCs. They notably differ in composition from cloud to cloud. Some example values are 4 × 10−10 cm−3 in B68 (Di Francesco et al. 2002)or 2 × 10−8 cm−3 in L134N and TMC-1 (Ohishi et al. 1992). Applying the Bonnor-Ebert sphere model (Ebert 1955; Bonnor 1956) and using observational data from ten different Bok globules, Kandori et al. (2005) determined an effective temperature of Teff = 13.2 K and an averaged confining pressure of Pgas = 5.3 × 104 K cm−3. These two variables of the state equation are considered here alongside computational calculations of the formation mechanisms of H2CO for this kindof astrophysical scenario.

After the gravitational collapse and if the total mass of the individual cloud is approximately the mass of the Sun (2 × 1030 Kg; (van Dishoeck 2014), a new astrophysical system forms that is dominated gravitationally by a low-mass protostar known as a young stellar object (YSO). The protostar is in the center of the system and is surrounded by a Keplerian-rotating envelope of dust and gas, that is gravitationally connected and in which the angular momentum is conserved (Cassen & Moosman 1981). The distance to the star will define the surrounding energy, which is dominantly thermal and capable of heating the more distant dustgrain mantles. The processes progressively inject chemical constituents into the gas-phase (Ceccarelli et al. 2001)and determine what types of physical-chemical processes govern in every region of the disk. Circumstellar envelopes oflow-mass protostars (CELMP) are environments that are extraordinarily rich in organic molecules as H2CO or HNCO and in iCOMs such as CH3CN, CH3CHO, and C2 H5OH (Schöier et al. 2002), in addition to other O- and N-bearing complexes (Jørgensen et al. 2012). IRAS 16293-2422 (hereafter IRAS 16293) was chosen for this work as the quintessential example of such an object. IRAS 16293 is adequate for this role because formaldehyde has previously been detected with notable variations in abundances in three differentiated regions of its protodisk (Ceccarelli et al. 2001; Jørgensen et al. 2012; Jaber et al. 2014; van der Wiel et al. 2019). Additionally, glycolaldehyde (C2 H4O2), the simplest form of sugar and one of the first intermediates in the formose reaction (Larralde et al. 1995), has also been detected in this object for the first time in space in 2012 (Jørgensen et al. 2012). The more distant regions of the disk (>4000 AU) contain species such as CO, H2CO, CH3OH, or H2O (van Dishoeck et al. 1995) at low temperatures (10–20 K) where the gas is slightly warmer than the dust grains as they are tightly coupled to them. Energy from collisions among dust and gas is therefore considered to be the main heating mechanism in this region. Nevertheless, the thermal energy generated by those collisions is not enough to activate the formation of molecules with barriers at or below 20 K (0.0397 kcal mol−1). The gas column density of molecular hydrogen has been estimated to be N(H2) = 1.3 × 1023 cm−2 (Ward-Thompson et al. 1999), with a fractional abundance of formaldehyde – N(H2CO)/N(H2) – ~ 4 × 10−10 cm−3 in the gas-phase (Ceccarelli et al. 2001). When the temperature rises above ~ 20 K, CO starts to desorb from the ice grains and enters the gas-phase with an increase of ~ 103 cm−3 in detected densities with respectto H2 (Cassen & Moosman 1981; Aikawa et al. 2015). Formaldehyde starts to deplete from frozen grains at around ~ 40 K and it is fully desorbed at ~ 60 K (Ceccarelli et al. 2001). The additional H2CO mixes with the existing circumstellar mass of gas, which may justify why at ~ 700 AU from the core and at gas temperatures between 80 and 100 K ~ 50 kcal mol−1), the detected fractional abundances of formaldehyde reach N(H2CO)/N(H2) = ~ 4.0 × 10−9 cm−3 (Ceccarelli et al. 2001). This implies an H2 column density that is estimated to be N(H2) = ~ 5.0 × 1021 cm−2 (Bottinelli et al. 2014). The inner part of the envelope at ~ 150 AU in a region with temperatures of 100–150 K has a higher density of formaldehyde with an N(H2CO)/N(H2) of ~ 10 × 10−7 cm−3, as well as an increase in fractional abundances for H2O (which desorbs from iced mantles at ~80 K) (Ceccarelli et al. 2001). This region also produces new molecules principally due to the thermal energy emitted from the YSO (99.99 kcal mol−1). One example is trans-HONO. This chemical compound has recently been detected for first time in space and in this part of the disk (Coutens et al. 2019). Its proposed formation has inspired some reactions proposed in this work that may lead to H2CO. In this region, a column density for molecular hydrogen is considered like that in region II (N(H2) = ~ 5.0 × 1021 cm−2). The three regions of IRAS 16293 dictate the physical parameters for the presentation of our computations which are defined as follows:

  • Region I/d ~ 4000 AU, Tgas = 20 K, Pgas = 2.29 × 107 K cm−3;

  • Region II/d~ 700 AU, Tgas = 80 K, Pgas = 7.06 × 107 K cm−3;

  • Region III/d~150 AU, Tgas = 150 K, Pgas = 6.08 × 108 K cm−3.

3 Method and computational details

Ninety-six known molecular reaction mechanisms that lead to ground electronic state H2CO in the gas-phase, in turn composed of smaller chemical species that are reported in the literature (NIST; Linstrom & Mallard 1997) or are defined as in Table 1 were considered as the starting point of our computational calculations. The possible selected reaction mechanisms themselves are composed of chemical species that have already been detected in the ISM (McGuire 2018), except for the likely H2+, CH2+, CH3+, cis-HONO and NO2 species. Themolecular energy (atomic when required) of chemical species, reactants, and products is computed using quantum chemical computational methods at two different levels of theory and basis set. Explicitly correlated, coupled-cluster theory at the single and double level, with perturbative triple corrections (CCSD(T)-F12; Adler et al. 2007; Knizia et al. 2009) and second-order Møller–Plesset perturbation theory (MP2; Möller & Plesset 1934) were selected. The basis set for CCSD(T)-F12 is the explicitly correlated correlation-consistent triple-zeta basis set (Dunning 1989; Peterson et al. 2008; Hill & Peterson 2010), or abbreviated cc-pVTZ-F12. This combination provides a similar accuracy as but a computation two orders of magnitude faster than CCSD(T)/aug-pV5Z (Knizia et al. 2009). On the other hand, MP2 (Möller & Plesset 1934), with a double-zeta basis set (aug-cc-pVDZ) was selected because it offers a suitable compromise between accuracy and computational cost (Kaminsky et al. 2008).

The Molpro quantum chemistry package, version 2020.01 (Patch Level 3; Werner et al. 2020) was employed for CCSD(T)-F12/cc-pVTZ-F12 computations, while Gaussian 16, Revision B.01 (Frisch et al. 2016) was used for the MP2/aug-cc-pVDZ computations. Restricted references for closed-shell molecules were used, while restricted open-shell references were employed for species with non-spin-paired electrons. To proceed with initial computations, for every chemical compound, spare reactants, and (spare) products, the spin multiplicity (SpM) belonging to every spatial electronic wave function and net charge must be defined (Hund 1925, 1927). For the thermodynamic function, calculations at different temperature and pressure profiles and the Maxwell-Boltzmann (Muncaster 1979, and ref. therein; Rowlinson 2005) through related partition functions from statistical quantum mechanics theory were applied. This was used to determine reactions as exothermic (ΔH° < 0) and exergonic (ΔG° < 0) in the three astrophysical scenarios we studied. The reaction and thermodynamic energies contained in this research are corrected for the corresponding zeropoint energy (ZPE) and vibrational frequencies from the harmonic approximation.

