# Observations and chemical modeling of the isotopologues of formaldehyde and
the cations of formyl and protonated formaldehyde in the hot molecular core
G331.512-0.103

Edgar Mendoza Dept. Ciencias Integradas, Facultad de Ciencias Experimentales,
Centro de Estudios Avanzados en Física, Matemática y Computación, Unidad
Asociada GIFMAN, CSIC-UHU, Universidad de Huelva, Spain  Miguel Carvajal Dept.
Ciencias Integradas, Facultad de Ciencias Experimentales, Centro de Estudios
Avanzados en Física, Matemática y Computación, Unidad Asociada GIFMAN, CSIC-
UHU, Universidad de Huelva, Spain  Instituto Universitario Carlos I de Física
teórica y computacional, Universidad de Granada, Spain  Manuel Merello
Departamento de Astronomía, Universidad de Chile, Casilla 36-D, Santiago de
Chile, Chile  Leonardo Bronfman  Departamento de Astronomía, Universidad de
Chile, Casilla 36-D, Santiago de Chile, Chile  Heloisa M. Boechat-Roberty
Observatório do Valongo, Universidade Federal do Rio de Janeiro, Ladeira do
Pedro Antônio, 43, Rio de Janeiro, RJ

###### Abstract

In the interstellar cold gas, the chemistry of formaldehyde (H2CO) can be
essential to explain the formation of complex organic molecules. On this
matter, the massive and energetic protostellar object G331 is still unexplored
and, hence, we carried out a comprehensive study of the isotopologues of H2CO
and formyl cation (HCO+), and of protonated formaldehyde (H2COH+) through the
APEX observations in the spectral window $\sim$159–356 GHz. We employed
observational and theoretical methods to derive the physical properties of the
molecular gas combining LTE and non-LTE analyses. Formaldehyde was
characterized via 35 lines of H2CO, H${}_{2}^{13}$CO, HDCO and H2C18O. The
formyl cation was detected via 8 lines of HCO+, H13CO+, HC18O+ and HC17O+.
Deuterium was clearly detected via HDCO, whereas DCO+ remained undetected.
H2COH+ was detected through 3 clean lines. According to the radiative
analysis, formaldehyde appears to be embedded in a bulk gas with a wide range
of temperatures ($T\sim$20–90 K), while HCO+ and H2COH+ are primarily
associated with a colder gas ($T\lesssim$ 30 K). The reaction
H2CO+HCO${}^{+}\rightarrow$ H2COH+ \+ CO is crucial for the balance of the
three species. We used Nautilus gas-grain code to predict the evolution of
their molecular abundances relative to H2 which values at time scales
$\sim$103 yr matched with the observations in G331: [H2CO] = (0.2–2)
$\times$10-8, [HCO+] = (0.5–4) $\times$10-9 and [H2COH+] = (0.2–2)
$\times$10-10. Based on the molecular evolution of H2CO, HCO+ and H2COH+, we
hypothesized about the young lifetime of G331, which is consistent with the
active gas-grain chemistry of massive protostellar objects.

ISM: molecules — astrochemistry — molecular data — methods: data analysis —
line: identification

††facilities: Atacama Pathfinder Experiment (APEX).††software: Based on
analysis carried out with the CASSIS software (Vastel et al., 2015), RADEX
(van der Tak et al., 2007), Nautilus (Wakelam et al., 2015).

##  1 Introduction

Formaldehyde was one of the first organic molecules detected in the
Interstellar Medium (ISM) (Snyder et al., 1969; Gardner & Whiteoak, 1974).
This compound has been observed both in the gas and solid phase of the ISM
(Meier et al., 1993; Schutte et al., 1996; Féraud et al., 2019). Formaldehyde
also plays a key role in the interstellar synthesis of pre-biotic and complex
organic molecules (Ferus et al., 2019; Layssac et al., 2020; Paiva et al.,
2023).

In observational studies, Mangum et al. (1990) used maps and spectral lines of
H2CO to constrain the different components of the Orion Kleinmann–Low (KL)
star forming region, which has been largely discussed in the context of cold,
hot cores and molecular outflows (Wootten et al., 1984; Sutton et al., 1995;
Zapata et al., 2011). Pegues et al. (2020) analyzed maps of H2CO toward a
sample of discs surveyed with ALMA. They highlighted the importance of H2CO to
understand the production mechanisms of O-bearing molecules in discs. In hot
corinos and low-mass protostellar objects, H2CO transitions have been useful
to estimate its physical and chemical conditions (Maret et al., 2004; Sahu et
al., 2018; Martín-Doménech et al., 2019). In the Horsehead photodissociation
region, Guzmán et al. (2013) analyzed observations of H2CO and CH3OH to
investigate their chemistry and dominant formation routes. In massive star
forming regions, H2CO transitions have been observed in hot molecular cores.
In addition, an H2CO maser at 6 cm has been proposed as an exclusive tracer of
high-mass star formation (Pratap et al., 1994; Araya et al., 2015). In the
IRAS 16562-3959 high-mass star-forming region, Taniguchi et al. (2020)
observed H2CO and investigate its formation pathways; they also analyzed the
emission of (CH3)2CO and CH3OCHO. The identification of both simple and
complex organic molecules contributes to our understanding of the chemical
network of reactions that interplay these species under interstellar
conditions (e.g. Horn et al. 2004; Singh et al. 2022). In evolved stellar
objects, Ford et al. (2004) observed H2CO toward the carbon star IRC+10216 and
discussed the H2CO formation in the context of solar system comet comae. In
sources outside the Milky Way, Tang et al. (2021) observed H2CO to construct a
map of the kinetic temperature of two massive star-forming regions in the
Large Magellanic Cloud (LMC). Shimonishi et al. (2016) analyzed observations
of H2CO, among other chemical species, to diagnose and claim the first
detection of a hot molecular core in the LMC.

The physical conditions of the formaldehyde isotopologues are also
investigated here. In a study about deuterated molecules in Orion KL, Neill et
al. (2013) estimated D/H ratios from formaldehyde and other molecules using
data from Herschel/HIFI. As a rare isotopic species, Turner 1990 detected D2CO
via four transitions toward Orion-KL. Nonetheless, Neill et al. (2013) did not
confirm such detection. In a sample of Galactic molecular sources, Yan et al.
(2019) performed a study about the 12C/13C ratio using transitions from the
H${}_{2}^{12}$CO and H${}_{2}^{13}$CO isotopologues. With respect to H2C18O, a
few works have reported its detection. Using ALMA observations, Persson et al.
(2018) not only detected H2C18O, but also H2C17O, D${}_{2}^{13}$CO and HDC18O,
toward IRAS 16293-2422 B.

In chemical association with H2CO, we also present results about the molecular
ions HCO+ and H2COH+ (Fig. 1). HCO+ and its isotopologues have been observed
in hot molecular cores and outflows (Sánchez-Monge et al., 2013; Shimonishi et
al., 2016). In previous works, Merello et al. (2013) and Hervías-Caimapo et
al. (2019) reported preliminary results about H13CO+ in G331.512-0.103, from
now on G331. H2COH+ is a rare molecular ion whose detection has been scarcely
reported in the literature. It has been observed toward Sgr B2, Orion KL, W51
and the prestellar core L1689B (Ohishi et al., 1996; Bacmann et al., 2016).

H2CO, HCO+ and H2COH+ are chemical species that participate and compete in
various chemical reactions, for instance:

$\text{H_{2}CO+HCO^{+}\longrightarrow H_{2}COH^{+}+CO~{}~{}~{}.}$ (1)

The kinetic constants were experimentally studied in an early work by Tanner
et al. (1979). Bacmann et al. (2016) observed H2COH+ in the prestellar source
L1689B and discussed the reaction presented in Eq. 1 in the context of cold
prestellar cores. The three species studied here, H2CO, HCO+ and H2COH+, also
play a key role in the gas-grain chemistry to form complex organic molecules,
such as glycolaldehyde and sugar related molecules (Halfen et al., 2006; Woods
et al., 2012; Eckhardt et al., 2018; Layssac et al., 2020). Additionally, it
is worth mentioning that H2CO is closely associated with methanol (CH3OH), a
pivotal molecule for interstellar chemical complexity. Studies have
demonstrated that successive hydrogenation reactions of carbon monoxide (CO)
can lead to the formation of both H2CO and CH3OH (Watanabe et al., 2003; Tsuge
et al., 2020).

The present work is based on observations of G331, a massive protostellar
object embedded in the G331.5-0.1 giant molecular cloud. The central object
drives a powerful outflow with a flow mass and momentum of $\sim
55~{}M_{\odot}$ and $\sim 2.4\times 10^{3}~{}M_{\odot}$ km s-1, respectively
(Bronfman et al., 2008). The source is located in the tangent region of the
Norma spiral arm, at an heliocentric distance of $\sim$ 7.5 kpc. Regarding the
ambient core of G331, ALMA observations evidence a lukewarm gas with
$n_{\text{H}_{2}}\sim 5\times 10^{6}$ cm-3 and $T\sim$ 70 K (Hervías-Caimapo
et al., 2019). In association with those gas conditions, Canelo et al. (2021)
and Santos et al. (2022) reported recent results about the physical and
chemical conditions of the isocyanic acid (HNCO) and methyl acetylene
(CH3CCH). Here, we then contribute with new results about the lukewarm and
cold gas conditions of G331, which are based on a broad spectral survey
collected with the Atacama Pathfinder Experiment (APEX).

In massive protostellar objects, the chemical scenario about the formation of
hydrocarbons, carbon chain and organic compounds is not well understood yet
(Taniguchi et al., 2018; Kalvāns, 2021). In view of the importance to
understand how gas-grain processes occur in massive protostellar objects, this
work also presents a chemical model to understand the cold chemistry that
connects two abundant species, as H2CO and HCO+, with a low abundant ion as
H2COH+. In addition, this study allowed us to venture the age of the
protostellar object G331. This article is organized as follows: Section 2
describes the observations carried out by APEX of the source G331 and the
methodology used for this study. Section 3 presents the results of the
spectral analysis and the estimated physical conditions of G331. In Section 4,
the astrophysical and astrochemical implications of the present results are
discussed. Final remarks and conclusions are summarized in Section 5.
Additionally, an appendix section (A) has been included to provide
supplementary information and data.

C(-[:-150]H)(-[:-30]H)=[:90]Oa) H-C O⊕b)
C(-[:-150]H)(-[:-30]H)=[:90]O⊕-[:30]Hc)

Figure 1: Representation of the neutral molecule a) formaldehyde, and the
molecular ions b) formyl cation and c) protonated formaldehyde.

##  2 Methodology

The APEX telescope (Güsten et al., 2006) was used to perform the observations,
adopting the single point mode, toward the source coordinates RA:DEC =
16h12m10.1s, $-$51∘28′38.1′′. The APEX-1 and APEX-2 receivers, of the Swedish
Heterodyne Facility Instrument (SHeFI) (Vassilev et al., 2008), were used to
collect spectral setups in the frequency ranges 213–275 GHz and 267–378 GHz,
respectively. The spectral resolution of the dataset was adjusted to be
between $\sim$0.15 and 0.25 km s-1 for a noise level of about 30 mK across the
bands.

