Rotational excitation of interstellar benzonitrile by helium atoms

Interstellar aromatic molecules such as polycyclic aromatic hydrocarbons and polycyclic nitrogen and oxygen bearing molecules are thought to be abundant in the interstellar medium. In this class of molecules, benzonitrile ($c$-C$_6$H$_5$CN) plays an important role as a proxy for benzene. It has been detected through rotational emission in several astrophysical sources and is one of the simplest N-bearing polar aromatic molecules. Even in the cold ISM, the population of the rotational levels of benzonitrile might not be at equilibrium. Consequently, modeling its detected emission lines requires a prior computation of its quenching rate coefficients by the most abundant species in the ISM (He or H$_2$). In this paper, we focus on the excitation of c-C$_6$H$_5$CN by collision with He. We compute the first potential energy surface (PES) using the explicitly correlated coupled cluster method in conjunction with large basis sets. The PES obtained is characterized by a potential well depth of -97.2 cm$^{-1}$ and an important anisotropy. Scattering computations of the rotational (de-)excitation of c-C$_6$H$_5$CN by He atoms are performed by means of the coupled states approximation that allow to obtain collisional rates for rotational states up to $j$ = 9 and temperatures up to 40 K. These rate coefficients are then used to examine the effect of C$_6$H$_5$CN excitation induced by collisions with para-H$_2$ in molecular clouds by carrying out simple radiative transfer calculations of the excitation temperatures and show that non-equilibrium effects can be expected for H$_2$ densities up to 10$^5$-10$^6$ cm$^{-3}$.


