Interaction of C₂H with molecular hydrogen: Ab initio potential energy surface and scattering calculations Free

The ethynyl radical, C₂H(⁠$X̃2Σ+$⁠), is one of the smallest unsaturated hydrocarbon radicals and is an important interstellar molecule. It also plays a role in the combustion of acetylene and appears as an intermediate in the combustion of fuel-rich mixtures of larger hydrocarbons. Laufer and Fahr¹ have reviewed what is known about the reaction kinetics of the ethynyl radical.

This radical was detected in the interstellar medium through the observation of radiative transitions² before its laboratory rotational spectrum was observed.^3–5 Subsequently, C₂H has been detected in a great variety of astrophysical environments, including cold dark clouds,⁶ photon dominated regions,⁷ protoplanetary disks,⁸ pre-stellar cores,⁹ and high-mass star-forming regions.¹⁰ Ethynyl is found to be a very abundant molecule in the interstellar medium and is by far the most abundant unsaturated hydrocarbon. The abundance of ethynyl is sufficiently high that the ¹³CCH and C¹³CH isotopologues have been observed in the interstellar medium.¹¹ Even though the interstellar H to D ratio is only ∼10⁻⁵, the ratio of interstellar C₂D to C₂H is enhanced by a large factor and can be diagnostic for the chemical evolution of the cloud.

Laboratory studies of the rotational spectrum C₂H radical^3–5 have confirmed that the ground electronic state is a ²Σ⁺ state. Because of the nonzero nuclear spin of H (⁠$I=12$⁠), as well as of D (I = 1), the rotational lines of both C₂H and C₂D have a hyperfine structure, which is usually resolved in the astronomical spectra.

There has been a considerable spectroscopic study of its low-lying $Ã2Π$ electronic state^12–14 and excited vibronic levels of the ground state.^15–22 The $Ã2Π$ state is only ∼3600 cm⁻¹ above the ground $X̃2Σ+$ electronic state. Because of the mixing of the ground state with the A′ component of the Π state as the molecule bends from its linear equilibrium structure,²³ there is considerable vibronic mixing, even around the origin of the $Ã−X̃$ transition. The vibronic levels of the ground state near the origin of this transition have considerable mixed character²⁴ and have led to difficulties in the assignment of the bands.¹⁹ It should be noted that two $Ã2Π−X̃2Σ+$ bands near 4000 cm⁻¹ have been detected in the outer region of the circumstellar envelope of CW Leo.²⁵ 

Determination of the abundance of C₂H in the interstellar medium is of particular interest for understanding the chemistry of carbon containing species. Under typical conditions, the rotational levels of C₂H are not in thermodynamic equilibrium because of the low density of the interstellar medium.⁹ Hence, the modeling of the molecular emission intensities requires application of a radiative transfer model (see, for example, Ref. 26). Such modeling requires the availability of radiative transition rates, as well as rate coefficients for rotational transitions induced by collisions with the dominant interstellar species, in particular, the H₂ molecule.

In the absence of rate constants for rotationally inelastic collisions with H₂, rate constants involving collisions with helium are often used as surrogates in radiative transfer calculations.²⁷ Spiefiedel et al.²⁸ computed a potential energy surface describing the interaction of C₂H with helium and employed this surface to compute rate constants as a function of temperature for rotational transitions in C₂H induced by collisions with helium. Since the hyperfine structure is spectrally resolved in most C₂H astronomical observations, these workers employed the recoupling method²⁹ to calculate cross sections for hyperfine-resolved C₂H collisional transitions.

These C₂H–He rate constants provided estimates of the rate constants for C₂H–H₂ inelastic collisions, with a scaling factor to account for the difference in the masses of He and H₂. These scaled C₂H–H₂ rate constants have been employed in the application of radiative transfer models describing C₂H in various astrophysical environments.^30–34 However, based on comparisons of rate constants for collisions of He and H₂ with other molecular species, this scaling leads only to qualitatively correct rate constants for H₂ collisions.^27,35,36

