# State-to-state rovibrational transition rates for CO2 in the bend mode in
collisions with He atoms

Taha Selim<EMAIL_ADDRESS>Theoretical Chemistry
Institute for Molecules and Materials, Radboud University
Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands  Ad van der Avoird
Theoretical Chemistry
Institute for Molecules and Materials, Radboud University
Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands  Gerrit C. Groenenboom
<EMAIL_ADDRESS>Theoretical Chemistry
Institute for Molecules and Materials, Radboud University
Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

###### Abstract

Modeling environments that are not in local thermal equilibrium, such as
protoplanetary disks or planetary atmospheres, with molecular spectroscopic
data from space telescopes requires knowledge of the rate coefficients of
rovibrationally inelastic molecular collisions. Here, we present such rate
coefficients in a temperature range from 10 to 500 K for collisions of CO2
with He atoms in which CO2 is (de)excited in the bend mode. They are obtained
from numerically exact coupled-channel (CC) calculations as well as from
calculations with the less demanding coupled-states approximation (CSA) and
the vibrational close-coupling rotational infinite-order sudden (VCC-IOS)
method. All of the calculations are based on a newly calculated accurate ab
initio four-dimensional CO2-He potential surface including the CO2 bend
($\nu_{2}$) mode. We find that the rovibrationally inelastic collision cross
sections and rate coefficients from the CSA and VCC-IOS calculations agree to
within 50% with the CC results at the rotational state-to-state level, except
for the smaller ones and in the low energy resonance region, and to within 20%
for the overall vibrational quenching rates except for temperatures below 50 K
where resonances provide a substantial contribution. Our CC quenching rates
agree with the most recent experimental data within the error bars. We also
compared our results with data from Clary et al. calculated in the 1980’s with
the CSA and VCC-IOS methods and a simple atom-atom model potential based on ab
initio Hartree-Fock calculations and found that their cross sections agree
fairly well with ours for collision energies above 500 cm-1, but that the
inclusion of long range attractive dispersion interactions is crucial to
obtain reliable cross sections at lower energies and rate coefficients at
lower temperatures.

##  I Introduction

The evolution of interstellar molecular clouds to protostars to protoplanetary
disks to planetary systems can be followed by observing spectra emitted or
absorbed by small molecules, such as CO, CO2, HCN, C2H2, etc. roueff:13 .
Microwave spectra are generated by rotational transitions in these molecules,
infrared spectra by rovibrational transitions. The shapes of the lines in the
spectra depend on the rotational and vibrational level populations, and these
populations are determined both by radiative transitions and by transitions
caused by molecular collisions with abundant species: hydrogen and helium
atoms, H2 molecules, and electrons. The information from the spectra is used
by astronomers in modeling the processes taking place in the various stages of
the evolution. Two situations are distinguished: (1) environments from which
the spectra originate are in local thermal equilibrium (LTE) and the line
shapes in the spectra only depend on the temperature, and (2) the spectra
originate from non-LTE environments. In the first situation it is sufficient
to know the Einstein $A$ and $B$ coefficients for spontaneous and stimulated
emission and absorption. In the non-LTE case one also needs to know state-to-
state transition rate coefficients from molecular collisions. This paper is
concerned with the latter case, and in particular with the calculation of
collisional transition rates from first principles. Cross sections and rate
coefficients for rotationally inelastic collisions have already been
calculated for several astronomically relevant molecular systems including
CO2-He roueff:13 ; palluet:22 , but the additional inclusion of vibrational
(de)excitations is more demanding. In earlier work song:13a ; song:15a ;
walker:15 ; song:15b  we studied rovibrational transitions in CO in collisions
with H atoms and rovibrational transitions in the stretch modes of CO2 in
collisions with He atoms selim:21 ; selim:22 . Here, we investigate the bend
mode of CO2 in collisions with He atoms.

Spectra of the CO2 bend mode in the 15 $\mu$m range originating from different
astronomical environments were observed by the Infrared Space Observatory
(ISO) boonman:03 ; boonman:03a  and the Spitzer Space Telescope
sonnentrucker:08  and, recently, also by the James Webb Space Telescope (JWST)
grant:23 . Another process in which collisions of CO2 in the stretch and bend
modes with He atoms are important is the CO2 laser action patel:64 ;
offenberger:70 ; clary:83 ; kasner:08 . Furthermore, the CO2 bend mode is
relevant because it yields the dominant contibution to the terrestial
greenhouse effect allen:79 ; lopez:92 , and collisions with (oxygen) atoms in
the mesosphere are important as well feofilov:12 . Finally, CO2 occurs in the
atmospheres of other planets and exoplanets jwst:23  and its spectra are
important in modeling these atmospheres.

The CO2 molecule has three vibrational modes: a twofold degenerate bend mode
with experimental frequency 667 cm-1, a symmetric stretch mode at 1333 cm-1,
and an asymmetric stretch mode at 2349 cm-1 shimanouchi:72 . Here, we
concentrate on the bend mode. In their pioneering theoretical work on rate
coefficients for vibrational transitions in CO2 induced by collisions with
rare gas (Rg) atoms Clary et al. clary:81 ; clary:81b ; clary:82 ; clary:83 ;
banks:87b ; wickham:87b  also investigated the bend mode. In scattering
calculations they used the coupled-states approximation (CSA), as well as the
VCC-IOS method, a vibrational coupled-channel (CC) method for the vibrations,
combined with the infinite-order sudden (IOS) approximation for the rotations.
Although they included the rotational states both in their VCC-IOS
calculations clary:83  and in their CSA calculations banks:87b , they only
provided some illustrative data for state-to-state rovibrational transitions.
And they used a model potential based on ab initio self-consistent field (SCF)
calculations, which can nowadays be calculated much more accurately.
Experimental data are available only for overall vibrational transition rates
lepoutre:79 ; siddles:94 . The more advanced models currently being developed
by astronomers bosman:17 ; bosman:19  and the availability of data from JWST
require rovibrational state-to-state collisional rate coefficients, which we
here present.

Section III describes the ab initio calculation of the 4-dimensional CO2-He
intermolecular potential depending on the CO2 bend coordinate. Also the
analytical representation of the potential is defined and the potential is
illustrated. The bend mode of a linear molecule like CO2 is twofold degenerate
and generates vibrational angular momentum, which makes the theory more
complicated than it is for the stretch modes or the bend modes in nonlinear
molecules. It is outlined in Sec. IV as part of the CO2-He scattering
approach. In Sec. V we present and discuss our results and compare the overall
vibrational transition rates with the available experimental data and with the
results of Clary et al. Section VI summarizes our conclusions.

##  II Coordinates and frames

The Jacobi scattering vector points from the center-of-mass of CO2 to the
helium atom. Its Cartesian coordinates with respect to a space-fixed (SF)
frame are given by the column vector ${\bm{R}}$. The spherical polar
coordinates of this vector are $(R,\Theta,\Phi)$, with $R=|{\bm{R}}|$. In the
scattering calculation we express the wave function in a two-angle embedded
body-fixed (BF) frame with the vector ${\bm{R}}$ as its $z$-axis, which
isdefined by the rotation matrix ${\mathbb{R}}(\Phi,\Theta,0)$. This matrix is
written in $zyz$-Euler angle parameterization using the active rotation
convention, see Biedenharn and Louck biedenharn:81  page 23, i.e.,

${\bm{R}}={\mathbb{R}}(\Phi,\Theta,0){\bm{R}}^{\mathrm{BF}}={\mathbb{R}}_{z}(\Phi){\mathbb{R}}_{y}(\Theta){\bm{R}}^{\mathrm{BF}},$
(1)

where ${\bm{R}}^{\mathrm{BF}}=(0,0,R)^{T}$ are the BF coordinates of the
vector ${\bm{R}}$. The rotation matrices ${\mathbb{R}}_{z}(\Phi)$ and
${\mathbb{R}}_{y}(\Theta)$ represent rotations around the $z$\- and $y$-axes,
respectively, see, e.g., Ref. biedenharn:81  or Eq. (5) in dhont:04 .

We also define a molecule-fixed (MF) frame, which has its $z$-axis parallel to
the vector that connects the two O atoms and which has the bent CO2 molecule
in the $xz$-plane. The MF coordinates of the Jacobi vector are related to its
BF coordinates through

${\bm{R}}^{\mathrm{BF}}={\mathbb{R}}(\alpha,\beta,\gamma){\bm{R}}^{\mathrm{MF}},$
(2)

where ${\mathbb{R}}(\alpha,\beta,\gamma)$ defines the MF frame with respect to
the BF frame ${\mathbb{R}}^{\mathrm{BF}}(\Phi,\Theta,0)$. Inverting this
equation gives

${\bm{R}}^{\mathrm{MF}}={\mathbb{R}}(\alpha,\beta,\gamma)^{T}{\bm{R}}^{\mathrm{BF}}=R\begin{pmatrix}-\sin\beta\cos\gamma\\\
\sin\beta\sin\gamma\\\ \cos\beta\end{pmatrix}.$ (3)

The superscript $T$ on the rotation matrix means transpose, which gives its
inverse, since the matrix is orthonormal. The angle $\alpha$ drops out of the
equation because the vector ${\bm{R}}^{\mathrm{BF}}$ is invariant under
rotations around the BF $z$-axis. This equation shows that the angles $\beta$
and $\gamma$ are related to the spherical polar angles
$(\beta^{\prime},\gamma^{\prime})$ of the vector ${\bm{R}}^{\mathrm{MF}}$ by
$\beta^{\prime}=\beta$ and $\gamma^{\prime}=\pi-\gamma$. In Sec. III we use
these angles to define the potential.

Combining Eqs. (1) and (2) we find that the MF frame is given with respect to
the SF frame by

${\mathbb{R}}(\Phi,\Theta,0){\mathbb{R}}(\alpha,\beta,\gamma)={\mathbb{R}}(\Phi,\Theta,\alpha){\mathbb{R}}(0,\beta,\gamma),$
(4)

where on right-hand-side we have a three-angle embedded BF frame
${\mathbb{R}}(\Phi,\Theta,\alpha)$ and a two-angle embedded MF frame
${\mathbb{R}}(0,\beta,\gamma)$. We use these frames in the VCC-IOS
calculations, see Sec. IV.4.

##  III Four-dimensional CO2(bend)-He potential

The coordinates in the CO2-He potential
$V(\tilde{Q},R,\beta^{\prime},\gamma^{\prime})$ are the spherical polar
coordinates $(R,\beta^{\prime},\gamma^{\prime})$ of the helium atom in the MF
frame defined above and the dimensionless amplitude $\tilde{Q}$ of the bend
mode in the harmonic approximation, which is defined in Sec. IV.1 below; the
classical turning points are $\tilde{Q}=\pm 1$ for $v=0$ and
$\tilde{Q}=\pm\sqrt{3}$ for $v=1$.

The potential was calculated with the ab initio coupled-cluster method with
single and double excitations and perturbative triples, CCSD(T), using the
MOLPRO package molpro . The basis was the augmented triple-zeta correlation-
consistent polarized (aug-cc-pVTZ) basis of Dunning and coworkers dunning:89 ,
supplemented with a set of $3s3p2d1f$ midbond functions. These midbond
functions were placed on the intersection of the vector $\bm{R}$ and an
ellipsoid around CO2, as described in Refs. selim:21 ; groenenboom:03 . This
ellipsoid is chosen such that it corresponds to the midpoint of $\bm{R}$ at
T-shaped configurations with $\beta^{\prime}=90^{\circ}$ and to the midpoint
of the vector connecting He with the nearest O atom in linear configurations
with $\beta^{\prime}=0^{\circ}$ and $180^{\circ}$. This choice of the location
of midbond functions and the use of geometry dependent exponents prevents
overcompleteness of the basis, especially in the short range. In all CCSD(T)
calculations the $T_{1}$ diagnostic was less than $0.018$, which indicates the
reliability of the CCSD(T) method. The interaction energies were computed
using the Boys and Bernardi boys:70  counterpoise method to correct for the
basis set superposition error (BSSE).