Known favorable reactions involving reactants and their relative abundances for each scenario were incorporated into a detailed search on the energetically easiest path to pass from reactants to products on their potential energy surface (PES) and possible transition states (TS). There is no unique or unified method to find TS structures with success, and to avoid intuition as much as possible, we used two different scientific approaches. The first approach consists of obtaining the TS with the PM7 semi-empirical quantum mechanical method (SQM; Stewart 2013), confirmed by an intrinsic reaction coordinate (IRC) process (Fukui 1981; Zhixing 1989). For this purpose, the automated reaction mechanisms and kinetics program (AutoMeKin) (Martínez-Núñez 2015a,b; MOPAC2016 2016; Rodríguez et al. 2018) was employed. Before going to high-level calculations,every suggested trajectory was analyzed in detail to determine whether it produces a physically meaningful pathway on the grid of the reaction PES. After a TS structure was confirmed at SQM, geometries and energies were reoptimized via DFT (Hohenberg & Kohn 1964; Kohn & Sham 1965) with the global hybrid B3LYP method (Becke 1988) and the aug-cc-pVTZ basis set before we proceeded with the application of the higher-level calculations.Additionally, a direct TS search in Gaussian 16 (Frisch et al. 2016) by applying the synchronous transit-guided quasi-Newton method (STQN; Schlegel 1982) using B3LYP/aug-cc-pVTZ was executed. After a definitive TS structure given was obtained and confirmed by IRC (Fukui 1981; Zhixing 1989) through B3LYP/aug-cc-pVTZ, single-point energy (SPE) calculations on that specific saddle point of the PES were computed at the CCSD(T)-F12/cc-pVTZ-F12 and MP2/aug-cc-pVDZ levels of theory and basis sets. On the other hand, some reactions that are able to proceed are composed of excited electronic state reagents; their abundances are discussed in a different section. As result, three types of reactions were obtained composed of reactants that were previously detected in the astronomical objects under study. These include (i) barrier-less reactions, (ii) reactions with outer transition states, and (iii) reactions with inner transitions states. With respect to (i) barrier-less reactions, it must be emphasized that they rely upon radiative association (RA) processes, a feature that will significantly slow their kinetic progression down without the presence of catalysts. (ii) Reactions with outer transition states that are composed of a “real” energy barrier(s) where the transition state energy is entirely or partially above than that of the separated reactants. This feature notably penalize the formation of the products in cold conditions of space. In contrast, reactions (iii) with an inner transition state(s), in which the energy is totally submerged along the reaction path below the asymptotic reactant’s energy, are the most likely reactions to proceed. We subsequently focus the discussions and conclusions on them. As computations consider gas-phase structures at LTE, the effect of a third physical phenomenon beyond temperature and pressure (e.g., the state of ionization of the medium, the presence of magnetic fields, hyper-gas turbulence, the morphological or chemical composition thereof, effects of gravity, existence of catalyst agents during reactions, or the quantum tunneling effect) are considered beyond the scope this research.

Table 1

Chemical reactions of formaldehyde (H2CO) proposed in this research.

4 Results and discussions

Ninety-six molecular reaction mechanisms have been proposed at various points in history to form formaldehyde in the gas-phase (Table 1). These reactions in turn are composed of 45 different chemical species, including excited electronic states. Initial computations return the thermodynamic function values for each reaction mechanism at T = 298.15 K and P = 1 atm, for the two levels of theory and basis set, presented in Table 2 as an initial reference. Under the cold conditions of the ISM the most favorable reactions are those that are exothermic and exergonic. After applying TS searching methods on the reactions presented in Table 2, (R18) ⋅CH + ⋅ 4OH →1H2CO, (R23) ⋅ 3CH2 + ⋅O(3P) →1H2CO and (R75) ⋅CHO + 4OH →1H2CO + O(1D) are found not to converge in the computed pathway to the desired products, and therefore removed from the reactions we consider.

4.1 Diffuse molecular clouds

In DMCs where Tgas = 100 K and Pgas = 6.9 × 103 K cm−3 for this study, 56 reactions from the preliminary reaction list presented in Table 1 at the CCSD(T)-F12/cc-pVTZ-F12 level of theory and basis set return favorable thermodynamics. Again, the nonviable R18, R23, and R75 reactions and then undetected ionized species were removed from consideration. R1 (CI + H2 O →1H2CO) cannot be considered either because interstellar radiation in DMCs is capable of photoionizing any existing atomic carbon (CI), with its predominant form being ionized (C+; Snow & McCall 2006). Additionally, CH4, CO2, and trans-HONO do not form in these environments. Subsequently, 28 reactions composed of reactants already present in DMCs might contribute to the detected amount of formaldehyde (N(H2CO)/N(H2) = ~ 10−10 cm−3; Millar et al. 1979; Liszt & Lucas 1995) in DMCs. The computed values of the thermodynamic functions for these reactions are presented in Table 3. As a result, candidate reactions to produce formaldehyde in this environment are comprised of 11 reagents in addition to neutral atomic (⋅HI) and molecular hydrogen (H2). To return toFig. 1, the fractional abundances for these reagents (detected in the ground electronic state) with respect to H2 (N(×)/N(H2)) as well as the abundance of H2CO can be understood. This diagram shows that ⋅CHO, HNO, and NH3 in their ground electronic states are less abundant than H2CO which affects the reactions composed containing them. From the transition-state energy perspective on PES profiles and after detailed computational work under astrophysical conditions characteristic of DMCs, the emerged (+) / submerged (−) Gibbs free energies belonging to transition states (in kcal mol−1) are presented in Table 4. In this way and after reactions composed of NH3, ⋅CHO, and HNO were excluded because of their low abundance in this environment, 13 reactions (R9, R17, R24, R25, R26, R27, R28, R30, R33, R44, R45, R46, and R53) with an inner transition state were considered for further discussion.

Table 2

Thermodynamic function values of initial reactions (Table 1).

Table 3

Thermodynamic functions values of reactions able to form H2CO in DMCs.

thumbnail Fig. 2

Fractional abundances N(X)/N(H2) (cm−3) of reactants vs. H2CO as product detected in DCDMCs, and Bok globules. References: 1. Agúndez et al. (2015). 2. Martin & Barrett (1978). 3. Ohishi et al. (1992). 4. Polehampton et al. (2005). 5. Feuchtgruber et al. (2000). 6. Meyer et al. (1997). 7. Cecchi-Pestellini et al. (2001). 8. Sakai et al. (2009). 9. Di Francesco et al. (2002). 10. Aikawa et al. (2008). 11. Marka et al. (2011).

4.2 Dark, cold, and dense molecular clouds

The drop in temperature and increase in gas pressure by one order of magnitude (Tgas = 13.2 K and Pgas = 5.3 × 104 K cm−3) representative of DCDMCs promises interesting disparities in molecular energetics compared to DMCs. The main source of energy (UV radiation) in these objects decreases to 4.48 eV (Klessen & Glover 2014), limiting the energy able to ionize, excite, and/or brake or produce new molecular ligands. This energy is in accordance with our computed calculations related to H2 dissociation (4.4855 eV at CCSD(T)-F12/cc-pVTZ-F12 with corrected ZPE). Because there is scant evidence that nitrosyl-hydride (HNO) will be present in DCDMCs and H2O remains mostly frozen on the dust grain surfaces, R9, R19, and R84 cannot be considered as part of H2CO production. These reactions were subsequently removed from consideration in this environment. One of the affected species is CI, which mightsurvive as a neutral in DCDMCs (Cecchi-Pestellini et al. 2001). However, as H2O abundance (the second necessary reactant) is negligible in the gas-phase, R1 cannot be considered either. In Table 5, we list the 25 most probable reactions and their thermodynamic function values, typical of DCDMCs, that are able to produce some quantities offormaldehyde. This means that the 25 reactions mentioned and presented in Table 1 are possible ways to form H2CO in DCDMCs in the gas-phase.

Their viability will depend upon the density of the reagents, as displayed in Fig. 2, driven again, by a maximum photonic energy of 4.48 eV. In the same figure we show the notably increasing abundances of oxygen (atomic and molecular) in DCDMCs, as well as the remarkable reduction of ⋅CH and ⋅ 3CH2 below the H2CO abundance. This will undeniably affect formaldehyde production through the most promising channels R27 & R28. In contraposition, DCDMCs are richer in ⋅CHO, exceeding H2CO abundances,reinforcing the role of R71 in this scenario from the perspective of the molecular density. Exothermic and exergonic reaction mechanisms (as is the case presently) are expected to be more spontaneous and highly favor increasing kinetic feasibility. This is the case of reactions that correspond in particular to reaction mechanisms with submerged TS(s), and thus, with a higher energy respect to the reactants energy. The emerged (+) / submerged (–) Gibbs free energies belonging to transition states to respect the asymptotic energy of the reactants, at the two levels of theory and basis sets per reaction ID, are presented for this type of astronomic environment in Table 6. This shows that the most probable mechanisms with submerged TS able to produce H2CO are R24, R25, R26, R27, R28, R30, R33, R44, R45, R46, R53, and R82, when reactions with methylidyne (⋅CH; see Fig. 2) are excluded.

