---
author:
- 'M. Wernli'
- 'L. Wiesenfeld'
- 'A. Faure'
- 'P. Valiron'
bibliography:
- 'cyano\_v3.bib'
date: 'Received / Accepted '
title: 'Rotational Excitation of HC$_3$N by H$_2$ and He at low temperatures'
---

Introduction
============

Cyanopolyyne molecules, with general formula HC$_{2n+1}$N, $n\ge
1$, have been detected in a great variety of astronomical environments and belong to the most abundant species in cold and dense interstellar clouds [@bell97]. One of these, HC$_{11}$N, is currently the largest unambiguously detected interstellar molecule [@bell85]. The simplest one, [$\mathrm{HC_3N}$]{}(cyanoacetylene), is the most abundant of the family. In addition to interstellar clouds, [$\mathrm{HC_3N}$]{}has been observed in circumstellar envelopes [@pepe04], in Saturn satellite Titan [@kunde81], in comets [@bockelee00] and in extragalactic sources [@mauersberger90]. Furthermore, [$\mathrm{HC_3N}$]{}has been detected both in the ground level and in excited vibrational levels, thanks to the presence of low-lying bending modes [e.g. @wyrowski03]. Owing to a low rotational constant and a large dipole moment, cyanoacetylene lines are thus observable over a wide range of excitation energies and [$\mathrm{HC_3N}$]{}is therefore considered as a very good probe of physical conditions in many environments.

Radiative transfer models for the interpretation of observed [$\mathrm{HC_3N}$]{}spectra require the knowledge of collisional excitation rates participating to line formation. To the best of our knowledge, the only available collisional rates are those of @green78 for the rotational excitation of HC$_3$N by He below 100 K. In cold and dense clouds, however, the most abundant colliding partner is H$_2$. In such environments, para-[$\rm H_2$]{}is only populated in the $J=0$ level and may be treated as a spherical body. @green78 and @dickinson82 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 while rates involving ortho-[$\rm H_2$]{}were not considered.

In the present study, we have computed new rate coefficients for rotational excitation of [$\mathrm{HC_3N}$]{}by He, para-[$\rm H_2$]{}($J=0$) and ortho-[$\rm H_2$]{}($J=1$), in the temperature range 5$-$20 K for He and 5$-$100 K for H$_2$. 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 Section \[sec:cross\]. A discussion and a first application of these rates is given in Section \[sec:disc\]. Conclusions are drawn in Section 5. The following units are used throughout except otherwise stated: bond lengths and distances in Bohr; angles in degrees; energies in cm$^{-1}$; and cross-sections in $\AA^2$.

Potential energy surfaces {#sec:pot}
=========================

Two accurate interatomic potential energy surfaces (PES) have recently been calculated in our group, for the interaction of [$\mathrm{HC_3N}$]{}with He and H$_2$. Both surfaces involved the same geometrical setup and similar *ab initio* accuracy. An outline of those PES is given below, while 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 [@jeziorska00; @jankowski05] that properly averaged molecular geometries provide a better description of experimental data than equilibrium geometries ($r_e$ geometries). For the $\rm H_2O$ – [$\rm H_2$]{}system, geometries averaged over ground-state vibrational wave-functions ($r_0$ geometry) were shown to provide an optimal approximation of the effective interaction [@faure05; @wernlithese].

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 [@hodges04; @faure05; @wernli06]. For [$\mathrm{HC_3N}$]{}, as vibrational wave-functions are not readily available from the literature, we resorted to experimental geometries deduced from the rotational spectrum of [$\mathrm{HC_3N}$]{}and its isotopologues (@thor00; see also Table 5.8 in @gordy). The resulting bond separations are the following: $r_{\mathrm{HC_1}}= 1.998385$; $r_{\mathrm{C_1C_2}}=2.276364$;$r_{\mathrm{C_2C_3}}= 2.606688$; $r_{\mathrm{C_3N}}= 2.189625$, and should be close to vibrationally averaged values.

