Issue 
A&A
Volume 567, July 2014



Article Number  A22  
Number of page(s)  5  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201423957  
Published online  04 July 2014 
Collisional excitation of O_{2} by H_{2}: the validity of LTE models in interpreting O_{2} observations
^{1} LOMC – UMR 6294, CNRSUniversité du Havre, 25 rue Philippe Lebon, BP 1123, 76063 Le Havre, France
email: francois.lique@univlehavre.fr
^{2} Department of Optics and Spectroscopy, Tomsk State University, 36 Lenin av., 634050 Tomsk, Russia
^{3} Université de Bordeaux, Institut des Sciences Moléculaires, CNRS UMR 5255, 33405 Talence Cedex, France
^{4} Radboud University Nijmegen, Institute for Molecules and Materials, Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
Received: 7 April 2014
Accepted: 28 May 2014
Context. Oxygen molecules (O_{2}) are of particular interest because of their crucial role in astrochemisty. Modelling of O_{2} molecular emission spectra from interstellar clouds requires the calculation of rate coefficients for excitation by collisions with the most abundant species.
Aims. Rotational excitation of O_{2}(X^{3}Σ^{−}) by H_{2} is investigated theoretically and experimentally and we check the validity of the local thermodynamic equilibrium (LTE) approach for interpreting O_{2} observations.
Methods. Using a new ab initio potential energy surface, collisional excitation of O_{2} is studied using a full closecoupling approach. The theoretical calculations are validated by comparison with crossed beam scattering experiments. We also performed calculations for the excitation of O_{2} from a large velocity gradient (LVG) radiative transfer code using the new rate coefficients.
Results. Statetostate rate coefficients between the 27 lowest levels of O_{2} were calculated for temperatures ranging from 5 K to 150 K. The critical densities of the O_{2} lines are found to be at ≳10^{4} cm^{3} for temperatures higher than 50 K. This value is slightly larger than the one previously determined using previous He rate coefficients.
Conclusions. The new rate coefficients will help in interpreting O_{2} emission lines observed where LTE conditions are not fully fulfilled and enable an accurate determination of the O_{2} abundance in the interstellar medium.
Key words: molecular processes / molecular data / ISM: abundances / ISM: molecules
© ESO, 2014
1. Introduction
Oxygen is the third most abundant element in the Universe after hydrogen and helium, which makes it very important in terms of understanding the formation and evolution of the chemistry in astronomical sources. Earlier gasphase chemical models predicted O_{2} to be a major reservoir of elemental oxygen in dense molecular clouds (Herbst & Klemperer 1973). Searches for interstellar O_{2} have been one of the major goals for molecular astrophysics during the last decades. However, until recently, attempts to detect interstellar O_{2} line emission with ground and spacebased observatories have failed (Combes & Wiklind 1995; Marechal et al. 1997; Pagani et al. 2003).
In 2007, Larsson et al. (2007) eventually reported the first observation of O_{2} in the interstellar medium (ISM) from the satellite ODIN mission. Surprisingly, the O_{2} abundance in the cores of molecular clouds was found to be ~1000 times lower than predicted by the astrochemical models. The low O_{2} abundance has since been confirmed by other O_{2} observations (Goldsmith et al. 2011; Liseau et al. 2012; Yıldız et al. 2013).
Such a low abundance for the O_{2} molecule pushed physical chemists and astrochemists to review the oxygen chemistry in dense molecular clouds (Lique et al. 2009). The low abundance of O_{2} in dark clouds can now be explained by gasgrain chemical models that conclude that oxygen depletion is important (Hincelin et al. 2011).
Although low, the abundance of O_{2} has still to be determined accurately. Since collisions compete with radiative processes in altering populations of molecular rovibrational levels, the estimation of molecular abundances in the ISM from spectral line data usually requires collisional rate coefficients with the most abundant interstellar species like He or H_{2}. Rate coefficients for O_{2}–He inelastic collisions were provided some time ago by Lique (2010). Goldsmith et al. (2011) used these rate coefficients to study the excitation of O_{2} in cold molecular clouds. They concluded that O_{2} abundance determination is possible assuming local thermodynamic equilibrium (LTE) (i.e. without using rate coefficients) because O_{2} radiative rates are small and because the excitation of O_{2} is dominated by collisions.