In our calculations of the potential surface we kept the C-O bond lengths
fixed and used the O-C-O angle as the bend coordinate, whereas in the harmonic
approximation used in Sec. IV.1 below the displacements of the C and O atoms
in the CO2 bend coordinate $\tilde{Q}$ are rectilinear by definition. We
tested both alternatives; some results are displayed in Fig. S1 of the
Supplementary material. They show that the differences are very small, because
the bend amplitude remains small. Only at very short distances $R$ where the
potential is strongly repulsive, small differences are visible.

The ab initio potential was calculated on a grid of 16 000 symmetry-unique
points. This grid contained 25 points for $R$, with step size $0.25\,a_{0}$
for $3.5\leq R\leq 7\,a_{0}$ and step size $0.5\,a_{0}$ for $7\leq R\leq
10\,a_{0}$, and 4 logarithmically spaced points for $10\leq R\leq 15\,a_{0}$.
For $\beta^{\prime}$ we used 8 Gauss-Legendre quadrature points, ranging from
$0$ to $\pi/2$ because of symmetry. For $\gamma^{\prime}$ we used 16 Gauss-
Chebychev equidistant quadrature points between $0$ and $\pi$, again because
of the symmetry. For the bend coordinate $\tilde{Q}$ we used five points:
$\tilde{Q}=0,0.5,1,1.5,2$. For a number of near-linear geometries with
$\beta^{\prime}=8.35^{\circ}$ and the smallest distance $R=3.5\,a_{0}$ the He
atom is very close to the nearest O atom and the ab initio calculations could
not be converged. The missing data points were provided by exponentially
extrapolating the interaction energies at $R=4.0$ and $3.75\,a_{0}$. The
potential is extremely repulsive in this region and the interaction energies
are much higher than the highest collision energy, so these data points do not
play a role in the scattering calculations.

The angular dependence of the 4D potential is represented by the expansion

$V(\tilde{Q},R,\beta^{\prime},\gamma^{\prime})=\sum_{\lambda=0}^{\lambda_{\rm
max}}\sum_{m_{\lambda}=0}^{\lambda}v_{\lambda
m_{\lambda}}(\tilde{Q},R)\,S_{\lambda
m_{\lambda}}(\beta^{\prime},\gamma^{\prime})$ (5)

in cosine type tesseral harmonics $S_{\lambda m_{\lambda}}=\left[C_{\lambda
m_{\lambda}}+(-1)^{m_{\lambda}}C_{\lambda-
m_{\lambda}}\right]/\sqrt{2(1+\delta_{m_{\lambda}0})}$ with $m_{\lambda}\geq
0$, which are real-valued linear combinations of the Racah-normalized
spherical harmonics $C_{\lambda m_{\lambda}}(\beta^{\prime},\gamma^{\prime})$.
The potential is symmetric with respect to reflection in the $xy$-plane, which
implies that only terms with even values of $\lambda+m_{\lambda}$ occur in the
expansion.

The expansion coefficients $v_{\lambda m_{\lambda}}(\tilde{Q},R)$ were
obtained at each grid point $(\tilde{Q}_{k},R_{l})$ by numerical integration
over $\beta^{\prime}$ and $\gamma^{\prime}$ using Gauss-Legendre and Gauss-
Chebyshev quadratures, respectively

$v_{\lambda
m_{\lambda}}(\tilde{Q}_{k},R_{l})=\frac{2\lambda+1}{4\pi}\sum_{i=1}^{16}\sum_{j=1}^{32}w_{i}w^{\prime}_{j}S_{\lambda
m_{\lambda}}(\beta^{\prime}_{i},\gamma^{\prime}_{j})V(\tilde{Q}_{k},R_{l},\beta^{\prime}_{i},\gamma^{\prime}_{j})$
(6)

with weights $w_{i}$ and $w^{\prime}_{j}$. The expansion has converged at
$\lambda_{\rm max}=15$. The expansion coefficients $v_{\lambda
m_{\lambda}}(\tilde{Q}_{k},R_{l})$ were fitted to a fourth-degree polynomial
in $\tilde{Q}$

$v_{\lambda m_{\lambda}}(\tilde{Q},R_{l})=\sum_{p=0}^{4}v_{p,\lambda
m_{\lambda}}(R_{l})\,\tilde{Q}^{p}.$ (7)

at each grid point $R_{l}$. It follows furthermore that $m_{\lambda}\leq p$
and since the potential is invariant under overall rotation about the
$z$-axis, that the sum of $p$ and $m_{\lambda}$ must be even. Finally, the
$R$-dependence of the coefficients $v_{p,\lambda m_{\lambda}}(R)$ was
represented by the Reproducing Kernel Hilbert space (RKHS) method ho:96 ;
ho:00 , which uses two parameters: a smoothness parameter $n$ and a parameter
$m$ which ensures that the potential decays as $1/R^{m+1}$ beyond the largest
$R$ value in the grid. We chose $n=2$ and $m=5$.

Figure 1:  The first few coefficients in the expansion of the potential in
Racah-normalized tesseral harmonics at $\tilde{Q}=1$.

In the repulsive short range of the potential at near-linear configurations
extremely high peaks occur at some specific geometries. These highly repulsive
peaks are not physically important because the system cannot reach these
geometries even at high collision energies. However, they cause a problem with
the convergence of the spherical expansion. In order to avoid this problem we
damped the interaction energies for values larger than $V_{0}=0.1$ Eh with the
smooth damping function

$V_{\rm
damped}(\tilde{Q},R,\beta^{\prime},\gamma^{\prime})=V_{0}+\tanh\left[\zeta\\{V(\tilde{Q},R,\beta^{\prime},\gamma^{\prime})-V_{0}\\}\right]/\zeta,$
(8)

where $\zeta=1/V_{0}$ and $2V_{0}$ is the maximum value of the damped
potential. The range of $R$ values for which the damping is effective depends
on the angles $\beta^{\prime}$ and $\gamma^{\prime}$. At near-linear
structures which $\beta^{\prime}$ close to 0 and $180^{\circ}$, damping was
applied for $3.5\leq R\leq 4.75\,a_{0}$ for all $\gamma^{\prime}$. For
T-shaped structures the interaction energies are less than $V_{0}$ and damping
was not needed.

The strong anisotropy of the 4D CO2-He potential is obvious already from the
large value of $\lambda_{\rm max}=15$ needed to converge its spherical
expansion. It is further illustrated by showing the first few expansion
coefficients $v_{\lambda m_{\lambda}}(\tilde{Q},R)$ at $\tilde{Q}=1$ in Fig.
1. The leading anisotropic term $V_{2,0}$ is larger than the isotropic term
$V_{0,0}$, and $V_{3,1}$ is larger than $V_{1,1}$. At the linear configuration
with $\tilde{Q}=0$, only terms with even $\lambda$ and $m_{\lambda}=0$
contribute to the expansion of the potential. Hence, the terms with odd
$\lambda$ and $m_{\lambda}>0$ which are due only to the CO2 bend are smaller.
A view of the CO2-He potential for linear CO2 and the effect of the bending is
shown for planar geometries in Fig. 2.

Figure 2: Contour plots of the CO2-He potential for $\tilde{Q}=0$ (upper
panel) and the difference between the potential for $\tilde{Q}=1$ and
$\tilde{Q}=0$ (lower panel). The contours in the lower panel are not drawn in
the region where the repulsive interaction energy is much higher than the
highest collision energy.

Finally we note that the potential
$V(\tilde{Q},R,\beta^{\prime},\gamma^{\prime})$ calculated with coordinates
$\beta^{\prime},\gamma^{\prime}$ relative to the MF frame can simply be
expressed relative to the BF frame used in the scattering calculations by the
substitution $\beta^{\prime}=\beta$ and $\gamma^{\prime}=\pi-\gamma$. This
yields an additional factor $(-1)^{m_{\lambda}}$ in the expansion of the
potential in Eq. (5) when applied to the coordinates $\beta$ and $\gamma$.

##  IV Scattering theory

We obtain the cross sections and rate coefficients for rovibrationally
inelastic collisions of CO2 with He by means of scattering calculations with
the numerically exact coupled-channel or close-coupling (CC) method. Since the
theory for the bend mode of CO2 is more complicated than described for the two
stretch modes in Refs. selim:21 ; selim:22 , it is outlined below.

###  IV.1 CO2 monomer Hamiltonian and wave functions

The bend mode of the CO2 monomer is twofold degenerate and the molecule bends
not only in the $xz$-plane but also in the $yz$-plane, with the coordinates
$(Q_{x},Q_{y})$. The atoms are labeled by the indices $i=1,2,3$ with $i=1$ and
3 for the two O atoms and $i=2$ for the C atom. The equilibrium positions in
the MF frame are $x_{i}^{e}=y_{i}^{e}=0$, $z_{1}^{e}=-z_{3}^{e}=2.196\,a_{0}$,
and $z_{2}^{e}=0$. We use rectilinear normal coordinates $Q_{x}$ and $Q_{y}$,
which involve the atomic displacements $\Delta x_{i}$ and $\Delta y_{i}$ from
their equilibrium positions: $\Delta x_{i}=c_{i}Q_{x}$ and $\Delta
y_{i}=c_{i}Q_{y}$. The coefficients $c_{1}$ and $c_{3}$ are equal, while
$c_{2}$ is fixed by the condition that the center of mass should not move.

Our CO2 monomer Hamiltonian is based on Watson’s isomorphic Hamiltonian for
the bend mode of a linear triatomic molecule watson:70 . In the Cartesian
monomer coordinates defined above and with the use of the harmonic-oscillator
rigid-rotor approximation it reads

$\hat{H}_{\textrm{CO}_{2}}(Q_{x},Q_{y})=-\frac{\hbar^{2}}{2\mu_{Q}}\left(\frac{\partial^{2}}{\partial
Q_{x}^{2}}+\frac{\partial^{2}}{\partial
Q_{y}^{2}}\right)+\frac{1}{2}f\left(Q_{x}^{2}+Q_{y}^{2}\right)+\frac{\hbar^{2}}{2I_{0}}\left(\hat{j}_{x}^{2}+\hat{j}_{y}^{2}\right),$
(9)

where $\mu_{Q}=2m_{\rm C}m_{\rm O}/(m_{\rm C}+2m_{\rm O})$ is the reduced mass
associated with the bend vibration, $m_{\rm C}$ and $m_{\rm O}$ are the masses
of the C and O atoms, $f$ is the harmonic force constant, and
$I_{0}=\sum_{i=1}^{3}m_{i}(z_{i}^{e})^{2}$ is the rigid rotor moment of
inertia for rotation about the $x$\- or $y$-axis. The harmonic frequency is
given by $\omega=\sqrt{f/\mu_{Q}}$. In the sequel we use dimensionless normal
coordinates $\tilde{Q}_{x}=aQ_{x}$ and $\tilde{Q}_{y}=aQ_{y}$ obtained by
scaling with $a=\sqrt{\mu_{Q}\omega/\hbar}$.

