Radiative Association of Atomic and Ionic Carbon

We present calculated cross sections and rate coefficients for the formation of the dicarbon cation ( ) by the radiative association process in collisions of a atom and a ion. Molecular structure calculations for a number of low-lying doublet and quartet states of are used to obtain the potential energy surfaces and transition dipole moments coupling the states of interest, substantially increasing the available molecular data for . Using a quantum-mechanical method, we explore a number of allowed transitions and determine those contributing to the radiative association process. The calculations extend the available data for this process down to the temperature of 100 K, where the rate coefficient is found to be about . We provide analytical fits suitable for incorporation into astrochemical reaction databases.


Abstract
We present calculated cross sections and rate coefficients for the formation of the dicarbon cation ( + C 2 ) by the radiative association process in collisions of a ( ) P C 3 atom and a ( ) ion. Molecular structure calculations for a number of low-lying doublet and quartet states of + C 2 are used to obtain the potential energy surfaces and transition dipole moments coupling the states of interest, substantially increasing the available molecular data for + C 2 . Using a quantum-mechanical method, we explore a number of allowed transitions and determine those contributing to the radiative association process. The calculations extend the available data for this process down to the temperature of 100 K, where the rate coefficient is found to be about´--2 10 cm s 18 3 1 . We provide analytical fits suitable for incorporation into astrochemical reaction databases.

Introduction
The dicarbon cation + C 2 is an important molecule in astrochemistry, as it is one of the species participating in hydrocarbon chemistry in, for example, interstellar clouds (Solomon & Klemperer 1972) and photon-dominated regions (Guzmán et al. 2015). In environments with hydrogen the dication is rapidly consumed, but it is of interest to securely characterize + C 2 formation mechanisms as the subsequent production of larger molecules + C H 2 , + C H 2 2 , ... will depend on the available + C 2 . In the present paper, we consider the formation of the dicarbon cation ( + C 2 ) by the radiative association process in collisions of a carbon atom and a carbon ion, The process in Equation (1) is a mechanism for dicarbon cation formation, which is viable in astrochemical environments because of the applicability of two-body kinetics and because of the abundance of carbon. (Another, more recent, application is to carbon plasma chemistry can be found in Hirooka et al. (2014)). We note-we will discuss the chemistry in more detail below-that in typical astrophysical applications with relatively abundant hydrogen (in atomic and molecular form) the radiative reaction (Equation (1)) will generally be outpaced by the ionic reactions such as (Solomon & Klemperer 1972) +  + + + CH C C H 2 (Federman & Huntress 1989;Guzmán et al. 2015) or +  + + + CH C C H 2 (Chabot et al. 2013;Rampino et al. 2016) and Equation (1) is likely to be a minor process. Nevertheless, there is one previous calculation of Equation (1) that was carried out to a lowest temperature of 300 K, and which is listed in astrochemical reaction databases for applications below this temperature. To remove the uncertainty for astrochemical applications, it is necessary to calculate the rate coefficient at lower temperatures. In the present paper, we compute cross sections and rate coefficients for Equation (1), and provide a new calculation of the process valid down to 100 K. We find that the rate coefficient is larger for < < T 100 300 K than the values assumed in astrochemical databases.
The theory of the formation of diatomic species by radiative association is generally established. A recent review summarizes the theoretical methodologies and lists 73 diatomic species 3 for which calculations have been carried out (Nyman et al. 2015). There is, however, only one previous calculation for + C 2 (Andreazza & Singh 1997), and that calculation does not extend below 300 K. (The Rice-Ramsperger-Kassel-Marcus (RRKM) theory applied to radiative association of + C and C n for = ¼ n 1, ,8 yielded a value of zero for the rate coefficient of process in Equation (1) at 30 K, but the theory was focused on the polyatomic case.) In a molecular description of the collisions that describe Equation (1), there are numerous possible approach channels and transitions leading to + C 2 and the theoretical treatment of Equation (1) becomes quite complicated, as we will show. The rate coefficients for Equation (1) were calculated by Andreazza & Singh (1997) using a semi-classical description of the collisions in which the atom and ion approach in the quartet S -B u 4 state, yielding a rate coefficient of about --3 10 cm s 18 3 1 at 300 K. In this paper, we will show by calculating the cross sections using a quantum-mechanical method with improved molecular data for + C 2 that the most significant channels are those in which the colliding atom and ion approach in molecular doublet states.
We present calculations for a number of molecular states of + C 2 , cross sections and rate coefficients for the process in Equation (1), discuss the present rate coefficients and the earlier determination, and consider in more detail the potential significance of Equation (1).

