# Quantum nature of molecular vibrational quenching: Water - molecular
hydrogen collisions

Laurent Wiesenfeld  Laboratoire Aimé-Cotton, CNRS & Université Paris-Saclay,
Orsay, France<EMAIL_ADDRESS>
###### Abstract

Rates of conversions of molecular internal energy to and from kinetic energy
by means of molecular collision allows to compute collisional line shapes and
transport properties of gases. Knowledge of ro-vibrational quenching rates is
necessary to connect spectral observations to physical properties of warm
astrophysical gasses, including exo-atmospheres. For a system of paramount
importance in this context, the vibrational bending mode quenching of H2O by
H2, we show here that exchange of vibrational to rotational and kinetic energy
remains a quantum process, despite the large numbers of quantum levels
involved and the large vibrational energy transfer. The excitation of the
quantized rotor of the projectile is by far the most effective ro-vibrational
quenching path of water. To do so, we use a fully quantum first principle
computation, potential and dynamics, converging it at all stages, in a full
coupled channel formalisms. We present here rates for the quenching of the
first bending mode of ortho-H2O by ortho H2, up to 500 K, in a fully converged
coupled channels formalism.

Collisional exchange of energy (internal and kinetic) is a fundamental process
leading to the equilibrium between internal temperatures of molecules and
kinetic temperature of the gas in which they are immersed Hartmann, Boulet,
and Robert (2008). There exists many attempts to compute those rates for
transport properties, all the way from classical computation (recent examples
are Hellmann (2020); Brown, Bastien, and Price (2011)) to full quantum ab
initio computations Dagdigian and Alexander (2013). While a large body of
investigation deals with collisional rates for astrophysics and planetary
science, as described in the next paragraph, the role of internal to kinetic
energy exchanges is also crucial in cold molecules physics
($T\sim\mathrm{K\,\text{to}\,mK}$). In an cold molecular gas, energy exchanges
between the molecules observed and with the buffer gas, possibly causing
heating of the cold molecules. Carr  _et al._ (2009); Hu  _et al._ (2021)

The recent surge in interest towards elastic and inelastic collisions stems
mainly from precise atmospheric/astrochemical needs. Roueff and Lique (2013);
Karman  _et al._ (2018); van der Tak  _et al._ (2020) Most data pertaining to
the physical and chemical-physical states of gaseous matter in the Universe
come from atomic or molecular spectroscopy. For decades, most of the
quantitative information was gained from the rotational lines of ground state
molecules, thanks to the high precision, high specificity of rotational
spectra in the cm to sub-mm spectral regions Cernicharo  _et al._ (2019);
Manigand  _et al._ (2020).

Because of newer instruments on the ground and thanks to plane- or space-born
telescopes, the opacity of the atmosphere to FIR-IR light is progressively
overcome, and opens up new opportunities: Microwave (THz frequencies) to Infra
Red spectra carry information otherwise unattainable, on the warm interstellar
matter and on planetary/cometary atmospheres in the Solar System or in
exoplanets. Many molecules are thus nowadays observed in their vibrationally
excited states, whether by rotational spectroscopy within those states, or
else by observing the FIR-IR lines connecting different ro-vibrational levels
Greenwood, A. J.  _et al._ (2019). Among the molecules observed, water is
prominent van Dishoeck  _et al._ (2021), being the third most abundant (after
H2 and CO) and the first polyatomic one. The relevance of the ro-vibrational
quenching of water is further enhanced by the relatively recent discovery of
masing transition of ortho-water, in the transition $v_{b}=1;1_{10}\rightarrow
1_{01}$, at $\nu=658.00$ GHz Menten and Young (1995); Nesterenok (2015);
Baudry  _et al._ (2018a) [Rotational levels of water are labelled
$j_{k_{a}k_{c}}$; ortho ${}^{1}\mathrm{H_{2}O}$ (nuclear spin triplet) has
$k_{a}+k_{c}$ odd; $v_{b}$ is the water bending mode quantum number] .

