Collisional excitation of HC3N by para- and ortho-H2
aa r X i v : . [ a s t r o - ph . GA ] M a y Mon. Not. R. Astron. Soc. , 1–8 (2016) Printed 10 October 2018 (MN L A TEX style file v2.2)
Collisional excitation of HC N by para- and ortho-H Alexandre Faure , ⋆ , Fran¸cois Lique † and Laurent Wiesenfeld , ‡ Univ. Grenoble Alpes, IPAG, F-38000 Grenoble, France CNRS, IPAG, F-38000 Grenoble, France LOMC-UMR 6294, CNRS-Universit´e du Havre, 25 rue Philippe Lebon, BP 1123 – 76063 Le Havre Cedex, France
Accepted 2016 May 11. Received 2016 May 10; in original form 2016 May 04
ABSTRACT
New calculations for rotational excitation of cyanoacetylene by collisions with hy-drogen molecules are performed to include the lowest 38 rotational levels of HC N andkinetic temperatures to 300 K. Calculations are based on the interaction potential ofWernli et al.
A&A , 464, 1147 (2007) whose accuracy is checked against spectroscopicmeasurements of the HC N–H complex. The quantum coupled-channel approach isemployed and complemented by quasi-classical trajectory calculations. Rate coeffi-cients for ortho-H are provided for the first time. Hyperfine resolved rate coefficientsare also deduced. Collisional propensity rules are discussed and comparisons betweenquantum and classical rate coefficients are presented. This collisional data shouldprove useful in interpreting HC N observations in the cold and warm ISM, as well asin protoplanetary disks.
Key words:
ISM: molecules, molecular data, molecular processes, scattering.
Cyanopolyyne molecules, with general formula HC n +1 N( n > N (cyanoacetylene) is themost abundant of the family. It has been first detected to-wards the giant galactic molecular cloud Sgr B2 by Turner(1971) and in comet Hale-Bopp by Bockel´ee-Morvan et al.(2000). Because of its low rotational constant and largedipole moment, HC N is considered as a very good ther-mometer and barometer in the ISM. It has been detected inthe ground level and in excited vibrational levels, thanks tothe presence of low-lying bending modes (below 1000 cm − ).Cyanoacetylene is also an abondant nitrogen bearing speciesand it has been proposed as a precursor of prebioticmolecules such as cytosine, one of the four nitrogen basesfound in DNA and RNA (e.g Orgel 2002).In most astrophysical regions where HC N is observed,the low frequency of collisions (due to low density) cannotmaintain local thermodynamical equilibrium (LTE). This isbest exemplified by the natural occurrence of the maser phe-nomenon (microwave amplification by stimulated emissionof radiation) in the 1 → ⋆ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] terpretation of HC N emission spectra, in terms of density,temperature and column density, thus requires to solve theequations of statistical equilibrium, which in turn necessi-tates a good knowledge of the collisional rate coefficients.In the cold and dense ISM, the most abundant colliders arehydrogen molecules and helium atoms. In warmer regions,such as photodissociation regions (PDRs) or comets, elec-tron collisions may also play an important role, due to thelarge HC N dipole (3.7 D) (see e.g. Gratier et al. 2013), aswell as hydrogen atoms.The collisional excitation of HC N by He and H was first studied by Morris et al. (1976) using the “hard-ellipsoid” classical approximation. This pioneering work wassoon followed by the first determination of the HC N-He potential energy surface (PES), using an approximate“electron gas” model, combined with Monte-Carlo quasi-classical trajectory (QCT) calculations (Green & Chapman1978). Classical mechanics was employed instead of quan-tum mechanics due to the difficulty of expanding the PESin terms of Legendre polynomials. Indeed, for such ex-tremely anisotropic interactions (HC N is ∼ nine bohr long),the Legendre polynomial expansion is very slowly conver-gent and subject to severe numerical problems, as first dis-cussed by Chapman & Green (1977). These problems werecircumvented by a novel approach proposed by Wernli et al.(2007a) who derived the first PES for HC N–H (hereafterdenoted as W07). These authors were thus able to performboth classical and quantum calculations for HC N collidingwith para-H ( j = 0, where j is the H angular momen- c (cid:13) Alexandre Faure, Fran¸cois Lique and Laurent Wiesenfeld tum). They provided rate coefficients for the lowest 51 ro-tational levels of HC N ( j = 0 −
50, where j is the HC Nangular momentum) and kinetic temperatures in the range10-100 K. Quantum results were however restricted to thelowest 16 levels, due to the high computational cost of thesecalculations. In addition, numerical errors in the implemen-tation of the PES routine were found to introduce small in-accuracies ( . N–H complex. It is then employed to derive newrate coefficients for HC N colliding with para-H ( j = 0)and, for the first time, with ortho-H ( j = 1). The quan-tum coupled-channel approach is employed for the lowest31 rotational levels of HC N ( j = 0 −
