Collisional excitation of H₂S by molecular hydrogen

ABSTRACT

Accurate estimates of the abundance of H₂S, and inferences about the unmeasured H₂ density, require accurate knowledge of radiative and collisional rate coefficients. Time-independent close-coupling quantum scattering calculations have been employed to compute rate coefficients for (de-)excitation of para- and ortho-H₂S in collisions with para- and ortho-H₂. These calculations utilized a potential energy surface for the interaction of H₂S with H₂ recently computed by the explicitly correlated CCSD(T)-F12a coupled-cluster method. Rate coefficients for temperatures ranging from 5 to 500 K were calculated for all transitions among the first 19 rotational levels of H₂S, whose energies are less than or equal to 405 K. These rate coefficients are compared with previous estimates of these quantities.

molecular data, molecular processes, ISM: abundances

1 INTRODUCTION

The hydrogen sulfide (H₂S) molecule, an isovalent analogue of water, was first detected in the interstellar medium by Thaddeus et al. (1972). In ground-based observations, only the 1₁₀ – 1₀₁ transition in ortho-H₂S has been observed. Nevertheless, this molecule has been detected in a variety of environments, for example cold, dark clouds (Minh, Irvine & Ziurys 1989), low-density molecular clouds (Tieftrunk et al. 1994), and pre-stellar cores (Vastel et al. 2018).

Crockett et al. (2014) recorded a wide spectral scan of the Orion KL core, using the orbiting Herschel/HIFI instrument. This object is a rich source of H₂S lines, and they observed lines up to j = 9 of the main H|$_2\,^{32}$|S isotopolog, as well lines of the rarer H|$_2\, ^{34}$|S and H|$_2\, ^{33}$|S isotopologs. This source of H₂S is a dense gas, and they estimated a H₂ density of ≥9 × 10⁷ cm⁻³ and a gas kinetic temperature of approximately 120 K.

In several studies (Turner 1996; van der Tak et al. 2003; Crockett et al. 2014; Vastel et al. 2018), non-LTE radiative transfer calculations were performed on the observed H₂S lines in order to estimate the column density and the physical conditions of the clouds, including the H₂ column density, and the kinetic temperature. These calculations require radiative transition rates, which are well-known (Müller et al. 2005), as well as state-to-state rate coefficients for collision-induced rotational transitions induced by the dominant H₂ molecule. Since accurate rate coefficients for H₂S–H₂ inelastic collisions have not been available until this work, different methods have been employed to estimate these rates, as described below.

As a surrogate for H₂S–H₂ rate coefficients, Turner (1996) used rate coefficients computed by Green, Maluendes & McLean (1993) for the H₂O–He system. They assumed that the H₂S–H₂–He rate coefficients would be five times larger than those for collisions of H₂O with helium.

Two groups used the rate coefficient measured by Ball et al. (1999) in a microwave double resonance experiment for the 1₁₀ ← 1₀₁ transition of ortho-H₂S in collisions with helium to calibrate their approximated H₂S–H₂ rate coefficients. Using the measurement of Ball et al. (1999), van der Tak et al. (2003) scaled H₂O–He rate coefficients computed by Green, Maluendes & McLean (1993) and also corrected for the different reduced mass of the H₂S–H₂ system. Crockett et al. (2014) scaled the collision rates to be proportional to radiative line strengths so that the sum of the downward rates equaled a base rate determined by the rate coefficient measured by Ball et al. (1999). They also obtained a different set of approximate H₂S–H₂ rate coefficients using H₂O–H₂ rate coefficients computed by Faure et al. (2007), again scaling with the rate coefficient measured by Ball et al. (1999).

Vastel et al. (2018) have investigated sulphur chemistry in the L1544 pre-stellar core. They find a rich chemistry and detected a total of 21 sulphur-bearing species. The H₂S 1₁₀–1₀₁ transition was the only line for which a double-peaked velocity profile was observed. They were unable to reproduce this profile using the H₂S abundance as a function of radius from chemical modelling with a 3D radiative transfer treatment and employing H₂O–H₂ collisional rate coefficients from Dubernet et al. (2006) as a surrogate for H₂S–H₂ collisional rate coefficients. It would be interesting to repeat this analysis with accurate H₂S–H₂ collisional rate coefficients.

