# Elastic and inelastic cross sections for 12C+CO2 and 13C+CO2 scattering at
superthermal energies

Marko Gacesa,1,2,3
1Department of Physics, Khalifa University, Abu Dhabi, United Arab Emirates
2Center for Catalysis and Separation (CeCaS), Khalifa University, Abu Dhabi,
United Arab Emirates
3Bay Area Environmental Research Institute, Moffett Field, CA, USA email:
<EMAIL_ADDRESS>
###### Abstract

We report elastic and inelastic cross sections for fast superthermal
12C(${}^{3}P$) and 13C(${}^{3}P$) atoms scattering on 12CO2. The cross
sections were computed using quantum-mechanical rotationally close-coupling
formalism with the electronic interaction described by a newly constructed
potential energy surface correlating to the lowest energy asymptote of the
complex. State-to-state cross sections, differential cross sections, and
derived transport properties of interest for energy relaxation are also
reported. The computed elastic cross sections are strongly anisotropic, show
significant energy dependence, and differ by up to 2% between the two isotopes
of carbon.

###### keywords:

scattering – molecular processes – planets and satellites: atmospheres –
planets and satellites: terrestrial planets – planetary nebulae: general

††pubyear: 2023††pagerange: Elastic and inelastic cross sections for 12C+CO2
and 13C+CO2 scattering at superthermal energies–Elastic and inelastic cross
sections for 12C+CO2 and 13C+CO2 scattering at superthermal energies

##  1 Introduction

Carbon is one of the most abundant elements in the Universe and an active
participant in chemistry of dense interstellar clouds in star-forming regions,
protostellar disks, and (exo)planetary atmospheres (Tielens, 2013; Sandford et
al., 2020). There is evidence that chemical reactions of hot C(3P) atom with
interstellar ices, such as CO2 ice found in protoplanetary disks, produce
complex organic molecule (Bergin, 2011; Suhasaria et al., 2017) that could be
brought to surfaces of newly-formed planetary bodies where they might play a
key role in the origin of life.

In atmospheres of terrestrial planets and exoplanets, carbon dioxide acts as a
significant coolant molecule (Sharma & Wintersteiner, 1990) and participates
in photochemical reactions driven by stellar radiation and stellar wind. In
CO2-dominated atmospheres of Mars and Venus, photochemistry of CO and CO2
molecules driven by UV solar photons produces hot carbon atoms in both ground
and excited electronic states, some of which can have translational energies
as high as several eVs, making them strongly superthermal with respect to the
background atmospheric gases (Fox & Bakalian, 2001; Lo et al., 2021). At Mars,
the photochemical processes appear to be a major driver of non-thermal
atmospheric escape of atomic species to space, estimated to be responsible for
the loss of hundreds of millibars of CO2 in the present epoch (Jakosky et al.,
2018; Jakosky & Treiman, 2023). Photochemical escape of atomic oxygen, where
the production of energetic O atoms largely occurs through dissociative
recombination of O${}_{2}^{+}$ with electrons, has long been regarded as the
dominant such process (Fox & Hać, 2009; Lillis et al., 2017; Fox & Hać, 2018)
and studied extensively in the context of climate evolution on Mars and
recently confirmed by NASA’s Mars Atmosphere and Volatile EvolutioN (MAVEN)
mission (Deighan et al., 2015; Lillis et al., 2017; Leblanc et al., 2017).
Similarly, photochemical escape of atomic carbon is regarded as a possible
explanation for a fraction of the observed deficit of carbon-bearing minerals
at Mars (Fox & Bakalian, 2001; Tian et al., 2009; Gröller et al., 2014; Cui et
al., 2019; Lo et al., 2021; Yoshida et al., 2023), especially since the
isotopes of Xe do not offer too much insight into the ancient mechanisms of
carbon escape (Zahnle, 1993; Cassata et al., 2022). Measurements of carbon
isotope ratios, such as 13C/12C, carry additional information about the
evolution of planetary atmospheres (Krasnopolsky et al., 2007; Hu et al.,
2015; Aoki et al., 2023) and other carbon-rich cosmic regions (Salak et al.,
2019).

In these environments, the collision cross sections, excitation rates, and
derived transport properties of superthermal carbon atoms and their isotopes
against a thermal CO2 background play an important role. These quantities are
not available in the literature for the kinetic energies of several eV. The
closest related studies focused on chemical kinetics and reaction rates of the
C(1D)+CO2 reaction at temperatures between 50 K and 300 K (Nuñez-Reyes &
Hickson, 2018) and found them to be in good agreement with previously reported
measurements (Husain & Kirsch, 1971). Consequently, for purposes of modeling
astrophysical systems and planetary atmospheres, 12,13C(3P)+CO2 cross sections
are commonly estimated by mass-scaling elastic and inelastic cross sections
for better known pairs, such as O(${}^{3}P$)+CO2 (Gacesa et al., 2020), O+H2
(Gacesa & Kharchenko, 2014), O+N2 (Balakrishnan et al., 1998), or O+O
(Kharchenko et al., 2000). A possible mass-scaling approach based on a number
of atom-molecule cross sections known at eV energy range and capable of
estimating differential cross sections has been proposed by Lewkow &
Kharchenko (2014) and adopted in some studies. Nevertheless, such approaches
generally cannot reliably estimate the cross sections and derived observables,
nor can they reproduce reliable inelastic or state-to-state cross sections for
internal excitations of rotational and vibrational modes of carbon dioxide
molecule, resulting in significant uncertainties in atmospheric escape models
(Kharchenko et al., 2000; Bovino et al., 2011; Fox & Hać, 2014, 2018; Lee et
al., 2020).