Table 4

Emerged (+) / submerged (−) TS Gibbs free-energy (ΔG) relative to thereactants free-energy (ΔG°) in DMCs.

Table 5

Thermodynamic function values of reactions able to form H2CO in DCDMCs.

Table 6

Emerged (+) / submerged (−) TS Gibbs free-energy (ΔG) relative to thereactants free-energy (ΔG°) in DCDMCs.

4.3 Circumstellar envelopes of low-mass protostars

Applying the P&T profiles defined in Sect. 2 for the three regions of IRAS 16293 our computations imply that 56 mechanisms possess valid reaction thermodynamic function values to proceed with the formation of H2CO. Twenty-six of these can produce H2CO in region I, 32 in region II, and 33 in region III (please see Table 7). These are the amended reactions with detected reagents in every region of the disk. The presence of methane (CH4) and neutral atomic carbon (CI) in this astronomical object, in addition to trans-HONO in region III (detected at ~ 150 AU from the core – Coutens et al. 2019), confirms the hypothetical feasibility of these mechanisms. Moreover, because nitrosyl hydride (HNO) has not been detected in IRAS 16293, R84 cannot be included in any circumstellar envelope region, as it was in DCDMCs. In Fig. 3 we show the abundance of reactants and formaldehyde itself already detected in the three regions of the disk. They have been calculated for the methylene (⋅ 3CH2) (Vasyunin & Herbst 2013), methyl radical (⋅CH3) (Sakai et al. 2012; Woon 2002), and methane (CH4) (Sakai et al. 2012) ground electronic state abundances rather than detected. The reason is that they are planar (or linear) symmetric-top molecules and therefore have a small permanent electric dipole moment, which makes them undetectable through terrestrial radio telescopes by rotational spectroscopy. On the other side of the spectrum, neither high-resolution infrared absorption nor ultraviolet-visible spectroscopy telescopes can identify them in the gas-phase because they lack of a powerful enough source of electromagnetic radiation. The abundance values for the triplet ground-state of methylene (⋅CH2(X3B1)) have therefore been estimated by using Monte Carlo algorithms (Vasyunin & Herbst 2013) which in turn are based on known abundances of derived species. The presence of the methyl radical (⋅CH3) in region I just before the CO depletion zone from icy grains should be negligible because ⋅CH3 is thought tobe mainly produced in this region by CH4 photodissociation (Sakai et al. 2012) and CH3OH (Woon 2002) desorbs from the grains in region II. Consequently, reactions based on ⋅CH3 were considered only for regions II and III, assuming for both an abundance of 6.40 × 10−9 cm−3 (Sakai et al. 2012).

Methane (CH4) is known to play a crucial role in the chemical processes belonging to envelopes of low-mass protostars. Hence, the CH4 abundances considered in this work were determined from CH3D abundances, tentatively detected in IRAS 16293 by Sakai et al. (2012). As described above, Fig. 3 shows the chemical species and their associated abundances that are detected or calculated in three regions of IRAS 16293. In addition, the figure shows three desorption zones (upright gray bars) in which CO, H2CO, and H2O are thought that are returned into the gas-phase due to grain mantles heating by radiation of YSO. This in turn allows us to better understand the increase in the abundance with rising temperature, and to determine the species that accompany these molecules. ⋅CHO is the only species below the formaldehyde abundance that is detected in all regions of IRAS 16293, but ⋅OH, H2 O, CO, and CI remain above the H2CO abundance in the three regions. The abundances of excited chemical compounds present in the reactions in Table 7 are considered below. In Fig. 3, we show an increase of one order of magnitude increase in H2CO abundance in region II compared to region I. This value increases up to three orders of magnitude in region III. This growth in abundance is normally attributed to formaldehyde desorption of H2CO from the ice grain mantles. By applying the deducted OTP of 3:1 (Ceccarelli et al. 2000, 2001) in evaporation regions II and III, an abundance of 1.33 × 10−9 cm−3 for para-H2CO in region II can therefore be estimated, while for region III, this would be 3.33 × 10−8 cm−3. This is the H2CO amount to be formed in the gas-phase for reactions proposed in regions II and III after the destruction processes that may occur. As in previous scenarios, according to computations and as shown in Table 8, an initial classification of the reactions according to the values of their emerged or submerged TS Gibbs free-energy with respect to the reactants per region of the disk is presented.

Table 7

Thermodynamic function values of reactions able to form H2CO in CELMPs.

4.4 Considerations about electronically excited reactants

All chemical compounds illustrated in Figs. 13 detected in DMCs, DCDMCs, and CELMPs, respectively, correspond to species in ground electronic states. However, some of the reactions proposed and presented in Table 3, Table 5, and Table 7 are composed of reactants which are in an excited electronic state. These reagents are CH2(a1A1), O2 (1Δg), OI (1D), ⋅OH(4Π), ⋅CO(3Π), and H2 (3Σ). As a first approach, we computationally calculated excitation enthalpies with CCSD(T)-F12/cc-pVTZ-F12 with corrected ZPE for a temperature range between 13.2–150 K. The results are presented in Table 9. We list the most notable observations below:

  • i. According to our calculations, the ⋅CH2(X3B1) ground-state requires a photonic energy of only 0.38 eV (or 8.87 kcal mol−1) to access the singlet-excited level CH2(a1A1), which agrees well with experiments (Leopold et al. 1985). However, methylene CH2(a1A1) could have a second formation route from methane photolysis (Gans et al. 2011; Lodriguito et al. 2009; Park et al. 2008; Wang et al. 2000) which produces ~50% of CH2(a1A1) in its photodissociation. The energy required for a primary decomposition CH4 is 5.01 eV, (or 115.53 kcal mol−1) according to Blitz & Seakins (2012). Consequently, the abundances for CH2(a1A1) incorporating the second formation route are amended, which mainly affects IRAS 16293 region III.

  • ii. The principal accepted source for O(1D) in cold gas clouds is also produced by photodissociation through reactions O2(3Σg) + hv → OI (3P) + OI (1D) and CO2(1 Σ +) + hv → CO(1) Σ + OI(1D) at 2424 Å (5.15 eV) and 2275 Å (5.45 eV), respectively (d’Hendecourt et al. 1986; from von Hagen 1982). Therefore its abundance in this regard is accommodated in scenarios in which the required energy might be present.

  • iii. O2(1Δg) formation from triplet ground-state species by direct radiation is highly unlikely because this state corresponds to a forbidden transition. Subsequently, and after pathways from H2O2 or from more complex molecules were discarded, the most probable way of forming singlet molecular oxygen in the cold ISM is in icy grain mantles of amorphous solid water (ASW) through the reaction O(3P) + O(3P) → O2(1Δg) as Pezzella et al. (2020) experimentally demonstrated.

  • iv. ⋅CO(3Π) abundances were revised, considering that one-third of the singlet species might be applicable according to Ridgway et al. (1976) and Burke et al. (1996), in studies of the cold regions of the ISM.

  • v. ⋅OH excited radicals form on ice-grain surfaces after applying a high-energy photonic radiation of 2 MeV, as empirically demonstrated (Miyazaki et al. 1993). However, this is an unlikely event to occur inside dense clouds and in envelopes beyond the CO snow line (<20 K), which is mainly reserved for interactions with small fractions of highly energetic CRs. In this way, an ⋅OH(4Π) column density of 8 × 1015 cm−2, as determined by Goldman et al. (1981), was applied for the three regions left (DMC, IRAS 16293 regions II and III).