In the ISM, the main collisional partner is H_{2} and it is well known that He and H_{2} rate coefficients differ significantly, especially in the case of orthoH_{2} (Roueff & Lique 2013). It is then very important to also provide collisional data for the O_{2}–H_{2} systems.
In this paper, we present a new study of the collisional excitation of O_{2} by H_{2} and we compare the theoretical results with experimental ones obtained from crossedbeam scattering experiments. Using the new data, we check the validity of the LTE approach for interpreting O_{2} observations.
2. Methodology of calculations
2.1. Potential energy surface (PES)
The PES used in this work was described in detail in Kalugina et al. (2012, hereafter Paper I). The PES was calculated in the supermolecular approach with a partially spinrestricted coupled cluster with single, double, and perturbative triple excitations [RCCSD(T)] (Knowles et al. 1993, 2000). The collision partners were considered to be rigid [r_{O − O} = 2.297 bohr and r_{H − H} = 1.449 bohr]. The four atoms were described by augmented correlationconsistent quadruple zeta (augccpVQZ) basis set of Dunning (1989) augmented with bond functions. Details of computations, analytical representation, and plots of the PES are given in Paper I. For the scattering calculations, we used the original PES of Paper I and not the scaled one of Chefdeville et al. (2013).
2.2. Scattering calculations
As noticed in Paper I and confirmed by the experiments of Chefdeville et al. (2013), the rotation of H_{2} has almost no influence on the magnitude of O_{2}–H_{2} integral cross sections (ICS). Thus, we performed calculations only for the j = 0 level of H_{2} (i.e. the PES was averaged over both orientation θ′ and dihedral φ angles describing the H_{2} geometry) and considered that our results are valid for both para and orthoH_{2}. Collisions with H_{2}(j = 0) are equivalent to collisions with a structureless atom. Then, the scattering problem is equivalent to atomdiatom inelastic collisions.
In the O_{2}() electronic ground state, the rotational levels are split by spinrotation coupling. In the intermediate coupling scheme, the rotational wave function of O_{2} can be written for j ≥ 1 as (Lique et al. 2005) (1)where  N,Sjm ⟩ denotes pure Hund’s case (b) basis functions and the mixing angle α is obtained by diagonalisation of the molecular Hamiltonian. In this relation corresponding to Hund’s case (b), the total molecular angular momentum j is defined by: (2)where N and S are the nuclear rotational and the electronic spin angular momenta. The rotational energy levels of the O_{2} molecule were computed with the use of the experimental spectroscopic constants of Endo & Mizushima (1982).
All calculations were carried out with the exact energy levels including the fine structure interaction. However, in the present paper, we will use the usual astrophysical level labelling N_{j} where N = j − 1, N = j, and N = j + 1 correspond to the F_{1}, F_{2}, and F_{3} levels as defined in Eq. (1). The quantal coupled equations were solved in the intermediate coupling scheme using the MOLSCAT code (Hutson & Green 1994) modified to take into account the fine structure of the energy levels. We used a fine collisional energy grid for the calculations so that the resonances (shape and Feshbach) that appear in the ICS at low and intermediate energies were well represented.
By averaging over a Maxwellian distribution of collision velocities, we can use the calculated ICS to obtain thermal rate coefficients for (de)excitation transitions between O_{2} levels (3)where σ_{α → β} is the cross section from initial level α (i.e. N_{j}) to final level β (i.e. ), μ is the reduced mass of the system and k_{B} is the Boltzmann constant.
In order to ensure convergence of the inelastic ICS, it was necessary to include in the calculations several energetically inaccessible (closed) levels. At the largest energies considered in this work, the O_{2} rotational basis was extended to N = 29 to ensure convergence of the rotational ICS of O_{2}.
3. Results
3.1. Cross sections
We have obtained the (de)excitation ICS between the first 27 fine structure levels of O_{2} by H_{2}. Figure 1 presents the typical kinetic energy variation of the ICS for deexcitation transitions from the N_{j} = 5_{6} level of O_{2}. As expected, the deexcitation ICS decrease with increasing energy. Noticeable resonances appear at low collisional energies. This is related to the presence of an attractive potential well, which allows for O_{2}–H_{2} bound and quasibound states to be formed before the complex dissociates (Smith et al. 1979; Christoffel & Bowman 1983).
Fig. 1 Collisional excitation ICS of O_{2} by H_{2} out of the N_{j} = 5_{6} state. 

