Collisional excitation of methylene by molecular hydrogen

ABSTRACT

Accurate estimates of the abundance of methylene (CH₂) in the interstellar medium require knowledge of both the radiative and collisional rate coefficients for the transfer of population between rotational levels. In this work, time-independent quantum close coupling calculations have been carried out to compute rate coefficients for the (de-)excitation of ortho- and para-CH₂ in collisions with ortho- and para-H₂. These scattering calculations have employed a recently computed, high-quality potential energy surface, based on the coupled cluster level of theory [RCCSD(T)-F12a], for the interaction of CH₂ in its ground |$\tilde{X} ^3B_1$| electronic state with H₂. The collisional rate coefficients were obtained for all fine-structure transitions among the first 22 and 24 energy levels of ortho- and para-CH₂, respectively, having energies less than 277 cm⁻¹. These rate coefficients are compared with previous calculated values, obtained by scaling data for CH₂–He. In the case of ortho-CH₂, whose levels display hyperfine structure, rate coefficients for transitions between hyperfine levels were also computed, by the M_J randomization approximation. Finally, some simple radiative transfer calculations are presented.

molecular data, molecular processes, ISM: abundances

1 INTRODUCTION

Methylene (CH₂) is an important astrophysical molecule since it is a small organic molecule that is present in the early stage of interstellar chemistry that concerns the formation of large organic molecules (Godard, Falgarone & Pineau des Forêts 2014; Gong, Ostriker & Wolfire 2017; Herbst 2017). Equilibrium models of dense clouds in Orion (Prased & Huntress 1980) have predicted CH₂ abundances to be comparable to, or possibly greater than, CH, a simpler organic molecule that is widely observed. Despite its importance, there have been relatively few astronomical observations of CH₂.

The first definitive detection of CH₂ was accomplished by Hollis, Jewell & Lovas (1995), who observed in the fine-structure components of the ortho-CH|$_2\, 4_{04}-3_{13}$| line in the spectral range 68–71 GHz. These transitions were observed in emission towards the dense molecular ‘hot cores’ associated with Orion KL and W51 and arise from levels with energies of ∼225 K. Observation of a transition involving a level with this high energy is surprising. Lyu, Smith & Bruhweiler (2001) reported tentative detection in absorption of several electronic bands of CH₂ in the vacuum ultraviolet towards HD 154368 and ζ Oph.

Polehampton et al. (2005) significantly extended observations on CH₂ with observation of lines of low rotational levels of both ortho- and para-CH₂. These lines were observed in absorption towards the molecular cloud complexes Sagittarius B2 and W49N using the orbiting ISO Long Wavelength Spectrometer. Recently, Jacob et al. (2021) observed emission in the 4₀₄−3₁₃ transition towards nine star-forming regions. They conclude that the 4₀₄−3₁₃ transition of CH₂ traces the hot but low-density component of the interstellar medium (ISM) present in PDR layers, with its abundance peaking at the edges of dense clouds.

A challenge for the detection of CH₂ arises from its small moments of inertia and the type-b rotational selection rule. As noted by Jacob et al. (2021), there are only four lines that appear below 1000 GHz. One of these lines falls within atmospheric window, namely the 4₀−3₁₃ line of ortho-CH₂. In addition, the 2₁₂−3₀₃ line of para-CH₂, which falls in the range of 440–445 GHz, should be observable from a high-mountain site. Jacob et al. (2021) searched for this transition in Orion with the APEX 12-m telescope but were not successful in detecting this transition.

The analysis of Jacob et al. (2021) was facilitated by a non-local thermodynamic equilibrium (non-LTE) radiative transfer modelling using collisional rate coefficients computed by Dagdigian & Lique (2018). This work employed the potential energy surface (PES) for the interaction of CH₂ in its ground |$\tilde{X}^3B_1$| electronic state with helium computed by Ma, Dagdigian & Alexander (2012), and rate coefficients between rotational/fine-structure levels were obtained by the recoupling method (Alexander & Dagdigian 1985; Offer, van Hemert & van Dishoeck 1994). The computed CH₂–He rate coefficients were scaled by the collision-reduced mass to estimate rate coefficients for CH₂–H₂. In a recent review of the status of calculations of collisional rate coefficients, van der Tak et al. (2020) conclude that this scaling procedure does not yield highly accurate rate coefficients for collisions with H₂ as the collision partner.