Following textbooks bunker:98  and previous work banks:87b  we may re-express
this Hamiltonian in polar coordinates $(\tilde{Q},\gamma)$ with
$\tilde{Q}_{x}=\tilde{Q}\cos\gamma$ and $\tilde{Q}_{y}=\tilde{Q}\sin\gamma$

$\hat{H}_{\textrm{CO${}_{2}$}}(\tilde{Q},\gamma)=-\frac{\hbar\omega}{2}\left(\frac{\partial^{2}}{\partial\tilde{Q}^{2}}+\frac{1}{\tilde{Q}}\frac{\partial}{\partial\tilde{Q}}+\frac{\hat{l}_{v}^{2}}{\tilde{Q}^{2}}\right)+\frac{\hbar\omega}{2}\tilde{Q}^{2}+\frac{\hbar^{2}}{2I_{0}}\left(\hat{j}^{2}-\hat{j}_{z}^{2}\right),$
(10)

where $\tilde{Q}$ is the bend amplitude and $\gamma$ is the angle between the
plane in which CO2 bends and the $xz$ plane. The operator
$\hat{l}_{v}=-i\frac{\partial}{\partial\gamma}$ is the dimensionless
vibrational angular momentum operator with eigenvalues
$l_{v}=-v,-v+2,\ldots,v-2,v$. The eigenvalues of this harmonic-oscillator
rigid-rotor Hamiltonian are

$\varepsilon_{v,l_{v},j}=(v+1)\hbar\omega+\frac{\hbar^{2}}{2I_{0}}\left[j(j+1)-l_{v}^{2}\right]$
(11)

and its eigenfunctions, including rotation with respect to the BF frame, are

$|\mskip 3.0muv,l_{v},j,\Omega\mskip
3.0mu\rangle=\sqrt{\frac{2j+1}{8\pi^{2}}}N_{v,l_{v}}\tilde{Q}^{|l_{v}|}\,\exp(-\tilde{Q}^{2}/2)\,L_{(v-|l_{v}|)/2}^{|l_{v}|}(\tilde{Q})\,D_{\Omega,l_{v}}^{(j)}(0,\beta,\gamma)^{\ast},$
(12)

where $D_{\Omega,l_{v}}^{(j)}$ are Wigner $D$-matrices brink:93  depending on
the rotation angles $(\beta,\gamma)$ of the MF frame relative to the BF frame,
and $\Omega$ is the projection of the monomer angular momentum ${\bm{j}}$ on
the BF $z$-axis. Note that in the definition of the polar coordinates
$(\tilde{Q},\gamma)$ above $\gamma$ is the angle of the plane in which CO2
bends and the functions $\exp(il_{v}\gamma)$ contained in the basis of Eq.
(12) actually belong to the vibrational part. The functions
$L_{(v-|l_{v}|)/2}^{|l_{v}|}(\tilde{Q})$ are associated Laguerre functions in
the convention of Abramowitz and Stegun abramowitz:64  and the normalization
constants

$N_{v,l_{v}}=\sqrt{2\frac{\left(\frac{v-|l_{v}|}{2}\right)!}{\left(\frac{v+|l_{v}|}{2}\right)!}}.$
(13)

The functions in Eq. (12) may be adapted to the permutation symmetry
$\hat{P}_{13}$ that interchanges the two O nuclei, which has the following
effect

$\hat{P}_{13}|\mskip 3.0muv,l_{v},j,\Omega\mskip 3.0mu\rangle=(-1)^{j}|\mskip
3.0muv,{-l_{v}},j,\Omega\mskip 3.0mu\rangle.$ (14)

A basis adapted to $\hat{P}_{13}$ with parity $\epsilon$ reads

$|\mskip 3.0muv,\tilde{l}_{v},j,\Omega,\epsilon\mskip
3.0mu\rangle=\frac{1}{\sqrt{2(1+\delta_{l_{v},0})}}\left[|\mskip
3.0muv,\tilde{l}_{v},j,\Omega\mskip 3.0mu\rangle+\epsilon(-1)^{j}|\mskip
3.0muv,{-\tilde{l}_{v}},j,\Omega\mskip 3.0mu\rangle\right],$ (15)

with $\tilde{l}_{v}=|l_{v}|\geq 0$. Since 16O nuclei are bosons with spin
$I=0$ only functions with $\epsilon=1$ are allowed, which implies that for
$l_{v}=0$ the basis only contains functions with even $j$.

###  IV.2 Coupled-channel calculations

As outlined in earlier papers selim:21 ; selim:22 , our version of the CC
method is based on the CO2-He Hamiltonian in BF coordinates

$\hat{H}=-\frac{\hbar^{2}}{2\mu R}\frac{\partial^{2}}{\partial
R^{2}}R+\hat{H}_{\textrm{CO}_{2}}(\tilde{Q},\gamma)+\frac{\hat{J}^{2}+\hat{j}^{2}-2\hat{\bm{j}}\cdot\hat{\bm{J}}}{2\mu
R^{2}}+V(\tilde{Q},R,\beta,\gamma),$ (16)

where
$\mu=m_{\textrm{CO}_{2}}m_{\textrm{He}}/(m_{\textrm{CO}_{2}}+m_{\textrm{He}})$
is the reduced mass of the complex, $\hat{\bm{j}}$ the CO2 monomer rotational
angular momentum operator, $\hat{\bm{J}}$ the total angular momentum operator
of the complex, and $\hat{J}^{2}+\hat{j}^{2}-2\hat{\bm{j}}\cdot\hat{\bm{J}}$
represents the end-over-end angular momentum operator $L^{2}$ in the BF frame
brocks:83 ; avoird:94 . The Hamiltonian $\hat{H}_{\textrm{CO}_{2}}$ for the
CO2 bend mode is defined in Eq. (10).

The BF channel basis is

$|\mskip 3.0muv,\tilde{l}_{v},j,\Omega,\epsilon;J,M_{J}\mskip
3.0mu\rangle=\sqrt{\frac{2J+1}{4\pi}}|\mskip
3.0muv,\tilde{l}_{v},j,\Omega,\epsilon\mskip 3.0mu\rangle
D_{M_{J},\Omega}^{(J)}(\Phi,\Theta,\alpha)^{\ast},$ (17)

where $\Omega$ is also the projection of the total angular momentum ${\bm{J}}$
on the intermolecular axis ${\bm{R}}$. The permutation symmetry-adapted
monomer eigenfunctions $|\mskip 3.0muv,\tilde{l}_{v},j,\Omega,\epsilon\mskip
3.0mu\rangle$ are defined in Eq. (15). The quantum numbers $J$ and $M_{J}$ are
good quantum numbers, while functions with different $\Omega$ are mixed by the
Coriolis coupling operator $\hat{\bm{j}}\cdot\hat{\bm{J}}$ so that $\Omega$ is
an approximate quantum number.

Another symmetry is the parity of the complex under overall inversion
$E^{\ast}$. The BF channel basis adapted also to inversion is

$|\mskip 3.0muv,\tilde{l}_{v},j,\tilde{\Omega};\epsilon,P,J,M_{J}\mskip
3.0mu\rangle=\frac{1}{\sqrt{2(1+\delta_{\tilde{\Omega},0})}}\left[|\mskip
3.0muv,\tilde{l}_{v},j,\tilde{\Omega};\epsilon,J,M_{J}\mskip
3.0mu\rangle+P\epsilon{(-1)}^{\tilde{l}_{v}+j+J}|\mskip
3.0muv,\tilde{l}_{v},j,{-\tilde{\Omega}};\epsilon,J,M_{J}\mskip
3.0mu\rangle\right],$ (18)

with $\tilde{\Omega}=|\Omega|\geq 0$ and $P=\pm 1$ being the overall parity.
Since $P$ is an exact quantum number the calculations can be made separately
for $P=\pm 1$, which is a considerable simplification.

In terms of the symmetry-adapted BF channel basis the scattering wave
functions are

$\Psi^{\epsilon,P,J,M_{J}}=\frac{1}{R}\sum_{v,\tilde{l}_{v},j,\tilde{\Omega}}|\mskip
3.0muv,\tilde{l}_{v},j,\tilde{\Omega};\epsilon,P,J,M_{J}\mskip
3.0mu\rangle\psi_{v,\tilde{l}_{v},j,\tilde{\Omega}}^{\epsilon,P,J,M_{J}}(R).$
(19)

The radial functions
$\psi_{v,\tilde{l}_{v},j,\tilde{\Omega}}^{\epsilon,P,J,M_{J}}(R)$ can be
obtained in the usual way by solving a set of coupled second order
differential equations, the coupled-channel equations. As in Refs. selim:21 ;
selim:22  we do this with the aid of the renormalized Numerov propagator
johnson:78 ; johnson:79 . The required matrix elements of the potential
$V(\tilde{Q},R,\beta,\gamma)$ defined in Sec. III over primitive BF channel
basis functions $|\mskip 3.0muv,l_{v},j,\Omega;J,M_{J}\mskip 3.0mu\rangle$ are
given by

$\displaystyle
V_{v^{\prime}l^{\prime}_{v}j^{\prime}\Omega^{\prime};vjl_{v}\Omega}^{JM_{J}}(R)$
$\displaystyle=$
$\displaystyle\delta_{\Omega^{\prime}\,\Omega}\;{(-1)}^{\Omega-l^{\prime}_{v}}\frac{1}{\sqrt{2(1+\delta_{m_{\lambda}0})}}\langle\mskip
3.0muv^{\prime},l^{\prime}_{v}\mskip 3.0mu|\mskip 3.0muv_{\lambda
m_{\lambda}}(\tilde{Q},R)\mskip 3.0mu|\mskip 3.0muv,l_{v}\mskip 3.0mu\rangle$
(26) $\displaystyle\times$
$\displaystyle\left(\begin{array}[]{ccc}j^{\prime}&\lambda&j\\\
-\Omega&0&\Omega\end{array}\right)\left\\{{(-1)}^{m_{\lambda}}\left(\begin{array}[]{ccc}j^{\prime}&\lambda&j\\\
-l^{\prime}_{v}&m_{\lambda}&l_{v}\end{array}\right)+\left(\begin{array}[]{ccc}j^{\prime}&\lambda&j\\\
-l^{\prime}_{v}&-m_{\lambda}&l_{v}\end{array}\right)\right\\}.$

The expressions in large round brackets are Wigner $3j$ symbols brink:93 .
With the symmetry-adapted BF basis of Eq. (18) one obtains linear combinations
of these matrix elements.

The expansion coefficients $v_{\lambda m_{\lambda}}(\tilde{Q},R)$ are defined
by Eq. (5) in Sec. III. Their matrix elements over the monomer eigenfunctions
$|\mskip 3.0muv,l_{v}\mskip 3.0mu\rangle$ defined in Eq. (12) and containing
associated Laguerre functions are calculated by numerical integration over the
bend normal coordinate $\tilde{Q}$ with the aid of a 5-point Gauss-Laguerre
quadrature. Transformation of the matrix elements in Eq. (26) to the symmetry-
adapted channel basis in Eq. (18) is easy.