The present approach constitutes, for the author’s best knowledge, a first ab
initio study of quantum ro-vibrational quenching for a polyatomic molecule,
that includes rotational states of the projectile. Integrating these new
rotational channels increases the magnitude of the vibrational quenching rates
(and cross sections) by several orders of magnitude, when compared with
earlier work, either classical Faure  _et al._ (2007) or quantum Stoecklin
_et al._ (2019, 2021) . It has always been expected that quantum effects would
dominate the landscape for low collisional energies, comparable to the minimum
of the van der Waals potential $|V_{\text{min}}|$ (here,
$V_{\text{min}}\sim-250\,\mathrm{cm^{-1}}$)Drouin and Wiesenfeld (2012); Faure
_et al._ (2013); Bergeat  _et al._ (2020), but quantum aspects are supposed to
disappear progressively as the collision energy increases In a very different
context, combustion modelling Brown, Bastien, and Price (2011), the same
assumption is made that there exists a critical temperature
$T^{*}=\mathrm{k}T/|V_{\text{min}}|$, above which classical picture is
sufficient to describe collisional transport.

We show here that the very different energy scales of water rotations,
molecular hydrogen rotation, and water vibration lead to the resurgence of a
strong dependance of the cross sections to the quantum levels of H2 and H2O
involved. This resurgence does not happen for purely rotational energy
redistribution: even classical and/or statistical approaches Loreau, Lique,
and Faure (2018) are relevant when a large number of coupled levels interact.
It is most significant here, and, to a lesser extent, to collisions involving
water and a heavier partner, like N2, which is described by a toy model in
this work.

While classical and semi-classical methods are certainly of relevance in
rotational quenching, it must be recalled that the semi-classical quantization
of an asymmetric rotor is problematic Faure and Wiesenfeld (2004), at least
for $j$ small enough so that the two associated levels $j_{k_{a},k_{c}}$ and
$j_{k_{a}-1,k_{c}+1}$ are clearly distinct.

Computing collisional coefficients for purely rotational transitions has been
a nearly continuous endeavour since the pioneering work of Delgarno Arthurs
and Dalgarno (1960); Green and Thaddeus (1974); Green (1977); Phillips,
Maluendes, and Green (1995); Flower, Bourhis, and Launay (2000); Dubernet  _et
al._ (2013); Hutson and Green (1994); M. H. Alexander and D. E. Manolopoulos
and H.-J. Werner and B. Follmeg and P. J. Dagdigian . The scenario is always
the same, and it is the one we pursue here: (i) Computing ab initio the
interaction potential $V_{n}(\bf R)$ of the polar molecule and the projectile
($\bf R$ denotes collectively all necessary coordinates, $n$ are the points
where computing is performed) , (ii) fitting the $n$ computed points onto a
functional form acceptable for dynamics, (iii) running quantum dynamics of the
collision pair, and (iv) computing cross-section as a function of collisional
energy $\sigma(E)$ or rates as a function of the kinetic temperature of the
collider gas $k(T)$.

Thanks to the experience gained in recent precise rotational inelastic
scattering computations and of very convincing comparisons of theory with
several types of experiments Ziemkiewicz  _et al._ (2012); Drouin and
Wiesenfeld (2012); Bergeat  _et al._ (2020), it is safe to deal with ro-
vibrational collisional excitation with the same types of methods and basis
sets that were used previously, and conduct a fully quantum dynamical
approach.

