# Rotational state-changing collisions of C2H- and C2N- anions with He under
interstellar and cold ion trap conditions: a computational comparison.

Jan Franz  Department of Theoretical Physics and Quantum Informatics, Faculty
of Applied Physics and Mathematics, Gdańsk University of Technology, ul.
Narutowicza 11/12, 80-233 Gdańsk, Poland  Barry Mant  Institute for Ion
Physics and Applied Physics, University of Innsbruck, Technikerstr. 25/3, 6020
Innsbruck, Austria  Lola González-Sánchez  Departamento de Química Física,
University of Salamanca, Plaza de los Caídos sn, 37008 Salamanca, Spain
Roland Wester  Institute for Ion Physics and Applied Physics, University of
Innsbruck, Technikerstr. 25/3, 6020 Innsbruck, Austria  Franco A. Gianturco
<EMAIL_ADDRESS>Institute for Ion Physics and Applied Physics,
University of Innsbruck, Technikerstr. 25/3, 6020 Innsbruck, Austria

###### Abstract

We present an extensive range of quantum calculations for the state-changing
rotational dynamics involving two simple molecular anions which are expected
to play some role in evolutionary analysis of chemical networks in the
Interstellar environments, C2H-($X^{1}\Sigma^{+}$) and C2N-
($X^{3}\Sigma^{-}$) but for which inelastic rates are only known for C2H-. The
same systems are also of direct interest in modelling selective photo-
detachment (PD) experiments in cold ion traps where the He atoms function as
the chief buffer gas at the low trap temperatures. This study employs
accurate, ab initio calculations of the interaction potential energy surfaces
(PESs) for these anions, treated as Rigid Rotors (RR) and the He atom to
obtain a wide range of state-changing quantum cross sections and rates at
temperatures up to about 100 K. The results are analysed and compared for the
two systems, to show differences and similarities between their rates of
state-changing dynamics.

††preprint: AIP/123-QED

##  I Introduction

The discovery of carbon chain anions in interstellar and circumstellar media
has triggered and stimulated a large number of theoretical and experimental
studies on these species (e.g. see references 1; 2; 3). Their structures and
spectral features, as well as the clarification of their importance and of
their role in interstellar chemistry, and in gas phase ion-molecule reactions
in general, have therefore also attracted many specific studies Herbst and
Osamura (2008); Walsh  _et al._ (2009); Eichelberger  _et al._ (2007) on their
behaviour. Recent examples have been our experimental work on the absolute
photodetachment cross-section measurements for hydrocarbon chain anions Best
_et al._ (2011) which has been nicely matched and confirmed by a computational
study of the same systems Douguet, Kokoouline, and A.E. (2014).

The possible, and likely, existence of anions in astrophysical sources was
first predicted theoretically and considered in earlier chemical models
Dalgarno and McCray (1973); Herbst (1981) although the first negative
hydrocarbon C6H- was only detected in 2006 McCarthy  _et al._ (2006), thereby
also solving the problem of the unidentified lines discovered by Kawaguchi et
al. Kewaguchi  _et al._ (1995). That identification was soon followed by the
detection of other negatively charged species such as C4H- Cernicharo  _et
al._ (2007), C8H- Remijan  _et al._ (2007); Brünken  _et al._ (2007a), C3N-
Thaddeus  _et al._ (2008), C5N- Cernicharo  _et al._ (2008), and CN- Agúndez,
M.  _et al._ (2010). The majority of these species were first detected in a
well observed Circumstellar Envelope (CSE) IRC+10216, although these and other
hydrocarbon anions were also discovered later on in other molecular clouds
Sakai  _et al._ (2010). As it is to be expected, the study and search for
interstellar anions of both simple and increasingly more complex structural
properties is still current and relevant. More specifically, a simple species
like C2H- ($X^{1}\Sigma^{-}$), has been expected to be amenable to observation
with the new astrophysical instruments such as ALMA, especially since it had
been already observed as a stable molecule in laboratory experiments Brünken
_et al._ (2007b); Amano (2008). Furthermore, its parent neutral form C2H
($X^{2}\Sigma^{+}$) has already been a well-known astrophysical molecule
discovered by observation as early as 1974 Tucker, Kutner, and Thaddeus
(1974), thereby suggesting that the corresponding C2H- anion should also be
present, although perhaps as only a very low-abundance species, a fact
justified in terms of its high chemical reactivity and therefore expected
rapid destruction upon formation. Actual current numerical models predict, in
fact, that the formation of larger carbon chain anions should be in any case
more probable than that for similar, but smaller chains as the C2H-
Barckholtz, Snow, and Bierbaum (2001), hence somehow supporting the
difficulties for its observation.

The detection of carbon chain anions and parent neutral species in various
astronomical environments has stimulated many computational studies of their
collisions with He and H2 to obtain rotationally inelastic excitation and de-
excitation rates. Examples include the series C2H/C2H- Spielfiedel  _et al._
(2012); Dagdigian (2018); Dumouchel  _et al._ (2017, 2012) , C4H- Senent  _et
al._ (2019), C6H/C6H- Walker  _et al._ (2017, 2016) and CN/CN- Lique  _et al._
(2010); Lique and Kłos (2011); Kalgina, Lique, and Kłos (2012); Kalgina, Kłos,
and Lique (2013); Kłos and Lique (2011); González-Sánchez  _et al._ (2020),
C3N/C3N- Lara-Moreno, Stoecklin, and Halvick (2017, 2019); Tchakoua, Motapon,
and Nsangou (2018).

In spite of the fact that the actual origin of many of the hydrocarbon anions
has not yet been solved, it is generally accepted that gas-phase processes are
crucial for their formation and therefore the many observed hydrocarbon
radicals CnH may also be the main precursors of the formation of CnH- anions
through electron attachment or association processes, whereas associative
detachment processes would contribute to the generation of initial, neutral
CnH species. As an example, a new mechanism for the formation of C2H- from
C2H2 has been proposed not long ago Cordiner  _et al._ (2008) from laboratory
experiments. Recent experiments in our group have also investigated reactions
of C2H- with C2H2 as a means of forming larger C2nH- and C2n chains Bastian
_et al._ (2019).

Along similar lines, another small C-bearing molecule, the neutral C2N radical
($X^{2}\Pi_{r}$) has been also detected earlier on in the interstellar medium
Anderson and Ziurys (2014), where the molecule was observed at the 1 – 2 mK
level toward the same circumstellar envelope (CSE) of IRC + 10216 already
mentioned earlier, using the facilities of the Arizona Radio Observatory
(ARO). Lambda doublets of the $J=4.5\rightarrow 3.5$ and the $J=6.5\rightarrow
5.5$ transitions at 106 GHz and 154 GHz in the $\Omega=1/2$ ladder were
measured with the ARO 12 m telescope, as well as the $J=9.5\rightarrow 8.5$
lines near 225 GHz, using the ARO Sub-Millimeter Telescope (SMT) Anderson and
Ziurys (2014). Considering other species in the same environment, it is
interesting to note that the [CN]/[C2N]/[C3N] abundance ratio was found to be
$\sim 500/1/50$, thus indicating again one could expect a rather low abundance
for the neutral radical C2N. For the corresponding anionic counterpart,
however, no observational evidence within the same CSE has been reported thus
far, also suggesting a low abundance of the latter once formed within that
chemical network.