The asymptotic boundary conditions to which we need to match the scattering
wave functions at large distance $R$ are defined in terms of a SF channel
basis with partial wave quantum numbers $L$. The monomer wave functions
$|\mskip 3.0muv,l_{v},j,m_{j}\mskip 3.0mu\rangle$ in the SF frame are the same
as those in the BF frame in Eq. (12), except that the Euler angles
$(\alpha,\beta,\gamma)$ now define the orientation of the MF frame relative to
the SF frame and the angular momentum component $\Omega$ on the BF $z$-axis
becomes the component $m_{j}$ on the SF $z$-axis. The SF monomer basis adapted
to $\hat{P}_{13}$ with parity $\epsilon$ is

$|\mskip 3.0muv,\tilde{l}_{v},j,m_{j};\epsilon\mskip
3.0mu\rangle=\frac{1}{\sqrt{2(1+\delta_{l_{v},0})}}\left[|\mskip
3.0muv,\tilde{l}_{v},j,m_{j}\mskip 3.0mu\rangle+\epsilon(-1)^{j}|\mskip
3.0muv,{-\tilde{l}_{v}},j,m_{j}\mskip 3.0mu\rangle\right]$ (27)

and the symmetry adapted SF channel basis is

$|\mskip 3.0muv,\tilde{l}_{v},j,L;\epsilon,P,J,M_{J}\mskip
3.0mu\rangle=\sum_{m_{j}M_{L}}|\mskip
3.0muv,\tilde{l}_{v},j,m_{j};\epsilon\mskip
3.0mu\rangle\,Y_{LM_{L}}(\Theta,\Phi)\,\langle\mskip 3.0mujm_{j}LM_{L}\mskip
3.0mu|\mskip 3.0muJM_{J}\mskip 3.0mu\rangle,$ (28)

with $\langle\mskip 3.0mujm_{j}LM_{L}\mskip 3.0mu|\mskip 3.0muJM_{J}\mskip
3.0mu\rangle$ being a Clebsch-Gordan coefficient brink:93  and
$Y_{LM_{L}}(\Theta,\Phi)$ a normalized spherical harmonic depending on the
polar angles of ${\bm{R}}$ with respect to the SF frame. This basis is also
adapted to overall inversion with parity $P=\epsilon(-1)^{\tilde{l_{v}}+L}$.
The primitive BF and SF channel bases are related by the unitary
transformation

$|\mskip 3.0muv,l_{v},j,\Omega;J,M_{J}\mskip 3.0mu\rangle=\sum_{L}|\mskip
3.0muv,l_{v},j,L;JM_{J}\mskip 3.0mu\rangle U_{L\Omega}^{Jj}$ (29)

with

$U_{L\Omega}^{Jj}=\langle\mskip 3.0muj\Omega L0\mskip 3.0mu|\mskip
3.0muJ\Omega\mskip 3.0mu\rangle\sqrt{\frac{2L+1}{2J+1}}.$ (30)

The unitary transformation between the parity-adapted BF and SF channel bases
then follows from Eqs. (15), (18), and (27). At the end of the propagation of
the BF scattering functions to large $R$ we transform them to the SF basis
with partial wave quantum numbers $L$ and match them to spherical Bessel
asymptotic boundary conditions to obtain the $S$-matrix. We then compute the
state-to-state scattering cross sections
$\sigma_{{v^{\prime},\tilde{l}^{\prime}_{v},j^{\prime}\leftarrow
v,\tilde{l}_{v},j}}(E)$ in the usual way from the $S$-matrix for a large range
of energies $E$ and compute state-to-state rate coefficients with

$k_{v^{\prime},\tilde{l}^{\prime}_{v},j^{\prime}\leftarrow
v,\tilde{l}_{v},j}(T)=\left(\frac{8k_{B}T}{\pi\mu}\right)^{1/2}\int_{0}^{\infty}\sigma_{{v^{\prime},\tilde{l}^{\prime}_{v},j^{\prime}\leftarrow
v,\tilde{l}_{v},j}}(E)\left(\frac{E}{k_{B}T}\right)\,\exp\left(-\frac{E}{k_{B}T}\right)\,d\left(\frac{E}{k_{B}T}\right),$
(31)

where $k_{B}$ is the Boltzmann constant. The vibrational quenching rate from a
rovibrational initial state $(v,\tilde{l}_{v},j)$ to a final vibrational state
$(v^{\prime},\tilde{l}^{\prime}_{v})$ is defined as the sum over all final
rotational states $j^{\prime}$ in $(v^{\prime},\tilde{l}^{\prime}_{v})$

$k_{v^{\prime}\tilde{l}^{\prime}_{v}\leftarrow
v\tilde{l}_{v},j}(T)=\sum_{j^{\prime}}k_{v^{\prime}\tilde{l}^{\prime}_{v}j^{\prime}\leftarrow
v\tilde{l}_{v}j}(T).$ (32)

At thermal equilibrium the total vibrational quenching rate coefficient is
computed by Boltzmann averaging over the thermally populated initial states
$j$ in the initial vibrational state $(v,\tilde{l}_{v})$ with energies
$\epsilon_{v\tilde{l}_{v}j}$

$k_{v^{\prime}\tilde{l}^{\prime}_{v}\leftarrow
v\tilde{l}_{v}}(T)=\frac{\sum_{j}(2j+1)\exp(-\epsilon_{v\tilde{l}_{v}j}/k_{B}T)\,k_{v^{\prime}\tilde{l}^{\prime}_{v}\leftarrow
v\tilde{l}_{v}j}(T)}{\sum_{j}(2j+1)\exp(-\epsilon_{v\tilde{l}_{v}j}k_{B}T)}.$
(33)

###  IV.3 Coupled-states approximation

In the coupled-states approximation (CSA) used in the paper by Banks and Clary
banks:87b  one neglects the Coriolis coupling terms in the kinetic energy
operator that couple BF basis functions with different $\Omega$. This makes
$\Omega$ an exact quantum number, so that the CC equations can be separated
into subsets of equations for each value of $\Omega$ that are much smaller
than the full set of CC equations. Moreover, the absolute value of $\Omega$ is
limited to the smallest of the initial or final $j$ value in the scattering
process, which reduces the number of subsets to be included. Altogether, this
makes CSA calculations much faster than full CC calculations.

In our version of the CSA method we use the full diagonal part of the BF
kinetic energy $\big{[}J(J+1)+j(j+1)-2\Omega^{2}\big{]}/2\mu R^{2}$, just as
in Ref. banks:87b . As mentioned in Sec. IV.2, we use the renormalized Numerov
method johnson:78 ; johnson:79  to obtain the radial scattering wave functions
$\psi_{v,\tilde{l}_{v},j,\tilde{\Omega}}^{\epsilon,P,J,M_{J}}(R)$ in Eq. (19).
This method propagates the matrices $\bm{Q}_{i}$ defined by

$\bm{\psi}(R_{i-1})=\bm{Q}_{i}\bm{\psi}(R_{i}),$ (34)

over a radial grid with points $R_{i}$ with $i=1,\ldots,n$. The column vectors
$\bm{\psi}(R_{i})$ contain the radial functions
$\psi_{v,\tilde{l}_{v},j,\tilde{\Omega}}^{\epsilon,P,J,M_{J}}(R_{i})$ at grid
point $R_{i}$. As mentioned in Sec. IV.2 about the CC method, the BF
Q-matrices at the end of the propagation to large $R$ are transformed to the
SF basis with partial wave quantum numbers $L$ with the aid of Eqs. (29) and
(30) and matched to the proper asymptotic boundary conditions. In the CSA
method we obtain matrices $\bm{Q}_{n}^{\Omega}$ for all $\Omega$ values, put
them together as diagonal blocks into a large matrix $\bm{Q}_{n}$ over all
channels, transform this large Q-matrix from the BF basis to the SF basis with
partial wave quantum numbers $L$ in the same way as in the full CC method, and
use the asymptotic boundary conditions to obtain the full $S$-matrix from
which the CSA cross sections are calculated.

###  IV.4 Rotational infinite-order sudden approximation

We also calculated vibrational and rovibrational (de-)excitation cross
sections and rate coefficients for CO2-He collisions with the VCC-IOS
approximation. This method was extensively used in the 1980’s by Clary and co-
workers clary:81 ; clary:81b ; clary:82 ; clary:83 ; banks:87b ; we briefly
outline the theory for collisions with CO2 in the bend mode.

In the IOS approximation the centrifugal term in the Hamiltonian is replaced
by $J(J+1)/2\mu R^{2}$, so that $\Omega$ becomes a good quantum number, and
the rotational energy of CO2 is set to zero pack:74 . As a result, the
vibrational coupled-channels problem can be solved for fixed orientations of
the molecule. Calculations are done for a set of $J$ values ranging from $J=0$
to $J=J_{\mathrm{max}}$. The vibrational wave functions for CO2 in the bend
mode included in Eq. (12) are

$|\mskip 3.0muv,l_{v}\mskip
3.0mu\rangle=N_{v,l_{v}}\tilde{Q}^{|l_{v}|}\,\exp(-\tilde{Q}^{2}/2)\,L_{(v-|l_{v}|)/2}^{|l_{v}|}(\tilde{Q})\,\frac{\exp(il_{v}\gamma)}{\sqrt{2\pi}},$
(35)

with ${N}_{v,l_{v}}$ defined in Eq. (13). When the factor $\exp(il_{v}\gamma)$
contained in the Wigner matrix $D_{\Omega,l_{v}}^{(j)}(0,\beta,\gamma)^{\ast}$
in Eq. (12) is removed from it, the rotational part of the basis
$D_{\Omega,l_{v}}^{(j)}(0,\beta,0)^{\ast}$ depends only on the angle $\beta$,
which is the angle between the CO2 $z$-axis and the $z$-axis ${\bm{R}}$ of the
BF frame. The interaction potential is symmetric with respect to reflection in
the BF $xz$-plane ($\gamma\rightarrow-\gamma$). Vibrational wave functions
that are even or odd for this symmetry can be obtained by writing
$\exp(il_{v}\gamma)=\cos(l_{v}\gamma)+i\sin(l_{v}\gamma)$. The normalized
cosine and sine type monomer eigenfunctions are denoted by $|\mskip
3.0muv,\tilde{l}_{v},p\mskip 3.0mu\rangle$ with $\tilde{l}_{v}=|l_{v}|$ and
$p$ being even ($\mathrm{e}$) and odd ($\mathrm{o}$), respectively. Only the
cosine functions with $p=\mathrm{e}$ are needed to calculate vibrational cross
sections for transitions involving $v=0,l_{v}=0$.

The vibrational coupled-channel equations are solved for a grid of fixed
orientations $\beta_{i}$ of the CO2 monomer $z$-axis in the BF frame for each
value of $J$. The coupling contains the matrix elements

$V^{(p)}_{v^{\prime}\tilde{l}^{\prime}_{v},v\tilde{l}_{v}}(R,\beta_{i})=\langle\mskip
3.0muv^{\prime},\tilde{l}^{\prime}_{v},p\mskip 3.0mu|\mskip
3.0muV(\tilde{Q},R,\beta_{i},\gamma)\mskip 3.0mu|\mskip
3.0muv,\tilde{l}_{v},p\mskip 3.0mu\rangle$ (36)

over the potential $V(\tilde{Q},R,\beta,\gamma)$ defined in Sec. III.
Integration over the angle $\gamma$, which determines the plane in which CO2
bends, can be done analytically since the potential is expanded in cosine type
tesseral harmonics containing $\cos(m_{\lambda}\gamma)$, see Eq. (5), and the
basis contains functions $\cos(\tilde{l}_{v}\gamma)$ or
$\sin(\tilde{l}_{v}\gamma)$. The $\tilde{Q}$-dependent part of the basis
contains associated Laguerre functions and the integration over $\tilde{Q}$ is
done numerically with a Gauss-Laguerre quadrature, as in Sec. IV.2.

The VCC problem is solved using the same procedure as outlined for the full CC
equations in Sec. IV.2. This yields a scattering matrix for each $\beta_{i}$
and $J$ with elements
$S_{v^{\prime},\tilde{l}^{\prime}_{v};v,\tilde{l}_{v}}^{(J,p)}(\beta_{i},E)$.
The vibrational (de)excitation cross sections can be calculated from these
$S$-matrices by integration over $\beta$

$\sigma_{v^{\prime},\tilde{l}^{\prime}_{v}\leftarrow
v,\tilde{l}_{v}}(E)=\frac{\pi}{2k^{2}_{v}}g_{v}\sum_{J,p}(2J+1)\,\int_{0}^{\pi}\left|S^{(J,p)}_{v^{\prime},\tilde{l}^{\prime}_{v};v,\tilde{l}_{v}}(\beta,E)\right|^{2}\,\sin\beta\,d\beta$
(37)

with $k_{v}^{2}=2\mu(E-\epsilon_{v})$, $\epsilon_{v}$ being the energy of the
initial vibrational state $v$, and $g_{v}$ a degeneracy factor which depends
on $\tilde{l}_{v}$ of the initial state: $g_{v}=1$ for $\tilde{l}_{v}=0$ and
$g_{v}=1/2$ for $\tilde{l}_{v}>0$. In our case $g_{v}=1/2$ and only
$p=\mathrm{e}$ contributes. We chose a Gauss-Legendre quadrature grid for the
angles $\beta_{i}$, so that the integral over $\beta$ can be calculated by
numerical quadrature.