We employ the full dimensional (9 degrees of freedom) water-molecular hydrogen
potential energy surface Valiron  _et al._ (2008) (hereafter denoted val08),
$V(R,\Omega,\delta r_{q})$. This surface includes the 5 intermolecular degrees
of freedom (intermolecular distance $R$ and 4 angles $\Omega$ to orient one
molecule with respect to the other), as well as 4 normal coordinates
describing motion around equilibrium position of the molecules, $\delta
r_{q}$, $q=1,\ldots,4$. Details of this surface are in the original paper.
Since quantum dynamics is performed with a quantum time-independent
computation of the S-matrix, we need to fit the ab initio points on the
relevant S. Green type of mixed coordinates Phillips, Maluendes, and Green
(1995). Molscat code was used Hutson and Green (1994), duly modified to
include vibrational modes in the potential expression. We did not include the
modification of the kinetic energy due to rovibrational motion Faure  _et al._
(2005), but restricted ourselves to potential effects, and expressed the
potential in coordinates satisfying Eckart conditions. Since we deal only with
the first bending mode $v_{b}=1$ and low lying rotational levels in the
excited vibrational mode, we made the assumption:
$\left|v_{b};j_{k_{a},k_{c}}\right>=\left|v_{b}\right>\otimes\left|j_{k_{a},k_{c}}\right>_{v_{b}}$.
The rotational functions are parameterized by the rotational parameters of
either $v_{b}=0$ Kyrö (1981) or $v_{b}=1$, but they are not rovibrational
functions. We also averaged the dependance of the potential on the vibrational
ground state of H2.

Time-independent quantum dynamics is performed here on two coupled sets of
potential energy surfaces with full dimensional (5D) potentials, including H2
rotation (denoted by the quantum number $j_{2}$), in a converged coupled
channels approach. The potentials were obtained by computing the averages of
the 9-D potential :
$C_{v^{{}^{\prime}}_{b},v_{b}}(R,\omega)=\left<v^{\prime}_{b}\left|V(R,\Omega,\delta
r_{q})\right|v_{b}\right>$ functions, with $v_{b},v^{\prime}_{b}=0,1$ [see
val08]. The coupling $\mathsf{W}(R)$ matrix is thus written blockwise as (with
$R$ the intermolecular distance) Manolopoulos (1986):

$\Psi^{\prime\prime}(R)=\left[\begin{pmatrix}\mathsf{W}_{11}(R)&\mathsf{V}_{01}(R)\\\
\mathsf{V}_{01}(R)&\mathsf{W}_{00}(R)\end{pmatrix}-\begin{pmatrix}k^{\\!2}_{\;2}\\\
k^{\\!2}_{\;1}\end{pmatrix}\right]\Psi(r)$ (1)

where the $\mathsf{W}_{00}(R)$ and $\mathsf{W}_{11}(r)$ matrices are formed by
using the potential of each vibrational level (including the diagonal
contributions), and the non-diagonal rotational terms, bracketed with he
relevant spherical harmonic functions Phillips, Maluendes, and Green
(1995),[val08]. The $\mathsf{V}_{01}(R)$ matrix is also given by duly
bracketing the $C_{v^{\prime}_{b},v_{b}}(R,\omega)$ function with the same
spherical harmonic functions.

In order to be able to deal with representative (symmetric) matrices of sizes
less than $12,000\times 12,000$ (the practical limit of OpenMP computing), we
further split the computation by treating each total angular momentum value
$J$ (recoupling $j$, $j_{2}$, $\ell$, the orbital angular momentum) and, if
necessary, each inversion symmetry, separately. We used a relatively coarse
grid of total energies ($E_{\text{tot}}$ from threshold to about 3,000
$\mathrm{cm^{-1}}$), as we are not interested in the detailed resonance
behaviour occurring typically at
$1\,\mathrm{cm^{-1}}<E_{\text{collision}}<\left|V_{\text{min}}\right|$.

We present results for ortho 1H2O - ortho 1H2 collisions. The present analysis
could serve as a basis for the modelling the masing transition of water at 658
GHz Baudry  _et al._ (2018b); Menten and Young (1995); also, ortho H2 is more
abundant than para H2 at the higher temperatures that we examine. With a large
rotational basis set we computed all inelastic cross sections with initial
levels $v_{b}=1$; $j_{k_{a}k_{c}}=1_{01}$ up to $3_{21}$. Rotational basis for
water is $j(v=0)\leq 14$; $j(v=1)\leq 6$; only rotational levels with $E\leq
3500\,\mathrm{cm}^{-1}$ are included. Convergence was reached at higher
energies for total angular momentum $J=28$ to 32.

