# Computed Rotational Collision Rate Coefficients for Recently Detected
Anionic Cyanopolyynes

L. González-Sánchez  Departamento de Química Física, University of Salamanca,
Plaza de los Caídos sn, 37008, Salamanca, Spain  A.Veselinova  Departamento de
Química Física, University of Salamanca, Plaza de los Caídos sn, 37008,
Salamanca, Spain  A. Martín Santa Daría  Departamento de Química Física,
University of Salamanca, Plaza de los Caídos sn, 37008, Salamanca, Spain  E.
Yurtsever  Department of Chemistry, Koc University Rumelifeneriyolu, Sariyer
TR 34450, Istanbul, Turkey  R.Biswas  School of Chemical Sciences, National
Institute of Science Education and Research (NISER) Bhubaneswar  K.Giri
Department of Computational Sciences, Central University of Punjab, Bathinda,
Punjab 151401, India  N. Sathyamurthy  Indian Institute of Science Education
and Research Mohali, SAS Nagar, Punjab 140306, India  U. Lourderaj  School of
Chemical Sciences, National Institute of Science Education and Research
(NISER) Bhubaneswar  R. Wester  Institut für Ionenphysik und Angewandte
Physik, Universität Innsbruck Technikerstr. 25 A-6020, Innsbruck, Austria  F.
A. Gianturco  Institut für Ionenphysik und Angewandte Physik, Universität
Innsbruck Technikerstr. 25 A-6020, Innsbruck, Austria
<EMAIL_ADDRESS>
###### Abstract

We report new results from quantum calculations of energy-transfer processes
taking place in interstellar environments and involving two newly observed
molecular species: C5N- and C7N- in collision with He atoms and the p-H2
molecules. These species are part of the anionic molecular chains labeled as
cyanopolyynes which have been observed over the years in molecule-rich
Circumstellar Envelopes (CSEs) and in molecular clouds. In the present work,
we first carry out new $ab$ $initio$ calculations for the C7N- interaction
potential with He atom and then obtain state-to-state rotationally inelastic
cross sections and rate coefficients involving the same transitions which have
been observed experimentally by emission in the interstellar medium (ISM) from
both of these linear species. For the C5N-/He system we extend the
calculations already published in our earlier work (see reference below) to
compare more directly the two molecular anions. We extend further the quantum
calculations by also computing in this work collision rate coefficients for
the hydrogen molecule interacting with C5N-, using our previously computed
interaction potential. Additionally, we obtain the same rate coefficients for
the C7N-/H2 system by using a scaling procedure that makes use of the new
C7N-/He rate coefficients, as discussed in detail in the present paper. Their
significance in affecting internal state populations in ISM environments where
the title anions have been found is analyzed by using the concept of critical
density indicators. Finally, similarities and differences between such species
and the comparative efficiency of their collision rate coefficients are
discussed. These new calculations suggest that, at least for the case of these
longer chains, the rotational populations could reach local thermal
equilibrium (LTE) conditions within their observational environments.

##  1 Introduction

The study of the molecular complexity of the interstellar medium (ISM)
provides us with fundamental information on the chemical storage of species
and on the energy available for the formation of stars and planets. It also
gives us useful indication about the chemical inventory that the primitive
Earth might have inherited and about the formation paths of many of the
species increasingly discovered by direct observations.

In recent years, several linear C-bearing and (C,N)-bearing chains of
molecular anions have been detected at various sites in the circumstellar
envelopes (CSE) and in molecular clouds. Specifically: CN-Agúndez, M. et al.
(2010), C3N- Thaddeus et al. (2008), C5N- Cernicharo et al. (2008);
Cernicharo, J. et al. (2020), C4H- Cernicharo, J. et al. (2007); Agúndez, M.
et al. (2008), C6H- McCarthy et al. (2006), and C8H- Brünken et al. (2007);
Remijan et al. (2007). Those reported so far constitute different terms of the
general linear chains associated with the polyynes and cyanopolyynes species.
Interestingly, very recently two additional (C,N)-bearing chains have been
detected. They involve the two longest sequences of the linear-chain series
observed so far : C10H- Remijan et al. (2023) and C7N- Cernicharo et al.
(2023).

The chemistry of formation of the cyanopolyyne chains has also been the object
of several studies and speculations (e.g. see: Khamesian et al. (2016);
Cernicharo, J. et al. (2020)). The possible formation of anions from the
neutral radical via a Radiative Electron Attachment (REA) process has been
considered in some detail (as quoted in: Jerosimić et al. (2018)) while a more
direct chemical route by reaction of the HC5N with H- has also been put
forward by our group Satta et al. (2015). Another possible mechanism of
formation could be the reaction between N atom and C${}_{n}^{-}$, as discussed
in: Cernicharo, J. et al. (2020). For the present case of the C5N- our
calculated rates were found to be large enough to be relevant within the
chemical evolution producing these anions. As also discussed by Cernicharo, J.
et al. (2020) the much larger rate constant for the REA formation path
expected for C5N compared to C3N is the main reason why C5N- has been
estimated to be much more abundant with respect to the neutral than C3N- (the
abundance ratios of the CnN /CnN- are $\sim$ 140 and 2.3 for $n=3$ and $n=5$,
respectively, in the Taurus molecular cloud 1 (TMC-1)). For the case of the
longer chain of the C7N- anion its abundance ratio with the next smaller chain
discussed in the present work, the C5N- anion, was found in the molecular
cloud TMC-1 to be around 5 in Cernicharo et al. (2023). On the other hand, the
same C5N-/C7N- ratio in the IRC -10216 was indicated in the same study to be
around 2.4, i.e. not very different from the situation in the TMC-1
environment. In the final analysis, however, the various options put forward
by the current literature about the chemical paths to the formation of the
cyanopolyynes have not yet coalesced into a unique proof of the chemical
mechanism for the title anions or for their smaller partners in the series.

