# On the robustness of the ammonia thermometer

###### Abstract

Ammonia inversion lines are often used as probes of the physical conditions in the dense ISM. The excitation temperature between the first two para metastable (rotational) levels is an excellent probe of the gas kinetic temperature. However, the calibration of this ammonia thermometer depends on the accuracy of the collisional rates with H. Here we present new collisional rates for ortho-NH and para-NH colliding with para-H () and we investigate the effects of these new rates on the excitation of ammonia. Scattering calculations employ a new, high accuracy, potential energy surface computed at the coupled-cluster CCSD(T) level with a basis set extrapolation procedure. Rates are obtained for all transitions involving ammonia levels with and for kinetic temperatures in the range 5100 K. We find that the calibration curve of the ammonia thermometer – which relates the observed excitation temperature between the first two para metastable levels to the gas kinetic temperature – does not change significantly when these new rates are used. Thus, the calibration of ammonia thermometer appears to be robust. Effects of the new rates on the excitation temperature of inversion and rotation-inversion transitions are also found to be small.

###### keywords:

molecular data — molecular processes — ISM: molecules^{†}

^{†}pagerange: On the robustness of the ammonia thermometer–References

^{†}

^{†}pubyear: 2009

## 1 Introduction

Since its discovery in the interstellar medium forty years ago by
Cheung et al. (1969), ammonia has been widely used as a probe of the
physical conditions in a variety of interstellar environments, ranging
from pre-stellar cores, molecular clouds, to external galaxies
(see Ho &
Townes, 1983, for a review). The peculiar structure of the
molecule makes ammonia lines excellent tracers of the density and
temperature in these environments. NH is a symmetric top
molecule, whose rotational levels can be denoted by two quantum
numbers, the total angular momentum , and its projection along
the molecular axis. Owing to the possible relative
orientations of the hydrogen spins, two distinct species exists:
ortho-NH (, with an integer; hereafter o-NH) and
para-NH (; hereafter p-NH). As both radiative and
non-reactive collisional transitions do not change the spin
orientations, transitions between o-NH and p-NH are forbidden.
Each of the rotational energy levels (with the exception of those with
) is further split into two sub-levels which can be denoted
either by the inversion symmetry of the vibrational wave functions or
by the symmetry index (see Fig. 1 and Eq. 29
in Rist
et al., 1993)^{1}^{1}1In this paper, we denote each level by its
symmetry index , with a sign for and a
sign for . We refer the reader to the Fig. 1 of
Rist
et al. (1993) for an energy diagram of the molecule.. This
splitting is caused by the inversion motion of the molecule, and the
corresponding inversion transitions fall in the range cm. Electric dipole transitions rules () prevent radiative transitions between different ladders to
occur. Consequently, the lowest inversion doublets in each ladder
(i.e. with ) are metastable; they can be relaxed only through
collisions.

For this reason, the relative population of the first two metastable inversion doublets of p-NH, and , depends solely on the kinetic temperature. Considering only the first three doublets, , and , and assuming that the population of the doublet is much smaller than that of the , Walmsley & Ungerechts (1983) showed that the excitation temperature between the two lowest doublets is given by the analytic formula:

(1) |

where is the energy difference between the first two metastable doublets ( K), is the kinetic temperature, is the collisional excitation rate (averaged over the symmetry index ) between the and rotational levels, and is the collisional de-excitation rate between the and levels. Thus, if one knows the excitation temperature, one can derive the kinetic temperature of the gas, effectively using ammonia inversion lines as a “thermometer”.

Observationally, can be determined by observing the hyperfine components of the and inversion transitions. The inversion doublets have indeed hyperfine components, which are due to the interaction between the electric quadrupole moment of the N nucleus and the electric field gradient created by the electrons. If one assumes that the excitation temperature of each hyperfine components (within a given rotational state) is the same, then one can derive the opacity and of the and multiplets from the relative intensity of each hyperfine component (Barrett et al., 1977). is then calculated from the following formulae (Ho et al., 1979; Hotzel et al., 2002):

(2) |

In order to “calibrate” the ammonia thermometer, i.e. to compute the kinetic temperature from the measured excitation temperature, a good knowledge of the collisional rates of NH colliding with H is necessary (see Eq. 1). Although a large number of measurements have been made on the NHH system, using in particular double resonance (e.g. Daly & Oka, 1970), crossed beam (e.g. Schleipen et al., 1993) and pressure broadening (e.g. Willey et al., 2002) experiments, laboratory data generally do not directly provide state-to-state rate coefficients. As a result, radiative transfer models can exclusively rely on theoretical estimates. On the other hand, laboratory measurements are crucial to establish the predictive abilities of theory and, in particular, of the potential energy surfaces (PES).