Next (Fig. 1), we compute quenching cross sections as a function of scattering
energy and quenching rates as a function of temperature (inset of Fig. 1 ). In
both figures, we show the total quenching section (or rate) from a series of
excited ro-vibrational levels (as described in the caption of Fig. 1) to all
$v_{b}=0$ levels and constraining to de-excitation (Excluding the $v_{b}=0$
levels that are above the $v^{\prime}_{b}=1$ original levels):

$\sigma_{v^{\prime}=0\leftarrow
v=1,j_{k_{a}k_{c}}}(E)=\sum_{j^{\prime},k^{\prime}_{a},k^{\prime}_{c}}\sigma_{v^{\prime}=0;j^{\prime}_{k^{\prime}_{a}k^{\prime}_{c}}\leftarrow
v=1,j_{k_{a}k_{c}}}(E)\;,$ (2)

and rates obtained therefrom by averaging over the collisional energy. Fig. 1
shows results for three rotational bases of the H2: (i) para-H2, $j_{2}=0$, to
be compared with the results of Stoecklin  _et al._ (2019) (ii) $j_{2}=1$, the
usual basis for many computations for collisions between a molecule and
ortho-H2, (iii) a larger basis for ortho-H2, $j_{2}=1,3$. Full results with
all detailed cross sections and rates will be published elsewhere.

Figure 1: Left panel: Quenching cross section (log scale) as a function of
collision energy. Individual transitions denoted by colors and line types.
Right panel: corresponding rates $k(T)$. Colors of the lines designate the
original $v=1$ rotational level. Line types designate the rotational basis of
H2: Lower group: $\sigma^{0}(E)$; middle group, $\sigma^{1}(E)$, upper group
$\sigma^{13}(E)$, see text.

Results are striking, as the main channel of quenching appears to go via a
_simultaneous ro-vibration quenching of_ H2O _and a rotational excitation_ of
H2, from $j_{2}=1$ upwards to $j_{2}=3$, an excitation of about 600 cm-1 (to
be compared with a vibrational threshold at 1585 $\mathrm{cm^{-1}}$), Fig. 1.
Note that the $j_{2}=5$ level has an excitation threshold at 1620 cm-1, above
the $j_{2}=1$ levels. We made some preliminary computations of the influence
of the full $j_{2}=1,3,5$ basis. The increase at a collisional energy of about
1500 cm -1 is of the order of 15%. Full investigation will be presented in
future papers. The relevance of the $\Delta j_{2}=+4$ is marginal at the
energies considered here.

Cross sections are very significantly larger for collisions with ortho-H2 than
with para-H2, $j_{2}=0$, an expected result , due to the non isotropy of H2
(static quadrupole and anisotropic polarizabilty of H2, allowing for
anisotropic long-distance interaction). The effect is particularly large, but
remains compatible with other collisions with a polar molecule Yang  _et al._
(2006); Daniel  _et al._ (2010); Bouhafs  _et al._ (2019). Here, however,
vibrational quenching cross sections gain another order of magnitude _with
including both $j_{2}=1$ and $j_{2}=3$ ortho rotational levels_ of H2,
remaining with initial conditions at $j_{2}=1$. An earlier classical work
Faure  _et al._ (2005) did not see such a strong effect at all (see their
figure 4, with a monotonous decrease of rotational energy of H2 after
collision). The recent quantum paper Stoecklin  _et al._ (2019) did not
consider these possibilities. Some hints of the importance of the rotational
transitions of the projectile are seen in Yang  _et al._ (2020).

Before trying to interpret the surprising results of Fig. 1, let us compare
those results with previous work. We compared present rate results with
earlier ones by Daniel et al., Daniel, Dubernet, and Grosjean (2011); Daniel
_et al._ (2011), for pure rational quenching inside the $v_{b}=0$ levels, for
temperatures up to 500 K, see Fig. 2. Agreement is not perfect, as expected,
since Daniel et al. had a different definitions of rates and convergence
(including the $j_{2}>0$ channels in their initial conditions), unrealistic
for the computations here, but the important features are present, validating
the code and the method.