For the [$\mathrm{HC_3N}$]{}– 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 [$\mathrm{HC_3N}$]{}and He. The two coordinates are the distance $R=|\vec{R}|$ and the angle $\theta_1$ between the [$\mathrm{HC_3N}$]{}rod and the vector ***R***. In our conventions, $\theta_1 = 0$ corresponds to an approach towards the H end of the [$\mathrm{HC_3N}$]{}rod. For the collision with H$_2$, 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 [$\mathrm{HC_3N}$]{}– He PES has thus two degrees of freedom, the [$\mathrm{HC_3N}$]{}– [$\rm H_2$]{}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 [$\mathrm{HC_3N}$]{}– He system, this expansion is in the form: $$\label{eq:pot}
     V_{}(R,\theta_1) = \sum_{l_1}
     v_{l_1}(R)\,P_{l_1}(\cos\theta_1)\quad ,$$ where $P_{l_1}(\cos\theta_1)$ is a Legendre polynomial and $v_{l_1}(R)$ are the radial coefficients.

For the [$\mathrm{HC_3N}$]{}– [$\rm H_2$]{}system, the expansion becomes: $$\label{eq:pot2}
  V(R,\theta_1, \theta_2, \phi) = \sum_{l_1 l_2 l} v_{l_1 l_2 l}(R)
    s_{l_1 l_2 l}(\theta_1, \theta_2, \phi),$$ where the basis functions $s_{l_1 l_2 l}$ are products of spherical harmonics and are expressed in Eq. (A9) of @green75. Two new indices $l_2$ and $l$ are thus needed, associated respectively with the rotational angular momentum of [$\rm H_2$]{}and the total orbital angular momentum, see also eq. (A2) and (A5) of @green75.

Because the Legendre polynomials form a complete set, such expansions should always be possible. However, @chapman77 failed to converge above expansion (\[eq:pot\]) due to the steric hindrance of He by the impenetrable [$\mathrm{HC_3N}$]{}rod, and @green78 abandoned quantum calculations, resorting to quasi classical trajectories (QCT) studies. Similar difficulties arise for the interaction with H$_2$. Actually, as can be seen on figure \[fig:PES\] 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. Let us summarize this first step for He, then for H$_2$.

For the [$\mathrm{HC_3N}$]{}– 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 to 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 H$_2$, 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 with 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 $m_2=0,1,2,\ldots$, $|m_2|\leq l_2$. Terms in $Y_{l_2m_2}$ and $Y_{l_2 -m_2}$ are equal by symmetry. Previous studies [@faure05a; @wernli06] have shown that terms with $l_2> 2$ are small, and we consequently truncated the $Y_{l_2m_2}$ series to $l_2\leq 2$. Hence, only four basis functions remain for the orientation of [$\rm H_2$]{}: $Y_{00},Y_{20},Y_{21}$ and $Y_{22}$.

Under this assumption, the whole [$\mathrm{HC_3N}$]{}– [$\rm H_2$]{}surface can be obtained knowing its value for four sets of $\left\{\theta_2, \phi\right\}$ angles at each value of $R$. We selected actually five sets, having thus an over-determined system allowing for the monitoring of the accuracy of the $l_2$ truncation. Consequently, we determined five independent PES, each being constructed similarly to the [$\mathrm{HC_3N}$]{}– 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 [$\mathrm{HC_3N}$]{}– 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. Finally, the [$\mathrm{HC_3N}$]{}– H$_2$ interaction can be readily reconstructed from these five PES by expressing its analytical dependence over $\left\{\theta_2, \phi\right\}$ [@wernlithese].

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 [@jansen69; @boys70]. We used augmented correlation-consistent atomic sets of triple zeta quality (Dunning’s aug-cc-pVTZ) to describe the [$\mathrm{HC_3N}$]{}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, H$_2$) targets, respectively. All calculations employed the direct parallel code <span style="font-variant:small-caps;">Dirccr12</span> [@dirccr12].