Following the pioneering work of Morris et al. (1973), Walmsley &
Ungerechts (1983)
performed statistical equilibrium calculations based on the
theoretical NHHe collisional rates of Green (1980).
Danby et al. (1988) then determined more accurate collisional rates for
collisions of NH with p-H() and used them to
recalibrate the ammonia thermometer. The scattering calculations of
Danby et al. (1988) were based on the *ab initio* NHH PES of
Danby et al. (1986). This latter was subsequently improved to investigate
propensity rules at selected collisional energies for ammonia
colliding with both p- and o-H (Offer &
Flower, 1990; Rist
et al., 1993). More
recently, high accuracy *ab initio* calculations have been
performed by Mladenović et al. (2008) to explore the topographical
features of the NHH interaction. In the present paper, we
present new collisional rates based on the determination of a new,
highly accurate, NHH *ab initio* PES. Ammonia and
hydrogen molecules are treated as rigid rotors and H is further
constrained in the scattering calculations to be in the (spherically
symmetrical) para- state, as in Danby et al. (1988). Hence, the main
difference between the present collisional rates and those of
Danby et al. (1988) arises from the PES. These new rates are then used to
estimate the robustness of the ammonia thermometer and the excitation
of ammonia lines under the condition that prevails in cold molecular
clouds, prestellar cores and protostars ( 100 K). For this, we
compute the excitation of both o-NH and p-NH using a
non-LTE radiative transfer code. The paper is organized as follows. In
§2 we present both the new NHH PES and the scattering
calculations. Our excitation computations and the comparison with
earlier computations are presented in § 3,
and §4 concludes this article.

## 2 Potential energy surface and collisional rates

### 2.1 Potential Energy Surface

The ammonia and hydrogen molecules were both assumed to be rigid. This assumption is adequate here because i) the investigated collisional energies are below the first vibrational excitation threshold of ammonia ( Rajamäki et al., 2004) and ii) the corresponding collision time scales are much faster than the inversion motion of ammonia. Monomer geometries were taken at their ground-state average values, as recommended by Faure et al. (2005). The average structure of NH was derived from the high-accuracy calculations of Rajamäki et al. (2004): , and . The ground-state average geometry of H was taken as (e.g. Faure et al., 2005). The conventions of Phillips et al. (1994) were employed in defining the NH-H rigid-rotor (five-dimensional) coordinate system (one H atom lies in the plane).

The NHH PES was constructed using the following two step
procedure: (i) a reference PES was computed from a large set
(89,000 points) of CCSD(T)^{2}^{2}2CCSD(T) stands for the coupled
cluster method with noniterative evaluation of triple excitations.
calculations using the Dunning’s correlation consistent aug-cc-pVDZ
basis set; (ii) this reference surface was calibrated using a
complete basis set (CBS) extrapolation procedure based on a smaller
set (29,000 points) of CCSD(T)/aug-cc-pVTZ calculations. A CBS-type
extrapolation was applied to the correlation part of the interaction
energy and was performed using a two-point type
extrapolation, where is the cardinal number corresponding to the
basis set, as described in Jankowski &
Szalewicz (2005). The self-consistent
field (SCF) contribution was not extrapolated but was taken at the
aug-cc-pVTZ level. All basis sets were supplemented with midbond
functions and all calculations were counterpoise corrected as in
Jankowski &
Szalewicz (2005). The same strategy was recently applied to
HCOH (Troscompt et al., 2009).