INTRODUCTION
Up to 10% to 25% of carbon present in astrophysical clouds is estimated to be in the form of polycyclic aromatic hydrocarbons (PAHs) (Dwek et al. 1997;Chiar et al. 2013).PAHs are the proposed carriers of the unidentified infrared emission lines, i.e a set of line identified at mid-infrared wavelengths (from 3 to 13 m), that are usually observed in the interstellar medium (ISM) and photodissociation regions of our Galaxy and numerous external ones (Leger & Puget 1984;Low et al. 1984;Allamandola et al. 1985).As an important carbon reservoir, these aromatic species play a crucial role in the formation of other complex organic molecules in the ISM.Nevertheless, the formation paths of PAHs in astrophysical clouds are still debated.To account for their existence in dense and diffuse regions of the ISM, two formation mechanisms were suggested : PAHs could form in the dense outflows of carbon-rich evolved stars during the destruction of carbonaceous solids (Pilleri et al. 2015;Martín-Doménech et al. 2020;Berné et al. 2015), or they could be formed from small and simple aromatic molecules and from smaller hydrocarbons (Woods et al. 2002;Cernicharo 2004).In dense and cold regions of the ISM, which are not subject to ultraviolet field and shocks, other pathways must exist to synthesize these species from smaller precursor molecules (McGuire et al. 2018).In the bottom-up scenario, the cyclization of small hydrocarbons leads to the first aromatic hydrocarbon, benzene (c-C 6 H 6 ), which is ★ E-mail: malek.benkhalifa@kuleuven.be † E-mail: jerome.loreau@kuleuven.beconsidered a probable precursor in the formation of PAHs.Understanding and characterizing its presence in the dense regions is thus highly relevant for the study of the PAHs formation process but its non-polar nature prevents its detection through radio astronomy.As a substitute, radio astronomers observed for the first time the hyperfineresolved transitions of the N-bearing aromatic benzonitrile, with CN substituted for one of H atoms (c-C 6 H 5 CN, with a large permanent dipole moment of 4.3 debye (Woods et al. 2002)) toward the TMC-1 dense and hot core of the Taurus Molecular Cloud (McGuire et al. 2018).Benzonitrile was subsequently observed toward four other prestellar sources (Serpens 1A, Serpens 1B, Serpens 2 and MC27/L1521F), which shows that aromatic chemistry is widespread (Burkhardt et al. 2021).Recently, Cernicharo et al. (2023) reported a large set of lines of benzonitrile and produced maps of its spatial distribution that indicate that C 6 H 5 CN is formed from bottom-up chemistry.These detections, in combination with theoretical work and modelling, show that C 6 H 5 CN can be used as a convenient observational proxy to characterize the existence of C 6 H 6 in the cold, starless ISM.Several theoretical and experimental studies have indeed established that the formation of benzonitrile from benzene and a source of chemically active nitrogen, commonly the CN radical, is efficient and rapid under typical interstellar conditions.The relevant formation pathway of benzonitrile is thus expected to be the barrierless, exothermic neutral-neutral reaction CN + C 6 H 6 → C 6 H 5 CN + H (Woods et al. 2002;Trevitt et al. 2010;Cooke et al. 2020).
In astrophysical clouds, molecular abundances are obtained from the modeling of molecular lines.For complex organic molecules, lo-cal thermodynamic equilibrium (LTE) conditions are often assumed to hold, either because all molecular lines can be fit with a single excitation temperature or because the rate coefficients for collisional excitation are unknown.However, in many cases such as in the low density ISM, the LTE conditions are not completely fulfilled.The population of molecular levels is then determined by the competition between radiative and collisional processes and it is crucial to determine accurate collisional data for the excitation of the involved molecules by the most abundant interstellar species (He atoms and H 2 molecules), in order to obtain reliably modeled spectra and column densities.
While there has been a dramatic increase in the detection of interstellar complex organic molecules (COMs) in recent years, their collisional properties are mostly unknown as the required computations are particularly challenging.This is notably the case for non-linear COMs, although quantum or mixed quantum/classical calculations have been performed for the excitation of species such as propylene oxide (Dzenis et al. 2022) and benzene (Mandal et al. 2022) by He atoms.
In the present paper, we present the first set of rate coefficients for the collisional excitation of the aromatic molecule C 6 H 5 CN with He atoms at kinetic temperature up to 40 K, based on an accurate potential energy surface and quantum scattering calculations.The overall structure of this paper is as follows: first, we present in section 2 the ab initio study of the C 6 H 5 CN-He system, leading to a new 3D-PES corresponding to the interaction between C 6 H 5 CN and He atoms.In section 3 we describe the study of the dynamics, where we illustrate the inelastic cross sections in C 6 H 5 CN-He collisions.We present the corresponding excitation rate coefficients in section 4. Conclusions and future outlooks are drawn in section 5.

Ab initio calculations
In the following, we compute the potential energy surface (PES) of the rigid molecule c-C 6 H 5 CN interacting with an helium atom in their respective ground electronic states.Benzonitrile is an asymmetric top molecule and its lowest vibrational mode has an energy of only 141.5 cm −1 (Császár & Fogarasi 1989).Consequently, the PES should depend on this vibrational coordinate to study the collisional excitation except in the low temperatures regime that is relevant to the study of the ISM ( ≤ 100 K).However, previous works (Faure et al. 2005;Stoecklin et al. 2013Stoecklin et al. , 2019) ) have shown that the inclusion of the vibrational motion has only a small impact on the pure rotational excitation cross sections, so that the rigid-rotor approximation should remain valid for the system studied here.The c-C 6 H 5 CN geometry was taken at its ground vibrational state averaged values (Császár & Fogarasi 1989) To describe the interaction potential, we used three Jacobi coordinates (, , ) as presented in Figure 1.The origin of the coordinate system coincides with the center of mass of benzonitrile. is the length of the intermolecular vector  between the helium atom and the mass center of benzonitrile,  is the polar angle between the  vector and the C 2 principal inertia axis of the c-C 6 H 5 CN molecule, and  is the azimuthal angle.A total of 15,000 ab initio energy points were calculated.These .Such a large number of points was found to be necessary to obtain an accurate representation of the PES (see below) and is due to the strong anisotropy of the PES.The energies were determined at the explicitly correlated coupledcluster level of theory with single, double, and non-iterative triple excitation (Knizia et al. 2009) in conjunction with an augmented correlation-consistent triple zeta basis set (Dunning Jr 1989) (CCSD(T)-F12a/aug-cc-pVTZ) implemented in the MOLPRO code (Werner et al. 2015).The BSSE correction was taken into consideration using the Boys & Bernardi (1970) counterpoise scheme, hence, the interaction potential is defined as : where  Mol−He represents the global electronic energy of c-C 6 H 5 CN-He and the last two terms are the energies of the two fragments, all calculations being performed using the full basis set of the complex.Due to the inclusion of non-iterative triple excitations, the CCSD(T)-F12 method is not size consistent, therefore, the asymptotic value of the potential (-6.21 cm −1 at =50a 0 ) was subtracted from these interaction energies.

