Rotational excitation of 20 levels of paraH_{2}O by orthoH_{2} (j_{2} = 1, 3, 5, 7) at high temperature
F. Daniel^{1}  M.L. Dubernet^{2,3}  F. Pacaud^{2}  A. Grosjean^{4}
1  Departamento Molecular and Infrared Astrophysics, Consejo Superior de Investigaciones
Cientificas, C/ Serrano 121, 28006 Madrid, Spain
2 
Université Pierre et Marie Curie, LPMAA UMR CNRS 7092, Case 76, 4 Place Jussieu, 75252 Paris Cedex 05, France
3 
Observatoire de Paris, LUTH UMR CNRS 8102, 5 Place Janssen, 92195 Meudon, France
4 
Institut UTINAM, UMR CNRS 6213, 41 bis avenue de l'Observatoire, BP 1615, 25010 Besançon Cedex, France
Received 26 November 2009 / Accepted 16 February 2010
Abstract
Aims. The objective is to obtain the best possible set of
rotational (de)excitation statetostate and effective rate
coefficients for temperatures up to 1500 K. Statetostate rate
coefficients are presented among the 20 lowest levels of paraH_{2}O with H_{2}(j_{2}=1) and
,
and among the 10 lowest levels of paraH_{2}O with H_{2}(j_{2}=3) and
.
Methods. Calculations are performed with the close coupling (CC)
method over the whole energy range, using the same 5D potential energy
surface (PES) as the one employed in our latest publications on water.
We compare our CC results both with thermalized quasiclassical
trajectory (QCT) calculations using the same PES and with previous
quantum calculations obtained between T=20 K and T=140 K with a different PES.
Results. Comparisons with thermalized QCT calculations show
factors from 1 to 3. Until recently the only other available set of
rate coefficients were scaled collisional rate coefficients obtained
with He as a collision partner, and differences between CC and scaled
results are shown to be greater than with QCT calculations. The use of
the CC accurate sets of rate coefficients might lead to reestimation
of water abundance in the astrophysical whenever models include the
scaled H_{2}OHe rate coefficients.
Key words: molecular data  molecular processes  ISM : molecules
1 Introduction
This is the second publication from the largescale effort carried out to obtain the highest possible accuracy for collisional excitation rate coefficients of H_{2}0 with rotationally excited H_{2}. Our previous paper (Dubernet et al. 2009) provided statetostate rate coefficients among the 45 lowest levels of orthoH_{2}O with paraH_{2} (j_{2} =0) and , as well as with j_{2} = 2 and . In addition to and only for the 10 lowest energy levels of orthoH_{2}O did Dubernet et al. (2009) obtain statetostate rate coefficients involving j_{2}=4 with and j_{2}=2 with .
Those efforts for the excitation of water are justified by the importance of water in various astrophysical media. Water is a key molecule for the chemistry and the energy balance of the gas in cold clouds and starforming regions, thanks to its relatively large abundance and large dipole moment.
The Heterodyne Instrument for the FarInfrared (HIFI) was launched in May 2009 on board the Herschel Space Observatory, publication of the water rate coefficients becomes urgent. The instrument will observe spectra of many molecules with unprecedented sensitivity with an emphasis on water lines in regions such as low or highmass starforming regions, protoplanetary disks, and AGB stars. The interpretation of these spectra will rely upon the accuracy of the available collisional excitation rate coefficients that enter into the population balance of the emitting levels of the molecules. In the temperature range from 5 K to 1500 K, the most abundant collider likely to excite molecules in media with weak UV radiation fields is the hydrogen molecule, followed by the helium atom.
The present paper investigates the excitation of the 20 lowest paraH_{2}0 rotational levels by orthoH_{2} (j_{2}=1, 3, 5, 7) for kinetic temperatures from 5 K to 1500 K. It extends the work of Dubernet et al. (2006), which provided rate coefficients for deexcitation of the lowest 10 rotational levels of ortho/paraH_{2}O by collisions with paraH_{2} (j_{2}=0) and orthoH_{2}(j_{2}=1). Present and past Dubernet et al. (2009,2006) quantum dynamical calculations were carried out with the 5D potential energy surface (PES) determined by Faure et al. (2005). This accurate 5D PES, suitable for inelastic rotational calculations, was obtained from a 9D PES by averaging over H_{2}and H_{2}O ground vibrational states. As pointed out in Faure et al. (2005), this stateaveraged PES is actually very close to a rigidbody PES using stateaveraged geometries for H_{2}O and H_{2}.
