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Abstract

Rate coefficients for state-to-state rotational transitions of C3N− induced by collision with both ortho- and para-H2 are presented. Quantum calculations are performed at the close-coupling level using the uniform J-shifting method and a new potential energy surface specially developed for this purpose. Rate coefficients are obtained for state-to-state transitions among the first 28 rotational levels of C3N− and for temperatures ranging from 10 to 300 K. The para-H2 rate coefficients are shown to differ strongly from the mass-scaled He rate coefficients previously computed. The ortho- and para-H2 rate coefficients are very similar, as it was already observed for the rotational transitions of CN− and C6H−. There is also an unexpected similarity between the rate coefficients of the rotational de-excitations of CN−, C3N−, and C6H−. This may open door to quantitative extrapolations of the rate coefficients for larger anions.

molecular data, molecular processes, scattering, ISM: molecules

1 INTRODUCTION

Although the existence and possible detection of anions in the interstellar medium (ISM) were proposed long ago (Dalgarno & McCray 1973; Sarre 1980; Herbst 1981), it was not till last decade that the first molecular anion C6H− was detected in the circumstellar envelope (CSE) IRC + 10216 and in the dense molecular cloud TMC-1 (McCarthy et al. 2006). So far, only six anion (CN−, C3N−, C5N−, C4H−, C6H−, C8H−) have been detected in carbon-rich interstellar sources (Brünken et al. 2007; Cernicharo et al. 2007, 2008; Thaddeus et al. 2008; Agúndez et al. 2010). The abundances of these molecules could in principle be extracted from the spectroscopic observations through radiative transfer calculations. This procedure however requires the knowledge of rotational state-to-state radiative and collisional rates with H2 since the latter is by far the most abundant interstellar molecule. Such data are unfortunately scarce especially for the recently detected anions. Up to now, the only available data are limited to the collisions of H2 with the CN− (Kłos & Lique 2011) and C6H− (Walker et al. 2017) anions. The rotational excitation rates available involving collisions with C3N− are limited to those we recently determined for the collisions with He (Lara-Moreno, Stoecklin & Halvick 2017). Collisional rates with He are often used to estimate those for para-H2 using a scaling law based on the ratio of the reduced masses. The applicability of this approximation is unfortunately unpredictable since it may work well as for example for CS (Denis-Alpizar et al. 2013) while it fails reproducing known collisional rates with para-H2 for a few systems (Lique et al. 2008; Guillon & Stoecklin 2012; Walker et al. 2014).

In this study, we then focus on the calculation of the rotational relaxation rates of C3N− in collisions with both ortho- and para-H2. We calculate state-to-state transitions among the first 28 rotational levels of C3N− for temperatures ranging from 10 to 300 K. While the close coupling (CC) method offers the highest level of accuracy for the calculation of the collisional rates, its application to the H2 + C3N− collision is quite challenging owing to the very small value of the rotational constant of C3N−, the large reduced mass of the collisional system and the strong long-range anisotropic potential energy. Similar numerical difficulties were already met in the previous study dedicated to the collisions of C3N− with He (Lara-Moreno et al. 2017) and overcome by using the uniform J-shifting (UJS) method (Zhang & Zhang 1999). It was shown in this previous study that the UJS method gives a very good level of accuracy (less than 5 per cent relative error) while offering large computer time saving. We then use here again the UJS method and compare our results with those obtained previously for the collisions with He. We discuss the possible reason of the failure of the mass scaling approximation for this molecule and evaluate the possibility of using the present rates to predict those for larger anions. The manuscript is organized as follows. A brief account of the potential energy surface (PES) model and the CC method is provided in the following section while the results are discussed in Section 3 and the conclusions are presented in Section 4.

2 METHODS

2.1 Potential energy surface

The PES used in this work was developed using explicitly correlated coupled-cluster method with single and double excitations and using a perturbative treatment of triple excitations [CCSD(T)-F12] and the augmented correlation-consistent polarized valence triple-zeta basis set (aug-cc-pVTZ) (Kendall, Dunning & Harrison 1992). A brief description of the PES is given below while more details are given in Lara-Moreno, Stoecklin & Halvick (2019). The 4D PES is calculated in body-fixed Jacobi coordinates, namely R the intermolecular distance, θ1 and θ2 the rotation angle of H2 and C3N−, respectively, and ϕ the torsion angle between the two molecules. Since C3N− is a long molecule, the interaction energy is strongly anisotropic for small intermonomer separation and a large density of ab initio points was then necessary to describe properly this region. A total number of 28 339 ab initio energies were calculated with R ranging from 2 to 50 a0, θ1 from 0° to 90°, θ2 from 0° to 180°, and ϕ from 0° to 180°. For all these calculations, carried out with the molpro package (Werner et al. 2012), the H2 bond length was fixed to its vibrationally averaged value in the rovibrational ground state, namely rHH  = 1.448736 a0, while the C3N− bond lengths were set to their equilibrium values obtained from CCSD(T)/aug-cc-pV5Z calculations (Kołos, Gronowski & Botschwina 2008), namely |$r_\mathrm{C_{1}C_{2}}$|  = 2.3653 a0, |$r_\mathrm{C_{2}C_{3}}$|  = 2.5817 a0, and |$r_\mathrm{C_{3}N}$|  = 2.2136 a0. The basis superposition error was then corrected by means of the counterpoise procedure (Boys & Bernardi 1970) applied to the rigid monomer case.