Najar et al.³⁷ carried out a study of the collisional excitation of C₂H with H₂ in which they treated the hydrogen molecule as a spherical collision partner. The potential energy surface for the interaction of C₂H with spherical H₂, i. e., H₂(j = 0), was computed by taking the average energy for three orientations of H₂ with respect to C₂H. Cross sections and rate constants could then be computed in a pseudo-atom molecule scattering calculation; this was much simpler than treating the full molecule–molecule interaction. This potential energy surface was recently employed with the recoupling method²⁹ to compute hyperfine-resolved rate constants for collisions of C₂H and C₂D (with a shifting of the center of mass) rotational/fine-structure levels with para-H₂(j = 0).³⁸ 

In the present study, the interaction of C₂H with H₂ is treated explicitly as a molecule–molecule interaction, and cross sections and rate constants for collisions with both para-H₂(j = 0) and ortho-H₂(j = 1) are computed. As in the previous studies of inelastic collisions of C₂H,^28,37 the molecule is treated as rigid. It should be noted that C₂H can react with hydrogen to yield acetylene through the reaction C₂H + H₂ → C₂H₂ + H. This reaction has been investigated both experimentally and theoretically.^39–43 The reaction proceeds through an early transition state barrier that has a linear HCC–H₂ geometry. A range of barrier heights (2.3–3.1 kcal mol⁻¹, including zero-point energy corrections) has been computed.^39–42 Despite the low barrier, the reaction is inhibited by the early barrier, and the reaction probability was computed in a quantum mechanical calculation⁴² with an 8-dimensional PES⁴¹ to be <0.01 for collision energies up ∼1000 cm⁻¹. The rate constant for this reaction is small at room temperature and below (k₂₉₈ = 5.8 × 10⁻¹³ cm³ molecule⁻¹ s⁻¹).¹ The reactive part of the C₂H–H₂ potential energy surface will not be accessed in the present calculation with the assumption of rigid structures for the collision partners.

Section II describes the calculation of the C₂H–H₂ potential energy surface; Sec. II A presents the ab initio calculation of the surface, while Sec. II B describes our fit of the calculated ab initio points to a form suitable for scattering calculations, and Sec. II C displays various features of the fitted potential energy surface. The scattering calculations are briefly described in Sec. III. The computed cross sections and rate constants are presented in Sec. IV. The paper concludes with a discussion in Sec. V.

Despite the strong vibronic mixing between the $X̃2Σ+$ and $Ã2Π$ electronic states in C₂H when the molecule bends, Najar et al.³⁷ concluded that they could use coupled cluster theory to compute the interaction between linear C₂H(⁠$X̃2Σ+$⁠) and H₂. They carried out complete active space self-consistent field (CASSCF) calculations of the ground ²A′ state and found that the weight of the leading configuration was greater than 0.94 for the range of geometries investigated. We have thus employed coupled cluster theory to compute the full C₂H(⁠$X̃2Σ+$⁠)–H₂ potential energy surface. To take account of the zero point motion in the molecules,⁴⁴ we have taken the H₂ internuclear separation as the average, r_HH = 1.449a₀, over the probability distribution for the v = 0 vibrational level. For C₂H, we take r_CC = 2.299a₀ and r_CH = 1.968a₀ from the substitution structure derived experimental rotational constants.⁴⁵ 

It is possible that the assumption of strict linearity of the C₂H molecule in rotationally inelastic collisions is questionable. Denis-Alpizar et al.⁴⁶ have considered the possible effect of neglecting the bending motion in treating rotationally inelastic collisions of HCN with helium. They find that this assumption barely affects the rotational excitation of the molecule, even for collision energies above the threshold for bending excitation. We verified through calculations of the C₂H energy vs. bond angle that the molecule is linear in its equilibrium geometry. Since the equilibrium geometry, rotational constants, and vibrational modes of HCN and C₂H are similar, we assume that the bending motion of C₂H can be neglected in treating rotationally inelastic collisions. This assumption was also made by Najar et al.³⁷ in their treatment of collisions of C₂H with spherical H₂.

Figure 1 illustrates the coordinates describing the geometry of the C₂H–H₂ complex. The interaction depends upon four coordinates: the angle θ₁ between the C₂H molecular axis and the Jacobi vector R, the angle θ₂ between the H₂ molecular axis and R, the dihedral angle ϕ, and the separation R between the centers of mass of C₂H and H₂.