Dubernet et al. (2006) showed that the new PES of Faure et al. (2005) have led to a significant reevaluation of the rate coefficients for the excitation of H_{2}O by paraH_{2} (j=0) below 20 K and to a weak effect with a maximum change of 40% for collisions with orthoH_{2}(j=1) when their results were compared to the collisional calculations of Phillips et al. (1996); Dubernet & Grosjean (2002); Grosjean et al. (2003) obtained with the 5D PES of Phillips et al. (1994).
For the deexcitation of orthoH_{2}O by paraH_{2} (j=0), Dubernet et al. (2009) compared the effective rate coefficients of Phillips et al. (1996) for the first 10 levels of orthoH_{2}O with temperatures in the range 20 K to 140 K. It was shown that the ratios of effective rate coefficients converge slowly towards unity with temperature, certainly reflecting the decreasing influence of the difference between the two different PES (Faure et al. 2005; Phillips et al. 1994) as temperature increases.
In addition, Dubernet et al. (2009) carried out comparisons with thermalized QCT calculations of Faure et al. (2007) that showed large factors at intermediate temperature and factors from 1 to 3 at high temperature for the strongest rate coefficients. We showed also that scaled collisional rate coefficients obtained with He could not be used in place of collisional rate coefficients with paraH_{2}. The quantum calculations of Dubernet et al. (2009) pointed out the importance of internal energy transfer between excitation of H_{2} and deexcitation of orthoH_{2}O, which was at the origin of some large differences observed with QCT calculations of Faure et al. (2007) and with scaled collisional rate coefficients obtained with He (Green et al. 1993).
We recall that Faure et al. (2007) provides rate coefficients for rotational deexcitation among the lowest 45 rotational levels of ortho/paraH_{2}O colliding with thermalized ortho/paraH_{2} in the temperature range 202000 K. This set is a combination of various data: 1) data obtained with quasi classical trajectory (QCT) calculations with the H_{2} molecule assumed to be rotationally thermalized at kinetic temperature and calculated between 100 K and 2000 K, 2) the values at 20 K are CC calculations from Dubernet et al. (2006) for the first 5 levels and equal to values at 100 K for all other levels, 3) scaled H_{2}OHe results from Green et al. (1993) for the weakest rate coefficients.
It should be mentioned that calculations of collisional excitation by orthoH_{2} have been carried out solely on 4 diatomic molecules CO (Flower 2001; Mengel et al. 2001; Flower & Launay 1985; Wernli et al. 2006), H_{2} (Flower & Roueff 1999b), HD (Flower 1999; Flower & Roueff 1999a), SiS (Kos & Lique 2008), and on water (Phillips et al. 1996; Faure et al. 2007; Dubernet et al. 2006; Grosjean et al. 2003), and that all quantum calculations except those of Dubernet et al. (2006) restricted the H_{2} basis set to j_{2}=1.
Our current objective is to obtain the best possible set of rotational (de)excitation statetostate and effective rate coefficients for temperatures up to 1500 K. In particular, our basis set for H_{2} is extended to j_{2}=3, which allows for internal energy transfer between excitation of orthoH_{2} and deexcitation of paraH_{2}O. We provide statetostate rate coefficients among the 20 lowest levels of paraH_{2}O with H_{2}(j_{2}=1) and , and among the 10 lowest levels of paraH_{2}O with H_{2}(j_{2}=3) and . We predict the effective rate coefficients for j_{2} = 5,7.
2 Methodology
2.1 Collisions with H_{2}
Our calculations provide statetostate collisional rate coefficients involving changes in both the target and the perturber rotational levels, i.e. where and represent the initial and final rotational levels of water, j_{2} and j'_{2} the initial and final rotational levels of H_{2}, and T is the kinetic temperature.
The statetostate collisional rate coefficients are the Boltzmann
thermal averages of the statetostate inelastic cross sections:
where E is the kinetic energy, the Boltzmann constant and the reduced mass of the colliding system.
These statetostate collisional rate coefficients follow the principle of detailed balance, and reverse rate coefficients can be obtained from forward rate coefficients by the usual formula:
(2) 
where g_{j1} and g_{j2} are the statistical weights related to rotational levels of H_{2}O and H_{2} respectively, and the different are the rotational energies of the species.
Some astrophysical applications might use the socalled effective rate coefficients
,
which are given by the sum of the statetostate rate coefficients (Eq. (1)) over the final j_{2}' states of H_{2} for a given initial j_{2}:
These effective rate coefficients do not follow the detailed balance principle, and both excitation and deexcitation should be explicitly calculated.