30) while the QCTapproach is employed for higher levels, up to j = 37 whichis the highest level below the first excited bending mode ν at 222 cm − . Hyperfine selective collisional cross sectionsare also deduced using the almost exact (but computation-ally demanding) recoupling method and the infinite-order-sudden (IOS) approximation. In Section 2, the spectra of theHC N–H complex is computed and it is compared to ex-perimental data. In Section 3, scattering calculations are de-scribed and the procedures used to derive cross sections andrate coefficients are briefly introduced. Results are presentedin Section 4. We discuss in particular the propensity rulesand a comparison between quantum and classical rate coef-ficients is made. Conclusions are summarized in Section 5. All bound-state and scattering calculations presented inthe following sections were performed with the W07 PES.This PES is briefly described below. The lowest bound-state rovibrational energy levels supported by the PES arecomputed and the corresponding HC N–H transitions fre-quencies are compared to the recent experimental results ofMichaud, Topic & J¨ager (2011). The W07 PES was computed at the coupled-cluster with sin-gle, double, and perturbative triple excitations [CCSD(T)]level of theory with an augmented correlation-consistenttriple zeta [aug-cc-pVTZ] basis set for HC N and quadru-ple zeta [aug-cc-pVQZ] for H , as described in Wernli et al.(2007a). The molecules were both treated as rigid rotors.Owing to the computational cost of the calculations (HC Nis a 26 electron system), the angular sampling of the PESwas performed by using only five independent H orien-tations. The resulting five PES were interpolated using abicubic spline function and the full HC N–H interactionpotential was reconstructed analytically (Wernli 2006). Dueto the steric hindrance caused by the HC N rod, a stan-dard spherical harmonics expansion of the full PES, asrequired by quantum calculations, was not possible (seeChapman & Green (1977)). This problem was solved byWernli et al. (2007a) using a novel approach called “regu-larization”. This technique consists in scaling the PES when it is higher than a prescribed threshold so that the potentialsmoothyl saturates to a limiting value in the highly repulsiveregions of the PES. The “regularized” PES thus retain onlythe low energy content of the original PES but it is accu-rately adapted to low collisional energies, in practice lowerthan ∼ − here. The final spherical harmonics fitincludes terms up to l = 24 and l = 2, resulting in 97 an-gular functions. Full details can be found in Wernli (2006)and Wernli et al. (2007a).The equilibrium structure of the HC N–H complexis linear with the nitrogen of HC N pointing towards H at an intermolecular separation of 9.58 bohr. The corre-sponding well depth is -193.7 cm − , which compares wellwith the PES of Michaud et al. (2011) (hereafter denotedas M11), also computed at the CCSD(T) level, whose gridminimum is at -188.8 cm − . When averaged over H orien-tation (corresponding to para-H in its ground state j = 0),the W07 PES has a shallower well of -111.2 cm − , also ingood agreement with the M11 averaged PES minimum of-112.6 cm − . It should be noted, however, that the calcula-tions of Michaud et al. (2011) were performed for only threeorientations of the hydrogen molecule within the complexand no global fit of the PES was provided by the authors. A spectroscopic and theoretical study of the HC N–H dimer was reported by Michaud et al. (2011). The bound-state energies of the complex were computed by these au-thors using scaled and unscaled versions of their PES. Thesecalculations were performed in two-dimensions (2D) usingthe Lanczos iterative procedure applied to the averagedM11 PES for para-H and to the three PES (correspondingto the three hydrogen orientation) separately for ortho-H .The results were compared with high-resolution microwavespectroscopy measurements. Theoretical and experimentaltransition frequencies were shown to be in good agreement,within a few percent or better (see below).We have also computed the bound-state energies sup-ported by the W07 PES using the coupled-channel approach,as implemented in the BOUND program (Hutson 1994). Thefull spherical harmonics expansion of the W07 PES wasemployed with 97 functions. The coupled equations weresolved using the diabatic modified log-derivative method.These calculations were performed in four-dimensions (4D)for both para- and ortho-H . Both molecules were taken asrigid rotors with the rotational constants B =59.322 cm − for H and B =0.151739 cm − for HC N. A total of 46 rota-tional states (i.e. up to j = 45) were included in the HC Nbasis set while the two lowest rotational states of para-H ( j = 0 ,