In this article, we present a fully quantum-mechanical study of C(${}^{3}P$) +
CO2 scattering on a newly constructed potential energy surface correlating to
the electronic ground state of the complex for two isotopes of carbon, 12C and
13C. The cross sections and derived quantities were computed for the collision
energies between 0.03 eV and 5 eV. The anticipated areas of application of the
results are in modeling of astrophysical environments where non-thermal C
atoms are produced, such as planetary and cometary atmospheres, dense photo-
dissociating regions, and protoplanetary disks (Kao et al., 2019).

##  2 Methods

###  2.1 Electronic interactions of C(${}^{3}P$)-CO2 complex

Figure 1: Coordinate system used to parameterize the potential energy surfaces
of the C+CO2 complex. Planar symmetry is assumed. The bending angle
$\alpha=180^{\circ}$ and $r_{\mathrm{CO}}=1.167$ $\AA{}$ at equilibrium
distance for a non-vibrating CO${}_{2}(00^{0}0)$ molecule.

In order to compute quantum-mechanical cross sections for C(${}^{3}P$) + CO2
scattering, we follow the approach used in our earlier study of O(${}^{3}P$) +
CO2, a complex with a similar molecular symmetry (Gacesa et al., 2020). We
first construct the lowest-energy electronic potential energy surface (PESs)
correlating to the lowest asymptote of the C$(^{3}P)-\mathrm{OCO}$ complex. It
is worth mentioning that we are only interested in the electronic interaction
of the complex and not in the ground state configuration of C2O2 molecule:
ethylenedione, C2O2, is a rather curious molecule that has eluded attempts at
synthesis for more than a century (Staudinger & Anthes, 1913; Krasnokutski et
al., 2019). Regardless of being an oxocarbon with a closed-shell Kekulé
structure (O=C=C=O), its ground state is a triplet, ${}^{3}\Sigma_{g}^{-}$
(Schröder et al., 1998; Bao et al., 2012; Davis & Sajeev, 2017), which makes
it a transient long-lived diradical whose electronic configuration is similar
to O2 molecule.

We further assume that the target CO2 molecule remains in its ground
vibrational state, CO${}_{2}(00^{0}0)$=CO${}_{2}(v_{s}v^{l}_{b}v_{a})$, where
$v_{s}=0$, $v^{l}_{b}=0$ and $v_{a}=0$ are symmetric, bending, and asymmetric
vibrational quantum numbers, respectively. Thus, in our model,
CO${}_{2}(00^{0}0)$ molecule is linear and symmetric, with both C-O
internuclear distances fixed at its equilibrium value, $r_{\mathrm{CO}}$ =
1.162 Å (Johnson, 2020). The rigid rotor assumption preserves the rotational
symmetry of the scattering complex with respect to the O-C-O internuclear axis
and allows us to use a two-dimensional (2D) potential energy surface to model
the electrostatic interactions. An intuitive parametrization of the PES for
purposes of scattering is given by the internuclear distance $R$, defined as
the distance between the impacting C atom and the C atom of the CO2 molecule,
and the angle $\beta$ between the CO2 internuclear axis and the vector $R$,
defined such that $0^{\circ}\leq\beta\leq 90^{\circ}$ (Fig. 1). Finally, the
bending angle $\alpha$ is set to $\alpha=180^{\circ}$. Thus, the symmetry
group of the system is at least $C_{S}$ and depends on the angle of approach
$\beta$. The special cases when the symmetry of the system is higher are the
T-approach ($\beta=0^{\circ}$, symmetry group C2v), and the collinear geometry
($\beta=90^{\circ}$, symmetry group C∞v).

We computed the electronic potential energies of the complex at the single-
reference locally-correlated coupled-cluster singles and doubles with
perturbative triples (LCCSD(T)) level of theory on restricted open-shell local
orbitals (ROHF) with the frozen core approximation (Rolik & Kállay, 2011;
Rolik et al., 2013; Nagy et al., 2018) on Dunning’s aug-cc-pV5Z correlation-
consistent basis sets (Dunning, 1989; Kendall et al., 1992). We settled for
this particular choice of the method to obtain a satisfactory computational
accuracy and numerical stability while keeping the computational costs
manageable. To benchmark the accuracy of the method, we repeated the
calculation using a partially spin-restricted explicitly-correlated theory
with MP2-F12 approximation (RCCSD(T)-F12) (Knizia et al., 2009) with the cc-
pCVTZ-F12 basis set (Hill et al., 2010). All calculations were carried out
using the MRCC suite of ab initio and density functional quantum chemistry
programs (Kállay et al., 2016, 2020), configured to use the recent
implementation of the local correlation algorithms (Nagy et al., 2016, 2018).

Figure 2: Ab initio potential energy points computed using LCCSD(T)/aug-cc-
pV5Z and RCCSD(T)-F12/cc-pVTZ-F12 methods for three different geometries.
Corresponding cuts through the constructed PES are given (solid curve).

