Issue 
A&A
Volume 657, January 2022



Article Number  A55  
Number of page(s)  8  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/202142434  
Published online  06 January 2022 
Deexcitation rate coefficients of C_{3} by collision with H_{2} at low temperatures^{★}
^{1}
Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile,
Santiago
837.0415, Chile
^{2}
Instituto de Ciencias Químicas Aplicadas, Facultad de Ingeniería, Universidad Autónoma de Chile,
Av. Pedro de Valdivia 425,
7500912
Providencia,
Santiago, Chile
email: otoniel.denis@uautonoma.cl
^{3}
Departamento de Física, Facultad de Ciencias, Universidad de Chile,
Casilla 653,
Santiago
7800024,
Santiago, Chile
^{4}
Centro para el Desarrollo de la Nanociencia y la Nanotecnología (CEDENNA),
Avda. Ecuador 3493,
Santiago
9170124, Chile
email: cardena@uchile.cl
Received:
14
October
2021
Accepted:
10
November
2021
Context. An accurate analysis of the physicalchemical conditions in the regions of the interstellar medium in which C_{3} is observed requires knowing the collisional rate coefficients of this molecule with He, H_{2}, electrons, and H.
Aims. The main goals of this study are to present the first potential energy surface for the C_{3} +H_{2} complex, to study the dynamics of the system, and to report a set of rate coefficients at low temperature for the lower rotational states of C_{3} with para and orthoH_{2}.
Methods. A large grid of ab initio energies was computed at the explicitly correlated coupledcluster with single, double, and perturbative tripleexcitation level of theory, together with the augmented correlationconsistent quadruple zeta basis set (CCSD(T)F12a/augccpVQZ). This grid of energies was fit to an analytical function. The potential energy surface was employed in closecoupling calculations at low collisional energies.
Results. We present a highlevel fourdimensional potential energy surface (PES) for studying the collision of C_{3} with H_{2}. The global minimum of the surface is found in the linear HHCCC configuration. Rotational deexcitation statetostate cross sections of C_{3} by collision with para and orthoH_{2} are computed. Furthermore, a reduced twodimensional surface is developed by averaging the surface over the orientation of H_{2}. The cross sections for the collision with paraH_{2} using this approximation and those from the fourdimensional PES agree excellently. Finally, a set of rotational rate coefficients for the collision of C_{3} with para and orthoH_{2} at low temperatures are reported.
Key words: astrochemistry / molecular data / molecular processes / scattering / ISM: molecules
Tables 5 and 6 are only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/cat/J/A+A/657/A55
© ESO 2022
1 Introduction
Small carbon chains, such as C_{2}, C_{3}, and C_{5}, have been observed in several regions of the interstellar medium (ISM) (Souza & Lutz 1977; Giesen et al. 2020; Maier et al. 2001; Hinkle et al. 1988; Bernath et al. 1989). They are expected to be the building blocks of more complex organic molecules that could explain the origin of life. Furthermore, these molecules could form larger chains, such as polycyclic aromatic hydrocarbon (PAH) and fullerene, which are candidates for explaining some of the diffuse interstellar bands (DIBs) (Omont et al. 2019).
In typical molecular clouds, the density and temperature are low (10^{2}–10^{4} cm^{−3} and 10–15 K). Therefore, the rotational population of the molecules detected cannot be described by a Boltzmann distribution. In these cases, an accurate analysis of the physicalchemical conditions of these regions should be made using nonlocal thermal equilibrium (nonLTE) models. These models require knowing the Einstein coefficients and statetostate rate coefficients of the observed molecules with the most common colliders in the ISM. However, the rate coefficients for the collision with H_{2}, He, H, and e are not always available.
The C_{3} molecule has no dipole moment, but it can be detected through its rovibrational transitions. Even in cases like this, knowing the rotational statetostate rate coefficients is valuable. For example, due to the absence of specific rotational rate coefficients for C_{3}, the molecular abundance and excitation temperatures of this species in Sgr B2 and IRC+10216 were estimated by Cernicharo et al. (2000) using the rotational rates for OCS with H_{2} within a given vibrational level, while those between the ground and bending states were the same rotational rates divided by 10. Furthermore, due to the lack of data for C_{3} at that moment, Roueff et al. (2002) assumed the collisional rates to be proportional to the radiative line strengths in the analysis of the observation of C_{3} in the translucent molecular cloud toward HD 210121, and concluded that it will be essential to understand the inelastic collisions with C_{3} better.