2) and ortho-H ( j = 1 ,
3) were considered. Thecalculations were performed with a propagator step size of0.01 bohr and the other propogation parameters were takenas the default
BOUND values.The transition frequencies in the HC N–H complexwere deduced from the computed bound-state energies.They are compared to the spectroscopic and theoretical dataof Michaud et al. (2011) in Tables 1 and 2 below. The as-signment of the (approximate) quantum numbers J K a K c wasperformed by Michaud et al. (2011). These authors reporteda set of experimental line frequencies including the splittinginto several hyperfine components due to the nuclear spin of c (cid:13) , 1–8 ollisional excitation of HC N by H the nitrogen atom ( I = 1). In the Tables below, the theoreti-cal line frequencies are compared to the unsplit experimentalfrequencies of Michaud et al. (2011). The agreement betweenthe present values and the experimental data is within 0.5%(i.e. better than 0.01 cm − ) for HC N–para-H and withina few percent (better than 0.1 cm − ) for HC N–ortho-H .The corresponding average differences are 0.32% and 4.2%,respectively. This very good agreement is similar, althoughslightly better, than the (nonscaled) theoretical results ofMichaud et al. (2011). This was expected since the level of ab initio theory is similar and the improvement is likely dueto the 4D approach. We note that Michaud et al. (2011)were able to obtain an even better agreement by applyingsimple scaling techniques (radial shifts) to the M11 PES.In summary, as observed previously for smaller systemssuch as CO-H (Jankowski, McKellar & Szalewicz 2012)and HCN-H (Denis-Alpizar et al. 2013), the comparisonbetween spectroscopic and theoretical data for HC N–H confirms that van der Waals rigid-rotor PES computed atthe CCSD(T) level with large basis sets can be used withconfidence for analyzing van der Waals complexes spec-troscopy. Scattering calculations for HC N-H were performed usingboth quantum and classical approaches. The objective wasto obtain cross sections for all transitions among the first38 rotational levels of HC N, i.e. up to j = 37 which lies213.3 cm − above j = 0 and just below the first excitedbending mode ν which opens at only 222 cm − . In prac-tice, for levels higher than j = 30, the basis set necessaryfor converged quantum calculations would require an exces-sive number of (open and closed) states, including vibra-tionally excited states. In addition to the substantial CPUcost, such calculations would require the determination ofa flexible, nonrigid-rotor, HC N–H PES, which is beyondthe scope of this work. For the high-lying levels close to thevibrational threshold ( j = 31 − The scattering quantum calculations were conducted atthe full coupled-channel level using the
MOLSCAT program(Hutson & Green 1995). The spherical harmonics expansionof the W07 PES, including 97 angular functions, was em-ployed and the coupled differential equations were solvedusing the hybrid modified log-derivative Airy propagator.Total energies up to 2120 cm − were considered using a finegrid below 200 cm − (with increments as small as 0.01 cm − )for the correct assessment of resonances close to thresh-olds. The total number of collision energies was 324 forpara-H ( j = 0) and 344 for ortho-H ( j = 1). The high-est rotational level of HC N in the basis set was j = 44while one single rotational level of H was included for boththe para ( j =0) and ortho ( j =1) modifications. The level j =2 of para-H was neglected because it was found to affectthe cross sections by less than 10-20% on average (see alsoWernli et al. 2007b) while the CPU cost was increased by large factors. Thus, at a total energy of 100 cm − , the num-ber of channels was increased by a factor of ∼ ∼
200 in CPUtime. We note that we performed a few high-energy calcu-lations with a basis set j = 0 , ( j = 2) with those for H ( j = 1), seeSection 4.3 below. The neglect of HC N levels j >
44 wasfound to change cross sections by less than a few percentfor all transitions among the first 31 levels of HC N. Themaximum value of the total angular momentum J used inthe calculations was J =130 at the highest collisional energy.The rotational constants were taken as B e =60.853 cm − forH and B =0.151739 cm − for HC N, as in Wernli et al.(2007a). Finally, convergence was checked as a function ofthe propagator step size (parameter STEPS) and the otherpropogation parameters were taken as the default
MOLSCAT values.Rate coefficients were obtained for kinetic temperaturesin the range 10-300 K by integrating the product of thecross section by the velocity over the Maxwell-Boltzmanndistribution of velocities at each temperature. Cross sectionsand rate coefficients are presented in Section 4 below.