Analytic fit
The efficient implementation of the interaction potential in scattering codes requires an expansion of the PES over a basis of suitable angular functions.In general, we use the spherical harmonics functions for the asymmetric top molecule-atom collisional systems: Taking into consideration the property of spherical harmonics, this can be rewritten as:  where   () and    (, ) denote the radial coefficients and the normalized spherical harmonics respectively, and  ,0 is the Kronecker symbol.The C 2 symmetry of benzonitrile should restrict the allowed terms in equation 2 to those with  being a multiple of 2, ( = 2 ,  integer), and the other terms were found to be negligible.For each intermolecular distance, the interaction potential was developed over the angular expansion and the   were obtained using a standard least-squares fit procedure in order to provide continuous expansion coefficients suitable for the collision dynamics.From a PES grid containing 25 values of  and 10 values of , we were able to include radial coefficients up to   = 19 and  = 18, resulting in 110 expansion terms with a final accuracy better than 1 cm −1 for  ≥ 5 bohr.For  ≥ 30 bohr, we extrapolated the longrange potential using an inverse exponent expansion implemented in the MOLSCAT code Hutson & Green (1994).We illustrate in figure 2 the variation of the first six radial coefficients ( 00 ,  10 ,  20 ,  22 ,  30 and  32 ) along the  Jacobi coordinate.Besides the isotropic term  00 , which shows a well of 47 cm −1 , a close examination reveals that for  = 0, the term with  = 2 outweighs the other anisotropic terms.This term is responsible for rotational transitions with Δ  = 2, which will have consequences on the propensity rules of cross sections and the magnitude of rate coefficients, as shall be further discussed below.and  = 4.75 Å as well as  = 130 • and  = 4.5 Å with depths of −49.6, −51.6 and −48.8 cm −1 , respectively.The positions of these wells correspond to the helium atom approaching between two hydrogen atoms or between an hydrogen atom and the CCN bond.These local minima are separated by barriers of −23.6 and −28.5 cm −1 that are located at  = 55 • ,  = 5.75 Å and  = 107 • ,  = 4.95 Å, respectively.We also illustrate in figure 3 (bottom panel) a two-dimensional cut of the PES along the angular coordinates  and  at = 4.75 Å.

Description of the potential energy surface
It illustrates that the interaction potential between c-C 6 H 5 CN and He is strongly anisotropic in short/medium range of distances.This anisotropy will drive the energy transfer to the benzonitrile molecule during the collision and suggests that rotational (de-)excitation of c-C 6 H 5 CN in collisions with He will be efficient.