Comparison of the [$\mathrm{HC_3N}$]{}– He PES with existing surfaces [@akinojo03; @topic05] showed an excellent agreement. The [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{}($J=0$) interaction (obtained by averaging the [$\mathrm{HC_3N}$]{}– H$_2$ PES over $\theta_2$ and $\phi$) is qualitatively similar to the [$\mathrm{HC_3N}$]{}– He PES with a deeper minimum (see values at the end of present Section). As illustrated in Figure \[fig:PES\], these PES are largely dominated by the rod-like shape of [$\mathrm{HC_3N}$]{}, implying a prolate ellipsoid symmetry of the equipotentials.

In a second step, let us consider how to circumvent the difficulty of the angular expansion of the above PES, in order to obtain reliable expansions for He and H$_2$ (eqs \[eq:pot\] and \[eq:pot2\]).

Using the angular spline representation, we first expressed each PES over a fine $\theta_1$ mesh suitable for a subsequent high $l_1$ expansion. As expected from the work of @chapman77, high $l_1$ expansions (\[eq:pot\]) resulted in severe Gibbs oscillations for $R$ in the range 5–7 Bohr, spoiling completely the description of the low energy features of the PES. Then, having in mind low energy scattering applications, we regularized the PES by introducing a scaling function $S_f$. We replaced $V(R,\theta_1,...)$ by $S_f(V(R,\theta_1,...))$, where $S_f(V)$ returns $V$ when $V$ is lower than a prescribed threshold, and then smoothly saturates to a limiting value when $V$ grows up into the repulsive walls. 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. @hodges04] 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 H$_2$ 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 [$\mathrm{HC_3N}$]{}– He, $V=-40.25$ for $R=6.32$ and $\theta_1=95.2^{\circ}$; for [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{}($J=0$), $V=-111.24$ for $R=6.41$ and $\theta_1=94.0^{\circ}$; and for [$\mathrm{HC_3N}$]{}– [$\rm H_2$]{}, $V=-192.49$ for $R=9.59$, $\theta_1=180^{\circ}$, and $\theta_2=0^{\circ}$.

![The [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{}PES. The [$\mathrm{HC_3N}$]{}molecule is shown at 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 [$\mathrm{HC_3N}$]{}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 [$\mathrm{HC_3N}$]{}rod. []{data-label="fig:PES"}](PES.eps){height="0.55\textheight"}

Inelastic cross section and rates {#sec:cross}
=================================

In the following $J_1, J^\prime_1$, denote the initial and final angular momentum of the [$\mathrm{HC_3N}$]{}molecule, respectively, and $J_2$ denote the angular momentum of H$_2$. We also denote the largest value of [$J_1, J^\prime_1$]{} as $J_{1\rm up}$.

The most reliable approach 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 [$\mathrm{HC_3N}$]{}[$B=4549.059\mbox{~MHz}$, see e.g. @thor00], quantum calculations become soon intractable, because of the large number of open channels involved. While observations at cm-mm wavelengths culminates with $J_{1\rm up}
\lesssim 24$ [@kahane94], sub-mm observations can probe transition as high as $J_{1\rm up} = 40$, at a frequency of 363.785 GHz and a rotational energy of $202.08 \:\rm
cm^{-1}$ [@pepe04; @charnley04; @cauxpc]. It is thus necessary to compute rates with transitions up to $J_1=50$ ($E=
386.8$ cm$^{-1}$), in order to properly converge radiative transfer models. Also, we aim at computing rates up to a temperature of 100 K for H$_2$. We resorted to two methods in order to perform this task. For $J_{1\rm up}\leq 15$, we performed quantum inelastic scattering calculations, as presented in next subsection \[par:molscat\]. For $J_{1\rm up} > 15$, we used the QCT method, as presented in subsection \[par:rates\].

