Quenching rates for formaldehyde

# Rotational quenching of H2CO by molecular hydrogen: cross-sections, rates, pressure broadening.

L. Wiesenfeld , A. Faure
UJF-Grenoble 1/CNRS-INSU, Institut de Planétologie et d’Astrophysique de Grenoble (IPAG) UMR 5274, Grenoble F-38041, France
E-mail : laurent.wiesenfeld@obs.ujf-grenoble.fr
Accepted xxx. Received xxxx; in original form July 27, 2019
###### Abstract

We compute the rotational quenching rates of the first 81 rotational levels of ortho- and para-HCO in collision with ortho- and para-H, for a temperature range of 10-300 K. We make use of the quantum close-coupling and coupled-states scattering methods combined with the high accuracy potential energy surface of Troscompt et al. (2009a). Rates are significantly different from the scaled rates of HCO in collision with He; consequently, critical densities are noticeably lower. We compare a full close-coupling computation of pressure broadening cross sections with experimental data and show that our results are compatible with the low temperature measurements of Mengel & De Lucia (2000), for a spin temperature of H around 50 K.

###### keywords:
Astrochemistry, molecular processes, molecular data, ISM: molecules
pagerange: LABEL:firstpageLABEL:lastpagepubyear: xxxx

## 1 Introduction

In order to relate quantitatively the observed rotational spectra and the molecular abundances, knowledge of the relative importance of photon induced transitions and collision induced transitions is imperative. While the Einstein and coefficients can be obtained experimentally (see e.g. the JPL and CDMS databases Drouin (2012); Müller et al. (2005)), the rates of molecular collisional excitation/quenching need to be calculated with help of precise microscopic frameworks. Many molecular quenching rates have been put forward in the last 40 years, either for molecular collisions with He or for collisions with H, electrons and H. In many types of interstellar regions, especially so when molecular complexity is present, the main collider is molecular hydrogen, H. Recently, a renewed large effort has been devoted to compute the rotational quenching rates of molecules by H (see references in the review by Van der Tak (2011)) and most collisional data are available in the BASECOL Dubernet et. al. (2012) and LAMDA Schöier et al. (2005) databases. Special emphasis was put on molecules specific to the Herschel Space Observatory, and among them, water. However, complex organic molecules continue to play a prominent role in the understanding of proto-stellar evolution as well as being probes for various interstellar environments.

Among all organic molecules, formaldehyde (HCO) is especially abundant, since it is the first stable molecule resulting from the hydrogenation of the ubiquitous CO molecule (Peters et al., 2011). Being abundant and displaying a large range of both transition frequencies and energy levels, formaldehyde is a tool of choice to probe the physical conditions of the gaseous interstellar matter (Maret et al., 2004; Kama et al., 2013). By using models that do not suppose Local Thermodynamical Equilibrium (LTE), it is possible to reach reliable estimates of the molecular abundance of HCO together with the other parameters of the gas, its temperature and density (Ceccarelli et al., 2003; Van der Tak et al., 2007). The famous formaldehyde “anomalous” absorption was also shown to provide a probe of the ortho-to-para ratio of H (Troscompt et al., 2009b) and a distance-independent tracer of the cosmic star formation history (Darling & Zeiger, 2012).

Several computations of the excitation rates of HCO have been proposed in the literature. Excitation by He atoms, being easier to perform, has been used for long as a model for excitation by H, even if it is known for being an underestimation of unknown precision (Green et al., 1978; Green, 1991; Sharma et al., 2012). In our group, we computed (Troscompt et al., 2009a) the quenching rates of HCO by H, but only for the low levels of excitation of ortho-formaldehyde (only the first 10 levels) and for low kinetic temperatures (K). These computations were based on a high precision Potential Energy Surface (PES) for the van der Waals interaction . In the present paper, we extend our previous computations to a much broader range of rotational energies and temperatures, for both ortho- and para-HCO, using the same PES. Also, in order to assess the precision of this PES, we compare experimental pressure broadening cross sections measured by Mengel & De Lucia (2000) to our own computations, in a manner similar to our recent works on the rotational excitation of HO and CO (Drouin & Wiesenfeld, 2012b; Faure et al., 2013).

We organize the paper as follows. Section 2 describes the details of the scattering computation. Section  3 shows the results of our computations including cross sections, rates and pressure broadening. We end (section 4) with a discussion and a conclusion.