We find the two methods to agree well for the geometries for which the
convergence was achieved. In Fig. 2, we compare calculated ab initio points
for three representative geometries, given by $\beta=90^{\circ}$,
$\beta=40^{\circ}$, and $\beta=10^{\circ}$. The convergence of the
RCCSD(T)-F12/cc-pCVTZ-F12 calculation was not achieved for all values of $R$
for the angles $\beta=50^{\mathrm{o}}$ through $\beta=80^{\mathrm{o}}$, while
for the collinear approach ($\beta=0^{\circ}$) the convergence of
LCCSD(T)/aug-cc-pV5Z level of theory was generally achieved if $C_{\infty v}$
symmetry was explicitly given, unlike for $\beta=90^{\mathrm{o}}$ where either
$C_{2v}$ or $C_{S}$ symmetry converged.

We constructed the potential energy surface by varying the distance $R$ and
angle $\beta$ between the approaching C atom and the center-of-mass of the CO2
molecule while keeping the CO2 molecule linear and in equilibrium. A non-
equidistant grid in $R$, with the highest ab initio point density around the
global surface minimum, and an equidistant grid in angle $\beta=0^{\circ}\dots
90^{\circ}$ in steps of 10∘, were used. A total of 278 points were calculated
per method. When constructing the PESs, we used the RCCSD(T)-F12/cc-pCVTZ-F12
points, except for the angles $\beta=50^{\circ}\dots 80^{\circ}$, for which we
used the LCCSD(T)/aug-cc-pV5Z points in order to assure the convergence for
all geometries.

Figure 3: A contour plot of the constructed C(${}^{3}P$)-CO2 potential energy
surface. The collinear approach ($\beta=0^{\circ}$) and the T-approach
($\beta=90^{\circ}$) correspond to the top and bottom of the $y$-axis,
respectively.

The final potential energy surface was constructed by fitting the ab initio
points to a functional form introduced by Bukowski et al. (1999) and further
adapted by Sun et al. (2012) (Fig. 3). We note that the counterpoise technique
by (Boys & Bernardi, 1970), commonly used to compensate for the basis set
superposition errors, did not result in significant modification of the final
surface. All calculated interaction energies and fitting parameters are
available on request.

###  2.2 Collisional Dynamics

We computed state-to-state quantum mechanical elastic and inelastic cross
sections for two isotopes of impacting carbon atoms: 12C(${}^{3}P$) +
CO${}_{2}(j)\rightarrow$ 12C(${}^{3}P$) + CO${}_{2}(j^{\prime})$, and
13C(${}^{3}P$) + CO${}_{2}(j)\rightarrow$ 13C(${}^{3}P$) +
CO${}_{2}(j^{\prime})$, where the initial and final rotational quantum numbers
of the CO2 molecule are labeled as $j$ and $j^{\prime}$, respectively.
Throughout the text, the isotope 12C will be assumed if the atomic mass
superscript is omitted in the notation. The scattering process was modeled
using time-independent coupled-channel formalism with the rigid rotor
approximation applied to CO${}_{2}(j)$ molecule according to Arthurs &
Dalgarno (1960) theory. The calculation was carried out using MOLSCAT code
(Hutson & Green, 1994) with the atom-rigid rotor interaction described using
the analytic representation of our newly constructed PES.

For each of the two isotopes of the impacting C atom, namely 12C and 13C, the
scattering cross sections were calculated for a total of 41 energy points
between $E_{\mathrm{col}}=240$ cm-1 [$2.976\times 10^{-2}$ eV] and
$E_{\mathrm{col}}=40,240$ cm-1 [4.9891 eV] in steps of 103 cm-1 [0.124 eV]. We
confirmed that the numerical convergence was achieved at all energy points by
performing convergence tests with respect to all relevant parameters,
including the number of partial waves, basis size given by the largest value
of the rotational quantum number $j_{\mathrm{max}}$, and numerical integration
parameters(Hutson & Green, 1994).

As in case of O(${}^{3}P$)+CO2 scattering (Gacesa et al., 2020), the elastic
cross sections were found to converge to within 5% for a basis set cutoff in
the rotational quantum number set as low as $j_{\mathrm{max}}=30$. A
significantly larger cutoff value was required to ensure the convergence of
inelastic cross sections. To keep the computation time and memory requirements
realistic, we used the centrifugal sudden (CS8) approximation (McGuire &
Kouri, 1974), where the dipole transitions between the rotational states whose
difference is larger than $\Delta j=8$ were not computed. A satisfactory
(relative uncertainties less than 5%) convergence of the inelastic cross
sections was achieved for $j_{\mathrm{max}}=90$ and above. The convergence
with respect to the total rotational quantum number of the complex,
$J_{\mathrm{max}}$, was achieved for $J_{\mathrm{max}}=200-1300$, with more
partial waves needed at higher collision energies. In all cases, numerical
integration of the coupled-channel system was carried out using modified log-
derivative method (Manolopoulos, 1986) replaced with the hybrid modified log-
derivative/Airy (LDA) propagator (Alexander & Manolopoulos, 1987) at large
separation projectile-target separations, where the Airy propagator was set up
to automatically detect integration region boundaries. In such approach, using
the faster LDA propagator did not result in an observable loss of numerical
accuracy. The parameters used to obtain the reported results are given in
Table 1.