There is clearly a need for rate coefficients for the H₂S–H₂ system to be computed with an accurate potential energy surface. The present author has computed the full H₂S–H₂ PES through explicitly correlated coupled cluster calculations with inclusion of single, double, and (perturbatively) triple excitations [CCSD(T)-F12a], using an aug-cc-PVTZ basis set (Dagdigian 2020). Good agreement was found for the global and local minima found in a previous investigation of the stationary points on the H₂S–H₂ PES by Bartucci et al. (2014).

In this work, this PES has been used in time-independent quantum scattering calculations to compute integral cross-sections and rate coefficients for transitions between para- and ortho-H₂S rotational levels in collisions with para- and ortho-H₂. Section 2 briefly presents the details of the scattering calculations to determine the state-to-state rate coefficients, which are presented and discussed in Section 3. The paper concludes with a discussion in Section 4, in which the rate coefficients computed in this work are compared with rate coefficients for the H₂O–H₂ system (Faure et al. 2007; Dubernet et al. 2009; Daniel et al. 2010; Daniel, Dubernet & Grosjean 2011). Comparison is also made with the H₂S–He system (Ball et al. 1999).

2 SCATTERING CALCULATIONS

Time-independent quantum scattering calculations for collisions of an asymmetric top with a diatomic molecule were carried out with the H₂S–H₂ PES determined by Dagdigian (2020), following the formalism presented previously (Phillips et al. 1994; Valiron et al. 2008). Rotational constants for the ground vibrational levels of H|$_2\, ^{32}$|S and H₂ were obtained from Belov et al. (1995) and Huber & Herzberg (1979), respectively. The scattering calculations were carried out with the Hibridon suite of programs (Hibridon 2012).

In this work, the total angular momenta are designated as j and j₂ for H₂S and H₂, respectively. Rotational levels of H₂S are denoted as |$j_{k_a k_c}$|⁠, where k_a and k_c are the prolate-limit and oblate-limit projection quantum numbers, respectively. As with H₂, H₂S has two nuclear spin modifications that do not interconvert in molecular collisions. The para and ortho levels have the sum k_a + k_c even and odd, respectively. An energy level diagram of H₂S rotational levels has been displayed by Dagdigian (2020). Since the hyperfine splittings in H₂S are small (Viswanathan & Dyke 1984) and not resolved in astronomical spectra, we do not consider this splitting of rotational levels.

State-to-state integral cross-sections for both para- and ortho-H₂S in collisions with para-H₂(j₂ = 0) and ortho-H₂(j₂ = 1) were computed in close-coupling calculations. For the determination of rate coefficients, the cross-sections for each pair of H₂S–H₂ nuclear spin modifications were computed at a total of 931 energies up to a total energy of 2000 cm⁻¹. The convergence of the cross-sections with respect to the spacing of the radial grid, the size of the H₂S rotational basis, and the number of partial waves was verified. H₂S rotational levels up to j = 10, depending upon the total energy, were included in the scattering basis. The scattering basis included H₂ rotational levels j₂ = 0 and 2 for para-H₂ and j₂ = 1 only for ortho-H₂. The integral cross-sections contained contributions from partial waves up to total angular momentum J = 182ℏ, depending upon the total energy.

Collisional rate coefficients for state-to-state H₂S–H₂ transitions were computed for temperatures from 5 to 500 K using the calculated integral cross-sections by averaging over the collision energy E_c:

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

(1)

where μ is the collision reduced mass, and k_B is the Boltzmann constant. The integral in equation (1) was directly computed from the calculated cross-sections up to E_c = 2000 cm⁻¹, and the high-energy tail was obtained by extrapolating the cross-sections out to 3500 cm⁻¹; this upper limit is sufficient for temperatures up to 500 K.

3 RATE COEFFICIENTS

Rate coefficients were computed using equation (1) for transitions between the 19 lowest ortho-H₂S and between the 19 lowest para-H₂S rotational levels in collisions with para-H₂(j₂ = 0) and ortho-H₂(j₂ = 1). For both nuclear spin modifications of H₂S the 19 levels have energies ≤282 cm⁻¹. The complete set of computed de-excitation rate coefficients has been provided in the supplementary information in this article. These data have been prepared in the format of the LAMDA data base (van der Tak et al. 2007). The accuracy of the computed rate coefficients is estimated to be 5 per cent or less.