As part of the observational runs, the SEPIA B5 instrument (Belitsky et al.,
2018) was used to collect setups between 159–211 GHz. SEPIA B5 is a dual-
polarisation sideband-separated receiver. The lower and upper sideband (LSB
and USB) are separated by 12 GHz. Each sideband is recorded by two XFFTS units
of 2.5 GHz with a 1 GHz overlap. The Half Power Beam Width (HPBW) values,
covered by the APEX receivers, are ranged from $\sim$ 17 to 39″, were
estimated for each transition using 7.8″$\times$ 800/$\nu$[GHz] (Güsten et
al., 2006; Quénard et al., 2017)111https://www.apex-
telescope.org/ns/instruments/, where $\nu$ is the rest frequency of the
spectra. With respect to the calibration uncertainty, Dumke & Mac-Auliffe
(2010) discussed that lines of C18O (3–2) and 13CO (3–2) might reach
uncertainties of $\sim$ 13 %, and lines of H2CO and CH3OH of up to $\sim$ 33
%. Considering that we observed different species through a multi-wavelength
analysis, it was adopted an overall calibration uncertainty of 30 %.

The data reduction was carried out with the CLASS package of
GILDAS.222https://www.iram.fr/IRAMFR/GILDAS/ The spectra obtained from the
APEX real-time calibration tool (Muders et al., 2006), in the CLASS data
architecture, are in the corrected antenna temperature
scale.333https://www.apex-telescope.org/telescope/efficiency/ For further
clarity, we exhibit spectra in units of antenna temperature (K) with respect
to the systemic velocity of G331 ($V_{\rm lsr}$ = -90 km s-1). Additionally,
all the spectra were smoothed to exhibit a common channel of 1 km s-1. The
Weeds444https://www.iram.fr/IRAMFR/GILDAS/doc/html/weeds-html/weeds.html
extension of CLASS (Maret et al., 2011), along with the spectroscopic
databases CDMS555https://cdms.astro.uni-koeln.de/ (Endres et al., 2016) and
JPL666https://spec.jpl.nasa.gov/ (Pickett et al., 1998), were also used here.
The CASSIS777http://cassis.irap.omp.eu/ software was used to estimate the
physical conditions from the spectral lines, which were utilized in the main
beam temperature ($T_{mb}$) scale adopting the main-beam efficiencies
$\eta_{mb}$ = 0.80, 0.75 and 0.73 for SEPIA180, APEX-1 and APEX-2,
respectively. In order to evaluate beam dilution effects, a source size of 5″
was adopted for the G331 core, and of 15″ for an expanded region. Using the
CASSIS tools, Local Thermodynamic Equilibrium analyses were performed using
calculations and the population diagram method (Goldsmith & Langer, 1999;
Mangum & Shirley, 2015; Roueff et al., 2021). Non-LTE calculations were
carried out with the RADEX code (van der Tak et al., 2007) using rate
coefficients from the LAMDA
database.888http://home.strw.leidenuniv.nl/~moldata/

###  2.1 Chemical models

The gas grain code Nautilus was used to compute abundances as a function of
time from a network of gas and grain chemical reactions (Semenov et al., 2010;
Reboussin et al., 2014; Ruaud et al., 2015, 2016). We focused on computing the
abundances of H2CO, HCO+ and H2COH+ considering similar conditions to those
obtained from the observations. Nautilus uses the
KIDA999https://kida.astrochem-tools.org/ database (Wakelam et al., 2015),
which includes rate coefficients for a total of about 7509 reactions and 489
chemical species. For the solid state chemistry, it considers mantle and
surface mechanisms as those investigated by Hasegawa & Herbst (1993) and
Ghesquière et al. (2015). For the neutral and molecular ions analyzed in this
work, there is a network of no more than 50 chemical reactions connecting
them. The chemistry of HCO+ and H2CO is significant and the detection of
various of their isotopologues suggests gas-grain processes in cold
environments. Previous studies unveiled a rich chemistry in lukewarm regions
of G331 ($T_{g}\thickapprox$70 K) evidenced from various spectral lines of
HNCO and CH3CCH (Canelo et al., 2021; Santos et al., 2022). We expanded here
the chemical models of G331 adopting a cold gas condition
($T_{g}\thickapprox$50 K), which might not only explain the abundances of
H2CO, HCO+ and H2COH+ but also the eventual formation of more complex
molecules.

##  3 Results

###  3.1 Line analysis

####  3.1.1 Formaldehyde and its isotopologues

Figure 2: Observed (solid) and modelled (dashed) spectra, from the LTE (in
blue) and non-LTE (in red) simulations (§ 3.2), of the formaldehyde
isotopologues in G331: a) H2CO, b) H${}_{2}^{13}$CO, c) HDCO and d) H2C18O.
For each spectrum, the $y$-axis represents the intensity in a scale of antenna
temperature (K) while the $x$-axis represents the velocity relative to the
local standard of rest (Systemic $V_{\rm lsr}=$ -90 km s-1). Table 1: Spectral
line analysis of H${}_{2}^{12}$CO, H${}_{2}^{13}$CO, HDCO and H2C18O.

Species and transitions | FrequencyaaRest frequency values, obtained from databases such as CDMS (Endres et al., 2016) and JPL (Pickett et al., 1998), are accessed using the CASSIS software (Vastel et al., 2015)(see § 2 for more details) | Beamwidth | $E_{u}$ | $g_{u}$ | $A_{ul}$ | Line area | $V_{\rm lsr}$ | Linewidth | rms
---|---|---|---|---|---|---|---|---|---
| [MHz] | [″] | [K] |  | [10-5s-1] | [K km s-1] | [km s-1] | [km s-1] | [mK]
p-H2CO (3 0 3 – 2 0 2) | 218222.192 | 28.56 | 20.96 | 7 | 28.2 | 12.99 $\pm$ 0.07 | -90.07 $\pm$ 0.02 | 6.4 $\pm$ 0.04 | 47
p-H2CO (3 2 2 – 2 2 1) | 218475.632 | 28.56 | 68.09 | 7 | 15.7 | 5.5 $\pm$ 0.3 | -90.4 $\pm$ 0.1 | 5.9 $\pm$ 0.3 | 35
p-H2CO (3 2 1 – 2 2 0) | 218760.066 | 28.52 | 68.11 | 7 | 15.8 | 5.04 $\pm$ 0.04 | -90.33 $\pm$ 0.02 | 5.85 $\pm$ 0.05 | 36
o-H2CO (3 1 2 – 2 1 1) | 225697.775 | 27.65 | 33.45 | 21 | 27.7 | 17.88 $\pm$ 0.06 | -90.2 $\pm$ 0.01 | 7.25 $\pm$ 0.03 | 38
p-H2CO (4 0 4 – 3 0 3) | 290623.405 | 21.47 | 34.9 | 9 | 69 | 16.1 $\pm$ 0.1 | -90.39 $\pm$ 0.03 | 7.29 $\pm$ 0.08 | 42
p-H2CO (4 2 3 – 3 2 2) | 291237.78 | 21.43 | 82.07 | 9 | 52.1 | 10.92 $\pm$ 0.03 | -90.578 $\pm$ 0.009 | 7.13 $\pm$ 0.02 | 33
o-H2CO (4 3 2 – 3 3 1) | 291380.488 | 21.41 | 140.94 | 27 | 30.4 | 9.653 $\pm$ 1E-3 | -90.36 $\pm$ 0.02 | 6.73 $\pm$ 0.04 | 28
o-H2CO (4 3 1 – 3 3 0) | 291384.264 | 21.41 | 140.94 | 27 | 30.4 | 12 $\pm$ 0.1 | -90.38 $\pm$ 0.04 | 7.75 $\pm$ 0.09 | 29
o-H2CO (5 1 5 – 4 1 4) | 351768.645 | 17.74 | 62.45 | 33 | 120 | 14.7 $\pm$ 0.2 | -90.88 $\pm$ 0.04 | 7.1 $\pm$ 0.1 | 46
o-H${}_{2}^{13}$CO (3 1 3 – 2 1 2) | 206131.626 | 30.27 | 31.62 | 21 | 21.1 | 1.68 $\pm$ 0.02 | -90.91 $\pm$ 0.03 | 4.37 $\pm$ 0.08 | 18
o-H${}_{2}^{13}$CO (3 1 2 – 2 1 1) | 219908.525 | 28.37 | 32.94 | 21 | 25.6 | 1.14 $\pm$ 0.04 | -91.01 $\pm$ 0.06 | 3.6 $\pm$ 0.2 | 34
o-H${}_{2}^{13}$CO (4 1 4 – 3 1 3) | 274762.112 | 22.71 | 44.8 | 27 | 54.7 | 2.49 $\pm$ 0.04 | -90.94 $\pm$ 0.03 | 4.47 $\pm$ 0.08 | 32
o-H${}_{2}^{13}$CO (4 3 2 – 3 3 1) | 284117.45 | 21.96 | 140.47 | 27 | 28.2 | 1.55 $\pm$ 0.05 | -92.1 $\pm$ 0.1 | 7.3 $\pm$ 0.3 | 21
o-H${}_{2}^{13}$CO (4 3 1 – 3 3 0) | 284120.62 | 21.96 | 140.47 | 27 | 28.2 | 1.1 $\pm$ 0.1 | -90.5 $\pm$ 0.4 | 6.3 $\pm$ 0.7 | 18
p-H${}_{2}^{13}$CO (4 2 2 – 3 2 1) | 284632.42 | 21.92 | 81.43 | 9 | 48.6 | 0.59 $\pm$ 0.02 | -91.54 $\pm$ 0.05 | 3.4 $\pm$ 0.1 | 17
o-H${}_{2}^{13}$CO (4 1 3 – 3 1 2) | 293126.515 | 21.29 | 47.01 | 27 | 66.4 | 2.25 $\pm$ 0.08 | -91.2 $\pm$ 0.07 | 4.2 $\pm$ 0.2 | 42
o-H${}_{2}^{13}$CO (5 1 5 – 4 1 4) | 343325.713 | 18.17 | 61.28 | 33 | 112 | 2.45 $\pm$ 0.09 | -90.93 $\pm$ 0.07 | 4.3 $\pm$ 0.2 | 30
p-H${}_{2}^{13}$CO (5 0 5 – 4 0 4) | 353811.872 | 17.64 | 51.02 | 11 | 127 | 1.54 $\pm$ 0.04 | -91.22 $\pm$ 0.05 | 3.8 $\pm$ 0.1 | 33
p-H${}_{2}^{13}$CO (5 2 4 – 4 2 3) | 354898.595 | 17.58 | 98.41 | 11 | 108 | 0.83 $\pm$ 0.05 | -91.3 $\pm$ 0.1 | 4.0 $\pm$ 0.3 | 34
o-H${}_{2}^{13}$CO (5 3 3 – 4 3 2) | 355190.9 | 17.57 | 157.52 | 33 | 82.5 | 1.39 $\pm$ 0.04 | -90.81 $\pm$ 0.08 | 5.9 $\pm$ 0.2 | 29
o-H${}_{2}^{13}$CO (5 3 2 – 4 3 1) | 355202.601 | 17.57 | 157.52 | 33 | 82.5 | 1.21 $\pm$ 0.03 | -90.79 $\pm$ 0.07 | 5.1 $\pm$ 0.2 | 32
p-H${}_{2}^{13}$CO (5 2 3 – 4 2 2) | 356176.243 | 17.52 | 98.52 | 11 | 109 | 1.02 $\pm$ 0.03 | -91.04 $\pm$ 0.08 | 5.2 $\pm$ 0.2 | 32
HDCO (3 0 3 – 2 0 2) | 192893.27 | 32.35 | 18.53 | 7 | 19.4 | 1.03 $\pm$ 0.02 | -90.73 $\pm$ 0.06 | 5.2 $\pm$ 0.1 | 21
HDCO (3 1 2 – 2 1 1) | 201341.35 | 30.99 | 27.29 | 7 | 19.6 | 0.956 $\pm$ 0.008 | -91.02 $\pm$ 0.02 | 5.25 $\pm$ 0.05 | 18
HDCO (4 1 4 – 3 1 3) | 246924.6 | 25.27 | 37.6 | 9 | 39.6 | 1.06 $\pm$ 0.04 | -90.73 $\pm$ 0.09 | 4.3 $\pm$ 0.2 | 47
HDCO (4 0 4 – 3 0 3) | 256585.43 | 24.32 | 30.85 | 9 | 47.4 | 1.25 $\pm$ 0.05 | -91.11 $\pm$ 0.08 | 4.6 $\pm$ 0.2 | 50
HDCO (4 1 3 – 3 1 2) | 268292.02 | 23.26 | 40.17 | 9 | 50.8 | 0.85 $\pm$ 0.03 | -91.06 $\pm$ 0.09 | 4.1 $\pm$ 0.2 | 37
HDCO (5 1 5 – 4 1 4) | 308418.2 | 20.23 | 52.4 | 11 | 80.8 | 1.54 $\pm$ 0.02 | -91.14 $\pm$ 0.02 | 4.72 $\pm$ 0.07 | 20
HDCO (5 0 5 – 4 0 4) | 319769.68 | 19.51 | 46.19 | 11 | 93.7 | 1.41 $\pm$ 0.03 | -91.31 $\pm$ 0.05 | 4.5 $\pm$ 0.1 | 37
HDCO (5 1 4 – 4 1 3) | 335096.739 | 18.62 | 56.25 | 11 | 104 | 1.43 $\pm$ 0.02 | -91.04 $\pm$ 0.04 | 4.76 $\pm$ 0.09 | 28
o-H2C18O (3 1 3 – 2 1 2) | 201614.256 | 30.95 | 31.22 | 21 | 19.7 | 0.473 $\pm$ 0.009 | -90.83 $\pm$ 0.05 | 4.8 $\pm$ 0.1 | 18
p-H2C18O (3 0 3 – 2 0 2) | 208006.441 | 29.99 | 19.97 | 7 | 24.4 | 0.134 $\pm$ 0.007 | -91.17 $\pm$ 0.09 | 3.4 $\pm$ 0.2 | 17
o-H2C18O (4 1 4 – 3 1 3) | 268745.789 | 23.22 | 44.12 | 27 | 51.1 | 0.76 $\pm$ 0.04 | -90.7 $\pm$ 0.1 | 4.9 $\pm$ 0.3 | 48
o-H2C18O (4 1 3 – 3 1 2) | 286293.96 | 21.79 | 46.23 | 27 | 61.8 | 0.64 $\pm$ 0.02 | -91.01 $\pm$ 0.06 | 4.3 $\pm$ 0.2 | 23
o-H2C18O (5 1 5 – 4 1 4) | 335816.025 | 18.58 | 60.24 | 33 | 105 | 0.85 $\pm$ 0.02 | -90.99 $\pm$ 0.06 | 4.8 $\pm$ 0.1 | 24