##  3 Results

###  3.1 Elastic and inelastic cross sections

Figure 4: Elastic cross sections $\sigma_{j=0}^{\mathrm{el}}$ for 12C(${}^{3}P$) + CO2($j$=0) and 13C(${}^{3}P$) + CO2($j$=0) scattering as a function of the collision energy $E$. The elastic cross section of Lewkow & Kharchenko (2014) (LK2014) and mass-scaled from O(${}^{3}P$)+CO2 (Gacesa et al., 2020) are given for 12C(${}^{3}P$) + CO2($j$=0). Inset: The ratio of the elastic cross sections (this work) for the two isotopes, $r_{13}=\sigma_{j=0}^{\mathrm{el}}[^{12}\mathrm{C}]/\sigma_{j=0}^{\mathrm{el}}[^{13}\mathrm{C}]$.  Table 1: Numerical parameters used in MOLSCAT calculation. Parameter | Value
---|---
$j_{\mathrm{max}}$ | 90
$J_{\mathrm{max}}$ | 200-1300
JZCSMX | 8
RMIN (Å) | 1.7a
RMAX (Å) | 15a
RVFAC | 1.3
STEPS | 12
aAutomatic scaling algorithms used.
Figure 5: Total cross sections $\sigma_{j=0}^{\mathrm{tot}}$ for 12C(${}^{3}P$) + CO2($j$=0) and 13C(${}^{3}P$) + CO2($j$=0) scattering as a function of the collision energy $E$. Inset: The ratio of the total cross sections for the two isotopes, $r_{13}=\sigma_{j=0}^{\mathrm{tot}}[^{12}\mathrm{C}]/\sigma_{j=0}^{\mathrm{tot}}[^{13}\mathrm{C}]$. Table 2: Integral cross sections for 12,13C(${}^{3}P$) + CO2($j$=0) scattering (units of $10^{-16}$ cm2) for selected collision energies. |  12C(${}^{3}P$) + CO2 |  13C(${}^{3}P$) + CO2
---|---|---
$E$ (eV) | elastic | total | elastic | total
0.03 | 401.33 | 535.78 | 435.62 | 569.01
0.15 | 284.40 | 398.40 | 300.37 | 415.75
0.28 | 257.28 | 358.62 | 268.59 | 371.50
0.40 | 242.28 | 336.67 | 239.05 | 334.65
0.53 | 234.76 | 323.37 | 239.86 | 329.70
0.65 | 212.62 | 296.95 | 219.20 | 304.50
0.77 | 201.44 | 282.30 | 203.77 | 286.37
0.90 | 198.72 | 277.67 | 198.06 | 278.92
1.02 | 199.60 | 276.73 | 200.13 | 276.92
1.27 | 199.91 | 273.74 | 200.45 | 275.23
1.52 | 194.63 | 265.50 | 197.10 | 269.02
1.77 | 180.66 | 253.76 | 189.51 | 258.98
2.01 | 175.50 | 241.56 | 180.14 | 247.34
2.26 | 166.30 | 230.54 | 171.04 | 236.11
2.51 | 159.16 | 221.50 | 163.03 | 226.50
2.76 | 153.76 | 214.55 | 157.04 | 218.68
3.01 | 150.20 | 209.35 | 152.13 | 212.34
3.25 | 147.56 | 205.28 | 148.48 | 207.32
3.50 | 146.14 | 202.68 | 146.12 | 203.66
3.75 | 145.41 | 200.85 | 144.50 | 200.91
4.00 | 146.85 | 199.30 | 143.65 | 198.68
4.25 | 146.01 | 198.42 | 142.32 | 197.28
4.49 | 146.25 | 198.11 | 144.34 | 195.57
4.74 | 147.39 | 197.14 | 142.73 | 194.75
4.87 | 146.31 | 196.75 | 142.73 | 194.36

For the title processes, following the approach described in the previous
section, we calculated the state-to-state cross sections
$\sigma_{j,j^{\prime}}$ as a function of the collision energy $E$ given in the
center-of-mass frame for all combinations of initial and final rotational
quantum numbers $j$ and $j^{\prime}$. For each $j$ and the collision energy
$E$, the elastic scattering cross section is simply
$\sigma_{j}^{\mathrm{el}}(E)=\sigma_{j,j}(E)$ and the total scattering cross
section is given by the sum over all rotational states of the CO2 molecule
after the collision

$\sigma_{j}^{\mathrm{tot}}(E)=\sum_{j^{\prime}}\sigma_{j,j^{\prime}}(E),$ (1)

where $j^{\prime}=0\ldots j_{\mathrm{max}}$.

In Figs. 4 and 5 the elastic and total scattering cross sections for the
initially non-rotating CO2 molecules, $\sigma_{j=0}^{\mathrm{el}}$ and
$\sigma_{j}^{\mathrm{tot}}$, respectively, are shown as a function of the
collision energy $E=0.03$ eV to 5 eV for the two isotopes of impacting carbon
atoms as well as tabulated for selected energies in Table 2. The computed
elastic cross sections show significant differences from the velocity-
dependent cross sections available in the literature. When compared to the
elastic cross sections of Lewkow & Kharchenko (2014) (LK2014), the reported
cross sections exhibit stronger dependence on the collision energy. The
difference is more pronounced for energies below 1.5 eV, where our cross
sections are about 20% (for $E$=1.5 eV) to more than 100% (for $E$=0.07 eV)
larger than LK2014 values. At higher energies, at $E>3$ eV, our cross sections
converge to a constant value that is about 10% smaller than LK2014 cross
sections. Moreover, within the considered collision energy range, the reported
elastic cross sections are about 80% greater than predicted by simple mass-
scaling from the O(${}^{3}P$)+CO2 (Lo et al., 2021; Gacesa et al., 2020).