In recent work, Dagdigian (2021) has calculated a PES for the interaction of CH|$_2(\tilde{X}^3B_1)$| with H₂. It should be noted that CH₂ possesses a low-lying electronic state (⁠|$\tilde{a}^1A_1$|⁠) that lies ca. 3000 cm⁻¹ above the |$\tilde{X}^3B_1$| state. Under the conditions of the ISM, the population of the |$\tilde{a}^1A_1$| state is negligible. To take account of the floppy nature of CH|$_2(\tilde{X}^3B_1)$|⁠, the interaction was averaged over the v = 0 bending vibrational probability distribution. In the present work, the recoupling method (Alexander & Dagdigian 1985; Offer et al. 1994) is applied to obtain accurate CH₂–H₂ collisional rate coefficients for transitions among rotational/fine-structure levels. The M_J randomization approximation (Alexander & Dagdigian 1985) was employed to compute rate coefficients for hyperfine-resolved transitions in ortho-CH₂, which displays hyperfine splittings.

Section 2 briefly describes the calculation of the PES for CH₂–H₂ and the recoupling and M_J randomization methods to compute collisional rate coefficients for transitions among the CH₂ rotational/fine-structure and hyperfine levels. Section 3 reports the calculated fine-structure-resolved rate coefficients for collisions with ortho- and para-H₂. These rate coefficients are compared with the previously derived (Dagdigian & Lique 2018) CH₂–H₂ rate coefficients, which were obtained by rescaling CH₂–He rate coefficients by the collision-reduced mass. In Section 4, a simple astrophysical application of these newly computed rate coefficients is presented. The paper finishes in Section 5 with a brief conclusion.

2 THEORETICAL METHOD

Dagdigian (2021) employed the spin restricted explicitly correlated coupled cluster method with inclusion of single, double, and (perturbatively) triple excitations [RCCSD(T)-F12a] method (Adler, Knizia & Werner 2007; Knizia, Adler & Werner 2009) with a correlation consistent basis (aug-cc-pVTZ) to calculate the PES for the interaction of CH|$_2(\tilde{X}^3B_1)$| with H₂. The equilibrium structure of the ground |$\tilde{X}^3B_1$| electronic state is bent (Jensen, Bunker & Hoy 1982); however, the barrier to linearity is small (∼2000 cm⁻¹), and the bending vibrational wavefunctions extend over a wide spread of bending angles (Ma et al. 2012). As was done in the previous calculation of the PES for the CH|$_2(\tilde{X}^3B_1)$|–He interaction (Ma et al. 2012), the CH₂–H₂ PES was computed for several values (a total of four) of the bending angle (Dagdigian 2021). The PES was then determined by weighting the calculations at the different bending angles by the probability distribution of the v = 0 bending vibrational level. The computed energies as a function of the geometry of the complex were fit to a form appropriate for use in scattering calculations (Phillips et al. 1994).

Since the CH|$_2(\tilde{X}^3B_1)$| state has triplet spin multiplicity (s = 1), each rotational level, labelled |$n_{k_ak_c}$|⁠, where n is the total rotational angular momentum and k_a and k_c are the prolate- and oblate-limit projection quantum numbers, respectively, is split into three fine-structure levels, j = n + 1, j = n, j = n − 1, except for n = 0. The hydrogen nuclei have nuclear spin I = 1/2. Consequently, CH₂ has two nuclear spin modifications, labelled ortho (I_tot = 1) and para (I_tot = 0). Because of the non-zero nuclear spin, the ortho fine-structure levels are further split into three (except for j = 0 levels) hyperfine levels; the hyperfine splittings are much smaller than the fine-structure splittings. Since the para (I_tot = 0) levels have zero nuclear spin, there is no hyperfine splitting for this nuclear spin modification. The rotational angular momentum of the H₂ collision partner is denoted as j₂.

Fig. 1 presents the low-lying rotational energy levels of ortho- and para-CH₂, as well as the radiative transitions connecting the levels. The first observed transition, detected by Hollis et al. (1995) and recently by Jacob et al. (2021), is denoted in red in the figure, while the transitions observed by Polehampton et al. (2005) with an orbiting telescope are marked in blue. As mentioned in Section 1, Jacob et al. (2021) also attempted, unsuccessfully, to detect the 444-GHz transition in para-CH₂.

Figure 1.

Rotational levels of ortho- and para-CH₂. The quantum numbers |$n_{k_ak_c}$| are indicated for each level, and the approximate frequencies (in GHz) connecting the levels are also given. The first transition detected with a ground-based telescope (at 69 GHz) is marked in green, while the transitions observed by an orbiting telescope are indicated in red.