The functional form of the PES is defined as a linear combination of products of radial functions by angular functions, the latter being a product of associated normalized Legendre polynomials |$P_l^m(x)$| and cosine functions:
(1)
with λ1 ∈ {0, 2, 4, 6},  λ2 ∈ [0, 18], and μ ∈ [0, 4]. The radial part is defined as the summation of two terms associated with the short-range and long-range contributions:
(2)
where the switching function w(R) is an hyperbolic tangent. The short-range contribution fSR(R) was interpolated using a cubic spline method, whereas the long-range part fLR(R) was fitted to reciprocal power functions.

The PES is characterized by large anisotropies and deep potential wells. The global minimum, De = 769.75  cm−1, corresponds to a linear geometry where the H2 molecule faces the C-end of the C3N− molecule. A secondary minimum, De = 561.77  cm−1, was found for another linear geometry where the H2 molecule faces the N-end of the C3N− molecule. These features are depicted in Fig. 1.

2D contour plots of the PES in the (R, θ2) polar coordinates frame for θ1 = 0° and ϕ  = 0°. Contour levels are equally spaced by 100  cm−1 and labelled by the energy in  cm−1. The contours are blue for negative interaction energy and red for positive one. The origin of coordinates corresponds to the centre of mass of C3N−.
Figure 1.

2D contour plots of the PES in the (R, θ2) polar coordinates frame for θ1 = 0° and ϕ  = 0°. Contour levels are equally spaced by 100  cm−1 and labelled by the energy in  cm−1. The contours are blue for negative interaction energy and red for positive one. The origin of coordinates corresponds to the centre of mass of C3N−.

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2.2 Dynamics

The CC equations for the collision of C3N− and H2 are solved in the space-fixed frame using an improved version of the didimat code (Guillon et al. 2008) originally developed to study the H2-HF collision. This code that uses the log-derivative propagator developed by Manolopoulos (1986) was also applied to several other collisional systems like H2-CO (Chefdeville et al. 2015) or H2-CS (Denis-Alpizar et al. 2018). By choosing the usual boundary conditions for the radial part of the scattering wavefunction
\begin{eqnarray*} \boldsymbol{\sf G}^J(0)=\boldsymbol{\sf 0} \nonumber \end{eqnarray*}
and
(3)
and using the familiar definition of the |$\boldsymbol{\sf K}^J$| and |$\boldsymbol{\sf T}^J$| matrices
(4)
where |$\boldsymbol{\sf N}^J(R)$| and |$\boldsymbol{\sf J}^J(R)$| are diagonal matrices of spherical Ricatti–Bessel functions of the first and second kinds |$\tilde{j}_l$| and |$\tilde{n}_l$| while the superscript J accounts for the total angular momentum quantum number. We then calculate the standard state-to-state transition probabilities as a function of the |$\boldsymbol{\sf T}^J$| matrix elements,
(5)
where j1, j1′, j2, j2′ denote the initial and final rotational states of H2 and C3N−, respectively. Subsequently one obtains the state-to-state inelastic cross-sections:
(6)
Finally, the state-selected rotational relaxation rates of C3N− in collisions with H2 are obtained by thermally averaging the inelastic cross-sections |$\sigma _{j_1j_2\rightarrow {j_1}^{\prime }{j_2}^{\prime }}$| over a Boltzmann distribution of the collision energy E,
(7)
where μ is the reduced mass and kB is the Boltzmann’s constant. The excitation rate coefficients |$k_{{j_1}^{\prime }{j_2}^{\prime }\rightarrow j_1j_2}$| can then be obtained from the relaxation rate coefficients by the usual detailed balance formula,
(8)
where |$\epsilon _{j_1}$| and |$\epsilon _{j_2}$| are the rigid rotor energies of the two linear molecules

2.3 Critical density

The critical density is the density of interstellar gas for which the rate of de-excitation by spontaneous emission is equal to the rate of de-excitation by collision. At temperature T, it is defined by (Tielens 2005)
(9)
where Aji is the Einstein coefficient for the spontaneous emission induced by the transition from state j to state i and kji(T) is the rate coefficient for the transition j → i induced by collision.
The Einstein coefficient Aji for spontaneous emission is defined by
(10)
where νji is the transition frequency, ϵ0 the vacuum permittivity, c the speed of light, and Sji the matrix element of the dipole operator. If we consider the spontaneous transitions between the states of a linear rigid rotor, then only the Δj = −1 transitions are possible. In that case, the matrix elements of the dipole operator are obtained as
(11)
where μ = 3.10 D (Kołos et al. 2008) is the dipole moment of C3N− at equilibrium geometry.