Fig. 1 presents plots of rate coefficients as a function of temperature for de-excitation from lower rotational levels to the lowest para-H₂S level, namely 0₀₀. Fig. 1(a) displays rate coefficients for collisions with para-H₂(j₂ = 0), while Fig. 1(b) displays rate coefficients for the corresponding transitions in collisions with ortho-H₂(j₂ = 1). As with the corresponding state-to-state cross-sections (Dagdigian 2020), the rate coefficients for collisions with ortho-H₂ are much larger than for the corresponding H₂S transition in collisions with para-H₂.

Figure 1.

Rate coefficients as a function of temperature for de-excitation transitions to the final 0₀₀ level of para-H₂S in collisions with (a) para-H₂(j₂ = 0) and (b) ortho-H₂(j₂ = 1). The initial levels are indicated on the plots.

Comparing Figs 1(a) and (b), we also see that the relative magnitudes of the rate coefficients are notably different for collisions with the two H₂ nuclear spin modifications. For example, the transitions from the 1₁₁ and 3₁₃ levels have similar magnitudes for collision with para-H₂, while the former transition is much larger than the latter in the case of collision with ortho-H₂.

The greater magnitudes of the rate coefficients for collision with ortho-H₂(j₂ = 1) as compared to those for collisions with para-H₂(j₂ = 0) mirrors behaviour for collisions of H₂ with other molecules (Kalugina, Kłos & Lique 2013; Hernández-Vera et al. 2014; Ma et al. 2015; Schewe et al. 2015; Dagdigian 2018). Because of its spherical nature, the j₂ = 0 level of H₂ has a vanishing electric quadrupole moment, unlike the higher j₂ > 0 levels. The full anisotropy of the PES is thus experienced by the higher j₂ levels, leading to larger rate coefficients for these levels, as compared to those for j₂ = 0. Indeed, rate coefficients for collisions of j₂ = 2 should be similar to those for j₂ = 1, as was found for OH–H₂ collisions (Schewe et al. 2015).

Fig. 2 presents plots of rate coefficients as a function of temperature for de-excitation from lower rotational levels to the lowest ortho-H₂S level, namely 1₀₁. Fig. 2(a) displays rate coefficients for collisions with para-H₂(j₂ = 0), while Fig. 2(b) displays rate coefficients for collisions with ortho-H₂(j₂ = 1). As in the case of collisions of para-H₂S, we see in Fig. 2 that the rate coefficients for collisions of ortho-H₂S levels are much larger for collisions with ortho-H₂(j₂ = 1) than for the corresponding transitions in collisions with para-H₂(j₂ = 0). We also observe that the relative magnitudes of the rate coefficients for the transitions to the 1₀₁ level are different in Figs 2(a) and (b).

Figure 2.

Rate coefficients as a function of temperature for de-excitation transitions to the final 1₀₁ level of ortho-H₂S in collisions with (a) para-H₂(j₂ = 0) and (b) ortho-H₂(j₂ = 1). The initial levels are indicated on the plots.

Comparing the plots in Figs 1(a) and 2(a) and the plots in Figs 1(b) and 2(b), we see that for these lower rotational levels the rate coefficients for collision-induced transitions in ortho-H₂S are somewhat larger than those in para-H₂S.

4 DISCUSSION

Since H₂O–H₂ rate coefficients have been used as surrogates for H₂S–H₂ rate coefficients (Crockett et al. 2014; Vastel et al. 2018), it is interesting to compare the overall magnitudes of these two sets of rate coefficients. Fig. 3 compares the rate coefficients for de-excitation transitions in ortho-H₂O and ortho-H₂S induced by collisions with para-H₂(j₂ = 0) and ortho-H₂(j₂ = 1) at temperatures of 5 and 100 K. Fig. 4 presents a similar comparison for para-H₂O and para-H₂S rate coefficients. The rate coefficients for H₂S–H₂ collisions are taken from this work, while those for H₂O–H₂ collisions were obtained from the work by Daniel et al. (2011).

Figure 3.

Comparison of de-excitation rate coefficients between the 12 lowest rotational levels of ortho-H₂O and ortho-H₂S in collisions with (a) para-H₂(j₂ = 0) at 5 K, (b) para-H₂(j₂ = 0) at 100 K, (c) ortho-H₂(j₂ = 1) at 5 K, and (d) ortho-H₂(j₂ = 1) at 100 K. The solid line denotes equal rate coefficients for ortho-H₂O and ortho-H₂S, while the dashed lines represent a factor of 2 difference in the rate coefficients.