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The quality of the theoretical results strongly depends on the accuracy of the PES employed for the quantum scattering calculations, especially at very low energy and for low O_{2} quantum numbers where the ICS can be significantly dominated by resonances. The energy position, width and magnitude of the resonances are so dependent on the calculated potential well that their experimental observation offers a stringent test for the theory. We already obtained a strong positive sign in previous combined theoretical and experimental work on the ICS of the 1_{0} → 1_{1} transition with complete resolution of the partialwave resonances (Chefdeville et al. 2013). In this work, we extend the study to the 1_{2} → 3_{2} transition.
Using the same apparatus as the one described in Chefdeville et al. (2013), we measured the ICS corresponding to the 1_{2} → 3_{2} transitions by probing the collision induced population in the 3_{2} level as a function of the collision energy using the 2+1(←) Q_{23}(2) resonanceenhanced multiphoton ionisation transition at 88.874 cm^{1} (Yokelson et al. 1992). The experiments were performed with normalH_{2}; they can nevertheless be directly compared with the theoretical results since the O_{2}–H_{2} ICS have been proven not to depend on the rotational levels of H_{2}. The initial residual population in 3_{2} level was offset by shottoshot background subtraction when triggering the probe laser and the O_{2} beam at 10 Hz with the H_{2} beam at 5 Hz. The ICS were recovered from the measured intensities using the procedure described in the Supplementary Materials of Chefdeville et al. (2013).
Fig. 2 Collisional energy dependence of the integral cross sections for O_{2} excitation . a) Theoretical ICS calculated with modified PES; b) theoretical ICS convoluted with the experimental collision energy spread calculated with the original PES (dashed line) and the modified PES (solid line); experimental data (open circles), obtained when crossing a beam of normalH_{2} from a first pulsed nozzle at ≃45 K propagating at velocity v_{1} = 951 ms^{1} with a beam of 0.5% O_{2} in Ne from a second pulsed nozzle at 300 K propagating at v_{2} = 803 ms^{1}; crossing angle χ was varied from 22.5° to 45°, resulting in a collision energy where μ stands for the reduced mass of the collision partners. 