2.4 Parameters of the calculations

In the present calculation, we neglect the vibration of both H2 and C3N−, thus considering both molecules as rigid rotors whose rotational constants are set to their experimental values B1 = 60.853 cm−1 (Herzberg & Howe 1959) and B2 = 0.1618 cm−1 (Kołos et al. 2008). As the potential well depth is relatively large and the value of the rotational constant B2 of C3N− is small, a large number of rotational levels of this molecule needs to be included in the rotational basis set used for the dynamics. This basis set includes 31 rotational levels of C3N− (0 ≤ j2 ≤ 30) and 2 rotational levels of H2 for both para-H2 (j1 = 0, 2) and ortho-H2 (j1 = 1, 3).

The calculations are performed for a grid of collision energies ranging from 0.1 to 2000  cm−1. For each collision energy, the convergence of the results as a function of the maximum value of the intermolecular coordinate R was checked. Owing to the strength of the long-range potential, a maximum value of 200 a0 was found to be necessary for the lowest collision energies. Furthermore, a 10−4 relative criterion was enforced for the convergence of the state-selected quenching cross-section as a function of the maximal value of the total angular momentum quantum number J. Because of the large reduced mass of the system, as much as 157 values of J were required to reach this level of relative convergence of the cross-section for the highest energies.

2.5 Computational methodology

The aforementioned convergence requirements make the CC calculation prohibitively expensive, even at very low collision energy. We hence developed an MPI version of the didimat code using an asynchronous task parallelization scheme. This MPI version of the code distributes N tasks over M processors where each task is associated with a propagation of the wavefunction for a given collision energy and a given value of J.

In order to further reduce the CPU time while keeping a good accuracy of the state-to-state rate coefficients, we also decided to use the UJS method developed by Zhang & Zhang (1999). As a matter of fact, we showed in our recent study dedicated to the rotational excitation of C3N− in collisions with He (Lara-Moreno et al. 2017) that this method offers a very good level of accuracy when compared to the exact calculations. The strategy of this method is to compute the state-to-state transition probabilities for selected values of J and then interpolate the missing probabilities in such a way that the rate coefficient at a given temperature can be calculated as accurately as possible. The shifting constant Bi(T) is determined by requiring that the following equation is satisfied
(12)
where PJ(E) is the probability of some collisional process. The shifting constant Bi(T) for a given value of J ∈ [Ji, Ji + 1] is obtained as
(13)
where QJ(T) is defined as
(14)
As a result equation (7) can be rewritten as
(15)
In the previous equations, the indexes j1, j1′, j2, j2′ have been removed for simplicity.

In the present case, we computed the transition probabilities for the values of J ∈ {0, 5, 10, 15, 20, 25, 30, 40, 60, 80} using the CC method. We then interpolated or extrapolated the missing transition probabilities and summed all contribution from J = 0 up to 200. In order to check the accuracy of the method, we also performed exact CC calculations for the collisions involving para-H2 and compared the results with those obtained using the UJS procedure in Fig. 2.

Error in the calculation of para-H2 rate coefficients by using the UJS procedure as a function of temperature. The error populations are depicted by Tukey boxplots. The mean relative error is depicted by dashed green lines. The solid orange line indicates the median (i.e. the second quartile) of the error population, the bottom and top of the box are the first and third quartiles, respectively, and the end of the dashed lines are distant from the box by 1.5 times the height of the box.
Figure 2.

Error in the calculation of para-H2 rate coefficients by using the UJS procedure as a function of temperature. The error populations are depicted by Tukey boxplots. The mean relative error is depicted by dashed green lines. The solid orange line indicates the median (i.e. the second quartile) of the error population, the bottom and top of the box are the first and third quartiles, respectively, and the end of the dashed lines are distant from the box by 1.5 times the height of the box.

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As it can be seen in this figure, the agreement between the two kinds of calculation is remarkably good, especially for the high temperatures. The mean relative error is smaller than |$3{{\ \rm per\ cent}}$| for the whole temperature range while the largest relative error, |${\sim } 10{{\ \rm per\ cent}}$|⁠, is reached at low temperature [10−50] K. In this range of low temperatures, the magnitude of the rate coefficients is anyway very small and the origin of the error is essentially numerical.

3 RESULTS

3.1 Rate coefficients

Selected state-to-state rate coefficients for rotational excitation and de-excitation of C3N− by collisions with ortho- and para-H2 are shown in Table 1 and Fig. 3. As can be seen in this figure, the rate coefficients increase slowly as a function of temperature while they monotonously decrease as a function of the transferred rotational angular momentum |Δj2|. Another important result for this system lies in the striking similarity between ortho- and para-H2 rates, especially at high temperatures.