Astrophysical observation of these molecular anions relies heavily on the
spectroscopic investigation of their properties in the laboratory and on
matching sighted lines with those observed in the earthly experiments.
Additionally, to carry out astrophysical modeling of molecular population
evolution from their distributions over a variety of internal states,
important indicators are provided by the rate coefficients for the probability
of rotational state-changes in molecular species induced by their interaction
with He and H2, both partners being present in substantial amounts in the
observational environments of these anions. The collision-induced occurrence
of non-LTE (Local Thermal Equilibrium) population distribution for rotational
states in molecular partners could, in fact, be a significant path to produce
radiative emissions from different excited spectral lines in the observational
microwave regions. Measurement of these rate coefficients in the laboratory is
still challenging, while available quantum methods can be used to model such
processes and then enter their results within networks of the kinetics of the
underlying chemistry or of the internal-state populations in different
environments. For example, Botschwina and Oswald Botschwina & Oswald (2008)
had computationally determined the structural characteristics of some of these
species to help observational sighting in the ISM or in the laboratory of
specific emission lines

In the present work we focus more specifically on the collision-induced
rotational state-changing processes associated with the observed emission
lines involving the C5N-/C7N- anions, those discussed in the recent
publications of (Cernicharo, J. et al., 2020; Cernicharo et al., 2023)
respectively. We thus compute the inelastic rate coefficients activated by
interaction with He atoms and with p-H2 molecules over the same range of
transitions, and further calculate the Einstein A-coefficients of spontaneous
emissions between the same rotational levels of these molecular anions. We
then analyze the behaviour of the critical densities over the temperature
range in the molecular clouds where the observations were made and discuss the
consequences that such indicators suggest for the local distributions among
rotational states in both molecules. The state-changing collision rate
coefficients are obtained using exact quantum computations in the cases of
both the C5N- and the C7N- anions in collision with the He atoms. For the H2
collision partner exact calculations are carried out for the C5N-/H2 system,
while a scaling procedure is implemented for the H2 collision rates with the
C7N- longer anion, making use of the new rates from the C7N-/He system and
from the C5N-/H2/He systems, as discussed in detail later in this paper. The
details of the calculations and the analysis of the obtained results will be
presented in the following Sections, while our conclusions will be given in
the last Section.

##  2 The Ab initio Computed Interaction Potential

The calculations of the new Potential Enery Surface (PES) for the C7N-/He
system were carried out using the GAUSSIAN09 set of codes as in Frisch et al.
(2009) with the UCCSD(T) approach based on initial UHF orbital expansion using
the aug-cc-pVTZ for the carbon and nitrogen atoms, while for the He atomic
partner the more extended basis set was of QZ-Vpp quality.For further details
on the atomic basis we have employed see: Dunning Jr (1989) and Weigend &
Ahlrichs (2005) . The usual BSSE correction was applied: Frisch et al. (2009),
Deega & Knowles (1994), Woon & Dunning Jr (1994). No convergence problems were
encountered in the calculations. In all the $ab$ $initio$ calculations we used
an angular grid of 5∘ intervals. The optimized bond distances in the C5N-
partner were taken from our previous calculations described in our earlier
publication Biswas, R. et al. (2023) where all the details of this anion’s
interaction with both He and H2 were reported. Hence, we shall not be
repeating this information in this Section, where we shall discuss instead the
new calculations we have carried out for the longer anionic chain.

In the case of the C7N- partner, the optimized bond distances were obtained
from such new calculations. Hence, if one takes the carbon atom at the extreme
left as the origin of the $x$-coordinate, then the C1-C2 distance was (all
values in Å ): 1.2648, the C2-C3 distance: 1.4062; the C3-C4 bond: 1.2413; the
C4-C5 bond: 1.2784; the C5-C6 bond: 1.2318; the C6-C7 bond: 1.3643; the N-C7
distance: 1.1707 for a total length of the linear cyanopolyyne of 8.9575 and
the c.o.m. (center-of-mass) location at 4.6002. The resulting interaction with
the He atom was described via the distance $R$ from the c.o.m. and the polar
angle $\theta$, both being the usual 2D Jacobi coordinates which describe the
present PES.

The radial range of the raw points for the variable $R$ was taken from initial
values of around 6.2 Å depending on the particular angular orientation, and
extended out to 20.0 Å . Radial intervals between 0.1 to 1.0 Å were used
depending on the distance and angle regions from the c.o.m. A variable number
of radial values for each angular orientation were obtained as raw points, for
a total range of 37 angular values. Note that the long, linear structure of
the anionic species, and hence the location of the center-of-mass at the large
distance mentioned earlier, are responsible for the interaction to become
highly repulsive at fairly large distances for the atomic partner’s approach
to the c.o.m. We shall further show below that the actual radial range of the
fitted potentials came closer to the c.o.m positions for some of the angles
needed in the scattering calculations. It is important to note at this point
that the _ab initio_ computed points are not symmetrically distributed along
the distance coordinate due to the large anisotropy of the system, depending
on the entrance angle of the He atom. This specific feature of the forces at
play will be further discussed below.

This lack of points at short distances was therefore improved and completed by
extrapolating with an exponential function for every angle up to 2 Å with
respect to the c.o.m. In order to generate a uniformly regular grid of points,
spline functions were used on the _ab initio_ and on the extrapolated data
sets to produce a total of 82 distance ($R$) points along the 37 angles
($\theta$), summing up to a total of 3034 points which were further fitted to
generate the final PES. The intrinsic problem of having ’ghost points’ at very
short distances, even without them having physical sense because of their
being inside the anionic atoms’ chain itself, is their extremely large energy
values which accumulate over short radial intervals. Such huge values cause
mathematical instabilities when trying to fit them via an analytical function.
To overcome this problem, we scaled the points by applying the simple
procedure implemented by Faure _et al._ Faure et al. (2011) and already used
in similar systems Khadri et al. (2020); Sahnoun et al. (2018); Massó &
Wiesenfeld (2014). It has the advantage that one can now reproduce the
original _ab initio_ points in the lower energy range of the PES (the most
relevant range for the scattering calculations) while one only scales the
higher energy values that would not be sampled during the scattering events.
Thus, one can write the following sequence,

