Rotational excitation of C₂(X¹Σ_g⁺) by para- and ortho-H₂

Abstract

A new four dimensional (4D) potential energy surface for the C₂(X¹Σ_g⁺)–H₂ van der Waals system is generated. The potential was obtained from a multi-reference internally contracted configuration-interaction method including the Davidson correction (MRCI+Q). The four atoms were described using the augmented correlation-consistent quadruple zeta (aug-cc-pVQZ) basis sets. Both molecules were treated as rigid rotors. Close-coupling calculations of the inelastic integral cross sections of C₂ in collisions with para-H₂(j_H2 = 0) and ortho-H₂(j_H2 = 1) were also carried out at low energies. After Boltzmann thermal averaging, rate coefficients were obtained for temperatures ranging from 5 to 100 K. The rate coefficients for collisions with ortho-H₂ are significantly larger than the rate coefficients for collisions with para-H₂.

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1 Introduction

The C₂ molecule is one of the best known constituents of the interstellar medium. It was first detected by Souza and Lutz.¹ Subsequently, many absorption lines have been detected in the near-infrared region for the A¹Π_u–X¹Σ_g⁺ Phillips system,^2–8 and in the UV region for the D¹Σ_u–X¹Σ_g⁺ Mulliken system⁹ and the F¹Π_u–X¹Σ_g⁺ band.^10,11 The C₂ molecule is used as a diagnostic probe of the physical condition in diffuse molecular clouds. The populations of its rotational levels are determined by competition between collisional excitation and radiative excitation through absorption into excited electronic states followed by fluorescence to various vibrational levels of X¹Σ_g⁺.² The analysis of rotationally excited C₂ provides useful information about the density of collision partners, the kinetic temperature and the strength of the radiation field permeating the gas.¹² In order to perform a more accurate analysis of C₂ excitation, rotational rate coefficients of C₂ excited by its principal interstellar collisional partner, H₂, are required. In ref. 13, the full dimensional PES was calculated by the CCSD(T)-F12a/cc-pCVTZ-F12 method. However, we suppose that there is a lack of static correlation because of the multi configurational character of the PES for interacting C₂ and H₂. In this paper, we present a new complete four dimensional (4D) PES for the ground electronic state of the C₂–H₂ collisional system calculated at the MRCI+Q level. Then, this accurate PES was used to perform rotational excitation of C₂ with both para-H₂ and ortho-H₂.

The activation energy of the C₂ + H₂ ⇌ C₂H + H reaction being about 900 cm⁻¹,¹⁴ inelastic cross sections of C₂ in collisions with H₂(j_H2 = 0) and H₂(j_H2 = 1) are reported for total energies of up to 1000 cm⁻¹ for the first 9 even rotational levels, yielding converged rate coefficients up to 100 K.

Excitation rate coefficients of C₂ excited by para-H₂(j_H2 = 0) were provided by Najar et al.¹⁵ In this study, the PES was derived by averaging only over three specific H₂ orientations. These calculations gave a first estimate of C₂–para-H₂ collisional rate coefficients. However, rotational excitation rate coefficients for C₂ in collision with ortho-H₂ are also needed for a complete description of C₂ excitation.¹⁶

The paper is organized as follows: Section 2 describes the calculation of the potential energy surface, Section 3 contains a concise description of the scattering calculations, and in Section 4, we present and discuss our results.

2 Potential energy surface

As mentioned in previous work,¹⁷ the ground electronic state of the C₂–H₂ system is described by two dominant configurations. Because of this multi configurational character, the potential energy surface (PES) of this van der Waals complex was accomplished by first performing full valence complete active space (CASSCF) calculations^18,19 followed by multi-reference internally contracted configuration-interaction (MRCI) calculations,^20,21 including all single and double excitations from the CASSCF reference function. The contributions of higher-order excitations were estimated using the normalized multi-reference Davidson correction.^22,23 In the CASSCF-MRCI calculations, the active space is defined by distributing 10 electrons in 10 molecular orbitals. The 1s² orbitals of the carbon atoms were kept frozen. To achieve a good description of weak interactions, the calculations were performed using a rather large aVQZ basis set of Dunning.^24–26 All ab initio calculations were carried out using the MOLPRO program package.²⁷

The body-fixed coordinate system presented in Fig. 1 was used. The center of the coordinates is placed in the middle of the bond of the C₂ molecule. Vector R is directed along the z-axis and connects the C₂ and H₂ centers of mass. The rotation of the C₂ molecule is defined by angle θ_C2 and the rotation of the H₂ molecule is defined by angle θ_H2, while ϕ is the dihedral angle.