.

The isotopologues H${}_{2}^{12}$CO, H${}_{2}^{13}$CO, HDCO and H2C18O were
identified in the frequency interval $\sim$ 190–357 GHz through 35 transition
lines whose assignments, specifying the ortho or para character of the states
for the symmetric isotopologues (e.g. Clouthier & Ramsay 1983; Chapovsky
2001), spectroscopic data and Gaussian fit parameters are given in Table 1.
The spectral analysis was primarily performed on lines with significantly
stronger signals than the limit of detection (3$\sigma$ level), which are
exhibited in the different panels of Fig. 2. In agreement with the expected
isotopic abundances, the most and least abundant isotopologues,
H${}_{2}^{12}$CO and H2C18O, respectively, exhibited the strongest and weakest
line intensities, respectively. Despite H2C18O is a rare isotopologue (e.g.,
Müller & Lewen 2017), we could detect it via 5 spectral lines.

The main isotopologue (o, p)-H2CO was detected via 9 lines although the lines
at $\sim$ 291380.48 and 291384.26 MHz, corresponding to the transitions
43,2-33,1 and 43,1-33,0, respectively, can be considered partially resolved
(see Fig 2(a)). Various of the H2CO transitions detected in this work were
also reported in sources as OMC-1, Orion-KL and in high-mass star-forming
regions (e.g. Loren 1984; Wootten et al. 1984; Mangum et al. 1990; Taniguchi
et al. 2020).

13 spectral lines of (o, p)-H${}_{2}^{13}$CO were identified although two of
them at the rest frequencies $\sim$ 284117.45 and 284120.62 MHz were deemed as
partially resolved (see Fig 2(b)). As it is expected, the intensities of the
H${}_{2}^{13}$CO lines are lower than those of H2CO. By comparing the
integrated areas of neighbor lines, e.g., o-H${}_{2}^{13}$CO at $\sim$
219908.52 MHz and o-H2CO at $\sim$ 225697.77 MHz, the H2CO/H${}_{2}^{13}$CO
ratio is about 16. In previous works, Jewell et al. (1989) and Helmich & van
Dishoeck (1997) observed various of the H${}_{2}^{13}$CO transitions in G331.

The HDCO isotopologue was detected via 8 spectral lines (see Fig. 2(c)). It
can be noted that the HDCO line profiles exhibit a partial asymmetry with a
blue-shifted emission wing. In an investigation on HNCO in G331, Canelo et al.
(2021) not only observed similar spectral asymmetries but also found them to
be more pronounced in some specific $K$-ladder transitions of HNCO. These
findings were discussed in the context of molecular outflows (e.g. Canelo et
al. 2021 and references therein). In this work, we expect that the HDCO
emission may be linked to an expanded gas region influenced by the molecular
outflow. However, to better understand the emission of spectral tracers
potentially associated with the molecular outflow, it is crucial to conduct
further investigations, including the development of models that consider the
core and outflow of G331.

In addition, 5 spectral lines of (o, p)-H2C18O isotopologue were clearly
identified and shown in Fig. 2(d). From this study toward G331, we confirm the
observation of the transitions of the rare isotopologue H2C18O previously
reported toward other sources (e.g., Mangum et al. 1990; Sutton et al. 1995).

The successful detection of HDCO prompted further investigation into the
presence of the doubly deuterated formaldehyde (D2CO), which has been scarcely
observed in objects of the ISM (e.g., Turner 1990; Ceccarelli et al. 1998). As
a result, we report the tentative detection of p-D2CO 31,3–21,2 at the rest
frequency $\sim$ 166102.74 MHz previously detected in prestellar cores
(Bacmann et al., 2003). Fig. 3(a) shows this spectral line whose emission is
above the limit of detection although shifted from the source systemic
velocity ($V_{\rm lsr}=-90$ km s-1). This makes us think that a possible
candidate for this line might also be OC34S (14–13) at 166105.75 MHz. A second
tentative identification is displayed in Fig. 3b, the o-D2CO 50,5–40,4
transition was also tentatively identified but a dominant line, likely blended
with SO2 v=0 at $\sim$ 287485.44 MHz, avoided a clear identification.

Figure 3: Tentative identification of D2CO: a) p-D2CO 31,3–21,2 at the
frequency $\sim$ 166102.74 MHz and appearing at $V_{\rm lsr}\thickapprox-96$
km s-1; b) o-D2CO 50,5–40,4 at the frequency $\sim$ 287485.65 MHz but likely
dominated by emission of SO2 at 287485.44 MHz. Under the upper limit condition
$N$(D2CO)$<N$(HDCO), and FWHM=5 km s-1, the dashed lines indicate hypothetical
LTE models of D2CO.

####  3.1.2 The formyl cation and its isotopologues

The isotopologues of formyl cation (HCO+, H13CO+, DCO+, HC18O+ and HC17O+)
were also sought in G331. Eight lines from these isotopologues were
identified, except the deuterated one DCO+, of which their spectroscopic and
fitted parameters are given in Table 2. The two rotational lines of the main
isotopologue HCO+ and their fits are shown in Fig. 4. The HCO+ spectra
exhibited broad spectral wings, from $-$150 km s-1 to $-$30 km s-1, and they
were better described by Lorentzian than Gaussian functions. The Lorentzian
profile suggests that the HCO+ emission is likely affected by the molecular
outflow (Hervías-Caimapo et al., 2019).

Figure 4: Spectral lines (solid lines) and Lorentzian fits (dotted lines) of
the transitions 2–1 and 3–2 of HCO+ identified at the rest frequencies $\sim$
178375.05 MHz and 267557.62 MHz, respectively. Figure 5: Legend similar to
that of Fig. 2 but for the isotopologues of the formyl cation, with their LTE
models: a) H13CO+, b) HC18O+ and c) HC17O+.

The spectral lines of H13CO+, HC18O+ and HC17O+ did not exhibit Lorentzian
profiles with broad wings. Thus, Gaussian functions, instead of Lorentzian
ones, were used to fit the spectra. The spectra of these isotopologues are
exhibited in the different panels of Fig. 5. For all the identified HCO+
isotopologues, the $J$=2–1 transition was observed. Considering the velocity
integrated temperatures for the line profiles of this transition, the ratios
HCO+:H13CO+:HC18O+:HC17O${}^{+}\thickapprox$ 64:21:3:1 were obtained.
Nevertheless, the results based on LTE and non-LTE methods will be given in
the next section.

It is worth highlighting that the deuterated formyl cation DCO+ was not
detected in spite of the isotopologue HDCO was identified through several
lines. In contrast, the 17O isotopologue of formyl cation was detected but not
the isotopologue of formaldehyde H2C17O. Those aspects demand follow-up
studies due to their implications on the understanding of the isotopic
fraction and the evolution of protostellar objects.

####  3.1.3 Protonated formaldehyde

The search for H2COH+ was performed across the frequency range $\sim$ 159–356
GHz. As a result, the identified lines of H2COH+ are exhibited in Fig. 6 and
were adjusted by means of Gaussian functions. The spectroscopic and fit
parameters are listed in Table 2. In Fig. 6, the first three spectra were
identified without blended emission at the rest frequencies $\sim$ 190079.13,
207964.75 and 252870.34 MHz; on the fourth display, a tentative identification
was made at $\sim$348102.33 MHz, the detection is unclear due to a spectral
signature which dominates the whole spectrum, it is likely associated with
34SO2 at $\sim$ 348117 MHz (e.g. Jewell et al. 1989).

In this work, the detection of H2COH+ is reported for the first time in a hot
molecular core as G331. This cation was identified by Ohishi et al. (1996),
across the frequency range $\sim$ 31–174 GHz, toward massive star-forming
regions but not in cold and dark clouds. Bacmann et al. (2016) detected
H2COH+, across the frequency range $\sim$ 102–168 GHz, in a cold ($\sim$ 10 K)
source, the prestellar core L1689B. By means of a different technique of
observation, Meier et al. (1993) detected H2COH+ in the coma of comet P/Halley
through a Neutral Mass Spectrometer on the Giotto spacecraft.

Table 2: Spectral line analysis of the molecular ions HCO+, H13CO+, HC18O+,
HC17O+ and H2COH+.