The functional form describing the PES was employed in time-independent quantum-scattering calculations to determine spin-free cross-sections for transitions between the rotational levels of CH₂ induced by collisions with H₂ (Dagdigian 2021). Since the nuclear spin is unaffected by molecular collisions, collisions cannot interconvert ortho- and para-CH₂. Similar considerations apply to the H₂ collision partner. Hence, four separate sets of scattering calculations need to be carried out to describe CH₂–H₂ collisions. Full close-coupling calculations were carried out with the Hibridon suite of programs (Alexander et al. 2011).

The scattering calculations were carried out up to a total energy of 1000 cm⁻¹. Convergence of the integral cross-sections with respect to the CH₂ rotational basis, spacing of the radial grid, and the number of partial waves was checked. The calculations included total angular momenta up to J = 76 ℏ, depending on the total energy. The H₂ rotational basis included j₂ = 0, 2 for para-H₂ and j₂ = 1 only for ortho-H₂. Spin-free integral cross-sections were computed between CH₂ rotational levels up to the 5₁₅ level (289 cm¹) for ortho-CH₂ and the 5₁₄ level (306 cm⁻¹) for para-CH₂.

Because of the floppy nature of the CH₂(³B₁) electronic state, its rotational energies are poorly described by conventional rotational energy formulas. Jensen (1988) developed a Morse oscillator-rotational bender internal dynamics Hamiltonian to describe such floppy triatomic molecules. This model was fit to spectroscopic data and used to compute non-rigid rotational energies. These energies were employed as the asymptotic rotational energies in the scattering calculations.

The fine-structure splittings of the rotational levels are approximately 0.5 cm⁻¹ or less and are much smaller than the rotational spacings. The cross-sections for transitions between rotational/fine-structure levels are thus computed using the recoupling method (Alexander & Dagdigian 1985; Offer et al. 1994), which has been widely used to compute fine- and/or hyperfine-resolved cross-sections and rate coefficients (Faure & Lique 2012). Here, the T matrix elements with inclusion of the electron spin can be obtained from spin-free T matrix elements as described below. The recoupling method is an essentially exact method for computing cross-sections between fine-structure levels, with only the assumption that the fine-structure splittings of the rotational levels are negligible compared to the rotational spacings.

To compute cross-sections for transitions between CH₂ rotational/fine-structure levels, the T matrix elements with inclusion of the electron spin are obtained in the recoupling method from spin-free T matrix elements. The total angular momentum J_T of the collision complex when the electron spin s = 1 of CH₂ is included equals J + s, where J is the total angular momentum of the spin-free collision pair. The T matrix elements describing the collision of two molecules including the electron spin is given by the following expression (Offer et al. 1994):

$$\begin{eqnarray*} T^{J_\mathrm{ T}}_{j^{\prime } n^{\prime } k^{\prime }_a k^{\prime }_c j^{\prime }_2 j^{\prime }_R l^{\prime },j n k_a k_c j_2 j_R l} &=& \sum _{Jj^{\prime }_{12}j_{12}} (-1)^{j_R+j^{\prime }_R+l+l^{\prime }+j_2+j^{\prime }_2} \nonumber \\ &&\times \, ([j][j^{\prime }][j_{12}][j^{\prime }_{12}][j_R][j^{\prime }_R])^{1/2} (2J+1) \nonumber \\ &&\times \left\lbrace {\begin{array}{*{20}{c}}n &\quad j_2 &\quad j_{12} \\ l &\quad J &\quad j_R \end{array}} \right\rbrace \left\lbrace {\begin{array}{*{20}{c}}n^{\prime } &\quad j^{\prime }_2 &\quad j^{\prime }_{12} \\ l^{\prime } &\quad J &\quad j^{\prime }_R \end{array}} \right\rbrace \nonumber \\ &&\times \left\lbrace {\begin{array}{*{20}{c}}j_R &\quad n &\quad J \\ s &\quad J_T &\quad j \end{array}} \right\rbrace \left\lbrace {\begin{array}{*{20}{c}}j^{\prime }_R &\quad n^{\prime } &\quad J \\ s &\quad J_T &\quad j^{\prime } \end{array}} \right\rbrace \nonumber \\ &&\times \, T^J_{n^{\prime } k^{\prime }_a k^{\prime }_c j^{\prime }_2 j^{\prime }_{12} l^{\prime },n k_a k_c j_2 j_{12} l} \end{eqnarray*}$$

(1)

where [x] = 2x + 1. The quantum numbers l, j₁₂, and j_R denote the orbital angular momentum of the collision complex, the vector sum of n + j₂, and the vector sum of j₂ + l, respectively. The initial and final levels are indicated by unprimed and primed quantum numbers, respectively.