Spectroscopy of benzonitrile
Benzonitrile is an asymmetric top molecule whose rotational Hamiltonian can be written as: where , , ,   ,   and    are the rotational constants and the first order centrifugal distortion constants of benzonitrile and   ,   and   are the projection of the angular momentum  along the different inertia principal axes that satisfies the relation : ).The small values of the rotational constants lead to a complex rotational structure and a high density of rotational levels even at low energy.There are for example 13 rotational levels at energies below 1 cm −1 .Due to the permutation symmetry of the hydrogen atoms and their nuclear spin  = 1/2, the rotational levels of C 6 H 5 CN are divided into two groups, ortho-C 6 H 5 CN and para-C 6 H 5 CN.The nuclear spin symmetry implies that para and ortho levels can not interconvert through radiative or inelastic collisional processes.Consequently, the scattering calculations will be performed separately for each nuclear spin species.

Scattering Calculations
Given that benzonitrile has been observed only in the cold ISM, our goal here is to provide rate coefficients for temperatures up to 10 -1 10 0 10 1 0 1 2 3 4 5 6 7 8 9 40 K.This requires the computation of scattering cross sections up to kinetic energies of 200 cm −1 .In order to investigate the impact of possible non-LTE effects on the emission lines of benzonitrile (McGuire et al. 2018;Burkhardt et al. 2021), one requires the rate coefficients for rotational levels up to at least  = 9.The most accurate approach to investigate the collision dynamics is the quantum close-coupling (CC) method (Arthurs & Dalgarno 1960).However, despite the possibility of performing calculations for ℎ− and para-C 6 H 5 CN separately, it proved impossible to perform accurate CC calculations for the required range of energies and rotational transitions due to the small rotational constants of c-C 6 H 5 CN and the high density of levels as well as the strong anisotropy of the PES.Alternative theoretical methods were explored to limit the computational cost.We found that the coupled states (CS) approximation provided a good compromise between accuracy and computational cost, as detailed below.We note here that Cernicharo et al. (2023) (Hutson & Green 1994), in a quantum timeindependent framework.To do so, the radial terms   () of our PES were implemented into the MOLSCAT code.The diabatic log-derivative propagator (Manolopoulos 1986) was used in order to solve the coupled equations.The reduced mass of the colliding system is taken as =3.85294au (isotopes 12 C, 14 N, 1 H and 4 He).Several tests were carried out in order to select the integration boundaries of the propagator,   and   which were fixed at 2.5 and 50 bohr, respectively.The number of integration steps was taken as 100 for  ≤ 30 cm −1 , 90 for 30 <  ≤ 50 cm −1 , 80 for 50 <  ≤ 100 cm −1 and 30 for 100 <  ≤ 200 cm −1 .The convergence with respect to the rotational basis set was also investigated, and preliminary tests showed that  max = 19 for A good description of the behavior of cross sections requires a sweep of the resonant region that takes place at low energies, typically for energies lower than the depth of the PES.In other words, we have used a grid of energy that carefully describes the rotational cross sections above the different rotational thresholds as follows:  = 0.2 cm −1 for 0 <  ≤ 50 cm −1 , 0.5 cm −1 for 50 <  ≤ 100 cm −1 and 2 cm −1 for 100 <  ≤ 200 cm −1 .The maximum value of the total angular momentum  was selected so that the inelastic cross sections were all converged to within 0.05 Å 2 .Finally, since the total angular momentum  is a conserved quantity, the total cross section is a sum of partial wave contributions.
While full CC calculations were not possible, it is still important to assess the accuracy of the (more approximate) CS method.In Table 1 we report converged cross sections obtained with the CC and CS methods for selected transitions, at the total energy of 50 cm −1 .This energy was chosen as it high enough that we are outside the resonance regime (see Fig. 5) but low enough that the CC calculation can be converged with  max = 17.A further illustration of the accuracy of the CS method for the present system is provided in the appendix.We observe that the relative error between CC and CS approximation for the transitions reported in table 1 does not exceed 14% for E tot = 50 cm −1 , while the average of the relative error over all cross sections is found to be 15% and with a CPU time and disk occupancy for CS computations that are much lower than CC ones.Furthermore, the number of channels needed to converge cross sections using the CC method exceed 3550 channels, while for the CS calculations, we need 200 channels for ortho-C 6 H 5 CN and 220 channels for para-C 6 H 5 CN.Therefore, the CS approximation dramatically reduced the computational cost without an important loss of the precision.
We illustrate in figure 5 the state-to-state rotational excitation cross sections for collisions of C 6 H 5 CN with He atoms as a function of the collisional energy for para-C 6 H 5 CN (left panel) and ortho-C 6 H 5 CN (right panel) for Δ  = 1 and Δ  = 2 transitions with Δ  = 0 and Δ  = 1 and 2. One can see that all cross-sections possess the same typical low energy features.A dense structure of shape and Feshbach resonances is observed for  ≤ 50 cm −1 .These features in the cross sections are due to the presence of attractive potential well depth of -97.2 cm −1 that allows He atom to be temporary confined so that the complex is formed in quasi-bound state before the dissociation of the complex.The impact of these resonances on the cross sections decreases as the collision energy increases, and disappears around  = 50 cm −1 .The cross sections display a monotonous decrease at higher energies.One can also see that state-to-state cross sections exhibit a strong even Δ  propensity rule at almost all collision energies.Cross sections associated to Δ  = 2 are larger than those with Δ  = 1, and the largest cross sections are found for the transitions 3 03 − 1 01 , 4 04 − 2 02 and  5 05 −3 03 for ortho-C 6 H 5 CN and for 3 03 −1 01 , 4 04 −2 02 and 5 05 −3 03 for para-C 6 H 5 CN.This behavior can be understood on the basis of the radial terms in equation 2, and in particular the fact that the term  20 that drives Δ  = 2 transitions is larger than the term  10 that drives Δ  = 1 transitions.Such results were already observed for other collisional systems, in particular cyanides colliding with He atoms such as SiH 3 CN (Naouai et al. 2021), HCN (Dumouchel et al. 2010), CCN (Chefai et al. 2018) and CH 3 CN (Ben Khalifa et al. 2022Khalifa et al. , 2023)).