Fig. 1 Geometry describing the C₂–H₂ complex.

The ab initio calculations were performed using the approximation of rigid interacting molecules with the H₂ bond length corresponding to the ground vibrational state: r_H–H = 1.449 Bohr.²⁸ For C₂, we used the experimental equilibrium distance r_C–C = 2.348 Bohr,²⁹ which differs by less than 1% from the calculated value obtained by averaging over the ν = 0 wavefunction. We note that employing state-averaged geometries is a reliable approximation for including zero-point vibrational effects within a rigid-rotor PES.^30–32

In order to determine the interaction potential V(R,θ_C2,θ_H2,ϕ), the basis set superposition error (BSSE) was corrected for all geometries with the Boys and Bernardi counterpoise scheme:³³

V(R,θ_C2,θ_H2,ϕ) = E_C2–H2(R,θ_C2,θ_H2,ϕ) − E_C2(R,θ_C2,θ_H2,ϕ) − E_H2(R,θ_C2,θ_H2,ϕ) (1)

where the energies of the C₂ and H₂ subsystems are computed using a full basis set of the complex. The size inconsistency was corrected for all geometries by adding the energy −33.75 cm⁻¹.

At each point R the calculations were carried out for 122 angular orientations with θ_C2 varying from 0° to 180°, θ_H2 varying from 0° to 90°, and ϕ varying from 0° to 90°. R-distances were varied from 4.8 to 20 Bohr, giving 29 grid points.

For the solution of the close-coupling scattering equations, it is most convenient to expand the interaction potential V(R,θ_C2,θ_H2,ϕ) into the angular functions at each value of R. For the scattering of two linear rigid rotors, we used the expansion of Green:³⁴

The basis functions A_l1,l2,l(θ_C2,θ_H2,ϕ) are products of the associated Legendre polynomials P_lm:

where 〈…∣…〉 is a Clebsch–Gordan coefficient. The P_lm functions are related to coupled spherical harmonics through Y_lm(θ,ϕ) = P_lm(θ)e^(imϕ). Here l₁ and l₂ are associated respectively with the rotational motion of C₂ and H₂, and l = |l₁ − l₂|, …, |l₁ + l₂|. In eqn (2), the homonuclear symmetry of the interacting molecules forces the indexes l₁ and l₂ to be even.

To obtain the expansion described above, at each point of the radial grid R we fitted the ab initio points to expression (2) using a linear least squares method, with l^max₁ = 6 and l^max₂ = 4. The choice of such expansion parameters limits the results to a total of 28 radial expansion coefficients v_l1,l2,l(R). The root-mean-square error (rmse) of the fits in the long and (attractive) medium range was of the order of 10⁻³ to 10⁻² cm⁻¹. The medium repulsive wall (energies of the order of 10² to 10³ cm⁻¹) was reproduced with rmse of 1–10 cm⁻¹.

Each radial coefficient was interpolated using cubic splines for 4.8 < R ≤ 16a₀. For R > 16a₀ the energy (not the coefficients) was extrapolated using the c₅/R⁵ + c₆/R⁶ functional form based on its values at R = 16a₀ and R = 18a₀. For R < 4.8a₀ the energy was extrapolated using the functional form A exp(BR) based on its values at R = 4.8a₀ and R = 5a₀. Thus, our PES covers the whole range of R. However, we suggest to use it in the range from 3.0 to 100a₀.

The global minimum of the 4D PES corresponds to a T-shaped structure with θ_C2 = 0°, θ_H2 = 90°, ϕ = 0° and R = 6.92a₀ with V = −109.22 cm⁻¹. In Fig. 2–4 we present the contour plots of the interaction energy surface for fixed equilibrium parameters. It is seen that there is a weak anisotropy of the PES for the Jacoby angle ϕ and a strong anisotropy for the angle θ_C2. The local minimum of the PES occurs for another T-shaped structure with θ_C2 = 90°, θ_H2 = 0°, ϕ = 0° and R = 6.70a₀. The interaction energy at this point is V = −101.63 cm⁻¹.

Fig. 2 Contour plot of the cut of the 4D PES for fixed ϕ = 0° and R = 6.92a₀. Energy is in cm⁻¹.

Fig. 3 Contour plot of the cut of the 4D PES for fixed θ_H2 = 90° and R = 6.92a₀. Energy is in cm⁻¹.

Fig. 4 Contour plot of the cut of the 4D PES for fixed θ_H2 = 90° and ϕ = 0°. Energy is in cm⁻¹. Red contour lines represent repulsive interaction energies.