Finally, averaged deexcitation rate coefficients for paraH_{2}O by rotationally thermalized orthoH_{2} can be obtained by averaging over the initial rotational levels of orthoH_{2}:
with , where Z is the partition function over either orthoH_{2} states. These averaged deexcitation rate coefficients are those directly calculated by Faure et al. (2007) with a QCT method.
2.2 Description of the calculations
In the current calculations we used the same expansion of the Faure et al. (2005) 5D PES as in Dubernet et al. (2006), where details can be found. For this PES, inaccuracies in inelastic cross sections might come from different sources: propagation parameters, description of the rotational Hamiltonians of the 2 molecules, sizes of H_{2}O and H_{2} rotational basis sets, and a level of approximation in quantum calculations where the coupled states (CS) approximation might be used instead of the exact close coupling (CC) method. Additional errors might be introduced in rate coefficients if the kinetic energy grid is not fine enough near thresholds, resulting in poor lowtemperature rate coefficients, or not extended to high enough energies, leading to wrong hightemperature results.
Our quantum calculations were carried out with modified versions of the sequential and parallel versions of the MOLSCAT code (Hutson & Green 1994; McBane 2004). Parameters of the propagation were optimized as in Dubernet & Grosjean (2002); Dubernet et al. (2006); Grosjean et al. (2003), and (Dubernet et al. 2009). Identical to Dubernet et al. (2009,2006), the H_{2} energy levels are the experimental energies of Dabrowski (1984), and the H_{2}O energy levels and eigenfunctions were obtained by diagonalisation of the effective Hamiltonian of Kyrö (1981), compatible with the symmetries of the PES (Dubernet & Grosjean 2002; Dubernet et al. 2006; Grosjean et al. 2003), and (Dubernet et al. 2009). The first 20 levels of paraH_{2}O are given in Table 1. The reduced mass of the system is 1.81277373 a.m.u.
Table 1: Energy levels of paraH_{2}O.^{a}
Table 2: Contribution of the statetostate rate coefficients with j_{2}=1 to j'_{2}=3 to the effective rate coefficients for the four largest transitions with .
2.2.1 Basis set convergence
The methodology in choosing an appropriate basis set is the same of the one described in Dubernet et al. (2009) which should be consulted for additionnal information. The basis set is a direct product of rotational wavefunctions of water, characterized by the rotational quantum number j_{1} (the lowest value of j_{1} is one for paraH_{2}O) and the pseudoquantum number which varies between j_{1} and j_{1} (alternatively, we may use the pseudoquantum numbers k_{a} and k_{c} with the correspondence ), and of rotational wavefunctions of hydrogen characterized by the rotational quantum number j_{2}. We call B(n,m) a basis set where n is the maximum value of j_{1}, m is the maximum value of j_{2}, and B(n,m) includes all the coupled states, with . The convergence of the basis set usually involves keeping a number of closed channels above the total energy at which the collisional crosssections are calculated. The potential couplings between ( j_{1} k_{a} k_{c}) energy levels decrease with increasing and and we find that a good convergence is reached with 10 energetically closed channels of water.
Another important question is the accuracy of crosssections with respect to the number of closed channels of the H_{2} molecule. Generally at least one to two closed H_{2} rotational channels would be required to ensure convergence to better than 5% of inelactic cross sections involving energy transfer in H_{2}. This would be particularly important if the purpose of our calculations was to find inelastic rate coefficients of H_{2} averaged over water transitions. For our calculations, it is sufficient to use a basis set with a maximum value of j_{2} =3, expecially as the j_{2} =5 level lies 1034.67 cm^{1} above the j_{2} =3 level of orthoH_{2}. The inclusion of j_{2} =3 leads to small differences for statetostate rate coefficients of paraH_{2}O with H_{2}(j_{2}=1) and , but it allows the possibility of internal energy transfer between excitation of orthoH_{2} and deexcitation of paraH_{2}O. Indeed Fig. 1 and Table 2 show examples of transitions of paraH_{2}O for which the statetostate rate coefficients with H_{2}(j_{2}=1) and is non negligible compared to the statetostate rate coefficients with H_{2}(j_{2}=1) and .
Figure 1: Statetostate rate coefficients (cm^{3}s^{1}) of the 6_{24} to 4_{22} and the 6_{15} to 4_{13} transition of paraH_{2}O as a function of temperature (Kelvin).The full line indicates the statetostate rate coefficients for , and the broken line corresponds to . 

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Finally our choice is a B(n,m) basis set with 10 energetically closed channels of water and an energy cutoff of 1309 cm^{1} for states connected to j_{2} =3.