RATE COEFFICIENTS AND APPLICATIONS
From the state-to-state cross sections, we obtain rate coefficients assuming a Maxwell-Boltzmann distribution of kinetic energies: Where =1/  , and   ,  and  denote the Boltzmann constant, the kinetic temperature and the reduced mass of the system, respectively.
The quenching rate coefficients of C 6 H 5 CN by collisions with He as a function of kinetic temperature are illustrated in Fig. 6 for selected Δ  = 1 and Δ  = 2 transitions accompanied by Δ  = 0 and Δ  = 1 and 2 (Panels a and b).These panels show that the collisional rates decrease with increasing temperature for transitions associated to Δ  = 1 while they increase when Δ  = 2.In addition, the magnitude of rate coefficients associated to transitions with Δ  = 2 is larger than those with Δ  = 1.The largest rate coefficients are found for the quadripolar transitions, i.e : 6 16 -4 14 , 5 15 -3 13 and 4 14 -2 12 for para-C 6 H 5 CN and 5 05 -3 03 , 4 04 -2 02 and 3 03 -1 01 for ortho-C 6 H 5 CN.Panels (c) and (d) present the rate coefficients for some selected transition associated to Δ  = Δ  =1 and Δ  = 0 (solid line) and Δ  = 2 (dashed line) as a function of the kinetic temperature.These rate coefficients show that transitions involving Δ  = 0 are more favorable compared to those with Δ  = 2. Finally, we illustrate in panels (e) and (f) , the variation of quenching rate coefficients for transitions with fixed Δ  = 1, Δ  = 0 and Δ  = 1 and 2. We note that for para-c-C 6 H 5 CN, transitions with Δ  = 1 dominate over the entire temperature range those with Δ  = 2, however, for ortho-c-C 6 H 5 CN, the predominance of collisional rates is associated for transitions with Δ  = 1 for  ≤ 20K and Δ  = 2 for  ≥ 20K, leading to propensity rules that depend on the temperature and on the nuclear spin symmetry.While figure 6 only presents the collisional rates for transitions with 4 ≤  ≤ 7, the same behaviour is also observed for other values of  for both ortho and para symmetries of benzonitrile.
To assess the potential impact of collisional excitation on the modelling of the spectra of benzonitrile, we carried out non-local thermodynamic equilibrium (non-LTE) radiative transfer calculations using the RADEX code (Van der Tak et al. 2007).The molecular data for 0 10 20 30 40 50 60 70 80 90 100 2 12 -3 13 3 13 -4 14 4 14 -5 15 2 12 -4 14 3 13 -5 15 4 14 -6 16 Cross sections (Å 2 C 6 H 5 CN are composed of collisional rate coefficients (from 5 to 40 K) scaled by a factor of 1.40 to model collisions with para-H 2 (by accounting only for the mass difference between He to H 2 ) completed by the Einstein coefficients, energy levels, as well as frequency lines.These spectroscopic data were extracted from the Cologne Database for Molecular Spectroscopy (CDMS) portal (Endres et al. 2016).We take into consideration both radiative and collisional processes, while the optical depth impacts are modeled within an escape probability formalism approximation.In the radiative transfer computation, we set the basic parameters as follows: a  CMB = 2.73 K cosmic microwave background as a radiation field, and a line width Δ of 0.4 km.s −1 (McGuire et al. 2018).We vary the molecular hydrogen density   2 between 10 2 and 10 8 cm −3 while the column density of c-C 6 H 5 CN was fixed at 4 × 10 11 cm −2 , a choice that is based on the estimated column density of benzonitrile in the dark molecular cloud TMC-1.Figure 7 illustrates the variation of the excitation temperature as a function of the H 2 density determined in our calculation for two detected transitions, namely the      = 7 07 → 6 06 and 7 25 → 6 24 .The excitation temperature   is similar for both transitions.For H 2 densities up to about 10 2 cm −3 , the   of observed lines is equal to the value of the background radiation field, and increases gradually as the collisional excitation processes becomes more important.For H 2 densities n  2 ≥ 10 6 cm −3 , the excitation temperature tends towards the kinetic temperature, at which point the LTE is achieved and the rotational levels populations no longer depend on the density of the medium and simply obey Boltzmann's law.For these two transitions, one can thus observe that the LTE is only achieved for densities above 10 6 cm −3 .These values are greater than the typical density of many regions of the ISM where benzonitrile has been observed (10 3 ≤ (H 2 ) ≤ 10 5 cm −3 ), which shows that the benzonitrile lines are likely not thermalized and that non-LTE models should be used to analyse emission spectra.