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 $J_1<10$).

Rotational inelastic cross sections with <span style="font-variant:small-caps;">Molscat</span> {#par: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 <span style="font-variant:small-caps;">Molscat</span> code [@molscat]. All quantum calculations for [$\mathrm{HC_3N}$]{}– ortho-[$\rm H_2$]{}were performed with $J_{\rm H_2}\equiv J_2=1$. Calculations for [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{}were performed with $J_2=0$. We checked at $E_{\rm
tot}=E_{\rm coll}+E_{\rm rot} = 30 \,\rm cm^{-1}$ that the inclusion of the closed $J_2=2$ channel led to 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 [@dubernet02; @dubernet03; @wernli06]. The energy grid and the quantum methods used are detailed in table \[tab:param\]. Using this grid, we calculated the whole resonance structure of all the transitions up to $J_1=15$ for the [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{}collisions. At least 10 closed channels were included at each energy to fully converge the [$\mathrm{HC_3N}$]{}rotational basis. We used the hybrid log-derivative/Airy propagator [@alexander87]. We increased the parameter <span style="font-variant:small-caps;">STEPS</span> 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 <span style="font-variant:small-caps;">molscat</span> default values.

[ccc]{}\
$E_{\rm tot}\:\rm (cm^{-1})$ & Energy step ($\rm cm^{-1}$) & Method\
$0.3 \rightarrow 60$ & $0.1$ & CC\
$60 \rightarrow 110$ & $10$ & CC\
$40 \rightarrow 200$ & $10$ & CS\
$50 \rightarrow 800$ & $10-100 $ & IOS\
\
$0\rightarrow 30 $& $1$ & CC\
\
$0\rightarrow 25 $& $0.1$ & CC\
$25\rightarrow 100 $& $5$ & CC\
$100\rightarrow 150 $& $10$ & CC\

Two examples of deexcitation cross-sections are shown in figure \[fig:sectionop\]. 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 by no means unexpected and is very similar to most earlier calculations is many different systems, see e.g. @dubernet02 [@wernli06] for a discussion. In a semi-classical point of view, those shape resonances manifest the trapping of the wave-packet between the inner repulsive wall and the outer centrifugal barrier, see @wie03 [@abrol01]. At energies higher than about 20 cm$^{-1}$ above threshold, all cross-sections become smooth functions of the energy.

Figure \[fig:sectionop\] also shows that ortho-[$\rm H_2$]{}inelastic cross-sections follow very closely the para-[$\rm H_2$]{}ones, including the position of resonances. Examination of all cross-sections reveals that the relative difference between $\sigma_{J_1J'_1}(E, \mbox{para})$ and $\sigma_{J_1J'_1}(E, \mbox{ortho})$ 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 $J_2 = 2$ closed para-[$\rm H_2$]{}channels. A detailed discussion of this behaviour is put forward in section \[sec:paraortho\].

Quantum rates and classical rates {#par: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 $kT$. The quantum calculations at the higher end of the energy range are approximated at the IOS level (see table \[tab:param\]), 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 $J_1 > 15$, 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-$J_1$ channels. In the meanwhile, the accuracy of classical approximations improves for higher collisional energies. For the energy range where $J_1 > 15$ channels are open *and* for deexcitation processes involving those channels, we employ a Quasi-Classical Trajectory (QCT) method, which has been shown at several instances to be a valid approximation for higher collisional energies and large rates [@chapman77; @lepp95; @mandy04; @faure06].