In spite of this absence of direct detection, both the above anionic
molecules, C2N- and C2H-, have been the object of several laboratory studies
which have analysed in some detail the photo-detachment (PD) mechanisms of
these species and several other structural properties Kawaguchi  _et al._
(1984); Garand, Yacovitch, and Neumark (2009); Huang and Lee (2009); Ervin and
Lineberger (1991); Zhou, Gerand, and Neumark (2007); Senent and Hochlaf
(2013). Furthermore, in general terms we should note here that possible
astrophysical abundances of either observed or not yet observed species have
to be understood in terms of molecular stabilities, reaction probabilities and
of both radiative and collisional excitations and relaxation of internal
molecular modes: the accurate knowledge of all these facts can indeed help us
to better explain the existence of a molecule and the probability of it being
observed. The molecular stability and the spectroscopic properties of both the
C2H- and C2N- anions have been studied in various earlier investigations
Kawaguchi  _et al._ (1984); Garand, Yacovitch, and Neumark (2009); Senent and
Hochlaf (2013), while the modelling of molecular emission in the ISM
environments where they could be expected to exist requires collisional rate
coefficients with the most abundant interstellar species like He and H2.
Collisional data have been already presented for the C2H- anion interacting
with He Dumouchel  _et al._ (2012), while no corresponding calculations, as
far as we know, exist for the collisional rotation-state changes of C2N-
interacting with He. The actual ab initio PES is also not known for that
system. The quantum dynamics of both these anions, besides not being
extensively studied under CSE conditions, has also been only partially
discussed under the operating low temperatures of ion cold traps Gianturco
_et al._ (2019). The present work is therefore directed to acquiring novel
knowledge about the quantum dynamics of C2N- in collision with He under both
astrochemical and cold ion trap conditions, further implementing a comparison
between its dynamical behaviour and that of the C2H- polar anion under similar
conditions.

The following section will only briefly remind readers about the features of
the anisotropic potential energy surface involving C2H- and He atoms, since it
has been presented and analysed already in earlier work Dumouchel  _et al._
(2012). It will instead present in greater detail the new calculations of the
ab initio points related to the Rigid Rotor (RR) PES associated to the
interaction between C2N- ($X^{3}\Sigma^{-}$) and the neutral He atom. The two
interaction potentials will then be compared and their level of spatial
anisotropy will be analysed and discussed.

Section III will present the state-changing rotationally inelastic cross
sections for both systems and discuss the effects from spin-spin and spin-
rotation structural features on the C2N- system via à vis the C2H-
($X^{1}\Sigma^{+}$) system. The corresponding inelastic rates at the
temperatures of interest will also be presented, compared and discussed.
Section IV shall examine the possible evolutionary dynamics of C2N- anions
under the conditions of a cold trap when they are undergoing laser-driven
photo-detachment processes. A comparison with the corresponding behaviour of
the C2H- system will also be presented and discussed. Conclusions will be
given in section V.

##  II Features of the ab initio interactions of C2H- and C2N- with He atoms

The electronic ground state of C2N- was taken to be ($X^{3}\Sigma^{-}$) as
indicated in earlier work Garand, Yacovitch, and Neumark (2009). The molecular
geometry of this linear anion is $r_{CC}=1.344$ Å and $r_{CN}=1.207$ Å and was
given in the experiments from Garand et al. Garand, Yacovitch, and Neumark
(2009). We have optimized the molecular geometry using the MRCI + Davidson
correction Werner  _et al._ (2012, 2019) employing the aug-cc-pVQZ Woon and
Dunning (1994) basis set obtaining $r_{CC}=1.360$ Å and $r_{CN}=1.212$ Å . The
ab initio calculations of the 2D grid of points in Jacobi coordinates (R,
$\theta$) were carried out with the program package MOLPRO 2012 Werner  _et
al._ (2012, 2019). For all the present calculations we have employed the
internally-contracted multi-reference configuration interaction method (IC-
MRCI) Werner and Knowles (1988); Shamasundar, Knizia, and Werner (2011), using
the aug-cc-pVQZ basis set Woon and Dunning (1994) on all atoms. The reference
space for the MRCI calculations consists of a complete active space by
distributing 14 electrons in 15 orbitals. All single and double excitations
from the reference configurations are included in the variational calculation.
The effect of quadruple excitations is estimated via the Davidson correction
Werner  _et al._ (2012, 2019). Since the MRCI approach does not provide a
size-consistent method, we have included a correction for size-consistency by
using the Davidson correction that estimates the contributions of quadrupole
excitations as discussed in references Langhoff and Davidson (1974); Buenker
and Peyeremhoff (1982). The Basis-Set-Superposition-Error (BSSE) correction
was not included in this study because this procedure is not well defined when
multi-reference methods are employed. In any event, our previous experience
with small anions interacting with He has shown us that the corresponding
rigid-rotor PES is only marginally modified by BSSE corrections in the well
regions and on the onset of the repulsive walls. We therefore expect that it
would also be not very significant in the present case. We further checked the
quality of the asymptotic form of the interaction by calculating the
corresponding potential term for the situation where He approaches with an
angle of 90°to the molecular axis. In this situation the terms of the
potential involving the higher order Legendre polynomial should be zero (by
symmetry) and the spherical part should dominate. The computed value of the
coefficient for the asymptotic potential gives us the value of the spherical
polarizability of the He atom, which comes out to be 1.379 $a_{0}^{3}$. This
value is in good agreement with a recent result from QED calculationsPuchalski
_et al._ (2020) of 1.38376078 .

To further test possible differences coming from our different post-Hartree-
Fock methods we have used, we have carried out a a comparison between MRCI(Q)
and CCSD-T calculations for the approach of the He partner at an angle of 90
degrees with respect to the axis of the C2N- ion. In the comparison the
potential energy curves for CCSD-T and MRCI(Q) are set to zero at 25 Angstrom.
We found that both curves have their minimum at 3.5 Angstrom, with the largest
difference between the two curves occurring at the location of the minimum
configuration: -59.9 cm-1 for CCSD-T compared to -54.5 cm-1 for MRCI(Q).

The angular grid involved calculations of radial ‘cuts’ every 5°, from 0°to
180°. The radial grid included a higher density of points around the various
minima regions at each selected angle. A total of 44 radial points were used
between 2.3 and 26 Å. We have not included the basis-set-superposition-error
procedure because such a procedure turns out not to be well defined for multi-
reference methods. The global minimum for the complex was found at a distance
of around 3.545 Å from the centre-of-mass, located at an angle of around 80°,
with a well depth of around 58 cm-1. The data shown by the two panels of
Figure 1 provide a pictorial view of the new PES calculated for the C2N-/He
system (upper panel), while it also shows for comparison the PES already
calculated in earlier work for the similar anion of C2H- interacting with He
Dumouchel  _et al._ (2012). One sees in the two panels similarities between
the two interaction potentials since both exhibit the presence of a minimum
energy region of their complex with He. The more shallow well of the lower
panel is extensively off the C2v geometry, while the deeper well on the upper
panel appears to be more localized and closer to the center-of-mass than it is
the case in the lower panel. Given the larger number of electrons in the C2N-
case, the corresponding well is therefore deeper than that for the C2H-/He
complex Dumouchel  _et al._ (2012) and more localized in space. Both
potentials will be asymptotically driven by the polarizability term involving
the spherical, dipole polarizability of the He partner and therefore will
behave very similarly in their long-range regions because of it. The
differences in their short-range features, however, will be reflected later in
the size differences between their collisional state-changing probabilities
discussed in the next Section.