Grid points were chosen for 29 fixed intermolecular distances (in the range a) via random sampling for the angular coordinates of H relative to NH. At each intermolecular distance, the interaction energy was then least square fitted using a 120 terms expansion for the angular coordinates, using Eq. (3) of Phillips et al. (1994) adapted to the C symmetry of NH. This expansion includes anisotropies up to and , where the integer indices and refer to the tensor ranks of the angular dependence of the NH and H orientation, respectively. The CBS correction was fitted over a subset of only 46 angular terms with and . We note that the expansion restricted to p-H (), in which all terms with are eliminated, includes only 24 terms. The accuracy of the angular expansions was monitored using a self-consistent Monte Carlo error estimator. A cubic spline interpolation was finally employed over the whole intermolecular distance range and was smoothly connected with standard extrapolations to provide continuous radial expansion coefficients suitable for scattering calculations. Technical details on the fitting strategy can be found in Valiron et al. (2008). The accuracy of the final five-dimensional fit was found to be better than 1 cm in the long-range and minimum region of the interaction ( Bohrs). The accuracy of the above procedure was also checked against a moderate set (1,200 points) of “high-cost” CCSD(T)-R12 calculations which offer a direct way of reaching the basis set limit value within a single calculation, that is without extrapolation (Noga & Kutzelnigg, 1994). The RMS error between the final fit and the benchmark CCSD(T)-R12 values was found to be lower than 1 cm in the whole attractive part of the interaction ( Bohrs). We emphasize that the intrinsic accuracy of CCSD(T) calculations at the basis set limit is cm.

Constraining H in its lowest para level () is strictly equivalent to averaging the PES over the H rotational motion. The global minimum of this averaged PES lies at -85.7 cm for =6.3 Bohrs, with H in an almost equatorial location, equidistant from the two closest H atoms of ammonia. A similar location was found for the global minimum of the NHHe interaction, but with a significantly more shallow potential well at cm (Hodges & Wheatley, 2001). The five-dimensional PES, including the anisotropy of H, is of course qualitatively different: the global minimum, as deduced from our fit, lies at cm for Bohrs, with H colinear with the C axis of ammonia at the nitrogen end. It is interesting to compare this result with the recent calculations of Mladenović et al. (2008): these authors found the global minimum of the NHH interaction at a similar location with a comparable, although significantly smaller, binding energy ( cm). As their calculations were performed at a similar level of accuracy (CCSD(T) method and aug-cc-pVQZ basis sets), this difference most likely reflects monomer geometry effects. Detailed comparisons will be investigated in dedicated future works.

Now, in order to compare the present NHH PES with that
employed by Danby et al. (1988), we present in Fig. 1 a
comparison of the angular expansion coefficients .
The definition of these coefficients is given in Eq. (2.1) of
Danby et al. (1986) and their values are listed in their
Table 4^{3}^{3}3 corresponds to in
Eq. (2.1) of Danby et al. (1986).. Only the first four are plotted for
clarity in the figure. Despite significant differences at short-range
( Bohrs), the overall agreement between the two sets of
coefficients is quite reasonable. This comparison i) indicates
the good quality of the *ab initio* calculations of
Danby et al. (1986) and ii) suggests moderate effects of the new PES
on the dynamics, as shown below.

Finally, it should be noted that the PES of Danby et al. (1986) has been previously checked against laboratory measurements: Schleipen et al. (1993) and Willey et al. (2002) reported, respectively, symmetry-resolved state-to-state and broadening cross sections. In both cases, a good overall agreement was obtained between theory and experiment, suggesting the adequacy of the PES. Discrepancies were however noted, in particular strong propensity rules predicted by theory for NHp-H were observed in experiment to a much lower extent (see below).

### 2.2 Scattering calculations

The quantal theory for scattering of a symmetric top with an atom or a
structureless molecule like H (=0) can be found in
Green (1976). The extension of the formalism to the scattering of a
symmetric top with a linear molecule can be found in Offer &
Flower (1990)
and Rist
et al. (1993). In the present work, calculations were performed
using the (nonreactive) scattering code MOLSCAT
(Hutson &
Green, 1994)^{4}^{4}4http://www.giss.nasa.gov/tools/molscat
in which the extension to allow for the rotational structure of H
is not yet implemented. Hence, the present calculations were
restricted to collisions between NH and p-H (. Extension
to p-H() and o-H() is under way and is further
discussed below.

All calculations were performed at the close-coupling (CC) level. Inversion doubling was neglected and the inversion-tunneling wavefunction was simply taken as a linear combination of two delta functions centered at the equilibrium position (Green, 1976; Davis & Boggs, 1978). We actually tested this approximation on the NHHe system by taking the inversion coordinate explicitly into account, as done previously by Davis & Boggs (1981). To this aim, we employed the high quality NHHe PES of Hodges & Wheatley (2001), which does include the inversion dependence of the interaction. The inversion motion was found to have a negligible effect (less than 10 %) on the rigid-body interaction potential and on the cross sections (Scifoni et al., 2007), as was concluded by Davis & Boggs (1981) from a lower quality potential. We note that van der Sanden et al. (1992) obtained a comparable result for the NHAr interaction. A similar conclusion is therefore expected for the NHH interaction, although the inversion dependence of this PES is yet not known.

