Inelastic rate coefficients for collisions of C₄H⁻ with H₂

ABSTRACT

Carbon-chain anions were recently detected in the interstellar medium. These very reactive species are used as tracers of the physical and chemical conditions in a variety of astrophysical environments. However, the local thermodynamic equilibrium conditions are generally not fulfilled in these environments. Therefore, collisional as well as radiative rates are needed to accurately model the observed emission lines. We determine in this work the state-to-state rate coefficients of C₄H⁻ in collision with both ortho- and para-H₂. A new ab initio 4D potential energy surface was computed using explicitly correlated coupled-cluster procedures. This surface was then employed to determine rotational excitation and de-excitation cross-sections and rate coefficients for the first 21 rotational levels (up to rotational level j₁ = 20) using the close-coupling method, while the coupled-state approximation was used to extend the calculations up to j₁ = 30. State-to-state rate coefficients were obtained for the temperature range 2–100 K. The differences between the ortho- and para-H₂ rate coefficients are found to be small.

scattering, molecular data, molecular processes, ISM: abundances

1 INTRODUCTION

Although the presence of anions in the interstellar medium (ISM) was predicted decades ago (Dalgarno & McCray 1973; Herbst 1981), the first detection of a negative ion, the C₆H⁻ linear carbon chain, is relatively recent (McCarthy et al. 2006). This is certainly due to the lack of laboratory spectroscopic data. Between 2006 and 2010, five new anions were detected: C₄H⁻ (Cernicharo et al. 2007; Agúndez et al. 2008; Sakai, Sakai & Yamamoto 2008), C₈H⁻ (Brünken et al. 2007; Kawaguchi et al. 2007; Remijan et al. 2007), CN⁻(Agúndez et al. 2010), C₃N⁻ (Thaddeus et al. 2008), and C₅N⁻ (Cernicharo et al. 2008), all in IRC+10216 and TMC-1.

Carbon-chain anions play an important role in the chemical evolution in the ISM. However, the abundance of anions was found to be smaller by several orders of magnitude compared to their neutral parents, and nowadays chemical models cannot reproduce the observed anion-to-neutral ratio. In the cool low-density regions where anions are observed, the local thermodynamic equilibrium approximation is generally not valid and a proper modelling of the emission lines requires radiative and collisional rates with the most abundant species, which is H₂.

Collisional rate coefficients were published for the anionic carbon chains CN⁻ (Kłos & Lique 2011; collision with H₂), C₂H⁻ (Dumouchel et al. 2012; collision with He), C₆H⁻ (Walker et al. 2016, 2017; collision with He and H₂), and C₃N⁻ (Lara-Moreno, Stoecklin & Halvick 2019; collision with H₂). Very recently, collisional rate coefficients for C₄H⁻ in collision with para-H₂ (j₂ = 0) were calculated using a 2D potential energy surface (PES) built as an average of the 4D C₄H⁻–H₂ interactions over different orientations of the H₂ molecule (Senent et al. 2019). However, no collisional data were generated for the ortho-H₂ colliders, whereas the C₄H⁻ anion is detected in media where the ortho-H₂ abundance is found to be relatively large (Cernicharo et al. 2007). Therefore, it seems useful to calculate C₄H⁻–H₂ rate coefficients using a 4D PES, necessary to obtain both ortho- and para-H₂ rate coefficients.

In this paper, we report the first 4D C₄H⁻–H₂ interaction PES and inelastic rate coefficients for C₄H⁻ in collision with both ortho- and para-H₂. The ab initio PES was computed using explicitly correlated coupled-cluster-based procedures. Quantum-scattering calculations based on the close-coupling (CC) method were performed to compute rotational excitation rate coefficients for the first 21 rotational levels of C₄H⁻ while the coupled-states (CS) approximation was used to extend the calculations to the first 30 rotational levels. Cross-sections were calculated for collisional energies up to 1000 cm⁻¹, leading to de-excitation rate coefficients for temperatures up to 100 K.

Details of the electronic structure calculations are given in Section 2. Section 3 provides a brief description of the scattering calculations. Section 4 presents collisional cross-sections and rate coefficients, and conclusions are given in Section 5.