The localization of the excess charge provided by the extra bound electron of
the anion is also an interesting item for the new C2N-/He PES presented here.
The Mulliken charges in the asymptotic situation (e.g. with the helium at a
distance of 26.55 Å from the center-of-mass of the anionic target) turn out to
be as follows (when computed with the MRCI / aug-cc-pVQZ of the post-HF
treatment): C1 = -0.59049; C2 = +0.22532 ; N1 = -0.63483. Here C2 is the
central carbon atom, so that the geometry of C2N- is given as: C1-C2-N1 , with
no extra charge on the interacting He atom, as expected. The direction and
value of the dipole moment are therefore given as: 2.1818 Debye, placed along
the positive direction of the molecular z-axis from the C1-end of the
molecule, a result from the delicate balance between excess charges in this
molecule.One should also note that the approach of the He atom along the
direction of 0°is on the N-side of the C2N- anion. The same angle corresponds
to the approach on the H-side of the C2H-. Another type of presentation of the
PESs for both systems, to be used in the scattering calculations below, is
obtained by numerically generating the radial coefficients of the multipolar
expansion of the Rigid Rotor (RR) 2D potential energy surfaces:

$V(r=r_{e},R,\theta)=\sum_{\lambda}V_{\lambda}(R)P_{\lambda}(\cos\theta)$ (1)

where $r_{e}$ is the geometry of the equilibrium structure of the anion and
the sum over the contributing $\lambda$ values went up to 19, although only
the dominant, stronger terms are shown in Figure 2. The panels of that figure
also compare the present findings for C2N- with the earlier data for C2H-
Dumouchel  _et al._ (2012). The expansion coefficients for the C2N--He PES are
provided in the supplementary information.

Figure 1:  Computed PESs for the two molecular anions of the present study.
The data are presented as in-plane maps in 2D with $r\cos{\theta}$ and
$r\sin{\theta}$ as coordinates. Energy levels in cm-1. Upper panel: C2N- from
present calculations; lower panel: C2H- from ref. Dumouchel  _et al._ (2012).
Figure 2:  Computed multipolar coefficients calculated from the initial PES
data. Upper panel: lower values of the dominant radial coefficients for the
C2N- anion; lower panel: same data but for the C2H- anion. Energy values in
cm-1 and distances in Å .

The radial coefficients presented in Figure 2 underline once more both the
general similarities between the two systems and important differences which
will vary their dynamical behaviour we shall analyse below. If we look at the
spatial anisotropy of the multipolar coefficients we see that the spherical
terms ($\lambda$ = 0) are very similar to each other in both strength and
spatial extension around their well regions. On the other hand, the radial
coefficients with $\lambda$ = 1, 2 and 4 are all uniformly repulsive in the
short-range region for C2H-, while only those for $\lambda$= 2 and 4 are so
for the C2N- anion. These differences indicate that the $\Delta N$ transitions
between rotational levels, those for which the acting torques from the
anisotropic PES coefficients will be stronger, could be those with $\Delta N$
with odd values for C2N- and with even values for the C2H-. The presence of
such possible propensity effects will be further discussed when analysing the
computed inelastic cross sections and rates in the next section. Another
important element of distinction/similarity are the relative spacings in
energy between the rotational states for both systems which are presented in
Figure 3.

Figure 3:  Schematic location of the rotational energy levels for the
molecular anion of the present study. For the case of C2N- ($X^{3}\Sigma^{-}$)
only the pseudo-singlet levels without splitting effects are shown. See main
text for further details.

To simplify the comparison we report only the energy spacing between $N$-level
separations, without showing the spin-spin and spin-rotation splitting for
C2N-. The sizes of those splitting constants are not known or available as yet
and we will be discussing further in the next Section how their effects will
be included in the dynamics of the present study. Their energy splittings
would in any event not be visible on the chosen energy scale in that Figure.
We clearly see, however, that the density of states over the examined range of
about 60 cm-1, which should cover most of the relevant energy range of the CSE
environments and of the cold ion traps operation, is dramatically different
between the two systems, with the C2N- anion showing twice as many states
being accessible within that energy range. The markedly higher ‘crowding’ of
rotational states per unit energy for the latter molecule translates into a
higher number of them being significantly populated at equilibrium
temperatures in either environments.

The use of a Local Thermal Equilibrium (LTE) population scheme is often
employed as a starting condition of ISM kinetic modellings. Additionally,
photo-detachment experiments in cold traps usually start, as we shall further
discuss in a later Section, with thermal equilibrium of the molecular internal
states being achieved at the temperature of the trap. A comparison between
steady-state rotational level populations, over the range of temperatures of
interest, is reported for the two anions by the two panels in Figure 4. We
clearly see in them that, as an example, at temperatures in a cold trap around
25 K the C2N- anion in the left panel significantly populates levels up to
$N=10$, while the C2H- anion on the right panel only has states up to $N=6$
significantly populated. Such differences of behaviour will be further
discussed in the next two Sections when the inelastic cross sections and rates
will be computed first to numerical convergence and then the possible dynamics
of laser-induced PD processes in traps will be analysed at low temperatures.

Since the efficiency of state-changing collisional processes for rotational
states is affected, among other features also discussed below, by the size of
the energy gaps between the involved levels, we shall therefore expect that
the marked differences shown by Figure 3 will be reflected in differences
between inelastic rates for the two anions.

Figure 4:  Steady-state distributions of relative molecular populations of
rotational levels for temperatures up to 30 K. Left panel: the C2N- anion;
right panel: the C2H- anion. See main text for further discussion.

##  III  Quantum calculations of state-changing rotationally inelastic cross
sections and rates

The inelastic cross sections involving collisional transitions in both the
title molecules were calculated using our in-house multichannel quantum
scattering code ASPIN, which we have already described in many earlier
publications of our group López-Durán, Bodo, and Gianturco (2008); Martinazzo,
Bodo, and Gianturco (2003); Hernández Vera  _et al._ (2017); González-Sánchez,
Bodo, and Gianturco (2006) and therefore will not be repeated here in detail.
Interested readers are referred to the above publications for consultation.

The ground state of the C2N- anion is ${}^{3}\Sigma^{-}$, the same as the OH+
molecule for which we have previously studied low-temperature collisions with
He in detail González-Sánchez, Bodo, and Gianturco (2007); González-Sánchez,
Wester, and Gianturco (2018). The conclusions of this previous work inform the
calculations presented here. In the OH+ (${}^{3}\Sigma^{-}$) state, we have
three levels for each total angular momentum $\geq 1$: the rotational levels
are in fact split by spin-spin and spin-rotation coupling effects. In the pure
Hund’s case (b) the electronic spin momentum $S$ couples with the nuclear
rotational angular momentum $N$ to form the total angular momentum $j$, given
by $j=N+S$ González-Sánchez, Bodo, and Gianturco (2007); González-Sánchez,
Wester, and Gianturco (2018). As a consequence of that coupling term, the
rotational levels of the molecule may be labelled not only by the above
quantum numbers defined in the Hund’s case (b) but also by their parity index
$\varepsilon$. The levels in molecules of odd multiplicity with parity index
equal to +1 are labelled e, and those with parity index equal to -1 are
labelled f González-Sánchez, Bodo, and Gianturco (2007). However, we shall
omit this index in the following discussion since we shall not be using it in
our analysis.