Integral cross-sections for transitions between CH₂ rotational/fine-structure levels induced by collisions with para-H₂ (j₂ = 0) and ortho-H₂(j₂ = 1) can be computed using equation (1):

$$\begin{eqnarray*} \sigma _{j^{\prime } n^{\prime } k^{\prime }_a k^{\prime }_c j^{\prime }_2, j n k_a k_c j_2} &=& \frac{\pi }{k_\mathrm{ i}^2 (2j+1)(2j_2+1)} \sum _{J_\mathrm{ T}}(2J_\mathrm{ T}+1) \nonumber \\ &&{\times \, \sum _{ll^{\prime }j_\mathrm{ R}j^{\prime }_\mathrm{ R}} \left|T^{J_\mathrm{ T}}_{j^{\prime } n^{\prime } k^{\prime }_a k^{\prime }_c j^{\prime }_2 j^{\prime }_\mathrm{ R} l^{\prime },j n k_a k_c j_2 j_\mathrm{ R} l } \right|^2 } \end{eqnarray*}$$

(2)

Here, k_i denotes the initial wave vector.

To compute cross-sections for hyperfine transitions between ortho-CH₂, the M_J randomization approximation (Alexander & Dagdigian 1985; Faure & Lique 2012) was employed. In this approach, the hyperfine-resolved cross-section is assumed to be proportional to the degeneracy (2F + 1) of the final level and is obtained from the cross-section involving the related rotational/fine-structure levels:

$$\begin{eqnarray*} \sigma _{nk_ak_cjF,n^{\prime }k^{\prime }_ak^{\prime }_cj^{\prime }F^{\prime }} = \frac{(2F^{\prime }+1)}{(2I+1)(2j^{\prime }+1)}\sigma _{nk_ak_cj,n^{\prime }k^{\prime }_ak^{\prime }_cj^{\prime }}. \end{eqnarray*}$$

(3)

Equation (3) cannot be used to compute cross-sections for quasi-elastic transitions since the cross-sections for elastic transitions are approximately two orders of magnitude larger than those for elastic transitions. Hence, the cross-sections for quasi-elastic transitions have been set to zero (Faure & Lique 2012).

Rate coefficients for fine- or hyperfine-resolved state-to-state CH₂–H₂ transitions were calculated for temperatures from 5 to 300 K using the calculated integral cross-sections by averaging over the collision energy E_c:

$$\begin{eqnarray*} k_{\mathrm{ i}\rightarrow \mathrm{ f}}(T) = {\left(\frac{8}{\pi \mu (\mathrm{ k_B} T)^3} \right)^{1/2}} \int _0^{\infty } \sigma _{\mathrm{ i}\rightarrow \mathrm{ f}}(E_\mathrm{ c}) E_\mathrm{ c} \mathrm{ e}^{-E_\mathrm{ c}/\mathrm{ k_B} T} \mathrm{ d}E_\mathrm{ c}. \end{eqnarray*}$$

(4)

Here, μ is the collision-reduced mass, and k_B is the Boltzmann constant. The integral in equation (4) was directly computed from the calculated cross-sections up to E_c = 1000 cm⁻¹, and the high-energy tail was obtained by extrapolating the cross-sections out to 2000 cm⁻¹.

3 RATE COEFFICIENTS

In previous work (Dagdigian 2021), integral cross-sections for transitions between rotational/fine-structure levels of ortho-CH₂ and para-CH₂ induced by collisions with para-H₂ (j₂ = 0) and ortho-H₂ (j₂ = 1) were presented and compared with cross-sections for corresponding transitions in helium. The cross-sections for collisions of both ortho-CH₂ and para-CH₂ were generally found to be much larger than for CH₂–He.

In this work, rate coefficients for collisions of CH₂ with H₂ are presented and discussed. These rate coefficients are also compared with the previously determined (Dagdigian & Lique 2018) CH₂–H₂ rate coefficients. Specifically, rate coefficients between the 22 and 27 rotational/fine-structure levels of ortho-CH₂ and para-CH₂, respectively, having energy ≤306 cm⁻¹, in collisions with para-H₂ (j₂ = 0) and ortho-H₂(j₂ = 1) were computed using equations (1)–(4). These rate coefficients have been prepared in the format of the LAMDA data base (van der Tak et al. 2007) and are available in the supporting information (files CH2o-H2-nonuc.dat and CH2p-H2.dat).