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 @faure04], 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 [$\mathrm{HC_3N}$]{}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: $$\label{eq:rate}
k_{J_1J'_1} = \left(\frac{8kT}{\pi\mu}\right)^{1/2}\,\pi
b_{max}^2\,\frac{N}{N_{\rm tot}}$$ where $b_{max}$ 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: $$\label{eq:error}
    \frac{\delta k_{J_1J_1'}}{k_{J_1J_1'}} = \left(\frac{N_{\rm
    tot}-N}{N_{\rm tot}N}\right)^{1/2} \quad ,$$ showing that the accuracy of the method improves for larger rates. The $b_{max}$ parameter was determined by sending small batches of $500$ to $1,000$ trajectories for fixed $b$ values; values of $20 \leq
b_{max} \leq 26$ Bohr were found. We then sent batches of $10,000$ trajectories for each temperature in the range $5-100$K, with a step of 5K. Trajectories are integrated by means of a Bürlich-Stoer algorithm [@numrec92], with a code similar to that of @faure05a. Precision is checked by conservation of total energy and total angular momentum.

Some illustrative results are shown in tables \[tab:rates\] and  \[tab:rates2\] and are illustrated in figures \[fig:ratecompare\] and \[fig:rate12\].

As an alternative to QCT calculations, we tested J-extrapolation techniques, using the form of @depristo79 generally used by astrophysicists (see for example @lamda, section 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 in the average.

  --------- ------------------ ------------------ ------------- -------------------
                                                                

   $J_{1}$   $T = 10 \rm K\;$   $T = 20 \rm K\;$   $T = 50 \rm   $T = 100 \rm K\;$
                                                      K\;$      

                                                                

                                                                

      1         2.03(-11)          1.59(-11)        1.32(-11)        1.24(-11)

      2         4.94(-11)          4.83(-11)        6.23(-11)        8.04(-11)

      3         1.20(-11)          1.04(-11)        8.23(-12)        7.43(-12)

      4         2.25(-11)          2.57(-11)        2.85(-11)        2.87(-11)

      5         7.01(-12)          6.80(-12)        5.62(-12)        4.77(-12)

      6         9.15(-12)          1.18(-11)        1.42(-11)        1.38(-11)

      7         3.14(-12)          3.40(-12)        3.46(-12)        3.26(-12)

      8         2.45(-12)          3.71(-12)        5.92(-12)        6.61(-12)

      9         1.63(-12)          1.63(-12)        1.95(-12)        2.18(-12)

     10         5.35(-13)          8.13(-13)        2.00(-12)        2.96(-12)

     11         7.81(-13)          7.01(-13)        9.42(-13)        1.36(-12)

     12         1.37(-13)          1.58(-13)        6.17(-13)        1.32(-12)

     13         2.74(-13)          2.51(-13)        4.17(-13)        8.26(-13)

     14         4.14(-14)          4.65(-14)        2.24(-13)        6.28(-13)

     15         7.63(-14)          8.26(-14)        1.76(-13)        4.85(-13)
  --------- ------------------ ------------------ ------------- -------------------

  : [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{} s($J=0$)collisions. Quantum deexcitation rates in $\mathrm{cm^3\,s^{-1}}$, for $J_1'=0$, for successive initial $J_1$ and for various temperatures. Powers of ten are denoted in parenthesis.[]{data-label="tab:rates"}