The interaction of C_{3} with He has been widely studied. Abdallah et al. (2008) computed a PES at the coupledcluster with single, double, and perturbativetripleexcitation level of theory (CCSD(T)) and reported the rate coefficients at low temperature, below 15 K, considering C_{3} as a rigid rotor. More recently, Smith et al. (2014) presented a new PES calculated with the CCSDT(Q) method, using the rigidmonomer approximation. They extended the set of rate coefficients up to 100 K. Two PESs at the CCSD(T) and CCSD(T)F12 level considering the bending angle of C_{3} in the C_{3} + He complex were also developed (DenisAlpizar et al. 2014; Al Mogren et al. 2014). Furthermore, the inclusion of the bending of C_{3} in the close coupling calculations was investigated by Stoecklin et al. (2015). In this work, the cross sections computed using the rigidrotor approximation and including the bending motion showed differences only for collisions energies higher than the bending frequency of C_{3}.
After the first studies of the collision C_{3} with He, Schmidt et al. (2014) reported the detection of this molecule toward HD 169454. In the analysis of the data, these author employed the same approximation as Roueff et al. (2002) and also included the rates for C_{3} + He (Abdallah et al. 2008), scaled to represent the collision with H_{2}, and the deexcitations from higher levels were fixed at values from rotational state j = 10 in the RADEX code (Van der Tak et al. 2007). The model using the new rates showed that a higher destruction rate is required to fit the observed column densities well. These authors also proposed that detailed research of collisional rates is necessary before the physical conditions of the molecular gas can be deduced in detail from the C_{3} column densities.
The collision of C_{3} with atomic hydrogen was studied recently by Chhabra and Dhilip Kumar (Chhabra & Dhilip Kumar 2019). These authors build a twodimensional PES for the ground states of C_{3} + H at the MRCI level and reported the rate coefficients up to 100 K. Even if the collision of C_{3} with H_{2} is nonreactive at low temperature (Mebel et al. 1998; Costes et al. 2006), no collisional rate coefficients have been reported so far. The rates for C_{3} + paraH_{2}(j = 0) can be estimated from those with He (Schöier et al. 2005). However, it has been shown that this approximation is not valid in all cases, and in general, it is not valid for collisions with orthoH_{2} (Kłos & Lique 2008; Vera et al. 2014; DenisAlpizar et al. 2018). Therefore, the collision of C_{3} with H_{2} deserves to be investigated.
The main goals of this study are to present the first potential energy surface for the C_{3} + H_{2} complex, to study the dynamics of the system, and to report a set of rate coefficients at low temperature for the lower rotational states of C_{3} with para and orthoH_{2}. This paper is organized as follows. In the next section, we present the methods we employed, and the results are discussed in Sect. 3. Finally, the main conclusions are summarized in Sect. 3.2.
2 Calculations
2.1 Ab initio calculations
The coordinates we employed to study the C_{3} + H_{2} complex are shown in Fig. 1. R connects thecenters of mass of the H_{2} and C_{3} molecules, and θ_{1}, θ_{2}, and φ describe the angular orientations of both systems. C_{3} and H_{2} are considered rigid rotors in this work. The bond length of H_{2} was taken as the vibrationally averaged value in the rovibrational ground state r_{H −H} = 0.76664 Å (Jankowski & Szalewicz 1998), while the distances between C=C were set to the equilibrium value of the C_{3} molecule, r_{C −C} = 1.277 Å (Van Orden & Saykally 1998).