The objective of the classical calculations was to provide ratecoefficients for all (classically allowed) transitions among thelevels j = 31 −
37. Kinetic temperatures were restricted tothe range 100-300 K since rate coefficients at lower tem-peratures are available in Wernli et al. (2007a), where fulldetails on the QCT approach can be found. Briefly, it con-sists in solving the classical Hamilton equations of motioninstead of the Schr¨odinger equation. In the Monte-CarloQCT approach, batches of trajectories are sampled with ran-dom (Monte-Carlo) initial conditions and they are analyzedthrough statistical methods. In the canonical formalism em-ployed here, the initial collision energies are selected ac-cording to the Maxwell-Boltzmann distribution of velocitiesand rate coefficients, instead of cross sections, are directlyobtained (see Eqs. (3) and (4) in Wernli et al. (2007a)).State-to-state rate coefficients are obtained by use of thecorrespondence principle combined with the bin histogrammethod for extracting the final j quantum number.Preliminary calculations had shown that QCT calcu-lations for para-H ( j = 0) and ortho-H ( j = 1) givevery similar cross sections (Wernli 2006). As in Wernli et al.(2007a), we therefore computed QCT cross sections for para-H ( j = 0) only, using the W07 PES averaged over H ori-entation and fitted with a bicubic spline function. The clas-sical equations of motion were numerically solved using anextrapolation Bulirsh-Stoer algorithm and numerical deriva-tives for the potential. The total energy and total angularmomentum were checked and conserved up to seven digits,i.e. within 0.01 cm − for the energy. The maximum impactparameter b max was found to range between 20 and 21 bohrfor temperatures in the range 100-300 K. Batches of 10,000trajectories were run for each initial level j = 31 −
37 and This value was used for consistency with our previous calcula-tions. Using B =59.322 cm − instead of B e would not modify thecross sections since a single rotational level of H was included.c (cid:13) , 1–8 Alexandre Faure, Fran¸cois Lique and Laurent Wiesenfeld
Table 1.
Transition frequencies (in MHz) in the HC N–para-H complex from experiment, from the calculations of Michaud et al. (2011)and from the present calculations. Deviations from the experiment are also given in percent. J ′ K a K c − J ′′ K a K c experiment Michaud et al. error (%) present error (%)1 − − − − − − − − − Table 2.