Figure 2:  Rates of de-excitation of ortho-H2O levels in $v_{b}=0$, by
ortho-H2. Diamonds, Daniel et al.Daniel, Dubernet, and Grosjean (2011); line,
this work.

The $j_{2}=0$ cross sections found here (lower manifold of Fig. 1) is readily
found to be comparable in magnitude to those found by Stoecklin et al.,
Stoecklin  _et al._ (2019). Secondly, in Faure and Josselin (2008), a large
set of transitions and temperatures was proposed, based on classical and
statistical assumptions. It is difficult to compare their results with the
ones presented here, as their methodology is more suited for higher $T$,
higher initial $j$ and makes no assumption on ortho or para state of H2. To
work qualitatively, at 300 K, we find a total rate
$k(300\,\mathrm{K})=3.5\,10^{-12}\,\mathrm{cm^{3}\,sec}$ (summed over all
initial levels considered, $1\leq j\leq 3$) and their rates amounts to
$9.7\,10^{-12}$ approximately, as taken from the LAMDA database,
https://home.strw.leidenuniv.nl/ moldata/ (also summed on all initial levels,
$j\leq 3$). Experiment points to $1.3\,10^{-12}$, with no very clear
definition of the initial state Zittel and Masturzo (1991). Clearly, the
experiment-theory comparison must be made more precise, probably by examining
pressure broadening of IR/Raman spectroscopy.

The magnitude of the rates allow us to infer the critical densities. Allowed
water IR transitions are the following: $\Delta v_{b}=\pm 1$; $\Delta j=0,\pm
1$, and $\Delta k_{a}+\Delta k_{c}=0,\pm 2$. IR transitions of the low
$j,\,v_{b}=1$ occur at larger IR frequencies, hence, a relatively large
_average_ spontaneous emission rate $A=24.6\,\mathrm{sec^{-1}}$ Barber  _et
al._ (2006). Critical densities of H2 are thus of the order of
$n^{*}=7\,10^{12}\,\mathrm{cm^{-3}}$, well within the range of atmospheric
densities and outflows of aging stars. Please note that critical densities (as
well as rates) are not unambiguously defined here, as many temperatures may
coexist. We made use of one kinetic temperature for H2, and summed over all
quenching rates.

To disentangle various possible effects leading to the results of just
outlined, we devised four different scenarios for total quenching cross
sections $\sigma\equiv\sigma(v^{\prime}_{b}=0\leftarrow v_{b}=1)$, summed for
all final rotational levels. They differ by the H2 rotational bases and the
way to treat the rotational excitation of H2: (i) $\sigma^{0}(j_{2}=0)$, (ii)
$\sigma^{1}(j_{2}=1)$, (iii) $\sigma^{13*}(j_{2}=1,3)$, but forbidding the
$\Delta j_{2}=+2$ transitions, and (iv) $\sigma^{13}(j_{2}=1,3)$, allowing for
$\Delta j_{2}=+2$ transitions. They all are shown in Fig.1. Wee see that the
$\sigma^{1}$ and $\sigma^{13*}$ cases are very similar, with an order of
magnitude difference with $\sigma^{0}$ and $\sigma^{13}$ To understand the
dynamics at hand, we took a twofold approach.

Firstly, we compared, for similar quenching energies, the $\Delta j_{2}=0$
($\sigma^{13*}$)to the $\Delta j_{2}=+2$ ($\sigma^{13}$), for ro-vibrational
quenching or pure rotational quenching (Fig. 3). We present results summed
over all partial waves (total angular momentum), either as a function of the
total collision energies (panels (a) and (c)) or as a function of the final
rotational state $j$ of (panels(b) and (d)).Initial conditions are chosen as
follows: For panels (a) and (b), initial states are $v=1,j_{1}=1,2,j_{2}=0$,
that is initial total energy $1737.2<E<1861.5$ cm-1. For panels (c) and (d),
we take initial levels at $v=0$, total $1737.2<E<1900$ cm-1, and $9\leq j\leq
11$.In this way, energy gap effects are kept more or less the same, whether
with or without vibrational quenching.