Figure 4.

Comparison of de-excitation rate coefficients between the 12 lowest rotational levels of para-H₂O and para-H₂S in collisions with (a) para-H₂(j₂ = 0) at 5 K, (b) para-H₂(j₂ = 0) at 100 K, (c) ortho-H₂(j₂ = 1) at 5 K, and (d) ortho-H₂(j₂ = 1) at 100 K. The solid line denotes equal rate coefficients for para-H₂O and para-H₂S, while the dashed lines represent a factor of 2 difference in the rate coefficients.

We see in both Figs 3 and 4 that the transitions in H₂O with the largest rate coefficients are greater in magnitude than the corresponding transitions in H₂S, that is to say, the points in upper right-hand corner of these figures are to the right of the solid diagonal line. These differences reflect the different topologies of the H₂O and H₂S PES.

By contrast, for H₂S–H₂ transitions with smaller rate coefficients, i.e. those lying on the left-hand side of the plots in these figures, are larger than the rate coefficients for the corresponding transitions in H₂O–H₂. It should be noted that the energy spacings in H₂S are smaller than in H₂O since the rotational constants for the former molecule are smaller. We expect the magnitude of a cross-section, and hence the corresponding rate coefficient, to scale roughly inversely with the size of the energy gap. This is the likely explanation of why the H₂S–H₂ rate coefficients of small magnitude are larger than the corresponding H₂O–H₂ rate coefficients.

It is also interesting to compare individual rate coefficients for H₂S–H₂ versus H₂O–H₂ collisions. H₂S is a near-oblate asymmetric top, while H₂O is a near prolate asymmetric top. The state-to-state propensities for this two systems might be different. Fig. 5 compares rate coefficients for de-excitation transitions for collisions of ortho-H₂S and ortho-H₂O in the initial 3₃₁ level with para- and ortho-H₂. We see that in all cases the 3₂₁ → 3₁₂ transition has the largest rate coefficient. In contrast to collisions of ortho-H₂O, the 3₂₁ → 2₂₁ transition has nearly the same magnitude as the 3₂₁ → 3₁₂ transition for collisions of ortho-H₂S with both para-H₂(j₂ = 0) and ortho-H₂(j₂ = 1). In the latter transition, the oblate-limit projection quantum k_c is conserved, and this may be related to the near-oblate character of the H₂S asymmetric top.

Figure 5.

Rate coefficients as a function of temperature for de-excitation of the 3₂₁ level to lower rotational levels in collisions of (a) ortho-H₂S with para-H₂(j₂ = 0), (b) ortho-H₂O with para-H₂(j₂ = 0), (c) ortho-H₂S with ortho-H₂(j₂ = 1), and (d) ortho-H₂O with ortho-H₂(j₂ = 1). The final levels are indicated on the plots.

It should also be noted that there are significant differences in the PES’s for the H₂S–H₂ and H₂O–H₂ systems. The well depth D_e is much larger for the latter system [235.1 cm⁻¹ (Valiron et al. 2008; van der Avoird & Nesbitt 2011; Wang & Carrington Jr. 2011) versus  146.8  cm ⁻¹ (Dagdigian 2020) for the former]. In addition, the geometries at the global minima are different. For H₂O–H₂, the H₂ axis lies along the C₂ axis of H₂O on the oxygen side of the molecule. By contrast, in H₂S–H₂ the centre of mass of H₂ lies in the plane of the H₂S molecule, the H₂ axis is perpendicular to this plane, and the Jacobi vector R makes an angle of 71.4° with respect to the C₂ axis of H₂S.

Since several studies (van der Tak et al. 2003; Crockett et al. 2014) employed the rate coefficient for the H₂S 1₁₀ ← 1₀₁ transition induced by collisions with helium determined in a microwave double resonance experiment by Ball et al. (1999) for scaling their estimates of H₂S–H₂ rate coefficients, it is interesting to compare the H₂S–H₂ and H₂S–He systems.

In addition to investigating experimentally pressure broadening of the 1₁₀ ← 1₀₁ line and collisional energy transfer in H₂S–He collisions, Ball et al. (1999) also computed a PES for the interaction of H₂S with helium using Møller-Plesset perturbation theory and correlation-consistent basis sets. They employed this PES in quantum scattering calculations to compute pressure broadening of the 1₁₀ ← 1₀₁ line and cross-sections for some transitions out of the lowest level (1₀₁) of ortho-H₂S.