Transition frequencies (in MHz) in the HC N–ortho-H complex from experiment, from the calculations of Michaud et al.(2011) and from the present calculations. Deviations from the experiment are also given in percent. J ′ K a K c − J ′′ K a K c experiment Michaud et al. error (%) present error (%)1 − − − − − − − − − each temperature, resulting in a total of 770,000 trajectories.It should be noted that a particular advantage of the QCTmethod is that the computational time decreases with in-creasing collision energy, in contrast to quantum methods.QCT calculations however ignores purely quantum effectssuch as interference and tunneling, as will be shown below. Due to the presence of a nitrogen atom, the HC N moleculepossess a hyperfine structure. Indeed, the coupling betweenthe nuclear spin ( I = 1) of the nitrogen atom and the molec-ular rotation results in a splitting of each rotational level j , into 3 hyperfine levels (except for the j = 0 level).Each hyperfine level is designated by a quantum number F ( F = I + j ) varying between | I − j | and I + j . Asalready discussed (e.g. Faure & Lique 2012; Lanza & Lique2014, and references therein), if the hyperfine levels are as-sumed to be degenerate, it is possible to simplify consid-erably the hyperfine scattering problem. Almost exact hy-perfine resolved cross sections can then be obtained fromnuclear spin-free S -matrices using the so-called recouplingapproach. This approach requires to store the S -matricesand to compute hyperfine resolved S -matrices by properlycombining the nuclear spin-free S -matrices. However, in thecase of HC N–H collisions, the recoupling approach be-comes rapidly prohibitive in terms of both memory and CPUtime because of the small rotational constant of the targetmolecule, i.e. the large number of channels to include.Alternatively, hyperfine resolved rate coefficients can bedirectly obtained from rotational rate coefficients using thescaled-infinite-order-sudden limit (S-IOS) method first in- troduced by Neufeld & Green (1994) in the case of HCl-Hecollisions and then extended to the hyperfine excitation oflinear molecules by para- and ortho-H by Lanza & Lique(2014). This adiabatic method is expected to be reliable athigh collision energies and/or for target molecules with smallrotational constants so that the rotational period is smallcompared to the collision time scale. In practice, the hy-perfine rate coefficients (or cross sections) are obtained byscaling the coupled-channel rotational rate coefficients bythe ratio of hyperfine and rotational IOS rate coefficients asfollows: k S − IOSj ,F ,j → j ′ F ′ ,j ′ = k IOSj F ,j → j ′ F ′ ,j ′ k IOSj ,j → j ′ ,j ′ k CCj ,j → j ′ ,j ′ , (1)where k IOSj F ,j → j ′ F ′ ,j ′ and k IOSj ,j → j ′ ,j ′ can be found inEqs. (6) and (7) of Lanza & Lique (2014). We note thatfor quasi-elastic transitions (those with j ′ = j ) no scalingis applied, as explained in Faure & Lique (2012). We notealso that Eq. (1) guarantees that the summed hyperfine ratecoefficients are identical to the coupled-channel pure rota-tional rate coefficients.The S-IOS approach was applied recently to the HCl–H collisional system (Lanza & Lique 2014). Despite the largerotational constant of HCl, the S-IOS method was found tobe accurate to within a factor of 2-3 in the case of collisionswith para-H ( j = 0) at intermediate and high kinetic ener-gies ( >
200 cm − ). It was however found to fail by factorslarger than 3 in the case of ortho-H ( j = 1). As HC N ismuch heavier than HCl and hence more adapted to IOS typemethods, one can expect the accuracy of the S-IOS approachto be substantially better. c (cid:13) , 1–8 ollisional excitation of HC N by H C r o ss s e c t i on ( Å ) -1 )0.11101001000 → → → → para-H ortho-H Figure 1.
Cross sections for rotational excitation out of theground state of HC N j = 0 into j ′ = 1 , , ( j = 0) and ortho-H ( j = 1), respec-tively. In Fig.1, cross sections for the rotational excitations j =0 → , , , ( j = 0) and ortho-H ( j = 1). It is firstnoticed that these cross sections show prominent resonancesat low energies, especially in the case of para-H ( j = 0).These resonances, of both shape and Feshbach types, arisefrom purely quantum effects. They have been observed ex-perimentally only recently (see e.g. Chefdeville et al. 2012,2015, in the case of CO–H ). We also observe in the caseof para-H ( j = 0) that transitions with even ∆ j (i.e.∆ j = 2 ,
4) have the largest cross sections, with ∆ j = 4being even dominant in the energy range ∼ − . Thisresult was interpreted by Wernli et al. (2007a) as originat-ing from the shape of the HC N–para-H PES, being nearlya prolate ellipsoid. It should be noted that these results areunchanged when the basis set on H include the j = 2 level(see Fig. 1 in Wernli et al. 2007b). In contrast, in the case ofortho-H , cross sections follow an energy-gap law behaviourwith ∆ j = 1 > ∆ j = 2 > ∆ j = 3, etc. We note, still,that above ∼
400 cm − , the transition 0 → . It was previously observed for manyother systems where the target has a large dipole, e.g. forHCN–H (Vera et al. 2014). In the case of formaldehyde(H CO), it was even used to indirectly constrain the ortho-to-para ratio of H (Troscompt et al. 2009). Indeed, the gen-eral distinction between para- and ortho-H is attributableto the permanent quadrupole moment of H , which vanishesfor j = 0 but not for j >
0. When the dipole of the targetis significant, the long-range dipole-quadrupole interactionterm is large and the difference between para-H ( j = 0)and ortho-H ( j = 1) is substantial (at the quantum level). j R a t e c oe ff i c i en t ( c m s - ) Quantum, para-H Quantum, ortho-H QCT, para-H Figure 2.
