A&A 464, 1147-1154 (2007)
DOI: 10.1051/0004-6361:20066112

Rotational excitation of HC₃N by H₂ and He at low temperatures

M. Wernli - L. Wiesenfeld - A. Faure - P. Valiron

Laboratoire d'Astrophysique de l'Observatoire de Grenoble, UMR 5571 CNRS/UJF, Université Joseph-Fourier, Boîte postale 53, 38041 Grenoble Cedex 09, France

Received 26 July 2006 / Accepted 20 November 2006

Abstract
Aims. Rates for rotational excitation of ${\rm HC_3N}$ by collisions with He atoms and H₂ molecules are computed for kinetic temperatures in the range 5-20 K and 5-100 K, respectively.
Methods. These rates are obtained from extensive quantum and quasi-classical calculations using new accurate potential energy surfaces (PES). The ${\rm HC_3N}$-He PES is in excellent agreement with the recent literature. The ${\rm HC_3N}$-H₂ angular dependence is approximated using 5 independent H₂ orientations. An accurate angular expansion of both PES suitable for low energy scattering is achieved despite the severe steric hindrance effects by the ${\rm HC_3N}$ rod.
Results. The rod-like symmetry of the PES strongly favours even $\Delta J$ transfers and efficiently drives large $\Delta J$ transfers. Despite the large dipole moment of ${\rm HC_3N}$, rates involving ortho-H₂ are very similar to those involving para-H₂, because of the predominance of the geometry effects. Except for the even $\Delta J$ propensity rule, quasi classical calculations are in excellent agreement with close coupling quantum calculations. As a first application, we present a simple steady-state population model that shows population inversions for the lowest HC₃N levels at H₂ densities in the range 10⁴-10⁶ cm^-3.
Conclusions. The ${\rm HC_3N}$ molecule is large enough to present an original collisional behaviour where steric hindrance effects hide the details of the interaction. This finding, combined with the fair accuracy of quasi classical rate calculations, is promising in view of collisional studies of larger molecules.

Key words: molecular data - molecular processes

1 Introduction

Cyanopolyyne molecules, with general formula HC_2n+1N, $n\ge 1$, have been detected in a great variety of astronomical environments and belong to the most abundant species in cold and dense interstellar clouds (Bell et al. 1997). One of these, HC₁₁N, is currently the largest unambiguously detected interstellar molecule (Bell & Matthews 1985). The simplest one, ${\rm HC_3N}$ (cyanoacetylene), is the most abundant of the family. In addition to interstellar clouds, ${\rm HC_3N}$ has been observed in circumstellar envelopes (Pardo et al. 2004), in the Saturn satellite Titan (Kunde et al. 1981), in comets (Bockelée-Morvan et al. 2000) and in extragalactic sources (Mauersberger et al. 1990). Furthermore, ${\rm HC_3N}$ has been detected both in the ground level and in excited vibrational levels, thanks to the presence of low-lying bending modes (e.g. Wyrowski et al. 2003). Because of its low rotational constant and large dipole moment, cyanoacetylene lines are observable over a wide range of excitation energies and ${\rm HC_3N}$ is therefore considered as a very good probe of physical conditions in many environments.

Radiative transfer models for the interpretation of observed ${\rm HC_3N}$ spectra require the knowledge of collisional excitation rates participating in line formation. To the best of our knowledge, the only available collisional rates are those of Green & Chapman (1978) for the rotational excitation of HC₃N by He below 100 K. In cold and dense clouds, however, the most abundant colliding partner is H₂. In such environments, para-${\rm H_2}$ is only populated in the J=0 level and may be treated as a spherical body. Green & Chapman (1978) and Bhattacharyya & Dickinson (1982) postulated that the collisional cross-sections with para-${\rm H_2}$ (J=0) are similar to those with He (assuming thus an identical interaction and insensitivity of the scattering to the reduced mass). As a result, rates for excitation by para-${\rm H_2}$ were estimated by scaling the rates for excitation by He; rates involving ortho-${\rm H_2}$ were not considered.

In the present study, we have computed new rate coefficients for rotational excitation of ${\rm HC_3N}$ by He, para-${\rm H_2}$ (J=0) and ortho-${\rm H_2}$ (J=1), in the temperature range 5-20 K for He and 5-100 K for H₂. A comparison between the different partners is presented and the collisional selection rules are investigated in detail. The next section describes details of the PES calculations. The cross-section and rate calculations are presented in Sect. 3. A discussion and a first application of these rates is given in Sect. 4. Conclusions are drawn in Sect. 5. The following units are used throughout except where otherwise stated: bond lengths and distances in Bohr; angles in degrees; energies in cm^-1; cross-sections in $\AA^2$.

  
2 Potential energy surfaces

Two accurate interatomic potential energy surfaces (PES) have recently been calculated in our group, for the interaction of ${\rm HC_3N}$ with He and H₂. Both surfaces involved the same geometrical setup and similar ab initio accuracy. An outline of those PES is given below and a detailed presentation will be published in a forthcoming article.