[llcccc]{} &\
$J_{1}'$ &$J_{1}$& $T = 10 \rm K\;$ & $T = 20 \rm K\;$ & $T = 50 \rm K\;$ & $T = 100 \rm K\;$\
&\
&\
0 & 1 & 2.03(-11) & 1.59(-11) & 1.32(-11) & 1.24(-11)\
0 & 2 & 4.94(-11) & 4.83(-11) & 6.23(-11) & 8.04(-11)\
0 & 3 & 1.20(-11) & 1.04(-11) & 8.23(-12) & 7.43(-12)\
0 & 4 & 2.25(-11) & 2.57(-11) & 2.85(-11) & 2.87(-11)\
\
5 & 6 & 6.34(-11) & 5.48(-11) & 4.80(-11) & 4.64(-11)\
5 & 7 & 1.30(-10) & 1.38(-10) & 1.72(-10) & 2.04(-10)\
5 & 8 & 3.93(-11) & 3.66(-11) & 3.27(-11) & 3.21(-11)\
5 & 9 & 6.83(-11) & 7.61(-11) & 8.63(-11) & 8.93(-11)\
\
10 & 11 & 5.77(-11) & 5.35(-11) & 4.75(-11) & 4.61(-11)\
10 & 12 & 1.50(-10) & 1.53(-10) & 1.81(-10) & 2.11(-10)\
10 & 13 & 3.91(-11) & 3.80(-11) & 3.50(-11) & 3.40(-11)\
10 & 14 & 8.51(-11) & 8.84(-11) & 9.42(-11) & 9.47(-11)\
\
\
15 & 16 $^\dag$ & 1.45(-10) & 1.49(-10) & 1.80(-10) & 2.28(-10)\
15 & 17 $^\dag$ & 1.06(-10) & 1.03(-10) & 9.60(-11) & 1.18(-10)\
15 & 18 $^\dag$ & 8.71(-11) & 9.38(-11) & 7.93(-11) & 8.24(-11)\
15 & 19 $^\dag$ & 7.59(-11) & 6.81(-11) & 6.23(-11) & 7.10(-11)\
\
25 & 26 $^\dag$ & 1.14(-10) & 1.50(-10) & 1.83(-10) & 2.30(-10)\
25 & 27 $^\dag$ & 1.13(-10) & 1.05(-10) & 1.18(-10) & 1.31(-10)\
25 & 28 $^\dag$ & 8.55(-11) & 8.38(-11) & 8.34(-11) & 7.76(-11)\
25 & 29 $^\dag$ & 7.67(-11) & 7.45(-11) & 8.31(-11) & 7.02(-11)\
\
35 & 36 $^\dag$ & 1.16(-10) & 1.34(-10) & 1.73(-10) & 2.32(-10)\
35 & 37 $^\dag$ & 9.63(-11) & 1.12(-10) & 1.21(-10) & 1.11(-10)\
35 & 38 $^\dag$ & 8.33(-11) & 9.20(-11) & 8.56(-11) & 9.19(-11)\
35 & 39 $^\dag$ & 8.77(-11) & 8.51(-11) & 7.65(-11) & 7.51(-11)\
\

For H$_2$, all deexcitation rates $k_{J_1J_1'}(T)$, $J_1\neq J_1'\leq
50$, are fitted with the following formula [@wernli06]: $$\label{eq:fit}
    \log_{10}\left(k_{J_1J_1'}(T)\right)=\sum_{n=0}^{
    4}a^{(n)}_{J_1J_1'} x^n$$ where $x=T^{-1/6}$. As some transitions have zero probability within the QCT approach, the above formula was employed when rates were bigger than 10$^{-12}$ cm$^3$s$^{-1}$ for at least one grid temperature. For these rates, null grid values were replaced by a very small value, namely 10$^{-14}$ cm$^3$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 $a^{(n)}_{J_1J_1'}$ 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-H$_2$, since their difference is smaller than the uncertainty on the rates themselves. Rates with He were not fitted, but can be obtained upon request to the authors.

  ---------------------------------- ----------------------------------
  ![image](rate1.eps){width="8cm"}     ![image](rate2.eps){width="8cm"}
  ---------------------------------- ----------------------------------

Discussion {#sec:disc}
==========

Para and ortho [$\rm H_2$]{}cross-sections {#sec:paraortho}
------------------------------------------

A comparison of the $\sigma_{J_1J_1'}(E)$ cross sections for [$\mathrm{HC_3N}$]{}with ortho-[$\rm H_2$]{}and para-[$\rm H_2$]{}is given in figure \[fig:sectionop\]. It can be seen that the difference between the two spin species of [$\rm H_2$]{}may be considered as very small, 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 C$_2$ molecule [@phillips94], the CO molecule [@wernli06], the OH radical [@offer94], the NH$_3$ molecule [@offer89; @flower94] and the $\rm H_2O$ molecule [@phillips96; @dubernet02; @dubernet03; @dubernet06], due to the interaction between the dipole of the molecule and the quadrupole of [$\rm H_2$]{}(J$_2$ $>$ 0).