Rate coefficients versus temperature have also been computed for the transitions between the 70 ortho-CH₂ hyperfine levels having energies ≤306 cm⁻¹, in collisions with para-H₂ (j₂ = 0) and ortho-H₂ (j₂ = 1). These rate coefficients have been collected in the file CH2o-H2.dat in the supporting information. The transition frequencies were taken from the Cologne Database for Molecular Spectroscopy (Müller et al. 2005).

The upper left-hand panel of Fig. 2 presents plots of the rate coefficients versus temperature for the transitions from the 1₁₁ rotational/fine-structure level of ortho-CH₂ to the |$0_{00}\, j^{\prime }=1$| and the |$2_{02}\, j^{\prime }$| rotational/fine-structure levels. The rate coefficient for the |$1_{11}\, j^{\prime }$| levels to the |$0_{00}\, j^{\prime }=1$| level in collision with ortho-H₂ (j₂ = 1) are approximately twice as large as the corresponding rate coefficient for collision with para-H₂ (j₂ = 0). This is consistent with the results for other molecule–H₂ systems (Kalugina, Kłos & Lique 2013; Hernández-Vera et al. 2014; Ma et al. 2015; Dagdigian 2020b). This difference can be explained by the difference in the H|$_2\, j_2=0$| and 1 rotational wavefunctions: The j₂ = 0 wavefunction possesses no quadrupole moment, unlike the j₂ ≥ 1 wavefunctions. Hence, the latter can feel the full anisotropy of the PES.

Figure 2.

Rate coefficients as a function of temperature for de-excitation of the ortho 1₁₁ rotational/fine-structure levels to the |$0_{00}\, j^{\prime }=1$| and |$2_{02}\, j^{\prime }=1$| (solid lines), j′ = 2 (dashed lines), j′ = 3 (dot–dashed lines) levels in collisions with para-H₂ (j₂ = 0) (blue lines), ortho-H₂ (j₂ = 1) (red lines), and helium, scaled to the CH₂–H₂ collision-reduced mass (black lines).

Rate coefficients for the 1₁₁ – 0₀₀ transition for collisions with helium, rescaled to the CH₂–H₂ collision-reduced mass are also plotted in the upper left-hand panel of Fig. 2. These are seen to be much smaller than the rate coefficients for collisions with ortho-H₂ and para-H₂.

Fig. 2 also displays rate coefficients for transitions from the |$1_{11}\, j=1$| to the |$2_{02}\, j^{\prime }=1$|⁠, 2, and 3 rotational/fine-structure levels in collisions with para-H₂ (j₂ = 0) and ortho-H₂ (j₂ = 1). In all cases, the rate coefficient for collision with ortho-H₂ (j₂ = 1) is larger than the corresponding rate coefficient for collision with para-H₂ (j₂ = 0). Also plotted in Fig. 2 are rate coefficients for collisions with helium, rescaled to the CH₂–H₂ collision-reduced mass. It can be seen that the latter rate coefficients are generally much smaller than the corresponding rate coefficients for collisions with ortho-H₂ (j₂ = 1) or para-H₂ (j₂ = 0). There are two notable exceptions to this trend, namely the j = 0 – j′ = 3 and j = 1 → j′ = 3 lines.

Rate coefficients for rotational/fine-structure transitions connecting the levels associated with the 69-GHz transition (see Fig. 1), namely the |$4_{04}\, j$| – |$3_{13}\, j^{\prime }$| lines, are presented in Fig. 3 for collisions with ortho-H₂ (j₂ = 1), para-H₂ (j₂ = 0), and helium, rescaled to the CH₂–H₂ collision-reduced mass. The rate coefficients for these transitions are also seen to be much larger for collisions with ortho-H₂ (j₂ = 1) than with collisions with para-H₂ (j₂ = 0). We also see that for all three collision partners by far the largest rate coefficients involve transitions with Δj = Δn (j = 3 – j′ = 2, j = 4 – j′ = 3, |$j=5\, \rightarrow \, j^{\prime }=4$|⁠). We also see in Fig. 3 that the rate coefficients for the Δj = Δn lines with scaled helium as the collision partner are much smaller than for the corresponding rate coefficients for ortho-H₂ (j₂ = 1) as the collision partner but only slightly larger than for para-H₂ (j₂ = 0) as the collision partner.

Figure 3.