Table 1 shows the interaction energies computed using different methods and basis sets. The energies at the completedbasisset (CBS) limit were estimated with the twopoint extrapolation formula (Halkier et al. 1999), , where X = 5 and Y = 4 are the cardinality of the augccpV5Z, and augccpVQZ basis sets from CCSD(T) calculations. The CCSD(T)F12a/augccpVQZ method gives energies close to the CBS limit at a relatively low computational cost. Therefore, all ab initio calculations were performed with this method as implemented in MOLPRO package (Werner et al. 2012). The size inconsistency of the CCSD(T)F12a method was corrected by shifting up the ab initio interaction energies to make it vanish at R = 200 Å (Lique et al. 2010). The basis set superposition error was corrected using the counterpoise procedure of Boys and Bernardi (Boys & Bernardi 1970).
In total, 38 870 ab initio energies were computed. The radial coordinate R includes 23 values from 1.7 Å to 10.0 Å, while θ_{1} and θ_{2} vary from 0° to 180° in steps of 15°. The azimuthal angle φ varies in steps of 20° in the [0, 180]° interval.
Fig. 1
Internal coordinates used to describe the C_{3} + H_{2} system. The azimuthal angle φ is undefined when either θ_{1} or θ_{2} is equal to 0° or 180°. 
2.2 Analytical fourdimensional surface
The grid of ab initio energies was fit following the same procedure as was recently used to study the HCO^{+} + H_{2} complex (DenisAlpizar et al. 2020). The analytical function employed has the form (1)
where the angular part is represented by a product of normalized associated Legendre polynomials and a cosine function. The molecules H_{2} and C_{3} have a center of symmetry; thus, l_{1} and l_{2} are restricted to even values (Nasri et al. 2015). For each value of R, the ab initio energies were fit using a leastsquares method. Each coefficient was then fit using the reproducing kernel Hilbert space (RKHS) procedure (Ho & Rabitz 1996), (2)
where N is the number of radial points of thegrid (R_{k}). The q^{2,4}(R, R_{k}) is the onedimensional kernel, defined as (3)
R_{>} and R_{<} are the greater and lower value between the R_{k} and R. The coefficients were found by solving the system of linear equations where i and j label the differentradial geometrical configurations of the grid.
Interaction energy (in cm^{−1}) of H_{2} with C_{3} at the fixed angles (θ_{1} = 0° and θ_{2} = 90°) for several values of R using different methods and basis sets (the augccpVXZ basis sets are represented as AVXZ).
2.3 Averaged surface
The use of a twodimensional PES averaged over the orientation of H_{2} to study the collision of a molecule with paraH_{2}(j = 0) reduces computational cost in ab initio and dynamics calculations. This approximation has shown a reasonable agreement in the determination of the rate coefficients with those computed using the fourdimensional surface in several studies, for instance, SiS (Lique et al. 2008), HNC (Dumouchel et al. 2011), HCO^{+} (Massó & Wiesenfeld 2014), and CF^{+} (Desrousseaux et al. 2019). This is justified because in the dynamics of the collision with paraH_{2}(j = 0), the coupling matrix elements are nonzero if l_{1} = 0 (see Eq. (9) in Green 1975).
The fourdimensional PES can be averaged as (4)
and if only the first rotational state of paraH_{2} is considered, the rotational angular momentum of H_{2}, j_{1}, and its projection on the intermolecular axis, k, are equal tozero. This averaged PES can be obtained numerically by quadrature. However, this would require knowing the fourdimensional PES, and one of the goals of this approximation is to reduce the calculation times in the development of the PES. It has been shown that considering three orientations of H_{2} is a good approximation to determine the averaged PES (Najar et al. 2014), and this is what we used here. The averaged energies were then computed as (5)
A grid of energies from Eq. (5), in the same intervals for R and θ_{1} as were used for the fourdimensional PES, was then fit to the analytical expression (6)
For each value of R, the f_{l} (R) coefficients were computed using a leastsquares procedure. The f_{l}(R) were then fit using the RKHS method, such as (7)
where q^{2,5}(R, R_{k}) is the onedimensional kernel (Ho & Rabitz 1996), defined as (8)
The subscript k corresponds to the k–ith radial point of the grid, and N is the number of R. R_{>} and R_{<} are the greater and lower value between the R_{k} and R. The coefficients were obtained by solving the linear equations system , where , and k and k′ label the geometrical configurations of the grid.