The extensive work we have done on collision of He with the OH+
(${}^{3}\Sigma^{-}$) molecule González-Sánchez, Bodo, and Gianturco (2007);
González-Sánchez, Wester, and Gianturco (2018) have already shown that the
$\Delta N\neq 0$ transitions involve much larger energy values than those
which cause changes in the spin quantum number $S$ that is responsible for the
separations between $j$ states. Thus, one can say that rotational quenching
transitions have much larger energy gaps than those which simply cause spin-
flipping processes within each $N$-labelled manifold. In a full close-coupling
(CC) approach to the quantum dynamics the molecular Hamiltonian includes, in
addition to the rotational contribution, a nuclear coupling contribution which
induces hyperfine energy splitting. These splittings are however lower than
typically 10-3 cm-1, i.e. they are much lower than the rotational spacings
investigated in this work, as we shall further show later in this paper. In
such situation, a possible approach would be to neglect the hyperfine
splitting and to decouple the spin wave functions from the rotational wave
functions, using an angular momenta recoupling scheme (e.g. see: Alexander &
Dagdigian Alexander and Dagdigan (1983)) which simplifies considerably the
dynamics of the problem. The latter is then reduced to solving the simpler
spin-less CC equations associated to the more usual ${}^{1}\Sigma$ case. This
approximate approach of treating the target as a pseudo-singlet case has been
found in earlier works of our group Hernández Vera  _et al._ (2017); González-
Sánchez, Wester, and Gianturco (2018) to yield results very close to the exact
cross sections. Hence the ‘recoupling approach’ outlined above will be
considered as our reference approach in the following calculations, where our
results will be compared with those carried out for the exact ${}^{1}\Sigma$
case of the C2H- molecular anion.

For C2N--He (reduced mass $\mu=3.619$ amu), scattering calculations were
carried out for collision energies between 1 and 500 cm-1 using steps of: 0.05
cm-1 for 0 to 6 cm-1, 0.1-1, for 6-100 cm-1, 0.2 cm-1 for 100-200 cm-1, 1.0
cm-1 for 200-400 cm-1 and of 2 cm-1 for 400-500 cm-1. This fine energy grid
was used to ensure that important features such as the many resonances
appearing in the cross sections were accurately accounted for and their
contributions correctly included when the corresponding rates were calculated,
as discussed below. The CC equations were propagated between 1.7 and 100.0 Å
in 2000 steps using the log-derivative propagator Manolopoulos (1986) up to 60
Å and the variable-phase method at larger distances Martinazzo, Bodo, and
Gianturco (2003) up to 100 Å. The potential energy was interpolated between
calculated $V_{\lambda}(r_{\rm{eq}}|R)$ values using a cubic spline and
extrapolated below and above the ab initio grid using linear and polynomial
functions respectively as implemented in ASPIN López-Durán, Bodo, and
Gianturco (2008). As our ab initio grid goes to 26 Å and the scattering
energies of interest are not in the ultracold or high energy regimes, the
details of the potential extrapolation have a negligible effect on the
scattering cross sections. To test this point, however, we have always
extended the radial integration with an extrapolated potential of the R-4\-
type out to 100 Å or more to reach convergence of the final cross sections. We
have also found that to employ the correct dipole polarizability coefficient
for the extrapolated potential instead of the value produced by ASPIN from the
ab initio potential only changed the final cross sections by less than 1.0%,
and only at the lowest energies. The rotational basis set was increased with
increasing energy from $N=20$ to $N=36$ at the highest energy considered. The
number of partial waves was also increased with increasing energy reaching
$J=96$ at the highest energy. Inelastic cross sections were computed for all
transitions between $N=0$ to $N=15$, which was deemed to be sufficient to
model buffer gas dynamics in a cold trap up to about 50 K (see below), while
the same range of levels is expected to be the one most significantly
populated during low-energy collisional exchanges with He atoms within ISM
environments dynamical conditions. Scattering calculations for C2H- were
carried out by us in a former study Gianturco  _et al._ (2019) using the PES
of Dumouchel et al. as discussed above Dumouchel  _et al._ (2012).

Another quantity that we shall compute from the cross sections is the state-
to-state inelastic rotational rates over a range of temperatures from
thresholds up to about 100 K, to cover the range of $T$ values expected to be
significant for processes in the ISM environments we are discussing here.
Hence, once the state-to-state inelastic integral cross sections are known,
the rotationally inelastic rate constants $k_{N\rightarrow N^{\prime}}(T)$ can
be evaluated as the convolution of the cross sections over a Boltzmann
distribution of the relative collision energies. In the equation below, all
quantities are given in atomic units:

$k_{N\rightarrow N^{\prime}}(T)=\left(\frac{8}{\pi\mu
k_{b}^{3}T^{3}}\right)^{1/2}\int_{0}^{\infty}E\sigma_{N\rightarrow
N^{\prime}}(E)e^{-E/k_{B}T}dE$ (2)

Both the calculations of inelastic cross sections and of their corresponding
rates over the range of temperatures mentioned earlier have been carried out
using the new interaction PES for the C2N-/He system and via the earlier PES
already computed for the C2H-/He system Dumouchel  _et al._ (2012). The
comparison between our findings for these two systems will be carried out
below by presenting our computed data.

Figure 5:  Rotationally inelastic cross sections for the C2N- (solid lines)
and the C2H- (dashed lines) anions. The upper panels report excitation
processes while the lower panels describe de-excitation processes. See main
text for further discussion.

The present results for the inelastic cross sections, over a range of 100
cm-1, are reported in the four panels of Figure 5. The upper two panels show
excitation processes while the lower two panels present de-excitation cross
sections for both systems. The solid lines refer to the calculations for C2N-
while the dashed lines are those for C2H-.

It is interesting to notice that in the low-energy range up to about 20 cm-1
all of the cross sections reported in the four panels show evidence of
resonances effects in both systems. The relative strength of the interaction
potentials and the presence of many more rotational states available for the
C2N- makes the cross sections for the latter anion exhibit a much denser
sequence of resonant features, as expected from the larger mass of the
molecular partner (hence a larger number of partial waves contributing to
shape resonances) and form the stronger coupling with the nearby closed
channels during the scattering (hence the presence of Feshbach resonances).
Since no experiments are as yet available on the scattering behaviour of these
anions, we shall not carry out any detailed analysis of the features seen in
our calculations. They will however contribute to changing the size of the
related inelastic rates we shall further discuss later, and will be taken into
account in our calculations.

In the upper two panels where excitation processes are reported, we see that
in the left panel, showing $\Delta N$=1 transitions, at energies beyond the
resonance regions, the excitation from the lowest rotational state of C2H- are
the largest over the whole range, in spite of the fact that the amount of
energy transferred between levels is also the largest. Two structural factors
could contribute to this dynamical difference: (i) the smaller value by about
30% of the reduced mass of the C2H- which affects relative sizes of
wavevectors, and (ii) the presence for only this molecule of an attractive
well in the radial coefficient with $\lambda$= 1 which directly couples the
levels involved in that transition during collision. The marked increase in
energy gap when one considers transitions still with $\Delta N$=1 but between
excited states (i.e. between the $j=2$ and $j=3$ levels) for the C2H- system
in the same upper-left panel, is the dominant factor which is causes that
inelastic cross section to be the smallest. On the other hand, the two
excitation cross sections involving the same levels but for the C2N- anion
which are in that same panel are closer to each other. This is due to the much
smaller energy gap between its levels which does not offset the greater
strength of the interaction potential for the latter molecule compared to the
former, hence making the C2N- anion more efficiently excited by collisions
with He to its upper rotational states.