2 POTENTIAL ENERGY SURFACE

The coordinates used to define the 4D C₄H⁻–H₂ PES are illustrated in Fig. 1. |$\vec{\boldsymbol{ \mathit{ r}}}_1$| and |$\vec{\boldsymbol{ \mathit{ r}}}_2$| are vectors aligned with the fragments. Coordinate R is the centre-of-mass separation, while θ₁ and θ₂ are the angles between |$\vec{\boldsymbol{ \mathit{ R}}}$| and the vectors |$\vec{\boldsymbol{ \mathit{ r}}}_1$| and |$\vec{\boldsymbol{ \mathit{ r}}}_2$| (respectively). φ is the dihedral (out-of-plane) torsional angle. Note that for θ₁ = 0° the C₄H⁻ molecule aligns with the H atom pointing to the H₂ molecule.

Figure 1.

Coordinates used to describe the C₄H⁻–H₂ interaction. R: centre-of-mass separation (length of |$\vec{\boldsymbol{ \mathit{ R}}}$|⁠); θ₁ and θ₂: angles between |$\vec{\boldsymbol{ \mathit{ R}}}$| and the vectors |$\vec{\boldsymbol{ \mathit{ r}}}_1$| and |$\vec{\boldsymbol{ \mathit{ r}}}_2$|⁠; and φ: torsional angle. The employed bond distances for the linear C₄H⁻ molecule are given in  Å (see text).

2.1 Electronic structure calculations

All ab initio calculations were performed using the molpro electronic structure code package (Werner et al. 2012). The non-bonded interaction energies were computed using explicitly correlated coupled-cluster theory (Knizia, Adler & Werner 2009; Werner, Knizia & Manby 2011). Each energy was computed with the fragments interacting as a supersystem at the CCSD(T)-F12b/VTZ-F12 level, without basis extrapolation or counterpoise correction (on the basis of preliminary radial test cuts used to explore sensitivity to basis set completeness). In these tests, the Complete Basis Set (CBS) limit was estimated by extrapolating results from the VTZ-F12 and VQZ-F12 bases using the l⁻³ formula. At the global minimum (GM), the well depth for VTZ-F12 was found to be very close to the CBS value (−793.5 and −794.9 cm⁻¹, respectively), and differences at geometries extending towards the long range are considerably smaller. We rationalize the lack of strong basis set dependence for this anionic system as being due to the closed-shell nature of the anion. Core electrons were excluded from the correlation treatment.

As the lowest bending mode, ν₇, equal to 214 cm⁻¹ (Senent & Hochlaf 2010), is well above the highest C₄H⁻ rotational level considered in this work, the result of previous work (Denis-Alpizar et al. 2013; Balança et al. 2018) supports the expectation of a relatively small impact of (closed) vibrational channels for pure rotational excitation.

The bond distance for H₂ was fixed at r_HH = 0.766 65 Å, the vibrationally averaged bond distance for the ground ro-vibrational state of H₂. For C₄H⁻, the equilibrium geometry is linear and the structure employed in this study was that obtained by optimization at the same level as the high-level PES; the employed bond distances are shown in Fig. 1. To guide the placement of high-level data toward energetically accessible regions, a lower level guide surface was first constructed at the CCSD(T)-F12a/VDZ-F12 level of theory.

2.2 Analytical representation

As we have done in the past for other van der Waals linear dimers (Dawes et al. 2010; Brown et al. 2012; Dawes, Wang & Carrington 2013; Brown et al. 2014; Donoghue et al. 2016; Wang, Carrington & Dawes 2016; Barclay et al. 2018; Bop et al. 2019; Castro-Juárez et al. 2019; Desrousseaux et al. 2019, 2021a; Quintas-Sánchez et al. 2020), the PES analytical representation was constructed using an automated interpolating moving least squares methodology, freely available as a software package under the name autosurf (Quintas-Sánchez & Dawes 2019). This interpolative approach can accommodate arbitrary energy-surface topographies and is advantageous for PESs with large anisotropy, which are challenging for traditional quadrature-type expansions (Dawes & Quintas-Sánchez 2018). The shortest intermonomer centre-of-mass distance considered is R = 2.3 Å, with the additional restriction of a maximum repulsive energy of 6 kcal/mol (⁠|${\sim}2100$| cm⁻¹) above the separated monomers’ asymptote. The initial lower level guide surface was constructed using a set of |$3441$| symmetry-unique points, distributed using a Sobol sequence (Sobol 1976) biased to sample the short-range region more densely. For the high-level PES, the global estimated root-mean-squared fitting error tolerance was set to 0.33 cm⁻¹ for energies below the asymptote, and the total number of automatically generated symmetry-unique points needed to reach that target was |$5122$|⁠. The ab initio data coverage in the fitted PES extends to R = 22 Å.