In the present work, we focus on low-temperature collision rates, well below the threshold for the excitation of the lower bending mode $\nu_7$ at 223 cm^-1. The collision partners may thus safely be approximated to be rigid, in order to keep the number of degrees of freedom as small as possible. For small van der Waals complexes, previous studies have suggested that properly averaged molecular geometries provide a better description of experimental data than equilibrium geometries ($r_{\rm e}$ geometries) (Jeziorska et al. 2000; Jankowski & Szalewicz 2005). For the $\rm H_2O$-${\rm H_2}$ system, geometries averaged over ground-state vibrational wave-functions (r₀ geometries) were shown to provide an optimal approximation of the effective interaction (Faure et al. 2005a; Wernli 2006).

Accordingly, we used the ${\rm H_2}$ bond separation $r_{\rm HH}= 1.44876$Bohr obtained by averaging over the ground-state vibrational wave-function, similarly to previous calculations (Hodges et al. 2004; Faure et al. 2005a; Wernli et al. 2006). For ${\rm HC_3N}$, as vibrational wave-functions are not readily available from the literature, we resorted to experimental geometries deduced from the rotational spectrum of ${\rm HC_3N}$ and its isotopologues (Thorwirth et al. 2000; see also Table 5.8 in Gordy & Cook 1984). The resulting bond separations are the following: $r_{{\rm HC_1}}= 1.998385$; $r_{{\rm C_1C_2}}=2.276364$; $r_{{\rm C_2C_3}}= 2.606688$; $r_{{\rm C_3N}}= 2.189625$, and should be close to vibrationally averaged values.

For the ${\rm HC_3N}$-He collision, only two coordinates are needed to fully determine the overall geometry. Let $\vec{R}$ be the vector between the center of mass of ${\rm HC_3N}$ and He. The two coordinates are the distance $R=\vert\vec{R}\vert$ and the angle $\theta_1$ between the ${\rm HC_3N}$ rod and the vector R. In our conventions, $\theta_1 = 0$corresponds to an approach towards the H end of the ${\rm HC_3N}$ rod. For the collision with H₂, two more angles have to be added, $\theta_2$ and $\phi$, that respectively orient the ${\rm H_2}$ molecule in the rod-R plane and out of the plane. The ${\rm HC_3N}$-He PES has thus two degrees of freedom, the ${\rm HC_3N}$-${\rm H_2}$ four degrees of freedom.

As we aim to solve close coupling equations for the scattering, we need ultimately to expand the PES function V over a suitable angular expansion for any intermolecular distance R. In the simpler case of the ${\rm HC_3N}$-He system, this expansion is in the form:

$$% V_{}(R,\theta_1) = \sum_{l_1} v_{l_1}(R)~P_{l_1}(\cos\theta_1),$$ (1)

where $P_{l_1}(\cos\theta_1)$ is a Legendre polynomial and  v_l1(R) are the radial coefficients.

For the ${\rm HC_3N}$-${\rm H_2}$ system, the expansion becomes:

$$% V(R,\theta_1, \theta_2, \phi) = \sum_{l_1 l_2 l} v_{l_1 l_2 l}(R) s_{l_1 l_2 l}(\theta_1, \theta_2, \phi),$$ (2)

where the basis functions s_{l1 l2 l} are products of spherical harmonics and are expressed in Eq. (A9) of Green (1975). Two new indices l₂ and l are thus needed, associated respectively with the rotational angular momentum of ${\rm H_2}$ and the total orbital angular momentum, see also Eqs. (A2) and (A5) of Green (1975).

Because the Legendre polynomials form a complete set, such expansions should always be possible. However, Chapman & Green (1977) failed to converge the above expansion (1) due to the steric hindrance of He by the impenetrable ${\rm HC_3N}$ rod, and Green & Chapman (1978) abandoned quantum calculations, resorting to quasi classical trajectories (QCT) studies. Similar difficulties arise for the interaction with H₂. As can be seen in Fig. 1 for small R values, the interaction is moderate or possibly weakly attractive for $\theta_1 \sim 90^{\circ}$ and is extremely repulsive or undefined for $\theta_1 \sim 0, 180^{\circ}$, leading to singularities in the angular expansion and severe Gibbs oscillations in the numerical fit of the PES over Legendre expansions.

Accordingly, we resorted to a cautious sampling strategy for the PES, building a spline interpolation in a first step, and postponing the troublesome angular Legendre expansion to a second step. All details will be published elsewhere. We summarize this first step for He, then for H₂.

Figure 1: The ${\rm HC_3N}$-para-${\rm H_2}$ PES. The ${\rm HC_3N}$ molecule is shown to scale. Equipotentials (in  $\rm cm^{-1}$) : in dashed red, -100, -30 -10, -3; in solid black, 0; in blue, 10, 30, 100, 300, 1000, 3000. The dotted circle centered at the ${\rm HC_3N}$ center of mass with radius R=6.41 Bohr illustrates the angular steric hindrance problem occurring when the collider rotates from the vicinity of the minimum towards the ${\rm HC_3N}$ rod.