As explained by Clary in his 1983 paper clary:83 , VCC-IOS can also be used to
compute rovibrational cross sections. The theory in Clary’s paper is based on
the IOS treatment of atom - symmetric rotor collisions derived by Green
green:79b . In this case the rotational states of the molecule are labeled
with the quantum numbers $j,k$ and the IOS angle-dependent $S$-matrix
$\overline{S}^{(J)}_{j^{\prime}k^{\prime};jk}(\beta,\gamma,E)$ depends on two
angles $\beta$ and $\gamma$. These angles determine the orientation of the
molecule in the BF frame and in the IOS scattering calculations they are
fixed. In the VCC-IOS method applied to CO2-He collisions with CO2 in the bend
mode the matrix
$S^{(J,p)}_{v^{\prime},\tilde{l}^{\prime}_{v};v,\tilde{l}_{v}}(\beta,E)$ in
Eq. (37) only depends on the angle $\beta$, however. In order to make the
connection with the theory in Refs. green:79b, ; clary:83, , we introduce an
auxiliary angle $\chi$, which is similar to the extra azimuthal angle
$\chi^{\prime}$ defined by Watson in his derivation of the rovibrational
Hamiltonian for linear molecules watson:70 . As in the work of Watson, this is
a mathematical trick which does not require any additional physical
asumptions. The vibrational functions $\Psi_{v,l_{v}}(\tilde{Q},\gamma)$ in
Eq. (35) are multiplied by a phase factor $\exp(-il_{v}\chi)$ which gives

$\Psi_{v,l_{v}}(\tilde{Q},\gamma-\chi)=\Psi_{v,l_{v}}(\tilde{Q},\gamma)\exp(-il_{v}\chi)$
(38)

and we compensate for this factor by multiplying the pure rotational wave
function $D_{\Omega,l_{v}}^{(j)}(0,\beta,0)^{\ast}$ in Eq. (12) with
$\exp(il_{v}\chi)$, so we get
$D_{\Omega,l_{v}}^{(j)}(0,\beta,0)^{\ast}\exp(il_{v}\chi)=D_{\Omega,l_{v}}^{(j)}(0,\beta,\chi)^{\ast}$.

In the VCC part of the calculation, the rotational part of the basis is
omitted, the angle $\beta$ and the auxiliary angle $\chi$ are fixed, and the
$S$-matrix
$\overline{S}^{(J)}_{v^{\prime},l^{\prime}_{v};v,l_{v}}(\beta,\chi,E)$ from
the VCC calculations formally becomes a function of two angles, just as in the
IOS treatment of atom - symmetric rotor collisions green:79b . The vibrational
angular momentum $l_{v}$ takes the role of the rotational angular momentum
projection $k$ in the symmetric rotor. In reality one does not need to vary
the angle $\chi$ in the IOS calculations because it can be shown that this
$S$-matrix is related to the $\beta$-dependent $S$-matrix in Eq. (37) as

$\overline{S}^{(J)}_{v^{\prime},l^{\prime}_{v},\Omega^{\prime};v,l_{v},\Omega}(\beta,\chi,E)=S^{(J)}_{v^{\prime},l^{\prime}_{v};v,l_{v}}(\beta,E)\exp[i(l^{\prime}_{v}-l_{v})\chi].$
(39)

This requires the $\beta$-dependent VCC $S$-matrix in the complex basis. The
transformation of the VCC $S$-matrices in the symmetry adapted basis of Eq.
(37) to the complex basis is given by Eqs. (14) and (15) in Ref. clary:83 .
For $\tilde{l}^{\prime}_{v}=0$ (or $\tilde{l}_{v}=0$) it is

$S^{(J)}_{v^{\prime},\pm{\tilde{l}^{\prime}_{v}};v,\pm{\tilde{l}_{v}}}(\beta,\chi)=\frac{1}{\sqrt{2}}S^{(J,\mathrm{e})}_{v^{\prime},\tilde{l}^{\prime}_{v};v,\tilde{l}_{v}}(\beta,\chi)e^{\pm
i(\tilde{l}_{v}^{\prime}-\tilde{l}_{v})}.$ (40)

Then, following Green green:79b  and Clary clary:83 , we define rotation-
vibration $S$-matrices

$\overline{S}^{(J)}_{v^{\prime},l^{\prime}_{v},j^{\prime},\Omega^{\prime};v,l_{v},j,\Omega}(E)=\delta_{\Omega^{\prime},\Omega}\langle\mskip
3.0mul^{\prime}_{v},j^{\prime},\Omega\mskip 3.0mu|\mskip
3.0mu\overline{S}^{(J)}_{v^{\prime},l^{\prime}_{v};v,l_{v}}(\beta,\chi,E)\mskip
3.0mu|\mskip 3.0mul_{v},j,\Omega\mskip 3.0mu\rangle,$ (41)

containing matrix elements of these two-angle dependent $S$-matrices over the
rotational parts of the CO2 monomer basis functions

$|\mskip 3.0mul_{v},j,\Omega\mskip
3.0mu\rangle=\sqrt{\frac{2j+1}{4\pi}}D_{\Omega,l_{v}}^{(j)}(0,\beta,\chi)^{\ast}.$
(42)

The next step is to transform the $S$-matrix to a $\hat{P}_{13}$ adapted
basis. In general, each element of the $S$-matrix in the symmetry adapted
basis is a linear combination of four unadapted $S$-matrix elements (see Eq.
(12) in Ref. clary:83 ), but when the final state has
$\tilde{l}^{\prime}_{v}=0$ and the initial state has $\tilde{l}_{v}>0$, as in
our case, only two terms remain,

$S^{(J,\epsilon)}_{v^{\prime},\tilde{l}^{\prime}_{v},j^{\prime}\Omega;v,\tilde{l}_{v},j,\Omega}(E)=\frac{1}{\sqrt{2}}\left[\overline{S}^{(J)}_{v^{\prime},\tilde{l}_{v}^{\prime},j^{\prime},\Omega;v,\tilde{l}_{v},j,\Omega}(E)+\epsilon(-1)^{j}\,\overline{S}^{(J)}_{v^{\prime},\tilde{l}^{\prime}_{v},j^{\prime},\Omega;v,{-\tilde{l}_{v}},j,\Omega}(E)\right].$
(43)

Note that our $\hat{P}_{13}$ symmetry label $\epsilon$ is a good quantum
number, whereas the $\epsilon$ in Clary’s paper clary:83  differs from ours by
a factor $(-1)^{j}$ for $l_{v}\neq 0$. For $l_{v}=0$ Clary’s $\epsilon$ is set
to zero.

The rovibrationally inelastic cross sections can be obtained from the
$S$-matrices in Eq. (43)

$\sigma_{v^{\prime},\tilde{l}^{\prime}_{v},j^{\prime},\epsilon\leftarrow
v,\tilde{l}_{v},j,\epsilon}(E)=\frac{\pi}{k^{2}_{v}\,(2j+1)}\sum_{J,\Omega}(2J+1)\left|S^{(J,\epsilon)}_{v^{\prime},\tilde{l}^{\prime}_{v},j^{\prime},\Omega;v,\tilde{l}_{v},j,\Omega}(E)\right|^{2}.$
(44)

The calculation of the matrix elements in Eq. (41) can be carried out
analytically when the angle-dependent matrices
$\overline{S}^{(J)}_{v^{\prime},l^{\prime}_{v};v,l_{v}}(\beta,\chi,E)$ are
expanded in spherical harmonics $Y_{LM_{L}}$

$\overline{S}^{(J)}_{v^{\prime},l^{\prime}_{v};v,l_{v}}(\beta,\chi,E)=\sqrt{2\pi}\sum_{L,M_{L}}Y_{L,M_{L}}(\beta,\chi)S^{(J,L)}_{v^{\prime},l_{v}^{\prime};v,l_{v}}(E).$
(45)

The expansion coefficients are given by

$S^{(J,L)}_{v^{\prime},l_{v}^{\prime};v,l_{v}}(E)=\sqrt{2\pi}\int_{0}^{\pi}Y_{L,l^{\prime}_{v}-l_{v}}(\beta,0)S^{(J)}_{v^{\prime},l^{\prime}_{v};v,l_{v}}(\beta,E)\,\sin\beta\,d\beta.$
(46)

Only $M_{L}=l^{\prime}_{v}-l_{v}$ contributes, because of the phase factor
$\exp[i(l^{\prime}_{v}-l_{v})\chi]$ in Eq. (39). The $\sqrt{2\pi}$ normalizes
the spherical harmonics with the azimuthal angle set to zero. The general
expression of the cross sections in terms of the expansion coefficients is
rather complex (see Eq. (35) of Green green:79b ), but again, for our case
with $\tilde{l}_{v}^{\prime}=0$ and $\tilde{l}_{v}>0$ it is simpler. It
corresponds to Eqs. (17)-(20) of Ref. clary:83  and in our notation reads

$\sigma_{v^{\prime},\tilde{l}^{\prime}_{v},j^{\prime}\leftarrow
v,\tilde{l}_{v},j}(E)=\frac{\pi}{2k^{2}_{v}}\sum_{L}|\langle\mskip
3.0muL,\tilde{l}_{v}^{\prime}-\tilde{l}_{v},j,\tilde{l}_{v}\mskip 3.0mu|\mskip
3.0muj^{\prime}\tilde{l}_{v}^{\prime}\mskip
3.0mu\rangle|^{2}\sum_{J}(2J+1)\left|S_{v^{\prime},\tilde{l}^{\prime}_{v};v,\tilde{l}_{v}}^{(J,L,\mathrm{e})}\right|^{2},$
(47)

with the expansion coefficients
$S_{v^{\prime},\tilde{l}^{\prime}_{v};v,\tilde{l}_{v}}^{(J,L,\mathrm{e})}$ for
the symmetrized $p=\mathrm{e}$ basis obtained from Eq. (46) by writing
$S^{(J,\mathrm{e})}_{v^{\prime},\tilde{l}^{\prime}_{v};v,\tilde{l}_{v}}(\beta,E)$
instead of $S^{(J)}_{v^{\prime},l^{\prime}_{v};v,l_{v}}(\beta,E)$. For our
case with $l^{\prime}_{v}=0$ and only even $j^{\prime}$ the constant $B$ in
Eq. (19) of Ref. clary:83,  is 1 for terms with odd values of $j+L$, while it
is 0 for even values.

###  IV.5 Technical details

In this section we specify the parameters used in our scattering calculations.
The radial grid in the renormalized Numerov propagator ranges from $R=3$ to 15
$a_{0}$ in 224 equal steps. We started with an outer $R$ value of 35 $a_{0}$,
but found out that this value could be reduced to 15 $a_{0}$ without loss of
accuracy in the cross sections. We included bend vibrational states with
$v=0,1,$ and 2, with $\tilde{l}_{v}=0$, 1, and (0, 2), respectively.
Convergence tests were carried out in which we also included the $v=3$
functions with $\tilde{l}_{v}=(1,3)$ in the channel basis, but the differences
in the cross sections were only about 1% for low collision energies where
resonances occur and for the highest energies, and less in the intermediate
energy range. So in the final calculations we omitted the $v=3$ functions. The
channel basis contained CO2 monomer rotational states with a maximum $j$ value
of 50 for each $v$. The minimum $j$ value is $\tilde{l}_{v}$.