Fig. 2 shows a 2D cut of the PES as a function of θ₁ and θ₂ for planar configurations (φ = 0°). The plot describes the complete ranges of θ₁ and θ₂ relaxing coordinate R for each pair of angles. Stationary points and transition structures (TSs) are also highlighted in the figure. The PES is characterized by a single minimum (the two molecules aligned, with the H₂ at the C-end of the C₄H⁻ molecule), with a well depth of 793.5 cm⁻¹ and a centre-of-mass separation of R = 5.027 Å. The energies and geometric parameters of the GM and the TS between equivalent minima (cf. Fig. 2) are given in Table 1.

Figure 2.

R-optimized contour plot of the PES as a function of the angles θ₁ and θ₂ for planar configurations (φ = 0°). For each pair of angles, the energy (given in cm⁻¹) is optimized with respect to the centre-of-mass distance R. The positions of stationary points – and the corresponding molecular configurations – are also highlighted.

Table 1.

Geometric parameters and potential energy for the GM and TS of the C₄H⁻−H₂ complex. The corresponding parameters for the GM of the averaged 2D PES (GM_2D) are also included (see text). Units are Angstroms, degrees, and cm⁻¹.

	GM	TS	GM2D
R	5.027	3.136	3.197
θ1	180.0	85.2	90.0
θ2	0.0	103.1	–
φ	0.0	0.0	–
V	−793.5	−295.5	−336.2

Fig. 3 shows the potential as a function of R upon approach from different angular poses, giving some indication of the anisotropy of the interactions. It is worth highlighting that none of the ab initio energies plotted in Fig. 3 were used in the PES-construction process, and thus serve to confirm the overall quality of our fitted potential. In order to better appreciate the anisotropy of the interaction with respect to C₄H⁻ and H₂ rotations, additional 2D plots of the PES are shown in Fig. 4. The plots in Fig. 4 explore the end-on (θ₂ = 0°) and side-on (θ₂ = 90°) approach of the H₂ fragment for planar configurations (φ = 0°). A strong dependence of the PES on the orientation of the H₂ fragment could contribute to different scattering dynamics for para- and ortho-H₂.

Figure 3.

The energy along various radial cuts – defined by different relative orientations (θ₁, θ₂) of the monomers – is plotted for planar configurations (φ = 0°); thus illustrating the significant anisotropy of the PES. Lines represent the fitted PES, points represent ab initio calculations (not used in the fit).

Figure 4.

Contour plots of the PES (for energies below −40 cm⁻¹) showing the highly anisotropic potential for C₄H⁻ interacting with H₂. Upper panel: the plot of end-on approach of the H₂ fragment (θ₂ = 0°) as a function of the intermonomer distance R and the orientation of C₄H⁻ illustrates the collinear well for approach of H₂ to the C-atom end (θ₁ = 180°, see text). Lower panel: the plot of the side-on approach of H₂ (θ₂ = 90°, ϕ = 0°) shows, in contrast, the strong dependence of the interaction on the orientation of H₂.

In previous work by Senent et al. (2019), rotational rate coefficients were obtained for C₄H⁻ in collision with para-H₂ using a PES averaged over five H₂ orientations. When building their 2D PES and testing the electronic structure calculations, a minimum of −780.27 cm⁻¹ was found in the collinear orientation of the two species at a distance of 5.026 Å. This minimum has characteristics very close to those obtained for the GM of our 4D PES (cf. Table 1). However, the PES for positions of H₂ about the C₄H⁻ fragment changes qualitatively in the averaging, and a minimum with a depth of only −301.54 cm⁻¹ was found at θ₁ = 90° (at the side of C₄H⁻) in the resulting 2D PES at R = 3.17 Å. To facilitate comparison, we used our 4D PES to construct a 2D averaged PES using the same 5-point formula as Senent et al. (2019; plotted in Fig. 5). The well depth of our 2D averaged PES is slightly deeper (−336.2 cm⁻¹), while the geometry of the minimum is very similar (parameters are given in Table 1). These differences are due to the higher level of electronic structure theory employed presently. As scattering calculations for para-H₂ are restricted to H₂, j₂ = 0 (see Section 3), the differences between Senent et al. and present rate coefficients are only due to the differences in the PES, not in the scattering methodology. The difference in the well depths of the two 2D potentials could affect the position and the shape of the resonances present at low kinetic energies but the global energy variations of the cross-sections are very similar, leading to small differences in the respective rate coefficients upon energy averaging.