The panel in the upper-right part of Figure 5 shows now excitation processes
involving $\Delta N$=2 transitions. In the low-energy resonance regions the
cross sections are uniformly smaller than those discussed before for the C2N-
anion, while those for the C2H- anion are instead larger in size and also
larger than those pertaining to the other anion. This difference can be linked
to the differences in behaviour between their corresponding multipolar
coefficients, as shown in Figure 2. The C2H- partner presents both terms with
$\lambda$= 1 and $\lambda$= 2 with marked attractive wells and steeply
repulsive walls, while the other anion only shows that behaviour for the
$\lambda$= 2 coupling term. It therefore follows that both direct and indirect
potential coupling effects are dynamically efficient for the C2H- anion, while
only one is effective for C2N-.

When turning to the collisional de-excitation processes in the lower panels of
Figure 5, we see in the left panel that the transitions with $\Delta N=-1$ for
the C2N- and the C2H- anions yield markedly larger cross sections than those
for de-excitation transitions with $\Delta N=-2$, a behaviour attributed again
to the role played by the increase of the energy gaps in the case of the
latter processes. Also the differences in the coupling strength between the
two PESs discussed earlier cause the $\Delta N=-2$ transitions for C2N- to be
the smallest of them all. It is also interesting to note that, in the lower-
left panel, the smallest of the cross sections is that for the $\Delta N=-1$
transition of the C2H- partner. The much larger energy gain transferred into
relative energy between collision partners makes this system have a reduced
interaction time that in turn makes sudden collisions more effective. Since
such gain is larger for $\Delta N=-2$ processes than for the $\Delta N=-1$
transition, it makes sense that the cross sections for the latter are smaller
than those for the former process when C2H- is concerned. The differences in
energy gains are however much smaller for C2N-, where the differences between
multipolar coefficients play a more important role than the smaller changes in
the interaction times, thus making the $\Delta N=-1$ transitions to be the
larger ones.

On the whole, the present quantum calculations of the relative state-changing
collision probabilities for the two title systems indicate a clear similarity
of general behaviour and of relative efficiency for such processes at the
energies of interest for ISM conditions, although changes linked to their
structural features can distinguish between their interactions with the He
atom and play a significant role in their relative dynamics.

Figure 6:  Rotationally inelastic collisional rates for the C2N-( solid lines)
and the C2H- (dashed lines) anions. The upper panels report excitation
processes while the lower panels describe de-excitation processes. The range
of temperatures goes up to 50 K. See main text for further discussion.

The results reported by Figure 6 indicate in the behaviour of the rotationally
inelastic collisional rates over a range of temperature covering values of
interest for cold ion traps and also for ISM environments where the present
anions are expected to be present. As with the cross sections of Figure 5, the
upper two panels report excitation processes while the lower two panels show
the behaviour of the de-excitation processes. Solid lines refer to the C2N-
anion while the dashed lines are for the C2H- case.

It is interesting to note, when looking at the results in the two upper
panels, that the differences in size between rates for $\Delta N$=1 and those
for $\Delta N$=2 are fairly small in the case of the C2N- target, while are
markedly larger for C2H- as a target. This is again due to the greater effects
of the energy gaps between rotational states which exist between the two
anions, as shown by Figure 3. These types of differences are also present in
the de-excitation rates reported by the two lower panels , where again we see
that the differences in size of the rates for each of the anion are all within
less than one order of magnitude. As expected, we see that the rates follow
closely the indicated differences discussed earlier between cross sections for
both systems.

The data reported by the four panels of Figure 7 make a pictorial comparison
between sizes of specific rate values at two different temperatures: at 15 K
in the two upper panels and at 50 K in the two lower panels. They confirm what
was discussed earlier and underline the delicate interplay between the
structural and dynamical differences controlling the state-changing collisions
between rotational states in the two anions.

The upper two panels of Figure 7 compare two de-excitation rates for two
different initial states: $N$=2 in the left panel and $N$=4 in the right
panel. The following comments can be made :

(i) the rates for the C2N- anion are larger for $\Delta N=-1$ processes, while
becoming smaller than those for C2H- for $\Delta N=-2$ processes. The
switching is related to the differences in multipolar coefficients discussed
earlier.

(ii) for the same types of transitions at the higher temperature shown in the
lower two panels, we see that the inversion of relative magnitudes when going
from $\Delta N=-1$ processes to the $\Delta N=-2$ is still present and linked
to different coupling strengths of their state-changing dynamics.

Figure 7:  Rotationally inelastic collisional quenching rates for the C2N- and
the C2H- anions for $N=2$ (left panel) and $N=4$ (right panel) at 15 K (top
panels) and 50 K (bottom panels). See main text for further discussion.

Indicators on possible propensity rules for the size-dependence of the cooling
(de-excitation) rates from an initial rotational state and down to different
final levels are reported in the two panels of Figure 8. The variations in
size of the rates from the initial $N=8$ to all final levels below are shown
at two different temperatures.

Figure 8 verifies directly what we have already discussed when analysing the
cross sections and the rates, i.e. that in the collisional cooling (de-
excitation) cascades within each system are linked to the differences in the
structural features of the interactions between the anionic partners and the
He atom. It is immediately clear that temperature does not play a significant
role and that structural features are largely driving the differences. In the
case of the C2N- anion, increasing the $\Delta N$ values down the ladder
uniformly reduces the size of the inelastic rates as the energy gaps uniformly
increase. Such differences are smaller than those for C2H- where larger energy
spacings occur between levels. Within this decrease in size, however, we still
see that for C2N- the transitions involving odd $\Delta N$ values are larger
that those with even values, while the opposite is true for the C2H-. This is
again linked to the dynamical couplings induced by the attractive $\lambda$=
1, 3 and 5 odd terms of the multipolar expansion of the C2N-, as shown by
Figure 2. On the other hand, for the C2H- anion we see an even more marked
effect linked to the relative strength of the multipolar coefficients (see
again Figure 2), since the $\lambda$=2 and 4 potential coupling terms have a
shallow attractive well in the outer radial range and become strongly
repulsive around the turning point values at the energies of interest. We
therefore see that they induce rotational torques during dynamics which would
favour transitions into final states where even $\Delta N$ values occur.

It follows from the above considerations that collisional repopulation of
rotational levels will take different preferential paths in the two title
systems, a result which would suggest the final presence in the relevant
environments of different populations for their molecular rotational states.

Figure 8:  Size changes of rotationally inelastic collisional rates for the
C2N- (left panel) and the C2H- (right panel) anions. The two curves in each
panel refer to two different temperatures as reported in the captions. The
rates are calculated from the $N=8$ state in each molecule, at the two
temperatures shown, and the downward rates are given as a function of the
final state for each anion. See main text for further discussion.

Rates for all transitions between $N=0$ and $N=8$ for C2H- and between $N=0$
and $N=15$ for C2N- are provided in the supplementary information for
temperatures between 0 and 100 K.

##  IV  Modelling Quantum dynamics of laser photo-detachment processes in cold
traps

One of the important processes which can take place in cold ion traps
Gianturco  _et al._ (2019, 2018) where molecular anions can be confined,
involves studying the internal state evolution under laser-induced detachment
of the extra anionic electron. The study photo-detachment of molecular anions
is obviously more complicated than for atomic PD processes due to the
increasing number of accessible pathways which become available in the case of
molecular targets. Molecules exhibit a higher density of lower- lying
electronic states while one must further consider the increase in the density
of states due to the additional vibrational motion, i.e. ro-vibrational and
pre-dissociative states of the initial anion are actively present as the bound
electron is moved into the continuum by the laser beam. The analysis of the PD
mechanism for molecular negative ions also provides an important tool for the
studies of reaction dynamics, where one wants to employ the initial anionic
partner in a reaction to be specifically produced in a given ro-vibrational
state of its ground electronic state as the neutral is generated in the PD
reaction.