CONCLUSIONS
We constructed the first highly accurate PES for the C 6 H 5 CN-He collisional system.The three-dimensional PES was computed in Jacobi coordinates using the CCSD(T)-F12/aug-cc-pVTZ approach.The interaction between benzonitrile and helium atom is strongly anisotropic and presents a global minimum of 97.2 cm −1 at  = 3.1 Å and  = 78 • and for  = 90 • , while numerous local minima are also observed.
The PES was used to carry out computations of state-to-state inelastic cross-sections for the rotational (de-)excitation of orthoand para-C 6 H 5 CN by collision with helium atoms.The calculations were performed using the quantum scattering dynamics coupled states method, and tests were performed to assess the accuracy of the method.By thermally averaging the cross sections over a Maxwell-Boltzmann distribution of velocities, state-to-state rate coefficients were obtained for the 100 lowest-lying rotational levels of benzonitrile and for temperatures up to 40 K. Propensity rules which favor transitions with Δ =2 were found for both para and ortho symmetries of benzonitrile, but propensities depending on   and   were also observed.
A simple radiative transfer calculation was performed to model the excitation of benzonitrile under typical cold cloud conditions, which showed that benzonitrile is likely not at LTE in these environments.Based on these results, further modelling of observed emission lines is needed to assess the impact on the derived column density.If non-LTE effects are seen, an extension of the present work would be to investigate the collisional excitation by H 2 molecules, which presents additional complexities from a computational viewpoint.
To the best of our knowledge, the present set of state-to-state rate coefficients is the first one obtained by fully quantum methods for an aromatic molecule.Assessing the impact of collisional excitation on benzonitrile spectra could give an indication of the behaviour of other cyclic COMs.This would be an important step given that a large number of detections of complex organic molecules in the ISM have been reported over the past few years and that the collisional excitation properties of these molecules are unknown, leading to the assumption that LTE conditions apply.collisions, rate coefficients obtained with the CS approximation can be useful for temperatures as low as 5-10 K.