Rate coefficients as a function of temperature for de-excitation of the ortho 4₀₄ rotational/fine-structure levels to the |$3_{13}\, j^{\prime }=2,3,4$| levels in collisions with para-H₂ (j₂ = 0) (blue lines), ortho-H₂ (j₂ = 1) (red lines), and helium, scaled to the CH₂–H₂ collision-reduced mass (black lines). The small cross-sections are not labelled; transitions to the j′ = 2, j′ = 3, and j′ = 4 levels are plotted with solid, dashed, and dot–dashed lines, respectively.

A similar Δj = Δn propensity rule for fine-structure resolved transitions was derived by Alexander (1982) in collisions of a diatomic molecule in a ²Σ⁺ electronic state with a structureless target. This propensity rule is a consequence of the fact that the electron spin is a spectator in a molecular collision and cannot be collisionally reoriented. Hence, the coupling of the rotational and electron spin angular momenta remains essentially unchanged in a molecular collision. This propensity has also been observed in collisions of other open-shell triatomic molecules, e.g. HCO (Dagdigian 2020a). This propensity breaks down for transitions between rotational levels both with small angular momenta. Hence, the rate coefficients for transitions from the 1₁₁ to the 2₀₂ rotational/fine-structure levels, shown in Fig. 2, do not display this propensity.

In Fig. 4, rate coefficients for transitions in para-CH₂ are considered, specifically for fine-structure resolved transitions from the 1₁₀ to the 1₀₁ rotational levels. Rate coefficients involving of collisions with ortho-H₂ (j₂ = 1) are significantly larger than the corresponding rate coefficients involving collisions of para-H₂ (j₂ = 0). The j = 0 – j′ = 0 transitions for collision with para-H₂ (j₂ = 0) is missing in the upper left-hand panel; this transition is forbidden. In the case of the |$1_{10}\, j=0$| initial fine-structure level, the largest rate coefficients are for the j′ = 1 final level; the j′ = −2 final level has the largest rate coefficient for the other initial fine-structure levels (j′ = 1 and 2).

Figure 4.

Rate coefficients as a function of temperature for de-excitation of the para 1₁₀ rotational/fine-structure levels to the 1₀₁ levels in collisions with para-H₂ (j₂ = 0) (blue lines) and ortho-H₂ (j₂ = 1) (red lines).

It is also interesting to make a global comparison of the rate coefficients for collisions of CH₂ with para-H₂ and helium, the latter scaled by the CH₂–H₂ collision-reduced mass. Fig. 5 presents a comparison of the rate coefficients at a temperature of 150 K for ortho-CH₂ and para-CH₂ with para-H₂ (j₂ = 0) and helium scaled by the CH₂-H₂ collision-reduced mass. We see that the rate coefficients for ortho-CH₂–para-H₂ collisions are generally well predicted (to within a factor 2) by the scaled helium rate coefficients. The correlation of rate coefficients for para-CH₂–para-H₂ collisions with the corresponding scaled helium rate coefficients is much poorer.

Figure 5.

Comparison of the rate coefficients at 150 K for collisions of ortho-CH₂ and para-CH₂ with para-H₂ (j₂ = 0) and helium scaled by the CH₂-H₂ collision-reduced mass. The dashed lines represent a factor of 2 difference in the rate coefficients.

As discussed in Section 2, the ortho-CH₂ levels also have hyperfine structure. It is of interest to examine hyperfine-resolved rate coefficients for this nuclear spin modification. Fig. 6 presents rate coefficients versus temperature for selected hyperfine transitions from the |$1_{11}\, j^{\prime }=1\, F=2$| rotational/fine-structure level to hyperfine levels associated with the 0₀₀ and 2₀₂ levels. In addition, rate coefficients for transitions from the |$4_{04}\, j=3\, F=2$| hyperfine levels to hyperfine levels associated with the |$3_{13}\, j^{\prime }=2$| rotational/fine-structure levels are also presented. These rate coefficients may be compared with the rate coefficients for de-excitation of the corresponding rotational/fine-structure levels displayed in Figs 2–4.

Figure 6.

Rate coefficients as a function of temperature for de-excitation of the ortho|$1_{11}\, j=1\, F=2$| hyperfine level to hyperfine levels associated with the |$0_{00}\, j^{\prime }=1$| and |$2_{02}\, j^{\prime }=3$| rotational/fine-structure levels and de-excitation of the |$4_{04}\, j=3\, F=4$| hyperfine levels to hyperfine levels associated with the |$3_{03}\, j^{\prime }=2$| rotational/fine-structure level, in collisions with para-H₂ (j₂ = 0) (blue lines) and ortho-H₂ (j₂ = 1) (red lines).