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For the ${\rm HC_3N}$-He PES, we selected an irregular grid in the $\left\{R,\theta_1\right\}$ coordinates. The first order derivatives of the angular spline were forced to zero for $\theta_1=0,180^{\circ}$ in order to comply with the PES symmetries. For each distance, angles were added until a smooth convergence of the angular spline fit was achieved, resulting in typical angular steps between 2 and 15$^{\circ}$. Then, distances were added until a smooth bicubic spline fit was obtained, amounting to 38 distances in the range 2.75-25 Bohr and a total of 644 geometries. The resulting PES is perfectly suited to run quasi classical trajectories.

We used a similar strategy to describe the interaction with H₂, while minimizing the number of calculations. We selected a few $\left\{\theta_2,\phi\right\}$ orientation sets, bearing in mind that the dependence of the final PES on the orientation of ${\rm H_2}$ is weak. In terms of spherical harmonics, the PES depends only on $Y_{l_2m_2}(\theta_2,\phi)$, with $l_2=0,2,4,\ldots$ and $m_2=0,1,2,\ldots$, $\vert m_2\vert\leq l_2$. Terms in Y_l2m2 and  Y_{l2 -m2} are equal by symmetry. Previous studies (Faure et al. 2005b; Wernli et al. 2006) have shown that terms with l₂> 2 are small, and we consequently truncated the  Y_l2m2 series to $l_2\leq 2$. Hence, only four basis functions remain for the orientation of ${\rm H_2}$: Y₀₀,Y₂₀,Y₂₁ and Y₂₂.

Under this assumption, the whole ${\rm HC_3N}$-${\rm H_2}$ surface can be obtained knowing its value for four sets of $\left\{\theta_2,\phi\right\}$ angles at each value of R. We selected five sets, thus having an over-determined system allowing for the monitoring of the accuracy of the l₂ truncation. Consequently, we determined five independent PES, each constructed similarly to the ${\rm HC_3N}$-He one as a bicubic spline fit over an irregular grid in $\left\{R,\theta_1\right\}$ coordinates. The angular mesh is slightly denser than for the ${\rm HC_3N}$-He PES for small R distances to account for more severe steric hindrance effects involving ${\rm H_2}$. In total, we computed 3420 $\left\{R, \theta_1, \theta_2, \phi\right\}$ geometries. The ${\rm HC_3N}$-H₂ interaction can be readily reconstructed from these five PES by expressing its analytical dependence over $\left\{\theta_2,\phi\right\}$(Wernli 2006).

For each value of the intermolecular geometry  $\left\{R,\theta_1\right\}$ or $\left\{R, \theta_1, \theta_2, \phi\right\}$, the intermolecular potential energy is calculated at the conventional CCSD(T) level of theory, including the usual counterpoise correction of the Basis Set Superposition Error (Jansen & Ross 1969; Boys & Bernardi 1970). We used augmented correlation-consistent atomic sets of triple zeta quality (Dunning's aug-cc-pVTZ) to describe the ${\rm HC_3N}$ rod. In order to avoid any possible steric hindrance problems at the basis set level, we did not use bond functions and instead chose larger Dunning's aug-cc-pV5Z and aug-cc-pVQZ basis set to better describe the polarizable (He, H₂) targets, respectively. All calculations employed the direct parallel code DIRCCR12 (Noga et al. 2003-2006).

Comparison of the ${\rm HC_3N}$-He PES with existing surfaces (Akin-Ojo et al. 2003; Topic & Wolfgang 2005) showed an excellent agreement. The ${\rm HC_3N}$-para-${\rm H_2}$ (J=0) interaction (obtained by averaging the ${\rm HC_3N}$-H₂ PES over $\theta_2$ and $\phi$) is qualitatively similar to the ${\rm HC_3N}$-He PES with a deeper minimum (see values at the end of present section). As illustrated in Fig. 1, these PES are largely dominated by the rod-like shape of  ${\rm HC_3N}$, implying a prolate ellipsoid symmetry of the equipotentials.

In a second step, we consider how to circumvent the difficulty of the angular expansion of the above PES, in order to obtain reliable expansions for He and H₂ (Eqs. (1) and (2)).

Using the angular spline representation, we first expressed each PES over a fine $\theta_1$ mesh suitable for a subsequent high l₁ expansion. As expected from the work of Chapman & Green (1977), high l₁ expansions (1) resulted in severe Gibbs oscillations for R in the range 5-7 Bohr, making impossible the description of the low energy features of the PES. For low energy scattering applications, we regularized the PES by introducing a scaling function $S_{\rm f}$. We replaced $V(R,\theta_1,...)$ by  $S_{\rm f}(V(R,\theta_1,...))$, where  $S_{\rm f}(V)$ returns V when V is lower than a prescribed threshold, and then smoothly saturates to a limiting value when V increases to very repulsive values. Consequently, the regularized PES retains only the low energy content of the original PES, unmodified up to the range of the threshold energy; it should not be used for higher collisional energies. However, in contrast to the original PES, it can be easily expanded over Legendre functions to an excellent accuracy and is thus suitable for quantum close coupling studies. We selected a threshold value of 300 cm^-1, and improved the quality of the expansion by applying a weighted fitting strategy (e.g. Hodges et al. 2004) to focus the fit on the details of the attractive and weakly repulsive regions of the PES. Using $l_1\leq 35$, both the He and H₂ PES fits were converged to within 1 cm^-1 for $V \le 300$ cm^-1. These expansions still describe the range 300<V<1000 cm^-1 to within an accuracy of a few  $\rm cm^{-1}$.