Examining the results, for panels (a) and (b) - vibrational quenching-, we see
that initial collisional energy has no peculiar feature and that final water
angular momentum is weakly peaked at $j\sim 6,7$, for both final $j_{2}=0,2$.
For panels (c) and (d) -no vibrational quenching-, initial collisional energy
peaks at about 1300 cm-1 and diminishes strongly for smaller values. There is
a propensity towards $\Delta j_{2}=+2$ for small final $j$ (large $\Delta
j<0$) and the opposite for larger final $j$. This goes in favour of a spilt of
outgoing angular momentum between the two outgoing products, not seen in
vibrational quenching.

The $\Delta j_{2}=+2$ channel dominates the picture for vibrational quenching
by at least one order of magnitude when the channels are opened (right of blue
threshold line, panels (a), (c), $j\leq 10$, panels (b),(d)). It is not the
case for pure rotational quenching. It is all the more remarkable that for the
$v=0\rightarrow 0$ quenchings, the loss of angular momentum is much higher,
with $j_{\text{ini}}\geq 9$. H2 carrying away angular momentum is not the main
effect here. Also, energy loss is similar, hence energy gap effects are ruled
out in the comparison. Inside the $v_{b}=0$ vibrational level, we find a weak
dominance of the $\Delta j_{2}=0$ transitions, similar to what Daniel et al.
found (Figure 1 of reference Daniel  _et al._ (2010))

Figure 3:  Comparison between rotational quenching, for transition without
excitation of H2 ($\Delta j_{2}=0$, black bars), and with excitation of H2
($\Delta j_{2}=+2$, red bars). Total angular momentum summed, $0\leq J\leq
32$. Panels (a), (c), cross sections as a function of collision energy. Blue
vertical line indicate the $\Delta j_{2}=+2$ threshold. Panels (b), (d), cross
sections as a function of final H2O principal rotation quantum number $j$.
$j=11,12$ are above threshold for $\Delta j_{2}=+2$ excitation. Panels (a),
(b), rovibrational quenching (summed for all initial $v_{b}=1,\,j\leq 2$ and
all final $v_{b}=0$, $E_{i}>E_{f}$). Panels (c), (d), pure rotational
quenching (summed for all initial $v_{b}=0,\,E\geq 1585\,\mathrm{cm}^{-1}$ and
all final $v_{b}=0$, $E_{i}>E_{f}$). Last bars in panels (b), (d) sum all
contributions.

Secondly, we compute and draw the ro-vibrational channels that are used for
the dynamical propagation, as a function of $R$. We consider the diagonal part
of the $\mathsf{W}(R)$ matrix (equation 1) as a diabatic image and its
eigenvalues, $\mathrm{eig}\left[\mathsf{W}(R)\right]$ as an adiabatic image,
Fig. 4 . Plots are for total $J=2$, $j\leq 3$, a good compromise between
generality (the $J=6$ or 12 are very similar in appearance; adding $j$
channels do not change the picture) and as small a number of channels as
possible, for sake of readability. We see straightforwardly that in the case
of $j_{2}=0$, crossings between incoming black $v_{b}=1,j_{2}=0$ channels and
outgoing red $v_{b}=0,j_{b}=0$ channels are limited to the repulsive wall, at
very high collision energies. Cross sections are exceedingly small, but
increase rapidly once the crossings are open, energy-wise. Exchanges between
the various $v_{b}=0$ channels is possible, seemingly at all energies and
distances. The $j_{2}=1$ case does not fundamentally change the picture. There
are crossings (diabatic picture) or avoided crossings for energies lower than
threshold, at $R\sim 8$ Bohr, sufficiently effective to insure some transfer
of probability amplitude. Note that the higher energy, $R\sim 10$ avoided
crossings should be less efficient, in a Landau-Zener picture, because of the
large slope difference between the adiabats.