Molecular Structure
Potential energies correlating to ( ) P C 3 and ( ) + P C o 2 were calculated by Petrongolo et al. (1981), Rosmus et al. (1986), Bruna & Wright (1992), Ballance & McLaughlin (2001), and Shi et al. (2013). There are several experimental studies of band spectra including Maier & Rösslein (1988), Boudjarane et al. (1995), and Tarsitano et al. (2004), and calculations of transition dipole moment (TDM) functions (Rosmus et al. 1986). As far as we know, no new TDM calculations have appeared since Rosmus et al. (1986), yet for the present study we will require a larger set. Therefore, we calculated a number of molecular potential energies and corresponding TDMs substantially increasing the available data for + C 2 . The potential energy curves (PECs) and TDMs are calculated for a set of low-lying doublet and quartet electronic states that enter into the radiative association calculations. We treat the molecular states formed by the approach of ( ) within an MRCI+Q approximation: a state-averaged (SA) complete-active-space-self-consistent-field (SA-CASSCF) approach, followed by multi-reference configuration interaction (MRCI) calculations, together with the Davidson correction (MRCI+Q) (Helgaker et al. 2000), is used. The SA-CASSCF method is used as the reference wave function for the MRCI calculations. The basis sets used in the present work are the augmented correlation consistent polarized core valence quintuplet (aug-cc-pcV5Z (ACV5Z)) Gaussian basis sets, which were used in our recent work on the dicarbon molecule (Babb et al. 2019) and were found to give an excellent representation of the states as the molecule dissociated.
All the PEC and TDM calculations for + C 2 were performed with the quantum chemistry program package MOLPRO 2015.1 (Werner et al. 2015), running on parallel architectures. For molecules with degenerate symmetry, an Abelian subgroup is required to be used in MOLPRO. Thus, for ( ) with ¥ D h symmetry, it will be substituted by D h 2 symmetry with the order of irreducible representations being (A g , B 3g , B 2g , B 1g , B 1u , B 2u , B 3u , A u ). When the symmetry is reduced from ¥ D h to D h 2 (Herzberg 1950), the correlating relationships are s  a g g , To take account of short-range interactions, we employed the nonrelativistic SA-CASSCF/MRCI method available within the MOLPRO (Werner et al. 2012(Werner et al. , 2015 quantum chemistry suite of codes. For the dicarbon cation, molecular orbitals (MOs) are put into the active space, including (3a g , 1b 3u , 1b 2u , 0b 1g , 3b 1u , 1b 2g , 1b 3g , 0a u ), symmetry MOs. The MOs for the MRCI procedure were obtained using the SA-CASCSF method, for doublet and quartet spin symmetries. The averaging process was carried out on the two lowest states of the symmetries: (A g , B 3u , B 1g , B 1u ) and the lowest states of the symmetries; (B 2u , B 3g , B 2g and A u ). This gives an accurate representation of the doublet and quartet states.
There are 24 molecular electronic states formed from ( ) P C 3 and ( ) (Chiu 1973;Shi et al. 2013); however, we do not consider the states, because they are repulsive (Shi et al. 2013) and will not be important for process in Equation (1).
PECs and TDMs as functions of internuclear distance R were calculated over the range   R 1.5 20 Bohr for the 16 states listed in Table 1. In Figure 1, we present the calculated quartet PECs. In Figure 2, we present the calculated doublet PECs along with the S -X g 4 PEC included for reference. For  R 20 Bohr, the PECs were fit to the long-range form where C R 3 3 is the electric charge-atomic-quadrupole interaction potential energy (Gentry & Giese 1977) and ( ) a C d is the Notes. In column 2, the values of the term energies T e are listed for the potential energies fit to Equation (2)   static electric dipole polarizability of carbon (Miller & Kelly 1972;Das & Thakkar 1998) for each state. We utilized the ( ) a C d values calculated by Das & Thakkar (1998) with the finite field method and a coupled cluster CCSD(T) approach for atomic carbon in the , and P f g 2 states, and we used C 3 =1.55 and was fitted to the ab initio potential curves. The resulting T e values measured with respect to the minimum of the S -X g 4 state are given in Table 1. We also list the values of T e from the recent MRCI+Q/CV+DK+56 calculations of Shi et al. (2013). (Comprehensive tabulations of calculations by earlier workers (and some experimental data) were given by Shi et al. 2013.) For T e , there is generally very good agreement (within 300cm 1 ) between our calculations and those of Shi et al. (2013), which also include relativistic (Douglas-Kroll effective Hamiltonian) effects and which were obtained using extrapolation procedures to account for energy dependencies on basis sets. For example, we find  Table 3) and an MRCI +Q/CV+DK+56 value of -12179.74 cm 1 (their Table 5  PEC. In the adiabatic approximation, which we utilize, the P f g 2 and P 2 g 2 PECs do not cross (Herzberg 1950), and the P f g 2 PEC "turns over" as it approaches the P 2 g 2 state with increasing R yielding a second well. The avoided crossings in the present calculations between the P f g 2 and P 2 g 2 states and between the P 2 u 2 and P 1 u 2 states are in good agreement with earlier work (Petrongolo et al. 1981;Ballance & McLaughlin 2001). The accuracy of the present PECs is suitable for the present calculations of the radiative association process.
There are numerous allowed electric dipole transitions within the manifold of electronic states calculated here and listed in Table 1 (Herzberg 1950). The radiative association cross sections arise from spontaneous transitions between the vibrational continuum of the initial electronic state and a bound vibrational state of the final electronic state and depend roughly on the third power of the electronic transition energies and the square of the TDMs. Thus, we will investigate in further detail only the allowed transitions where the electronic transition energies and TDMs are comparatively large. In anticipation of the cross-section calculations presented in the next section, Table 2 gives a list of the transitions investigated here for the radiative association proccess.
Since the dominant configurations change as + C 2 dissociates -as highlighted in the early MRDCI work of Petrongolo et al. (1981), the valence-CI calculations of Ballance & McLaughlin (2001) Shi et al. (2013) declare that "no other theoretical T e is closer to the measurements" than theirs. While our value is closer to the experiment than theirs, without a detailed convergence study, our close agreement with the experiment may be coincidental. They also state "no other theoretical R e result is superior to (theirs) in quality when compared with the measurements." We obtain the same value of = R a 2.55  Rosmus et al. (1986).
In general, the quartet and doublet TDMs are well-behaved at the lower and upper limits of the ranges of R for which they are calculated. However, we note that for the quartet TDMs, 3.1. In the case of the doublet TDMs, rapid variation around R=2.9 in the functions involving the P a u 2 and P 2 u 2 states, Figures 5 and 6, reflects the avoided crossing between the PECs where the dominant configuration of the P a u 2 state changes (Petrongolo et al. 1981;Ballance & McLaughlin 2001). A more detailed analysis of the electronic state configurations and their effect on the TDMs should be carried out for spectroscopic applications, such as for calculations of band oscillator strengths. Such an analysis is unnecessary for our purposes, because, as we will show, the most important factors leading to significant radiative association cross sections are the lack of a barrier in the PEC and a significant TDM at large R. The P f g 2 entrance channel lacks a   The TDMs were fit to the form R transitions which were fit to R 2 at large nuclear separations.