## 2 Scattering calculations

As mentioned above, the PES for the interaction of HCO and H has been described in our previous paper (Troscompt et al., 2009a). The PES was computed for frozen monomer geometries (rigid-rotor approximation). The geometries were those of the averaged distances and angles, at the ground vibrational state for both HCO and H. Using average ground state geometries instead of equilibrium geometries have been shown to give more reliable results (Valiron et al., 2008). Since no modifications were performed on the PES, neither on the ab initio computations nor on the fits, the reader is deferred to Troscompt et al. (2009a) for all necessary details of the ab initio and fitting procedures (see also Rist & Faure (2011) for the fitting procedure).

Both HCO and H have each two identical H nuclei, of nuclear spin . Hence, both exist in para and ortho spin states. For H, the para state (total nuclear spin ) has even rotational states (We denote by , the rotational quantum number of H). The ortho state () on the opposite, has odd rotational states, . The H rotational constant is taken at . In the PES, the H-H distance is taken at  Å, its average value at ground vibrational level (Troscompt et al., 2009a). The HCO molecule is an asymmetric rotor, with rotational constants (in ) : , , and centrifugal parameters , , and . The rotational constant is along the electric dipole, on the symmetry axis 111This is the MOLSCAT convention, differing from usual spectroscopic conventions by the ordering of the axes.. Describing the rotational states by the usual rotational (pseudo-)quantum numbers, para states () of HCO correspond to even and ortho states () to odd (the main quantum rotational number for HCO is denoted by throughout this paper).

Scattering calculations were performed for all levels with a rotational energy , that is up to for ortho-HCO and for para-HCO. We computed the inelastic cross sections with a collision energy . This slightly extends the previous HCO-He computations of Green (1991) and extends our previous HCO-H computations by an order of magnitude. Experimental (Bocquet et al., 1996) and computed para and ortho rotational levels of HCO are given in table 1, along with the level numbering used in the results. Note that restricting energies to entail . Also, for high enough values, pairs of successive and levels tend to be degenerate, for all practical purposes. This degeneracy is all the more precise that the values are small. Inspection of table 1 shows that the experimental - theoretical energy level differences are very small: (mean value ).

All scattering calculations have been performed with the OpenMP version of the MOLSCAT code222Repository at http://ipag.osug.fr/afaure/molscat/index.html . The reduced mass for HCO-H is 1.889053 amu. The coupled-channel (CC) and coupled-states (CS) equations were integrated using the diabatic modified log-derivative propagator.

The rotational basis set is devised as follows. For all scattering energies, if is the last open channels, the values were added to the basis. However, since a given rotational number spans a large amount of rotational energy, rotational levels were capped, at for collision energies , and increasing progressively to for . These large values of are needed to converge cross sections; a similar effect was previously observed for methyl-formate colliding with Helium (Faure et al., 2011).

The rotational basis for ortho-H is . It has been shown for many systems that including in the basis does not have a noticeable influence for temperatures as low as 300 K (e.g. Daniel, Dubernet & Grosjean (2011)). The rotational basis for para-H proved difficult to settle. We were able to use the basis set for CC para-H - ortho-HCO collisions, with (Troscompt et al., 2009a). For para-HCO, because of the level structure, the basis for CC computations proved to be practically impossible for . As a result, we resorted only to a basis, both for ortho- and para-HCO, stretching the CC computations as high as possible and continuing with the CS approximation. In order to assess the importance of the channel, however, we complemented the ortho-HCO data with a full CS computation up to , using a coarse energy grid. This allowed us to check i) that cross sections for transitions with are negligible, ii) that cross sections for transitions with are very similar to those with (as observed for other systems, see e.g. Daniel, Dubernet & Grosjean (2011)) and iii) that the difference between the basis sets and decreases with increasing collision energy, from an average of to below 10 %.

Because of the large number of expansion terms of the potential function in the spherical harmonic basis (Troscompt et al., 2009a), several strategies have been devised in order to converge the inelastic scattering computations in a reasonable amount of time (We arbitrarily tried to limit ourselves to 72 hours of clock time, for one energy scattering point, on 12 CPU cores). The radial propagation used a step size parameter except at collision energies below 10 cm where was steps progressively increased up to 50. Also, rmax values were progressively increased in the low-energy regime from default to 100. Other propagation parameters were taken as the MOLSCAT default values.

For total energies above approximately 330 cm the coupled channels (CC) approach, exact in the fully converged limit, proved to be impractical. We thus had to resort to the usual coupled states (CS) approximation, with all its shortcomings. Like in our recent HDO-H rate calculations (Wiesenfeld, Scribano & Faure, 2011), we used an additive constant to scale appropriately the CS cross sections by their counterpart CC values. Overlap of the CC and CS calculations showed the validity of these corrections. The statistics of difference between the CC and CS cross-sections was also examined. We found the following values (in , , the standard deviation): , . Also, for collisions with ortho-H, the total angular momentum was not stepped by unit values, as is done usually. For CC calculation above , a step in value was used, with a careful checking of the convergence of the procedure. This procedure was used throughout the CS computations.