This process can therefore be outlined as having to follow this sequence of
events: (i) collisional population re-distribution among the rotational levels
of the anion loaded in the trap and interacting with He, as the prevalent
buffer gas, at a given trap temperature; (ii) switching on of the photo-
detaching laser after the previous rotational population equilibration has
been achieved. One could then decide to vary the laser’s operational
wavelength to selectively depopulate different rotational states of the anion
present in the trap after the step (i), and producing the ejected electron
either at threshold or away from it, or simply follow the total anion losses
in the trap during the photo-detachment (PD) step; (iii) test the dynamical
effects from changing other operating conditions such as: buffer gas density,
trap equilibration temperature, laser power and laser wavelength in order to
find the best choices for the PD processes tailored to the specific systems.

To follow the above steps in modelling the PD process, the state-changing
collisional rates for the molecular anion of interest need to be known at
different operating temperatures. Furthermore, we need to know as best as
possible the PD rates (obtained from the PD cross sections as functions of the
initial rotational state, as we have previously discussed Gianturco  _et al._
(2019, 2018)). As a comment about the calculated cross sections and rates for
state-to-state inelastic processes, it suffices to say that the dominant
inelastic processes we found for the present systems are those for which the
state-changing values of $\Delta N\pm 1,2$ and for which the rates at a
temperature of, say 20 K, are of the order of 10-10 in units of cm3 s-1. As a
comparison, the corresponding spontaneous emission Einstein A coefficients for
the C2H- anion ( in units of s-1) vary to be from about 10-3 to 10-5 times
smaller than the collisional rate constants for this system Gianturco  _et
al._ (2019). We shall therefore chiefly consider only collision-induced
effects in the present analysis.

For modelling the rotational population evolution dynamics, the master
equations need to be solved using the collisional thermal rates already
computed above, at each chosen trap temperature and for the range of selected
He density which are expected to be experimentally achieved:

$\frac{dn_{i}(t)}{dt}=\sum_{j\neq
i}n_{j}(t)C_{ji}(T)-n_{i}(t)\left(\sum_{i\neq
j}P_{ij}(T)+K_{i}^{PD}\right)\;\;,$ (3)

The quantities $P_{ij}(T)$ are the rates for the destruction of level $i$,
while its formation rates are given by the $C_{ji}(T)$ terms. During the
collisional step, i.e. before the laser is switched on, the coefficients are
given as: $P_{ij}(T)=\eta_{He}k_{i\rightarrow j}(T)$ and
$C_{ji}(T)=\eta_{He}k_{j\rightarrow i}(T)$. These relationships describe the
‘collision-driven’ time evolution process of thermalisation of the relative
populations of the rotational levels of the anion which are reached at the
selected buffer gas temperature and for a given density $\eta_{He}$ in the
trap. Once thermalisation is reached and the PD laser is turned on,
$K_{i}^{PD}$ is the additional destruction rate of the selected level $i$
caused by the PD laser. The set of rates $K_{i}^{PD}$ is critical in the
experiments in general and for the present numerical simulations because they
drive the destruction of both the population of one specific rotational level
$i$ and of all the molecular ions which have been populating that specific
state during the previous thermalisation step in the traps. As the trapped
anion is still undergoing collisions with the buffer gas, there is now
competition between the laser-induced de-population of a rotational level and
its collisional re-population in the trap.

In the experiments of this study these rates depend on the laser photon flux
and on the overlap between the laser beam and the ion cloud within the trap.
Since these parameters, as well as the absolute values of the state-to-state
PD cross sections, are presently unknown for the title molecular anions, we
shall introduce a scaling parameter which we have already discussed in our
earlier work Gianturco  _et al._ (2019, 2018); Mant  _et al._ (2020); Simpson
_et al._ (2020) according to which the relation between the required rate and
the estimated cross section is given by:

$K_{N}^{PD}=\alpha(\nu)\sigma_{N}^{PD}(\nu)$ (4)

The above scaling parameter $\alpha(\nu)$ accounts globally for modelling the
relative role of different features of the photo-detachment experiment which
are effectively the laser-driven features of the PD dynamics. They include
quantities such as laser-flux, spatial overlap, etc. which compete with the
pure collisional rates that repopulate the anion’s rotational levels via its
interaction with the buffer gas. The specific scaling of that parameter within
the modelling allows us to simulate either a ‘collision-dominated’ situation
or a ‘PD-dominated’ situation whenever the laser strength is markedly varied
by taking the $\alpha(\nu)$ to be either equal to 1.0 or one or more orders of
magnitude smaller, as shown in our earlier work Gianturco  _et al._ (2019,
2018); Mant  _et al._ (2020); Simpson  _et al._ (2020) and as discussed below.
The values of $\sigma_{N}^{PD}(\nu)$ can be obtained through an analysis which
we have presented before Gianturco  _et al._ (2019, 2018); Mant  _et al._
(2020); Simpson  _et al._ (2020). Briefly, the PD cross section is given as:

$\sigma_{N}^{PD}(\nu)\propto\sum_{N`=0}^{N_{\rm
max}}\left|C_{N010}^{N^{\prime}0}\right|^{2}(E-E_{th})^{p}\Theta(E-E_{th})\;\;.$
(5)

Here $\Theta(E-E_{th})$ is a step function so that only transitions with
$E_{th}<E$ are considered, and the Clebsch-Gordan coefficient enforces the
selection rule $\Delta J^{\prime\prime}=\pm 1$ for the specific PD process
under consideration. Since the rotational levels’ relative populations
initially also sum to one, they have limited effect on the relative sizes of
the PD cross sections and the PD curve identified by the above equation.
Hence, the most important factors are the selected value for the $p$
exponential parameter, a factor linked to the dipole selection rules of the
photon-induced electron detachment and to the dominant partialwave ( angular
momentum) of the ejected electron, as discussed in Gianturco  _et al._ (2019,
2018); Mant  _et al._ (2020); Simpson  _et al._ (2020). The actual wavelength
$\nu$ of the laser source with respect to the threshold of the specific
electron-detachment process is also an important feature in the discussion of
the PD processes, as we shall further show below.

Table 1:  Relative values of $\sigma^{\text{PD}}_{N}(\nu)$ for C2N- as a function of the rotational state N, given by the first column, and the different $p$ values reported by the next three columns. The value of $\nu$ is fixed to be that of the electron affinity (EA) at 2.7489 eV. $N$ | p=0.5 | p=1.5 | p=2.5
---|---|---|---
0 | 0.00 | 0.00 | 0.00
1 | 0.23 | 0.03 | 0.003
2 | 0.35 | 0.06 | 0.01
3 | 0.44 | 0.11 | 0.03
4 | 0.51 | 0.16 | 0.05
5 | 0.58 | 0.22 | 0.08
6 | 0.64 | 0.28 | 0.12
7 | 0.69 | 0.35 | 0.18
Table 2:  Same data as in the previous table.Relative values of $\sigma^{\text{PD}}_{N}(\nu)$ as a function of $p$ for $\nu$ = EA \+ 50 cm-1. $N$ | p=0.5 | p=1.5 | p=2.5
---|---|---|---
0 | 1.000 | 1.000 | 0.934
1 | 0.999 | 0.999 | 0.935
2 | 0.999 | 0.999 | 0.936
3 | 0.999 | 0.999 | 0.938
4 | 0.999 | 0.999 | 0.940
5 | 0.998 | 0.999 | 0.942
6 | 0.997 | 0.999 | 0.946
7 | 0.996 | 0.999 | 0.947

Following the prescription indicated in equation 5, we have calculated the
relative values of the photo-detachment cross sections for the C2N- anion. The
data for the C2H- anion we use here for comparison were presented before
Gianturco  _et al._ (2019); Mant  _et al._ (2020) and therefore we shall only
discuss them during the analysis of the compared data below.