Restricted coupled cluster calculations with inclusion of single, double, and (perturbatively) triple excitations [RCCSD(T)$$^47,48 were employed to compute points on the C₂H–H₂ potential energy surface. The MOLPRO 2012 suite of computer codes⁴⁹ was employed for these calculations. A counterpoise correction⁵⁰ was applied to the computed interaction energies for each configuration Ω = (θ₁, θ₂, ϕ),

where the energies of the C₂H and H₂ moieties are computed with the basis set for the full complex.

In preliminary calculations, the convergence of the basis set employed was examined. Radial cuts were computed for several orientations around the global minimum of the potential energy surface using the aug-cc-pVTZ, aug-cc-pVQZ, and aug-cc-pV5Z basis sets.⁵¹ It can be seen in Fig. 2 that for all geometries, the interaction is computed to be more attractive when computed with a successively larger basis set. The differences in the interaction energy computed with the aug-cc-pVQZ and aug-cc-pV5Z basis sets are less than 1 cm⁻¹ for R > ∼7.5a₀. As found by Najar et al.,³⁷ the most attractive orientation has θ₁ = 180°, θ₂ = 90°, ϕ = 0°, corresponding to a T-shaped structure with the hydrogen end of C₂H pointing toward the center of the H₂ moiety. For this radial cut, the minimum of the interaction energy equals −132.3 and −133.4 cm⁻¹ at R = 7.83 and 7.82a₀ when computed with the aug-cc-pVQZ and aug-cc-pV5Z basis sets, respectively. This well depth is greater than the computed well depth (−61.0 cm⁻¹) for the interaction of C₂H with spherical H₂,³⁷ as is expected.

The full C₂H–H₂ potential energy surface was computed with the aug-cc-pVQZ basis set. Some initial calculations were carried out with the addition of mid-bond functions.^52,53 However, for many geometries, the calculations did not converge, as the dummy atom would lie too close to a real atom. In the end, the calculations were carried out without the addition of mid-bond functions. The calculations were carried out on a 4-dimensional grid (R,θ₁,θ₂,ϕ), at 28 values of the intermolecular separation R ranging from 4.25 to 20a₀. The angular range covered was 0° < θ₁ < 180°, 0° < θ₂ < 90°, and 0° < ϕ < 180°. The initial angular grid included 858 orientations, with 11 values of θ₁ defined by cos θ₁ = −1 to 1 in steps of 0.2, 6 values of θ₂ defined by cos θ₂ = −1 to 0 in steps of 0.2, and 13 values of ϕ from 0° to 180° in steps of 15°. This choice of the angular grid provides a uniform sampling of the differential solid angle sin dθ₁ sin dθ₂dϕ.⁵⁴ 

Because of the significant anisotropy of the C₂H–H₂ potential energy surface and difficulties in fitting the surface, in particular for small values of R, discussed in Sec. II B, it was found necessary to compute the interaction energy at more orientations than specified above. Further calculations were carried out for 6 additional values of θ₁ over the same range of θ₂ and ϕ values, as well as other orientations. Interaction energies were computed for a total of 1591 and 1643 orientations for R ≥ 5.57a₀ and R ≤ 5a₀, respectively. The total number of geometries for which the interaction energy was computed equals 43 165.

It is convenient to expand the molecule-molecule interaction energy in a basis of bispherical harmonics for each value of R to facilitate the solution of the close-coupling scattering equations. The expansion derived by Green⁵⁵ for the interaction of two linear molecules both in Σ electronic states was employed here. The potential is expressed as

In Eq. (2), the basis functions are defined as

where (….|..) is a Clebsch-Gordan coefficient and the Y_lm are spherical harmonics. Equation (2) can also be written as⁵⁵ 

where $Plm′(θ)$ is an unnormalized Legendre function, defined in terms of the spherical harmonic Y_lm(θ, ϕ) as $Ylm(θ,ϕ)=Plm′(θ)eimϕ$⁠.