Species and transitions | Frequency | Beamwidth | $E_{u}$ | $A_{ul}$ | Line area | $V_{\rm lsr}$ | Linewidth | rms
---|---|---|---|---|---|---|---|---
| [MHz] | [″] | [K] | [10-5s-1] | [K km s-1] | [km s-1] | [km s-1] | [mK]
HCO+ (2 – 1)a | 178375.056 | 34.98 | 12.84 | 40.2 | 20 $\pm$ 2 | -91.56 $\pm$ 0.02 | 6.97 $\pm$ 0.07 | 120
HCO+ (3 – 2)a | 267557.626 | 23.32 | 25.68 | 145 | 31 $\pm$ 3 | -91.21 $\pm$ 0.03 | 9.2 $\pm$ 0.1 | 133
H13CO+ (2 – 1) | 173506.700 | 35.96 | 12.49 | 37 | 6.41 $\pm$ 0.02 | -89.631 $\pm$ 0.006 | 4.93 $\pm$ 0.01 | 13
H13CO+ (3 – 2) | 260255.339 | 23.97 | 24.98 | 134 | 7.05 $\pm$ 0.05 | -89.94 $\pm$ 0.02 | 5.14 $\pm$ 0.04 | 53
H13CO+ (4 – 3) | 346998.344 | 17.98 | 41.63 | 329 | 6.25 $\pm$ 0.02 | -90.205 $\pm$ 0.007 | 5.19 $\pm$ 0.02 | 22
HC18O+ (2 – 1) | 170322.626 | 36.63 | 12.26 | 35 | 1.05 $\pm$ 0.01 | -89.69 $\pm$ 0.03 | 4.37 $\pm$ 0.07 | 11
HC18O+ (3 – 2) | 255479.389 | 24.42 | 24.52 | 127 | 1.14 $\pm$ 0.04 | -89.86 $\pm$ 0.08 | 4.5 $\pm$ 0.2 | 44
HC17O+ (2 – 1) | 174113.169 | 35.83 | 12.53 | 37.4 | 0.31 $\pm$ 0.01 | -89.5 $\pm$ 0.1 | 5.1 $\pm$ 0.3 | 11
H2COH+ (3 0 3 – 2 0 2) | 190079.131 | 32.82 | 18.26 | 6.59 | 0.16 $\pm$ 0.01 | -89.9 $\pm$ 0.2 | 4.5 $\pm$ 0.5 | 12
H2COH+ (5 1 4 – 5 0 5) | 207964.754 | 30.00 | 55.51 | 14.7 | 0.12 $\pm$ 0.01 | -91.6 $\pm$ 0.2 | 5.6 $\pm$ 0.8 | 17
H2COH+ (4 0 4 – 3 0 3) | 252870.339 | 24.67 | 30.39 | 16.1 | 0.23 $\pm$ 0.05 | -90.7 $\pm$ 0.3 | 3.2 $\pm$ 0.9 | 44
H2COH+ (10 1 9 – 10 0 10)b | 348102.330 | 17.92 | 181.67 | 48.2 | 0.4 $\pm$ 0.1 | -91.6 $\pm$ 0.4 | 7 $\pm$ 2 | 30

a)The line fit parameters of the HCO+ lines were obtained using Lorentzian
functions. b)Line likely blended with 34SO2.

Figure 6: Legend similar to Fig. 2 but for the spectral lines of H2COH+ with
their LTE models (dashed lines). In the fourth panel, the Gaussian fit depicts
an H2COH+ transition tentatively identified at $\sim$ 348102.330 MHz. The
asterisk indicates a line likely blended with 34SO2 at $\sim$ 348117 MHz.

Figure 7: Population diagrams of the formaldehyde isotopologues a)
H${}_{2}^{13}$CO, b) HDCO and c) H2C18O; and of the molecular ions d) H13CO+
and e) H2COH+. The solid-line and shaded areas represent the linear regression
fits and their (95%) confidence bands for each species.

###  3.2 Physical conditions

The temperature and the column densities of the molecular species were
estimated using LTE and Non-LTE conditions. LTE methods were applied to
analyze the physical conditions of all the detected isotopologues of
formaldehyde, the formyl cation and the protonated formaldehyde (H2COH+). Non-
LTE hypotheses were considered to infer the physical conditions of the 12C
isotopologues, H2CO and HCO+. All the analyses were performed under the
general assumption of a source size of 5″. In the particular case of HCO+, a
second solution adopting 15″was tested too.

When the LTE approximation is assumed, the population diagram method was used
to estimate the excitation conditions of the molecular species, i.e., the
total column densities ($N$) and excitation temperatures ($T_{\rm exc}$) by
means of

$\ln\frac{N_{u}}{g_{u}}=\ln\frac{N}{Q}-\frac{E_{u}}{kT_{\rm exc}},$ (2)

where $N_{u}$, $g_{u}$, $E_{u}$ and $Q$ are the column density, the degeneracy
of the upper state, the energy of the upper level involved in the transition
and the partition function, respectively. In Table 1, the upper state
degeneracy $g_{u}=(2J_{u}+1)g_{ns}^{(u)}$ was given for each detected
transition, where $J_{u}$ and $g_{ns}^{(u)}$ are the rotational angular
momentum and the nuclear spin statistical weight of the upper state (Bunker &
Jensen, 1989). The optical depth ($\tau$) has been inferred using the
population diagram method. For a given transition, this can be estimated using
the following expression

$\tau=\frac{c^{3}A_{ul}N_{u}}{8\pi\nu^{3}\Delta v\sqrt{\pi}/2\sqrt{\ln
2}}[\exp(h\nu/kT_{\rm exc})-1],$ (3)

where $\Delta v$ and $A_{ul}$ represent the spectral linewidth (km s-1) and
the Einstein coefficients (s-1), respectively (Goldsmith & Langer, 1999;
Mangum & Shirley, 2015; Vastel et al., 2015).

####  3.2.1 Formaldehyde isotopologues: LTE analyses

The population diagrams of the 13C, D and 18O isotopologues of formaldehyde
have been obtained with updated values of the partition function (see A) and
Eq. (2) can be corrected taking into account the optical depth ($\tau$)
applied when the gas emission tends to be optically thick, and beam dilution
factors, which allows us to constrain the size of the emitting region with
respect to the antenna beam (Goldsmith & Langer, 1999). The population
diagrams of H${}_{2}^{13}$CO, HDCO and H2C18O are exhibited in Fig. 7 (panels
a, b and c, respectively) and were obtained after applying a beam dilution
correction for an emitting source assumed to be 5″ and observed with an
antenna beam ranged from $\sim$ 17 to 39″.

H${}_{2}^{13}$CO: Its population diagram was plotted using the 13 detected
lines corresponding to transitions in the $E_{u}\thickapprox 31-158$ K range
(Fig. 7a). The linear fit provided the result $T_{\rm exc}=82.9\pm 21.8$ K and
$N=(8.0\pm 2.0)\times 10^{14}$ cm-2 ($\chi^{2}_{\rm red}$=1.53). From this
result, synthetic spectra were computed and compared with the observed lines
in Fig. 2. The values of the optical depth were estimated $\tau\lesssim 1$ for
all the spectra, except for the partially blended lines at $\sim$ 284117.45
and 284120.62 MHz, in agreement with the LTE formalism.

In comparison with previous works, Schöier et al. (2002) carried out radiative
analyses of H${}_{2}^{13}$CO based on transitions in the $E_{u}\thickapprox
21-99$ K range toward the low-mass protostellar object IRAS 16293-2422. They
reported an excitation temperature around 90 K.

HDCO: Based on its 8 spectral lines, detected in the range $E_{u}\thickapprox$
18–56 K, the population diagram is displayed in Fig. 7(b). From the linear
fit, it was obtained $T_{\rm exc}$ = 20.8 $\pm$ 3.5 K and $N$ = (2.5 $\pm$
0.8) $\times$ 1014 cm-2 ($\chi^{2}_{\rm red}$=0.84). The observed spectral
lines and the simulated ones obtained from this result are exhibited in Fig.
2, the values of the optical depth were estimated $\tau\lesssim 1$.

Neill et al. (2013) performed a study about deuterated molecules in Orion KL.
They discussed for H${}_{2}^{13}$CO and HDCO different LTE scenarios with
temperatures of $\sim$ 40, 63, and 67 K as well as particular aspects about
the deprotonation of H2COH+ and HDCOH+. Bianchi et al. (2017) analyzed several
deuterated species toward the Class I protostar SVS13-A. From the population
diagrams of H${}_{2}^{13}$CO, HDCO and D2CO, with source sizes of $\sim$ 10″,
they derived temperatures of $\sim$ 23, 15, and 28 K, respectively.

D2CO: LTE calculations were carried out to estimate an upper limit on the D2CO
emission. Taking into account the excitation conditions of HDCO, the D2CO
column density was computed assuming that $N$(D2CO)
$<N$(HDCO)$\thickapprox$2.5$\times$1014 cm-2 and $T_{\rm exc}\thickapprox$ 21
K. The best solution gave $N$(D2CO) $\thickapprox$ 1.3 $\times$ 1014 cm-2
($\chi^{2}_{\rm red}$=0.8) providing a column density ratio of D2CO/HDCO
$\thickapprox$ 0.5.

D2CO has been observed in only few sources. In Orion KL, Neill et al. (2013)
established an upper limit on the D2CO abundance of [D2CO]/[HDCO]$\leq$0.1.
Turner (1990) detected three transitions of D2CO in Orion KL reporting
[D2CO]/[HDCO]=0.02. In IRAS 16293-2422, a source with several studies on
molecular deuteration, Ceccarelli et al. (1998) found [D2CO]/[HDCO]$\leq$0.5.

H2C18O: The population diagram of this isotopologue was obtained from 5
detected transition lines which covered the $E_{u}\thickapprox$ 19–60 K range.
The linear fit provided $T_{\rm exc}$ = 30.7 $\pm$ 10.3 K and $N$ = (6.8 $\pm$
3.2) $\times$ 1013 cm-2 ($\chi^{2}_{\rm red}$=1.23). From this solution,
synthetic spectra were simulated and compared with the observed lines (Fig.
2). The spectral lines of H2C18O are optically thin, as expected according to
its low abundance. It is worth mentioning that the population diagrams of HDCO
and H2C18O have similar $E_{u}$ ranges, between $\sim~{}18-60$ K, different
from those of H${}_{2}^{13}$CO within $E_{u}\sim$ 30–160 K. The excitation
temperatures of HDCO and H2C18O ($T_{\rm exc}\lesssim$ 30 K) are lower than
the obtained one for H${}_{2}^{13}$CO ($T_{\rm exc}\thickapprox$ 83 K). On the
one hand, such difference might come from the fact that more lines of
H${}_{2}^{13}$CO were observed than HDCO and H2C18O, providing a more extended
population diagram. On the other hand, there might be a scenario in which the
formaldehyde isotopologues trace different excitation conditions although, to
assume this hypothesis would require further studies including, e.g., gas-
grain calculations and chemical models.