We adopted the rotational constants cm and cm. The reduced mass of the system is 1.802289 a.m.u. As the ortho- and para-levels of ammonia do not interconvert in inelastic collisions, these were treated separately. The coupled-channel equations were integrated using the modified log-derivative propagator of Manolopoulos (1986). The radial propagation used a stepsize parameter STEPS=10 except at total energies below 30 cm where STEPS was increased up to 300 to constrain the step length of the integrator below 0.1 Bohrs. Other propagation parameters were taken as the MOLSCAT default values. Calculations were performed for collision energies between cm and 700 cm. The energy grid was adjusted to reproduce all the details of the resonances, with an energy step of 0.2 cm up to total energies of 150 cm and 0.5 cm from 150 to 300 cm. All calculations also included several energetically closed channels to ensure that cross sections were converged to within % for all transitions involving . Thus, at the highest investigated energies, the basis set incorporated all target states with and 12 for o- and p-NH, respectively.

Excitation cross sections for o-NH are presented in Fig. 2 for rotation-inversion transitions out of the ground state of o-NH. Prominent resonant features are observed in this plot. These are caused by both Feshbach and shape type resonances. It should be noted that only Feshbach type resonances are observed in NHHe collisions (e.g. Machin & Roueff, 2005). This difference between He and H() reflects the deeper potential well of the NHH PES, as discussed in Sect. 2.1. Resonances are found to significantly increase the cross sections at low energy and, therefore, the rate coefficients at low temperature. For example, at 10 K, the NHH rate coefficient for the ground-state transition of o-NH is a factor of 20 larger than the NHHe rate of Machin & Roueff (2005). We note, however, that this factor reduces to 2.5 for the transition at the same temperature. Willey et al. (2002) also reported significant differences (up to a factor of 4) between p-H and He broadening cross sections. It is also noticed that the cross section for the transition is much lower than for . This propensity rule was already observed in earlier calculations (Offer & Flower, 1990; Rist et al., 1993) but, interestingly, it was found to be considerably weakened for collisions with o-H () and it was observed experimentally only to a slight extent (Schleipen et al., 1993). On the other hand, it was found to be preserved when including the state in the p-H basis set (Offer & Flower, 1990; Rist et al., 1993). This inclusion was also found to change the absolute values of the cross sections, at a few selected energies, by up to a factor of 3 (Offer & Flower, 1990; Rist et al., 1993). Its effect on the average cross sections and rate coefficients is however expected to be moderate, typically 2030 %. This was indeed checked in the case of NDHH calculations employing the present PES (Scifoni et al., in preparation). As a result, the rate coefficients presented below are expected to be accurate within typically 30 %.

Cross sections were integrated over Maxwell-Boltzmann distributions of
collisional velocities and collisional rate coefficients were obtained
in the range 5100 K for all transitions involving ammonia levels
with (the lowest levels with lie at 177 and 237 K
above the ground states of p- and o-NH, respectively). Higher
levels and temperatures were not investigated in the present work
because collisional rates with o-H are required in models
considering temperatures above 100 K. These collisional rate
coefficients are made available in the
BASECOL^{5}^{5}5http://www.obspm.fr/basecol/ and
LAMBA^{6}^{6}6http://www.strw.leidenuniv.nl/~moldata/
databases as well as at the
CDS^{7}^{7}7http://cdsweb.u-strasbg.fr/.

In Fig. 3, downward rate coefficients are presented for rotation-inversion transitions towards the ground state. The resonant features are found to be completely washed out by the thermal average. The present results are compared with the data of Danby et al. (1988). As expected from the comparison of the expansion angular coefficients (see Fig. 1), the new rates agree within a factor of 2 with those of Danby et al. (1988). We see, however, that there is no particular trend, although the present rates are generally larger.

In Fig. 4, we show the symmetry-averaged rates and that appears in Eq. (1), as a function of the temperature. The new rates are found to be larger than those of Danby et al. (1988) by typically 15% but to follow closely the same temperature dependence. As a result, minor modifications of the ammonia thermometer are expected, as shown below.