This apparently null result deserves an explanation. We focus on equation (9) of @green75. 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 equation (9); the relevant angular coupling algebra is represented there as a sum of terms of the type $$\label{eq:green75}
    \left(\begin{array}[c]{ccc} l &L'& L \\ 0 & 0 & 0
    \end{array}\right)
    \;
 \left(\begin{array}[c]{ccc} l_1 &J'_1& J_1 \\ 0 & 0 & 0
    \end{array}\right)
    \;
     \left(\begin{array}[c]{ccc} l_2 &J'_2& J_2 \\ 0 & 0 & 0
    \end{array}\right)
    \;
     \left\{\begin{array}[c]{ccc} L'& L & l \\ J_{12} & J'_{12} &
     J
    \end{array}\right\}\quad ,
$$ where we have the potential function expanded in terms of Eqs. (4) and (A2) in @green75, by means of the coefficients $v_{l_1l_2
l}$. The symbol $(\ldots)$ are 3-$j$ symbols, the $\{\ldots\}$ is a 6-$j$ symbol, see @messiah69. 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 $J_2 = 0$ channel (the $J_2=2$ channel is closed till $E_{coll} \gtrsim 365.12
    \;\rm cm^{-1}$). Then, only the $l_2=0$ may be retained ($J_2=J'_2=0$), due to the third 3-$j$ symbol in eq.(\[eq:green75\]).

-   The ortho-[$\rm H_2$]{}remains in $J_2=1$, implying $l_2=0,2$.

-   For inelastic collisions, $J_1\neq J'_1$ implies potential terms with $l\neq 0$, because of the 6-$j$ term in Eq.(\[eq:green75\]). Indeed, $J_2=J'_2$ and $J_1\neq J'_1$ entail $J_{12}\neq J'_{12}$.

The key point is thus to compare the $v_{l_1l_2 l}(R)$ coefficients (eq. \[eq:pot2\]) with $l \neq 0$ in the two cases:

-   [$l_2=0$]{} para and ortho contributions;

-   [$l_2=2$]{} ortho contribution only.

Figure \[fig:comp\] displays such a comparison. We notice that the coupling is largely dominated by the $l_2=0$ 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 [$\rm H_2$]{}(J$_2=2,3$) channels.

Propensity rules
----------------

In figure \[fig:ratecompare\], we compare the various rates that we obtain here with the ones previously published by @green78. 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 @green78 are qualitatively comparable with the quantum rates obtained here, in an average way. However, as table \[tab:rates2\] and figures \[fig:rate12\] and \[fig:ratecompare\] 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 [$\mathrm{HC_3N}$]{}and *not dominated* by the large dipole of HC$_3$N molecule (3.724 Debye). Because of the very good approximate symmetry $\theta_1\leftrightarrow \pi - \theta_1$, the $l_1$ even terms (equation (\[eq:green75\]) and @green75) are the most important ones, directing the inelastic transition toward even $\Delta J_1$. This propensity has also been explained semi-classically by @miller77 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. \[fig:rate12\] for transitions with $\Delta J>10$. A similar propensity rule has been experimentally observed for CO–He collisions [@sims04].

Besides this strong $\Delta J = 2$ propensity rule, one can see from table \[tab:rates2\] and figures \[fig:ratecompare\], \[fig:rate12\] 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 [@bosanac80] and is of even greater importance for longer rods like $\rm HC_5N$, $\rm HC_7N$, $\rm HC_9N$, see @snell81 [@dickinson82].