Figure 5.

A plot is provided of a 2D averaged PES constructed by evaluating the full 4D PES using the same 5-point formula reported by Senent et al. (2019), see text.

Fundamentally, since asymptotically the rotational wavefunction for H₂ in j₂ = 0 is a constant (has no angular preference), the C₄H⁻ fragment really does see the approach of the H₂ molecule in an averaged way, which is much different than features of the 4D PES would suggest. Given this extreme quantum delocalization, an interesting aside would be to consider whether classical trajectories can realistically treat the scattering process (perhaps better in two dimensions than four?). Of course in a fully quantum treatment, in the close-interaction region this asymptotic representation starts to break down and higher channels (j₂ > 0) may need to be included for precise results.

3 SCATTERING CALCULATIONS

We consider in this article the collisions of C₄H⁻ with para- and ortho-H₂ described by:

$$\begin{eqnarray*} {\rm C_4H}^-(j_1)+{\rm H_2}(j_2)\rightarrow {\rm C_4H}^-(j^{\prime }_1)+{\rm H_2}(j_2), \end{eqnarray*}$$

(1)

where j₁ and j₂ are the rotational angular momenta of C₄H⁻ and H₂, respectively. The calculations were performed for collisional energies in the range 0.1 cm⁻¹ ≤ E_c ≤ 1000 cm⁻¹, which allows a converged calculation of the rate coefficients for temperatures up to 100 K. The rotational energy levels of C₄H⁻, shown in Table 2, were computed using the experimental rotational constants of Amano (2008): B₀ = 0.1552722, D₀ = 0.196 × 10⁻⁷, and H₀ = −3.1 × 10⁻¹⁵ cm⁻¹.

Table 2.

C₄H⁻ rotational energy levels (in cm⁻¹) obtained from the experimental rotational constants of Amano (2008).

j1	|$\epsilon _{j_{1}}$| (cm−1)	j1	|$\epsilon _{j_{1}}$| (cm−1)
0	0.000	21	71.732
1	0.311	22	78.563
2	0.932	23	85.704
3	1.863	24	93.156
4	3.105	25	100.919
5	4.658	26	108.851
6	6.521	27	117.216
7	8.695	28	125.889
8	11.179	29	134.871
9	13.974	30	144.162
10	17.080	31	153.760
11	20.496	32	163.667
12	24.222	33	173.881
13	28.259	34	184.403
14	32.606	35	195.233
15	37.264	36	206.370
16	42.233	37	217.814
17	47.511	38	229.565
18	53.101	39	241.623
19	59.000	40	253.988
20	65.211	–	–

Due to the strong anisotropy of the C₄H⁻–H₂ PES at short intermolecular distances – that leads to steric hindrance and singularities in the angular expansion – we adopted the regularization procedure proposed by Wernli et al. (2007). The anisotropy of the PES is thus artificially reduced at very high energies by applying a scaling factor such that the regularized PES remains identical to the original potential at low energies, but above a threshold energy V_a tends smoothly to a constant limit value based on the parameter V_b.

We have selected the parameters V_a = 300 cm⁻¹ and V_b = 2000 cm⁻¹. With these values, the regularized PES coincides with the original one for interaction energies below 300 cm⁻¹, it is extremely accurate up to 500 cm⁻¹, and smoothly saturates to a maximum value of 990 cm⁻¹. It should be noted that this regularized PES should not be used for high collision energy calculations. For R distances larger than 22 Å, the PES is smoothly connected to an R⁻³ extrapolation.