$\displaystyle
V_{s}=V_{a}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\quad\quad\quad\text{for}V_{a}\leq
V_{T1}$ $\displaystyle
V_{s}=V_{T1}+(V_{T2}-V_{T1})~{}S\bigg{(}\frac{V_{a}-V_{T1}}{V_{T2}-V_{T1}}\bigg{)}\quad\quad\quad\text{for}V_{T1}<V_{a}\leq
V_{T2}$ (1) $\displaystyle
V_{s}=V_{T1}+\frac{2}{\pi}(V_{T2}-V_{T1})\quad\quad\quad\quad\quad\,\,\,\,\,\quad\quad\quad\text{for}V_{a}>V_{T2}$

where $V_{s}$ is the scaled potential, $V_{a}$ is the _ab initio_ energy and
$V_{T1}$ and $V_{T2}$ refer to two threshold energy values, which for this
particular case are 2000 cm-1 and 10000 cm-1, respectively. The switching
function is given by the $S$ function defined as

$\displaystyle
S(x)=\frac{2}{\pi}\sin\bigg{[}\frac{\pi}{2}\sin\bigg{(}\frac{\pi}{2}x\bigg{)}\bigg{]}\;,$
(2)

and it guarantees the proper transition between the three different scaling
windows.

In the standard procedures employed for solving the Coupled-Channel (CC)
scattering equations, it is usually convenient, as discussed below, to expand
the interaction potential $V(R,\theta)$ into orthogonal angular functions
(Arthurs & Dalgarno, 1960; Secrest, 1979; Kouri & Hoffman, 1997; Gianturco,
1979). Hence, for a more direct, and quantitative evaluation of the spatial
anisotropy around the C5N- and the C7N- linear anions it is useful to
represent the raw, 2D grid of points from the $ab$ $initio$ calculations in
terms of the familiar Legendre polynomials in their standard ($R$,$\theta$)
form. In the case of the new calculations discussed above, using the 3034
‘scaled points’ we were able to expand the PES via the usual orthogonal
Legendre polynomials for the $\theta$ angle:

$V_{\rm FIT}=V(R,\theta)=\sum_{\lambda=0}^{\lambda_{\rm
max}}V_{\lambda}(R)P_{\lambda}(\cos\theta)$ (3)

The fitted potential is restricted to a finite range of $R$ values and
therefore an extrapolation to an asymptotic form is needed to describe the
long-range interaction forces. The analytical form used for the long-range is

$\displaystyle V_{\rm
LR}(\theta)=-\frac{\alpha_{0}}{2R^{4}}+\frac{2\alpha_{0}\mu}{R^{5}}\cos{\theta}\;,$
(4)

where $\alpha_{0}=1.41~{}a_{0}^{3}$, is the polarizability of the He atom and
$\mu=7.5D$, from Cernicharo et al. (2023), is the rigid rotor dipole moment of
the C7N- anionic partner.

To ensure the smooth transition between the $V_{\rm FIT}$ fitted PES (Eq. 3)
and the $V_{\rm LR}$ long-range term (Eq. 4) in the construction of the
$V_{\rm f}$ final PES, the switching procedure which was already used for
C5N-/He Biswas, R. et al. (2023) has also been implemented for the longer
chain anion:

$\displaystyle V_{\rm f}=f_{s}V_{\rm FIT}+(1-f_{s})V_{\rm LR}$ (5)

where the switching function is

$\displaystyle f_{s}(R)=\frac{1}{e^{\frac{(R-R_{0})}{\Delta R}+1}}\;,$ (6)

with $R_{0}=18.5$ and $\Delta R=1.2$ Å.

A pictorial view of the spatial shape of the interaction forces for C7N-/He is
reported in Figure 1, where the energy isolines are depicted around the
linear, rod-like structure of the anionic partner.

Figure 1: Pictorial representation of the spatial features of the interaction
between the C7N- partner and the He atom. The first C atom is on the left
while the end N atom is on the right. See main text for further details.

It is interesting to note from the structure of the isolines that the
interaction becomes, at all angles, very repulsive at rather large distances
from the c.o.m., a difficult feature to represent, as we have already
discussed earlier, and which reflects the unusually long, linear structure of
this anionic target. We further see that the attractive regions of its
interaction with a neutral He atom are located within a fairly localized
valley corresponding to a T-shaped configuration of the van der Waals complex.
Finally, the ‘oscillations‘ shown by the isolines, as one comes closer to the
chain of carbon atoms, indicate the detailed changes of the interaction forces
with the approaching He partner as one moves from the carbon atoms to the
bonding sections along the molecular chain.

The multipolar expansion was carried out to $\lambda$ values up to $\lambda$ =
93, to ensure numerical convergence, employing the procedure discussed more
extensively earlier in this Section, where we explain how a larger number of
raw points were obtained by adding to the initial selection. The actual
features of the multipolar coefficients for the C5N- partner with the He
projectile were discussed in detail in our earlier work reported by ref
Biswas, R. et al. (2023), hence we shall not be repeating it in the present
paper. We only report in Figure 2 a comparison between the lower six
coefficients newly obtained for the longer molecular anion, the C7N-, and
those we had found earlier for the C5N- anion, to indicate the relative
strength of the angular coupling that will be active for the various radial
regions of their interactions with the neutral He atoms. The multipolar
coefficients reported in that Figure are usually found to be the more
effective during the quantum dynamics couplings of target rotational levels,
as further discussed in the next Section.

Figure 2: A comparison of the lower radial multipolar coefficients for the
interaction potential energy surfaces between the C5N- and the C7N- anions and
the He atom. See main text for further comments.

As it is to be expected, the interactions with $\lambda$=0 exhibit the largest
well depths for their attractive regions, while the next strong interactions
come from the $\lambda$=1 and 2 coefficients for both partners. These
coefficients also present marked well depths. The significance of having these
radial coefficients providing large anisotropic couplings will be discussed
further in the next Section.