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The ICS obtained for O_{2}–H_{2} collisions are displayed in Fig. 2. The theoretical results obtained with a slightly modified PES (the original PES scaled by a factor of 1.05; see Chefdeville et al. 2013) for the three possible transitions from the N = 1 state (1_{j = 0 − 2} → 3_{2}) are given in the upper panel (Fig. 2a). The experimental data obtained between E_{k} = 11.4 cm^{1} and 37.8 cm^{1} are displayed in the lower panel (Fig. 2b) along with the theoretical ICS convoluted with the experimental collision energy spread to allow for comparison with the experimental values. Excitation of any 1_{j = 0 − 2} state can contribute to the signal. However, the observed signal is probably due mainly to the 1_{0} state for the following reasons: (i) the rotational temperature within the molecular beam for CO (Chefdeville et al. 2012) and O_{2} (Chefdeville et al. 2013) beams was proven to be below 1 K; a similar behaviour is then expected for the O_{2} source in the present work, resulting in relative populations (P(0) ≃ 0.8, P(2) ≃ 0.2 and P(1) ≤ 0.01); (ii) the calculated cross section for excitation of the N_{j} = 1_{1} state (Δj = ΔN − 1) is only slightly less intense than for the N_{j} = 1_{0} state (Δj = ΔN); a significant population of the N_{j} = 1_{1} level would result in a significant signal below the 1_{0} → 3_{2} threshold at 16.3 cm^{1}, which is not observed; and (iii) the calculated cross section for excitation from the 1_{2} level is much weaker (Δj = ΔN − 2); a 20% population results in a small contribution. The theoretical data displayed in Fig. 2b are the sum of the contributions of excitation from the 1_{0}, 1_{2}, and 1_{1} states, weighted to their relative populations at 1 K, and convoluted to experimental energy resolution.
The agreement between experimental values and present theory is good. The agreement becomes almost perfect as for the 1_{0} → 1_{1} transition when using the slightly modified PES. However, this modified PES, tailored for very low collisional energies, can hardly be used for the determination of rate coefficients for temperatures above 30–50 K. This PES accurately describes the bound state energies but the scaling factor of 1.05 leads to an overestimation of the interaction energies in the repulsive wall. For the original PES, the overall good agreement again confirms the accuracy of the calculated PES and of the theoretical approach. We found that rate coefficients obtained from both the original and the modified PES differ by less than 10% at 10K. It suggests that the rate coefficients obtained here are accurate enough to allow improved astrophysical modelling.
3.2. Rate coefficients
By performing a thermal average of the collision energy dependent ICS, we have obtained rate coefficients between the first 27 fine structure O_{2} levels for temperatures up to 150 K. This complete set of (de)excitation rate coefficients is available online from the LAMDA (Schöier et al. 2005) and BASECOL (Dubernet et al. 2013) websites. The thermal dependence of the statetostate O_{2}–H_{2} rate coefficients is illustrated in Fig. 3 for transitions out of the N_{j} = 5_{6} level.
Fig. 3 O_{2}–H_{2} rate coefficients out of the N_{j} = 5_{6} state. 

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A strong propensity rule exists for Δj = ΔN transitions. A propensity rule of this kind was predicted theoretically (Alexander & Dagdigian 1983) and is general for molecules in the electronic state. It was also observed previously for the O_{2}(X)–He (Lique 2010), SO(X)–H_{2} (Lique et al. 2007) or NH(X)–He (Toboła et al. 2011) collisions.
Collisions with He are often used to model collisions with H_{2}. It is generally assumed that rate coefficients with H_{2} should be larger than He rate coefficients; a scaling factor of 1.4 being often used (Schöier et al. 2005). In Table 1, we show, on a small sample, a comparison of our rotational O_{2}–H_{2} rate coefficients versus the O_{2}–He ones of Lique (2010).
Comparison between O_{2}–He and present O_{2}–H_{2} rate coefficients.
As one can see, differences exist between the two sets of data. Globally, the H_{2} results are larger by a factor of ≃2 than the He ones at low temperature and for transitions between the first rotational states. The ratio can even be larger (see e.g. the 1_{1} → 1_{0} transition). However, for transitions between higher energy levels and for temperature above 50 K, He rate coefficients can be equal to or larger than the H_{2} rate coefficients. Therefore, using He rate coefficients scaled by a factor of 1.4 results in an overestimation of the collisional process in modelling of O_{2} emission.
4. Discussion and astrophysical applications
We have presented quantum mechanical calculations of rate coefficients for the collisional excitation of O_{2} by H_{2} molecules. The accuracy of the theoretical calculations has been checked by comparing theoretical and experimental ICS. The rotationally inelastic energy transfer in O_{2} is more efficient with H_{2} molecules than with He atoms at low temperature, but is similar at high temperatures and for high rotational states.
As a first application and in order to test the impact of the new rate coefficients, we have performed radiative transfer calculations for typical physical conditions from where O_{2} emission is detected. Nonlocal thermodynamic equilibrium (nonLTE) calculations were performed with the RADEX code (van der Tak et al. 2007) using the large velocity gradient (LVG) approximation for an expanding sphere. Both collisional and radiative processes are taken into account.
We focus on the calculation of the excitation temperature determined for O_{2} transitions that have been already observed. Figure 4 shows some results of our calculations for a column density of O_{2} fixed at 10^{15} cm^{2}.
Fig. 4 Excitation temperature for the 1_{1} → 1_{0}, 3_{3} → 1_{2}, 5_{4} → 3_{4} and 7_{6} → 5_{6} lines of O_{2}, as a function of volume density for T = 10 and 100 K. The column density of O_{2} is 10^{15} cm^{2}. 