To simplify the comparison between the two isotopes of carbon atoms, the
energy-dependent ratios of their elastic and total cross sections,
$r_{13}^{\mathrm{el,tot}}=\sigma_{j=0}^{\mathrm{el,tot}}[^{12}\mathrm{C}]/\sigma_{j=0}^{\mathrm{el,tot}}[^{13}\mathrm{C}]$,
are given in Figs. 4 and 5 (inset), respectively. They exhibit a very similar
dependence on the collision energy, indicating that the elastic scattering
plays a dominant role. Within three energy ranges, namely 0.03 eV < $E$ < 0.35
eV, 0.5 eV < $E$ < 0.8 eV, and 1.5 eV < $E$ < 3 eV, cross sections for the
impacting 12C are 2% or more smaller than for 13C isotope. In contrast, for
$E>3.6$ eV, their ratio changes in favor of 12C isotope.

Figure 6: Partial cross sections $\sigma_{j=0,j^{\prime}}$ for 12C(${}^{3}P$)
+ CO${}_{2}(j=0)\rightarrow$ 12C(${}^{3}P$) + CO${}_{2}(j^{\prime})$
scattering given for selected collision energies as a function of rotational
quantum number $j^{\prime}$.

Inelastic cross sections $\sigma_{j=0,j^{\prime}}(E)$ for 12C(${}^{3}P$) +
CO${}_{2}(j=0)\rightarrow$ 12C(${}^{3}P$) + CO${}_{2}(j^{\prime})$ scattering
are shown in dependence of rotational quantum number $j^{\prime}$ in Fig. 6
for selected collision energies $E$. This scenario corresponds to
translationally hot carbon atoms colliding with thermal CO2 gas. Here, at low
collision energies, $E<0.3$ eV, the inelastic cross sections are larger by up
to an order of magnitude for low values of $j^{\prime}$, indicating that
rotational excitations are more efficient than at higher collision energies,
in agreement with general predictions from the scattering theory.

Table 3: State-to-state cross sections for 12C(${}^{3}P$) + CO2($j$=0) scattering (units of $10^{-16}$ cm2) for selected collision energies. | Center-of-mass collision energy $E$ (eV)
---|---
$j^{\prime}$ | 0.03 | 0.28 | 0.53 | 0.77 | 1.02 | 1.5 | 2.26 | 3.5 | 4.5
0 | 401.324 | 256.775 | 234.487 | 200.453 | 199.154 | 193.849 | 165.656 | 145.637 | 145.467
2 | 68.3663 | 34.6633 | 29.5356 | 27.1094 | 26.6361 | 26.047 | 24.0761 | 20.0731 | 17.3975
4 | 26.4247 | 14.491 | 12.6783 | 12.7031 | 12.0122 | 10.1052 | 7.91625 | 6.00399 | 5.18549
6 | 10.9098 | 8.88699 | 8.95512 | 7.6208 | 6.03346 | 4.05253 | 2.82757 | 2.23402 | 2.23083
8 | 6.52679 | 7.01532 | 5.38974 | 3.78105 | 3.04357 | 2.44427 | 1.96517 | 1.60265 | 1.55226
10 | 5.58287 | 5.3432 | 3.29722 | 2.21903 | 1.79568 | 1.59195 | 1.65628 | 1.97337 | 1.85149
12 | 3.12262 | 3.91948 | 2.33609 | 1.92948 | 1.66758 | 1.26329 | 1.05273 | 1.24388 | 1.25211
14 | 2.75617 | 2.9195 | 1.64655 | 1.4618 | 1.45524 | 1.38627 | 1.23449 | 1.11584 | 1.01313
16 | 3.47292 | 2.2306 | 1.57411 | 1.24859 | 1.08966 | 1.08873 | 1.25008 | 1.17158 | 1.10766
18 | 2.60125 | 1.82583 | 1.51856 | 1.33933 | 1.12798 | 0.894677 | 0.970187 | 0.941696 | 0.944952
20 | 2.21932 | 1.67722 | 1.22527 | 1.18633 | 1.13796 | 1.01698 | 0.982231 | 0.765726 | 0.763308
22 | 1.82226 | 1.65229 | 1.18317 | 0.95754 | 0.925198 | 0.947543 | 0.965936 | 0.836626 | 0.776433
24 | 0.64374 | 1.45597 | 1.26635 | 0.99916 | 0.869824 | 0.885191 | 0.871104 | 0.710202 | 0.775156
26 |  | 1.22009 | 1.10095 | 1.0165 | 0.94086 | 0.806275 | 0.756057 | 0.714643 | 0.688039
28 |  | 1.21936 | 0.970015 | 0.880804 | 0.858167 | 0.78822 | 0.733544 | 0.631119 | 0.723374
30 |  | 1.23765 | 0.989488 | 0.844094 | 0.772114 | 0.677686 | 0.65636 | 0.679467 | 0.556519
32 |  | 1.09768 | 0.99898 | 0.875506 | 0.775674 | 0.712077 | 0.609136 | 0.646915 | 0.763255
34 |  | 0.973927 | 0.905967 | 0.820596 | 0.741321 | 0.631615 | 0.566797 | 0.61629 | 0.652802
36 |  | 0.921907 | 0.897234 | 0.770033 | 0.712032 | 0.59541 | 0.56636 | 0.768978 | 0.707538
38 |  | 0.893776 | 0.870997 | 0.750954 | 0.657156 | 0.616715 | 0.621074 | 0.639651 | 0.480125
40 |  | 0.835861 | 0.827643 | 0.732733 | 0.642326 | 0.558326 | 0.617224 | 0.564043 | 0.650754
42 |  | 0.796883 | 0.779928 | 0.701964 | 0.613693 | 0.597033 | 0.554656 | 0.495692 | 0.572498
44 |  | 0.764573 | 0.747445 | 0.665384 | 0.634108 | 0.587729 | 0.582373 | 0.554466 | 0.526183
46 |  | 0.693603 | 0.712067 | 0.681205 | 0.590588 | 0.564234 | 0.51305 | 0.576965 | 0.671481
48 |  | 0.630077 | 0.654706 | 0.62518 | 0.592153 | 0.546247 | 0.555946 | 0.655523 | 0.737486
50 |  | 0.594434 | 0.644639 | 0.634405 | 0.576241 | 0.575339 | 0.627566 | 0.736257 | 0.747189
52 |  | 0.582099 | 0.588582 | 0.592116 | 0.648283 | 0.5846 | 0.658818 | 0.66313 | 0.640635
54 |  | 0.536851 | 0.522942 | 0.587642 | 0.538116 | 0.619776 | 0.513755 | 0.669838 | 0.523245
56 |  | 0.460184 | 0.499495 | 0.547018 | 0.684895 | 0.663933 | 0.72041 | 0.736676 | 0.586429
58 |  | 0.427049 | 0.501287 | 0.537004 | 0.507546 | 0.760488 | 0.760985 | 0.578092 | 0.469333
60 |  | 0.40175 | 0.472843 | 0.57758 | 0.757063 | 0.53975 | 0.709113 | 0.496145 | 0.459552
62 |  | 0.391413 | 0.463544 | 0.503444 | 0.440138 | 0.854023 | 0.545465 | 0.413908 | 0.456024
64 |  | 0.369847 | 0.434698 | 0.518835 | 0.798318 | 0.412657 | 0.518797 | 0.369499 | 0.358804
66 |  | 0.274164 | 0.415146 | 0.372572 | 0.396237 | 0.943628 | 0.507418 | 0.420325 | 0.48291
68 |  | 0.171155 | 0.439141 | 0.545257 | 0.704827 | 0.447355 | 0.55177 | 0.558242 | 0.487159
70 |  | 0.0925556 | 0.37876 | 0.38209 | 0.36033 | 0.634371 | 0.383843 | 0.374619 | 0.512453
72 |  | 0.0476278 | 0.346935 | 0.494899 | 0.589563 | 0.400621 | 0.434618 | 0.323787 | 0.518411
74 |  | 0.0167521 | 0.310003 | 0.340696 | 0.328313 | 0.480581 | 0.440139 | 0.514467 | 0.468069
76 |  |  | 0.297422 | 0.479285 | 0.56593 | 0.488623 | 0.319565 | 0.424642 | 0.378858
78 |  |  | 0.327916 | 0.30213 | 0.39251 | 0.518561 | 0.553353 | 0.585903 | 0.377246
80 |  |  | 0.304192 | 0.524157 | 0.566743 | 0.442019 | 0.336872 | 0.453667 | 0.342414
82 |  |  | 0.285694 | 0.41838 | 0.420498 | 0.499584 | 0.632724 | 0.299095 | 0.354913
84 |  |  | 0.170565 | 0.326018 | 0.465344 | 0.442403 | 0.399575 | 0.375898 | 0.357701
86 |  |  | 0.109612 | 0.484479 | 0.545315 | 0.576273 | 0.62991 | 0.366574 | 0.314384
88 |  |  | 0.0393414 | 0.165967 | 0.205915 | 0.269364 | 0.29006 | 0.332944 | 0.310817