The computed rotational constants for the two anions, treated at the level of
spherical, rigid rotors in ${}^{1}\Sigma$ states (see further discussion
below) were both found to be fairly small, as is to be expected for such long
molecular chains:the $B_{e}$ value is 0.046292 cm-1 for the C5N- molecule and
0.01942 cm-1 for the C7N- anion as discussed in Cernicharo, J. et al. (2020)
and in Cernicharo et al. (2023). For a comparison between the two rotational
energy ladders which involve the range of states which have been
experimentally detected in Cernicharo, J. et al. (2020) and in Cernicharo et
al. (2023), we report in Figure 3 the relative positioning of these states for
the two molecules. We clearly see there that the relevant levels cover in each
case about 20 or so cm-1 and that many more states are available in that range
for the longer anion on the right of the figure. Such features indeed suggest
the occurrence of strong coupling between the closely spaced levels during the
dynamical steps, as we shall further discuss below.

Figure 3: Energy ladders of the rotational levels of the two molecular anions
of this study within the range of the observed transitions in the ISM. See
main text for further comments

As a result of the relatively large number of states which can be available
over rather small ranges of collision energies, we can argue that at the
temperatures reported by the experimental observations of either anion and
further discussed below, the reaching of an LTE condition could involve states
up to about $j$ max = 50 or = 60 to be contributing to rotational state
populations up to 50 K. This structural feature will further be discussed
later when we shall examine the results from collision and radiative processes
involving both molecular anions.

##  3  Rotationally Inelastic Quantum Dynamics

Since in our present study no chemical modifications occur during the
collision events, the total scattering wave function can be expanded in terms
of asymptotic target rotational eigenfunctions (taken to be spherical
harmonics) whose eigenvalues are given by $B_{e}j(j+1)$ where $B_{e}$ is the
first rotational constant mentioned earlier for the present systems (Arthurs &
Dalgarno, 1960; Secrest, 1979; Kouri & Hoffman, 1997; Gianturco, 1979). The
channel components for the CC formulation are therefore expanded into products
of total angular momentum eigenfunctions and of radial functions (Arthurs &
Dalgarno, 1960). The latter are in turn the elements of the solutions matrix
which appear in the familiar set of coupled, second order homogeneous
differential equations:

$\left(\frac{d^{2}}{dR^{2}}+\mathbf{K}^{2}-\mathbf{V}-\frac{\mathbf{l}^{2}}{R^{2}}\right)\mathbf{\Psi}=0$
(7)

where $[\mathbf{K}]_{ij}=\delta_{i,j}2\mu(E-\epsilon_{i})$ are the matrix
elements of the diagonal matrix of the asymptotic (squared) wavevectors and
$[\mathbf{l}]_{ij}=\delta_{i,j}l_{i}(l_{i}+1)$ is the matrix representation of
the square of the orbital angular momentum operator. This matrix is block-
diagonal with two sub-blocks that contain only even values of ($l’+j’$) or
only odd values of $(l’+j’)$.

The scattering observables are thus obtained in the asymptotic region where
the Log-Derivative matrix has a known form in terms of free-particle solutions
and unknown mixing coefficients. For example, in the asymptotic region the
solution matrix can be written in the form:

$\mathbf{\Psi}=\mathbf{J}(R)-\mathbf{N}(R)\mathbf{K}$ (8)

where $\mathbf{J}(R)$ and $\mathbf{N}(R)$ are matrices of Riccati-Bessel and
Riccati-Neumann functions. Therefore, at the end of the propagation one uses
the Log-Derivative matrix to obtain the $\mathbf{K}$ matrix by solving the
following linear system:

$(\mathbf{N}^{\prime}-\mathbf{Y}\mathbf{N})=\mathbf{J^{\prime}}-\mathbf{YJ}$
(9)

and from the $\mathbf{K}$ matrix the S-matrix is easily obtained and from it
the state-to-state cross sections. We have already published an algorithm that
modifies the variable phase approach to solve that problem, specifically
addressing the latter point and we defer the interested reader to these
references for further details (López-Durán et al., 2008; Martinazzo et al.,
2003). In the present calculations we first generated the necessary state-to-
state rotationally inelastic cross sections and, once these quantities were
known, the required rotationally inelastic rate coefficients $k_{j\rightarrow
j^{\prime}}(T)$ were evaluated as the convolution of these cross sections
$\sigma_{j\rightarrow j^{\prime}}$ over a Boltzmann distribution of the
relative translational energy values between partners($E_{\textrm{trans}}$):

$k_{j\rightarrow j^{\prime}}(T)=\left(\frac{8}{\pi\mu
k_{\text{B}}^{3}T^{3}}\right)^{1/2}\\\
\times\int_{0}^{\infty}E_{\textrm{trans}}\sigma_{j\rightarrow
j^{\prime}}(E_{\textrm{trans}})e^{-E_{\textrm{trans}}/k_{\text{B}}T}dE_{\textrm{trans}}$
(10)

The reduced masses for the C5N-/He and C7N-/He systems were (in units of amu):
3.797372 and 3.8455, respectively. The individual rate coefficients were
obtained at intervals of 1 K, starting from 20 K and going up to 50 K. The new
scattering calculations for the larger C7N- anion were carried out using the
MOLSCAT suite of codes (Hutson & Sueur, 2019a, b).

Briefly, We have included 87 rotational states for the C5N-/He system and two
rotational states for H2 for our calculations on the C5N-/H2 system . The
maximum value of JTOT was 200, depending on the energy. The convergence checks
of the results indicated an accuracy of the cross sections around 2-3$\%$. The
spacing of the energy points (in cm-1) for the cross section calculations
varied from 0.1 up to 15.0, to 5.0 up to 180.0 and then to 25.0 up to the
largest energies of about 350.0 cm-1. In the case of the C7N-/He system we
found that, for the studied transitions,to include rotational states up to
jmax=60 were enough, and the maximum value of JTOT was 180, depending on the
energy. Regarding the energy points, we did not use a regular grid. To ensure
the convergence of the rate constants to the same values mentioned before in
the same interval of temperatures, we have considered, for all initial states,
collision energies from 0.001 to 2.5 cm-1 and total energies from 10 to 300
cm-1.

We should further note here that at the highest collision energies considered,
i.e. above 140 cm-1, we have switched from the full Coupled Channel (CC)
approach to the reduction scheme of the Coupled States (CS) approach as
described in (Hutson & Sueur, 2019a, b). Furthermore, at some of the selected
high energy values we additionally run calculations at the CC level for
comparison and found that the actual cross sections remained within 2-3$\%$ of
the CS values.