The energy grid was chosen to guarantee a good description of the resonances, including those pertaining to the high lying rotational states. Also, since we aim at rates for temperatures up to 300 K, a particular care was taken to ensure both some economy in the computational load and a good convergence of the rate computation. Let us recall that the quenching rate from state to state , (in cmsec) is related to the inelastic cross section (in Å, with , the collision energy) by the well known Boltzmann average :

 kj←i(T)=√8πμ1T3∫∞0σj←i(E)Eexp(−E/T)dE, (1)

where and are expressed in the same units, and is the collisional reduced mass. The probability density going very slowly down with energy, it is customary, for calculating , to compute up to (e.g., Dubernet et al. (2009)). This approach is prohibitively time-intensive for a heavy molecule like HCO, even within the CS approximation. For high lying rotational states, we inspected carefully the actual numerical convergence of equation 1 and stopped our energy grid as soon as the rate was saturated by 10% in general, and 25% for the highest levels, with .

The computation of the pressure broadening cross-sections (Wiesenfeld & Faure, 2010; Drouin & Wiesenfeld, 2012b) necessitates a very fine energy grid and a good control of the elastic cross sections, much more difficult to obtain than the corresponding inelastic sections. Hence, we had to re-calculate all the -matrices with collision energies from 2 to .Then, we compute the in order to compare it with the experimental results of Mengel & De Lucia (2000), in a temperature range of . Both collisions with para-H (basis set, ) and ortho-H (basis set, ) were performed on an identical fine energy grid and in the CC formalism.

## 3 Results and discussion

### 3.1 Cross sections

The inelastic cross sections have a general shape that is similar to all earlier findings, for collisions of a molecule with H. As usual, the inelastic scattering with may be markedly different from the scattering with . This was observed and thoroughly discussed for HO and HDO scattering computations (Dubernet et al., 2009; Daniel, Dubernet & Grosjean, 2011; Wiesenfeld, Scribano & Faure, 2011; Faure et al., 2012), and observed for a wide range of other collisions, SO and Cl atoms being recent examples (Cernicharo et al., 2011; Lique & Alexander, 2012). Experiments with H colliding with water molecules also extensively confirm this difference (Drouin & Wiesenfeld, 2012b; Yang et al., 2011). The situation with formaldehyde colliding with H, and is however less clear, as some ortho-H and para-H collisions are nearly identical, especially for small sections and large . Examples of astrophysical significative cross-sections are given in figure  1. Those three cases are representative of all these astrophysically relevant types of sections that we examined: even if the displays a richer resonance structure, because of the disappearance of the supplementary quantum coupling . In each case, we observe that , in structure and magnitude. Remember, however, that the threshold for scattering with is higher than for . A full and reliable computation of HCO scattering with para-H, is thus impossible for all practical purposes, in the present computer configurations.

### 3.2 Rates

All quenching rates for all levels of table 1 are computed for the same temperature grid as Green (1991), . The full table is deposited in the LAMDA database (Schöier et al., 2005) and BASECOL database (Dubernet et. al., 2012), and may be asked to the authors. The rates with ortho-H are based on the sections only. In all cases, rates with para-H are given as a Boltzman average over the populations of , with the further approximation of , when necessary (see above). The influence of the initial states may indeed be very large, as was also observed, in another context, for the pressure broadening of HO by H (Drouin & Wiesenfeld, 2012b).

Figure 2 compares globally present critical densities and critical densities from Green (1991), for electric-dipole allowed transitions. We compare our present rates with para-H with the properly scaled rates with He. We have the following definition on the critical density :

 n∗i=∑j′

where is the level under scrutiny and denotes all levels connected by a radiative transition, while spans all levels. Left panel shows the scatter of critical densities ratios at , while on the right panel, all ratios are averaged and plotted against temperature. The scatter is moderate, with no ratio exceeding 3. The right panels shows the evolution of the average as a function of . For higher temperatures, the effect of H being different of He diminishes, since higher collisions energies are more sensitive to the hard walls of the target, where the potential grows exponentially (repulsion of the wave functions). H and He become more similar, and the rates tend one towards the other.