Picture changes dramatically for the third case, $j=1,3,\,\Delta j_{2}=0,+2$,
two lower panels. If we detail the diabatic picture (lower left panel), we see
(i) that the $v_{b}=1$, $j_{2}=1$ channels (black) are allowed to cross the
$v_{b}=1$, $j_{2}=3$ channels (green) in the $R\sim 10$ region, allowed
energy-wise. Then, (ii), the probability amplitude, divided into many
channels, crosses a large manifold of $v_{b}=0,j_{2}=3$ (magenta) or
$v_{b}=0,j_{2}=1$ (red) channels, at $R\sim 7-10$, in the diabatic picture,
allowing for probability amplitude to flow out on the whole manifold. This
picture accounts for the increase of inelastic cross sections and rates, but
without any propensity rule apparent. If we turn to the adiabatic picture
(lower right panel), we see that the magenta levels have a series of avoided
crossings with the incoming channels at all distances. The larger number of
magenta adiabats, their avoided crossings at all distances, including the
larger $R$ distances just below threshold may be enough to point to the
propensity rules observed in the quantum dynamics.

Figure 4:  Diabats and adiabats for the $\mathrm{H_{2}O\cdots H_{2}}$
collision. Left panels, diabats (diagonal part of the W matrix, Eq.(1)), as a
function of intermolecular distance. Right panels, adiabats (eigenvalues of
the W matrix, Eq.(1)) as a function of intermolecular distance. Asymptotic
quantum numbers : black lines: $v_{b}=1$, $j_{2}=0$ or $j_{2}=1$; red lines
$v_{b}=0$, $j_{2}=0$ or $j_{2}=1$; light green $v_{b}=1$, $j_{2}=3$; magenta
$v_{b}=1$, $j_{2}=1$. Blue dashed line: $v_{b}=1$, ortho H2O threshold. See
discussion in the text.

To conclude the discussion, we found it worthwhile to have a first glimpse of
what the situation might be, for a heavy-heavy collision of utmost atmospheric
importance, $\mathrm{H_{2}O\cdots N_{2}}$. Even if a very recent rigid bender
PES exists, Wang  _et al._ (2020), fits are not in line of what is needed
here, and we look only fore some very qualitative results. We used the same
PES as before (same symmetry of the system); it is all the more justified that
$V_{\text{min}}$ remains similar (about 250-300 cm-1). This tN2 (toy-N2)
computation of diabats and adiabats is performed exactly similarly, with the
two following changes: $B_{\mathrm{N}_{2}}=2.01\,\mathrm{cm^{-1}}$ and reduced
mass $\mu=10.9565$. While ortho and para ${}^{14}\mathrm{N}_{2}$ have the
opposite meaning from ${}^{1}\mathrm{H}_{2}$ (because of the bosonic character
of 14N, of nuclear spin $I=1$), we did not change the $j_{2}$ values.

Fig. 5 show the results, with the same conventions as in Fig. 4. Without any
supposition for the dynamics, we see that the same effects persist, namely, a
large increase of avoided crossings, from $j_{2}=1$ to $j_{2}=1,3$. In this
tN2 case we have a large decrease (factor of 25) of the projectile rotational
constant (here below the target rotational constants), but the vibrational gap
remains. The large fans of adiabats should allow for higher cross sections,
because of the various energy scales involved.

Figure 5:  Diabats and adiabats for the $\mathrm{H_{2}O\cdots tN_{2}}$
collision. Left panels, diabats (diagonal part of the W matrix, Eq.(1)), as a
function of intermolecular distance. Right panels, adiabats (eigenvalues of
the W matrix, Eq.(1)) as a function of intermolecular distance. Asymptotic
quantum numbers : black lines: $v_{b}=1$, $j_{2}=0$ or $j_{2}=1$; red lines
$v_{b}=0$, $j_{2}=0$ or $j_{2}=1$; light green $v_{b}=1$, $j_{2}=3$; magenta
$v_{b}=1$, $j_{2}=1$. Blue dashed line: $v_{b}=1$, ortho H2O threshold. See
discussion in the text.