At low collision energies, $E<0.5$ eV, the basis set used is sufficiently
large to fully capture all rotational excitations (the highest excited
rotational quantum number $j^{\prime}$ is smaller than $j_{\mathrm{max}}=90$).
At higher collision energies, the highest rotational excitations are not
captured but the numerical solver converged in all cases due to the fact that
the contribution from $j^{\prime}>j_{\mathrm{max}}$ is not significant. Based
on the procedure described by Gacesa et al. (2020), the uncertainties in the
cross sections increase with the collision energy, from about 1-2% for
$20<j^{\prime}<40$ at $0.5<E<1$ eV to about 10% for $j^{\prime}>80$ at $E>4$
eV. A complete list of state-to-state cross sections for selected collision
energies of interest for planetary aeronomy is given in Table 3. The complete
datasets are available on Zenodo archive (Gacesa, 2023).

###  3.2 Differential cross sections

Figure 7: Differential cross sections for a low collision energy ($E$=0.0297
eV) in dependence of the scattering angle. Elastic,
$Q_{j=0,j^{\prime}=0}(\theta,E)$ (thin curves), and total,
$Q^{\mathrm{tot}}_{j=0}(\theta,E)$ (thick curves), DCSs are shown for
12,13C+CO${}_{2}(j=0)$ scattering. Inset: Zoom in on the large scattering
angles characteristic of backscattering. Figure 8: As above, for the collision
energy $E$=2 eV.

We constructed state-to-state differential cross sections (DCSs),
$Q_{j,j^{\prime}}(\theta,E)={\partial\sigma_{j,j^{\prime}}(\theta,E)}/{\partial\Omega}$,
for all considered collision energies using numerically evaluated S-matrices.
Here, $d\Omega=\sin\theta d\theta d\phi$ is the solid angle element and
$\theta$ is the scattering angle, defined as the relative angle between an
incoming and outgoing trajectory of the impacting C(${}^{3}P$) atom. In this
notation, $\theta=0^{\circ}$ corresponds to forward scattering without
deflection, $\theta=90^{\circ}$ to sideways scattering, and
$\theta=180^{\circ}$ to complete backscattering. The matrices were processed
using the utility code dcs_save.f included with the MOLSCAT v14 (Hutson &
Green, 1994) release. For each initial state $j$, we additionally constructed
the total differential cross section as a sum over all open channels populated
in a collision,
$Q^{\mathrm{tot}}_{j}(\theta,E)=\sum_{j^{\prime}}Q_{j,j^{\prime}}(\theta,E)$.