Rate coefficients for the HC N deexcitation j = 15 → j ′ by H at 100 K. Quantum coupled-channel results are givenby the solid black and light green lines for para-H ( j = 0) andortho-H ( j = 1), respectively. QCT results are represented bythe solid circles with error bars representing two standard devia-tions. A comparison between quantum and classical rate coeffi-cients is presented in Fig. 2 for HC N initially in level j = 15. The kinetic temperature is fixed at 100 K andthe rate coefficients are plotted as function of the finalHC N level j ′ . We first notice that the quantum resultsfor para-H ( j = 0) show a pronounced even ∆ j (“nearhomonuclear”) propensity, as expected from the cross sec-tions reported above. This propensity is not observed at theQCT level whereas the PES are identical, demonstratingthat this result indeed arises from a purely quantum ef-fect. In fact, this propensity was explained semi-classicallyby McCurdy & Miller (1977) in terms of an interference ef-fect related to the even anisotropy of the PES. Interferencesare of course ignored in QCT calculations so that the “zig-zags” are totally absent. Now when ortho-H ( j = 1) is theprojectile, the zig-zags are almost entirely suppressed and,interestingly, the quantum results are in excellent agreementwith the QCT calculations, within error bars (except forthe largest ∆ j = 14 transfer). This demonstrates that thequadrupole moment of H , which vanishes for H ( j = 0),plays a crucial role by breaking the even symmetry of thePES and suppressing the interference effect. But it doesnot modify the magnitude of the cross sections, which arewell reproduced by purely classical mechanics with para-H ( j = 0). This result also suggests that the scatteringprocess is dominated by the short-range part of the PES, i.e.by the ellipsoidal shape of the potential. It should be notedthat the interference structure is extremely sensitive to thePES anisotropy and rotationally state-selected experimentswhich resolve this structure would provide a critical test oftheory (see e.g. Carty et al. 2004, in the case of CO+He).These results show that QCT calculations can be em- c (cid:13) , 1–8 Alexandre Faure, Fran¸cois Lique and Laurent Wiesenfeld ployed with confidence to mimic collisions with rotationallyexcited H ( j > ∼ − cm s − . Smaller ratecoefficients have much larger uncertainties because they cor-respond to rare or “classically forbidden” transitions withsmall probabilities. The ortho-to-para ratio of H can be out of thermal equi-librium in the ISM (e.g. Faure et al. 2013, and referencestherein). It is thus crucial to consider the two nuclear spinspecies of H as two separate colliders. We can howevergenerally assume that each nuclear spin species has a ther-mal distribution of rotational levels. At temperatures below ∼
80 K, only the ground states are significantly populated sothat in practice only para-H ( j = 0) and ortho-H ( j = 1)must be considered. At higher temperatures, however, thelevels j = 2 , ∼
80 K, ∼
160 K and ∼
260 K, respectively. Due to the pro-hibitive CPU cost, we did not perform extensive scatteringcalculations for these excited levels but a few energy pointswere computed. We have thus found that rotational crosssections for H initially in j = 2 differ by less than ∼ in j = 1. As a result, it can be assumedthat all cross section, and by extension all rate coefficients,for H ( j >