Figure 1 .
Figure 1.Jacobi coordinates used to describe the interaction of the c-C 6 H 5 CN-He van der Waals complex.The origin of the reference frame is at the c-C 6 H 5 CN center-of-mass.

Figure 2 .
Figure 2. Dependence on  of the first   (R) components for c-C 6 H 5 CN-He with  ≤ 3.

Fig. 3 Figure 3 .
Fig. 3 displays two-dimensional contour plots of the c-C 6 H 5 CN-He van der Waals complex as a function of the two Jacobi coordinates  and  for  = 0 • (upper left panel) and  = 90 • (upper right panel) and as a function of  and  for  = 4.75 Å (bottom panel).The global minimum of the surface is found to occur at  = 90 • , corresponding to the He atom above the molecular plane.The geometry of the minimum is  = 78 • and  = 3.1 Å , with a well depth of   = −97.2cm −1 .For  = 0 • (rotation in the molecular plane), we observe three wells in the potential, located at  = 26 • and  = 5.4 Å ,  = 79 • The wave functions |  ⟩ of benzonitrile are characterized by three quantum numbers ,  and  and can be expressed as a linear combination of the rotational wave functions of a symmetric top molecule |  ⟩ such as(Townes & Schawlow 2013):  denotes the projection of  along the -axis of the body-fixed reference and  is its projection on the space fixed -axis.The rotational levels of the asymmetric top benzonitrile are labelled by   and   projections of the rotational angular momentum along the axis of symmetry in case of prolate and oblate symmetric tops limits.The relation between   and   is defined by  =   −   .For illustration, we present in Fig.4the energy level diagram corresponding to the c-C 6 H 5 CN rotational levels.This diagram is constructed using the rotational constants as given by Wohlfart et al. (2007), i.e.,  = 0.0516,  = 0.0405,  = 0.1886,   = 1.5205 × 10 −9 ,   = 1.6678 × 10 −8 and    = 3.1291 × 10 −8 (all values are in cm −1
(McGuire & Kouri 1974) of the spatial distribution of benzonitrile based on observation of transitions involving levels up to  = 19.It is unlikely that CS calculations can be used for transitions involving such highly-excited levels.The rotational excitation cross-sections   ′  ′   ′  ←      (  ) of para and ortho C 6 H 5 CN-He were thus computed using the CS approximation(McGuire & Kouri 1974)implemented in the MOLSCAT code

Table 1 .
Comparison between CC and CS cross sections (in Å 2 ) for the excitation of ortho c-C 6 H 5 CN by He for total energies  tot = 50 cm −1 with  max = 17.CN and  max = 20 for para-C 6 H 5 CN for collisional energies   ≤ 100 cm −1 are sufficient to converge cross-sections for the first 50 rotational ortho levels and 50 rotational para levels (i.e. up to level      = 9 81 for ortho-C 6 H 5 CN and 9 90 for para-C 6 H 5 CN), while for energies 100 <   ≤ 200 cm −1  max = 20 for ortho-C 6 H 5 CN and  max = 21 for para-C 6 H 5 CN.