####  3.2.2 H2CO: Non-LTE calculations

The physical conditions of H2CO were calculated using the non-LTE code RADEX,
which provides an alternative approach to the population diagram based on the
assumption of optically thin emission lines. Within this approach, the optical
depth effects are treated with an escape probability method assuming an
isothermal and homogeneous medium without large-scale velocity fields (van der
Tak et al., 2007). The non-LTE calculations were performed using available
excitation rates between o- and p-H2CO and o- and p-H2. Wiesenfeld & Faure
(2013) reported calculations for the first 81 rotational levels of
formaldehyde considering temperatures between $\sim$ 10 and 300 K. In an
earlier work, Troscompt et al. (2009) also discussed the rotational excitation
of H2CO by H2.

From the five lines of p-H2CO, the best solution ($\chi^{2}_{\rm red}$=4.5)
provided $T_{\rm kin}=87.4\pm 0.6$ K and $N=(6.2\pm 0.3)\times 10^{15}$ cm-2.
From the four lines of o-H2CO, it was obtained $T_{\rm kin}=95.1\pm 0.3$ K and
$N=(9\pm 3)\times 10^{15}$ cm-2 ($\chi^{2}_{\rm red}$=3.5). These results
provide a ortho-to-para ratio of H2CO of $\sim$ 1.4. The H2 density was also
estimated from the calculations $n({\rm H_{2}})\simeq 4\times 10^{7}$ cm-3
adopting a source size of 5″. The density of molecular hydrogen was computed
using conventional ortho-to-para ratios, whose value can fluctuate between
$\sim$0.1 and 3, contingent on the chemistry, the physical properties and the
evolutionary stage of sources. In prestellar cores, such a range was discussed
from 3 to values even smaller than 0.001 (e.g. Osterbrock 1962; Lacy et al.
1994; Pagani et al. 2013).

Synthetic lines were generated from the non-LTE approximations and compared
with the observations in Fig. 2(a). Concerning the optical depth of the p-H2CO
transitions, the highest and lowest values were $\sim$ 4.5 and 0.9 for the
lines at $\sim$ 290623.40 MHz and 218760.06 MHz, respectively. Similarly, for
the o-H2CO transitions, the highest and lowest optical depths were $\sim$ 5.2
and $\sim$ 0.7 for the transitions at $\sim$ 351768.64 MHz and 291380 MHz,
respectively.

A comparison of our results with other works is useful to show they are in
accordance. Schöier et al. (2002) analysed the physical conditions of H2CO in
IRAS 16293-2422. They estimated temperatures around 90 K and ortho-to-para
ratios of H2CO around 0.9. Guzmán et al. (2013) used non-LTE models of o- and
p-H2CO based on observations with the IRAM-30m toward the Horsehead photo-
dissociation region (PDR) to estimate ortho-to-para ratios around 3 at the
dense core and 2 in the PDR. In a pioneering work, Mangum et al. (1990)
stressed the importance of H2CO as a key tracer to estimate density,
temperature and molecular abundances. They performed LTE and non-LTE
calculations obtaining $n$(H2)=(0.5–1)$\times$107 cm-3 and $T_{\rm
kin}\thickapprox$100 K in Orion KL. In this work we detected various lines of
formaldehyde that were identified in Orion KL by Mangum et al. (1990).

####  3.2.3 Formyl cation isotopologues: LTE analyses

The population diagram obtained for H13CO+ using the three detected lines is
exhibited in Fig. 7(d). Assuming that the emitting region is 5″, the
excitation temperature and the column density were estimated, $T_{\rm exc}$ =
11 $\pm$ 1 K and $N$ = (2.1 $\pm$ 0.7) $\times$ 1014 cm-2 ($\chi^{2}_{\rm
red}$=1.30), respectively. In the case of HC18O+ and HC17O+, no population
diagram could be displayed because of the lack of enough detected lines.
Nevertheless, LTE calculations were performed to estimate their column
densities. Similar methodologies were discussed in Schöier et al. (2002), who
performed LTE and non-LTE calculations for HCO+ and its isotopologues in IRAS
16293-2422.

In this work the LTE calculations of HC18O+ and HC17O+ were performed fixing
the excitation temperature to the value obtained for H13CO+ and considering
the column densities as free parameters with the following condition
$N$(HC17O+) $<N$(HC18O+) $<N$(H13CO+). The estimates of the column densities
were $N$(HC18O+) $\thickapprox$ 1.8 $\times$1013 cm-2 ($\chi^{2}_{\rm
red}$=1.5) and $N$(HC17O+) $\thickapprox$ 1.4 $\times$1013 cm-2
($\chi^{2}_{\rm red}$=0.3). Synthetic spectra were included in the panels of
Fig. 5.

The low temperature estimated from the H13CO+ lines suggests that its emission
is likely associated with a cold and expanded region. In addition, we also
obtained results assuming an extended source size of 15″. Thus, the H13CO+
population diagram gave the values $T_{\rm exc}$ = 12 $\pm$ 2 K and $N$ = (2.6
$\pm$ 0.8) $\times$ 1013 cm-2, and for the 18O and 17O isotopologues
$N$(HC18O+) $\thickapprox$ 4.2 $\times$1012 cm-2 and $N$(HC17O+)
$\thickapprox$ 1.5 $\times$1012 cm-2.

####  3.2.4 HCO+: Non-LTE calculations

The main isotopologue HCO+ was observed through two intense and optically
thick lines (Fig. 4), they were analyzed via non-LTE calculations using
collisional excitation rates. The two transitions 2–1 and 3–2 were computed
using RADEX and the rate coefficients available in the LAMDA database (Flower,
1999) assuming $n$(H2)=(0.1–5)$\times$107 cm-3, $T<$ 30 K and 5″. The column
density resulted in $N$(HCO+) $\thickapprox$ 3 $\times$ 1015 cm-2
($\chi^{2}_{\rm red}$=6.5). In the literature, line analysis of HCO+ has been
reported with high reduced values of $\chi^{2}$ (Schöier et al., 2002). The
result reported in this work should be taken as a rough approximation.

####  3.2.5 Protonated formaldehyde: LTE analysis

The 3 unblended lines of H2COH+ allowed us to obtain the population diagram
shown in Fig. 7(e) and the estimates $T_{\rm exc}$ = 25 $\pm$ 4 K and $N$ =
(1.4 $\pm$ 0.3) $\times$ 1014 cm-2 ($\chi^{2}_{\rm red}$=0.41) assuming a
source size of 5″. Fig. 6 exhibits a comparison between the simulated and the
observed lines with the exception of the spectrum affected by contaminant
emission.

In comparison with other works, Ohishi et al. (1996) reported $T_{\rm
exc}=60-110$ K and $N$(H2COH+)$\thickapprox$1012–1014 cm-2 in surveys toward
Sgr B2, Orion KL and W51. In W51, since they could not detect a sufficient
number of lines of H2COH+, they used HCO+ to infer the physical conditions of
H2COH+ taking different excitation temperatures. In the ultra cold ($T\sim$10
K) source L1689B, Bacmann et al. (2016) estimated column densities between 3
$\times$ 1011 cm-2 and 1 $\times$ 1012 cm-2. They discussed that the H2COH+
formation can occur via:

$\text{H_{2}CO\ +\ x^{+}\longrightarrow H_{2}COH^{+}+H_{2}},$ (4)

where x+ represents a proton donor such as H${}_{3}^{+}$. Furthermore, in the
context of Eq. 1, the specific proton donor is HCO+ obtaining CO as a by-
product instead of H2. They predicted the abundance ratios
[H2COH${}^{+}]\thickapprox$ 0.007[H2CO], when H${}_{3}^{+}$ is the proton
donor, and [H2COH${}^{+}]\thickapprox$ 0.003 [H2CO], when it is HCO+.

In summary, we present in Table 3 the results obtained from the LTE and non-
LTE analyses of the isotopologues of formaldehyde and the formyl. The results
suggest that these species might trace different gas components. From the
estimated column densities of the formaldehyde isotopologues, it was obtained
the abundance ratios
H2CO:H${}_{2}^{13}$CO:HDCO:H2C18O$\thickapprox$223:12:4:1. In addition, Table
3 also presents the results obtained from the population diagram of H2COH+.

Table 3: Summary of the LTE and non-LTE analyses of the isotopologues of
formaldehyde and the formyl cation, and the protonated formaldehyde. The
temperatures and column densities come from the results obtained at a source
size of 5″.

Species | # Analysed | Method |  $N$ [cm-2] |  $T$ [K]
---|---|---|---|---
| lines |  | $\thickapprox$ | $\thickapprox$
H2CO | 9 | non-LTEa | 1.52 $\times$ 1016 | 91
H${}_{2}^{13}$CO | 13 | LTEb | 8.0 $\times$ 1014 | 83
HDCO | 8 | LTEb | 2.5 $\times$ 1014 | 21
H2C18O | 5 | LTEb | 6.8 $\times$ 1013 | 31
D2CO | 2 | LTEc |  $<$1.3 $\times$ 1014 | 21
HCO+ | 2 | non-LTE | 3$\times$1015 |  $<$30
H13CO+ | 3 | LTEb | 2.1$\times$1014 | 11
HC18O+ | 2 | LTEd | 1.8$\times$1013 | 11
HC17O+ | 1 | LTEd | 1.4$\times$1013 | 11
H2COH+ | 3 | LTEb | 1.4 $\times$ 1014 | 25

a) From the o-H2CO and p-H2CO results, the column density and temperature are
the sum and mean value, respectively. b) Obtained from the population diagram
analysis. c) Upper limit based on the HDCO analysis. d) Obtained from the LTE
calculation.

##  4 Discussion

###  4.1 Fractional abundances

The fractional abundances ([$X$]) were estimated with respect to molecular
hydrogen using [$X$]=$N_{i}$/$N_{\text{H$_{2}$}}$, where $N_{i}$ and
$N_{\text{H$_{2}$}}$ represent the column density of the species $i$ and H2,
respectively. The H2 column density was indirectly inferred from H13CO+ by
means of the ratio H13CO+/H2 = 3.3 $\times$ 10-11 (Blake et al., 1987;
Sánchez-Monge et al., 2013; Merello et al., 2013). Using that ratio and the
results obtained from the population diagrams of H13CO+, considering source
sizes of 5 and 15″, an interval of H2 column densities was estimated as
$N_{\text{H$_{2}$}}=(0.8-6)\times 10^{24}$ cm-2. In the literature, similar H2
column densities have been discussed in the context of high-mass star-forming
regions (e.g. Yu et al. 2018; Motte et al. 2018). In addition, values of the
order of $N_{\text{H$_{2}$}}\thickapprox 10^{24}$ cm-2 have been used in Orion
KL (e.g. Crockett et al. 2014). In Table 4, we summarize the abundances of the
main isotopologues H2CO, HCO+ and H2COH+ obtained with respect to the
estimated $N_{\text{H$_{2}$}}$ interval and are compared with values reported
for the sources NGC 7129 FIRS 2, Orion KL, Sgr B2 and W51, which are known
sources for exhibiting a rich chemistry in simple and complex organic
molecules (Blake et al., 1986; Ohishi et al., 1996; Fuente et al., 2014;
Crockett et al., 2014).

Table 4: Fractional abundances of H2CO, HCO+ and H2COH+ in G331 in comparison
with values reported in other sources.

Species | Fractional abundances
---|---
| This work | Other works | Ref.
H2CO | (0.2–2) $\times$10-8 | (2–8) $\times$ 10-8 | (1)
HCO+ | (0.5–4) $\times$10-9 | 2.3 $\times$ 10-9 | (2)
H2COH+ | (0.2–2) $\times$10-10 | (0.01–1) $\times$10-9 | (3)

(1) NGC 7129 FIRS 2 (Fuente et al., 2014). (2) OMC-1, extended ridge (Blake et
al., 1987). (3) Sgr B2, Orion KL, and W51 (Ohishi et al., 1996).