Cross Sections
The radiative association cross sections for each transition listed in Table 2 contributing to the process in Equation (1) were calculated for C 12 nuclei as a function of the collision energy using the quantum-mechanical formalism, which was previously applied to like-atom-ion radiative association for species without nuclear spin; for He and + He (Stancil et al. 1993;Augustovičová et al. 2013) and for O and + O (Babb et al. 1994).
From the 24 molecular electronic states formed from ( ) P C 3 and ( ) + P C o 2 there are 72 possible approach channels considering that due to the absence of nuclear spin only half of the possible lambda doubling levels are populated for L = 1 or 2. Thus the weight factor appearing in the cross section for an entrance collision channel labeled by i is . Because the collisional cross sections will be mainly due to partial waves with  N 1, we approximate J=N and use Hund's case(b) coupling, where N is the rotational quantum number, J is the total angular momentum | | + N S, and S is the total electronic spin, as in our previous work on radiative association of O and O (Babb & Dalgarno 1995). The reduced mass μ is 10937.35 in units of the electron mass. As discussed in Section 2, the PECs and TDM functions utilized were fit to appropriate long-range forms. The cross sections were evaluated using the methods detailed in Babb et al. (2019). For collisions in the S -B u 4 entrance channel only odd partial waves were treated, while for the S + 1 u 4 and Sc g 2 entrance channels only even partial waves were treated.
The calculated cross sections are shown in Figure 8. We do not plot the cross sections for the P 2 u which were found to be insignificant. Note that we also explored the transitions and found that the TDMs are comparable in magnitude to those shown in Figure 4.Therefore because the transition energies are comparable to the states shown in that figure, we expect the cross sections to be insignificant.