## 3 Non-LTE excitation computations

In order to estimate the effect of the new collisional rates on the calibration of the ammonia thermometer, we have computed the excitation of both o-NH and p-NH using the large velocity gradient code of van der Tak et al. (2007). We have used the o-NH and p-NH collisional rates with p-H presented in the previous section, as well as those from Danby et al. (1988), for comparison. The latter were taken from the LAMBA database (Schöier et al., 2005). Collision between NH and He were neglected, because, in addition to H being more abundant than He by a factor of 5, NH-H collision rates are typically a factor of 3 larger than the NH-He rates (Machin & Roueff, 2005). Energy levels, statistical weights and Einstein coefficients were taken from the JPL database for molecular spectroscopy (Pickett et al., 1998). For the calculations using Danby et al. (1988) collisional rates, the first 24 levels of o-NH and the first 17 levels of p-NH were considered (corresponding to energy levels up to 416 and 297 cm, respectively). For the calculations using the new rates, only the first 6 levels of o-NH and the first 10 levels of p-NH were considered (up to 118 and 115 cm, respectively). In both cases, we have neglected the hyperfine structure of the molecule, i.e. we have considered that each hyperfine level within a given inversion level corresponds to the same energy level. While this hypothesis will lead to an overestimate of the line opacity for optical depths greater than a few, it is valid if the line is optically thin (Daniel et al., 2006). We have therefore chosen a column density to velocity gradient ratio that is large enough for this approximation to be valid (; the same value adopted by Walmsley & Ungerechts (1983)).

Fig. 5 shows the excitation temperature between the and metastable levels computed using both sets of collisional rates and as a function of the kinetic temperature. This excitation temperature is obtained by summing the populations of the and within each rotational state. A p-H density of 10 cm was assumed. On this figure, we also show the excitation temperature computed from Eq. (1), i.e. assuming that only the first three rotational levels are populated. As seen on this figure, the excitation temperature computed using the rates of Danby et al. (1988) and the one computed using the rates presented in this paper agree extremely well; both values differ by less than 2%. We also notice that for kinetic temperatures lower than 20 K, the excitation temperature is well approximated by Eq. (1), but it underestimates it at larger temperature, because higher energy levels start to become populated. The good agreement between the kinetic temperature obtained using the rates of Danby et al. (1988) and those presented in this paper can be simply understood by examination of Eq. (1). In this approximation, the excitation temperature depends on . Although the rates computed in this paper differ by 15% with respect to those of Danby et al. (1988), their ratio (and a fortiori the logarithm of their ratio) differ much less. Therefore, the relation between the excitation temperature and the kinetic temperature – or in other words the calibration of the ammonia thermometer – appears to be robust.

Species | Transition | |||
---|---|---|---|---|

(GHz) | (K) | (cm) | ||

p-NH | 23.694496 | 1.1 | ||

p-NH | 23.722633 | 42.3 | ||

p-NH | 23.098819 | 58.3 | ||

p-NH | 22.834185 | 128.1 | ||

p-NH | 22.234506 | 144.0 | ||

o-NH | 23.870129 | 123.6 | ||

o-NH | 572.498068 | 27.5 |

Fig. 6 shows excitation temperature of several p-NH inversion transitions, as a function of the density, for a kinetic temperature of 10 K and the same column density and line velocity than in Fig. 5. Spectroscopic data and critical densities are given in Table 1. For a multi-level system, the critical density can be defined (in the optically thin case) as the density at which the sum of the collisional de-excitation rates out of a given level is equal to the sum of the spontaneous radiative de-excitation rates:

(3) |

where the summation is done over the levels (with energies smaller than that of the considered). With this definition, the critical density refer to a level, and not to a transition.

For densities lower than cm, the excitation temperature of the and inversion transitions computed using the rates presented here and those of Danby et al. show little differences. For these densities, collisional de-excitation is negligible, and the excitation temperature of these lines are close to the background temperature (2.73 K). For densities much greater than the critical density (i.e. cm), collisional de-excitation dominates, and lines are essentially thermalized. At intermediate densities, the and line excitation temperatures predicted using the rates from this work is slightly lower than the one predicted using those from Danby et al.. This is because the de-excitation rates from this work are smaller (by about a factor two) than those of Danby et al. for these lines. As a consequence, the critical densities of the corresponding levels is greater than previously estimated, and the transition thermalize at greater densities. Larger differences in the excitation temperatures of the and transitions – for which critical densities are a few cm – are seen. For example, at a density of cm, the excitation temperature of the transition computed using the rates of Danby et al. is almost a factor two larger than the one computed with the rates obtained here. From the observer point of view, this has no consequences because the energy of the upper level of the transition is 123.6 K. At low densities, for the kinetic temperature considered here, the fractional population of this level is extremely small, and the predicted antenna temperature is essentially zero.