  • vi. Many of the transitions from diatomic singlet ground-state hydrogen leading to triplet excited states mainly correspond to forbidden electronic transitions. However, some laboratory experiments have reported a dominance of triplet electronic states after intense photonic radiation in the 20–100 eV (Aguilar et al. 2008) or 15.1–16.7 eV (Jungen & Glass-Maujean 2016) ranges. This enables the existence of metastable triplet hydrogen in space after photon absorption by the singlet ground-state. Nevertheless, and as already mentioned, this is a rare event to occur in cold regions of the ISM and negatively impacts their abundances. We no longer consider triplet H2 here.

In Fig. 4, the final distribution (abundances) of reagents in cold regions of the ISM of all molecular species that were part of the reactions of these computations have returned as valid in the cold regions of the ISM. In this figure, the dotted black H2CO line represents the total formaldehyde detected by radio-astronomers, while the dotted orange line represents to para-H2CO formed in the gas-phase as determined by Ceccarelli et al. (2000) and (2001). The prequalified reactions to produce H2CO in every astronomic environment are analyzed below, together with their corresponding PES profiles.

thumbnail Fig. 3

Chemical species and detected fractional abundances in the three studied regions of IRAS 16293. Upright bars in gray correspond to desorption zones of H2 O, H2CO and CO (see text). The temperature is given in K, the molecular density in cm−3, and the distance in AU. References: 1. Ceccarelli et al. (2001). 2. Doty et al. (2004). 3. Bottinelli et al. (2014). 4. Carty et al. (2005). 5. Rivilla et al. (2018). 6. Sakai et al. (2012). 7. Aikawa et al. (2015). 8. Coutens et al. (2019). 9. Crimier et al. (2010). 10. Ceccarelli et al. (1998). 11. Parise et al. (2012). 12. Monte Carlo calculations from Vasyunin & Herbst (2013). 13. Sakai et al. (2012). 14. Hily-Blant et al. (2010). 15. Schwarz & Bergin (2014).

Table 8

Emerged (+) / submerged (−) TS Gibbs free-energy (ΔG) relative to the reactants free-energy (ΔG°) in CELMPs.

Table 9

Computed electronic enthalpies of excitation (ΔH°exc).

thumbnail Fig. 4

Abundances of reactants, detected H2CO (dotted black line), and gas-phase para-H2CO (dotted orange line) (in cm−3) in the three studied regions of IRAS 16293. The reactions composed of reactants above the orange dotted line might be the main contributors to para-H2CO production.

4.5 Potential energy profiles of prequalified reactions

In order to focus on the energy discussion, the following conventions and their respective units (if any) and signs are applicable:

  • a) For reactions with PES coordinates submerged below the reference reactant energies, a minimum in a PES profile corresponds to a maximum in energy required to dissociate the chemical compound.

  • b) As the energy released in an early-downhill surface from reactants to intermediates is given in the form of kinetic energy (all prequalified reactions are exothermic and exergonic), the change in enthalpy among them (− ΔH°) is employed, and the units used are kcal mol−1.

  • c) For the energetically easiest passage from reactants (or intermediates) to products through a saddle point on PES, the Gibbs free-energy of activation (ΔG°) was used, and the units, as in the previous case, will be kcal mol−1.

  • d) An “inner” transition state is one whose energy at the saddle point is greater in magnitude (submerged) than the energy of the reactants that constitute the input channel. For an “outer” transition state, the conceptis the opposite (emerged energy; see item a). The energy considered is the difference between the molecular enthalpy of the transition state and the input channel reactants. The units are kcal mol−1 (eV onlyas an astrophysical reference).

  • e) For the inner transition state, (or from reactants to products in a direct reaction), in which the energy is submerged (see item d)) with respect to the reactants energy, the enthalpy difference among them was used. The units are given in kcal mol−1.

In accordance with the previous discussion, the following six direct barrierless reactions were initially studied:

  • R4- ⋅3CO + 1H21H2CO

  • R20- 1CH2 + O(1D) →1H2CO

  • R21- 1CH2 + ⋅O(3P) →1H2CO

  • R22- ⋅3CH2 + O(1D) →1H2CO

  • R29- 1CH2 + ⋅4OH →1H2CO + ⋅H

  • R73- ⋅CHO + ⋅4OH →1H2CO + ⋅O(3P)

  • R24- 1CH2 + 1O21H2CO + O(1D)

Even though these reactions are likely going to progress in the three astrophysical objects studied, reactions R4, R20, R21, and R22 respond to radiative association (RA) processes. Consequently, they will proceed to form products but will be slower than reactions that trigger a leaving group from a kinetic point of view, even if they possess submerged transition states. The most efficient path that we computationally found for R4 is through two consecutive downhill steps, forming the formaldehyde triplet as an intermediate (⋅ 3CO + 1H23H2CO + hv11H2CO + hv2). In the first stage of the reaction, an average enthalpy (range 13.2–150 K) of −64 kcal mol−1, calculated at CCSD(T)-F12/cc-pVTZ-F12, is emitted. In the second reaction step, and to establish singlet H2CO from an excited triplet-state,an energy of −72 kcal mol−1 is reached. As a reference, the total average submerged energy computed for products from reactants for the cold regions is −137 kcal mol−1. Because R20, R21, and R22 are composed of a methylene group and atomic oxygen but are in different electronic states (see text above), different PES profiles from each other and differences in their yielded and increased energy are expected. R20 is an electronically adiabatic reaction with just one downhill step, which reaches an averaged submerged energy of −236 kcal mol−1 for the three regions we studied. However, the triplet reactions R21 and R22 can form singlet H2CO in two steps. Triplet formaldehyde again appears as the easiest way to pass from reactants to products in both reactions. In the first downhill step, R21 generates an additional averaged energy of −114 kcal mol−1, but in the case of R22, −155 kcal mol−1 is obtained. R21 and R22 complete their pathway from triplet to singlet H2CO, increasing their energy initially up to −72.34 kcal mol−1 (as in R4). The total averaged sunken enthalpy has a value of −186 kcal mol−1 for R21 and −227 kcal mol−1 for R22. Regarding R29, the computations return two downhill steps in the process of forming H2CO. In the first down course, an average enthalpy of −71 kcal mol−1 (just at the limit of UV and UVA radiation) is released forming a planar doublet radical intermediate. This intermediate detaches one hydrogen to reach the more stable H2CO + H form, spending an average −185.91 kcal mol−1 in cold regions. In the process a spin-flip must take place, which will likely slow down the molecular formation velocity. Finally, R73 flows directly in one step from ⋅CHO + 4OH to singlet formaldehyde, but it could be suggested or thought that there should be two transitions states. In the process and as a result of breaking ⋅OH(4Π) little bonds forming CHO + H + O + e+ → H2CO (1 A1) + H + e, detailed calculations returned convergence in one step, suggesting that the two processes could proceed simultaneously. Again, the energy data for these reactions are from our CCSD(T)-F12/cc-pVTZ-F12 calculations, which incorporate ZPE corrections. Detailed PES profiles of direct barrier-less reactions are available in Appendix B.1. In addition, the computationally calculated harmonic infrared-band positions, individual molecular and ZPE energies, as well as their spatial coordinates for the main excited species at the CCSD(T)-F12/cc-pVTZ level of theory and basis set are listed in Appendix C.1.

Table 10

Thermodynamic function values of prequalified reactions with their inner transition state per astrophysical scenario (AS).