3 Scattering study of excitation of C₂ by para- and ortho-H₂

In the present work, we consider collisions of ¹²C₂ with para- and ortho-H₂. Only even rotational levels j were considered here because, in homonuclear molecules with zero nuclear spins like C₂, rotational levels with odd values are not allowed. On the other hand, the H₂ molecule can exist in two forms denoted para- and ortho-H₂. In the former, the nuclear spins of the hydrogen atoms are opposed resulting in total I = 0, whereas, in the latter, the nuclear spins are aligned and I = 1. As the nuclear wavefunction must be asymmetric, in para-H₂, only even rotational states, j_H2 = 0, 2,… are allowed, and in ortho-H₂, only odd values, j_H2 = 1, 3,… are allowed.

Full close-coupling calculations³⁵ were carried out for total energies up to 1000 cm⁻¹, using C₂ and H₂ as rigid rotors with rotational and centrifugal distortion constants B_C2 = 1.820 cm⁻¹ and D_C2 = 6.92 × 10⁻⁶ cm⁻¹ (ref. 36), and B_H2 = 60.853 cm⁻¹ and D_H2 = 3.3 × 10⁻⁶ cm⁻¹,²⁹ respectively. Transitions among C₂ rotational levels up to j = 16 were computed for collisions involving para-H₂(j_H2 = 0) and ortho-H₂(j_H2 = 1). For convergence of the cross sections, the rotational states j_H2 = 0, 2 and j_H2 = 1, 3 were included in the rotational basis sets for para- and ortho-H₂, respectively. The inclusion of the j_H2 = 4 channel in para-H₂ affects the cross sections by less than 1 percent. For the C₂ molecule, the rotational basis set was extended up to j_max = 24 at the largest total energy considered (1000 cm⁻¹). The coupled scattering equations were solved by the MOLSCAT code³⁷ using the propagator of Manolopoulos.³⁸ For the C₂–H₂ system the reduced mass is μ = 1.8597 amu. The integration parameters were chosen to ensure convergence of the cross sections in this range. The minimum and maximum integration distances chosen were R_min = 3 Bohr and R_max = 50 Bohr, respectively. The maximum value of the total angular momentum J used in the calculations was chosen according to a convergence criterion of 0.01 Ų. The energy grid was adjusted to describe all the details of the resonances of the inelastic cross sections. The thermal rate coefficients at temperature T are obtained by averaging the rotational inelastic cross sections, σ_j→j′, over a Boltzmann distribution of collision energy E_c:

with , where k_B is the Boltzmann constant.

4 Results

First we compare the σ_0→2 cross sections for the excitation of C₂ with para-H₂(j_H2 = 0, 2) calculated with the 4D PES and including j_H2 = 0, 2 in the rotational basis set of H₂ with the cross sections for the excitation of C₂ with para-H₂ calculated previously by Najar et al.¹⁵ with a 2D effective PES averaged over three specific H₂ orientations using equal weights of 1/3 for the three geometries considered. As we can see in Fig. 5, the two cross sections are close, having the same form of resonances with a shift and small difference in amplitudes. The difference is appreciable at high energies, approaching 35 percent. To understand where the difference comes from, we present in Table 1 cross sections at different energies for C₂-paraH₂ calculations with a PES averaged over 3 orientations, C₂-paraH₂ 4D PES calculations with just the basis j_H2 = 0 and C₂-paraH₂ 4D PES calculations with the rotational basis involving j_H2 = 0 and j_H2 = 2 levels (j_H2 = 0, 2). We note that at low energies the major difference (∼15%) is due to lack of anisotropy in the PES averaged over 3 orientations. At high energies the inaccuracies of the PES have only a minor influence on the cross sections, and the differences stem mainly from neglect of the j_H2 = 2 rotational state. This result was also confirmed by Lique et al.³⁹ in the SiS–H₂ system.

Fig. 5 Cross sections as a function of collision energy for 0 → 2 rotational excitation of C₂ with para-H₂ obtained by a 2D effective PES (red curve) and by the 4D PES including (j_H2 = 2) in the rotational basis set of H₂ (black curve).