Rotational excitation (upper panel) and de-excitation (lower panel) rate coefficients of C3N− in collision with ortho- (dotted lines) and para-H2 (full lines). The curves are labelled by the final state of the transition j2 → j2′.
Figure 3.

Rotational excitation (upper panel) and de-excitation (lower panel) rate coefficients of C3N− in collision with ortho- (dotted lines) and para-H2 (full lines). The curves are labelled by the final state of the transition j2 → j2′.

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Table 1.
Open in new tab

State-to-state rate coefficients |$k_{j_1,j_2\rightarrow {j_1}^{\prime },{j_2}^{\prime }}$| (cm3 molecule−1 s−1) for the rotational excitation of C3N− in collision with para (j1 = 0) and ortho (j1 = 1) H2 for various temperature values. Power of 10 is denoted in parenthesis. The complete set of rate coefficient is available in the online supplementary material.

j1j2j1′j2′T = 10 KT = 100 KT = 300 K
00012.16(−10)3.95(−10)4.73(−10)
10114.28(−10)3.99(−10)4.69(−10)
00022.94(−10)3.65(−10)4.85(−10)
10122.70(−10)3.68(−10)5.41(−10)
00031.92(−10)2.97(−10)3.62(−10)
10132.22(−10)2.64(−10)3.35(−10)
00041.15(−10)1.84(−10)2.24(−10)
10141.46(−10)2.00(−10)2.38(−10)
05062.09(−10)3.51(−10)4.20(−10)
15162.83(−10)3.32(−10)3.94(−10)
05071.33(−10)2.46(−10)3.30(−10)
15171.46(−10)2.74(−10)3.57(−10)
05087.06(−11)1.97(−10)2.43(−10)
15188.13(−11)1.73(−10)2.15(−10)
05093.36(−11)1.27(−10)1.73(−10)
15194.46(−11)1.49(−10)1.73(−10)
0100111.69(−10)3.27(−10)3.89(−10)
1101111.80(−10)2.96(−10)3.65(−10)
0100128.30(−11)2.07(−10)2.93(−10)
1101128.78(−11)2.35(−10)3.21(−10)
0100133.43(−11)1.70(−10)2.11(−10)
1101133.41(−11)1.45(−10)1.88(−10)
0100141.17(−11)9.98(−11)1.43(−10)
1101141.49(−11)1.15(−10)1.43(−10)
0150161.38(−10)3.13(−10)3.77(−10)
1151161.32(−10)2.78(−10)3.50(−10)
0150175.18(−11)1.88(−10)2.73(−10)
1151175.46(−11)2.17(−10)3.04(−10)
0150181.81(−11)1.58(−10)1.99(−10)
1151181.65(−11)1.30(−10)1.73(−10)
0150194.92(−12)8.85(−11)1.30(−10)
1151195.97(−12)1.01(−10)1.32(−10)
0200211.24(−10)3.09(−10)3.72(−10)
1201219.64(−11)2.69(−10)3.46(−10)
0200223.08(−11)1.75(−10)2.63(−10)
1201223.52(−11)2.02(−10)2.95(−10)
0200239.52(−12)1.45(−10)1.90(−10)
1201237.59(−12)1.20(−10)1.69(−10)
0200242.09(−12)8.16(−11)1.27(−10)
1201242.43(−12)9.34(−11)1.32(−10)
j1j2j1′j2′T = 10 KT = 100 KT = 300 K
00012.16(−10)3.95(−10)4.73(−10)
10114.28(−10)3.99(−10)4.69(−10)
00022.94(−10)3.65(−10)4.85(−10)
10122.70(−10)3.68(−10)5.41(−10)
00031.92(−10)2.97(−10)3.62(−10)
10132.22(−10)2.64(−10)3.35(−10)
00041.15(−10)1.84(−10)2.24(−10)
10141.46(−10)2.00(−10)2.38(−10)
05062.09(−10)3.51(−10)4.20(−10)
15162.83(−10)3.32(−10)3.94(−10)
05071.33(−10)2.46(−10)3.30(−10)
15171.46(−10)2.74(−10)3.57(−10)
05087.06(−11)1.97(−10)2.43(−10)
15188.13(−11)1.73(−10)2.15(−10)
05093.36(−11)1.27(−10)1.73(−10)
15194.46(−11)1.49(−10)1.73(−10)
0100111.69(−10)3.27(−10)3.89(−10)
1101111.80(−10)2.96(−10)3.65(−10)
0100128.30(−11)2.07(−10)2.93(−10)
1101128.78(−11)2.35(−10)3.21(−10)
0100133.43(−11)1.70(−10)2.11(−10)
1101133.41(−11)1.45(−10)1.88(−10)
0100141.17(−11)9.98(−11)1.43(−10)
1101141.49(−11)1.15(−10)1.43(−10)
0150161.38(−10)3.13(−10)3.77(−10)
1151161.32(−10)2.78(−10)3.50(−10)
0150175.18(−11)1.88(−10)2.73(−10)
1151175.46(−11)2.17(−10)3.04(−10)
0150181.81(−11)1.58(−10)1.99(−10)
1151181.65(−11)1.30(−10)1.73(−10)
0150194.92(−12)8.85(−11)1.30(−10)
1151195.97(−12)1.01(−10)1.32(−10)
0200211.24(−10)3.09(−10)3.72(−10)
1201219.64(−11)2.69(−10)3.46(−10)
0200223.08(−11)1.75(−10)2.63(−10)
1201223.52(−11)2.02(−10)2.95(−10)
0200239.52(−12)1.45(−10)1.90(−10)
1201237.59(−12)1.20(−10)1.69(−10)
0200242.09(−12)8.16(−11)1.27(−10)
1201242.43(−12)9.34(−11)1.32(−10)
Table 1.
Open in new tab