2.4 Scattering calculations
The fourdimensional PES was included in the Didimat code (Guillon et al. 2008) to study the collision of C_{3} with paraH_{2} and orthoH_{2}. This code solves the closecoupling equations in the spacefixed frame. The logderivative propagator (Manolopoulos 1988) was employed starting from 3 a_{0}. The minimum value of the maximum propagation distance was set to 60 a_{0} and extended automatically. At each collision energy, the convergence of the quenching cross sections was checked as a function of the total angular momentum and maximum propagation distance. The rotational constants of H_{2} and C_{3} we used are cm^{−1} (Huber & Herzberg 1979) and cm^{−1} (Van Orden & Saykally 1998). The inclusion of 18 rotational states of C_{3} in the basis for the closecoupling calculation was enough to reach convergence. Table 2 shows the deexcitation cross sections for the collision with paraH_{2} including one () and two () rotational states in the basis at the collisional energies of 10 and 100 cm^{−1}. The cross sections with the two bases agree well at both energies. The average percent difference is lower than 4.2% at 10 cm^{−1} (this energy is in the region in which the resonances strongly affect the magnitude of the cross sections) and reduces to 2% at 100 cm^{−1}. Therefore, only one rotational state of H_{2} was included in the basis to study the dynamics of the collision.
Furthermore, the averaged PES was employed to study the dynamics of the collision of C_{3} with paraH_{2}(j = 0). In this case,the code for investigating the atommolecule collision, Newmat (Stoecklin et al. 2002), was used. The closecoupling equations were solved in the spacefixed frame, and the logderivative propagator was also employed (Manolopoulos 1988). The minimum value of the largest propagation distance was 50 a_{0} and was automatically extended. At each collisional energy, the convergence of the quenching cross section was checked as a function of the maximum intermolecular distance and the total angular momentum. In these calculations, 18 rotational states of C_{3} were also considered.
Finally,the statetostate deexcitation rate coefficients () were computed by the average of the rotational cross sections () over a MaxwellBoltzmann distribution at a given temperature T, as (9)
where j_{i} and j_{f} are the initial and final rotational states of C_{3}, E_{c} is the collisional energy, and k_{B} is the Boltzmann constant.
Rotational deexcitation cross sections (in ) of C_{3} by collision with paraH_{2} using one () and two () states of H_{2} in the dynamics calculations at collisional energies of 10 and 100 cm^{−1} for the lower transitions.
3 Results and discussion
3.1 Potential energy surface
The quality of the fit of the fourdimensional PES was checked by evaluating the rootmeansquare deviation (RMSD) of the ab initio energies of the grid and the fitted values. For E ≤ 0 cm^{−1}, the RMSD was 6.19 × 10^{−2} cm^{−1}. In the intervals 0 ≤ E ≤ 1000 cm^{−1}, 1000 ≤ E ≤ 5000 cm^{−1}, and 5000 ≤ E ≤ 10 000 cm^{−1}, the RMSD were 0.155, 0.286, and 2.143 cm^{−1}, respectively.
Furthermore, a set of 987ab initio energies (E_{ab}) not employed in the fitting procedure was computed. The averaged percentage difference between these energies and those (E_{ana}) computed with the analytical PES (100 × (E_{ab} − E_{ana})∕((E_{ab} + E_{ana})∕2)) for E_{ab} < 5000 cm^{−1} is lower than 2.6%, while in the 5000 ≤ E_{ab} < 20 000 cm^{−1} interval, thisvalues is 2.1%. Figure 2A shows the computed energies and the ab initio energies at geometrical configurations outside the grid that was used in the fit, which confirms that the agreement is very good. This figure also gives an indication of the anisotropy of the interaction.
The long range of the PES is also analysed from the multipolar expansion. The interaction energy can be written as the sum of the electrostatic (E_{elec}), induction (E_{ind}), and dispersion (E_{disp}) contributions, such as (Pullman 1978) (10) (11)
where α = (2α_{xx} + α_{zz})∕3 is a mean polarizability. Superscripts A and B denote the H_{2} and C_{3} molecules, respectively,and U is the ionization energy ( eV and eV) (Johnson 2002). The computed molecular properties of C_{3} and H_{2} are listed in Table 3. Figure 2B shows the excellent agreement between the analytical multipolar energies and those from our PES. This makes this surface suitable for studying cold molecular collisions.