Computation of collisional cross-sections was performed using the molscat nonreactive scattering code (Hutson & Green 1994). The modified log-derivative Airy propagator of Alexander & Manolopoulos (1987) was used to solve the scattering equations from R = 2 to R = 42 Å at low energies, with a decrease in maximum propagation to R = 22 Å for the highest energies. The rotational excitation rate coefficients |$k_{j_1 \rightarrow j^{\prime }_1}(T)$| were obtained for temperatures up to 100 K by averaging the computed cross-sections |$\sigma _{j_1 \rightarrow j^{\prime }_1}(E)$| as:

$$\begin{eqnarray*} k_{j_1 \rightarrow j^{\prime }_1}(T) &=& \left(\frac{8k_{\mathrm{ B}}T}{\pi \mu }\right)^{\frac{1}{2}}\left(\frac{1}{k_{\mathrm{ B}}T}\right)^{2} \nonumber\\ &&\times \, \int _{0}^{\infty }E_{\mathrm{ k}}\,\, \sigma _{j_1 \rightarrow j^{\prime }_1} (E_\mathrm{ k})\,\, \mathrm{ e}^{\frac{-E_k}{k_{\mathrm{ B}}T}}\mathrm{ d}E_\mathrm{ k}, \end{eqnarray*}$$

(2)

where |$E_\mathrm{ k}=E-\epsilon _{j_1}$| is the relative kinetic energy, μ = 1.9336023 is the reduced mass of C₄H⁻–H₂, and k_B is the Boltzmann constant. The rate coefficients for the reverse transitions are given by the detailed balance relation.

The interaction potential V(R, θ₁, θ₂, φ) was expanded over a suitable basis of angular functions for any intermolecular distance R (Green 1975):

$$\begin{eqnarray*} V(R,\theta _1,\theta _2,\varphi)=\sum _{l_{1}l_{2}l}v_{l_{1}l_{2}l}(R) A_{l_1 l_2 l} (\theta _1, \theta _2 ,\varphi), \end{eqnarray*}$$

(3)

where |$\nu _{l_{1}l_{2}l}(R)$| are the radial coefficients, and the angular basis functions |$A_{l_1 l_2 l} (\theta _1, \theta _2, \varphi)$| are contracted normalized spherical harmonics. The angular expansion of the PES was carefully analysed. It was found that a PES expansion up to order |$l_1^\text{max}=40$| for C₄H⁻ and up to |$l_2^\text{max}=4$| for H₂ reproduces correctly the interaction. This |$l_1^\text{max}$| value was retained as it leads to cross-sections converged to better than 5 per cent from the results obtained with |$l_1^\text{max}=50$| and avoids prohibitively expensive calculations.

The fully quantal CC approach of Green (1975) was used to compute state-to-state cross sections between all rotational levels up to j₁ = 20. For 21 ≤ j₁ ≤ 30, the CS approximation (McGuire & Kouri 1974) was employed since the numerical solution of CC calculations become prohibitive due to the large basis set needed for convergence. The validity of the CS approach was also tested. Fig. 6 displays CS C₄H⁻–para-H₂ rate coefficients as a function of the CC ones at 10, 50, and 100 K. It was found that the CS approach can lead to inaccuracies of 20–50 per cent at low temperatures (10 K) whereas the two methods tend to agree within 10–30 per cent at higher temperatures (50 K).

Figure 6.

Comparison of the rate coefficients between the 15 first C₄H⁻ rotational levels for collisional de-excitation rate coefficients of C₄H⁻ by para-H₂ obtained at the CS approximation and those obtained at the CC level. The two dashed lines delimit the region where the rate coefficients differ by less than a factor of 1.5.

The large well depth of the PES and the small value (0.155 cm⁻¹) of the rotational constant of C₄H⁻ imply that large basis sets are needed for convergence. Thus, rotational levels up to j₁ = 25 and j₁ = 40 were included into the C₄H⁻ basis set to obtain converged excitation cross-sections between the 21 and 30 first rotational levels, respectively. The H₂ rotational basis included only para-H₂ (j₂ = 0) and ortho-H₂ (j₂ = 1) levels. For different C₄H⁻ rotational transitions, Table 3 displays the convergence of the para-H₂ cross-sections calculated at a total energy of 100 cm⁻¹ as a function of the (⁠|$j_1^\text{max}$|⁠, |$j_2^\text{max}$|⁠) values in the basis sets. The addition of the H₂ (j₂ = 2) level in the basis set, led to cross-sections that are within 10 per cent (on average) of the values obtained using only the H₂(j₂ = 0) level. Hence, only H₂ (j₂ = 0) and H₂ (j₂ = 1) were included in the basis. Hereafter, para- and ortho-H₂ refer to H₂ (j₂ = 0) and H₂ (j₂ = 1), respectively.