State-to-state rate coefficients |$k_{j_1,j_2\rightarrow {j_1}^{\prime },{j_2}^{\prime }}$| (cm3 molecule−1 s−1) for the rotational excitation of C3N− in collision with para (j1 = 0) and ortho (j1 = 1) H2 for various temperature values. Power of 10 is denoted in parenthesis. The complete set of rate coefficient is available in the online supplementary material.

j1j2j1′j2′T = 10 KT = 100 KT = 300 K
00012.16(−10)3.95(−10)4.73(−10)
10114.28(−10)3.99(−10)4.69(−10)
00022.94(−10)3.65(−10)4.85(−10)
10122.70(−10)3.68(−10)5.41(−10)
00031.92(−10)2.97(−10)3.62(−10)
10132.22(−10)2.64(−10)3.35(−10)
00041.15(−10)1.84(−10)2.24(−10)
10141.46(−10)2.00(−10)2.38(−10)
05062.09(−10)3.51(−10)4.20(−10)
15162.83(−10)3.32(−10)3.94(−10)
05071.33(−10)2.46(−10)3.30(−10)
15171.46(−10)2.74(−10)3.57(−10)
05087.06(−11)1.97(−10)2.43(−10)
15188.13(−11)1.73(−10)2.15(−10)
05093.36(−11)1.27(−10)1.73(−10)
15194.46(−11)1.49(−10)1.73(−10)
0100111.69(−10)3.27(−10)3.89(−10)
1101111.80(−10)2.96(−10)3.65(−10)
0100128.30(−11)2.07(−10)2.93(−10)
1101128.78(−11)2.35(−10)3.21(−10)
0100133.43(−11)1.70(−10)2.11(−10)
1101133.41(−11)1.45(−10)1.88(−10)
0100141.17(−11)9.98(−11)1.43(−10)
1101141.49(−11)1.15(−10)1.43(−10)
0150161.38(−10)3.13(−10)3.77(−10)
1151161.32(−10)2.78(−10)3.50(−10)
0150175.18(−11)1.88(−10)2.73(−10)
1151175.46(−11)2.17(−10)3.04(−10)
0150181.81(−11)1.58(−10)1.99(−10)
1151181.65(−11)1.30(−10)1.73(−10)
0150194.92(−12)8.85(−11)1.30(−10)
1151195.97(−12)1.01(−10)1.32(−10)
0200211.24(−10)3.09(−10)3.72(−10)
1201219.64(−11)2.69(−10)3.46(−10)
0200223.08(−11)1.75(−10)2.63(−10)
1201223.52(−11)2.02(−10)2.95(−10)
0200239.52(−12)1.45(−10)1.90(−10)
1201237.59(−12)1.20(−10)1.69(−10)
0200242.09(−12)8.16(−11)1.27(−10)
1201242.43(−12)9.34(−11)1.32(−10)
j1j2j1′j2′T = 10 KT = 100 KT = 300 K
00012.16(−10)3.95(−10)4.73(−10)
10114.28(−10)3.99(−10)4.69(−10)
00022.94(−10)3.65(−10)4.85(−10)
10122.70(−10)3.68(−10)5.41(−10)
00031.92(−10)2.97(−10)3.62(−10)
10132.22(−10)2.64(−10)3.35(−10)
00041.15(−10)1.84(−10)2.24(−10)
10141.46(−10)2.00(−10)2.38(−10)
05062.09(−10)3.51(−10)4.20(−10)
15162.83(−10)3.32(−10)3.94(−10)
05071.33(−10)2.46(−10)3.30(−10)
15171.46(−10)2.74(−10)3.57(−10)
05087.06(−11)1.97(−10)2.43(−10)
15188.13(−11)1.73(−10)2.15(−10)
05093.36(−11)1.27(−10)1.73(−10)
15194.46(−11)1.49(−10)1.73(−10)
0100111.69(−10)3.27(−10)3.89(−10)
1101111.80(−10)2.96(−10)3.65(−10)
0100128.30(−11)2.07(−10)2.93(−10)
1101128.78(−11)2.35(−10)3.21(−10)
0100133.43(−11)1.70(−10)2.11(−10)
1101133.41(−11)1.45(−10)1.88(−10)
0100141.17(−11)9.98(−11)1.43(−10)
1101141.49(−11)1.15(−10)1.43(−10)
0150161.38(−10)3.13(−10)3.77(−10)
1151161.32(−10)2.78(−10)3.50(−10)
0150175.18(−11)1.88(−10)2.73(−10)
1151175.46(−11)2.17(−10)3.04(−10)
0150181.81(−11)1.58(−10)1.99(−10)
1151181.65(−11)1.30(−10)1.73(−10)
0150194.92(−12)8.85(−11)1.30(−10)
1151195.97(−12)1.01(−10)1.32(−10)
0200211.24(−10)3.09(−10)3.72(−10)
1201219.64(−11)2.69(−10)3.46(−10)
0200223.08(−11)1.75(−10)2.63(−10)
1201223.52(−11)2.02(−10)2.95(−10)
0200239.52(−12)1.45(−10)1.90(−10)
1201237.59(−12)1.20(−10)1.69(−10)
0200242.09(−12)8.16(−11)1.27(−10)
1201242.43(−12)9.34(−11)1.32(−10)