For each value of the intermolecular separation R, the computed values of the interaction energy computed on the grid of orientations were fit in a linear least squares procedure to Eqs. (2) and (3). Fits were carried out with different numbers of expansion coefficients, determined by the maximum values of l₁ and l₂ in the sum in Eq. (2). The allowed values of the l are given by the vector addition of l₁ and l₂, with the restriction that l₁ + l₂ + l must be even.⁵⁶ 

For certain orientations at small R, in particular with θ₁ ∼ 180°, the potential is very repulsive. At these geometries, the H end of the C₂H moiety, which is considerably displaced from the C₂H center of mass, comes close to the H₂ molecule. To facilitate the fits, we followed the procedure of Wormer et al.⁵⁶ and damped the interaction energy with a hyperbolic tangent function up to a maximum value of V_max. In the present study, V_max was chosen to equal 20 000 cm⁻¹.

Least squares fits for values of R in the radial grid of the ab initio calculations were carried out with different values of maximum values of l₁ and l₂. A acceptable fit was obtained for $l1max=12$ and $l2max=6$⁠. The choice of these parameters leads to a total of 174 radial expansion coefficients $vl1l2l(R)$ in Eq. (2). Fits with larger values of $l1max$ (=13 and 14) yielded RMS deviations at small R essentially the same as for $l1max$ = 12.

As an indication of the quality of the fit, we compare in Fig. 3 the RMS deviation of the fit to the absolute value of the average of the potential as a function of R. The kink in the absolute value of the average of the potential near R = 5.8a₀ is due to a change in the sign of the average value of the potential. We see in Fig. 3 that the RMS deviation of the fit is very small at large R and increases as R is decreased. Except for the repulsive region for R ≤ 5a₀, the RMS deviation of the fit is well under 1% of the average of the potential.

For large R, a multipole expansion was employed to describe the long-range electrostatic interaction in terms of the multipole moments of C₂H and H₂, as was carried out for HCl–H₂ and OH–H₂.^56,57 We considered the dipole, quadrupole, and octopole moments of C₂H and the quadrupole moment of H₂. We computed the multiple moments of C₂H through MRCI calculations with an aug-cc-pV5Z basis set and obtained the values q₁₀ = −0.2897 a.u., q₂₀ = 3.3693 a.u., and q₃₀ = −5.5812 a.u. For the quadrupole moment of H₂, we employed the value (q₂₀ = 0.5235 a.u.) computed by Ma et al.⁵⁷ 

These multipole moments were used to compute the following expansion terms at long-range: CH-dipole/H₂ quadrupole (V₁₂₃), quadrupole-quadrupole (V₂₂₄), and octopole-quadrupole (V₃₂₅). For these expansion coefficients, we switched from the values obtained from the least squares fits to the ab initio points at smaller R with a hyperbolic tangent function. The isotropic term (v₀₀₀) was switched to a C₆R⁻⁶ radial dependence at long range, with the C₆ coefficient determined by the value of V₀₀₀ at R = 12a₀. The remaining expansion coefficients were damped to zero at large R with a hyperbolic tangent function.

The global minimum of the potential energy surface corresponds to a T-shape structure, with the hydrogen end of C₂H pointing toward the center of H₂. The radial cut for this orientation (θ₁ = 180°, θ₂ = 90°, ϕ = 0°) is displayed in Fig. 2(d). The dissociation energy D_e of the C₂H complex is computed to equal 133.4 cm⁻¹ at an equilibrium intermolecular separation R_e = 7.82a₀ (computed with the aug-cc-pV5Z basis).

Figure 4 presents contour plots as a function of the polar angles θ₁ and θ₂ of the C₂H–H₂ potential energy surface for various planar geometries (ϕ = 0°), with the intermolecular separation R ranging from the repulsive to attractive regions. We see that the most repulsive orientations for all the values of R displayed have θ₁ near 180°, while the potential at small R for θ₁ near 0° is more repulsive then for the intermediate values of θ₁. These geometries correspond to having the hydrogen end of C₂H pointing toward the H₂ moiety. This significant repulsion arises because the H atom in C₂H is a considerable distance from the center of mass of C₂H. Indeed, the potential is repulsive even at R = 7.5a₀ for geometries near θ₁ = 180° and θ₂ = 0° (see Fig. 4). By contrast, the potential for CH–H₂ at R = 7.5a₀ is attractive for all orientations; in the case, the hydrogen atom in displaced from the CH center of mass by a much smaller distance.⁵⁸ 