##  4 Einstein Coefficients For Radiative Emission

Another important process for characterizing the internal state distribution
of the anions is their interaction with the surrounding radiative field, also
known as black-body radiation. The total emission (em) transition rates from
an excited state $X$ can be written as sum of stimulated (sti) and spontaneous
(spo) emission rate coefficients, as discussed in detail in, e.g.: Brown &
Carrington (2003) and as defined below:

$X^{em}_{k\to i}=X^{sti}_{k\to i}+X_{k\to i}^{spo}=A_{k\to
i}(1+\eta_{\gamma}(\nu,T))$ (11)

where $A_{k\to i}$ is the Einstein coefficient for spontaneous emission and
$\eta_{\gamma}(\nu,T)=(e^{(h\nu/k_{B}T)}-1)^{-1}$ is the Bose-Einstein photon
occupation number for the stimulated processes. In the present study the most
important role is played by the spontaneous emission step, since in the fairly
low photon densities of the regions of the diffuse molecular clouds, which we
are considering relevant here, make the stimulated processes less likely to
occur.

The Einstein coefficient for dipole transitions is given as

$A_{k\to i}=\frac{2}{3}\frac{e^{2}\omega^{3}_{k\to
i}}{\epsilon_{0}c^{3}h}\left|\int\Psi^{*}_{k}\cdot\overrightarrow{r}\cdot\Psi_{i}\cdot
dr^{3}\right|^{2}$ (12)

where $\omega_{k\to i}\approx 2B_{0}(j_{i}+1)$ is the transition’s angular
frequency, $\overrightarrow{r}$ is the charge displacement vector and
$\Psi_{k/i}$ are the wavefunctions for states $k$ and $i$ respectively. For
pure rotational transitions, eq. 12 simplifies to

$A_{k\to i}=\frac{2}{3}\frac{\omega^{3}_{k\to
i}}{\epsilon_{0}h}\mu_{0}^{2}\frac{j_{k}}{2j_{k}+1}$ (13)

where $\mu_{0}$ is the permanent electric dipole moment of the molecule. Both
the dipole values and the chosen rotational constants for the present anions
were discussed earlier and given in the previous Section on the structural
calculations.

The results from the present calculations, which involve the same transitions
as in the emission lines detected in the Interstellar medium by Cernicharo, J.
et al. (2020); Cernicharo et al. (2023), are given by the data in Table 1.

Transition $j$ $\leftarrow$ $j^{\prime}$ | (C5N-) | (C7N-)
---|---|---
27 $\leftarrow$ 28 | 7.26 | 1.12
28 $\leftarrow$ 29 | 8.07 | 1.24
29 $\leftarrow$ 30 | 8.94 | 1.38
30 $\leftarrow$ 31 | 9.87 | 1.52
31 $\leftarrow$ 32 | 10.9 | 1.67
32 $\leftarrow$ 33 | 11.9 | 1.83
33 $\leftarrow$ 34 | 13.0 | 2.01
34 $\leftarrow$ 35 | 14.2 | 2.19
35 $\leftarrow$ 36 | 15.5 | 2.38
36 $\leftarrow$ 37 | 16.8 | 2.59
37 $\leftarrow$ 38 | 18.2 | 2.81
38 $\leftarrow$ 39 | 19.7 | 3.03
39 $\leftarrow$ 40 | 21.3 | 3.28
40 $\leftarrow$ 41 | 22.9 | 3.53
Table 1: Einstein Emission coefficients ($A_{ij}$) for C5N- and C7N- with
$\Delta j=-1$. In units of 105 s-1.

The emission lines for the C5N- anion were observed in the IRC +10216 at an
estimated temperature of (37 +/- 6) K. For the case of the C7N- anion the
emission lines were observed in either the TMC-1 or the IRC +10216 CSEs at an
estimated temperature of 26(+/-1.8) K. Given the very small spacing between
the levels involved, if the two anions were to be considered under local
thermal equilibrium, then the C5N- molecule would have its rotational states
with $j$ values from 25 and up to 35 with a fractional population around
25$\%$. For the case of the C7N- anion, its rotational states with $j$ values
up to 41 would have a fractional population around 20$\%$. We have therefore
computed the rate coefficients involving that range of populated internal
levels of the two title anions.

##  5 Inelastic rate coefficients

Following the numerical evaluation of the relevant inelastic cross sections
indicated in the previous Section, we further implemented the integration
process of eq.(10) and employed the computed cross sections to generate
inelastic rate coefficients over a range of about 50 K, the expected
temperatures relevant to the CSE environments where detection of the present
radical occurred, as discussed in our Introduction.

Examples for the case of the C5N- anion are displayed in the Figures reported
in the following. Figure 4 shows excitation processes which start from the
$j$=25 rotational state of this anion and end up in one of the higher
rotational states involved in the emission lines experimentally detected in
(Cernicharo, J. et al., 2020). This example is meant to show how large the
rates would be when considering excitation with fairly large $\Delta$ $j$
values beyond the more efficient cases with $\Delta$ $j$= $+/-$1 discussed by
our earlier work in (Biswas, R. et al., 2023).

Figure 4: Computed excitation rate coefficients from the $j$=25 rotational
state for the C5N- for transitions into the observed range of rotational
states, over a temperature range of 50 K. See main text for further details.

Because the excitation processes shown in Fig. 4 involve large values of the
$\Delta$j transition index, we can see that the corresponding rates have
fairly small values at the lower temperatures and remain small up to the
highest $T$ values considered.

On the other hand, if we now consider the excitation processes shown by the
data in Figure 5, we clearly see that the rate coefficients associated with
the $\Delta$j=+1 value are much larger than the ones corresponding to the
larger $\Delta$j reported in Figure 4. This is indeed to be expected from the
changes in the relative energy gaps involved in the excitation processes, and
as discussed in detail by us in our earlier work in ref (Biswas, R. et al.,
2023). It follows from these findings that the efficient collisional re-
populations of the levels in the observational emission lines of ref.
Cernicharo, J. et al. (2020) would more easily occur from the contiguous
levels that are active during those emission processes.