Sample DCSs for two collision energies, 0.0297 eV and 2.5 eV, representative
of low- and high-energy scattering, are illustrated in Figs. 7 and 8 for both
carbon isotopes considered, respectively, and CO2 in the initial state $j=0$.
The DCSs are strongly anisotropic, with the forward-scattering peak
($\theta<2^{\circ}$) contributing more than 90% of the integral cross section,
similar to other cases of atom-atom and atom-molecule scattering at eV
energies (Balakrishnan et al., 1998; Kharchenko et al., 2000; Gacesa et al.,
2012). For backscattering ($\theta=180^{\circ}$), the cross sections are
enhanced by a factor of 3 to 6, as compared to the large angle scattering,
with almost the entire contribution coming from inelastic excitations (Figs. 7
and 8, inset). Moreover, the total DCSs for backscattering of
12C+CO${}_{2}(j=0)$ are larger by nearly a factor of two than for
13C+CO${}_{2}(j=0)$. A similar enhancement is present for other initial
rotational states $j$ of CO2, suggesting that different transport properties
for the two isotopes may be present for geometries where the backscattering is
important. The total DCSs are nearly identical to the elastic ones for small
scattering angles ($\theta<1^{\circ}$). The elastic DCSs rapidly decrease with
the scattering angle and for $\theta>5^{\circ}$ become on average about 2
orders of magnitude smaller than the total DCSs for scattering energies larger
than about 0.5 eV (Fig. 8). The difference becomes smaller at low energies
(Fig. 7). Except for the forward- and backscattering, the differences in DCSs
between the isotopes are more subtle and change with the scattering energy. A
common trend present for higher scattering energies is that for
$\theta>90^{\circ}$ the elastic DCS for the 13C scattering decreases with
respect to the 12C scattering (see Fig. 8), suggesting that the heavier
isotope is preferentially scattered at smaller angles. The complete datasets
containing state-to-state DCSs are available online (Gacesa, 2023).

###  3.3 Momentum transfer cross sections

Momentum transfer cross sections (MTCS; also called momentum transport or
diffusion cross sections) are effective quantities suitable for transport
studies, such as energy exchange, diffusion, and transport of a particle in a
background gas at non-thermal conditions encountered in astrophysical
environments and planetary atmospheres (Dalgarno, 1962; Hirschfelder et al.,
1964; Kharchenko et al., 1997).

For the title scattering process, C+CO${}_{2}(j)\rightarrow$
C+CO${}_{2}(j^{\prime})$, the state-to-state momentum transfer cross section
(MTCS) for inelastic collisions is given by (Parker & Pack, 1978)

$\sigma^{\mathrm{mt}}_{j,j^{\prime}}(E)=2\pi\int_{0}^{\pi}\mathrm{d}\theta\sin\theta\left(1-\sqrt{1-\gamma_{j^{\prime}}}\cos\theta\right)Q_{j,j^{\prime}}(\theta,E)\;,$
(2)

where $\gamma_{j^{\prime}}=\epsilon_{j^{\prime}}/\epsilon$, is the ratio of
$\epsilon_{j^{\prime}}$ and $\epsilon$, the translational kinetic energies in
the center-of-mass frame after and before the collision, respectively. We
define the total MTCS for the initial state $j$ as

$\sigma^{\mathrm{mt,tot}}_{j}(E)=\sum_{j^{\prime}}\sigma^{\mathrm{mt}}_{j,j^{\prime}}(E).$
(3)

Figure 9: Total ($\sigma_{j=0}^{\mathrm{mt,tot}}$) and elastic
($\sigma^{\mathrm{mt}}_{j=0,j^{\prime}=0}$) momentum transfer cross sections
for 12,13C + CO2($j=0$) as a function of the collision energy for the two
considered isotopes of carbon. The elastic cross sections are multiplied by a
factor of ten for visual comparison.

Using the above expressions, we calculated the total and elastic MTCSs,
$\sigma^{\mathrm{mt,tot}}_{j}$ and
$\sigma^{\mathrm{mt,el}}_{j=0}=\sigma^{\mathrm{mt}}_{j=0,j^{\prime}=0}$,
respectively, for the range of collision energies considered for both isotopes
of carbon (Fig. 9).

As observed for other atom-molecule pairs (Parker & Pack, 1978; Dagdigian,
2015; Gacesa et al., 2020; Chhabra et al., 2023), the MTCSs at low energies
are dominated by the elastic cross sections and rapidly decrease monotonically
as the energy increases and converge to a near-constant value for the energies
greater than about 3 eV. The elastic MTCSs exhibit oscillatory periodic
features between about 1.5 eV and 3 eV, reflecting the character of the
oscillations present in the elastic and total cross sections (Figs. 4 and 5).
The oscillations appear to be out of phase for the two carbon isotopes,
resulting in a relative difference of up to a factor of two around the peaks
at about 1.8 eV and 2.6 eV. The oscillations are not present in total MTCSs
due to the summation over the state-to-state contributions.