This ensures very good convergence for the significant statetostate rate coefficients, i.e., among the 20 lowest levels of paraH_{2}O with H_{2}(j_{2}=1) and , among the 10 lowest levels of paraH_{2}O with H_{2}(j_{2}=3) and , among the 10 lowest levels of paraH_{2}O with H_{2}(j_{2}=1) and , from the 11th20th levels to the first 13th levels of paraH_{2}O with H_{2}(j_{2}=1) and . A worse convergence is reached for statetostate rate coefficients among levels between the 14th of 20th levels of paraH_{2}O with H_{2}(j_{2}=1) and . Fortunally, most of the last statetostate rate coefficients are negligible or very small compared to statetostate rate coefficients among levels between the 14th of 20th levels of paraH_{2}O with H_{2}(j_{2}=1) and , so the corresponding effective rate coefficients with H_{2}(j_{2}=1) have good accuracy. Uncertainties linked to convergence of the basis set are part of the total uncertainties indicated in Table 3.
Table 3: Summary of the sets of statetostate rate coefficients (STSR) available in BASECOL (sets (1), (2), (3), (4) below 10th level), and effective rate coefficients (ER) .
2.2.2 Choice of total energy points
The CC calculations were carried out over essentially the whole energy range spanned by the Boltzmann distributions (Eq. (1)). The highest energy point calculated is at 8000 cm^{1} and cross sections are extrapolated at higher energy in order to achieve convergence for deexcitation from the highest water energy levels. These extrapolations do not degrade the accuracy of rate coefficients because the concerned cross sections behave regularly. We carefully spanned the energy range above the inelastic channels and added more points in the presence of resonance structures. The energy steps were fixed to 0.1 cm^{1} for the total energy below 205 cm^{1}. Between 205 cm^{1} and 720 cm^{1} energy steps vary from 0.5 to 1 cm^{1}, from 720 to 815 cm^{1}they range from 2 to 5 cm^{1}, from 815 to 920 cm^{1} they range from 13 to 30 cm^{1}, and above 920 they vary up to 500 cm^{1}. We paid particular attention to having a fine description of lowenergy behaviors of cross sections connected to j_{2}=3for the first 10 rotational levels of paraH_{2}O. Some of the additionnal thresholds cross sections connected to j_{2}=3 were calculated with the breathing sphere approximation (Agg & Clary 1991b,a), i.e., we averaged the PES over j_{2}=3, similarly to what had been done for j_{2}= 4 in Dubernet et al. (2009). This procedure is fully justified by the small magnitude of cross sections involving energy transfer with from j_{2}=3. We checked that the resulting breathing sphere crosssections agreed with sparser cross sections obtained with the accurate basis set described above. Nevertheless, for j_{2}=3 at high energy, the energy grid is coarser than for j_{2}=1, though still allowing adequate precision (see Table 3). Overall there are about 1192 energy points.
3 Discussion
3.1 Results for j_{2} = 1, 3, 5, 7
We used the methodology described above to calculate sets of statetostate rate coefficients (Eq. (1)) in the temperature range from 5 K to 1500 K for deexcitation among the 20 lowest levels of paraH_{2}O with H_{2}(j_{2}=1) and , and among the 10 lowest levels of paraH_{2}O with H_{2}(j_{2}=3) and . The statetostate rate coefficients involving energy transfers from j_{2}=3 to j_{2}=1 contribute only 4% or less to the effective rate coefficients; nevertheless, we provide them for the sake of completeness. They should only be used to calculate the effective excitation and deexcitation rate coefficients of paraH_{2}O corresponding to the j_{2}=3 level of orthoH_{2} as their accuracy is very low. In addition we obtain deexcitation rate coefficients from the 11th20th levels of paraH_{2}O with H_{2}(j_{2}=3) and , but we did not try to obtain accurate values so those rate coefficients will not be published.
From the calculated statetostate rate coefficients, the effective rate coefficients corresponding to j_{2}= 1, 3 can be calculated using Eq. (3). The ratios of effective deexcitation rate coefficients (Eq. (3)) over effective deexcitation rate coefficients for the first 10 levels of paraH_{2}O are given in Fig. 2. Table 4 provides labels of first and last deexcitation transitions from a given level as given in Table 1. This table is the key for reading the figures giving ratios of rate coefficients as a function of transitions labels. The first transition from level n is the transition , the last transition is the transition .
Table 4: Labels of transitions.
Ratios are generally close to 1 within a maximum variation of 20%, except at low temperature for the weakest transitions from levels 6, 7, 8 (see Table 1) to the ground state. Because of the weakness of transitions and because those stronger ratios occur at low temperature where j_{2}=3 is unlikely to be important in most application cases, the difference between j_{2}=1 and j_{2}=3 is not significant for astrophysical applications. We can assume that effective rate coefficients , will be close to .