Figure 3 shows the contour plot of the analytical PES at several geometrical configurations. A strong dependence of the energies on the angular orientation is shown in this figure. The global minimum of the surface, named − 135.64 cm^{−1}, was found at the R = 4.76 Å, in the linearconfiguration H–H–C–C–C, see panel A and D. This well is around 5 times deeper than the one found in C_{3} + He (− 25.87 cm^{−1} Abdallah et al. 2008, − 25.54 cm^{−1} Zhang et al. 2009, − 31.29 cm^{−1} Zhang et al. 2009, − 26.73 cm^{−1} DenisAlpizar et al. 2014 and − 27.91 cm^{−1} Al Mogren et al. 2014).
Figure 3 also shows apparent local minima; see panels B, C, E, and F. However, Fig. 4 shows a contour plot in which R relaxes from 3.0 to 5.0 Å for φ = 0° and θ_{1} = 90°. Only two secondary minima are observed in these figures: E = − 123.7 cm^{−1} at (θ_{1} = 69°, θ_{2} = 122°, φ = 0°) and at symmetric configuration; and E = − 103.4 cm^{−1} at (θ_{1} = 90°, θ_{2} = 90°, φ = 0°). A Fortran subroutine of this PES is available upon request to the authors or at the github link^{1}.
We also evaluated an averaged twodimensional PES using the energies from Eq. (5). Figure 5 shows a contour plot of this surface. The minimum of this surface, − 78.62 cm^{−1} was found at θ_{2} = 90°, and R = 3.6 Å. This value is also higher than the value for the collision with He, but at the same T configuration. Additionally, an averaged PES was computed from Eq. (4) using a GaussChebyshev quadrature of 20 points for the integration over φ and a Gauss–Legendre quadrature with 20 points over θ_{1}. This surfaceshows the same behavior as that of Fig. 5 (without visible differences). The minimum of this surface (− 78.70 cm^{−1}) was found at the same geometrical configuration as the averaged PES shown in Fig. 5.
Fig. 2
Interaction energy along R of the C_{3} + H_{2} complex. Panel A: energies computed using the fourdimensional PES (solid lines) and from ab initio calculations (points) at several angular configuration (θ_{1}, θ_{2}, φ) of C_{3} and H_{2} that are notincluded in the grid used for the fit. Panel B: energies computed from the PES (solid lines) and analytical multipolar expansion (dashed line) at several angular configuration. 
Fig. 3
Contour plot of the PES for the C_{3} + H_{2} complex at several geometrical configurations. Negative energies are represented as blue lines in steps 20 cm^{−1}. Positive energies (red lines) are logarithmically spaced between 1 cm^{−1} and 10^{4} cm^{−1}. 
Molecular properties of C_{3} and H_{2} calculated at the CCSD(T)/augccpV5Z level using the finitefield method of Cohen and Roothaan^{1}.
3.2 Dynamics
The fourdimensional PES presented in the previous section was employed in closecoupling calculations for collision energies from 10^{−2} up to 300 cm^{−1}. These calculations were performed for energies higher than the bending frequencies of C_{3} (ω = 63 cm^{−1}). In the study of the dynamics of C_{3} in collision with He including the bending motion, the rotational deexcitation cross sections in the groundvibrational state of the triatomic system agreed very well with those computed using the rigidrotor approximation for energies lower than ω (Stoecklin et al. 2015). With increasing collisional energy, this good agreement between the two sets of cross sections decreases. For example, for the transition 4 → 0 at 200 cm^{−1}, the rigidrotor approximation overestimated the cross section by 20% (DenisAlpizar 2014). This difference is also observed in the rate coefficients. At 50 K, the rates computed by Stoecklin et al. (2015) considering the molecule as a rigid rotor and including the bending motion of C_{3} showed a percent difference of 21% (averaged over all deexcitation transitions for j ≤ 10). This percent difference increases with increasing temperature. Therefore, we expect that considering the monomers as rigid rotors is a reasonably good approximation for collisional energies up to 300 cm^{−1}. In the case of C_{3} + H_{2}, the PES is deeper than for C_{3}+He, and the bendingrotation coupling could be stronger. The cross sections at high collisional energies should be considered with caution.