Similarities between ortho and para rates have also been observed for the collisions of H2 with anions such as CN− (Kłos & Lique 2011) and C6H− (Walker et al. 2017). These authors suggested that these similarities indicate that the effects from the long-range interaction outweigh those from the short-range interaction. This explanation however does not seem to hold for C3N− since the similarities disappear at the lowest temperatures, i.e. when the effects of the long-range potential become more important than those of the short-range potential. Furthermore, the same similarities have been also observed in the collisions of H2 with HC3N (Wernli et al. 2007), which involves a long-range potential significantly weaker than in the case of collisions with molecular anions. In this last work, it was shown that the leading term of the short-range potential is common to collisions with para and ortho conformations while the leading term of the long-range potential contributes only to collisions with the ortho conformation. Therefore, the resemblance between ortho and para rates seems to rather result from the features of the short-range interaction, such as angular anisotropy.

Following this line of thought, we analysed the matrix elements of the potential as given by the equation (9) of the seminal paper of Green (1975). These quantities are linear combinations of the radial space-fixed expansion coefficients |$A^\lambda _{\lambda _1\lambda _2}(R)$| of the potential and are related to their body-fixed counterparts |$v_{\lambda _1\lambda _2}^{\mu }(R)$|⁠, defined in equation (1) of this paper, by
(16)

For the collisions involving para-H2 (j1 = 0), the coupling matrix elements are non-zero only if λ1 = 0 while for those of ortho-H2 (j1 = 1) both λ1 = 0 and 2 are possible.

As can be seen in Fig. 4 where the first six |$A_{\lambda _1\lambda _2}^{\lambda }(R)$| coefficients are represented, the short-range interaction appears to be dominated by the attractive |$A_{00}^0$| term that gives non-zero contributions for collisions involving both ortho- and para-H2. This term is isotropic and its minimum has an intermolecular distance R close to those of the minima of the PES. The major contribution to this term is the charge-induced dipole interaction. In contrast, the strongest contribution to the long-range part of the potential is associated with the charge-quadrupole |$A^2_{20}$| term that gives non-zero potential matrix elements only for the ortho states of H2. This may explain why the difference between para and ortho rates represented in Fig. 3 are larger at low temperatures, while decreasing at higher temperature. This explanation however holds only for the lowest para and ortho states of H2 and another explanation would need to be found if further calculations show that the similarities are also observed for the higher rotational states of ortho-H2 (j1 = 3, 5, …) and para-H2 (j1 = 2, 4, …).

First six expansion coefficients $A^\lambda _{\lambda _1\lambda _2}$ as a function of the intermolecular distance. The curves are labelled by their corresponding indexes.
Figure 4.

First six expansion coefficients |$A^\lambda _{\lambda _1\lambda _2}$| as a function of the intermolecular distance. The curves are labelled by their corresponding indexes.