##  4 Summary and Discussion

In this paper, we have carried out non-reactive quantum-mechanical scattering
calculations for 12C(${}^{3}P$) + CO2 and 13C(${}^{3}P$) + CO2 pairs at
thermal and superthermal collision energies up to 5 eV in the center-of-mass
frame, using the ground state electronic potential energy surface computed in
this work. The PES was constructed in a 2D planar geometry, with the CO2
molecule in the ground electronic state and assumed to be linear with both C-O
bonds fixed at the equilibrium, corresponding to the rigid rotor
representation. Within this approximation, we computed velocity-dependent
state-to-state cross sections, initial state-dependent elastic and total cross
sections, and corresponding differential and momentum transfer cross sections.

We have found that the computed cross sections differ significantly from the
estimates obtained by mass-scaling the cross sections for different atom-
molecule systems (Lewkow & Kharchenko, 2014; Lo et al., 2021), in particular
at low energies, below 1 eV. When compared to the values obtained by mass-
scaling the cross sections for O(${}^{3}P$)+CO2 (Gacesa et al., 2020), the
computed cross sections are more than two times larger, suggesting that the
differences in the electronic structure between these complexes play an
important role in dynamics. Moreover, we have found that the state-specific
elastic and total cross sections differ by up to 2% for the two isotopes of
carbon colliding with 12CO2. The relative differences are the most significant
below 0.7 eV and between 1.5 eV and 3 eV, where the 13C yields mostly larger
cross sections, while the lighter isotopes yields a larger cross section at
the collision energies above 4 eV. As observed for other atom-molecule pairs,
the scattering is strongly anisotropic and the excitations of rotational
degrees of freedom of the CO2 molecule is significant. This is in agreement
with Yeung et al. (2012) who concluded that in O$(^{3}P)$ \+ CO2 collisions,
about 40% of the collision energy remained as the translation energy of the
products, while a simple energy partitioning based on statistical distribution
of the level energies gives an even larger estimate (Fox & Hać, 2018). Such a
high efficiency of the kinetic-to-internal energy transfer may be explained
the formation of a metastable C2O2 complex during the collision (Yeung et al.,
2012).

The main contributions to the uncertainties in the computed cross sections
come from the linear rigid rotor approximation of the CO2 molecule and the PES
used to model the C(${}^{3}P$)-CO2 electronic interaction. The vibrational and
rotational transitions have been studied extensively in O(${}^{3}P$)+CO2
collisions at superthermal energies (Schatz & Redmon, 1981; Mullaney Harvey,
1982; Castle et al., 2012; Upschulte & Caledonia, 1992; Feofilov et al.,
2012). These and related studies agree that the vibrational excitations due to
the collisions are significant, in particularly of the bending mode of CO2,
and likely the leading inelastic process at collision energies greater than
about 4 eV (Schatz & Redmon, 1981). From the energy partitioning arguments,
the CO${}_{2}(010)$ bending mode has a low excitation energy of about 667 cm-1
(about 960 K or 0.083 eV), followed by the symmetric stretching mode (1388
cm-1 or 0.172 eV) and the asymmetric stretching mode (2349 cm-1 or 0.291 eV),
making its density of states the largest of the three. At lower energies, the
vibrational transitions are likely to be at least an order of magnitude
smaller than the rotational transitions (Mullaney Harvey, 1982; Upschulte &
Caledonia, 1992; Dagdigian, 2015). A more extensive discussion is given in
Gacesa et al. (2020). Including the bending mode in the collisional dynamics
would require at least a 3-D or, preferably, a full-dimensional PES(s) and
their couplings, none of which are currently available.

Finally, our study did not include possible reaction channels of the title
process. Here, energetically permitted reaction product channels are isotope
exchange [12C(${}^{3}P$)+13CO${}_{2}\rightarrow$ 13C(${}^{3}P$)+12CO2], and
carbon atom abstraction [C+CO2 -> CO+CO]. Neither process is particularly
likely to occur, even though carbon isotope exchange could be relevant for
isotope cycling in the middle and upper atmosphere of Mars. Yeung et al.
(2012) studied these reaction channels for a hot O$(^{3}P)$ impacting on CO2
and concluded that the oxygen isotope exchange reaction yield is less than
1.6%, while the non-reactive scattering yield is greater than 98.4%.
Similarly, our earlier computational study of O$(^{3}P)$+H2 reactive
scattering (Gacesa & Kharchenko, 2014) over a similar collision energy range
predicted that reactions occur in less than 4% of all processes, in good
agreement with earlier results.

The computed scattering cross sections are directly applicable to the
transport of superthermal atomic carbon encountered in dense astrophysical
gases, such as CO2-rich atmospheres of Mars (Wordsworth & Pierrehumbert, 2013;
Jakosky & Treiman, 2023) and Venus (Gillmann et al., 2022), exoplanetary
atmospheres (JWST Transiting Exoplanet Community Early Release Science Team et
al., 2023), insterstellar molecular clouds (Sandford et al., 2020), and
photodissociation dominated regions (Wolfire et al., 2022).

## Acknowledgements

MG was partially funded by the NASA MAVEN mission grant and Khalifa University
grant #8474000362-KU-FSU-2021. The author is grateful to R. Lillis, J. Fox, R.
Yelle, and H. Ali Lulu for helpful suggestions and discussion.

## Data Availability

The data underlying this article are available in Zenodo
(doi:10.5281/zenodo.8224848), at https://doi.org/10.5281/zenodo.8224848. The
data will be shared on reasonable request to the corresponding author.

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