Figure 6 shows the rotational cross sections for the collision of C_{3} with para and orthoH_{2}. The cross sections decrease with increasing collision energy and Δj. The typical shape and Feshbach resonances, associated with the formation of quasibound states due to the PES centrifugal barrier and as a result of the coupling between open and closed channels (Yazidi et al. 2014), can also be observed in this figure.
The cross sections for the collision with paraH_{2} from the averaged PES are also included in Fig. 6. These values agree very well with the cross sections computed using the fourdimensional PES. The use of a reduced PES has shown to be a reasonably good approximation for determining the rate coefficients in the study of the collision of other systems (SiS Lique et al. 2008, HNC Dumouchel et al. 2011, C_{2}H Dagdigian 2018, CF^{+} Desrousseaux et al. 2019, N_{2}H^{+} Balança et al. 2020, and HCO^{+} DenisAlpizar et al. 2020). The C_{3} molecule is another case that shows that this is an attractive approximation for estimating the rate coefficients with paraH_{2} at reduced computational cost.
A set of rate coefficients for the lower rotational states of C_{3} (up to j = 20) for the collision with para and orthoH_{2} was determined from the computed cross sections at temperatures lower than 50 K. These rates are reported in Table 4 and in the supplementary material. The difference between the rates with para and orthoH_{2} is small. This behavior has recently been observed in the study of the collisions with H_{2} of several systems, for instance, C_{4}H^{−} (Balança et al. 2021), HCO^{+} (DenisAlpizar et al. 2020), DCO^{+} (DenisAlpizar et al. 2020), N_{2}H^{+} (Balança et al. 2020), SH^{+} (Dagdigian 2019), CF^{+} (Desrousseaux et al. 2019), C_{3}N^{−} (LaraMoreno et al. 2019), C_{6}H^{−} (Walker et al. 2017), HC_{3}N (Wernli et al. 2007), and CN^{−} (Kłos & Lique 2011). At low temperatures, this is associated with the effects of the longrange part of the interaction outweighing those of the short range (Balança et al. 2021).
Furthermore, the rates for the collision with He determined by Abdallah et al. (2008) that are available in the Basecol database (Dubernet et al. 2013) and the rates for the collision with H, computed from the excitation rates from Chaabra and Dhilip Kumar (Chhabra & Dhilip Kumar 2019) using the principle of detailed balance, are also included in Table 4. The rates for the collision with He are lower than those for the collision with para and orthoH_{2}. The massscaling approximation for determining the rates for the collision with paraH_{2} from those with He was evaluated. The ratio of the rates with paraH_{2} and He is 1.40 at 5 K, 1.54 at 10 K, and 1.64 at 15 K. These values are close to the 1.4 scaling factor that is commonly employed (Schöier et al. 2005), but this ratio varies from 0.8 up to 2.4, and this approximation should be considered with cation for C_{3}. In the case of the collision with H, our rates are also higher. Chhabra and Dhilip Kumar (Chhabra & Dhilip Kumar 2019) found that the rates for H were lower than those for He, and they associated this behavior with the different PES and colliders. Finally, we expect that the new rate coefficients reported here will be useful in determining the interstellar conditions in the regions in which this molecule has been detected.
Fig. 4
Contour plot of the PES for the C_{3} + H_{2} complex. R is relaxed from 3.0 to 5.0 Å for φ = 0° (panel A) and θ_{2} = 90° (panel B). Inthe range [−125, 0] cm^{−1}, the space between the lines is 15 cm^{−1} and 5 cm^{−1}. 
Fig. 5
Contour plot for the averaged PES of the C_{3} + H_{2} complexcomputed from Eq. (5). The space between blue lines is 20 cm^{−1} in the range [−70, 0] cm^{−1}, and the red lines are logarithmically spaced between 1 cm^{−1} and 10^{4} cm^{−1}. 
Fig. 6
Rotational deexcitations cross sections of C_{3} from the initial rotational state j = 4 (panel A) and j = 6 (panel B), incollision with paraH_{2} (solid lines), orthoH_{2} (dashed lines), and paraH_{2} from averaged PES (dotted black lines). Rotational transitions of C_{3} are labeled j_{i} → j_{f}. 