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We now turn our attention to the comparison of the rotational transition rate coefficients of the anions CN−, C3N−, and C6H− in collision with H2. The PES of these three collisional systems are not much different. In all three PES, the global minimum is a colinear structure and the dissociation energies are in the range 700–900  cm−1. For H2–CN− and H2–C3N−, the secondary minimum is also a colinear structure, while it is a T-shape structure for H2–C6H−. Moreover, the |$A_{00}^0$|⁠, |$A_{20}^2$|⁠, and |$A_{02}^2$| expansion coefficients are the leading terms for all three PES, as shown in Figs 4, S1, and S2 (see supplementary material). Fig. 5 shows the rate coefficients at 100 K for the de-excitation transitions from the initial rotational state j2 = 10 of the three anions through collision with para-H2. Fig. 6 shows the same comparison, but only for C6H− and C3N−, at 100 K and from the initial state j2 = 20. Very similar figures can be obtained for the de-excitation rate coefficients in the whole common range of temperature for which data are available, namely 10–100 K, and for both para- and ortho-H2.

De-excitation rate coefficients of CN−, C3N−, and C6H− in initial state j2 = 10 by collision with para-H2 at 100 K as a function of the magnitude of the transferred angular momentum |Δj2|.
Figure 5.

De-excitation rate coefficients of CN−, C3N−, and C6H− in initial state j2 = 10 by collision with para-H2 at 100 K as a function of the magnitude of the transferred angular momentum |Δj2|.

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De-excitation rate coefficients of C3N− and C6H− in initial state j2 = 20 by collision with para-H2 at 100 K as a function of the magnitude of the transferred angular momentum |Δj2|.
Figure 6.

De-excitation rate coefficients of C3N− and C6H− in initial state j2 = 20 by collision with para-H2 at 100 K as a function of the magnitude of the transferred angular momentum |Δj2|.

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As can be seen on these figures, the rate coefficients for C6H− are in remarkably good agreement with those for C3N− for |Δj2| ≲ 10, and for larger values of |Δj2|, the rate coefficients for C3N− becomes progressively smaller than the one for C6H−. The agreement between the rate coefficients for CN− and those of C3N− and C6H− is also relatively good for the transitions with |Δj2| ≲ 5 while for larger values of |Δj2|, the CN− rates become much smaller than the other ones. A simple explanation involving the size of the molecules can be proposed. The three anions are linear molecules, but they are of significantly different lengths. A longer molecule imply a stronger anisotropy of the short-range interaction potential and also a smaller rotational constant. Both features are expected to enhance the transitions with large transferred rotational angular momentum, as observed in Figs 5 and 6.

The most interesting result is nevertheless the remarkably good agreement between the rate coefficients for the lowest values of |Δj2|. This suggest that collisional rates of H2 with the larger chains of the two C2n + 1N− and C2nH− families could be extrapolated from those now available for C3N− and C6H−. One reason for such a good agreement might be the nature of the long-range interaction. For anion–H2 collisional systems, the leading long-range interactions are the charge-quadrupole interaction, which scales as R−3, and the charge-induced dipole that scales as R−4. These interactions are independent of the nature of the ionic monomer. Cations–H2 collisional systems are also expected to show the same behaviour.

We conclude this study by checking the validity of the He/para-H2 scaling law for the rotational transitions of C3N−. In this approximation, the inelastic collisional rates of any molecule with para-H2 (j1 = 0) are derived from those with He by using the following reduced mass ratio:
(17)
where μA − B stands for the reduced mass of the A + B collisional system. For C3N−, the ratio between the previously calculated rates for the collisions with He (Lara-Moreno et al. 2017) and those calculated in this work for para-H2 is in average equal to 3 and can reach values as large as 6 at low temperature. The averaged value is then more than twice larger than the expected value of r ≈ 1.41, thus leading to a relatively large average error |$(\lt 50{{\ \rm per\ cent}})$| for the scaled para-H2 rate coefficients. In addition, we found that the use of the He/para-H2 scaling law would introduce a spurious propensity rule |Δj2| = 2 that is not observed in the accurately computed rate coefficients with para-H2.

In order to better understand the origin of the failure of the He/para-H2 scaling law for this system, we compare the H2–C3N− PES averaged over the rotational ground state of para-H2 with the He–C3N− PES in Fig. 7. Let us note that the averaging washes out the global and secondary potential wells seen in Fig. 1 since the rotation of H2 involves to pass through a large energy barrier located around θ1 = 90° (Lara-Moreno et al. 2019). We see at first that the long-range potential is stronger in the case of para-H2. This is mainly due to the large difference between the isotropic polarizabilities1 of He (αHe = 1.38 a03) and H2 (⁠|$\alpha _{\mathrm{H_2}}=5.4~{a_0}^3$|⁠) since the charge-induced dipole interaction, proportional to |$-\frac{\alpha }{R^4}$|⁠, is the leading contribution to both interaction potentials at large distance. As a result, the magnitude of the long-range potential of C3N− with H2 is almost five times larger than its He counterpart. This may explain the quite large value r ≈ 6 found at low collision energy. A second difference that could be significant is related to the potential wells that are both associated with a distorted T-shape geometry, but the para-H2–C3N− well is slightly deeper than the He–C3N− one by a few  cm−1. A third difference stems from the symmetry of the two PES with respect to the inversion of the C3N− molecule. Both PES are not symmetric, but the He–C3N− PES is slightly more symmetrical than the para-H2–C3N− one if we compare the -10  cm−1 contour lines. This last point may explain the differences observed in the propensity rules.