Figure 5: Computed excitation rate coefficients for$\Delta$j=+1 transitions
over the observed range of rotational states for the C5N-. The data cover a
temperature range up to 50 K See main text for further details.

Looking at the data in Figure 5 which involve excitations associated with
$\Delta$j=+1 transitions between the levels detected by the experimental work
of Cernicharo, J. et al. (2020), we clearly see that all the rate coefficients
are uniformly much larger than those associated with $\Delta$j values from +4
to +8 reported in Figure 4. These rates are therefore several orders of
magnitude larger than those shown by Figure 4.

If we now turn to the de-excitation ( collisional cooling) processes
(presented in Figure 6), and involving transitions with $\Delta$ $j$=-1, we
see that the rate coefficients are all of the order of 10-11 cm3molecule-1s-1,
a size which makes them comparable with the corresponding excitation rates
discussed in Figure 5. These will be the quantities that we shall further
compare with the competing radiative processes, those driven by the large
dipole moments of the present anions.

Figure 6: Computed de-excitation (cooling) rate coefficients from a variety of
initial rotational states for C5N- the range of temperature values is up to 50
K. See main text for further details.

The set of calculations for the larger anionic chain, detected just recently
by Cernicharo et al. (2023), and also involving excitation and de-excitation
processes for the levels observed in their emission lines (see: Cernicharo et
al. (2023)), are given in Figures 7 and 8.

Figure 7: Computed excitation rate coefficients from a variety of initial
rotational states for C7N- in collision with He atoms. The range of
temperature values is up to 50 K. See main text for further details. Figure 8:
Computed de-excitation (cooling) rate coefficients from a variety of initial
rotational states for C7N- in collision with He. The range of temperature
values is up to 50 K. See main text for further details.

The computed excitation rate coefficients are now nearly one order of
magnitude larger than those pertaining to the shorter linear anion (see the
previous Figure 5). Their range of values indicate how its smaller energy
spacings shown by the Figure 3 are causing the corresponding excitation
processes to be more efficient, hence producing larger collision rate
coefficients. A similar behaviour can also be gleaned by the results for the
de-excitation (cooling) processes, reported in Figure 8. In this case,
however, the differences with the results we had found for the C5N- at lower
temperatures turn out to be smaller and therefore the two sets of de-
excitation processes are very similar in size over the same range of
temperatures.

It is interesting at this point to also compare the inelastic rate
coefficients just obtained for the long chains discussed here with those which
we have calculated in the past for smaller anions with larger energy spacing
between rotational states: the C2H- and C2N- interacting with He atoms (Franz
et al., 2020) and the CN- interacting with He and H2 partners: (González-
Sánchez et al., 2020). In the case of the former anions, in fact, our earlier
calculations of the excitation rate coefficients involving the lower
rotational states (Franz et al., 2020) found such rates to be about one order
of magnitude larger than those reported here by Figures 5 and 7. Further, in
the case of the CN- anion, our earlier calculations in (González-Sánchez et
al., 2020) also showed that the excitation rate coefficients were again about
one order of magnitude larger than those reported here for the longer chains.
Such findings suggest that the transitions involving the higher rotational
states of the present study yield smaller rate coefficients than when the
lower-lying rotational states are involved.

In the Introduction Section we have also pointed out that, besides He atoms,
the simple hydrogen molecule is also one of the most abundant partner species
within the environments where these anions have been observed. Hence, it is
also useful to look at the collision-driven processes where the hydrogen
molecule is the relevant partner. In our earlier work on the C5N- we have
already considered excitation and de-excitation transitions induced by
collision with the H2 partner (Biswas, R. et al., 2023). Hence, the relevant
collision processes among the experimentally detected levels are reported in
Figure 9.

Figure 9: Comparing the computed collision-driven excitation processes into
the higher rotational states of C5N-, for the He projectile and the H2(j=0)
molecular partner. See main text for further details. Figure 10:  Same
comparison of collision-driven transitions as in the previous Figure 9 but
involving here the de-excitation processes in C5N- induced by both the He and
the H2 projectiles. See main text for further details.

We clearly see from the comparison reported by Figure 9 that the excitation
rate coefficients from collisions with the $p-H_{2}$ molecule are much larger,
by about one order of magnitude on average, than those obtained for the He
projectile, thus indicating that both baryonic partners could substantially
contribute to a possible, collision-driven, local termal equilibration between
the rotational levels of this target anion.

The de-excitation processes involving the j=28, 29 and 30 rotational states of
the same anion, induced by collision with p-hydrogen molecules, are reported
by Figure 10, where comparison is made with the same transitions induced by
collision with He atoms. We see once more that such rates are now about one
order of magnitude larger in comparison with those obtained for the He
partner. Given the expected abundances of the H2 molecule, we therefore also
see that collision-driven population changes could be fairly significant for
this partner, as they have already been shown to be important for the case of
the He partner.

To acquire a more quantitative understanding of the size differences between
the state-changing collision rate coefficients for the two types of partners
with the C5N- anion, we report in Figures 11 and 12 the ratios between the two
coefficient values for a sampling of three of the levels involved and over the
relevant range of temperatures.

Figure 11:  Computed numerical ratios between the two sets of rate
coefficients discussed in this work for the C5N- anion in collision with H2
and with He. The ratios involve, as examples, three of the relevant rotational
states observed experimentally. They are given over a range of temperatures in
line with that experimentally attributed to the relevant molecular clouds. See
main text for further details. Figure 12: Same ratios as those discussed in
Figure 11 but involving instead the de-excitation rate coefficients between
the same rotational states. See main text for further details.

It is clear from the data in those Figures that the rate coefficients
pertaining to the H2 projectile are uniformly larger than those obtained from
the He partner: their dependence on $T$ at the lowest temperatures above
threshold peaks at factors of 10 and larger while becoming fairly constant in
the range of $T$ values between 15 K and 50 K, covering the observed
temperatures in the molecular clouds where the title anions have been recently
detected: Cernicharo, J. et al. (2020); Cernicharo et al. (2023). This is an
interesting result since, in the case of a linear anion of similar structure,
e.g. for the C7N- discussed in this work, one could conceivably resort to
scaling its present data just obtained for the He partner to estimate the rate
coefficients needed for the H2. This option and its results will be further
discussed in the following Section 6.