The two tables report three different choices for the exponential parameter
already defined in the equation for the photo-detachment cross section. They
were chosen according to what is known about the experimental findings on the
C2N- anion as discussed, for example, by ref. Garand, Yacovitch, and Neumark
(2009). Briefly, the photo-detachment process originates from the linear anion
in its ($X^{3}\Sigma^{-}$) electronic ground state with a molecular orbital
(MO) configuration given as [core] $\pi^{4}$ $\sigma^{2}$ $\pi^{2}$. Removal
of an electron from the highest occupied $\pi$ or $\sigma$ MO produces the
neutral ($X^{2}\Pi$) or the neutral ($a^{4}\Sigma^{-}$) final states since
they are both accessible from the initial anionic states. As already shown in
previous work Garand, Yacovitch, and Neumark (2009); Zhou, Gerand, and Neumark
(2007), photo-detachment from a $\pi$ or $\sigma$ MO at threshold energies can
be mainly described as proceeding via either $s+d$ partial-wave contributions
from the ejected electron or via a $p$-wave scattering contribution.
Accordingly to the previous definition of the PD cross section, we can then
assign to each of the outgoing electron channels different values of the
exponential parameter $p=0.5$ for s-wave scattering, 1.5 for $p$-wave
scattering and 2.5 for $d$-wave scattering. The two tables therefore report
our relative estimates of the cross sections for all three possible values of
the exponential parameter. The data in Table 1 present our results for the
case in which the laser frequency is chosen to be resonant with the value of
the electron affinity (EA) for the present anion, while the data in Table 2
indicate the cross section values when the laser frequency is increased by 50
cm-1 above threshold. The data clearly indicate that the relative values of
the PD cross sections are affected by both the initial value of the anion
rotational state $N$ and the specific dominant partial-wave for the outgoing
electron. Such difference will clearly reflect in the dynamical modelling of
the loss rates in the trap that we will further present below.

The computational results for the case of the C2H- anion in the trap were
already reported earlier by us in previous work Gianturco  _et al._ (2019);
Mant  _et al._ (2020) and will not be repeated here. New calculations have
been carried out for comparison and are given for the C2N- anion by the four
panels shown in Figure 9. The laser action is switched on after 5 s, which is
well beyond the time needed for the collisional equilibration of the
populations of the rotational levels to the temperature of the trap of 15 K.
Earlier experiments Garand, Yacovitch, and Neumark (2009) indicate that two
peaks are associated to the transitions to the neutral ($X^{2}\Pi$) state due
to the splitting into two spin-orbit components. In the case of the C2N that
splitting was found to be 38 cm-1, close to earlier estimates and calculations
of 40 cm-1. We shall not include this value in estimating the energy of the
ejected electron since the EA value is much larger ( i.e. around 2.75 eV) and
the location of the upper ($a^{4}\Sigma^{-}$) is about 9,000 cm-1 above the
lower ($X^{2}\Pi$) neutral state. The photo-detachment process into the latter
neutral product will therefore involve at threshold the $p$ values of either
0.5 or 2.5. The data presented by Figure 9 in three of the panels show changes
in trap operation for the case of $s$-wave scattering, while all three options
for the $p$ parameter are reported in the lower-right panel in that same
figure to compare their effects on the total anion loss rates.

The most dramatic difference with the earlier studies we have done on the PD
population evolutions in small molecular anions in cold traps Gianturco  _et
al._ (2019, 2018); Mant  _et al._ (2020); Simpson  _et al._ (2020) is the far
higher number of populated rotational states of C2N- even at the low
temperature of 15 K in the trap. In the data shown in all panels the laser
frequency has been kept resonant with the molecular EA of 2.7489 eV and the
density of the buffer gas kept constant at 1010 cm-3. The upper left panel
employs the $p$ exponent at 0.5, a value corresponding to the expected
threshold behaviour of the ejected electron from the ground
($X^{3}\Sigma^{-}$) state of the anion undergoing photo-detachment of its
outer $\pi$ MO into a free electron at threshold energy and the ($X^{2}\Pi$)
state of the neutral. We see that all rotational states up to $N=8$ are
populated after collisional equilibration of the molecular rotational state
population within the trap. On the reported time scale, the PD laser is
switched on at t=0 and the first five second evolution of fractional
population losses are shown. We clearly see that essentially a uniform
exponential decay if followed separately by each of the populated states in
the trap.

As a different $p$ exponent value is chosen, however, we see that the photo-
detachment loss rates on the upper right panel correspond now to a threshold
electron being ejected from an outer $\sigma$ orbital of the initial anion
into a free electron plus the next ($a^{4}\Sigma^{-}$) state of the final
neutral molecule. Here the loss rates evolve much more slowly and on the same
time scale we see that their relative populations remain much higher in the
trap. Such differences of behaviour should be amenable to detection in
possible experiments. When the $p$ exponent is further changed to the value
corresponding to the higher energy component in the partial-wave expansion of
the ejected electron from an outer $\pi$ MO into the ($X^{2}\Pi$) ground
electronic state of the neutral molecule, we see from the changes of
fractional populations of the rotational states populated initially at 15K
(lower-left panel in the figure) that the decay rates become even smaller and
some of the higher rotational states are hardly depleted during the shown time
interval that was essentially emptying them in the data of the upper-left
panel. This clearly indicates that the dominant partial-wave component for the
ejected electron markedly affects the photo-detachment efficiency described by
the value of the PD rates included in the previous set of coupled evolutionary
equations.

A summary of such effects is presented in the lower-right panel of Figure 9,
where the total values of the fractional population losses are given for the
three different choices of the $p$ exponent in the expression for the PD cross
section shown earlier. We clearly see marked differences in the rate losses
within the allotted time interval, indicating again that such differences in
trap depletion behaviour could be experimentally detectable.

Figure 9:  Computed population evolution results for the PD process for the
case of the C2N- anion. Different operating conditions are shown in the
panels, while the lower right panel reports all the changes in the total loss
rates as the exponent $p$ is modified in estimating the PD cross sections used
to generate the rates. See main text for further details. Figure 10:  A
comparison of computed results for the PD-induced rotational population decays
for three different polar anions reported in the captions, under different
trap conditions. The trap temperature is set to 15 K, while the two top panels
select the laser frequency to be resonant with each EA value of the anion. In
the lower two panels the laser frequency is more energetic by 50 cm-1. See
main text for further details.