To explore further the features of the C₂H–H₂ potential energy surface, we present in Fig. 5 contour plots for two nonplanar geometries (ϕ = 90°) at intermolecular separations R = 5.5 and 7.5a₀. These contour plots can be compared with the plots in Figs. 4(c) and 4(d), which present plots planar geometry (ϕ = 0°) at the same values of R. For R = 5.5a₀, the potential for some orientations is slightly less repulsive for ϕ = 90° than for ϕ = 0°. For R = 7.5a₀, the potential is still repulsive for orientations around θ₁ = 180°, θ₂ = 0°, as is also the case for planar geometries. Comparing Figs. 4(d) and 5(b), the attractive part of the potential is slightly less strong for ϕ = 90°.

The contour plots presented in Figs. 4 and 5 show that the potential for the interaction of C₂H with H₂ is strongly anisotropic. This is a result by the large displacement of the H atom in C₂H from the center of mass of the molecule. We may anticipate that collisions of C₂H with H₂ will exhibit significant inelasticity. This has already been partly demonstrated in the scattering calculations carried out by Najar et al.,³⁷ on collisions of C₂H with spherical H₂. We may anticipate larger cross sections for inelastic cross sections for collisions of ortho-H₂(j = 1), for which the full anisotropy of the potential energy surface can be accessed.

Time-independent quantum scattering calculations were carried with the HIBRIDON suite of programs.⁵⁹ The theory of scattering of two linear molecules both in ¹Σ⁺ electronic states has been worked out by Green.⁵⁵ We have extended this treatment to describe collisions of a molecule in a ²Σ⁺ electronic state with a molecule in a ¹Σ⁺ state. We present in the  Appendix the expression for the matrix element of the potential between space-frame scattering channel basis functions for this system. Hereafter, we denote the angular momenta of C₂H and H₂ with subscripts 1 and 2.

Full close-coupling calculations were carried out to compute state-to-state integral cross sections describing inelastic collisions of C₂H rotational/fine-structure levels with para-H₂(j₂ = 0) and ortho-H₂(j₂ = 1). Spectroscopic constants for C₂H were taken from the work of Müller et al.⁵ In our calculations, the Hamiltonian for the isolated C₂H(⁠$X̃2Σ+$⁠) molecule was described by the rotational constant B, centrifugal distortion constant D, and fine-structure constant γ. The rotational constant B for H₂ was taken from Huber and Herzberg.⁶⁰ The total angular momentum exclusive of electron spin is denoted for the C₂H(⁠$X̃2Σ+$⁠) molecule as n₁. The coupling of n₁ with the electron spin $s1=12$ yields two closely spaced fine-structure levels labeled F₁ and F₂, with total angular momentum $j1=n1+12$ and $j1=n1−12$⁠, respectively. In this work, we do not consider the hyperfine splitting in C₂H.

Scattering calculations were carried out for 410 energies up to 1000 cm⁻¹ for collisions with para-H₂(j₂ = 0) and 729 total energies up to 1200 cm⁻¹ for collisions with ortho-H₂(j₂ = 1). The calculations were checked for convergence of the cross sections with respect to the C₂H rotational basis, spacing in the radial grid, and total number of partial waves included. Depending upon the collision energy, C₂H rotational levels up to n₁ = 25 were included in the channel basis. The H₂ rotational basis included j₂ = 0 and 2 for para-H₂ and j₂ = 1 only for ortho-H₂. Partial waves up total angular momentum $J=93.5ℏ$ were included in the calculations.

The computed state-to-state integral cross sections were employed to compute the corresponding rate constants between 15 and 300 K for collisions of C₂H rotational/fine-structure levels with H₂ j₂ = 0 and 1 rotational levels by averaging over the collision energy E_c,⁶¹ 

where μ is the collision reduced mass and k_B is the Boltzmann constant. The integral was evaluated out to collision energies of 1000 cm⁻¹ for collisions of H₂(j₂ = 0) and 1100 cm⁻¹ for collisions of H₂(j₂ = 1). The high-energy tail of the integrand in Eq. (5) was evaluated by exponentially extrapolating the cross sections out to 1500 cm⁻¹; this upper limit is sufficient for temperatures up to 300 K. State-to-state cross sections and rate constants were computed for collisions of the 20 lowest C₂H rotational/fine-structure levels (j₁ ≤ 9.5) with both para-H₂(j₂ = 0) and ortho-H₂(j₂ = 1).