###  4.2 Chemical modelling

In the gas phase, there are several ion-molecule reactions that can explain
the H2COH+ formation. In Eq. (1), one of the most important mechanism
involving the chemical species H2CO and HCO+ is described (Tanner et al.,
1979; Woon & Herbst, 2009). Concerning surface reactions, Song & Kästner
(2017) performed calculations about the hydrogenation of H2CO in amorphous
solid water surfaces and found some implications about the protonation of CH3O
isomers. H2COH+ is a major product from the ionization and fragmentation of
CH3OH, CH3CH2OH and CH3OCH3 (Mosley et al., 2012).

We observed that the reaction between H2CO and HCO+ is one of the most
important channels to produce H2COH+, as well as the general scheme described
in Eq. 4. In addition, it is observed that other cations can also react with
H2CO to produce H2COH+, for instance:

$\displaystyle\text{H$_{2}$CO+H$_{3}^{+}$}$ $\displaystyle\longrightarrow$
$\displaystyle\text{H$_{2}$COH$^{+}$+H$_{2}$,}$ (5)
$\displaystyle\text{H$_{2}$CO+H$_{3}$O$^{+}$}$ $\displaystyle\longrightarrow$
$\displaystyle\text{H$_{2}$COH$^{+}$+H$_{2}$O,}$ (6)
$\displaystyle\text{H$_{2}$CO+N$_{2}$H$^{+}$}$ $\displaystyle\longrightarrow$
$\displaystyle\text{H$_{2}$COH$^{+}$+N$_{2}$.}$ (7)

and the major reactions of destruction are

$\displaystyle\text{HCO$^{+}$+e$^{-}$}$ $\displaystyle\longrightarrow$
$\displaystyle\text{CO+H,}$ (8) $\displaystyle\text{H$_{2}$COH$^{+}$+e$^{-}$}$
$\displaystyle\longrightarrow$ $\displaystyle\text{HCO+2H,}$ (9)
$\displaystyle\text{H$_{2}$COH$^{+}$+e$^{-}$}$ $\displaystyle\longrightarrow$
$\displaystyle\text{CO+H+H$_{2}$.}$ (10)

In this work, we performed a gas-grain model to study the chemistry of H2CO,
HCO+ and H2COH+ (§ 2.1). As a result, the evolution of the gas abundances were
simulated for these species. Therefore, we adopted the initial chemical
abundances of Vidal & Wakelam (2018) (and references therein), but considering
the physical parameters obtained in this work from the line observations of
H2CO, HCO+ and H2COH+ in G331. The model was computed using different values
of gas temperature and density, assuming the ranges $T$=10–90 K and
$n_{\text{H}_{2}}$ = (0.05–1) $\times$ 107 cm-3, respectively, and using a
visual extinction of $A_{V}=10$ mag and a cosmic ray ionization rate of
$\zeta$=1.3$\times$10-17 cm-3. The model that yielded the most accurate
abundance prediction, consistent with the observed abundances within an order
of magnitude, was obtained at $T$=30 K and $n_{\text{H}_{2}}=1\times$ 106
cm-3. However, at temperatures and densities above these values, the model’s
chemistry become unpredictable. The result is exhibited in Fig. 8, where we
marked the mean values of the fractional abundances of H2CO, HCO+ and H2COH+
(given in Table 4) with horizontal lines (black dashed, blue dash-dotted and
red dotted respectively). Here it is observed that H2CO reaches its maximum
abundance at $\sim$ 103 yr (the time scale $\sim$ 103 yr is pointed by a
vertical line) in agreement with the results shown in Table 4. This timescale
would be optimal for the formation of neutral species in G331. Furthermore, at
the time scale of $\sim$ 102–103 yr, the predicted abundances of HCO+ and
H2COH+ are in agreement with the observational mean values, in line with the
general assumption of massive protostellar objects (i.e., young sources with
an active chemistry with abundant molecular emission). These agreements make
us develop the hypothesis that the lifetime of the source G331 might be $<$104
yr.

To explore the effects of the major reactions of destruction, we carry out a
second simulation shown in Fig. 8. While Fig. 8(a) represents a model computed
with the whole network of chemical reactions, Fig. 8(b) depicts a model that
excludes the destruction reactions described in Eq. 8, 9, and 10. In general,
the major destructive reactions reduce noticeably the abundances of the
molecular ions, but the abundance of neutral H2CO remains relatively stable.
In addition, in the models of Fig. 8(a) and Fig. 8(b), it is observed that
H2COH+ follows the abundance curves of HCO+ and H2CO, respectively. In the
case of Fig. 8(a), HCO+ is a major precursor of H2COH+. Thus, changes in the
abundance of HCO+ affect the production of H2COH+. In the case of Fig. 8(b),
the increase of the abundance of HCO+, which reacts with H2CO to form H2COH+
(see Eq. 1), make rise the abundance of H2COH+.

The dissociative recombination of molecular ions with electrons is one of the
most complex, destructive and least understood mechanism in the ISM (Hamberg
et al., 2007; Meier et al., 1993). Hamberg et al. (2007) carried out
experiments to measure the cross-sections and branching ratios of various
protonated and deuterated molecular ions. Meier et al. (1993) detected H2COH+
in the coma of the comet P/Halley and performed an ion chemical model to
estimate the H2CO production from H2COH+. The chemical models presented in
this work provide a preliminary framework for understanding the chemistry of
molecular ions in cold regions of G331.

Figure 8: Predicted abundances of H2CO, HCO+ and H2COH+ as a function of time,
represented by the black, blue and red solid lines, respectively, obtained
from a model with $T$=30 K and $n_{\text{H}_{2}}$=1 $\times$ 106 cm-3. Panel
a) represents a model with the whole chemical network whereas panel b) does
not include the major destruction channels of HCO+ and H2COH+. The dashed,
dash-dotted and dotted horizontal lines indicate the mean fractional
abundances observed in G331 for H2CO, HCO+ and H2COH+, respectively. The
vertical arrow depicts the abundance peak of H2CO at $t\thickapprox 10^{3}$
yr.

###  4.3 Isotopic fractionation

D/H ratios: The ratio was calculated using the neutral isotopologues H2CO and
HDCO, while molecular ions HCO+ and DCO+ were not used due to the non
detection of DCO+ in G331. From the LTE and non-LTE analyses of HDCO and H2CO,
respectively, it is reported HDCO/H2CO $\sim$ 0.02 for a bulk gas with $T<$90
K and, from the D2CO tentative detection, it was estimated D2CO/H2CO $\leq$
0.009. Roberts & Millar (2007) estimated HDCO/H2CO $\sim$ 0.01, 0.03, and
$<$0.002 in G34.26, G75.78 and G31.41, respectively. In addition, for the
G34.26 and G75.78 sources, they also reported D2CO/H2CO $<$ 0.001 and $<$0.01,
respectively, for gas components within a broad distribution of temperatures
($T<$ 100 K).

The HDCO detection demonstrated clear evidences of deuteration fractionation
in G331. In previous studies of G331 conducted with APEX, deuterated species
were searched but no conclusive results were obtained (Mendoza et al., 2018;
Duronea et al., 2019; Canelo et al., 2021; Santos et al., 2022).

12C/13C ratios: The H2CO/H${}_{2}^{13}$CO and HCO+/H13CO+ ratios were
estimated in G331 within $\sim$ 14–20. Wilson & Rood (1994) analyzed isotopic
carbon ratios from CO and H2CO reporting values of $\sim$ 20 and 77 for the
Galactic center and Local ISM, respectively. Yan et al. (2019) estimated
12C/13C ratios from formaldehyde. From 112 observations, the transition
(11,0–11,1) of H2CO was detected in 84 sources and from these 84,
H${}_{2}^{13}$CO (11,0–11,1) was detected in 38 sources (the entire
description is in § 2 of Yan et al. 2019). They found 12C/13C ratio values up
to 99 for the source G49.21-0.35, around 32 for G31.41+0.31, and $\sim$ 16–47
for W31.

16O/18O and 18O/17O ratios: The H2C16O/H2C18O and HC16O+/HC18O+ ratios
provided a rough estimate between $\sim$ 170–250. In order to estimate the
18O/17O ratio, from the integrated areas of the lines HC17O+ (2–1) and HC18O+
(2–1), the ratio HC18O+/HC17O${}^{+}\sim$ 3 is obtained, whose value
represents an upper limit compared with the abundance ratio
HC18O+/HC17O${}^{+}\sim$ 1.3, obtained for G331 from the LTE analysis. In the
Galactic center and local ISM, Wilson & Rood (1994) obtained ratios of 16O/18O
$\sim$ 250 and 560, respectively, as well as 18O/17O $\sim$ 3.2 for both of
them. Persson et al. (2018) claimed the first detection of H2C17O and
D${}_{2}^{13}$CO, among other isotopologues, in IRAS 16293-2422 B and reported
the values H2C16O/H2C18O$\sim$ 800, H2C16O/H2C17O$\sim$ 2596 and
H2C18O/H2C17O$\sim$ 3.2.

In summary, we present in Table 5 the isotopic ratios estimated for G331 which
are in good agreement compared to the values reported from Giant Molecular
Clouds (Orion, Sgr B2) and hot molecular cores as G31.41 and G34.26 (Guelin et
al., 1982; Turner, 1990; Mangum et al., 1990; Roberts & Millar, 2007; Neill et
al., 2013; Yan et al., 2019).

Table 5: Ratios between the isotopologues of H2CO and HCO+ estimated for G331
which are compared with the values reported in Giant Molecular Clouds (GMCs),
i.e., Orion and Sgr (B2), and hot molecular cores (HMCs), i.e., G31.41 and
G34.26.

Ratios | G331 | GMCs | G31.41 | G34.26
---|---|---|---|---
Formaldehyde |  |  |  |
H${}_{2}^{13}$CO/H2CO | 0.05 | 0.035a | 0.03e | 0.02e
HDCO/H2CO | 0.02 |  $\leqslant$ 0.005b |  $<$ 0.002f | 0.01f
D2CO/HDCO |  $\lesssim$0.5 | 0.02c | – |  $<$ 0.001f
H2C18O/H2CO | 0.004 | 0.003a | – | –
Formyl cation |  |  |  |
HCO+/H13CO+ | 14 | 21.2d | – | –
HC18O+/HC17O+ | 1.3 | 3.1d | – | –

a)Orion KL compact ridge (Mangum et al., 1990). b) Orion KL Hot Core (Neill et
al., 2013). c) Orion KL Compact Ridge (Turner, 1990). d) Sgr B2 (Stark, 1981;
Guelin et al., 1982). e) HMCs (Yan et al., 2019). f) HMCs (Roberts & Millar,
2007).