Another interesting comparison is reported in the four panel of Figure 10
where we compare the laser-induced population losses in the traps for two
different values of the buffer gas density: on the two left panels that value
is kept at 1010 cm-3, while the data in the two left panels is taken to be
1011 cm-3. The laser frequencies are selected to be resonant with each of the
chosen molecules in the data reported by the two panels on the left of the
figure, while the data in the two panels on the right show an increase of the
laser photo-detachment energy by 50 cm-1. Besides showing the variations of
the computed global losses when varying the operational conditions of the trap
for the two title anions, we have also added as a comparison the results we
have obtained for another system studied in photo-detachment experiments in
traps: the CN- anion which we have discussed in some of our recently submitted
work Simpson  _et al._ (2020); González-Sánchez  _et al._ (2020). The latter
anion is a closed-shell, (${}^{1}\Sigma$) molecule with a permanent dipole
moment of 0.709 Debye. Its EA value has been recently measured very accurately
to be 3.6843(2) eV Simpson  _et al._ (2020) and at threshold the process is
considered to chiefly give rise to an s-wave scattering electron modified by
the dipole of the anion, so that a value of the $p$ exponent of around 0.44
can be realistically chosen, as discussed in an upcoming publicationSimpson
_et al._ (2020).

The upper-left panel of Figure 10 presents two choices for the present C2N-
anion of the $p$ exponent describing its PD cross section energy dependence
and associated to dominant $s$ -wave and dominant $p$-wave behaviour of the
ejected electron at threshold, as already discussed earlier. For the C2H-
anion the threshold behaviour of the ejected electron is known to be chiefly
controlled by $p$-wave scattering Zhou, Gerand, and Neumark (2007) and
therefore the total loss rates reported in that panel employ the $p=1.5$
exponential value. It is also interesting to note that the permanent dipole
moments, involved in the evaluation of the transition moments defining the PD
cross sections, show the largest value (3.09 Debye) for the C2H- anion,
followed by the C2N- ( 2.18 Debye) and then by the CN- ( 0.709 Debye).
However, these values are not dominating the PD process in the trap since
collisional efficiency and density of rotational states in the three systems
are also important contributors. Thus, we see that the slowest anion losses in
time occur for the two largest exponents describing the dominant partial waves
of the ejected electrons. On the other hand, the two cases where at threshold
$s$-wave scattering dominates exhibit here faster anion losses in the trap.

This relative behaviour is also maintained by the data seen in the upper-right
panel in Figure 10: the trap setting which has been changed is the density of
the buffer gas atoms, thus making the PD process to be driven by the increased
collision numbers and the increased role of the collisional repopulations of
the rotational states of the trapped molecule, referred to earlier as a
‘collision-dominated’ photo-detachment process. We can therefore say that the
more important differences between similar molecular anions are linked to
their structural properties which decide the dominant partial-wave during the
threshold ejection of the extra electron.

As the photon energy is increased well above the EA thresholds in each system,
as shown by the lower two panels of Figure 10, we see how remarkably the anion
losses are driven by the energetics of the laser and not by the structural
features of the individual target: the present examples now exhibit markedly
higher rates of decaying fractional populations of their anions. Additionally,
the rates are the same for all of them without having their structural
differences playing any significant role during the trap processes. We also
clearly see that increasing the presence of collisional effects (in the lower-
right panel) does not have any significant role on modifying the anion loss
rates. In all the present systems we also note that a single exponential decay
can describe their rates, indicating that no structural features involving
relative re-population between different rotational states are important
enough to modify the rates during decay processes. This fact was also
indicated by the individual decay rates shown by different rotational states
of the C2N- anion shown in the panels of Figure 9. At the considered trap
conditions of temperature much fewer rotational states ( i.e. up to $j=3$) are
significantly populated for the C2H- anion and for the CN- anion ( also up to
$j=3$).

Finally, by changing the value of $\alpha$ in equation 4 the time for anion
loss for PD is modified. This accounts for details of a PD experiment as shown
in our previous study of PD processes involving the ortho-NH2- and the para-
NH2- anions Gianturco  _et al._ (2018) where the value of $\alpha$ was scaled
to experimental ion losses. An order of magnitude decrease in the value of
$\alpha$ leads to an order of magnitude increase in trap time. Thus, our
calculations also find that for $\alpha$ values scaled down to 0.1 or 0.01 the
survival of anions in the traps can be extended up hundreds of seconds for the
present systems.

##  V  Conclusions

In this work we have analysed in detail the structural features of the
interaction potential energy between He atoms and two small linear anions
which have been often considered to be present (albeit not yet detected) in
ISM when modelling the chemistry of such environment, C2H- ($X^{1}\Sigma^{-}$)
and C2N- ($X^{3}\Sigma^{-}$). We have employed the ab initio computed PESs to
further evaluate the network of state-changing cross sections involving their
lower rotational states in each of these systems which would be significantly
populated at the expected temperatures of interstellar conditions. The
computed cross sections were then employed to obtain the corresponding state-
changing rates over a range of temperatures also significant for interstellar
environments. The calculations found different relative sizes between rates in
both systems and the preference of transitions involving the $\Delta N$=2
selection rule for C2H- anion while transitions with $\Delta N$=1 are
dominating for the C2N- anion. These differences of behaviour can be linked to
the structural differences between coupling dynamical terms in the interaction
potentials which guide rotational state-changing collisions for either
molecular anion. Differences in propensity rules involving transitions with
larger $\Delta N$ values also indicate that the C2N- anion would favour
transitions with odd values of $\Delta N$ while the even values are those
favoured for C2H-. On the whole, however, our calculations find that both
anions behave similarly in terms of efficiency of collisional state-changing
probabilities while specific differences between processes are clearly present
and can be amenable to observations.

We have also modelled the behaviour of both molecular anions when uploaded to
cold temperature ion traps (e.g. for temperatures around 15 K as an example)
and using He as a buffer gas in the traps. In particular we have modelled the
photo-detachment processes of both anions after collisional equilibration of
their populated rotational states at the trap temperature. By modifying trap
operating conditions such as the density of the buffer gas, energy of the
photo-detaching laser and its power via scaling the corresponding PD rates
driven by the laser, we have found differences of behaviour between the two
systems and indicated their crucial role in controlling the global loss rates
of trapped anions of the dominant partial-wave of the ejected threshold
electron. The latter parameter depends on the selection rules of the PD
process and on the initial and final electronic states of the anionic and of
the final neutral molecule. The increase of the photon energy of the laser is
also significantly affecting the loss rates and is seen in our modelling that
can shift operating conditions from being either chiefly collision-driven
events or laser-driven events, as we have already discussed in our earlier
work Gianturco  _et al._ (2019); Mant  _et al._ (2020).

The availability of accurately determined state-changing rotational rates at
the temperature of the ISM for molecular ions employed in these networks are
expected to document more realistically the values for such quantities within
kinetic models of chemical processes and energy exchange processes in that
environment. We also suggest that our specific results for estimating the
rates of ion loss changes in cold ion traps under different trap conditions
could help in the preparation of future experiments on these anions.

##  VI  Supplementary Material

There are 4 files in the Supplementary material: The Vλ multipolar
coefficients for C2N-/He PES, the computed rates for C2H-/He, the computed
rates for C2N-/He and a $readme$ file explaining the format and units of the
rates.

###### Acknowledgements.

We are very grateful to professor M.L. Senent for having generously provided
us with her computed potential energy values for the C2H- system interacting
with He atoms. F.A.G. and R.W. acknowledge the financial support by the
Austrian Science Fund (FWF), Project P29558-N36 . One of us (J.F.) thanks
Europlanet for a travel grant, the European COST action Our Astro-Chemical
History for financing a short-term scientific mission (STSM) at the University
of Innsbruck, and the computer center WCSS in Wroclaw for computational
resources made available under grant number KDM-408.

##  VII  Data Availability Statement

The data that supports the findings of this study are available within the
article and its supplementary material.

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