Rotational deexcitation rate coefficients (×10^{−11} cm^{3} molecule^{−1} s^{−1}) of C_{3} in collision with orthoH_{2}, and paraH_{2} at several temperatures.
4 Conclusions
We developed the first fourdimensional PES for the C_{3} H_{2} complex. This surface was fit from a large grid of ab initio energies computed at the CCSD(T)F12a/augccpVQZ level of theory. The global minimum of the surface was found in the linear CCCHH configuration. This surface was employed in closecoupling calculations. The cross sections for the lower rotational states of C_{3} in collision with H_{2} were computed. Furthermore, an averaged PES over the orientation of H_{2} was used to study the relaxation of C_{3} with paraH_{2}(j = 0). The cross sections using this surface and those computed with the fourdimensional PES were found to agree very well. Finally, we reported a set of rate coefficients for the lower rotational states of C_{3} with para and orthoH_{2} at low temperature.
Acknowledgements
Support from projects CONICYT/FONDECYT/ REGULAR/ Nos. 1200732 and 1181121 is gratefully acknowledged. This research was partially supported by the supercomputing infrastructure of the NLHPC (ECM02). C.C. acknowledges Center for the Development of Nanoscience and Nanotechnology CEDENNA AFB180001
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All Tables
Interaction energy (in cm^{−1}) of H_{2} with C_{3} at the fixed angles (θ_{1} = 0° and θ_{2} = 90°) for several values of R using different methods and basis sets (the augccpVXZ basis sets are represented as AVXZ).
Rotational deexcitation cross sections (in ) of C_{3} by collision with paraH_{2} using one () and two () states of H_{2} in the dynamics calculations at collisional energies of 10 and 100 cm^{−1} for the lower transitions.
Molecular properties of C_{3} and H_{2} calculated at the CCSD(T)/augccpV5Z level using the finitefield method of Cohen and Roothaan^{1}.
Rotational deexcitation rate coefficients (×10^{−11} cm^{3} molecule^{−1} s^{−1}) of C_{3} in collision with orthoH_{2}, and paraH_{2} at several temperatures.
All Figures
Fig. 1
Internal coordinates used to describe the C_{3} + H_{2} system. The azimuthal angle φ is undefined when either θ_{1} or θ_{2} is equal to 0° or 180°. 

In the text 
Fig. 2
Interaction energy along R of the C_{3} + H_{2} complex. Panel A: energies computed using the fourdimensional PES (solid lines) and from ab initio calculations (points) at several angular configuration (θ_{1}, θ_{2}, φ) of C_{3} and H_{2} that are notincluded in the grid used for the fit. Panel B: energies computed from the PES (solid lines) and analytical multipolar expansion (dashed line) at several angular configuration. 

In the text 
Fig. 3
Contour plot of the PES for the C_{3} + H_{2} complex at several geometrical configurations. Negative energies are represented as blue lines in steps 20 cm^{−1}. Positive energies (red lines) are logarithmically spaced between 1 cm^{−1} and 10^{4} cm^{−1}. 

In the text 
Fig. 4
Contour plot of the PES for the C_{3} + H_{2} complex. R is relaxed from 3.0 to 5.0 Å for φ = 0° (panel A) and θ_{2} = 90° (panel B). Inthe range [−125, 0] cm^{−1}, the space between the lines is 15 cm^{−1} and 5 cm^{−1}. 

In the text 
Fig. 5
Contour plot for the averaged PES of the C_{3} + H_{2} complexcomputed from Eq. (5). The space between blue lines is 20 cm^{−1} in the range [−70, 0] cm^{−1}, and the red lines are logarithmically spaced between 1 cm^{−1} and 10^{4} cm^{−1}. 

In the text 
Fig. 6
Rotational deexcitations cross sections of C_{3} from the initial rotational state j = 4 (panel A) and j = 6 (panel B), incollision with paraH_{2} (solid lines), orthoH_{2} (dashed lines), and paraH_{2} from averaged PES (dotted black lines). Rotational transitions of C_{3} are labeled j_{i} → j_{f}. 

In the text 
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