It should be noted that cross sections for rotationally inelastic transitions could be slightly overestimated at high collision energies because of the chemical reaction, as was found for H–D₂ and H–HCl collisions.^62,63 However, the effect of chemical reaction on C₂H–H₂ inelastic rate constants should not be very significant for temperatures at or below 300 K because of the small reaction probability at low collision energies.⁴² 

Figure 6 presents energy dependent cross sections for the excitation of the C₂H n₁ = 0 F₁ rotational/fine-structure level to n₁ = 1–4 levels in collisions with para-H₂(j₂ = 0) and ortho-H₂(j₂ = 1). Figures 6(a) and 6(b) display cross sections for fine-structure conserving (Δj₁ = Δn₁) transitions, and Figs. 6(c) and 6(d) display cross sections for fine-structure changing (⁠$Δj1≠Δn1$⁠) transitions. We see that the latter are somewhat smaller than the former. As we show below, this trend becomes more pronounced for high initial rotational levels. This is consistent with the propensity rule derived by Alexander⁶⁴ for collisions of a molecule in a ²Σ⁺ electronic state with a structureless target and reflects the fact that the direction of the electron spin cannot be changed in a molecular collision.

For collisions of para-H₂, we compare in Figs. 6(a) and 6(c) cross sections calculated with a minimal H₂ rotational basis (maximum value of j₂ = 0) with a basis with a maximum value of j₂ = 2. There are slight differences in the cross sections computed with the two rotational bases, which become larger at higher collision energies. The cross sections computed with a maximum value j₂ = 0 are equivalent to the scattering calculations performed by Najar et al.³⁷ for collisions with spherical H₂, albeit with a different computed potential energy surface. Comparison of the cross sections displayed in Figs. 6(a) and 6(c) with those presented by Najar et al. in their Fig. 4 shows that there is quite good agreement in the two sets of computed cross sections. In fact, the contour plot for the interaction of C₂H with spherical H₂, presented in their Fig. 2, and a contour plot computed by spherically averaging the present potential energy surface [which is equivalent to taking only the l₂ = 0 terms in the angular expansion in Eq. (2)] are in very good agreement.

The energy dependent cross sections below 60 cm⁻¹ collision energy, presented in Fig. 6 for collisions of para-H₂(j₂ = 0), show a dense resonance structure consistent with the formation and decay of quasibound states of the collision complex, as was also seen by Najar et al.³⁷ We also see in Figs. 6(b) and 6(d) a structured, but less resolved, energy dependence for cross sections involving collisions of ortho-H₂(j₂ = 1). These cross sections have less structure because there are many more, and hence overlapping, resonances in the interaction of the molecule with H₂(j₂ = 1) than with H₂(j₂ = 0), as was observed for OH–H₂ and NH₃–H₂ collisions.^65,66

We also observe in Fig. 6 that the cross sections for collisions with ortho-H₂ are larger by a factor of approximately 2 than the cross sections for the same C₂H transitions in collisions with para-H₂. A similar enhancement of the cross sections for collisions of ortho-H₂ was also found in OH–H₂ and NH₃–H₂ collisions.^65,66 In the case of the HCN–H₂ collision pair,⁶⁷ the enhancement was roughly a factor of 3, suggesting that the anisotropy with respect to the H₂ orientation is greater in this system.

In order to explore the dependence of the cross sections upon the initial H₂ level, we have also computed cross sections for collisions of the C₂H(n₁ = 0 F₁) level with H₂(j₂ = 2); these cross sections at selected collision energies are plotted in Figs. 6(b) and 6(d). We see that the cross sections for collisions of H₂(j₂ = 1 and 2) are nearly identical. This behavior is quite similar to what was found for OH–H₂ and HCN–H₂ collisions.^66,67 In the interaction of C₂H with H₂, the full anisotropy of the potential energy surface is experienced in interactions of C₂H with H₂(j₂ > 0) rotational levels. In addition, the long-range interaction is stronger for this nuclear spin modification since the j₂ > 1 levels have a nonzero quadrupole moment while the j₂ = 0 level does not.