##  5 Conclusions and perspectives

We have identified isotopologues of formaldehyde, formyl cation and protonated
formaldehyde in G331, a hot molecular core and outflow system. The search for
those species were carried out using spectral setups collected with the APEX
telescope in the frequency range $\sim$ 159–356 GHz. A time dependent chemical
model, using the gas grain code Nautilus, was performed to study the chemistry
and predict the abundances of H2CO, HCO+ and H2COH+ as a function of time. The
conclusions and perspectives are:

  * •

Formaldehyde is an abundant molecule in G331. Several lines of H2CO,
H${}_{2}^{13}$CO, HDCO and H2C18O were detected in G331, 35 spectral lines in
total. D2CO was tentatively detected and was used to set an upper threshold
for its abundance. From non-LTE calculations, the o- and p-H2CO spectral lines
provided kinetic temperatures between $\sim$ 87–95 K. From the population
diagrams assuming the LTE approximation, the spectral lines of
H${}_{2}^{13}$CO, HDCO and H2C18O provided excitation temperatures of $\sim$
83, 21 and 31 K. From the estimates of column densities considering a source
size of 5″, the ratios
H2CO:H${}_{2}^{13}$CO:HDCO:H2C18O$\thickapprox$223:12:4:1 were obtained. The
formaldehyde isotopologues can trace different physical conditions, from cold
gas to lukewarm temperatures. These results agree with sublimation processes
and the active gas-grain chemistry expected in hot molecular cores.

  * •

The formyl cation is also an abundant species in G331. In total, 8 rotational
lines of HCO+, H13CO+, HC18O+ and HC17O+ were identified in G331. In contrast
with formaldehyde, the 17O isotopologue, HC17O+, was detected but not the
deuterated one, DCO+. Assuming the LTE approximation, the population diagram
of H13CO+ provided a temperature and column density $T_{\rm exc}\thickapprox$
11 K and $N$(H13CO+) $\thickapprox$ 2.1 $\times$ 1014 cm-2, whose values were
used to estimate the column densities of HC18O+ and HC17O+. From non-LTE
calculations, the spectral lines of HCO+ provided $N$(HCO+) $\thickapprox
3\times$1015 cm-2 assuming a temperature of about 30 K.

  * •

The protonated formaldehyde, H2COH+, should be present under interstellar
conditions when H2CO and HCO+ are detected (see Eq. 1). In fact, this cation
was also detected through four lines, of which one of them is likely blended
with 34SO2. From the population diagram considering the LTE formalism, it was
obtained $T_{\rm exc}\thickapprox$ 25 K and $N$(H2COH+) $\thickapprox$ 1.4
$\times$ 1014 cm-2 adopting a source size of 5″. H2CO and HCO+ can produce
H2COH+ under interstellar conditions. In addition, these species could play a
key role in the formation of complex organic molecules.

  * •

A gas grain chemical model was performed to predict the fractional abundances
of H2CO, HCO+ and H2COH+ and to study their evolution. The best model was
obtained adopting $T=$30 K and $n_{\text{H}_{2}}=1\times$ 106 cm-3. In a young
molecular stage of $\sim$ 103 yr, the H2CO, HCO+ and H2COH+ abundances reached
comparable values to those derived from the observations, [H2CO] = (0.2–2)
$\times$10-8, [HCO+] = (0.5–4) $\times$10-9 and [H2COH+] = (0.2–2)
$\times$10-10. The reaction between H2CO and HCO+ is one of the major channels
to produce H2COH+. On the other hand, it was noticed that dissociative
recombination mechanisms with electrons can rapidly destroy HCO+ and H2COH+
affecting their predicted abundances. The results obtained with the chemical
modeling of the three molecular species makes us to develop a hypothesis that
the evolution stage of G331 is around $10^{3}$ yr. Further studies in G331
should be carried out to confirm this hypothesis.

  * •

From the multi-line analysis of formaldehyde and the formyl cation, new
12C/13C, H/D, 16O/18O and 18O/17O ratios were inferred in G331 and agree with
the results of other works. In particular, deuterium was observed in
formaldehyde but not in the formyl cation, HDCO and DCO+, respectively. 17O
was observed in the formyl cation but not in formaldehyde, HC17O+ and H2C17O,
respectively. In perspective, along with new observational analysis, gas-grain
chemical models might shed light on the molecular processes that lead
different isotopic ratios.

We sincerely acknowledge the anonymous reviewer for their constructive and
careful revision of the manuscript. EM acknowledges support under the grant
”María Zambrano” from the UHU funded by the Spanish Ministry of Universities
and the ”European Union NextGenerationEU”. This project has also received
funding from the European Union’s Horizon 2020 research and innovation program
under the Marie Skłodowska-Curie grant agreement No 872081 and from grant
PID2019-104002GB-C21 funded by MCIN/AEI/10.13039/501100011033 and, as
appropriate, by ‘ERDF A way of making Europe’, by the ‘European Union’ or by
the ‘European Union NextGenerationEU/PRTR’. This work is also supported by the
Consejería de Transformación Económica, Industria, Conocimiento y
Universidades, Junta de Andalucía and European Regional Development Fund (ERDF
2014-2020) PY2000764. LB gratefully acknowledges support by the ANID BASAL
projects AFB17002 and ACE210002.

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##  Appendix A Update of the internal partition functions of the detected
isotopologues of formaldehyde

The internal partition functions of the isotopologues of formaldehyde
H${}_{2}^{13}$CO, H2C18O, HDCO and D2CO detected in this work have been
updated. Since the lists of energies, either experimental or theoretical, of
the rovibrational levels for these isotopologues are incomplete for computing
the internal partition functions for temperatures up to 500 K, the direct sum
expression does not reach the convergence. Therefore, the internal partition
functions of the isotopologues of formaldehyde have to be computed using some
other tested approximations (Carvajal et al., 2019), e.g., writing it in terms
of the product of the rotational contribution ($Q_{\rm rot}(T)$) and the
harmonic approximation of the vibrational contribution ($Q_{\rm vib}^{\rm
harm}(T)$) (Herzberg, 1991)

$Q_{\rm rv}(T)\approx Q_{\rm rot}(T)\,Q_{\rm vib}^{\rm harm}(T)~{}~{}.$ (A1)

where the rotational contribution $Q_{\rm rot}(T)$ is computed as a direct sum
(e.g., Herzberg (1991)) because there are enough rotational energies for the
typical temperatures of the ISM,

$Q_{\rm rot}(T)=\sum_{\rm i}g_{\rm ns}^{({\rm i})}\,(2\,J_{\rm
i}+1)\,e^{-{E^{({\rm rot})}_{\rm i}\over kT}}~{}~{}.$ (A2)

$E^{({\rm rot})}_{\rm i}$ represents the energy for the i-th rotational state
in the ground vibrational state assuming that uniquely the ground electronic
state is populated. The rotational energies of the isotopologues of
formaldehyde have been taken from the Cologne Database for Molecular
Spectroscopy catalog (CDMS) (Endres et al., 2016) and from JPL (Pickett et
al., 1998).

The nuclear spin degeneracy $g_{\rm ns}^{({\rm i})}$ is included in the
definition of the rotational contribution of the partition function (Eq. A2)
because, in general, the values of $g_{\rm ns}^{({\rm i})}$ can be different
depending on the symmetry of the rotational states. In the case of
formaldehyde, the isotopologues H${}_{2}^{13}$CO, H2C18O, and D2CO have
symmetry ${\cal C}_{2v}(M)$, whose rovibrational states are characterized by
the irreducible representations $A_{1}$, $A_{2}$, $B_{1}$ and $B_{2}$, whereas
the monodeuterated isotopologue HDCO has symmetry ${\cal C}_{s}(M)$ and its
states are labeled with the irreducible representations $A^{\prime}$ and
$A^{\prime\prime}$ (Bunker & Jensen, 1989). The labelings of the rotational
states of the different isotopologues as well as their nuclear spin
statistical weights are given in Table 6.

Although the validity of the approximation (A1) has been proven suitable for
the typical ISM temperatures (Carvajal et al., 2019), a new check is carried
out comparing the values of the internal partition function of the main
isotopologue of formaldehyde from 2.725 K to 500 K with those calculated as a
direct sum of a comprehensive data set for the rovibrational energy levels
provided by Al-Refaie et al. (2015). In Tab. 7, the values of the harmonic
approximation for the vibrational partition function $Q_{\rm vib}^{\rm
harm}(T)$ and the internal partition function $Q_{\rm rv}(T)$ (A1) for the
main isotopologue are given. $Q_{\rm vib}^{\rm harm}(T)$ is calculated using
the experimental fundamental vibrational energies taken from Perrin et al.
(2003, 2006) and the rotational partition function is computed from the JPL
rotational energies predicted up to J=99 and Ka=25 (Pickett et al., 1998). The
uncertainties of the rovibrational partition function (Carvajal et al., 2019)
are obtained considering the uncertainties of all the rotational energies
provided in the JPL database (Pickett et al., 1998) and the experimental
uncertainties of the vibrational fundamental energies (Perrin et al., 2003,
2006). These values are compared with those from CDMS catalog (Endres et al.,
2016) and from Al-Refaie et al. (2015). According to the relative differences
between the approximated internal partition function (A1) and the one
calculated more thoroughly (Al-Refaie et al., 2015), the two results are
comparable in the temperature interval from 2.725 K and 500 K. However, the
difference with the values provided in the CDMS catalog is of -8.8% at 500 K
and of -0.72% at 300 K. Therefore, as the approximation (A1) using a harmonic
vibrational partition function is acceptable for the interval of temperature
typical for the ISM and improves notably the values of JPL and CDMS catalogs
for higher temperatures, this is going to be used for the other isotopologues
of formaldehyde.

Tab. 8 evinces the values of the partition function of H${}_{2}^{13}$CO,
H2C18O, D2CO, and HDCO calculated in this work. As in Tab. 7, the values of
the calculated partition function are compared with those available in CDMS
database (Endres et al., 2016). For the case of H${}_{2}^{13}$CO, the values
and the uncertainties of $Q_{\rm rv}(T)$ are computed using experimental
fundamental vibrational energies (Ng & Tan, 2017; Wohar & Jagodzinski, 1991)
for the harmonic partition function and rotational energies up to J=99 and
Ka=25 taken from JPL database (Pickett et al., 1998). The highest relative
difference between this work and CDMS values is of -0.76% at T=300 K.
Therefore, at this temperature is not relevant to incorporate the vibrational
contribution to the partition function. Nevertheless, the vibrational
contribution is important at T=500 K increasing the rotational partition
function about 9%.

For the isotopologue H2C18O only the values of the rotational partition
function are evinced in Tab. 8. These values and their uncertainties have been
calculated using the CDMS rotational data (up to J=54, Ka=16) complemented
with those predicted for the excited rotational levels from JPL database (up
to J=20, Ka=20) when missing in the CDMS catalog. The vibrational partition
function could not be calculated because, as far as we know, there are neither
fundamental energies nor excited vibrational energies reported in the
literature at all. This hindrance could be overcome providing the vibrational
fundamental energies calculated by ab initio or other empirical approaches.
Therefore, the partition function in the present work is practically the same
reported in CDMS apart from the uncertainties and the new values of the
rotational partition function for temperatures from 300 K to 500 K.

The vibrational and the internal partition functions of the double deuterated
isotopologue D2CO are presented in Tab. 8. The vibrational contribution and
its uncertainty have been calculated using the available experimental
vibrational fundamental bands (Perrin et al., 1998; Lohilahti & Alanko, 2001;
Lohilahti et al., 2006). The values and the uncertainties of the rotational
partition function are calculated using the CDMS rotational energies (with
data up to J=66 and Ka=26) complemented with higher excited rotational energy
predictions (up to J=60 and Ka=60) given in JPL database. At T=300 K the
difference of the updated internal partition function with CDMS values is
around 2.6% and this will be around 20% at T=500 K.