The cross sections were calculated for collision energies from 1 to 2000 cm-1,
with a step size of 0.2 cm-1 for $E\leq 16$ cm-1, 1 cm-1 up to 40 cm-1, 5 cm-1
up to 50 cm -1, 10 cm-1 up to 100 cm-1, and 100 cm-1 up to 2000 cm-1. Sharp
peaks occur in the cross sections at low energies, due to resonances, so we
had to use a fine energy grid in this region. The maximum value of the total
angular momentum $J$ needed to converge the cross sections depends on the
energy; it was $J_{\rm max}=20$ for $E\leq 100$ cm-1, 35 for $100<E\leq 500$
cm-1, 50 for $500<E\leq 1000$ cm-1, 60 for $1000<E\leq 1500$ cm-1, and 70 for
$1500<E\leq 2000$ cm-1. Not all $J$ values were actually used; for energies
above 16 cm-1 $J$ was increased in steps of 4 and the cross sections were
obtained for all $J$’s by cubic spline interpolation over the available $J$
values. This was allowed because the cross sections summed over both parities
vary smoothly with $J$, except for the resonances at low energies. The
integration over energy in Eq. (31) for the rate coefficients was done by
first making a cubic spline interpolation of the cross sections at the
energies for which they were calculated and next applying the trapezoidal rule
on an energy grid with a spacing of 0.2 cm-1. These energies range up to 2000
cm-1, which is sufficient to obtain rate coefficients up to $T=300$ K. In
order to obtain reliable rates up to $T=500$ K we extrapolated the
exponentially decaying high energy tail of the integrand in Eq. (31) by a
simple exponential function $a\exp(-bR)$ with coefficients $a,b$ fitted to the
integrand at the highest two energies and extended the integration up to an
energy of 5000 cm-1.

In the VCC-IOS calculations the integration over $\beta$ in Eqs. (37) and (46)
was done numerically using Gauss-Legendre quadrature. When calculating
vibrational (de)excitation cross sections from Eq. (37) we used only 16 fixed
monomer angles $\beta_{i}$ ranging from 0 to $\pi$, but when calculating
rovibrationally inelastic cross sections from Eqs. (47) and (46) the number of
quadrature points had to be much larger (up to 100) in order to achieve
convergence.

Our computer codes were written in the free and open source script language
Scilab, version 6.1.1 scilab2022  and computations were done on a cluster of
linux servers.

##  V Results and discussion

###  V.1 Cross sections

Before presenting our results calculated with the potential described in Sec.
III, let us mention that we also calculated $v=1,\tilde{l}_{v}=1\rightarrow
v^{\prime}=0,\tilde{l}^{\prime}_{v}=0$ quenching cross sections with the model
potential used by Clary et al. clary:83 ; banks:87b , which exponentially
depends on the He-O and He-C distances with parameters based on Hartree-Fock
calculations. The CSA and VCC-IOS cross sections that we calculated with this
potential differ at most by a few percent from the results in Ref. banks:87b ,
which confirms the correctness of both their and our CSA and VCC-IOS programs.
The small differences are probably due to some technical differences between
our scattering calculations and those in Ref. banks:87b , which were not
specified in all detail.

Figure 3: Vibrational $v=1,\tilde{l}_{v}=1\rightarrow
v^{\prime}=0,\tilde{l}^{\prime}_{v}=0$ quenching cross sections sections from
CC, CSA, and VCC-IOS calculations with our potential, in comparison with CSA
and VCC-IOS results calculated with the atom-atom model potential of Clary et
al. clary:83 ; banks:87b . In the CC and CSA calculations the initial
rotational state has $j=1$ and the cross sections are summed over all final
$j^{\prime}$ states. Figure 4: Vibrational $v=1,\tilde{l}_{v}=1\rightarrow
v^{\prime}=0,\tilde{l}^{\prime}_{v}=0$ quenching cross sections sections from
CC, CSA, and VCC-IOS calculations with our potential. In the CC and CSA
calculations the initial rotational state has $j=10$, the VCC-IOS results are
the same as in Fig. 3.

Figure 3 shows the results of these calculations in comparison with those
obtained with the CC, CSA, and VCC-IOS methods and our potential in Sec. III.
The results are quite similar for collision energies above 500 cm-1. For the
highest energies Clary’s potential yields slightly larger cross sections,
which is probably due to this potential being more strongly repulsive than
ours and the atom-atom model producing a stronger coupling between the $v=1$
and $v=0$ bend modes of CO2. At energies below 500 cm-1 the cross sections
from our potential are much larger, with the difference increasing to more
than two orders of magnitude at collision energies below 10 cm-1. We note
incidentally that in Ref. banks:87b  the lowest energy at which the cross
sections were calculated was 0.01 eV, which corresponds to about 80 cm-1.
Clary’s potential based on SCF calculations completely lacks the attractive
dispersion interactions which are present in our potential and we think that
this explains the much smaller cross sections that it yields at low collision
energies. Another noticeable difference is the presence of scattering
resonances manifested by the peaks in the cross section at energies below 20
cm-1, which are completely missing in the results calculated with Clary’s
potential. The reason that the latter potential does not produce resonances is
that it is purely repulsive, so it has no Van der Waals (VdW) minimum, which
implies that it cannot give rise to bound or quasi-bound states. Our potential
based on presently available computational electronic structure methods is
accurate also in the region of the VdW minimum, and the rate coefficients of
rovibrationally inelastic collisions presented below can also be trusted at
low temperatures.

We see also in Fig. 3 that the total vibrational
$v=1,\tilde{l}_{v}=1\rightarrow v^{\prime}=0,\tilde{l}^{\prime}_{v}=0$
quenching cross section obtained from the approximate CSA and VCC-IOS methods
agrees fairly well with the result from the numerically exact CC method. Only
the subtle resonance structures at low energies are more different, but these
depend very sensitively on the potential and on the scattering method used.
The CSA and CC cross sections in this figure were computed with the lowest
initial $v=1,\tilde{l}_{v}=1$ rotational state with $j=1$. Figure 4 shows that
this holds also for an initial rotational state with $j=10$, with a resonance
structure that is less pronounced. Comparison of Figs. 3 and 4 shows,
moreover, that the total vibrational quenching cross section is quite similar
for different initial $j$ states.

Figure 5: Product $j^{\prime}$ distributions in
$v=1,\tilde{l}_{v}=1,j\rightarrow
v^{\prime}=0,\tilde{l}^{\prime}_{v}=0,j^{\prime}$ transitions with initial
$j=1$ from state-to-state CC, CSA, and VCC-IOS calculations at different
collision energies. Panels (a), (b), (c), and (d) correspond to collision
energies of 10, 100, 500, and 2000 cm-1, respectively. Figure 6: Same as Fig.
5, with initial rotational state $j=10$. Panels (a), (b), (c), and (d)
correspond to collision energies of 10, 100, 500, and 2000 cm-1, respectively.

The CC and CSA methods directly produce rotationally resolved
$v=1,\tilde{l}_{v}=1,j\rightarrow
v^{\prime}=0,\tilde{l}^{\prime}_{v}=0,j^{\prime}$ cross sections and we
explained in Sec. IV.4 how the VCC-IOS method has been extended clary:83  to
also yield such rotational state-to-state cross sections. The $j^{\prime}$
product distributions from the different methods are displayed for various
collision energies in Figs. 5 and 6 for initial $j=1$ and $j=10$,
respectively. In Fig. 5 we observe that for initial $j=1$ higher and higher
$j^{\prime}$ states are excited when the collision energy is increased, which
is natural of course, and that the CC, CSA, and VCC-IOS methods show similar
trends. The largest differences occur at the energy of 10 cm-1, but this is in
the region where scattering resonances occur. The differences also become
larger at higher collision energies, but they typically stay in the range of
20 to 50%, with the CSA method yielding smaller $j^{\prime}$ values than CC
and the VCC-IOS method yielding larger $j^{\prime}$. In Fig. 6 we observe that
for initial $j=10$ there is a very strong preference for rotationally elastic
$j=10\rightarrow j^{\prime}=10$ transitions, although also here the
$j^{\prime}$ distribution naturally becomes wider at higher collision
energies. Again, the different methods produce quite similar results, with the
CSA method yielding smaller $j^{\prime}$ values than CC and the VCC-IOS method
yielding larger $j^{\prime}$. Also here the largest deviations occur in the
region of the resonances at 10 cm-1.

Figure 7: State-to-state cross sections as functions of the energy for
$v=1,\tilde{l}_{v}=1,j\rightarrow
v^{\prime}=0,\tilde{l}^{\prime}_{v}=0,j^{\prime}$ transitions with initial
$j=1$ and different $j^{\prime}$ from state-to-state CC, CSA, and VCC-IOS
calculations. Panels (a), (b), (c), and (d) correspond to $v^{\prime}=0$ final
states with $j^{\prime}=0$, 4, 10, and 20, respectively. Figure 8: Same as
Fig. 7, with initial rotation state $j=10$. Panels (a), (b), (c), and (d)
correspond to $v^{\prime}=0$ final states with $j^{\prime}=0$, 4, 10, and 20,
respectively.

State-to-state $v=1,\tilde{l}_{v}=1,j\rightarrow
v^{\prime}=0,\tilde{l}^{\prime}_{v}=0,j^{\prime}$ cross sections from CC, CSA,
and VCC-IOS calculations are shown as functions of the collision energy for
different $j^{\prime}$ values in Figs. 7 and 8 for initial $j=1$ and $j=10$,
respectively. These figures confirm that the cross sections from the
approximate CSA and VCC-IOS methods agree to within 50% with those from the
full CC method also at the state-to-state level for energies higher than 30
cm-1 and that the deviations become larger for lower energies where the
resonances occur. Also for larger $\Delta j=|j^{\prime}-j|$ where the cross
sections become smaller the agreement gets somewhat worse. It is striking that
the more approximate VCC-IOS method yields cross ssections, even at the
rotational state-to-state level, that agree as well with the full CC data as
the CSA results, and in some cases even better.

Apart from the quality of the approximate CSA and VCC-IOS results in
comparison with the full CC data, we may also discuss some general trends in
the cross sections. One can observe in the above figures that $\Delta j$, the
difference between the rotational quantum numbers, is more significant in
determining the magnitude of the cross sections than the energy gap between
the initial $v=1,\tilde{l}_{v}=1,j$ and final
$v^{\prime}=0,\tilde{l}^{\prime}_{v}=0,j^{\prime}$ states. We note here that
the energy gap between the lowest vibrationally excited state with
$v=1,\tilde{l}_{v}=1,j=1$ and the
$v^{\prime}=0,\tilde{l}^{\prime}_{v}=0,j^{\prime}$ state is smallest for
$j^{\prime}=41$. Generally the cross sections are smallest in the energy range
between 30 and 100 cm-1 and increase by at least two orders of magnitude when
the energy is raised to 2000 cm-1. This increase is larger when $\Delta j$ is
larger.

A feature in Figs. 7 and 8 that is particularly striking is that the cross
sections from CSA calculations become very small for collision energies below
2 cm-1. This may be explained as follows. In the BF coordinates on which the
CSA method is based the centrifugal barrier is represented by a term that
includes diagonal and off-diagonal Coriolis couplings, see Eq. (16). The
latter are omitted in CSA, which effectively increases the height of the
centrifugal barrier. This implies that at low energies the colliding CO2
molecule and He atom are more strongly prevented from getting closer, where
the potential coupling is larger and the vibrational transitions occur. We
confirmed this explanation by CSA calculations in which we lowered the
centrifugal barrier by adding a term to the diagonal angular kinetic energy
that more or less compensates for the omission of the off-diagonal terms.
Indeed, we found that this brings the CSA cross sections at low energies
substially closer to the CC results. In Fig. 8(a) the CSA cross sections are
smaller than those from CC even at higher energies, but also the CC cross
sections are extremely small in that case.