To explore the dependence of the state-to-state cross sections upon the initial C₂H level, we present in Fig. 7 cross sections for transitions out of the C₂H n₁ = 2 F₁ level. A dense resonance structure is observed at low collision energies, as is the case for the initial n₂ = 0 F₁ level. We see a slight difference in the computed cross sections for collisions with H₂(⁠$j2=0$⁠) when the H₂ rotational basis is reduced to a maximum value j₂ = 0.

We also see that the cross sections for fine-structure changing transitions are much smaller than for fine-structure conserving transitions. Hence, the propensity rule for the dominance of fine-structure conserving transitions has become stronger than for the initial n₂ = 0 F₁ level. This is expected from the formal analysis by Alexander⁶⁴ of collisions of ²Σ⁺ molecule with a structureless target.

We again see that the cross sections for transitions involving collisions with ortho-H₂(j₂ = 1) are significantly larger than the cross sections for the corresponding transitions involving collisions with para-H₂(j₂ = 0). The magnitude for the fine-structure conserving transitions for a given change in rotational angular momentum n₁ is smaller for the initial level n₁ = 2 than for n₁ = 0; this reflects the larger energy gap between the initial and final levels for the former.

Finally, we see that for collisions of both para-H₂(j₂ = 0) and ortho-H₂(j₂ = 1), the fine-structure conserving transitions with the largest cross sections involve Δn₁ = +2. This contrasts with the HCN–H₂ collision pair, for which different propensity rules are observed for para-H₂(j₂ = 0) and ortho-H₂(j₂ = 1).⁶⁷ The potential energy surface for the interaction of C₂H with spherical H₂ (see Fig. 3 in Ref. 37) displays an approximate forward-backward (θ₁ < 90° vs. θ₁ > 90°) symmetry, indicative of larger angular expansion coefficients for even values of l₁, and implies larger cross sections for even Δn₁ collisional transitions. This propensity for even Δn₁ transitions carries over to the full C₂H–H₂ potential energy surface, as we see in Figs. 6 and 7 that the largest cross sections for C₂H transitions involve Δn₁ = +2 for collisions with H₂(j = 1).

Rate constants for transitions between the 20 lowest rotational/fine-structure levels of C₂H induced by collisions with H₂(j = 0) and H₂(j = 1) were computed for temperatures between 10 and 300 K, using the energy dependent integral cross sections, such as those displayed in Figs. 6 and 7. A table of de-excitation rate constants is included in the supplementary material. Computed rate constants for excitation transitions out of the C₂H n₁ = 0 F₁ initial level induced by collisions with H₂(j₂ = 0) and H₂(j₂ = 1) are presented in Fig. 8. Also plotted in panels (a) and (b) of Fig. 8 are rate constants computed by Najar et al.³⁷ for C₂H–para-H₂ collisions. There is quite reasonable agreement between these rate constants and the ones computed in the present work. The slight differences are due to small differences in the PESs employed and the difference in the H₂ rotational bases (⁠$j2max=2$ and 0 in the present work and in the work of Najar et al., respectively).

As with the energy dependent cross sections, the largest rate constants involve Δn₁ = +2 transitions in C₂H. We also see that the rate constants for collisions with H₂(j₂ = 1) are significantly larger than the corresponding rate constants involving collisions with H₂(j₂ = 0). The differing dependence of the cross sections upon the collision energy for collisions of H₂(j₂ = 0) and H₂(j₂ = 1) (see Fig. 6) is reflected in differences in the temperature dependence of the corresponding rate constants. As found by Najar et al.,³⁷ the rate constants for the H₂(j₂ = 0) collision partner rise gradually with temperature from their small values at low temperature. By contrast, the rate constants at low temperature change rapidly with temperature and then become relatively constant at higher temperature. The same propensity rules favoring transitions for which Δj₁ = Δn₁ are also found, as with the energy dependent cross sections.

In this work, we have reported the calculation of the potential energy surface describing the interaction between C₂H(⁠$X̃2Σ+$⁠) and H₂. These calculations were carried out using coupled cluster theory [RCCSD(T)]. We find that the interaction is strongly anisotropic, principally because of the repulsion between the H atom in the C₂H and H₂ molecule. The former is located at a considerable distance from the C₂H center of mass. Since the geometries of the collision partners are fixed, the computed potential energy surface does not describe the reactive portion of the surface.