The corresponding absolute minima are the following (in cm^-1 and Bohr): for ${\rm HC_3N}$-He, V=-40.25 for R=6.32 and $\theta_1=95.2^{\circ}$; for ${\rm HC_3N}$-para-${\rm H_2}$ (J=0), V=-111.24 for R=6.41 and $\theta_1=94.0^{\circ}$; and for ${\rm HC_3N}$-${\rm H_2}$, V=-192.49for R=9.59, $\theta_1=180^{\circ}$, and $\theta_2=0^{\circ}$.

  
3 Inelastic cross section and rates

In the following, $J_1, J^\prime_1$, denote the initial and final angular momentum of the ${\rm HC_3N}$ molecule, and J₂ denotes the angular momentum of H₂. We also denote the largest value of  $J_1, J^\prime_1$ as $J_{1\rm up}$.

The most reliable way to compute inelastic cross sections $\sigma_{J_1J^{~\prime}_1}(E)$ is to perform quantum close coupling calculations. In the case of molecules with a small rotational constant, like ${\rm HC_3N}$ ( $B=4549.059~{\rm MHz}$, see e.g. Thorwirth et al. 2000), quantum calculations soon become intractable, because of the large number of open channels involved. While observations at cm-mm wavelengths culminate with $J_{1\rm up} \la 24$ (Audinos et al. 1994), sub-mm observations can probe transitions as high as $J_{1\rm up} = 40$, at a frequency of 363.785 GHz and a rotational energy of $202.08~\rm cm^{-1}$ (Kuan et al. 2004; Caux 2006; Pardo et al. 2004). It is thus necessary to compute rates with transitions up to J₁=50 (E= 386.8 cm^-1), in order to properly converge radiative transfer models. Also, we aim to compute rates up to a temperature of 100 K for H₂. We used two methods. For $J_{1\rm up}\leq 15$, we performed quantum inelastic scattering calculations, as presented in next Sect. 3.1. For $J_{1\rm up} > 15$, we used the QCT method, as presented in Sect. 3.2.

For He, of less astrophysical importance ([He]/[H] $\sim$ 0.1), only quantum calculations were performed and were limited to the low temperature regime (T = 5-20 K and J₁<10).

  
3.1 Rotational inelastic cross sections with MOLSCAT

All calculations were made using the rigid rotor approximation, with rotational constants $B_{\rm HC_3N}=0.151739$ cm^-1 and $B_{\rm H_2}=60.853$ cm^-1, using the MOLSCAT code (Hutson & Green 1994). All quantum calculations for ${\rm HC_3N}$-ortho-${\rm H_2}$ were performed with $J_{\rm H_2}\equiv J_2=1$. Calculations for ${\rm HC_3N}$-para-${\rm H_2}$ were performed with J₂=0. We checked at $E_{\rm tot}=E_{\rm coll}+E_{\rm rot} = 30~\rm cm^{-1}$ that the inclusion of the closed J₂=2 channel had negligible effects.

The energy grid was adjusted to reproduce all the details of the resonances, as they are essential to calculate the rates with high confidence (Dubernet & Grosjean 2002; Grosjean et al. 2003; Wernli et al. 2006). The energy grid and the quantum methods used are detailed in Table 1. Using this grid, we calculated the whole resonance structure of all the transitions up to J₁=15 for the ${\rm HC_3N}$-para-${\rm H_2}$ collisions. At least 10 closed channels were included at each energy to fully converge the ${\rm HC_3N}$ rotational basis. We used the hybrid log-derivative/Airy propagator (Alexander & Manolopoulos 1987). We increased the parameter STEPS at the lowest energies to constrain the step length of the integrator below 0.1 to 0.2 Bohr, in order to properly follow the details of the radial coefficients. Other propagation parameters were taken as the MOLSCAT default values.

Table 1: Details of the quantum MOLSCAT cross section calculation parameters. $J_{1\rm up}({\rm HC_3N}) \leq 15$ (10 for He). Methods: CC, Close coupling, CS, Coupled states approx., IOS, Infinite Order Sudden approx.

Two examples of deexcitation cross-sections are shown in Fig. 2. We see that for energies between threshold and about 20 cm^-1 above threshold, the cross-section displays many shape resonances, justifying a posteriori our very fine energy grid. This behaviour is similar to most earlier calculations is many different systems, see e.g. Dubernet & Grosjean (2002); Wernli et al. (2006) for a discussion. From a semi-classical point of view, these shape resonances reflect the trapping of the wave-packet between the inner repulsive wall and the outer centrifugal barrier, see Wiesenfeld et al. (2003), Abrol et al. (2001). At energies higher than about 20 cm^-1 above threshold, all cross-sections become smooth functions of the energy.