4.6 Prequalified reactions with inner transition states

In a pure gas-phase reaction process and as a general description of reactions containing submerged transition state energy with respect to the reactant’s energy, entrance channels occur over PESs without barriers to overcome. However, this entry is featured (bounded) by radial and angular anisotropies that may be generated as result of the energies. For the most favored reactions this work, the most relevant radial and angular energies of interaction were computed, and will be shown in the following. Most reactions commonly start to descend into an initial associated complex valley, known as early-downhill surfaces, to directly form an intermediate molecular complex (as mentioned) without entrance barriers. This feature over an attractive PES releases energy (see Table 10) that is generated by the rearrangement of the atoms after the approach of their center of mass. When the related forces between them continue in the appropriate direction, a substantial decrease in the atomic distances will affect their electronic, vibrational, rotational, and translational energy. This excess of energy will accompany the reaction in the form of kinetic energy along the reaction pathway, being the most favored reactions these that require the least amount of energy to proceed to the products. In addition, Table 10 presents (i) the potentially qualified reactions to produce H2CO in the regions of cold space, and (ii) their relative energies after reactions have progressed through their PES to form the products. Additionally, in order to study their respective energy profiles in detail, we present the Gibbs free-energy computed at the CCSD(T)-F12/cc-pVTZ-F12 level of theory and basis sets under the astrophysical conditions of DMCs and DCDMCs (corrected for ZPE), inFig. 5. Following these two rules, the most favored reactions are R27 and R28 (methylene and 3O2 based reactions) in DMCs, followed by R44 and R45 (methyl radical and atomic oxygen), and R30 and R33 (methylene with hydroxyl radical). In this way, R46 despite its great exothermicity, is relegated to a less favorable position due to its high internal barrier. R53and R82 are also not as likely due to a combination of both factors.

R27 (1 CH2 + ⋅ 3O21H2CO + ⋅O(3P)) and R28 (⋅ 3CH2 + ⋅ 3O21H2CO + ⋅O(3P)) involve two electronic states of methylene, (ground-state -X3B1- and singlet-excited state -a1 A1-), combined with molecular oxygen in its triplet ground-state (3 P). The reaction starts to descend into an initial downhill surface to form a molecular Criegee intermediate (CI) complex, known as carbonyl oxide, in its triplet electronic state (3 H2COO). Independent of the initial trajectory followed by reactants, the entrance channel belonging to 1CH2 + ⋅ 3O2 is established when the relative energy of interaction (Eint) between the products and the intermediate are equalized radially and angularly. To determine more details about the entrance process, the radial and angular energies of interaction (Eint) were computed at the CCSD(T)-F12/cc-pVTZ-F12, in steps of 0.1 Å for radial and 10° for angular anisotropies. The complete results are presented in Fig. 6. Such information will provide some kinetic insights beyond the thermodynamics described thus far. This figure represents four different approaches of triplet molecular oxygen over two excited electronic states of methylene. Radially (along the C–O molecular bond) and as expected, the possible association of both channels initiates at different distances (R) about 1.90 Å in R28, while for R27, the association starts around 4.8 Å (beyond the graphic range, see Fig. 6-(1)). The radial interaction in R27 is dominated by dispersion forces (more so than in R28) that are due to the perturbative interactions produced by the excited spin-orbit of the CH2 singlet. As the molecules approach each other radially, the energy gradually increases through the attractive zone until the energetic equilibrium is reached (upright blue bar in Fig. 6-(1)). This point is located in both channels at R = 1.38 Å, producing a at minimum −51.27 kcal mol−1 for R27 and −41.98 mol−1 for R28. The new bond that is created has a harmonic stretching frequency (ω5 - A) of 1044 cm−1 (calculated at CCSD(T)-F12/cc-pVTZ-F12) after 3H2COO is formed. Angular interactions have different approaches and provide clues about where geometrical entry bounds might be spatially located, Fig. 6-(2), (3), and (4). When O = O accesses vertically (see Fig. 6–(2α)) with the right angle (108.48°), a wide asymmetric entry angle between ≈ −140° and +100° in R27 is exhibited, which is slightly narrower at ≈ −120° and +80° in R28. In both cases, this orientation is wide enough to facilitate an intermediate formation. In contrast, if O = O horizontally joins Fig. 6-(2-β) with the same angle, the access is restricted to ≈ ±50° in R27 and to ≈ ±40° in R28. This orientation becomes the most probable entrance bottleneck for reactants. Another option appears if molecular oxygen approaches carbon (atom) angularly with the same C-O-O angle (108.48°). The anisotropies of this anisotropy are shown in Fig. 6-(3)-, suggesting that in R27 an angle between −40° and +190° maintains this possibility. It is most favored in the regions centered at ≈ 0° and 140°. In this approach, the access for R28 is reduced to a zone between ≈ 58° and 88° (see Fig. 6-(3), orange curve). Finally, the possibility that molecular oxygen tangentially accesses the complex is considered, with a tilted angle of ≈108.48° confirming the anisotropies exposed in Fig. 6-(4) as a possible cone-shaped input trajectory. Surprisingly, the access to the complex is restricted in certain regions of the cone, more specifically in an angle between ≈90° to 135° and ≈ −90° to −135°, as shown in Fig. 6-(4). Continuing the reaction path along the PES, R27 confirms an electronically adiabatic reaction that flows over triplet electronic states on all molecular associations that appear during the transit from reactants to products. The initial fall releases an enthalpic energy of about −45.01 kcal mol−1 in this environment. This energy is enough to overcome the low free-energy barrier belonging to its inner transition state (3.77 kcal mol−1), which in addition is submerged by −41.19 kcal mol−1 with respect to the products energy. The exit channel from H2COO in the route the H2CO + O(3P) PES valley is conformed by the dissociation of the (O–O) bond, constrained along an imaginary vibrational mode that is calculated to be 552.8i cm−1. This reaches the stationary point at 1.34 Å from the oxygen attached to C. On the other hand, R28 has a similar PES profile as R27. They differ in energy values because both reactants in R28 are in the ground-state, forming an initial quintet complex that is more energetic (see Table 10) than the reactant complex compared to R27. This produces an initial first downhill enthalpic step in R28 of -36.14 kcal mol−1, which means a difference of 9.02 kcal mol−1 compared with the initial reaction step of R27. The remaining reaction path follows the same course as R27. Kinetically, both reactions have been studied for a long time because formaldehyde is important in atmospheric hydrocarbon combustion chemistry. For R27, a rate constant of 5 × 10−11 cm3 mol−1 s−1 (Tsang & Hampson 1986), and 0.93 ± 0.22 × 10−12 cm3 mol−1 s−1 (Lakshmanan et al. 2019) have been defined. For R28, a rate of 3.2 ± 0.3 × 10−12 cm3 mol−1 s−1 (e.g., Böhland et al. 1984; Darwin et al. 1989; Bley et al. 1992; Baulch et al. 1992) and more recently, 0.98 ± 0.28 × 10−12 cm3 mol−1 s−1 (Lakshmanan et al. 2019) have been determined at close to room conditions. These figures confirm in any case that R27 is favored with respect to the ground-state of R28 in the formation pathway to H2CO + O(3P). This is in line with what the thermodynamic profiles and entrance channel anisotropies indicate.

As in the previous case, the two channels R44 (⋅CH3 + O(1D) →1H2CO + ⋅H; radical-excite) and R45 (⋅CH3 + ⋅O(3P) →1H2CO + ⋅H; radical-radical) are characterized by a molecular junction without any entrance potential barrier. Radial anisotropies (see Fig. 7-(1)) indicate that the most probable entry along (R) for R44 is located at a distance of ≈4.1 Å, which differs from the entry calculated for R45, which is determined to be ≈2.5 Å. This interaction, also present in R27 and R28, is also dominated by dispersion forces. These forces will alter the radial trajectory between atomic oxygen and methyl radical in particular, in R44. From an angular entrance, the vertical and horizontal anisotropies show perturbations along the entry pathways (see Fig. 7), where limits of the trajectory are also governed by dispersion interactions, led by electron densities (H) as well as by rotational, vibrational, and translational dynamics. However, vertical-radial pathways in Fig. 7-(2) are defined by an angle of entry between ≈ −70° to +80° for R44 and −70° to +70° for R45. Their asymmetries are in the upper approach (H-interactions in α curves). On the other hand, horizontal-radial motions (β curves in Fig. 7-(2)) are conformed between ≈ −90° to +90° for R44 and ≈ −80° to +80°for R45, but maintain the symmetry of the β curves and widen the angular range of access to the molecule. The energy used in the downhill portion to form methoxy radical (CH3O) and the difference in the internal energy of (O2(3P) vs. O2 (1D)) produces a more pronounced initial enthalpy drop in R44 compared to R45 (−137.84 kcal mol−1 vs. −88.04 kcal mol−1 in DMCs). This is in favor of R44 (see Fig. 10). A proposed exit channel to produce H2CO + H must be determined from the C atom, assuming that in practice, there are three potential TSs (one per H) that statistically can be involved in the process. In this way, and in accordance with the previous intermediate energy, the transition state in R44 is submerged −115.39 kcal mol−1 or (−5.00 eV), below the reactant’s enthalpy; R45 is −65.60 kcal mol−1 or −2.84 eV, having a moderately submerged free-energy barrier to overcome of 22.37 kcal mol−1 under DMC condition (see Table 10 for more details and scenarios). The exit channel flows over a stationary point located along a H–C bond that is constituted by an imaginary frequency of 481.81i cm−1, which requires only 16.28% of the enthalpy released in the case of R44 and 25.49% in R45 to form the products. One of these two reactions has been kinetically studied, with rate coefficients that were obtained under normal atmospheric conditions of ~1 × 10−10 cm3 mol−1 for R44 (Atkinson et al. (2006) and references therein). No coefficient rates for R45 appear to be listed in astrochemical databases such as KIDA, UMIST, or NIST.