Another interesting observation regards the partial cross sections from
different total angular momenta $J$ and different parities. Instead of the
total parity $P$ with respect to overall inversion, we consider the
spectroscopic parity $P(-1)^{J}$. States with even and odd spectroscopic
parity are conventionally labeled $e$ and $f$, respectively. The partial cross
sections for different $J$ values summed over both parities are displayed in
Fig. S2. One observes, quite naturally, that for higher collision energies the
contributions from higher $J$’s become more important. And that the maximum
values of $J=20$, 50, and 70 used in different energy ranges produce well
converged cross sections. Since the initial monomer angular momentum is $j=1$
in this example, the total angular momentum $J$ is nearly equal to the orbital
angular momentum $L$, see Sec. IV.2, which is related to the impact parameter
$b$ as $L=\mu bv$. From the data shown in Fig. S2 it follows then that the
impact parameters are $b=3.24$, 2.05, and 2.08 $a_{0}$ for collision energies
of 100, 1000, and 2000 cm-1. These impact parameters are small, especially at
higher energies, which indicates that (ro)vibrational transitions mostly take
place for nearly head-on collisions and happen in the region where the
potential is strongly repulsive and the coupling potential is relatively
large, see Fig. 2.

Figure 9: Partial wave distributions of the $v=1,\tilde{l}_{v}=1,j$ to
$v^{\prime}=0,\tilde{l}^{\prime}_{v}=0$ cross sections computed at the CC
level at collision energy $E=100$ cm-1 summed over all final $j^{\prime}$
values and resolved by spectroscopic parity $e$ or $f$. Panels (a), (b), and
(c) correspond to the odd initial $j$ values 1, 5, and 9, panels (d), (e), and
(f) to the even $j$ values 2, 6, and 10. The closed black curves are the total
partial cross sections, the dashed red and dot-dashed blue curves are the
contributions from parities $e$ and $f$, respectively. Note that for odd
initial $j$ the contributions from parity $e$ are dominant, especially for
$j=1$, while for even $j$ the contributions from parities $e$ and $f$ are
nearly equal.

The relative contributions from the scattering states with spectroscopic
parities $e$ and $f$ to the total cross sections are less obvious, see Fig. 9.
For initial $j=1$ the contributions from the $e$ states are very dominant, and
also for higher odd $j$ values the $e$ states contribute more to the cross
sections than the $f$ states. For even initial $j$ values, on the other hand,
the contributions from the $e$ and $f$ states are practically equal. This can
be explained by considering the role of the approximate quantum number
$\Omega$ in the BF formalism explained in Sec. IV.2. The absolute value of
$\Omega$ is limited by the initial $j$ value, the final $j^{\prime}$ value,
and the total angular momentum $J$. When analyzing the state-to-state cross
sections, we found that the largest allowed absolute $\Omega$ values yield the
largest contributions. It follows from Eq. (15) that for the ground state with
$v^{\prime}=\tilde{l}^{\prime}_{v}=0$ only states with even $j^{\prime}$ are
allowed, and from Eq. (18) that for $j^{\prime}=\Omega=0$ only states with
parity $e$ occur. Furthermore, it follows from the second and third $3j$
symbols in Eq. (26) with $l_{v}=1$ and $l^{\prime}_{v}=0$ that $m_{\lambda}$
must be odd, from the reflection symmetry of the potential in Sec. III that
also $\lambda$ must then be odd, and from the first $3j$ symbol in Eq. (26)
that for $\Omega=0$ the sum of $j$ and $j^{\prime}$ must be odd as well. We
already derived that $j^{\prime}$ must be even, so only for odd initial $j$
values the $\Omega=0$ states contribute to the cross sections. Because these
$\Omega=0$ states are purely of parity $e$, this explains the observed
dominance of the $e$ states over the $f$ states in the cross sections with odd
initial $j$. For the initial states with even $j$ the cross sections do not
contain this $\Omega=0$ component with pure $e$ parity and therefore the $e$
and $f$ parity contributions are equal. The dominance of the $e$ states over
the $f$ states for odd initial $j$ values is most pronounced for initial $j=1$
and decreases with increasing $j$. This follows because for the initial
$v=\tilde{l}_{v}=j=1$ state $|\Omega|$ is limited to 0 and 1, and the
restriction to even $j^{\prime}$ implies that only the $\Omega=0$ state with
parity $e$ contributes to the cross section through the potential, see Eq.
(26). For larger $j$ and $j^{\prime}$ also $\Omega\neq 0$ states contribute
directly to the cross sections through the potential and the relative
contribution from the $\Omega=0$ states becomes less important.

Another feature that can be observed in Figs. 9(d), (e), and (f) where the
cross sections do not contain the $\Omega=0$ contribution is that for total
$J$ smaller than $j$ and $j^{\prime}$, where the maximum $|\Omega|$ is limited
by $J$, the partial cross sections remain small and rise steeply when $J$
becomes equal to the minimum of $j$ and $j^{\prime}$. This confirms that,
indeed, functions with $|\Omega|$ equal to min$(j,j^{\prime})$ yield the
largest contributions to the cross sections. Since $\Omega$ is the projection
of the CO2 angular momentum ${\bm{j}}$ on the intermolecular axis ${\bm{R}}$
this suggests that for rovibrational transitions involving the CO2 bend mode
sideways collisions are most effective. Figure S3 in the Supplementary
material shows the $e/f$ parity-resolved state-to-state cross sections for
initial states with $j=9$ and $j=10$ and final $j^{\prime}$ values ranging up
to 25. This figure shows that our findings regarding the different parity
contributions to the cross sections for odd and even initial $j$ values also
hold for the state-to-state cross sections: the $e$ parity contributions
dominate over those of $f$ parity for odd $j=9$, especially for low final
$j^{\prime}$, while the $e$ and $f$ contributions are equal for even $j=10$.
It also shows that the $j^{\prime}$ dependence of the state-to-state cross
sections is similar to the total $J$ dependence of the partial cross sections
in Fig. 9, which follows from the limitation of the maximum $|\Omega|$ by both
$j^{\prime}$ and $J$.

###  V.2 Rate coefficients

In Fig. 10 we display some state-to-state $v=1,\tilde{l}_{v}=1,j\rightarrow
v^{\prime}=0,\tilde{l}^{\prime}_{v}=0,j^{\prime}$ rovibrational transition
rates from CC calculations for different initial $j$ and final $j^{\prime}$
values as functions of the temperature. Such rovibrational state-to-state
rates are the data needed in radiative transfermodels. For lower initial $j$
the rates of transitions to different $j^{\prime}$ states differ by at most
one order of magnitude at low temperature and become more similar with
increasing temperature. For higher initial $j$ one clearly observes that
transitions to $j^{\prime}=j$ are the strongest and that especially
transitions to $j^{\prime}$ smaller than $j$ are weak. More generally, these
features indicate that transitions with small $\Delta j=|j^{\prime}-j|$ are
favored and that transitions with $j^{\prime}>j$ are stronger than those with
$j^{\prime}<j$. The latter propensity can be explained by the energy gap law:
the energy difference between the initial $v=1,j$ and final
$v^{\prime}=0,j^{\prime}$ states is smaller when $j^{\prime}>j$. We will
include a more complete set of rovibrational transition rates in the LAMDA
data base lamda:05 .

Figure 10: State-to-state transition rate coefficients from CC calculations
for different initial $j$ and final $j^{\prime}$ values. Panels (a), (b), (c),
and (d) correspond to $v=1$ initial states with $j=1$, 3, 6, and 10,
respectively.

Figures S4 and S5 in the Supplementary material show the state-to-state
$v=1,\tilde{l}_{v}=1,j\rightarrow
v^{\prime}=0,\tilde{l}^{\prime}_{v}=0,j^{\prime}$ rovibrational transition
rates for initial $j=1$ and 10 and different final $j^{\prime}$ values from
CC, CSA, and VCC-IOS calculations. Both the CSA and VCC-IOS rates agree to
within about 50% with the CC results, just as the corresponding cross
sections, except when $j^{\prime}$ is smaller than $j$ and the rates become
insignificant.

The rates increase monotonically with the temperature, except for transitions
involving low $j$ or low $j^{\prime}$ states for which the rates have minima
around 50 K. The relatively large rates at low temperature for these
transitions are due to contributions from scattering resonances, as shown in
Fig. 7. This figure shows the strongest scattering resonances in the cross
sections from CC calculations, weaker ones in the CSA results, and practically
none in the VCC-IOS cross sections, which clearly explains why the minima in
the rate coefficients from the CC, CSA and VCC-IOS methods in Figs. S6 and S7
of the Supplementary material are strongest for CC, weaker for CSA, and absent
for VCC-IOS.

In Figs. S6 and S7 of the Supplementary material we compare the total
vibrational quenching rates, i.e., the rovibrational transition rates summed
over all final $j^{\prime}$ values, calculated with the different methods,
with initial $j=1$ and 10 in the CC and CSA methods. No initial $j$ value is
defined in the VCC-IOS method, so the VCC-IOS quenching rates are identical in
the two figures. Also the CC and CSA quenching rates are quite similar for
$j=1$ and $j=10$, except at the lowest temperatures where the minima are more
pronounced for initial $j=1$, due to the stronger resonance contributions at
low $j$. We observe that the CSA method somewhat underestimates the rates from
CC calculations, while the VCC-IOS method yields slightly higher rates. The
differences are quite small, however, on the order of 20% or less. These
differences between the approximate CSA and VCC-IOS methods and the CC method
are smaller than in many of the state-to-state rate coefficients shown in
Figs. S4 and S5, which is because the smaller state-to-state rates are more
sensitive to the approximations made than the most significant ones. The
larger total vibrational quenching rates from VCC-IOS and smaller rates from
CSA, as compared to the CC results, are due to the contributions from higher
final $j^{\prime}$ values such as shown in Figs. S4(d) and S5(d). We already
concluded from Figs. 5 and 6 that these are favored by VCC-IOS relative to CC
and disfavored by CSA.

Figure 11: Vibrational quenching rates from CC calculations for
$v=1,\tilde{l}_{v}=1,j\rightarrow v^{\prime}=0,\tilde{l}^{\prime}_{v}=0$
transitions with different initial $j$ values ranging from 1 to 10 compared
with experimental results siddles:94 .

Figure 11 shows the total vibrational quenching rates for initial $j$ values
ranging from 1 to 10. It shows even more clearly that the total vibrational
quenching rate hardly depends of the initial $j$, except at temperatures below
50 K where the rates for low initial $j$ values are enhanced by resonance
contributions. This figure also shows that our calculated total vibrational
quenching rate agrees with the most recent experimental data within the error
bars, which confirms the accuracy of our results.

The rovibrational transition rate coefficients presented in this paper refer
to vibrational de-excitation from $v=1,\tilde{l}_{v}=1,j$ to
$v^{\prime}=0,\tilde{l}^{\prime}_{v}=0,j^{\prime}$. The corresponding
rovibrational excitation rates can be easily obtained from the detailed
balance relation heijmen:99a . In our previous papers on CO2-He collisions
with CO2 excited in the stretch modes selim:21 ; selim:22  we also
investigated the accuracy of cross sections and rate coefficients from the
multi-channel distorted-wave Born approximation (MC-DWBA) and the nearest-
neighbor Coriolis coupling (NNCC) method, which is an extension of the CSA
method, and we compared the computational efficiency of both methods relative
to CC. Our conclusion was that cross sections and rates from the MC-DWBA
method are practically equal to those from the full CC method, or from the
NNCC method when MC-DWBA was combined with the latter method. The extension of
CSA to NNCC resulted in better agreement with full CC results. We did not
investigate these methods in the present paper on the bend mode of CO2, but we
expect similar results in terms of accuracy and efficiency.