Figure 2 also shows that ortho-${\rm H_2}$ inelastic cross-sections very closely follow the para-${\rm H_2}$ ones, including the position of resonances. Examination of all cross-sections reveals that the relative difference between $\sigma_{J_1J'_1}(E, {\rm para})$ and $\sigma_{J_1J'_1}(E, {\rm ortho})$ is less than $5\%$. This justifies a posteriori the much smaller amount of computational effort devoted to ortho-${\rm H_2}$ collisions as well as the neglect of J₂ = 2 closed para-${\rm H_2}$ channels. A detailed discussion of this behaviour is given in Sect. 4.1.

Figure 2: ${\rm HC_3N}$-${\rm H_2}$ collisions. Quantum deexcitation cross sections for transitions $J_1=1\rightarrow 0$ (lower trace) and $J_1=4 \rightarrow 2$ (upper trace) as a function of the energy E = $E_{\rm coll}+E_{\rm rot(HC_3N)}$. Full line, para-${\rm H_2}$ (J=0) collisions (0.1 cm-1 energy spacing); open circles, ortho-${\rm H_2}$ (J=1). The fine energy grid emphasizes the resonances for $E \protect\la 10~\rm cm^{-1}$.

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3.2 Quantum and classical rates

The quantum collisional rates are calculated for $J_{1\rm up}\leq 15$ at astrophysically relevant temperatures, from 5 K to 100 K. We average the cross-sections described in the preceding section over the Maxwell distribution of velocities, up to a kinetic energy at least 10 times $\it kT$. The quantum calculations at the higher end of the energy range are approximated at the IOS level (see Table 1), which is justified at these energies by the smallness of the rotational constant  $B_{\rm HC_3N}$. Also we used a coarse energy grid for the IOS calculations because the energy dependence of the cross-sections becomes very smooth.

For values of J₁ > 15, the close coupling approach enters a complexity barrier due to the rapid increase of the number of channels involved in calculations, while memory and CPU requirements scale as the square and the cube of this number, respectively. Resorting to quantum CS or IOS approximations is inaccurate, because the energy is close to threshold for high-J₁ channels. In the meanwhile, the accuracy of classical approximations improves for higher collisional energies. For the energy range where J₁ > 15 channels are open and for deexcitation processes involving those channels, we employ a Quasi-Classical Trajectory (QCT) method, which has been shown to be a valid approximation for higher collisional energies and high rates (Lepp et al. 1995; Chapman & Green 1977; Mandy & Progrebnya 2004; Faure et al. 2006).

For Monte-Carlo QCT methods, we must devise a way of defining an ensemble of initial conditions for classical trajectories, on the one hand, and of analyzing the final state of each trajectory, on the other hand. Contrary to the asymmetric rotor case (like water, see Faure & Wiesenfeld 2004), the analysis of final conditions for a linear molecule is straightforward. Using the simplest quantization approximation, we bin the final classical angular momentum J'₁ of  ${\rm HC_3N}$ to the nearest integer. While the quantum formalism goes through a microcanonical calculation - calculating $\sigma_{J_1J_1'}(E)$ for fixed energies, then averaging over velocity distributions - it is possible for QCT calculations to directly resort to a canonical formalism, i.e. to select the initial velocities of the Monte-Carlo ensemble according to the relevant Maxwell-Boltzmann distribution and find the rates as:

$$% k_{J_1J'_1} = \left(\frac{8kT}{\pi\mu}\right)^{1/2}~\pi b_{\rm max}^2~\frac{N}{N_{\rm tot}}$$ (3)

where $b_{\rm max}$ is the maximum impact parameter used (with the impact parameter b distributed with the relevant  $b~\textmd{d}b$probability density) and N is the number of trajectories with the right final J'₁ value among all $N_{\rm tot}$ trajectories. The Monte-Carlo standard deviation is:

$$% \frac{\delta k_{J_1J_1'}}{k_{J_1J_1'}} = \left(\frac{N_{\rm tot}-N}{N_{\rm tot}N}\right)^{1/2},$$ (4)

showing that the accuracy of the method improves for higher rates. The $b_{\rm max}$ parameter was determined by running small batches of 500 to 1000 trajectories for fixed b values; values of $20 \leq b_{\rm max} \leq 26$ Bohr were found. We then ran batches of 10 000 trajectories for each temperature in the range 5-100 K, with a step of 5 K. Trajectories are integrated by means of a Bürlich-Stoer algorithm (Press et al. 1992), with a code similar to that of Faure et al. (2005b). Precision is checked by conservation of total energy and total angular momentum.

Some results are shown in Tables 2 and 3 and are illustrated in Figs. 3 and 4.

As an alternative to QCT calculations, we tested J-extrapolation techniques, using the form of DePristo et al. (1979) generally used by astrophysicists (see for example Schöier et al. (2005), Sect. 6). We found that even if it reproduces the interference pattern, the extrapolation systematically underestimates the rates, for $J_1\ge 20$. Hence, QCT rates are more precise on average.

Table 2: ${\rm HC_3N}$-para-${\rm H_2}$ (J=0) collisions. Quantum deexcitation rates in ${\rm cm^3~s^{-1}}$, for J₁'=0, for successive initial J₁ and for various temperatures. Powers of ten are denoted in parentheses.