Table 3.

Convergence of the rotational bases expressed as (⁠|$j_1^\text{max},j_2^\text{max}$|⁠). Cross-sections (× 10⁻¹⁶ cm²).

Transition	25,0	25,2	30,0	30,2	40,0	40,2
|${j_1 \rightarrow j^{\prime }_1}$|
1 → 0	18.77	18.70	18.88	16.53	18.89	16.70
2 → 0	5.38	4.30	5.66	5.50	5.68	5.28
2 → 1	25.90	28.27	25.20	28.07	25.21	27.52
5 → 4	33.16	35.16	32.70	33.98	32.84	35.22
10 → 5	6.28	7.02	6.38	7.52	6.26	7.02
10 → 9	37.34	38.43	38.11	39.43	37.45	41.75
15 → 10	7.61	9.42	7.94	9.67	7.55	8.97
15 → 14	42.08	40.81	41.62	43.25	43.47	45.23

4 RESULTS

State-to-state inelastic rate coefficients are displayed in Fig. 7 for collisions of C₄H⁻ with both para- and ortho-H₂. After a rapid decrease at very low temperature, the rate coefficients for these selected Δj₁ = −1 transitions exhibit rather constant values and then increase slowly with temperature. Such quasi-constant values are expected for para-H₂ collisions according to the Langevin theory for ion–neutral interactions. The propensity rules for these rotational transitions are also interesting to consider. Fig. 8 shows the de-excitation rate coefficients at T = 50 K out of initial state j₁ = 15 for collisions with para- and ortho-H₂. The rate coefficients decrease continuously with increasing Δj₁, showing a strong propensity for Δj₁ = −1.

Figure 7.

State-to-state rate coefficients for the rotational de-excitation of C₄H⁻ by para-H₂ and ortho-H₂ for selected Δj₁ = −1 transitions as a function of temperature. The rate coefficients are labelled according to the related |$j_1\rightarrow j^{\prime }_1$| transition between the rotational states of C₄H⁻.

Figure 8.

Propensity rule for transitions out of initial state j₁ = 15 for C₄H⁻ in collision with para- and ortho-H₂ at 50 K.

An important result is the high similarity shown in both Figs 7 and 8 between para- and ortho-H₂ rates. This similarity is observed at all temperatures, as shown in Fig. 9, which compares ortho- and para-H₂ de-excitation rate coefficients at 10, 50, and 100 K for all transitions involving the first 16 rotational levels. Difference between the two sets of data is rather small (less than 50 per cent). The agreement improves, with differences less than 10 per cent for the highest values, corresponding to transitions with the smallest Δj₁. Similar behaviour has also been observed in H₂ collisions with other ions, such as CN⁻ (Kłos & Lique 2011), C₆H⁻ (Walker et al. 2017), C₃N⁻ (Lara-Moreno et al. 2019), HCO⁺ (Massó & Wiesenfeld 2014), N₂H⁺ (Balança et al. 2020), and CF⁺ (Desrousseaux et al. 2021b). As previously discussed by Walker et al. (2017), this close agreement between para- and ortho-H₂ rates shows that at the considered temperatures, effects of the long-range part of the interaction outweigh those of the short-range part. Anisotropies in the PES at short intermolecular distances mostly affect transitions with large Δj₁, leading to relatively small rate coefficients.

Figure 9.

Comparison at 10, 50, and 100 K between rate coefficients for the collisional de-excitation of C₄H⁻ by para- and ortho-H₂. The two dashed lines delimit the region where the rate coefficients differ by less than a factor of 1.5.

Fig. 10 presents a comparison between rate coefficients of C₄H⁻ in collision with para-H₂ obtained in this work and those obtained with a PES averaged over H₂ orientation (Senent et al. 2019). The agreement between the two sets of data is good, with differences lower than 10 per cent for the most dominant transitions. Such agreement was also found for rotational excitation of HCO⁺ by para-H₂. It is shown that the 4D PES calculations (Massó & Wiesenfeld 2014; Denis-Alpizar et al. 2020) are in excellent agreement with those obtained by Yazidi, Ben Abdallah & Lique (2014) with an averaged 2D PES. This confirms the relatively small contribution of the H₂ orientation in the dynamics of collision for those ionic systems at low temperatures.