The values and uncertainties of the internal partition function of the
monodeuterated isotopologue HDCO as well as the vibrational contribution are
also included in Tab. 8. The values of the rotational partition function are
calculated using the CDMS rotational energies (up to J=56 and Ka=20)
complemented with JPL higher excited rotational energy predictions up to J=90
and Ka=50. Since the CDMS data are more accurate they were substituted in the
JPL predictions to have a more accurate internal partition function. The
vibrational partition function is computed using the available experimental
and calculated fundamental energies (Oka & Morino, 1961; Dangoisse et al.,
1978; Ellsworth et al., 2008; Morgan et al., 2018). The experimental
uncertainties of the fundamentals $\nu_{1}$, $\nu_{2}$, $\nu_{4}$, and
$\nu_{6}$ measured by Dispersed Fluorescence spectroscopy (Ellsworth et al.,
2008) are considered as 2 cm-1 according to the widths of the spectral lines
whereas the uncertainties of $\nu_{3}$ and $\nu_{5}$ bands obtained with DVR
calculations are assigned with 1.20 cm-1. By comparing with the CDMS data, the
relative difference of the updated internal partition function is 1.5% larger
at T=300 K and at least of 14% at T=500 K.

In general, the update of the internal partition functions incorporate the
vibrational contribution as well as the uncertainties and new values from 300
K to 500 K for the four isotopologues H${}_{2}^{13}$CO, H2C18O, D2CO, and
HDCO. As supplementary material, their rotational, vibrational and
rovibrational partition functions are reported up to T=500 K using a 1 K
interval. This update of the partition functions could be relevant for the
estimate of the abundances of the four isotopologues of formaldehyde.

Table 6: Nuclear spin statistical weights of the isotopologues of formaldehyde
H2CO, H${}_{2}^{13}$CO, H2C18O, D2CO and HDCO associated to the rotational
statesa.

Isotopologues |  $\Gamma_{\rm rot}$ b |  $K_{a}$ c |  $K_{c}$ c | $g_{\rm ns}^{({\rm i})}$ |  Type g
---|---|---|---|---|---
H2CO,H2C18O | $A_{1}$ | even | even | 1 | para
| $A_{2}$ | even | odd | 1 | para
| $B_{1}$ | odd | odd | 3 | ortho
| $B_{2}$ | odd | even | 3 | ortho
H${}_{2}^{13}$CO | $A_{1}$ | even | even |  2 d | para
| $A_{2}$ | even | odd |  2 d | para
| $B_{1}$ | odd | odd |  6 d | ortho
| $B_{2}$ | odd | even |  6 d | ortho
D2CO | $A_{1}$ | even | even |  6 e | ortho
| $A_{2}$ | even | odd |  6 e | ortho
| $B_{1}$ | odd | odd |  3 e | para
| $B_{2}$ | odd | even |  3 e | para
HDCO | $A^{\prime}$ | —- | even |  6 f | —-
| $A^{\prime\prime}$ | —- | odd |  6 f | —-

a The nuclear spin degeneracy $g_{\rm ns}^{({\rm i})}$ is computed according
to Bunker & Jensen (1989).

b The symmetry labeling of the rotational states.

c The rotational states of the asymmetric top, such as formaldehyde, are
labeled by the quantum numbers $J_{K_{a},K_{c}}$, where $J$ is the rotational
angular momentum and the $K_{a}$ and $K_{c}$ are the projections of the
rotational angular momentum along the $a$\- and $c$\- molecule-fixed axes. In
this case, the symmetry of the rotational states is characterized by their
even and odd values of $K_{a}$ and $K_{c}$. For the monodeuterated
isotopologue, the two symmetries are only characterized by the even and odd
values of $K_{c}$.

d For H${}_{2}^{13}$CO, the nuclear spin degeneracy is also considered in the
literature with a ratio 3:1 for the ortho:para states, respectively (see,
e.g., Endres et al. (2016)). As a warning in order to avoid wrong results,
before using data from a catalog, it should be checked whether the nuclear
spin weights agree with the partition function considered.

e For D2CO, the nuclear spin degeneracy is also considered in the literature
with a ratio 2:1 (ortho:para) (see, e.g., Endres et al. (2016)). Same warning
from footnote $d$ should be considered in this case.

f For the monodeuterated isotopologue, the nuclear spin degeneracy is
considered in this paper as 1 because the degeneracy is state independent.

g Only for the symmetric isotopologues of formaldehyde, this column shows
whether the transitions involving these rotational states are either ortho or
para.

Table 7: Vibrational and rotational-vibrational partition function of the main
isotopologue of formaldehyde (H2CO). Comparison between the values obtained in
the present study and those published before.a

$T$(K) |  $Q_{\rm vib}^{\rm harm}$ b |  $Q_{\rm rv}$(Present work)c |  $Q$(CDMS)d |  $Q$(Al-Refaie et al., 2015)e |  Rel. Diff.(%)f
---|---|---|---|---|---
2.725 | 1.000000 | 2.0166(0) | 2.0166 | 2.0165 | 0.00
5.000 | 1.000000 | 4.4832(0) | 4.4832 | 4.4833 | 0.00
9.375 | 1.000000 | 13.8009(0) | 13.8008 | 13.8010 | 0.00
18.750 | 1.000000 | 44.6813(0) | 44.6812 | 44.6835 | 0.00
37.500 | 1.000000 | 128.6496(0) | 128.6492 | 128.6581 | 0.01
75.000 | 1.000000 | 361.7207(0) | 361.7195 | 361.7053 | 0.00
150.000 | 1.000021 | 1019.9947(0) | 1019.9706 | 1019.9549 | 0.00
225.000 | 1.000996 | 1874.4927(0) | 1872.6221 | 1874.4679 | 0.00
300.000 | 1.007228 | 2903.8609(0) | 2883.0163 | 2904.1778 | 0.01
500.000 | 1.087521 | 6751.7086(3) | 6208.3442 | 6760.2315 | 0.13

a The nuclear spin degeneracy is given in Tab.6.

b Values of the vibrational partition function computed with the harmonic
approximation. For more details, see the text.

c $Q_{\rm rv}=Q_{\rm rot}(\mbox{\rm direct sum})\,Q_{\rm vib}^{\rm harm}$. An
estimate of the uncertainties is given in parentheses in units of the last
quoted digits. For more details, see the text.

d Rotational partition function computed as a direct sum with no vibrational
contribution. Their values are reported in CDMS catalog (Endres et al., 2016).

e Internal partition function computed as the direct sum using a comprehensive
set of rovibrational energies up to 18000 cm-1 and up to J=70 (Al-Refaie et
al., 2015).

f Relative difference of the partition function computed in the present study
with respect to the one reported by Al-Refaie et al. (2015).

Table 8: Vibrational and rotational-vibrational partition functions for the
isotopologues H${}_{2}^{13}$CO, H2C18O, D2CO and HDCO. Comparison between the
values obtained in the present study and those published in CDMS cataloga.

|  H${}_{2}^{13}$CO |  H2C18O
---|---|---
$T$(K) |  $Q_{\rm vib}^{\rm harm}$ b |  $Q_{\rm rv}$(Present work)c |  $Q$(CDMS)d |  Rel. Diff.(%)e |  $Q_{\rm rot}$(Present work)c |  $Q$(CDMS)d |  Rel. Diff.(%)e
2.725 | 1.000000 | 4.1136(0) | 4.1136 | 0.00 | 2.0944(0) | 2.0944 | 0.00
5.0 | 1.000000 | 9.1701(0) | 9.1700 | 0.00 | 4.6806(0) | 4.6805 | 0.00
9.375 | 1.000000 | 28.2638(0) | 28.2636 | 0.00 | 14.4425(0) | 14.4424 | 0.00
18.75 | 1.000000 | 91.5675(0) | 91.5672 | 0.00 | 46.8159(0) | 46.8156 | 0.00
37.50 | 1.000000 | 263.7391(0) | 263.7382 | 0.00 | 134.8810(0) | 134.8805 | 0.00
75.0 | 1.000000 | 741.6749(0) | 741.6726 | 0.00 | 379.3628(0) | 379.3616 | 0.00
150.0 | 1.000023 | 2091.5874(3) | 2091.5330 | 0.00 | 1069.8884(0) | 1069.8850 | 0.00
225.0 | 1.001065 | 3844.176(18) | 3840.0722 | -0.11 | 1964.3668(0) | 1964.3606 | 0.00
300.0 | 1.007605 | 5957.09(14) | 5912.1084 | -0.76 | 3024.3241(1) | 3024.3231 | 0.00
500.0 | 1.090530 | 13884.1(19) | —- | —- | 6508.862(13) | —- | —-
|  D2CO | HDCO
$T$(K) |  $Q_{\rm vib}^{\rm harm}$ b |  $Q_{\rm rv}$(Present work)c |  $Q$(CDMS)d |  Rel. Diff.(%)e |  $Q_{\rm vib}^{\rm harm}$ b |  $Q_{\rm rv}$(Present work)c |  $Q$(CDMS)d |  Rel. Diff.(%)e
2.725 | 1.000000 | 14.984964(0) | 14.9850 | 0.00 | 1.000000 | 2.2608(0) | 2.2608 | 0.00
5.000 | 1.000000 | 29.703636(0) | 29.7036 | 0.00 | 1.000000 | 4.6483(0) | 4.6483 | 0.00
9.375 | 1.000000 | 67.339744(0) | 67.3395 | 0.00 | 1.000000 | 11.2681(0) | 11.2680 | 0.00
18.750 | 1.000000 | 182.210324(0) | 182.2098 | 0.00 | 1.000000 | 31.0556(0) | 31.0555 | 0.00
37.500 | 1.000000 | 509.265756(0) | 509.2641 | 0.00 | 1.000000 | 86.7359(0) | 86.7356 | 0.00
75.000 | 1.000000 | 1432.775532(0) | 1432.7709 | 0.00 | 1.000000 | 243.8496(0) | 243.8488 | 0.00
150.000 | 1.000226 | 4044.511598(0) | 4043.5866 | 0.02 | 1.000093 | 688.0292(0) | 687.9631 | -0.01
225.000 | 1.005192 | 7464.6706(1) | 7426.0941 | -0.52 | 1.002701 | 1266.752(29) | 1263.3349 | -0.27
300.000 | 1.025736 | 11729.1102(6) | 11434.7820 | -2.57 | 1.015159 | 1974.73(18) | 1945.2333 | -1.52
500.000 | 1.202824 | 29624.2096(74) | —- | —- | 1.138623 | 4769.8(21) | —- | —-

a The nuclear spin degeneracy is considered according to Tab. 6.

b The vibrational partition function is computed with the harmonic
approximation. For more details, see the text.

c $Q_{\rm rv}=Q_{\rm rot}(\mbox{\rm direct sum})\,Q_{\rm vib}^{\rm harm}$. An
upward estimate of the uncertainties is given in parentheses in units of the
last quoted digits. For more details, see the text. For the isotopologue
H2C18O, it is presented the rotational partition function because there are
neither experimental nor theoretical vibrational levels available in the
literature.

d Rotational partition function computed as a direct sum with no vibrational
contribution. Their values are reported in CDMS catalog (Endres et al., 2016).
The CDMS partition functions of H${}_{2}^{13}$CO and D2CO are multiplied by 2
and 3, respectively, to consider the same nuclear spin statistical weight from
Tab. 6.

e Relative difference of the partition function given in the present study
with respect to the one reported in CDMS catalog.