Table 3: ${\rm HC_3N}$-para-${\rm H_2}$ collisions. Quantum or QCT ($^\dag$) deexcitation rates in ${\rm cm^3~s^{-1}}$, for J₁-J'₁=1,2,3,4, for various temperatures and representative values of J₁. Powers of ten are denoted in parentheses.

For H₂, all deexcitation rates k_J1J1'(T), $J_1\neq J_1'\leq 50$, are fitted with the following formula (Wernli et al. 2006):

$$% \log_{10}\left(k_{J_1J_1'}(T)\right)=\sum_{n=0}^{ 4}a^{(n)}_{J_1J_1'} x^n$$ (5)

where x=T^-1/6. As some transitions have zero probability within the QCT approach, the above formula was employed when rates were greater than 10^-12 cm³ s^-1 for at least one grid temperature. For these rates, null grid values were replaced by a very small value, namely 10^-14 cm³ s^-1, to avoid fitting irregularities. All rates not fulfilling this condition are set to zero. Note that below 20 K, QCT rates for low-probability transitions may show a non physical behaviour. All a⁽ⁿ⁾_J1J1' coefficients are provided as online material, for a temperature range $5~{\rm K}\leq T \leq 100~{\rm K}$. We advise to use the same rates for collisions with ortho-${\rm H_2}$ as for para-H₂, since their difference is smaller than the uncertainty on the rates themselves. Rates with He were not fitted, but can be obtained upon request from the authors.

	Figure 3: Collisional excitation rates for ${\rm HC_3N}$, from J1=0, at 20 K. Present quantum close-coupling rates: Open circles, ${\rm HC_3N}$-He collisions; filled squares, ${\rm HC_3N}$-para-${\rm H_2}$ (J=0). Gray lines serve as a guide. For comparison, QCT rates for the ${\rm HC_3N}$-He (Green & Chapman 1978) are shown as a dashed line.
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	Figure 4: ${\rm HC_3N}$-para ${\rm H_2}$ collisions. Deexcitation rates, from J1=15, at 10 K ( left panel) and 100 K ( right panel). Full black line, quantum calculations; dashed line, Monte-Carlo quasi-classical calculations with error bars.
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4 Discussion

  
4.1 Para and ortho H₂ cross-sections

A comparison of the $\sigma_{J_1J_1'}(E)$ cross sections for ${\rm HC_3N}$ with ortho-${\rm H_2}$ and para-${\rm H_2}$ is given in Fig. 2. The difference between the two spin species of ${\rm H_2}$ is tiny, in any case smaller than other PES and cross-section uncertainties. This is an unexpected result, as sizeable differences between para-${\rm H_2}$ and ortho-${\rm H_2}$ inelastic cross-sections exist for other molecules. These differences were expected to increase for a molecule possessing a large dipolar moment, in view of the results obtained for the C₂ molecule (Phillips 1994), the CO molecule (Wernli et al. 2006), the OH radical (Offer et al. 1994), the NH₃ molecule (Offer & Flower 1989; Flower & Offer 1994) and the $\rm H_2O$ molecule (Phillips et al. 1996; Dubernet & Grosjean 2002; Dubernet et al. 2006; Grosjean et al. 2003), due to the interaction between the dipole of the molecule and the quadrupole of ${\rm H_2}$ (J₂ > 0).

This apparently null result deserves an explanation. We focus on Eq. (9) of Green (1975). This equation describes the different matrix elements that couple the various channels in the close-coupling equations. Some triangle rules apply which restrict the number of terms in the sum of Eq. (9); the relevant angular coupling algebra is represented there as a sum of terms of the type

$$% \left(\begin{array}[c]{ccc} l &L'& L \\ 0 & 0 & 0 \end{arr... ...{ccc} L'& L & l \\ J_{12} & J'_{12} & J \end{array}\right\},$$ (6)

where we have the potential function expanded in terms of Eqs. (4) and (A2) in Green (1975), by means of the coefficients  v_{l1l2 l}. The symbol $(\ldots)$ are 3-j symbols, the $\{\ldots\}$ is a 6-j symbol, see Messiah (1969). We also define $\vec{J}_{12}=\vec{J}_1+\vec{J}_2$. We have the following rules:

• The para-${\rm H_2}$ inelastic collisions are dominated by the J₂ = 0 channel (the J₂=2 channel is closed until  $E_{\rm coll} \ga 365.12~\rm cm^{-1}$). Then, only the l₂=0 may be retained ( J₂=J'₂=0), due to the third 3-j symbol in Eq. (6).

• The ortho-${\rm H_2}$ remains in J₂=1, implying l₂=0,2.

• For inelastic collisions, $J_1\neq J'_1$ implies potential terms with $l\neq 0$, because of the 6-j term in Eq. (6). Indeed, J₂=J'₂ and $J_1\neq J'_1$ entail $J_{12}\neq J'_{12}$.

The key point is thus to compare the v_{l1l2 l}(R) coefficients (Eq. (2)) with $l\neq 0$ in the two cases:

• l₂=0 para and ortho contributions;
• l₂=2 ortho contribution only.