R30 corresponds to an excited-radical reaction, while R33 represents a radical-radical chemical interaction composed of CH2(a1A1) and ⋅CH2(X3B1) reagents, inaddition to the doublet hydroxyl radical ⋅OH(2Π). This union produces an initial doublet and quartet set of reactants that in both cases form the hydroxymethyl radical (CH2OH) as an intermediate. As a brief description of the possible entrance channels, when (intermolecular radial distance -R- Table 8(1)) R > 3 Å, radial interactions are characterized by the already mentioned dispersion forces that are more pronounced in excited-radical reaction R30. When the radial distance (R) between2OH-C is ≈ 3 Å, the joining process behaves gradually and progressively. The energetic equilibrium is reached at a distance R = 1.36 Å, acquiring a maximum energy (E) of −120.65 kcal mol−1 for R30 and −111.37 mol−1 for R33, as shown in Fig. 8-(1). The corresponding stretching vibrational harmonic mode belonging to the created intermediate (v4 - A) is 1207 cm−1 (calculated at CCSD(T)-F12/cc-pVTZ-F12). Vertical-angular interactions (Fig. 8-(2),(3), and (4)) demonstrate that the approach between ≈ ±110° enables molecular formation in this orientation, while horizontally angular access is limited to ≈ ±60° in R30. The horizontal and vertical molecular entrance energy difference for R33 in both cases is <10 kcal mol−1 below R30, is the lowest variation of all molecular systems studied so far. If the 2OH path faces angularly the carbon atom(C belongs to CH2), as is represented in Fig. 8-(3), the interaction will proceed satisfactorily because it presents a favored orientation with an angle of (108.98°). Finally, and as can be denoted by the anisotropies represented in Fig. 8-(4), the interactions show a less restrictive energy profile (compared to R27 and R28), where the most favored regions are those that lie alternatively between 45° centered at the 0° point of the truncated cone pathway. When the chemical union is realized, R30 produces an initial enthalpy in excess of −113.64 kcal mol−1 and −104.76 kcal mol−1 in R33 under astrophysical conditions such as those in DMCs. In order to complete the reaction to form H2CO + ⋅H, a Gibbs free-energy barrier to overcome of 38.07 kcal mol−1 is required in both cases. However, this maximum on the PES is submerged enthalpically with respect to the reagents −75.56 kcal mol−1 in R30 and −66.69 kcal mol−1 in R33. This featureis in favor of R30. The exit channel is located along the H-O bond, which requires a Gibbs free-energy of activation of 37.9 kcal to pass over the transitory state to reach products on PES. This detachment is produced through the stretching vibrational mode (ω9 = 3861 cm−1 - A) of hydroxymethyl radical (CH2OH) generatinga transition state characterized by an imaginary frequency of ts = −1092 cm−1.

From a kinetic viewpoint, and as in previous cases, reactions based on triplet methylene and the hydroxyl radical doublet have been studied from an atmospheric combustion perspective. At room conditions, a rate coefficient of 3.01 × 10−11 cm3 mol−1 s−1 was defined for R33 (Tsang & Hampson 1986; Jasper et al. 2007). Even though there are no published rates for R30, recent laboratory studies based on atmospheric conditions of TITAN by Douglas et al. (2018), indicate that the reactivity of 1CH2 tends to increase with decreasing temperature, which could denote a positive factor for reactions R27, R30, or R31 under extremely cold conditions (<20 K). Examining the qualified reactions in DCDMCs (R28, R33, R44, R45 and R53), the presence of reagents with densities above H2CO (see Sects. 2 and 4.2) is notably reduced. The surrounding energy in DCDMCs, compared to DMCs and the lower number of collisions typical of this environment (see Sect. 2), is a crucial physical factor for determining the formation of new chemical compounds. In this environment, the relative detected abundances of 1CH2, 4OH, and ⋅CHO are below those of formaldehyde. This means that the reactions that are composed of them are less relevant in H2CO formation. For energy profiles and details in DCDMC, we refer to Fig. 5, and for thermodynamic values to Table 10.

With respect to CELMPs, we present in Fig. 9, the Gibbs energy profiles for the most relevant reactions (defined in Table 10) for the three regions. IRAS 16293 is plotted for reference. In region I, R27, R28, R30 and R33 are presented as the most probable sources of H2CO. In region II, R27 and R28 disappear, which leaves the already discussed reactions R33, R44, and R45. R46 and R91 enter this scenario as reactions to consider because of the abundances of 4OH and CH4, respectively,but even though they possess inner barriers, the high free barriers (~ 90 kcal mol−1) relegate them to the background. Region III is slightly more complex, because only 1CO, 3CO, H2 O, CH4, and 1CH2 exceed formaldehyde abundances. Five reactions (R30, R31, R45, R46, and R92) are proposed here as the main avenues responsible for the para-H2CO formation, but the last two (R46 and R92) continue with high barriers as in the previous region, despite an increasing temperature and pressure (see Table 10). At this point, reactions from Table1 might be considered based on CH4 and 1CO (CH4 + 1CO →1H2CO + 1CH2 and CH4 + 1CO →1H2CO + 3CH2), but both cases show exergonic and exothermic behavior. However, reactions composed of methane and carbon monoxide, a) CH4 + 3CO → H2CO + 3CH2 and b) CH4 + 3CO → H2CO + 1CH2, are exothermic and exergonic under region III conditions, showing values of ΔH° = −36.76 kcal mol−1, and ΔG° = −37.60 kcal mol−1 for (a), and ΔH° = −21.31 kcal mol−1, and ΔG° = −21.93 kcal mol−1, for (b). To finalize this section, a new reaction is proposed for consideration. It is composed of singlet methylene and carbon dioxide. R31 (1 CH2 + CO21H2CO + 1CO; see the Gibbs free-energy profile in Fig. 9 and thermodynamic data in Table 10), has a moderate inner barrier of 27.39 kcal mol−1 that is submerged enthalpically −20.75 kcal mol−1 (−0.90 eV) below the reactants. According to Doty et al. (2004), CO2 can be present in IRAS 16293 with a column density (N) of 1.5 × 10−17 cm−2 which represents a fractional abundance (n) compared to H2 in this region of ~2 × 10−5 cm−3, above H2CO abundance. If this density is confirmed (CO2 has not been detected in this region yet), R31 may well be one of the most productive reactions to produce H2CO in region III ofIRAS 16293 and similar environments. The structures, reactants, intermediates, transition states, and products of the most relevant reactions for the astrophysical scenarios studied are depicted in Fig. 10.

thumbnail Fig. 5

Gibbs free-energy profiles (kcal mol−1) of prequalified reactions under DMC and DCDMC astrophysical conditions (Tgas = 100 K, Pgas = 6.90 × 103 K cm−3 and Tgas = 13.2 K, Pgas = 5.30 × 104 K cm−3 respectively). Energy values are obtained at CCSD(T)-F12/cc-pVTZ-F12 with the respective corrections for ZPE. In parenthesis we plotted charge and spin multiplicity.