2D polar contour plots of the interaction of C3N− with para H2(upper panel) and He (lower panel). The interaction of C3N− with H2 is represented by a potential averaged over the rotational wavefunction |j1 = 0, m1 = 0〉. Contour levels are equally spaced and labelled by the energy in  cm −1. The contours are blue for negative interaction energy and red for positive one. The origin of coordinates corresponds to the centre of mass of C3N−.
Figure 7.

2D polar contour plots of the interaction of C3N− with para H2(upper panel) and He (lower panel). The interaction of C3N− with H2 is represented by a potential averaged over the rotational wavefunction |j1 = 0, m1 = 0〉. Contour levels are equally spaced and labelled by the energy in  cm −1. The contours are blue for negative interaction energy and red for positive one. The origin of coordinates corresponds to the centre of mass of C3N−.

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3.2 Critical densities

The local thermodynamic equilibrium (LTE) is established if the population of excited levels is given by the Boltzmann’s law. This occurs only if the rate of spontaneous emission is significantly smaller than the rate of de-excitation by collision and therefore only if the density of gas is significantly larger than the critical density.

C3N− has been detected in the CSE of the carbon star IRC  + 10216 and in the Taurus molecular cloud 1 (TMC-1). In a CSE, the gas density and temperature are decreasing functions of the radius r. The shell in which the largest abundance of C3N− is predicted by chemical models is approximately defined by the radius range 1016–1018 cm (Cernicharo et al. 2008; Millar 2016). In this shell, the gas density is assumed to fall as r−2 from more than 105 cm−3 to less than 102 cm−3 while temperature is assumed to decrease from 40 to 10 K (Cordiner & Millar 2009). In the dark cloud TMC-1, the gas density is believed to be about 104 cm−3 with temperature in the range 10–30 K.

Fig. 8 shows that the critical densities for the lowest rotational levels (j ≲ 5) are relatively small and are reached in most of the interstellar environments where C3N− has been detected. For the higher rotational levels, the LTE cannot be taken for granted. It is clearly dependent of the temperature and gas density. Sufficiently hot (kinetic temperature T ≳ 40 K) and diffuse (n ≲ 104 cm−3) conditions will not allow LTE.

Critical densities for the rotational levels of C3N−. The curves are labelled by the rotational quantum number.
Figure 8.

Critical densities for the rotational levels of C3N−. The curves are labelled by the rotational quantum number.

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4 CONCLUSIONS

State-to-state rotational excitation and de-excitation rate coefficients of C3N− in collisions with ortho- and para-H2 in the temperature range [10, 300] K were obtained by combining CC calculations and the UJS procedure. We find that the relative error offered by the UJS method is always less than 10 |${{\ \rm per\ cent}}$| demonstrating once again that the UJS procedure is a very good approximation that reduces computational cost without losing much accuracy. On the other hand, we found once again that the application of the He/para-H2 scaling law to the He–C3N− rate coefficients would lead to relatively large error |$(\lt 50{{\ \rm per\ cent}})$| for the para-H2 rates.

The strongly repulsive nature of the H2-C3N− interaction at short range leads to close similarities between ortho and para rates and enhances transition associated with large transferred angular momentum. The rotational transition rates of C3N− in collision with para-H2 are quite close to those with para-H2. The same effect was previously observed for CN− and C6H−.

The rotational de-excitation rate coefficients of the three anions CN−, C6N− and C3N− are in close agreement for small values of the transferred rotational angular momentum |Δj2|, and a remarkably good agreement is observed between C6N− and C3N− for |Δj2| ≲ 10. This agreement arises probably from the fact that the leading long-range interactions, namely the charge-quadrupole and the charge-induced dipole interactions, are independent of the charged species. This opens the way for a quantitative extrapolation of the rotational transition rate coefficients of anions in collision with H2, such as for example, the carbon chains of the two C2n + 1N− and C2nH− families that have been discovered in the ISM.

SUPPORTING INFORMATION

Table 1. State-to-state rate coefficients |$k_{j_1,j_2\rightarrow {j_1}^{\prime },{j_2}^{\prime }}$| (cm3 molecule−1 s−1) for the rotational excitation of C3N− in collision with para (j1 = 0) and ortho (j1 = 1) H2 for various temperature values. Power of 10 is denoted in parenthesis.

Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.

ACKNOWLEDGEMENTS

This research has been supported by the Agence Nationale de la Recherche (Project ANR-AnionCosChem). Computer time for this study was provided by the Mésocentre de Calcul Intensif Aquitain that is the computing facilities of Université de Bordeaux et Université de Pau.

Footnotes

1

Calculated using finite field method at CCSD(T)/aug-cc-pVQZ level.

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