Figure 5 displays such a comparison. We notice that the coupling is largely dominated by the l₂=0 contribution, terms which are common to collisions with para and ortho conformations. This is particularly true for R<10 Bohr, the relevant part of the interaction for collisions at temperatures higher than a few Kelvin. At a higher intermolecular separation, terms implied only in collisions with ortho-${\rm H_2}$ become dominant, but in this regime the potential is also less than a few cm^-1. Sizeable differences in rates between ortho and para forms are thus expected only either at very low temperatures, or possibly at much higher temperatures, with the opening of ${\rm H_2}$ (J₂=2,3) channels.

	Figure 5: Comparison of the coupling terms ($l\ne 0$) in the ${\rm HC_3N}$-${\rm H_2}$ potential as a function of the intermolecular distance. CSS is $\left (\sum _{l\ne 0, l_1} v_{l_1 l_2 l}(R)^2\right )^{1/2}$. Terms with l2=0 are common to ortho and para ${\rm H_2}$, while the l2=2 curve represents purely ortho terms. See text.
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4.2 Propensity rules

In Fig. 3, we compare the various rates that we obtain here with the ones previously published by Green & Chapman (1978). These authors used a coarse electron-gas approximation for the PES, and computed rates by a QCT classical approach. Despite these approximations, we see that the rates obtained by Green & Chapman (1978) are qualitatively comparable with the quantum rates obtained here, on average. However, as Table 3 and Figs. 4 and 3 show clearly, only quantum calculations manifest the strong $\Delta J = 2$ propensity rule. This rule originates in the shape of the PES, being nearly a prolate ellipsoid, dominated by the rod shape of ${\rm HC_3N}$ and not dominated by the large dipole of HC₃N molecule (3.724 Debye). Because of the very good approximate symmetry $\theta_1\leftrightarrow \pi - \theta_1$, the l₁ even terms (Eq. (6) and Green 1975) are the most important ones, directing the inelastic transition toward even $\Delta J_1$ values. This propensity has also been explained semi-classically by McCurdy & Miller (1977) in terms of an interference effect related to the even anisotropy of the PES. These authors show in particular that the reverse propensity can also occur if the odd anisotropy of the PES is sufficiently large. This reverse effect is indeed observed in Fig. 4 for transitions with $\Delta J>10$. A similar propensity rule has been experimentally observed for CO-He collisions (Carty et al. 2004).

Besides this strong $\Delta J = 2$ propensity rule, one can see from Table 3 and Figs. 3, 4 that the rod-like interaction drives large $\Delta J$ transfers. For instance, for T > 20 K, rates for $\Delta J > 6$ are generally larger than rates for $\Delta J = 1$, and rates for $\Delta J > 8$ are only one order of magnitude below those for $\Delta J = 2$. This behaviour is likely to emphasize the role of collisional effects versus radiative ones. This effect, of purely geometric origin, has been predicted previously (Bosanac 1980) and is of even greater importance for longer rods like $\rm HC_5N$, $\rm HC_7N$, $\rm HC_9N$, see Snell et al. (1981), Bhattacharyya & Dickinson (1982).

We also observe that the ratio $k_{J_1J'_1}({\rm He})/k_{J_1J'_1}({\rm para{-}H_2})$ is in average close to $1/1.4 \sim 1/\!\sqrt{2}$, thus confirming the similarity of He and para-${\rm H_2}$ as projectiles, as generally assumed. But it is also far from being a constant, as already observed for H₂O (Phillips et al. 1996) or CO (Wernli et al. 2006). Our data shows that the $1/\!\sqrt{2}$ scaling rule results in errors of up to 50%.

4.3 Population inversion and critical densities

Because of the strong $\Delta J_1=0,2,4$ propensity rule, population inversion could be strengthened if LTE conditions are not met, even neglecting hyperfine effects (Hunt et al. 1999). In order to see the density conditions giving rise to population inversion, we solved the steady-state equations for the population of the $J=0,1,\dots,15$ levels of  ${\rm HC_3N}$, including collisions with ${\rm H_2}$ (densities ranging from 10² to $10^6~\rm cm^{-3}$), a black-body photon bath at 2.7 K, in the optically thin approximation, (Goldsmith 1972):

 

$$% \frac{\textmd{d}n_i}{\textmd{d}t}=0= +\sum_{j\neq i}n_j~\left[ ... ...} \left[A_{ij}+B_{ij}~n_\gamma\left(\nu_{ij}\right)+k_{ij}\; n_{\rm H_2}\right]$$ (7)

where i,j are the levels, $n_\gamma$ is the photon density at temperature $T_\gamma$ and  $n_{\rm H_2}$ is the hydrogen density at kinetic temperature  $T_{\rm H_2}$. Figure 6 shows the results at $T_{\rm H_2} = 40$ K. The lines show the population per sub-levels $\left\vert J_1, m_{J_1}\right>$. For a large range of ${\rm H_2}$ densities, $10^4\la n_{\rm H_2} \la 10^6$, population inversion does occur for $0\leq J_1\leq 2, 3, 4$. Our new rates are expected to improve the interpretation of the lowest-lying lines of  ${\rm HC_3N}$, especially so in the 9-20 GHz regions (cm-mm waves), see for example Walmsley et al. (1986), Takano et al. (1998), Hunt et al. (1999), and Kalenskii et al. (2004) for a recent study. Moreover, from the knowledge of both collision coefficients k_ij and Einstein coefficients A_ij, it is possible to derive a critical density of ${\rm H_2}$, defined as:

$$% n^{\star}_i(T)=\frac{\sum_{j<i}A_{ij}}{\sum_{j<i}k_{ij}}\cdot$$ (8)