Fig. 6 compares cross-sections computed for the 1₁₀ ← 1₀₁ transition in ortho-H₂S for collisions with para-H₂(j₂ = 0), ortho-H₂(j₂ = 1), and helium. Cross-sections for H₂S–H₂ and H₂S–He collisions are taken from this work and Ball et al. (1999), respectively. As noted above, the cross-sections for collision with ortho-H₂(j₂ = 1) is significantly larger than for collision with para-H₂(j₂ = 0). We also see in Fig. 6 that this cross-section for collision with helium is much smaller than the corresponding cross-section for collision with either para-H₂(j₂ = 0) or ortho-H₂(j₂ = 1). Indeed, the cross-section for helium is roughly an order-of-magnitude smaller than the corresponding cross-section for collision with ortho-H₂(j₂ = 1). Ball et al. (1999) also measured the rate of the 1₁₀ ← 1₀₁ transition for temperatures from 1.36 to 35.3 K. Cross-sections for this transition derived from the measured rate are in reasonable agreement with the cross-section plotted in Fig. 6.

Figure 6.

Cross-sections as a function of the collision energy for the 1₁₀ ← 1₀₁ transition in ortho-H₂S in collisions with para-H₂(j₂ = 0), ortho-H₂(j₂ = 1), and helium.

The smaller cross-sections for the ortho-H₂S 1₁₀ ← 1₀₁ transition in collision with helium as compared for that involving collision with para/ortho-H₂ can be reconciled by differences in the H₂S–He and H₂S–H₂ PESs. The anisotropy of the former PES is much less than for the latter PES, as evidenced by the much smaller well depth D_e for the latter. For example, D_e for H₂S–He is computed (Ball et al. 1999) to be only ∼15 cm⁻¹, while D_e for the H₂S–H₂ PES is equal to  146.8 cm ⁻¹ (Dagdigian 2020).

The approximate H₂S–H₂ rate coefficients estimated by van der Tak et al. (2003) and Crockett et al. (2014) were calibrated with the rate coefficient for the rate coefficient for the ortho-H₂S–He 1₁₀ ← 1₀₁ transition. The comparison of cross-sections for collisions of H₂S with para/ortho-H₂ and helium presented in Fig. 6 suggests that these approximate rate coefficients are significantly smaller than the present accurate values. This, in turn, implies that the critical density will be significantly overestimated with use of these approximate rate coefficients.

Using their two sets of approximate rate coefficients, Crockett et al. (2014) have carried out a non-LTE analysis of the H₂S lines observed in the Orion KL hot core in order to determine column densities, as well as estimating the kinetic temperature and H₂ density. The cloud was assumed to be irradiated by the observed continuum towards IRc2. The opacities of the H|$_2\, ^{32}$|S lines were large. So they fit the peak intensities of the weaker H|$_2\, ^{34}$|S and H|$_2\, ^{33}$|S lines. The best fit of the line intensities over the entire range of upper-level energies yielded the estimates of 130 K for the kinetic temperature and 7.0 × 10⁹ cm⁻³ for the H₂ density. This density was deemed to be unrealistically high since it would imply a very large mass for the cloud.

To address the derived very high derived H₂ density from their analysis, Crockett et al. (2014) considered that the observed continuum was an underestimate of the radiation field seen by the H₂S. They assumed that there was additional radiation at the short wavelength end of the far-IR (i.e. |$\lambda \le 100\, \mu$|m). In this case, their fit yielded a kinetic temperature of 120 K and a somewhat smaller H₂ density of 3.0 × 10⁹ cm⁻³.

As discussed above, the approximate H₂S–H₂ rate coefficients employed by Crockett et al. (2014) are significantly smaller than the accurate values reported here. This suggests that the H₂ density in the Orion KL hot core is actually considerably smaller than that estimated by Crockett et al. (2014). It may not be necessary to postulate a source of radiation impinging on the cloud beyond the observed continuum toward IRc2 in order to estimate a reasonable H₂ density.

SUPPORTING INFORMATION

h2so_h2.dat.

h2sp_h2.dat.

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ACKNOWLEDGEMENTS

The author gratefully acknowledges correspondence with Alex Faure on the H₂O–H₂ system and helpful conversations with David Neufeld.