1) are identical to those for H ( j = 1). This re-sult was observed previously for many other systems before,as a rather general rule (e.g. Daniel et al. 2014, and refer-ences therein). We note that at higher temperatures wherethe H levels can be (de)excited, this rule holds for “effec-tive” rate coefficients which are summed over the final H levels.For para-H , we have thus computed the “thermalized”rate coefficients by weighting the para-H ( j = 0) rate co-efficients, k j ,j =0 → j ′ ( T ) , by the thermal distribution ofthe j = 0 level and the ortho-H ( j = 1) rate coefficients, k j ,j =1 → j ′ ( T ), by the thermal distribution of all para levelswith j > k j ,p H → j ′ ( T ) = ρ k j ,j =0 → j ′ ( T )+(1 − ρ ) k j ,j =1 → j ′ ( T ) , (2)where ρ = 1 P j =0 , , ,... (2 j + 1) exp( − E j /k B T ) (3)is the thermal population of j = 0.For ortho-H , all H levels were assumed to have thesame rate coefficients so that the “thermalized” rate coeffi-cients were simply obtained as: k j ,o H → j ′ ( T ) = k j ,j =1 → j ′ ( T ) . (4)In Fig. 3, the thermalized rate coefficients for para-H and ortho-H are plotted as function of temperature for theground-state transition j = 1 → j = 2 is significant above ∼
80 K, as expected, andit is here amplified by the fact that the corresponding rate Since rotational transitions in H are neglected in our calcula-tions, the calculated rate coefficients are “effective” and the finalH level j ′ can be omitted from the notation. × -11 × -10 × -10 × -10 R a t e c oe ff i c i en t ( c m s - ) ortho-H para-H para-H ( j =0) Figure 3.
Rate coefficients for the HC N deexcitation j = 1 → and ortho-H in the temperature range 10-300 K.The thermalized rate coefficient for para-H ( j = 0 , , , ... ) iscompared to the contribution of the ground state j = 0. coefficient (set equal to k j ,j =1 → j ′ ( T )) is about a factor of10 larger than k j ,j =0 → j ′ ( T ) (see Eq. 2). At 300 K, thecontribution of j = 2 (and to much a lesser extent j = 4)increases the rate coefficient of para-H by a factor of ∼ N and kinetic tempera-tures in the range 10-300 K are available at the
LAMDA (Sch¨oier et al. 2005) and BASECOL (Dubernet et al. 2013)data bases. We note that for HC N levels between j = 31and j = 37, only QCT rate coefficients for para-H ( j = 0)are available and no thermal averaging was applied to thisset which is employed for both para-H and ortho-H . In order to evaluate the accuracy of the S-IOS method (seeSection 3.3) in the case of HC N–H collisions, we have com-puted recoupling and S-IOS hyperfine cross sections at twoselected collisional energies (10 and 50 cm − ). In the aboveEq. (1), cross sections were employed instead of rate coef-ficients. Fig. 4 shows a comparison between recoupling andS-IOS hyperfine cross sections for all the de-excitation tran-sitions from the initial level j =5, including quasi-elastictransitions (those with j = j ′ and F = F ′ ), for colli-sions with para-H ( j = 0) and ortho-H ( j = 1). We notethat the S-IOS approach imposes selection rules (throughWigner 6 j symbols) and some cross sections (not plotted)are strictly zero. The corresponding transitions are those be-tween the ( j = 1 , F = 0) level and levels with F = j , e.g.(1 , → (1 , ∼ . ∼ moldata http://basecol.obspm.fr c (cid:13) , 1–8 ollisional excitation of HC N by H σ REC (Å ) -1 σ S - I O S ( Å ) -1 E c = 10 cm -1 p-H σ REC (Å ) -1 σ S - I O S ( Å ) -1 E c = 50 cm -1 p-H σ REC (Å ) -1 σ S - I O S ( Å ) -1 E c = 10 cm -1 o-H σ REC (Å ) -1 σ S - I O S ( Å ) -1 E c = 50 cm -1 o-H Figure 4.