We also observe that the ratio $k_{J_1J'_1}(\textrm{He})/k_{J_1J'_1}(\mbox{para-H$_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 H$_2$O [@phillips96] or CO [@wernli06]. Our data shows that the $1/\sqrt{2}$ scaling rule results in errors up to 50%.

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[^1] [@hunt99]. 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 [$\mathrm{HC_3N}$]{}, including collisions with [$\rm H_2$]{}(densities ranging from $10^2$ to $10^6 \rm\; cm^{-3}$), a black-body photon bath at 2.7 K, in the optically thin approximation, [@goldsmith72] : $$\begin{aligned}
\label{eq:ss}
    \frac{\textmd{d}n_i}{\textmd{d}t}=0&=&
    +\sum_{j\neq i}n_j\,\left[ A_{ji} +
    B_{ji} \;n_\gamma\left(\nu_{ji}\right)+k_{ji}\;
    n_{\rm H_2}\right] \nonumber \\
    & & - n_i\,\sum_{j\neq i}
    \left[A_{ij}+B_{ij}\,n_\gamma\left(\nu_{ij}\right)+k_{ij}\; n_{\rm
    H_2}\right]\end{aligned}$$ 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 \[fig:invers\] shows the results at $T_{\rm H_2} = 40$ K. The lines show the population per sub-levels $\left|J_1, m_{J_1}\right>$. For a consequent range of [$\rm H_2$]{}densities, $10^4\lesssim n_{\rm H_2}
\lesssim 10^6$, 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 [$\mathrm{HC_3N}$]{}, especially so in the 9 - 20 GHz regions (cm-mm waves), see for example @walms86 [@takano98; @hunt99], and @kal04 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: $$\label{eq:nstar}
    n^{\star}_i(T)=\frac{\sum_{j<i}A_{ij}}{\sum_{j<i}k_{ij}}$$ 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 $J_1$ at $T= 40\;\rm K$ is given in figure \[fig:critical\]. It can be seen that for many common interstellar media, the LTE conditions are not fully met.

It must be underlined that 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 [@dickinson82]. 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.

Conclusion
==========

We have computed two [*ab initio*]{} surfaces, for the [$\mathrm{HC_3N}$]{}– He and [$\mathrm{HC_3N}$]{}– [$\rm H_2$]{}systems. The latter was built using a carefully selected set of [$\rm H_2$]{}orientations, limiting the computational effort to approximately five times the [$\mathrm{HC_3N}$]{}– 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 @chapman77 for such large molecules. The final accuracy of both PES is a few cm$^{-1}$ for potential energy below 1000 cm$^{-1}$.

Rates for rotational excitation of [$\mathrm{HC_3N}$]{}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 with @green78 QCT rates, indicating a weak dependance to the details of the PES. For para-H$_2$, rates are compatible in average with the generally assumed $\sqrt{2}$ scaling rule, with a spread of about 50 %. Despite the large dipole moment of $\mathrm{HC_3N}$, rates involving ortho-H$_2$ are very similar to those involving para-H$_2$, due to the predominance of the rod interactions.

A simple steady-state population model shows population inversions for the lowest [$\mathrm{HC_3N}$]{}levels at [$\rm H_2$]{}densities in the range 10$^4-$10$^6$ cm$^{-3}$. This inversion pattern manifests the importance of large angular momentum transfer, and is enhanced by the even $\Delta J_1$ quantum propensity rule.

The [$\mathrm{HC_3N}$]{}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.

This research was supported by the CNRS national program “Physique et Chimie du Milieu Interstellaire” and the “Centre National d’Etudes Spatiales”. LW was partly supported by a CNRS/NSF contract. MW 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). <span style="font-variant:small-caps;">Molscat</span> 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 from F. Roch.

[c c c c c c c]{}

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[^1]: Hyperfine effects in [$\mathrm{HC_3N}$]{}inelastic collisions will be dealt with in a forthcoming paper, @wie06