The $n^\star$ density is the ${\rm H_2}$ density at which photon deexcitation and collisional deexcitation are equal. The evolution of $n^\star$ with J₁ at $T= 40~\rm K$ is given in Fig. 7. It can be seen that for many common interstellar media, the LTE conditions are not fully met.

	Figure 6: Population per J1,mJ1-sub-level, following Eq. (7), for varying ${\rm H_2}$ densities (in  $\rm cm^{-3}$) at a kinetic temperature of 40 K.
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	Figure 7: ${\rm H_2}$ critical density $n^\star$ (in cm-3) for the ${\rm HC_3N}$-para-${\rm H_2}$ collisions, at $T= 40~\rm K$, Eq. (8). For $j\leq 15$, quantum rates, for j>15, classical rates. The change of method explains the small discontinuity between j=15,16. The increase of critical density at j=1 is related to the propensity rule, see Fig. 4.
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Similar effects should appear for the whole cyanopolyyne ( $\rm HC_{5,7,9}N$) family, where cross-sections should scale approximately with the rod length (Bhattacharyya & Dickinson 1982). It is expected that the propensity rule $\Delta J_1=2,4,\dots$ should remain valid. Also, the critical density should decrease for the higher members of the cyanopolyyne family, as the Einstein A_ij coefficients, hence facilitating the LTE conditions.

5 Conclusion

We have computed two ab initio surfaces for the ${\rm HC_3N}$-He and ${\rm HC_3N}$-${\rm H_2}$ systems. The latter was built using a carefully selected set of ${\rm H_2}$ orientations, limiting the computational effort to approximately five times the ${\rm HC_3N}$-He one. Both surfaces were successfully expanded on a rotational basis suitable for quantum calculations using a smooth regularization of the potentials. This approach circumvented the severe convergence problems already noticed by Chapman & Green (1977) for such large molecules. The final accuracy of both PES is a few cm^-1 for potential energies below 1000 cm^-1.

Rates for rotational excitation of ${\rm HC_3N}$ by collisions with He atoms and ${\rm H_2}$ molecules were computed for kinetic temperatures in the range 5 to 20 K and 5 to 100 K, respectively, combining quantum close coupling and quasi-classical calculations. The rod-like symmetry of the PES strongly favours even  $\Delta J_1$ transfers and efficiently drives large $\Delta J_1$ transfers. Quasi classical calculations are in excellent agreement with close coupling quantum calculations but do not account for the even $\Delta J_1$ interferences. For He, results compare fairly well with Green & Chapman (1978) QCT rates, indicating a weak dependance on the details of the PES. For para-H₂, rates are compatible on average with the generally assumed $\sqrt{2}$ scaling rule, with a spread of about 50%. Despite the large dipole moment of  ${\rm HC_3N}$, rates involving ortho-H₂ are very similar to those involving para-H₂, due to the predominance of the rod interactions.

A simple steady-state population model shows population inversions for the lowest ${\rm HC_3N}$ levels at ${\rm H_2}$ densities in the range 10⁴-10⁶ cm^-3. This inversion pattern shows the importance of large angular momentum transfer, and is enhanced by the even $\Delta J_1$ quantum propensity rule.

The ${\rm HC_3N}$ molecule is large enough to present an original collisional behaviour, where steric hindrance effects hide the details of the interaction, and where quasi classical rate calculations achieve a fair accuracy even at low temperatures. With these findings, approximate studies for large and heavy molecules should become feasible including possibly the modelling of large $\Delta J$ transfer collisions and ro-vibrational excitation of low energy bending or floppy modes.

Acknowledgements

This research was supported by the CNRS national program "Physique et Chimie du Milieu Interstellaire'' and the "Centre National d'Études Spatiales''. L.W. was partly supported by a CNRS/NSF contract. M.W. was supported by the Ministère de l'Enseignement Supérieur et de la Recherche. CCSD(T) calculations were performed on the IDRIS and CINES French national computing centers (projects no. 051141 and x2005 04 20820). MOLSCAT and QCT calculations were performed on local workstations and on the "Service Commun de Calcul Intensif de l'Observatoire de Grenoble'' (SCCI) with the valuable help of F. Roch.

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Online Material

Table A.1: Fitting coefficients of HC₃N-para-H₂(J₂ = 0) rates, following formula (5). J₁ and J'₁ are the initial and final rotation states, respectively. The rates thus obtained are in  ${\rm cm^3~s^{-1}}$. Rates are quantum for $J_{1 \rm up} \protect\le 15$, classical otherwise.