##  VI Conclusions

With the use of a newly computed four-dimensional CO2-He potential which
includes the CO2 bend coordinate we calculated the cross sections
$\sigma_{{v^{\prime},l^{\prime}_{v},j^{\prime}\leftarrow v,l_{v},j}}(E)$ and
rate coefficients $k_{{v^{\prime},l^{\prime}_{v},j^{\prime}\leftarrow
v,l_{v},j}}(E)$ of rovibrational transitions between different bend
vibrational and rotational ($v,j$) states of CO2 induced by collisions with He
atoms. The quantum number $l_{v}$ represents the vibrational angular momentum
generated by the bend mode of CO2, which is linear at equilibrium. In our
scattering calculations we used the numerically exact coupled-channels (CC)
method, but also the coupled-states approximation (CSA) and the vibrational
close-coupling rotational infinite-order sudden (VCC-IOS) approximation. The
effects of each of these approximations on the rovibrational cross sections
and rates was found to be less than 50% at the rotational state-to-state
level, except for the smaller ones and in the low energy resonance region, and
less than 20% on the overall vibrational quenching rates, except for
temperatures below 50 K where resonances provide a substantial contribution.
Our calculated collisional state-to-state transition rate coefficients can be
used in modeling interstellar non-LTE environments and our results show that
they may also be obtained from the computationally less demanding CSA or VCC-
IOS methods when the high accuracy achieved by CC calculations is not
required. Our CC quenching rates agree with the most recent experimental data
siddles:94  withing the error bars.

The fairly good performance of the VCC-IOS method seems surprising, since we
found in calculations on the symmetric and asymmetric stretch modes of CO2
selim:21  in collisions with He that the cross sections and rate coefficients
from VCC-IOS calculations differ by one to three orders of magnitude from full
CC results. The disagreement was worse for the asymmetric stretch mode with
frequency 2349 cm-1 than for the symmetric stretch mode at 1333 cm-1 and the
cross sections were larger for the latter mode, so we concluded in Ref.
selim:21  that the quality of VCC-IOS ameliorates with the magnitude of the
cross sections and we guessed already that VCC-IOS might perform better for
the bend mode at 667 cm-1. Figure S8 in the Supplement shows total quenching
cross sections from CC and VCC-IOS calculations of the bend mode at collision
energies 100 and 1000 cm-1 calculated for hypothetical frequencies in the
range from 333 to 1500 cm-1. This figure confirms our expectations. It shows
clearly that the VCC-IOS cross sections agree fairly well with CC results up
to a certain frequency, but deviate more and more for higher frequencies. For
collision energy 100 cm-1 this critical frequency is about 850 cm-1, which is
higher than the frequency of 667 cm-1 of the bend mode. For higher collision
energies the cross section is larger and the critical frequency, i.e., the
energy gap between the $v=1$ and $v=0$ states where the VCC-IOS cross section
starts deviating from the CC result, increases; at 1000 cm-1 it is about 1200
cm-1.

We also compared our cross sections and rates for CO2(bend)-He collisions with
data from Clary et al. clary:83 ; banks:87b  calculated in the 1980’s with the
CSA and VCC-IOS methods and a simple atom-atom model potential based on ab
initio Hartree-Fock calculations. Their cross sections agree fairly well with
ours for collision energies above 500 cm-1 but at lower energies the cross
sections from our potential are much larger, with the difference increasing to
more than two orders of magnitude at collision energies below 10 cm-1. This
shows that the inclusion of long range attractive dispersion interactions is
crucial to obtain reliable cross sections at lower energies and rate
coefficients at lower temperatures.

Finally we note that the computer programs and the knowledge about the
accuracy of the approximate but computationally simpler CSA and VCC-IOS
methods will also be useful to provide similar collisional data of
astronomical interest for collisions of CO2 with H2 and for other linear
molecules of which bend mode spectra have been observed and are used in
modeling, such as C2H2 and HCN.

## Data availability

The data that support the findings of this study are available within the
article and the Supplementary material.

## Supplementary material

The Supplementary material contains Figs. S1 to S8, which are discussed in the
main text. It also contains a copy of the Fortran program that calculates the
CO2-He potential for CO2 being deformed along the bend mode normal coordinate.

## Acknowledgements

We thank Ewine van Dishoeck and Arthur Bosman for stimulating and useful
discussions. The work is supported by The Netherlands Organisation for
Scientific Research, NWO, through the Dutch Astrochemistry Network DAN-II.

## References

  * (1) E. Roueff and F. Lique,  Chem. Rev. 113, 8906 (2013).
  * (2) A. G. Palluet, F. Thibault, and F. Lique,  J. Chem. Phys. 156, 104303 (2022).
  * (3) L. Song, A. van der Avoird, and G. C. Groenenboom,  J. Phys. Chem. A 117, 7571 (2013).
  * (4) L. Song, N. Balakrishnan, A. van der Avoird, T. Karman, and G. C. Groenenboom,  J. Chem. Phys. 142, 204303 (2015).
  * (5) K. M. Walker, L. Song, B. H. Yang, G. C. Groenenboom, A. van der Avoird, N. Balakrishnan, R. C. Forrey, and P. C. Stancil,  Astrophys. J. 811, 27 (2015).
  * (6) L. Song, N. Balakrishnan, K. M. Walker, P. C. Stancil, W.-F. Thi, I. Kamp, A. van der Avoird, and G. C. Groenenboom,  Astrophys. J. 813, 96 (2015).
  * (7) T. Selim, A. Christianen, A. van der Avoird, and G. C. Groenenboom,  J. Chem. Phys. 155, 034105 (2021).
  * (8) T. Selim, A. van der Avoird, and G. C. Groenenboom,  J. Chem. Phys. 157, 064105 (2022).
  * (9) A. M. S. Boonman, E. F. van Dishoeck, F. Lahuis, S. D. Doty, C. M. Wright, and D. Rosenthal,  Astron. Astrophys. 399, 1047–1061 (2003).
  * (10) A. M. S. Boonman, E. F. van Dishoeck, F. Lahuis, and S. D. Doty,  Astron. Astrophys. 399, 1063–1072 (2003).
  * (11) P. Sonnentrucker, D. A. Neufeld, P. A. Gerakines, E. A. Bergin, G. J. Melnick, W. J. Forrest, J. L. Pipher, and D. C. B. Whittet,  Astrophys. J. 672, 361 (2008).
  * (12) S. L. Grant et al.,  Astrophys. J. Lett. 947, L6 (2023).
  * (13) C. K. N. Patel,  Phys. Rev. A. 136, 187 (1964).
  * (14) A. A. Offenberger and D. J. Rose,  J. Appl. Phys. 41, 3908 (1970).
  * (15) D. C. Clary,  J. Chem. Phys. 78, 4915 (1983).
  * (16) W. H. Kasner and L. D. Pleasance,  Appl. Phys. Lett. 31, 82 (2008).
  * (17) D. Allen, J. Haigh, J. T. Houghton, and C. Simpson,  Nature 281, 660 (1979).
  * (18) F. W. T. M. Lopez-Puertas, M. A. Lopez-Valverde,  Royal Meteorological Society, Quarterly Journal 118, 499 (1992).
  * (19) A. G. Feofilov, A. A. Kutepov, C.-Y. She, A. K. Smith, W. D. Pesnell, and R. A. Goldberg,  Atmos. Chem. Phys. 12, 9013–9023 (2012).
  * (20) J. T. E. C. E. R. S. Team,  Nature 614, 649–652 (2023).
  * (21) T. Shimanouchi,  Tables of Molecular Vibrational Frequencies Consolidated, Volume I,  National Bureau of Standards, Gaithersburg, 1972.
  * (22) D. C. Clary,  J. Chem. Phys. 75, 209 (1981).
  * (23) D. C. Clary,  Mol. Phys. 43, 469 (1981).
  * (24) D. C. Clary,  Chem. Phys. 65, 247 (1982).
  * (25) A. J. Banks and D. C. Clary,  J. Chem. Phys. 86, 802 (1987).
  * (26) C. T. Wickham-Jones, C. J. S. M. Simpson, and D. C. Clary,  Chem. Phys. 117, 9 (1987).
  * (27) F. Lepoutre, G. Louis, and J. Taine,  J. Chem. Phys. 70, 2225 (1979).
  * (28) R. Siddles, G. Wilson, and C. Simpson,  Chem. Phys. 189, 779 (1994).
  * (29) A. D. Bosman, S. Bruderer, and E. F. van Dishoeck,  Astron. Astrophys. 601, A36 (2017).
  * (30) A. D. Bosman,  Uncovering the ingredients for planet formation,  PhD thesis, Leiden University, 2019.
  * (31) L. C. Biedenharn and J. D. Louck,  Angular Momentum in Quantum Physics, volume 8 of Encyclopedia of Mathematics,  Addison-Wesley, Reading, 1981.
  * (32) G. Dhont, W. B. Zeimen, G. C. Groenenboom, and A. van der Avoird,  J. Chem. Phys. 120, 103 (2004).
  * (33) Molpro is a package of ab initio programs written by H.-J. Werner and P. J. Knowles, with contributions from R. D. Amos, A. Berning, D. L. Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, C. Hampel, T. Leininger, R. Lindh, A. W. Lloyd, W. Meyer, M. E. Mura, A. Nicklaß, P.Palmieri, K. Peterson, R. Pitzer, P. Pulay, G. Rauhut, M. Schütz, H. Stoll, A. J. Stone and T. Thorsteinsson.
  * (34) T. H. Dunning,  J. Chem. Phys. 90, 1007 (1989).
  * (35) G. C. Groenenboom and N. Balakrishnan,  J. Chem. Phys. 118, 7380 (2003).
  * (36) S. F. Boys and F. Bernardi,  Mol. Phys. 19, 553 (1970).
  * (37) T.-S. Ho and H. Rabitz,  J. Chem. Phys. 104, 2584 (1996).
  * (38) T.-S. Ho and H. Rabitz,  J. Chem. Phys. 113, 3960 (2000).
  * (39) J. K. G. Watson,  Mol. Phys. 19, 465 (1970).
  * (40) P. R. Bunker and P. Jensen,  Molecular Symmetry and Spectroscopy,  NRC Research Press, Ottawa, 2nd edition, 1998.
  * (41) D. M. Brink and G. R. Satchler,  Angular Momentum,  Clarendon, Oxford, third edition, 1993.
  * (42) M. Abramowitz and I. A. Stegun,  Handbook of Mathematical Functions,  National Bureau of Standards, Washington, D.C., 1964.
  * (43) G. Brocks, A. van der Avoird, B. T. Sutcliffe, and J. Tennyson,  Mol. Phys. 50, 1025 (1983).
  * (44) A. van der Avoird, P. E. S. Wormer, and R. Moszynski,  Chem. Rev. 94, 1931 (1994).
  * (45) B. R. Johnson,  J. Chem. Phys. 69, 4678 (1978).
  * (46) B. R. Johnson,  NRCC Proceedings 5, 86 (1979).
  * (47) R. T. Pack,  J. Chem. Phys. 60, 633 (1974).
  * (48) S. Green,  J. Chem. Phys. 70, 816 (1979).
  * (49) Scilab Enterprises,  Scilab: Free and Open Source software for numerical computation.
  * (50) Rate coefficient data in the Leiden Atomic and Molecular Database (LAMDA, Schoier et al. 2005).
  * (51) T. G. A. Heijmen, R. Moszynski, P. E. S. Wormer, A. van der Avoird, A. D. Rudert, J. B. Halpern, J. Martin, W. B. Gao, and H. Zacharias,  J. Chem. Phys. 111, 2519 (1999).