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At very low volume densities, the excitation temperature of four lines tends to be equal to the adopted value of the background radiation field (2.7 K). This parameter rises to higher values as collisional excitation becomes more important. At volume densities above a critical value, the excitation temperature approaches the kinetic temperature, at which point the LTE approximation may be used. At low temperatures (10 K) the critical density lies at ≈10^{3} cm^{3}, while for higher temperatures, the critical density can be up to 2 − 3 × 10^{4} cm^{3} for transitions between highly excited states. This conclusion is in reasonable agreement with the one found by Goldsmith et al. (2011) although we found critical densities a bit larger than in their study. This can certainly be explained by the magnitude of H_{2} rate coefficients between high energy levels of O_{2} that is smaller than the corresponding He ones (scaled by a factor of 1.4) used by Goldsmith et al. (2011).
We also vary the column density of O_{2} (from 10^{14} to 10^{16} cm^{2}) and we have also found although not shown here that the column density has almost no impact on the magnitude of the excitation temperature. This can certainly be explained by the fact that the O_{2} lines are optically thin.
To conclude, at typical physical conditions of molecular clouds where O_{2} is observed (n(H_{2}) >3 × 10^{4} cm^{3}, T = 10 − 100 K), O_{2} lines are mainly thermalised. However, if O_{2} can survive in slightly lower density and/or higher temperature molecular clouds, nonLTE analysis will be required in order to enable an accurate determination of O_{2} abundance.
Acknowledgments
This research was supported by the CNRS national program “Physique et Chimie du Milieu Interstellaire”. We also thank the CPER HauteNormandie/CNRT/Énergie, Électronique, Matériaux. This work extends the objectives of ANR12BS05001102 contract with the Agence Nationale de la Recherche for which financial support is gratefully acknowledged. Scattering calculations were carried out using HPC resources of SKIF Cyberia (Tomsk State University). We thank Alexandre Faure for helpful discussion.
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All Tables
All Figures
Fig. 1 Collisional excitation ICS of O_{2} by H_{2} out of the N_{j} = 5_{6} state. 

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In the text 
Fig. 2 Collisional energy dependence of the integral cross sections for O_{2} excitation . a) Theoretical ICS calculated with modified PES; b) theoretical ICS convoluted with the experimental collision energy spread calculated with the original PES (dashed line) and the modified PES (solid line); experimental data (open circles), obtained when crossing a beam of normalH_{2} from a first pulsed nozzle at ≃45 K propagating at velocity v_{1} = 951 ms^{1} with a beam of 0.5% O_{2} in Ne from a second pulsed nozzle at 300 K propagating at v_{2} = 803 ms^{1}; crossing angle χ was varied from 22.5° to 45°, resulting in a collision energy where μ stands for the reduced mass of the collision partners. 

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In the text 
Fig. 3 O_{2}–H_{2} rate coefficients out of the N_{j} = 5_{6} state. 

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In the text 
Fig. 4 Excitation temperature for the 1_{1} → 1_{0}, 3_{3} → 1_{2}, 5_{4} → 3_{4} and 7_{6} → 5_{6} lines of O_{2}, as a function of volume density for T = 10 and 100 K. The column density of O_{2} is 10^{15} cm^{2}. 

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In the text 
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