This potential energy surface was employed in time independent quantum scattering calculations to compute cross sections for transitions between C₂H rotational/fine-structure levels induced by collisions with both para-H₂(j₂ = 0) and ortho-H₂(j₂ = 1) as a function of collision energy. The cross sections computed for collisions with para-H₂ are in reasonable agreement with the work of Najar et al.,³⁷ who treated H₂(j₂ = 0) as a spherical collision partner, by averaging the C₂H–H₂ interaction over three orientations of H₂.

In the present study, we have taken into account the full anisotropy of the C₂H potential energy surface and found that the cross sections and rate constants for collisions with ortho-H₂ are significantly larger than for para-H₂. This implies that the rate constants for collision of C₂H with the hydrogen molecule will depend strongly on the ratio of para-H₂ and ortho-H₂. The C₂H–H₂ collision pair is yet another example for which the rotational level of H₂ strongly affects the anisotropy of the interaction. This has recently been dramatically demonstrated in the presence or absence of resonances in low-energy collisions.⁶⁸ 

Our primary motivation for this work is to provide C₂H–H₂ rate constants for use in radiative transfer calculations to interpret astronomical observations of C₂H spectroscopic transitions. The rotational spectrum of C₂H has a resolvable hyperfine structure,^3–5 which can usually be resolved in the astronomical observations.^30–34 Hence, it would be desirable to compute rate constants for transitions between hyperfine-resolved rotational/fine-structure levels of C₂H in collisions with both para-H₂(j₂ = 0) and ortho-H₂(j₂ = 1). Dumouchel et al.³⁸ have computed rate constants for hyperfine-resolved transitions in C₂H with para-H₂(j = 0), using the PES of Najar e al.³⁷ They also computed hyperfine-resolved transitions in C₂D–para-H₂(j = 0) collisions. In this case, the PES was obtained from the C₂H–H₂ PES by shifting the C₂H/D center of mass. In future work, we plan to employ the recoupling method²⁹ to compute rate constants for transitions between hyperfine-resolved rotational/fine-structure transitions in C₂H induced by collisions with both H₂(j₂ = 0) and H₂(j₂ = 1), for use in radiative transfer simulations for the interpretation of astronomical observations of C₂H.

See supplementary material for the file PES.tar, which contains a Fortran program for computing the C₂H–H₂ interaction energy for entered values of R, θ₁, θ₂, ϕ, and a table of rate constants for de-excitation transitions between the 20 lowest rotational/fine-structure levels of C₂H induced by collisions with H₂(j = 0) and H₂(j = 1) for 7 temperatures between 10 and 300 K.

The author is grateful for support from the U.S. National Science Foundation (Grant No. CHE-1565872). François Lique kindly provided the rate constants computed in the work of Najar et al. (Ref. 37). The quantum chemistry calculations were performed at the Maryland Advanced Research Computing Cluster, which was funded by the state of Maryland and is jointly managed by Johns Hopkins University and the University of Maryland College Park.

This appendix presents the expression for the matrix element of the potential between space-frame scattering channel basis functions for the collision of a linear molecule in a ²Σ⁺ electronic state with a linear molecule in a ¹Σ⁺ electronic state. The angular momentum of the two molecules is denoted j₁ and j₂, respectively. The angular momentum j₁₂ is the vector sum of j₁ and j₂. The total angular momentum and orbital angular momentum of the complex are denoted J and L, respectively. Hence, the channel basis functions are defined by the quantum numbers j₁F_ij₂j₁₂LJM, where F_i, i = 1 or 2, labels the fine-structure level of the ²Σ⁺ molecule.

The matrix element of the potential, which is described by the angular expansion in Eqs. (2) and (3), equals

where [x] = 2x + 1, (:::), {:::}, and $⋮⋮⋮$ are 3-j, 6-j, and 9-j symbols, respectively.⁶⁹ The unprimed and primed quantum numbers in Eq. (A1) apply to the initial state i and final state f, respectively. The symmetry index⁶⁴ ϵ in Eq. (A1) equals +1 and −1 for the F₁ and F₂ fine-structure levels, respectively, of the ²Σ⁺ molecule.