In order to assess the reliability of the PES, we found it useful to compare measured and computed pressure-broadening cross sections, (Wiesenfeld & Faure, 2010; Drouin & Wiesenfeld, 2012b; Faure et al., 2013). The main advantage of pressure broadening is that experiments and computations held both absolute quantities, with no scaling involved, rending the comparison very meaningful. Unfortunately, only a very limited set of pressure broadening data exists for an H buffer gas (Mengel & De Lucia, 2000), for very low temperatures. The experimental results for the transition are depicted in figure 3, along with full CC calculations of . We see that the para-H and ortho-H computation bracket the experimental values, which are very well simulated by an ortho- to para-H ratio (opr) corresponding to a pseudo-equilibrium at 50 K, corresponding to a OPR of 0.27. That the opr of H may vary during the collisional cooling experiment has been proved with cell walls covered with amorphous water (Drouin & Wiesenfeld, 2012b). Nothing is known for formaldehyde, and discussion with the authors of Mengel & De Lucia (2000) could not settle the case. We remain thus with a good plausibility argument, as long as the pressure broadening experiments with H are not fully characterized.

## 4 Discussion - Conclusion

It is important to know which errors are to be expected, and to have some clues on how these errors might influence astrophysical modeling. We expect the error on the rates to be uniformly increasing from low lying levels to higher lying ones, and also from low temperature to high temperature. Quantifying this error is very risky, as errors may arise from all the phases of the rate computation: ab initio computation, fits and long distance behavior of the PES, convergence of the CC/CS procedures, convergence of the averaging procedure of the sections with collision energy.

Internal consistency with our earlier approach (Troscompt et al., 2009a) show differences less than 10 %, for , . This shows that the convergence error in this domain cannot exceed 20-30%. In the high temperature domain ( K), it is safe to assume errors much larger than 30%, because of the poor convergence of formula (1), but still within a factor 2. The magnitude of the values as well as the similarities between the and plead in favor of such a conservative value, as does the convergence of critical densities between our work and the previous collisions with He by Green (1991).

Accuracy of the PES is very difficult to assess, without any firm experimental comparison, like has been done for HO and to a lesser extent CO (Yang et al., 2011; Drouin & Wiesenfeld, 2012b; Faure et al., 2013; Chefdeville et al., 2012). Our results are, however, compatible with measurements of Mengel & De Lucia (2000). Because of the great importance and ubiquity of the formaldehyde molecule, further experiments would give indications on the precision of the PES and convergence procedures used in this paper.

The relevance of the ortho-to-para ratio of H has been stressed several times already, as it may be of crucial importance in order to correctly model the astrophysical environments. While the difference in rates for ortho-H and para-H is large for low lying levels at low energies, see Troscompt et al. (2009b) for an application, this difference decreases at larger transition energies, as the hard walls of the PES play a more important role than the long range behavior. Indeed, the main increase in with respect to is due to the averaging out of the quadrupolar moment and dipolar polarizability of H in its ground rotational states. The same is true, up to a global scale, for the difference in behavior between He and ortho-H.

Extension of these computations to higher levels and higher temperatures is by no means a difficult task, on the physics point of view, because of the rigid rotor structure of HCO, its first bending frequency (the out-of-plane bend) arising at 1167.3 cm (Clouthier & Ramsay, 1983). The true limiting factor arises from numerical load, with very large matrices to invert and propagate (). The same situation arises for the excitation of heavier complex organic molecules, like methyl formate or dimethyl ether, which display many spectral lines very far from LTE, in various spectral surveys like Caux et al. (2011). Unfortunately, quasi classical trajectories methods are limited for all those cases because of ambiguities arising in the subsequent quantization of rotational levels (Faure & Wiesenfeld, 2004). Use of very large grid of computers and combinations of OpenMP/MPI approaches may overcome these difficulties.

We have calculated an extensive set of low to medium temperature quenching rates, for all levels of HCO below . These rates are ready to be incorporated in the various non-LTE models for the interstellar medium. At low temperatures, differences with earlier rates of HCO colliding with Helium are very important and they remain noticeable at all temperatures, with ratios up to 50% at 300 K, where they are the most similar. Since Troscompt et al. (2009a) was tailored to be very precise at low temperatures for ortho-HCO, we still recommend to use those rates for applications at  K. The present rates should have a large importance on the inferring of HCO column densities, away from LTE conditions.

## acknowledgments

Many discussions with C. Ceccarelli and members of the CHESS team are gratefully acknowledged. The authors thank generous funding through the ANR and the CNES agencies, thanks to the FORCOMS contract (ANR-08-BLAN-022) and the CHESS Herschel Space Observatory Key Program funding. This work was also supported by the CNRS-INSU national program “Physique et Chimie du Milieu Interstellaire”. All calculations presented in this paper were performed at the “Service Commun de Calcul Intensif de l’Observatoire de Grenoble (SCCI)”.

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