One should not take those tN2 computations for more than indications. In
particular, they do not claim for any convergence. However, they should be a
warning before deciding whether a cold molecule collision vibrational
quenching proceeds classically for all the energy sectors considered. In low
$T$ physics, it could be fine structure, vibration, rotation, and each could
operate in a very different way.

Foseca dos Santos et al. Santos  _et al._ (2011) studied some time ago the
$\mathrm{H_{2}\cdots H_{2}}$ system, considering ro-vibrational de-excitation
cross sections, for both ortho-para and ortho-ortho collisions, at mostly low
collisions energy (up to 100 K). Since in the present work, molecules are
distinguishable, the comparison is more relevant in the ortho-H${}_{2}\cdots$
para-H2 case, as the ortho-ortho case presents resonances and symmetry effects
absent here. We found here that the dominant channel for vibrational quenching
is $j_{2}=1\rightarrow 3$, irrespective of the collision energy (Fig. 1),
thereby reducing the amount of energy to be transferred, independently of
angular momentum transfer. The situation is somewhat different for
$\mathrm{H_{2}\cdots H_{2}}$: for the 1001 collision,
(notation:$v_{1}j_{1}v_{2}j_{2}$; 1,2 the two H2 molecules), the dominant
channel by far is $1001\rightarrow 0011$, conserving the rotational angular
momentum. The analogous $1001\rightarrow 0201$ is about a factor 5 smaller. An
adiabatic analysis similar would sort out the differences, that may be due to
the paucity of adiabatic levels (two rods, $B=60\,\mathrm{cm^{-1}}$, far fewer
channels). The ortho-ortho case is quite different from the case analyzed
here, as resonances and symmetry effects may be dominant effects.

We have shown that the vibrational quenching behaves in a very different way
than the rotational quenching, because of the organization of the ro-
vibrational quantum levels. Approximations not allowing rotational (and
possibly vibrational) excitation of the projectile may be underestimating the
actuel rates by a large amount, here by a factor of 10 at least. Ignoring the
nuclear symmetry of the projectile would also lead to large over-estimation of
rates. The results presented here suggest that ro-vibrational quenching by
light atoms or by molecules at their zero angular momentum level is
fundamentally different from non-zero angular momentum, because _both_ long
distance and short distance behaviours of the diabatic or adiabatic channels.

The atmospheric vibrational quenching, driving the IR radiation redistribution
towards kinetic energy should thus be carefully conducted in a manner allowing
for all the essential angular momenta to be properly taken into account.
Probing line shapes via IR/Raman spectroscopy is all the more relevant and
will be the object of further theoretical investigations.The critical density
found for ro-vibrational quenching points to high atmospheres of stars or
planets, to dense parts of late stage stars, confirming that the 658 GHz maser
line finds its origin in the unbalance of collisions and optical rates for the
low $j$, $v_{b}=1$ levels.

As the similarity of rates for the various initial states in our limited range
of initial water angular momentum suggests, it is worthwhile to develop
specific ways of treating theoretically ro-vibrational quenching, as was done
long ago for rotational quenching, to ease the burden of computation time. In
this way, the all-important collision systems for our atmosphere
($\mathrm{H_{2}O/CO_{2}/CH_{4}\cdots N_{2}/O_{2}}$)will become doable by
relevant quantum methods.

###### Acknowledgements.

The author thanks J. Hutson and M. Gonzalez-Martinez (Durham U.) for setting
up the relevant basis sets for vibrational quenching with Molscat conventions,
and A. Faure (Grenoble U.) for providing details of the potential surface.
Enlightening discussion and support from O. Dulieu and N. Bouloufa are
grateful acknowledged. Computations presented in this paper were performed
with grants A0060810769/A0080810769 on the ADA and JEAN-ZAY CNRS super-
computers, hosted at IDRIS.

## Data Availability

The potential energy surface has been published earlier Valiron  _et al._
(2008). The Fortran subroutines for computing the full surface are available
from the author upon reasonable request. The quenching data presented here are
computed from a large body of raw data, which were generated at the CNRS-IDRIS
large facility. Derived data supporting the findings of this study are
available from the author upon reasonable request.

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