As a result of the M_J randomization approximation, the rate coefficients are proportional to the degeneracy (2F′ + 1) of the final level. In analogy with the Δj = Δn propensity rule for collision-induced rotational fine-structure transitions, a propensity rule ΔF = Δj applies to collision-induced hyperfine transitions (Alexander & Dagdigian 1985). This propensity should also become stronger as the rotational angular momenta of the initial and final levels are increased. In contrast to the strong propensity seen for changes of the rotational/fine-structure level seen in the 4₀₄–3₁₃ transition (see Fig. 3), the hyperfine transitions between the |$4_{04}\, j=3$| and the |$3_{13}\, j^{\prime }=2$| rotational/fine-structure levels, displayed in the lower left-hand panel of Fig. 6, do not show this propensity.

Faure & Lique (2012) have considered the impact of employing the M_J randomization approximation, and also the sudden approximation, in place of the almost exact recoupling method for the calculation of rate coefficients for hyperfine transitions. Radiative transfer calculations using the large velocity gradient approximation with rate coefficients for the CH–H₂ and HCN–H₂ systems computed with the recoupling method were compared with calculations using the approximate methods. At low-to-moderate optical depths (τ ≤ 10), the approximate methods provide reasonable results as compared to the recoupling approach.

4 EXCITATION OF CH₂ IN THE ISM

In previous work, Dagdigian & Lique (2018) and Jacob et al. (2021) carried out non-LTE radiative transfer calculations on several CH₂ lines in the ISM, with H₂ taken as the collision partner. Of special interest was the 4₀₄–3₁₃ transition of ortho-CH₂ since these lines are observed in emission towards hot cores despite the fact that the highly excited levels of these transitions are expected to be weakly populated at the typical temperatures of hot cores.

In a first application of their computed CH₂–H₂ rate coefficients obtained by CH₂–He rate coefficients rescaled to the CH₂–H₂ collision-reduced mass, Dagdigian & Lique (2018) found that the excitation temperature of the fine-structure lines of the 4₀₄–3₁₃ transition were negative up to a H₂ density of ∼10⁶ cm⁻³, with a kinetic temperature of 100 K, indicative of a population inversion. Moreover, the brightness temperatures were found to be relatively high (T_mb > 0.1 K). These calculations explain why the transition is observed in emission despite the high excitation energies of the upper and lower levels of the levels and show weak maser action.

Jacob et al. (2021) used the CH₂–H₂ rate coefficients obtained by Dagdigian & Lique (2018) to carry out a non-LTE analysis of ortho-CH|$_2\, 4_{04}$|–3₁₃ transition towards W3 IRS5. To constrain the ortho-CH₂ column density, they used the (CH)/(CH₂) ratio determined by Polehampton et al. (2005) from observation of absorption in low-j lines and CH column densities towards IRS5 obtained by Wiesenmeyer et al. (2018) and derived an ortho-CH₂ column density of (4.5 ± 1.7) 10¹⁴ cm⁻². With a range of assumed ortho-CH₂ column densities [(3–7) × 10¹⁴ cm¹⁴ cm⁻²], they compared their observed intensities of the 68- to 70-GHz lines (T_mb ∼ 8–39 mK) with those from the non-LTE analysis to constrain the H₂ density and kinetic temperature to ∼3.7 × 10³ cm⁻² cm⁻² and ∼163 K, respectively.

It is interesting comparing these non-LTE radiative transfer calculations with similar calculations using the CH₂–H₂ rate coefficients obtained in this work. These calculations have been performed with the RADEX code (van der Tak et al. 2007). We consider a uniform spherical geometry of the interstellar cloud, and the escape probability formalism approximation for an expanding sphere was assumed. We take into account radiative and collisional processes, with H₂ as the collision partner, and we assume a cosmic microwave background radiation of 2.73 K. The ratio of ortho- to para-H₂ is computed from the kinetic temperature. The velocity width Δv was taken to be 5 km s⁻¹, consistent with the observations of Jacob et al. (2021). We have taken the ortho-CH₂ column density as 5 × 10¹⁴ cm⁻², or the mid-point of the densities assumed by Jacob et al. (2021) in their calculations. The present set of rate coefficients for ortho-CH₂ involves transitions between hyperfine resolved levels, while the previous set of rate coefficients (Dagdigian & Lique 2018) only considered the fine-structure splittings as found previously.