##  6 Analysis of Critical Densities

The assumption of a local thermodynamic equilibrium in different regions of
the interstellar medium is expected, in general, to hold whenever the
population of the excited levels under consideration is likely to be given by
the Boltzman’s law. In the present situation, this might happen whenever the
rates of spontaneous emission from the internal levels of the polar anions are
smaller than the rates of state-changing by collision with the most abundant
partners present in that ISM region. This implies that the density in the
interstellar gas for the partners should be significantly larger than some
critical value so that the LTE assumption can be considered to be physically
viable. The definition of a critical density (e.g. see: Lara-Moreno et al.
(2019); Gianturco et al. (2021)) is given as follows:

$n_{\textrm{crit}}^{i}(T)=\frac{A_{ij}}{\sum_{j\neq i}k_{ij}(T)}$ (14)

where the critical density for any $i^{th}$ rotational level is therefore
obtained by giving equal weights to the consequences of either the collision-
induced or the spontaneous emission processes. We have taken the rate
coefficients discussed in Section 5, involving the rotational states which
have been identified by the detection of our present anions, as discussed in
the previous Section. We have also employed the computed spontaneous decay
Einstein coefficients discussed and presented in Section 4. It is also worth
mentioning here that there is no sum over $j$ values for the terms at the
numerator since the dipole-driven contribution is by far the largest from any
given initial $i$th state.

Although the values of the particle densities is considered to vary widely in
different ISM environments,so that several general models of the conditions in
the molecular clouds suggest variations from the situation of the diffuse
clouds (around 106 cm-3) to dense clouds (around 1010 \- 1012 cm-3) as
discussed, e.g., in (Snow & McCall, 2006; Agúndez & Cernicharo, 2006) and in
the references reported there. On the other hand, the specific conditions in
the molecular clouds where the present anions have been detected indicate that
their densities are expected to be around 104 \- 105 cm-3. Hence, from our
computed collision de-excitation rates of, say, Figure 9 for example, we can
observe that the above values of expected particle densities can provide a
specific range of collisional de-excitation efficiency in units of s-1. Hence,
in the present molecular clouds we obtain collision de-excitation quantities
of the order of about 10-5 s-1.

From the above considerations it follows that in such environments the
collision-induced state-changing processes are faster than the spontaneous
radiative emissions reported, for example, for the C5N- anion by our
calculations in Table 1, while in the more diffuse environments the
spontaneous decay channels could be comparable with the collision-induced
ones.

The results in Figure 13 were obtained using the collision-induced rate
coefficients calculated earlier by us for the C5N- partner (Biswas, R. et al.,
2023). The fairly large values obtained for the critical densities are mainly
controlled by the equally large spontaneous radiative emission coefficients,
as reported in the present Table 1, which appear in the numerator of eq. (14).

Figure 13: Computed critical densities for the C5N-/He system as obtained from
Eq. (14), for temperatures from 35 K to 45 K. Present results are reported for
the same rotational levels involved in the detection lines discussed in ref.
(Cernicharo et al., 2023). Figure 14:  Comparison between the critical
densities obtained for He as a collision partner (solid lines) and those
pertaining the molecular hydrogen as the collision partner of the C5N- anion
(dotted lines). The results are reported for some of the rotational levels
involved in the detection lines of ref. (Cernicharo et al., 2023).

A further example of the relative values of the critical densities for some of
the rotational states which have been experimentally detected are reported in
Figure 14, where we additionally present results obtained when molecular
hydrogen is the collision partner of the C5N- anion, and compare them with
those obtained for the He partner reported in Figure 13. As expected, the
larger collision rate coefficients we had found for that molecular partner are
now producing smaller values of the critical densities. Within the observed
temperature range around 30 K we therefore see that both partners are likely
to effectively compete with the radiative processes since the estimated
critical densities shown by Figure 14 vary between 105 $cm^{-3}$ and 106
$cm^{-3}$. Such range of values is well within the expected density values of
the partner baryonic species in the interstellar environments where these
anions have been detected. We have further extended the previous analysis of
the critical densities to the next larger anion recently detected: the C7N-
anion. In that case, we employ our accurate calculations of the collision rate
coefficients obtained for He as a partner atom, while we generate by a scaling
procedure those for the H2 molecular partner. Hence, we use the He/H2 scaling
factors we have obtained for the smaller C5N- anion and reported by Figures 11
and 12. That same ratio was thus applied to scale the data for the longer
chain anion in collision with He which we have been obtained in this work. The
results of the exact and scaled rate coefficients obtained for the longer
anion are given in Figures 15 and 16.

Figure 15: Computed critical densities for the C7N-/He system as obtained from
Eq. (14), for temperatures from 20 K to 50 K, using the exact quantum rates
discussed in the present paper. The results are reported for the same
rotational levels involved in the detection lines discussed in ref.
(Cernicharo et al., 2023). Figure 16:  Comparison between the critical
densities obtained for He as a collision partner (solid lines) and those
pertaining the molecular hydrogen as the collision partner of the C7N-
anion.The latter rates were obtained from the scaling procedure discussed in
the main text. The results are reported for some of the rotational levels
involved in the detection lines of ref. (Cernicharo et al., 2023).

We clearly see from the data in these two figures that the larger collision
rate coefficients, exhibited by the longest cyanopolyyne detected thus far,
generate critical densities which are down to vales around 104cm-3 for the He
partner and to around 103cm-3 for the H2 molecular partner. Given the baryonic
densities around 104cm-3 in the observational environments of these anions,
our results suggest that the rotational state population of the C7N- anions
exhibit critical density values which are large enough to allow the molecules
to undergo collision-driven thermalization probabilities which are in
competition with their radiative emission de-excitation paths. Under such
conditions, therefore, to accurately know the collision-driven rate
coefficients which we have computed would be important since their knowledge
would allow us to establish whether the LTE approximation would be valid
within their kinetic networks and whether it would provide reliable estimates
of the relative populations reached under the densities of the CSEs where
these anions are present. The knowledge of the actual collision rates with
abundant partners like He (and H2) is therefore relevant for more realistic
modelings of the energy flow processes.