Fig. 7 shows the excitation temperature of the
o-NH inversion transition as a
function of the density, for the same column density to velocity
gradient ratio than in Fig. 6, but – since the
upper level of the transition lies at K above the ground
level of o-NH – a kinetic temperature of 50 K ^{8}^{8}8The
computations with the rates from the present work do not include the
levels above . To make sure that these levels can be
neglected at a kinetic temperature of 50 K, we have computed the
excitation temperature for the transitions shown on Fig. 7 using
Dandy’s rates, but without considering the levels above . These
were found to be quasi-identical to those computed when the levels
above are considered. The behavior of the excitation
temperature is similar to that of p-NH inversion lines; at low
density, it is close to the background temperature, while it is
thermalized at densities greater than cm. We predict,
in agreement with Walmsley &
Ungerechts (1983), a population inversion for
densities ranging between and cm (note that the range in which the inversion occurs
is slightly different for the two collisional rate sets). This
population inversion was first predicted by Walmsley &
Ungerechts (1983), and
has been studied in detail by Flower
et al. (1990). The corresponding
maser transition has been since observed in several star forming
regions, e.g. NGC 6334I (Beuther et al., 2007). As explained by
Walmsley &
Ungerechts (1983), the lower level of the transition () is
de-populated by collisions to excited levels in the ladder,
while it is populated by radiative transitions from the
level. For densities greater than cm,
the collisional de-population rate is greater than the radiative
population rate of the lower level, and the inversion occurs. For
densities greater than cm, excited
levels in the ladder start to become populated and populate
the level collisionally. This limits the maser gain to
moderate values; for a density of cm, we
predict a negative opacity of only .

Fig. 7 also shows the excitation temperature of
the rotation-inversion transition of
o-NH. This line, at a frequency of GHz, was first
detected towards OMC-1 with the *Kuiper Airborne Observatory*
(Keene
et al., 1983). It was also detected towards -Oph A with
*Odin* space telescope (Liseau
et al., 2003), and it will be soon
observable with the *Heterodyne Instrument for the Far Infrared*
(HIFI) on board the *Herschel Space Observatory*. This line is
found to thermalize at densities of cm. Once
again, little differences between the excitation temperatures computed
using the rates of Danby et al. and those from this work are seen. No
significant difference were found between the excitation temperatures
of the other rotation-inversion transitions that will be observable
with HIFI either.

## 4 Conclusions

We have presented new collisional excitation rates of p-NH and o-NH with p-H(). With respect to older computations from Danby et al. (1988), the present rates were found to agree within a factor of 2. In order to investigate the effect of the new rates on the excitation of o-NH and p-NH, we have computed the excitation of these species under physical conditions that are typical of dense molecular clouds, prestellar cores, as well as the outer envelopes of embedded protostars, using an LVG code. We found that the excitation temperature between the and levels computed using the new rates is almost identical to that computed using older rates at the low temperatures considered here ( K). Thus, the calibration of the ammonia thermometer appears robust. The effect of the new rates on the inversion transitions (at cm wavelenght) or the rotation-inversion transitions that will be observable with Herschel-HIFI are also found to be small.

Future works include extension of the present calculations to p-H and o-H as well as to higher ammonia levels and kinetic temperatures. Comparisons with double resonance (Daly & Oka, 1970), crossed beam (Schleipen et al., 1993) and pressure broadening (Willey et al., 2002) experiments will be also investigated, with the objective to establish the predictive ability of the present PES and to distinguish between the predictions of the available PES.

## Acknowledgments

The authors wish to acknowledge their friend and colleague Pierre Valiron who initiated the work that lead to this paper. Pierre passed away on August, 31th 2008, and he is deeply missed. We also thanks Evelyne Roueff for her critical reading of this manuscript. This research was supported by the CNRS national program “Physique et Chimie du Milieu Interstellaire” and by the FP6 Research Training Network “Molecular Universe” (contract number MRTN-CT-2004-512302).

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