The left-hand panels of Fig. 7 display the excitation temperatures (T_exc) and brightness temperatures (T_mb) for the three strongest hyperfine lines of the 4₀₄–3₁₃ transition. The excitation temperatures are seen to be negative over the range of H₂ densities 10¹–10⁷ cm⁻³, consistent with the calculations by Dagdigian & Lique (2018), Jacob et al. (2021). The brightness temperatures are seen to reach a roughly constant value of ∼0.8–1.2 K over the H₂ density of 10⁴–10⁸ cm⁻³. The only significant difference with the calculations of Dagdigian & Lique (2018) is that here the brightness temperatures start to decrease for densities greater than |$ca.\, 10^8$| cm⁻³ cm⁻³, while the brightness temperatures remain approximately constant in the calculations of Dagdigian & Lique (2018).

Figure 7.

Computed excitation temperature (T_exc) and brightness temperature (T_mb) for the three strongest lines of the 4₀₄–3₁₃ transition of ortho-CH₂ and the 2₁₂–3₀₃ transition of para-CH₂. The three strongest lines of each transition are plotted, and the frequencies (in GHz) are indicated. For both transitions, the column density was set to 5 × 10¹⁴ cm⁻², and the kinetic temperature and line width fixed at 163 K and 0.5 km s⁻², respectively. The inset in the lower right-hand panel shows an expanded plot over 10¹–10⁶ cm⁻³ H₂ density.

We have also carried out non-LTE radiative transfer calculations for lines of the 2₁₂–3₀₃ transition of para-CH₂ at 444 GHz. The kinetic temperature was set to 163 K. We have followed Jacob et al. (2021) and set the para-CH₂ column density to 5 × 10¹⁴ cm⁻². The excitation temperatures (T_exc) are seen to be small and positive, and increase as a function of the H₂ density, reaching ∼50 K at 10⁹ cm⁻³. The computed brightness temperatures (T_mb) of the lines are seen to be very small (<5 × 10⁻³ K) for H₂ densities. These results are consistent with the calculations that Jacob et al. (2021) performed for this line with the scaled H₂ rate coefficients of Dagdigian & Lique (2018). This further provides an explanation for the fact that Jacob et al. (2021) were not able to detect this transition.

5 SUMMARY

In this work, rate coefficients for the (de-)excitation of ortho-CH₂ and para-CH₂ by para-H₂ (j₂ = 0) and ortho-H₂ (j₂ = 1) have been computed in close coupling time-independent quantum-scattering calculations. Rate coefficients for transitions between rotational/fine-structure levels were obtained using the recoupling method with spin-free cross-sections. Rate coefficients between hyperfine levels of ortho-CH₂ were derived from the rate coefficients between the corresponding rotational/fine-structure levels with the M_J randomization approximation. These rate coefficients were compared with the previously calculated (Dagdigian & Lique 2018) CH₂-H₂ rate coefficients, obtained by rescaling CH₂–He rate coefficients rescaled by the collision reduce mass. The rate coefficients determined in this work are found to be generally larger than the data of Dagdigian & Lique (2018).

As a further comparison of the two sets of CH₂–H₂ rate coefficients, non-LTE radiative transfer calculations were carried out with the present set of CH₂–H₂ rate coefficients on the ortho-CH|$_2\, 4_{04}$| – 3₁₃ transition at 69 GHz and the para-CH|$_2\, 2_{12}$| – 3₀₃ transition at 444 GHz.

The 69-GHz transition was found to have a negative excitation temperature and hence show a population inversion, suggestive of maser action. Despite the high excitation energy of the levels associated with this transition, the computed brightness temperatures were fairly high (T_mb ∼ 0.1 K). These findings are consistent with the non-LTE radiative transfer calculations carried out by Dagdigian & Lique (2018) and Jacob et al. (2021).

The excitation temperature for the 444-GHz transition was computed with the present set of CH₂–H₂ rate coefficients to be quite small (T_exc < 5 × 10⁻³ K) for H₂ densities of less than 10⁶ cm⁻³. The calculated brightness temperature for this transition was found to be very small (T_mb < 5 × 10⁻³ K) over this density range. These results are quite consistent with the calculation by Jacob et al. (2021) using the CH₂–H₂ rate coefficients obtained by Dagdigian & Lique (2018).

For these two lines, and presumably for other lines, non-LTE radiative transfer calculations using the rescaled rate coefficients by Dagdigian & Lique (2018) and the present set of rate coefficients yield similar results. The qualitative conclusions about the 69- and 444-GHz transitions remain the same when the present set of CH₂–H₂ rate coefficients are employed, despite differences in individual rate coefficients between the two sets.

SUPPORTING INFORMATION

CH2o-H2.dat

CH2p-H2.dat

CH2o-H2-nonuc.dat

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DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.