thumbnail Fig. 6

Radial (1) and angular (2), (3), and (4) interaction energy diagrams (Eint) of R27 (black) and R28 (orange) entrance channels. (1) Upright blue bar represent R at the equilibrium. (2) Dotted lines illustrates the entrance energy curve of the angular β anisotropies. Molecular energies were calculated at CCSD(T)-F12/cc-pVTZ-F12, units in kcal mol−1, angles in sexagesimal degrees, and molecular distances in Å.

thumbnail Fig. 7

Radial (1) and angular (2) energy of interaction diagrams (Eint) belonging to R44 and R45 entrance channels. Eint = Molecular energy of intermediate (E2) – molecular energy of reagents (E1). Molecular energies (E2 & E1) calculated at CCSD(T)-F12/cc-pVTZ-F12 level of theory and basis sets, units in kcal mol−1, angles in sexagesimal degrees, and molecular distances in Å.

thumbnail Fig. 8

Radial (1) and angular (2), (3), and (4) energy of interaction diagrams (Eint) belonging to the R30 and R33 entrance channels. Eint = Molecular energy of intermediate (E2) - molecular energy of reagents (E1). Molecular energies (E2 and E1) calculated atCCSD(T)-F12/cc-pVTZ-F12, units in kcal mol−1, angles in sexagesimal degrees, and molecular distances in Å.

thumbnail Fig. 9

Gibbs free-energy (kcal mol−1) profiles of pre-qualified reactions for three regions of IRAS 16293 astrophysical conditions at CCSD(T)-F12/cc-pVTZ-F12 (corrections for ZPE incorporated). Region I: Tgas = 20 K and Pgas = 2.29 × 107 K cm−3. Region II:. Tgas = 80 K and Pgas = 7.06 × 107 K cm−3. Region III: Tgas = 150 K and Pgas = 6.08 × 108 K cm−3. Energy unitsof ΔH°, ΔG° in kcal mol−1; ΔS° in cal mol−1 K−1. In parentheses, we plot the net charge and spin multiplicity.

thumbnail Fig. 10

Initial structures of reactants, intermediates, transition states (from IRC), and products of the most relevant reactions determined by this study. In parenthesis we plotted net charge, and spin multiplicity of the molecular complexes.

5 Conclusions and astrophysical implications

High-level ab initio quantum chemical computations (CCSD(T)-F12/cc-pVTZ-F12) applied to gas-phase molecular astrochemistry confirm that neutral H2CO in cold space at LTE conditions can be synthesized by a multi-channel system of reactions. Considering the relative abundance of reactants, the submersion and the internal barriers of inner transition states composed of three reaction paths and their thermodynamic functions at temperature and pressure profiles characteristic of the astrophysical scenarios proposed in this article, we obtained the most likely molecular formation routes of H2CO for the gas-phase. From this point of view, in diffuse molecular clouds the better positioned reactions are R27 - 1CH2 + ⋅ 3O21H2CO + ⋅O(3P) and R28 - ⋅ 3CH2 + ⋅3O21H2CO + ⋅O(3P), while in dark, cold, and dense molecular clouds H2CO is dominated by R28 - ⋅ 3CH2 + ⋅ 3O21H2CO + ⋅O(3P) and R44 - ⋅CH3 + ⋅O(1D) →1H2CO + ⋅H. In the hotter region III of circumstellar envelopes of low-mass, R45 - ⋅CH3 + ⋅O(3P) →1H2CO + ⋅H and R31 - 1CH2 + CO21H2CO + CO are likely the most efficient productors of H2CO, in region II, R44 - ⋅CH3 + O(1D) →1H2CO + ⋅H and R45 - ⋅CH3 + ⋅O(3P) →1H2CO + ⋅H are the most favored, while in the farthest and colder region I, R27 - 1CH2 + ⋅ 3O21H2CO + ⋅O(3P) and R28 - ⋅ 3CH2 + ⋅ 3O21H2CO + ⋅O(3P), likely represent the two dominant molecular mechanisms in the formation of formaldehyde for the gas-phase.

According to these computational results, in an early evolutive state of DMCs with a gas density ~ 100 cm−3 at a temperature of ~100 K, and a pressure of 6.9 × 103 K cm−3, in which UV radiation (13.60 eV at most) dominates the gas-phase reactions energetically, the main mechanisms to produce H2CO seem to be dictated by excited singlet (CH2(a1A1)) and triplet ground-state (⋅CH2(X3B1)) methylene, which both combine with the ground triplet-state oxygen of O2 (3Σg). When the cloud collapses and reaches particle densities from 103 to 106 cm−3 and an averaged gas temperature and pressure of 1̃3.2 K and 5.3 × 104 K cm−3 accompanied by a surrounding energy that does not exceed 4.18 eV, hydroxyl radical (⋅CH3) seems to be produced in the gas-phase, but from more elementary constituents, (Feuchtgruber et al. 2000) which generates a reduction of CH2(X1B1) abundance. Under these conditions, triplet methylene (⋅CH2(X3B1)) combined with the triplet ground-state oxygen O2 (3Σg) is probably the main mechanism responsible for H2CO(1A1) production followed by the union between methyl radical (⋅CH3) and singlet atomic oxygen (O(1D)). After the DMC gravitationally collapses, and if the cloud mass is one or two times the solar system mass (~ 2 × 1030 Kg), a new astrophysical solar-like system composed of one or two YSOs with the corresponding circumstellar envelope of gas and dust is created (the astrophysical conditions of IRAS 16293 were used as a model for our computations). At 150 AU from the core, with temperatures and pressures close to 150 K and 6.08 × 108 K cm−3, there is a region of the disk where abundances of o-H2CO the typical abundances of cold regions exceed by 2–3 orders of magnitude. This H2CO is composed of o-H2CO that is depleted from icy grain mantles and the para nuclear isomer, which is thought to be formed in a pure gas-phase. This zone is relatively poor in abundances of potential reagents to produce H2CO in the gas-phase. Consequently, the possibilities of its formation are quite limited. The most probable reactions that can produce para-H2CO in sufficientquantities are those composed of the methyl radical (⋅CH3(2A2)) and triplet ground-state OI (⋅O(3P)) followed bya hypothetical reaction between CH2(a1A1) and CO2(1Σg+). At 700 AU from the core, a new region appears in which formaldehyde is present. This time, the abundances decrease by approximately two orders of magnitude compared to the hotter region (see Fig. 4). This effect is thought to occur because o-H2CO is not fully evaporated from the grains, but a portion of it is present in the gas-phase. o-H2CO evaporation starts at about 40 K and stops at around 80–90 K (see Fig. 3). The thermal energy emanating from the YSO was determined to be ~50 kcal mol−1, which reduces the gas temperature and pressure to ~80 K and 7.06 × 107 K cm−3. According to our calculations, reactions based on the methyl radical (⋅CH3) with ⋅O(1D) and ⋅O(3P) might be the main gas-phase supplier site of H2CO in this region. Far from the core at ~4000 AU, there is a region in which dust grain mantles are abundant (candidates for protoplanet formation), the temperature and pressure do not exceed ~20 K and 2.29 × 107 K cm−3, and the action of thermal energy emanating from the core is practically negligible. Consequently, this environment contains not enough energy to activate even very low barriers for chemical reactions in the gas-phase. Therefore the H2CO formation in the gas-phase can be led by (CH2(a1A1)) and (⋅CH2(X3B1)), combined with the ground triplet oxygen O2 (3Σg), which does not require any external source of energy because their transition state is submerged, as we have already mentioned.

In this way and according to our computations, the most probable carbonaceous precursors for the gas-phase production of neutral H2CO in cold regions of space (ordered by importance) are: CH2(a1A1), ⋅CH2(X3B1) and ⋅CH3(2A2). The states of just before, during, and immediately after gravitational collapses and stellar main-sequence forming processes requires further studies.

Acknowledgements

J.C.R.O. wishes to thank all the staff of the Mississippi Center for Supercomputing Research (MCSR) at the University of Mississippi, in Oxford, MS, USA for their availability, and commitment. Additionally, thanks are given to Prof. Emilio J. Martínez-Núñez from Chemical Physics Department of Universidad de Santiago de Compostela (Spain) for his recommendations regarding the AUTOMEKIN program (priv. comm.). R.C.F. also acknowledges funding from NASA grant NNX17AH15G and start-up funds provided by the University of Mississippi. MCSR funding has been provided in part by NSF grant OIA-1757220.