Table 1 Cross sections σ_0→2 (Ų) for C₂–paraH₂ obtained using a 2D effective PES, a 4D PES with basis j_H2 = 0 and a 4D PES with the basis j_H2 = 0, 2

Energy (cm^−1)	2D effective PES	4D PES, jH2 = 0	4D PES, jH2 = 0, 2
50	11.249	13.307	14.160
100	5.613	6.580	7.087
300	2.622	2.650	2.515
500	2.978	2.819	2.379
1000	3.540	3.298	2.567

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Fig. 6 shows the collision energy dependence of the 0 → 2 inelastic integral cross sections of C₂ in collision with para-H₂(j_H2 = 0) (including j_H2 = 0, 2) and ortho-H₂(j_H2 = 1) (including j_H2 = 1, 3). The figure illustrates many resonances in the cross sections for collision energies below 100 cm⁻¹, which are a consequence of the temporary trapping of the H₂ molecule in the attractive potential well, involving the formation of quasi-bound states of the van der Waals C₂–H₂ complex. We note a difference in amplitude and shape of the resonances between the two cross sections. The collisions with ortho-H₂ are more strongly inelastic. This is expected because, unlike with ortho-H₂, the long-range interactions with para-H₂(j_H2 = 0) do not involve the permanent quadrupole moment of the H₂ molecule; only induction and dispersion forces participate.⁴⁰ This behavior was also found in H₂–H₂ (ref. 41) and SO₂–H₂ (ref. 42) collisions.

Fig. 6 Cross sections for rotational excitation of C₂ by para-H₂(j_H2 = 0, 2) and ortho-H₂(j_H2 = 1, 3) for the 0 → 2 transition as a function of collision energy.

Rate coefficients for rotational excitation of C₂ by para-H₂(j_H2 = 0) and ortho-H₂(j_H2 = 1) have been computed from cross sections generated from the grid of collision energies. The representative variation with temperature is illustrated in Fig. 7. The curves in Fig. 7 and the results in Table 2 show that the rates for ortho-H₂(j_H2 = 1) are significantly larger by a factor of 5 on average. This result, also found previously for some systems like CO–H₂,⁴³ H₂O–H₂ (ref. 44) and SiS–H₂,⁴⁵ confirms the expectation that ortho-H₂(j_H2 = 1) is a more efficient perturbant than para-H₂(j_H2 = 0).

Fig. 7 Temperature dependence of rotational de-excitation rate coefficients of C₂ excited by H₂ for Δj = 2 transitions. Dashed lines: para-H₂(j_H2 = 0); solid lines: ortho-H₂(j_H2 = 1).

Table 2 Rate coefficients (cm³ s⁻¹) for rotational de-excitation in C₂ induced by para-H₂(j_H2 = 0) and ortho-H₂(j_H2 = 1). The upper entry for each transition is for T = 50 K, and the lower entry is for T = 100 K

Transition	Para-H2	Ortho-H2
2 → 0	0.234(-10)	1.122(-10)
	0.188(-10)	1.100(-10)
4 → 2	0.322(-10)	1.191(-10)
	0.273(-10)	1.216(-10)
6 → 4	0.192(-10)	0.870(-10)
	0.201(-10)	0.949(-10)
8 → 6	0.110(-10)	0.610(-10)
	0.141(-10)	0.704(-10)

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In the paper of Lavendy et al.,⁴⁶ for construction of the PES the authors considered only 5 relative orientations of C₂ and H₂ for each R value. Later, Phillips⁴⁷ used this PES to compute the rate coefficients using the close-coupling approach. In our previous work,¹⁵ we constructed a 2D PES based on nearly 900 ab initio points. For each (R,θ) we considered 3 angular orientations of the H₂ molecule, the energies of which were then averaged. In that work,¹⁵ we compared our collision rates for C₂–para-H₂(j_H2 = 0) to those of Phillips⁴⁷ for the temperature range T < 100 K. We found that the rates calculated by Phillips are greater by a factor of 5. This difference is due to the fact that the interaction potential used by Phillips overestimates the anisotropy of the potential. In addition, Phillips found that para-H₂(j_H2 = 0) and ortho-H₂(j_H2 = 1) are nearly identical colliders, which doesn’t agree with our findings that there is a notable difference. This can also be explained by the fact that the potential of Phillips based on the ab initio energies of Lavendy et al. was constructed by neglecting the terms responsible for the quadrupole–quadrupole interactions.

5 Conclusions

We have applied full close-coupling calculations to investigate rotational excitation in the C₂ molecule for collisions involving para-H₂(j_H2 = 0) and ortho-H₂(j_H2 = 1). The calculations are performed using a highly accurate 4D potential energy surface. The inclusion of j_H2 = 2 in the rotational basis set of H₂ makes an appreciable contribution at high energies to the para-H₂ cross sections, approaching 35 percent. Rate coefficients for transitions among the levels of C₂ up to j = 16 were determined for the range of temperatures 5 ≤ T ≤ 100 K. Rate coefficients with ortho-H₂ are larger by a factor of 5 on average than rate coefficients with para-H₂.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

Y. K. acknowledges financial support from the Russian Science Foundation through grant no. 17-12-01395 and grant 8.1.03.2017, and ‘The Tomsk State University competitiveness improvement programme’.

Notes and references

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