Rate Coefficients
The rate coefficients were calculated by averaging the cross sections over a Maxwellian velocity distribution. The dominant entrance channels were found to be the doublet . The rate coefficients for these four transitions and the total rate coefficients are shown in Figure 9 with the rate coefficients for the S -    ) as as functions of temperature (in units of K) for the dominant channels contributing to the radiative association association process (1). The rate coefficients for the four transitions and the sum are plotted in comparison to the calculations of Andreazza & Singh (1997).
The agreement between our total rate coefficients and those from Andreazza & Singh (1997) is good but appears to be accidental. Andreazza & Singh (1997) transition based on a careful analysis of available TDM data from the literature. In particular, they determined that doublet transitions would not contribute significantly compared to their predictions for the strengths of the S -B u 4 -S -X g 4 transition. Andreazza & Singh (1997) considered the P f g 2 -P a u 2 transition but estimated that it was insignificant based on the relatively small TDM value at R=2.4 calculated by Rosmus et al. (1986). As discussed in Section 2, our calculated TDM is in agreement with the estimate given by Rosmus et al. (1986); however, as illustrated in Figure 5 we find that the TDM of the P f g 2 -P a u 2 transition grows linearly with internuclear distance and consequently this is the most significant contribution to the process Equation (1) at the lowest energies because there is no barrier in the P f g 2 approach channel. In contrast, while the P 2 g 2 and S -B u 4 approach channels have barriers and will not contribute at the lowest energies, for sufficient collisional energies the transitions rapidly diminish for collisional energies below 0.023eV and these transitions will be inefficient at temperatures below about 300 K, in marked contrast to the calculations of Andreazza & Singh (1997), which do not show a rapid decrease of the S -B u 4 -S -X g 4 transition rate coefficient below 1000 K. Therefore, any apparent agreement between the total rate coefficient obtained here and the calculations of Andreazza & Singh (1997), certainly for temperatures below 1000 K, must be accidental. Because of the similarity between the P 2 g 2 -P a u 2 and S -B u 4 -S -X g 4 TDMs and because the well depths of the P a u 2 and the S -X g 4 PECs are comparable the earlier calculations of Andreazza & Singh (1997) are in good agreement with the present results, though the identifications of the dominant formation channels for the formation of + C 2 via process (1) differ.
The total calculated rate coefficient ( ) a T is fit to better than 11% by the expression

Discussion
We find that the rate coefficient for process (1) is about --2 10 cm s 18 3 1 at 100 K. It is important to note that the only prior existing calculation (Andreazza & Singh 1997) is valid for temperatures greater than 300 K. As we showed, the calculation of Andreazza & Singh (1997) agrees fortuitously with our calculation over the range < < T 300 10,000 K and they provided the fit In the context of astrochemical models, the fit provided by Andreazza & Singh (1997), which was limited to < < T 300 14,700 K, is implemented in astrochemical databases that are applied to temperatures below 300 K. Moreover, applying the fit function from Andreazza & Singh (1997), for example, (incorrectly) to 100 K yields a rate coefficient that is a factor of two lower with an exponential decrease for < T 100 K. The fit, Equation (4), is listed as applicable for < < T 10 41,000 K in the UMIST RATE12 file, 5 and was used, for example, in the photon-dominated region code comparison study 6 with the same temperature validity listed (Röllig et al. 2007 Based on the trend of the P f g 2 -P a u 2 cross section with decreasing collision energy, see Figure 8, and estimating the rate coefficient using with μ the reduced mass, we can expect that the rate coefficient will be no greater than´--2 10 cm s 18 3 1 at 10 K, though a detailed calculation considering the fine structure of ( ) P C 3 and ( ) + P C o 2 should be performed (the fine structure splitting of ( ) is about -63.4 cm 1 or ∼91 K). The calculated rate coefficients for (1) are small, and the process will certainly be unimportant when hydrogen is present. Nevertheless, we conclude that the present rate coefficient will lead to an enhancement in + C 2 production, generally speaking, in astrochemical models, compared to using the fit (4), which is invalid for < T 300 K.

Conclusions
We calculated the rate coefficients for the formation of + C 2 by radiative association of ( ) P C 3 and ( ) + P C ; o 2 process (1). We substantially added to the available molecular data on + C 2 by calculating a large set of PECs and TDMs. We found that the dominant approach channels contributing to process (1) are the doublet P f g 2 and P 2 g 2 states and that the quartet S -B u 4 approach channel is of lesser importance. The rate coefficient for process (1) should not decrease exponentially as temperature decreases because there is no barrier in the dominant P f g 2 entrance channel. As a consequence, the present rate coefficients will lead to enhanced production of + C 2 in astrophysical models, though the net impact may be insignificant when hydrogen is present.