Appendix A Main H2CO detection in space.

Table A.1

Main detection of Formaldehyde (H2CO) and isotopologues in outer space in chronological order.

Appendix B PES profiles of barrier-less reactions.

thumbnail Fig. B.1

Barrier-less PES profile reactions. R4/R73 - Tgas = 13.2 K, Pgas = 5.30 x 104 K cm−3. R20/R21/R22 - Tgas = 100 K, Pgas = 6.90 x 103 K cm−3. R29 - Tgas = 80 K, Pgas = 7.06 x 107 K cm−3. Black lines CCSD(T)-F12/cc-pVTZ-F12, dotted blue lines MP2/aug-cc-pVDZ. ΔE°, ΔH°, ΔG°) units in kcal mol−1, ΔS° in cal mol−1 K−1. In parentheses, charge, and spin multiplicity.

Appendix C Main excited reactants and intermediates.

thumbnail Fig. C.1

Calculated infrared-band positions (cm−1), individual molecular and ZPE energies (Hartrees), and spatial coordinates of the main excited reactants and intermediates (as mentioned in sections 4.4 and 4.5). Calculations at the CCSD(T)-F12/cc-pVTZ-F12 level of theory and basis set, angles in sexagesimal degrees, and inter-atomic distances in Angstroms (Å).

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All Tables

Table 1

Chemical reactions of formaldehyde (H2CO) proposed in this research.

Table 2

Thermodynamic function values of initial reactions (Table 1).

Table 3

Thermodynamic functions values of reactions able to form H2CO in DMCs.

Table 4

Emerged (+) / submerged (−) TS Gibbs free-energy (ΔG) relative to thereactants free-energy (ΔG°) in DMCs.

Table 5

Thermodynamic function values of reactions able to form H2CO in DCDMCs.

Table 6

Emerged (+) / submerged (−) TS Gibbs free-energy (ΔG) relative to thereactants free-energy (ΔG°) in DCDMCs.

Table 7

Thermodynamic function values of reactions able to form H2CO in CELMPs.

Table 8

Emerged (+) / submerged (−) TS Gibbs free-energy (ΔG) relative to the reactants free-energy (ΔG°) in CELMPs.

Table 9

Computed electronic enthalpies of excitation (ΔH°exc).

Table 10

Thermodynamic function values of prequalified reactions with their inner transition state per astrophysical scenario (AS).

Table A.1

Main detection of Formaldehyde (H2CO) and isotopologues in outer space in chronological order.

All Figures

thumbnail Fig. 2

Fractional abundances N(X)/N(H2) (cm−3) of reactants vs. H2CO as product detected in DCDMCs, and Bok globules. References: 1. Agúndez et al. (2015). 2. Martin & Barrett (1978). 3. Ohishi et al. (1992). 4. Polehampton et al. (2005). 5. Feuchtgruber et al. (2000). 6. Meyer et al. (1997). 7. Cecchi-Pestellini et al. (2001). 8. Sakai et al. (2009). 9. Di Francesco et al. (2002). 10. Aikawa et al. (2008). 11. Marka et al. (2011).

In the text
thumbnail Fig. 3

Chemical species and detected fractional abundances in the three studied regions of IRAS 16293. Upright bars in gray correspond to desorption zones of H2 O, H2CO and CO (see text). The temperature is given in K, the molecular density in cm−3, and the distance in AU. References: 1. Ceccarelli et al. (2001). 2. Doty et al. (2004). 3. Bottinelli et al. (2014). 4. Carty et al. (2005). 5. Rivilla et al. (2018). 6. Sakai et al. (2012). 7. Aikawa et al. (2015). 8. Coutens et al. (2019). 9. Crimier et al. (2010). 10. Ceccarelli et al. (1998). 11. Parise et al. (2012). 12. Monte Carlo calculations from Vasyunin & Herbst (2013). 13. Sakai et al. (2012). 14. Hily-Blant et al. (2010). 15. Schwarz & Bergin (2014).

In the text
thumbnail Fig. 4

Abundances of reactants, detected H2CO (dotted black line), and gas-phase para-H2CO (dotted orange line) (in cm−3) in the three studied regions of IRAS 16293. The reactions composed of reactants above the orange dotted line might be the main contributors to para-H2CO production.

In the text
thumbnail Fig. 5

Gibbs free-energy profiles (kcal mol−1) of prequalified reactions under DMC and DCDMC astrophysical conditions (Tgas = 100 K, Pgas = 6.90 × 103 K cm−3 and Tgas = 13.2 K, Pgas = 5.30 × 104 K cm−3 respectively). Energy values are obtained at CCSD(T)-F12/cc-pVTZ-F12 with the respective corrections for ZPE. In parenthesis we plotted charge and spin multiplicity.

In the text
thumbnail Fig. 6

Radial (1) and angular (2), (3), and (4) interaction energy diagrams (Eint) of R27 (black) and R28 (orange) entrance channels. (1) Upright blue bar represent R at the equilibrium. (2) Dotted lines illustrates the entrance energy curve of the angular β anisotropies. Molecular energies were calculated at CCSD(T)-F12/cc-pVTZ-F12, units in kcal mol−1, angles in sexagesimal degrees, and molecular distances in Å.

In the text
thumbnail Fig. 7

Radial (1) and angular (2) energy of interaction diagrams (Eint) belonging to R44 and R45 entrance channels. Eint = Molecular energy of intermediate (E2) – molecular energy of reagents (E1). Molecular energies (E2 & E1) calculated at CCSD(T)-F12/cc-pVTZ-F12 level of theory and basis sets, units in kcal mol−1, angles in sexagesimal degrees, and molecular distances in Å.

In the text
thumbnail Fig. 8

Radial (1) and angular (2), (3), and (4) energy of interaction diagrams (Eint) belonging to the R30 and R33 entrance channels. Eint = Molecular energy of intermediate (E2) - molecular energy of reagents (E1). Molecular energies (E2 and E1) calculated atCCSD(T)-F12/cc-pVTZ-F12, units in kcal mol−1, angles in sexagesimal degrees, and molecular distances in Å.

In the text
thumbnail Fig. 9

Gibbs free-energy (kcal mol−1) profiles of pre-qualified reactions for three regions of IRAS 16293 astrophysical conditions at CCSD(T)-F12/cc-pVTZ-F12 (corrections for ZPE incorporated). Region I: Tgas = 20 K and Pgas = 2.29 × 107 K cm−3. Region II:. Tgas = 80 K and Pgas = 7.06 × 107 K cm−3. Region III: Tgas = 150 K and Pgas = 6.08 × 108 K cm−3. Energy unitsof ΔH°, ΔG° in kcal mol−1; ΔS° in cal mol−1 K−1. In parentheses, we plot the net charge and spin multiplicity.

In the text
thumbnail Fig. 10

Initial structures of reactants, intermediates, transition states (from IRC), and products of the most relevant reactions determined by this study. In parenthesis we plotted net charge, and spin multiplicity of the molecular complexes.

In the text
thumbnail Fig. B.1

Barrier-less PES profile reactions. R4/R73 - Tgas = 13.2 K, Pgas = 5.30 x 104 K cm−3. R20/R21/R22 - Tgas = 100 K, Pgas = 6.90 x 103 K cm−3. R29 - Tgas = 80 K, Pgas = 7.06 x 107 K cm−3. Black lines CCSD(T)-F12/cc-pVTZ-F12, dotted blue lines MP2/aug-cc-pVDZ. ΔE°, ΔH°, ΔG°) units in kcal mol−1, ΔS° in cal mol−1 K−1. In parentheses, charge, and spin multiplicity.

In the text
thumbnail Fig. C.1

Calculated infrared-band positions (cm−1), individual molecular and ZPE energies (Hartrees), and spatial coordinates of the main excited reactants and intermediates (as mentioned in sections 4.4 and 4.5). Calculations at the CCSD(T)-F12/cc-pVTZ-F12 level of theory and basis set, angles in sexagesimal degrees, and inter-atomic distances in Angstroms (Å).

In the text

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