It is interesting to note here that in a recent work by Agúndez et al. (2023)
excitation calculations were carried out using computed collision rate
coefficients for a series of cyanopolyynes and polyynes. Their results
indicate that the lines from anions accessible to radiotelescopes vary from
being subthermally excited to being thermalized as the sizes of the anions
increase and link the degree of non-LTE conditions on the H2 volume densities
and to the line frequencies. Since the present anions are the largest ones
ever observed and not yet discussed in that work, we can argue from our
present findings that the critical densities associated to the C5N- and the
C7N- anions are well within the range of values indicated for the n-H2 column
densities that are suggested to vary between 1.0 and 7.5x104cm-2. Hence, it
stands to reason to assume that the rotational populations of these longer
chains would be more likely to reach thermal conditions in the relevant
molecular clouds.

##  7 Present Conclusions

In the work reported in this paper we have analysed in some detail the
behaviour of the excitation and de-excitation (cooling) rate coefficients
associated with the rotational states of the two longest cyanopolyyne chain
anions, the C5N- and the C7N-, which have been detected so far in different
CSE environments, as we have discussed in the Introduction section. The
computed rate coefficients originate from the collisional interactions of
these anions with He atoms and H2 molecules, both species well known to be
present in significant abundances in the same ISM environments. More
specifically, the relative abundance of He with respect to H2 is taken to be
about 0.17 Agúndez et al. (2023) and usually considered to have about a 20$\%$
contribution to the state-changing probabilities in comparison with that of
the neutral molecular partner. The abundance of the latter is considered to
have a local density of the order of 104 cm-3 Agúndez et al. (2023) and
therefore relevant for its contribution to the rotational thermalization or
sub-thermalization of the emission lines observed for the two title anions.

To further investigate the actual collision rate coefficients for both
partners interacting with the two anionic chains, we have employed the
accurate _ab initio_ evaluation of both He and H2 interactions with the C5N-
already reported by us in our earlier work Biswas, R. et al. (2023). For the
longer anionic chain of C7N- we have carried out new _ab initio_ calculations
of its interaction with He atoms and estimated those pertaining to the H2
molecule by scaling the results for He with the same factors produced for the
same partners in the case of C5N-. We have therefore been able to obtain
realistic estimates of collision rate coefficients for both anions for the
rotational states which had been found to be responsible of their emission
lines recently observed in different CSE environments, as described in the
Introduction Section.

Using the information we have on the energy spacing between the relevant
rotational levels, and with the values we have computed for their permanent
dipole moments, we have calculated the Einstein radiative coefficients acting
in the cases of the observed emission lines. The combined use of such
coefficients, and of the computed collision rate coefficients involving the
same levels, has allowed us to obtain realistic estimates of the critical
density values associated with both He and H2 as partners of the title anions.
We know, in fact, that the critical density (and thus the degree of departure
from LTE) is very different depending on the dipole moment of the anion and on
the frequency of the transition involved. In the present case we have
specifically shown that the relevant emission lines which are at the detected
temperatures of the cold clouds can be associated, in the case of the C5N-
anion, with critical density values around 106 cm-3 for the He partner at 35 K
and around 105 cm-3 for the H2 partner at similar temperatures. The last
quantity is indeed comparable with the expected baryonic density of the
neutral molecular partner in the molecular clouds where these anions have been
observed and therefore suggests that collisional thermalization can be very
competitive with the radiative emission rates shown by this anion.

In the case of the next longer molecular cyanopolyyne of our present
calculations, we have further found larger collision rate coefficients but
comparable radiative coefficients, as shown by our data in Table 1. These
findings have produced even smaller critical densities around the same
temperatures discussed before: values between 104 and 105 cm-3 for the case of
the He partner and scaled values for the H2 partner of less than 104 cm-3.
These results further strengthen the suggestion that collision processes are
definitely competitive with the radiative emission processes and can lead to
final thermalization of the level populations of these anions in the observed
ISM environments. Naturally, the quantities which we have computed in the
present study are indeed very important to proceed with a more detailed
analysis of the relevant kinetics since, in order to study the abundances and
excitations of the present molecular anions in their interstellar sources we
need to further know the physical parameters of the clouds, like the gas
kinetic temperature and the H2 volume density, but also the emission size of
anions and the linewidth obtained from observation. In any event, our present,
accurate calculations are already able to establish, from the obtained
critical densities of both species at the observational temperatures, that
collision events are indeed very competitive with the radiative efficiency
indicated by their large dipole moment values and therefore that both long
chains should be present with a thermal population of their rotational
internal states.

##  8 Acknowledgements

One of us (L.G-S.) acknowledges the financial support by Ministerio de Ciencia
e Innovación (Spain) MCIN/AEI/10.13039/501100011033 (Ref. PID2020-113147GA-I00
and PID2021-122839NB-I00). A.V. acknowledges Grant. No. EDU/1508/2020 (Junta
de Castilla y León and European Social Fond). F.A.G. acknowledges the support
of the Computing Center of the Innsbruck University, where some of the present
calculations were carried out. This research has also made use of the high
performance computing resources of the Castilla y León Supercomputing Center
(SCAYLE, www.scayle.es), financed by the European Regional Development Fund
(ERDF). We are also grateful to Dr Marcelino Agundez for several discussions
and for providing us with a preprint of his recent work on ISM anions.

##  9 Supplementary Material

The following data have been prepared as Supplementary Material:

(i) original raw points from the $ab$ $initio$ calculations,

(ii) the multipolar coefficients for the Legendre expansion of the rigid rotor
potential for the C7N-/He system,

(iii) the computed inelastic cross sections,

(iv) the corresponding rate coefficients obtained from the 2D-RR-PES employed
in this work for the C7N- anion;

(v) The critical density values obtained for both anions and linked to their
collision data for either He or H2 are all provided as Supplementary Material.

All the above data are available at GREDOS:

10.14201/gredos.153158. (catalog 10.14201/gredos.153158)

##  10  Data Availability

The data that support the findings of this study are available within the
article and in its supplementary material, their accessibility is described
via the link reported in the previous Section.

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