Figure 10.

Comparison between the CC rate coefficients for collisional de-excitation of C₄H⁻ by para-H₂ obtained in this work (k_4D) and those (k_AV) obtained with a PES averaged over H₂ orientation (see text). The two dashed lines delimit the region where the rate coefficients differ by less than a factor of 1.5.

We consider now the comparison of the rotational transition rate coefficients obtained in this work for C₄H⁻–H₂ with those obtained for other large anionic chains such as C₆H⁻ and C₃N⁻. For all three systems, the PESs are not much different, with the GM observed in a collinear configuration and well-depth values in the range 700–800 cm⁻¹. Fig. 11 shows the para-H₂ rate coefficients at T = 50 K (upper panel) and T = 100 K (lower panel) for the de-excitation transition from the initial state j₁ = 15. As can be seen in the figure, the rate coefficients for the three systems are in good agreement for transitions with Δj₁ ≤ 5. For larger Δj₁ values, the C₄H⁻ and C₆H⁻ rate coefficients are very similar, with remarkable agreement for the even |$j^{\prime }_{1}$| values and small differences for the odd |$j^{\prime }_{1}$| values, whereas the C₃N⁻ rate coefficients become progressively smaller. An explanation for these different behaviours could come from differences in angular anisotropy of the PESs at short intermolecular distances.

Figure 11.

The para-H₂ de-excitation rate coefficients out of initial state j₁ = 15 of C₄H⁻ (this work), C₆H⁻ (Walker et al. 2017), and C₃N⁻ (Lara-Moreno et al. 2019) for T = 50 K and T = 100 K as a function of the final state |$j^{\prime }_{1}$|⁠.

5 SUMMARY AND CONCLUSION

A new accurate 4D PES for the C₄H⁻–H₂ system has been calculated at the (CCSD(T)-F12b/VTZ-F12) level of theory. This PES has a large well depth and exhibits strong anisotropies at short intermolecular distances. Rotational rate coefficients for collisions with para- and ortho-H₂ have been computed for temperatures ranging from 2 to 100 K for the first 21 rotational levels of C₄H⁻ using the CC method, while the CS approximation was used to extend the results for levels up to j₁ = 30.

The analysis of the results shows several important results: first of all, both para- and ortho-H₂ rate coefficients have dominant values for Δj₁ = −1 and decrease slowly for increasing Δj₁. Notably, strong similarities between the rate coefficients for collisions with para- and ortho-H₂ were found, in agreement with what was obtained for other ionic systems. Furthermore, the rotational rate coefficients computed for collisions with para-H₂ are in good agreement with those calculated by Senent et al. (2019) using a 2D PES. This behaviour has been also found for other ionic systems (Denis-Alpizar et al. 2020).

The rotational de-excitation rate coefficients for transitions out of the j₁ = 15 level compare well with those obtained for other large anionic chains (C₆H⁻ and C₃N⁻). The agreement is remarkably good for Δj₁ ≤ 5, and thus confirms the dominant effects of the long-range part of the interaction at the considered temperatures for these transitions.

ACKNOWLEDGEMENTS

This work was supported by the CNRS programs ‘Physique et Chimie du Milieu Interstellaire (PCMI)’ co-funded by the ‘Centre National d’Etudes Spatiales (CNES)’ and ‘Programme National de Physique Stellaire (PNPS)’. Parts of the calculations were performed using HPC resources from GENCI-(CINES/IDRIS) (grant no. 2010040883) and work stations at the ‘Centre Informatique’ of Paris observatory. This work was also granted access to the HPC resources of MesoPSL financed by the ‘Région Île de France’ and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the ‘programme Investissements d’Avenir’ supervised by the ‘Agence Nationale pour la Recherche’. RD and EQ-S are supported by the US Department of Energy (Award DE-SC0019740). Computing resources were supported by the National Science Foundation under grant no. OAC-1919789.

DATA AVAILABILITY

The resulting rate coefficients will be made available in the lambda (Schöier et al. 2005) and basecol (Dubernet et al. 2013) data bases.