Comparison between HC N–H recoupling and S-IOS hyperfine cross sections for all the de-excitation transitions from j =5at two different collisional energies. The vertical axis represents the hyperfine S-IOS cross sections and the horizontal axis represents thecorresponding hyperfine recoupling cross sections. The two dashed lines in each panel delimit the region where the cross sections differby less than a factor of 3. For collisions with para-H ( j = 0), the agreement be-tween recoupling and S-IOS calculations is very satisfac-tory. At low energies, the differences between the two setsof data are less than 20-30% (especially for the dominantcross sections). At kinetic energies greater than 50 cm − ,cross sections are almost identical. We conclude that theS-IOS approach can be used with confidence to computehyperfine rate coefficients in the case of collisions with para-H ( j = 0). For collisions with ortho-H ( j = 1), the agree-ment between recoupling and S-IOS calculations is less sat-isfactory. At low energies, the two sets of data agree withina typical factor of 3. At kinetic energies greater than 50cm − , the differences between the two sets decrease and theaverage difference is lower than a factor of two.To summarize, the S-IOS method can provide hyperfineresolved rate coefficients with an average accuracy betterthan 20-30% in the case of collisions with para-H ( j = 0)and within a factor of 2-3 in the case of collisions withortho-H ( j = 1). With respect to the memory and CPUcost of full recoupling calculations, the S-IOS approxima-tion therefore represents a suitable (and low-cost) alterna-tive for this system. Finally, in terms of radiative trans-fer application, it should be noted that at moderate andhigh opacities, where the relative hyperfine populations cansignificantly depart from the statistical weights, the S-IOSmethod is notably better than the statistical approach (seeFaure & Lique 2012).In practice, hyperfine resolved rate coefficients were ob-tained for the lowest 61 hyperfine levels of HC N, i.e. up to( j = 20 , F = 20) which lies 63.73 cm − above (0, 1), and for kinetic temperatures in the range 10-100 K. This set ofdata is available at the LAMDA and
BASECOL data bases.
We have reported in this paper rate coefficients for the ro-tational excitation of HC N by para- and ortho-H . Thelowest 38 rotational levels of HC N were included and ki-netic temperatures up to 300 K were considered. The scat-tering calculations were performed at the quasi-classical andquantum coupled-channel level using the interaction poten-tial of Wernli et al. (2007a). This potential was also em-ployed to compute the bound-states of the complex in orderto make comparisons with the spectroscopy measurementsof Michaud et al. (2011). Theory and experiment were foundto agree within 0.5% for para-H ( j = 0) and within a fewpercent for ortho-H ( j = 1), demonstrating the high ac-curacy of the potential. It appears from these comparisonsthat the calculated state-to-state rotational rate coefficientsare likely to be accurate to about 20-30%. Hyperfine re-solved rate coefficients were also deduced using the S-IOSapproximation, with a somewhat lower accuracy. The wholeset of data represent a significant improvement and exten-sion over the previous data of Green & Chapman (1978) andWernli et al. (2007a).The next step is to determine a flexible potential en-ergy surface in order to treat the ro-vibrational excitationof HC N. The lowest vibrational state is the ν bendingmode which lies at 222 cm − above the ground vibrational c (cid:13) , 1–8 Alexandre Faure, Fran¸cois Lique and Laurent Wiesenfeld state. Ro-vibrational excitation due to collisions is there-fore expected to play a role above ∼
300 K. Such calcula-tions are highly challenging due to excessively large num-ber of channels involved. This problem has been howevertackled recently at the quantum coupled-channel level us-ing the rigid-bender approximation (Stoecklin et al. 2013;Stoecklin, Denis-Alpizar & Halvick 2015). This latter ap-proximation was also employed previously in quasi-classicaltrajectories (Faure et al. 2005) which offer an (economi-cal) alternative to quantum computations, as shown in thepresent work.In summary, the present set of data, possibly com-plemented by electron-impact rate coefficients (as givenin Gratier et al. 2013), should help in modelling non-LTEHC N spectra in cold to warm regions of the ISM. Wenote in particular that HC N has recently been detectedin protopolanetary disks (Chapillon et al. 2012; ¨Oberg et al.2015). The collisional data provided here should prove veryuseful in interpreting such observations.
ACKNOWLEDGEMENTS
This research was supported by the CNRS national pro-gram ’Physique et Chimie du Milieu Interstellaire’. Most ofthe computations presented in this paper were performedusing the CIMENT infrastructure (https://ciment.ujf-grenoble.fr), which is supported by the Rhˆone-Alpes region(GRANT CPER07 13 CIRA).
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