Figure 2: Ratios of effective deexcitation rate coefficients / (Eq. (3)) for the first 10 levels of paraH_{2}O and for temperatures ranging from 20 K to 1600 K. The abscissae indicate the labeling of the deexcitation transitions as indicated in Table 4. 

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The BASECOL database (Dubernet et al. 2006) provides full tables of the rate coefficients sets mentioned in Table 3, i.e. sets (1), (2), (3), and (4).
3.2 Accuracy of results
Apart from the usual checks of convergence with respect to propagation parameters, basis set, and total angular momentum, the statetostate rate coefficients have been carefully checked by detailed balance. It should be recalled that the quality of rate coefficients at low temperature is linked to the number of energy points close to the molecular thresholds, and we have an excellent energy grid for both j_{2}=1 (20 levels of paraH_{2}O) and j_{2} = 3 (10 levels of paraH_{2}O). The maximum values of the estimated errors are given in Table 3 for transitions starting from different levels of paraH_{2}O (T1, T2) and for the various sets of statetostate rate coefficients (1 to 4). Set (2a) shows very good accuracy for transitions among the first 10 levels of paraH_{2}O and for transitions from the 11th20th levels to the first 10 levels of paraH_{2}O. Set (2b) has lower accuracy, but this is no concern because this set does not contribute significantly to the effective rate coefficients. Set (3) has very low accuracy, which is not a cause for concern for the present application since they provide a very negligible contribution to the effective rate coefficients. The accuracy of the effective rate coefficients (ER) reflects the accuracies of set (1) and of sets (2). The accuracy of the effective rate coefficients (ER) is mainly connected to set (4) because set (3) rate coefficients give a small contribution to the effective rate coefficients.
3.3 Thermalized rate coefficients
We do not explicitly provide deexcitation rate coefficients of paraH_{2}O with thermalized H_{2} (Eq. (4)). j_{2}=3 makes a significant contribution around 300400 K and that j_{2}=5 starts to contribute significantly around 1000 K. Therefore our calculations can provide highly accurate averaged deexcitation rate coefficients (Eq. (4)) up to 400 K for transitions from the 20 first levels of paraH_{2}O, up to 1000 K for transitions from the first 10th level of paraH_{2}O, the accuracy being connected to accuracies listed in Table 3. We showed that j_{2}=3 significant effective rate coefficients are very close to j_{2}=1, therefore we can safely assume all unknown j_{2}= 3,5, 7significant effective rate coefficients to be close to j_{2}=1 in calculating the thermalized rate coefficients. Another possibility at high temperature and high levels of water is to directly use the QCT rate coefficients of Faure et al. (2007), but to be aware of their limitation. Comparisons between our averaged deexcitation rate coefficients and the QCT results are given in the following sections.3.4 Comparison with QCT and scaled He calculations
To compare the quantum and QCT results obtained with the same PES, we must remove the scaled H_{2}OHe results of Green et al. (1993) from their published set since they cover a large fraction of the transitions over a range of temperatures. The averaged CC rate coefficients in Fig. 3 include our unpublished statetostate rate coefficients corresponding to the extension of sets (3) and (4) of Table 3 above the 11th level of paraH_{2}O. These unpublished statetostate rate coefficients have a maximum accuracy of 50%. The deexcitation rate coefficients of H_{2}O + He of Green et al. (1993), scaled by a factor of 1.344 to correct for the differing colliding system masses, have systematically been used in astrophysical applications to mimic rate coefficients of H_{2}O + H_{2}. Phillips et al. (1996) pointed out that this method is not valid for temperatures up to 140 K. We note that the CC and QCT rate coefficients are within a factor of 3 over the temperature range with ratios around 1, whereas scaled He calculations mostly underestimate CC rate coefficients. The agreement between CC and He scaled calculations gets better with increasing temperature. These figures do not include He scaled transitions that lead to very high ratios, i.e., transitions starting from some level n to level 1.
Figure 3: Ratios of CC averaged deexcitation rate coefficients (Eq. (4)) of paraH_{2}O with orthoH_{2} over data sets published by Faure et al. (2007) as a function of the CC averaged deexcitation rate coefficients of paraH_{2}O with orthoH_{2} (in cm^{3}s^{1}). These data sets include both the quasiclassical calculations (crosses in squares) and scaled waterHe rate coefficients (crosses without squares) published in Faure et al. (2007). 

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Figures 4 to 6 compare our averaged rate coefficients with those of Faure et al. (2007) and with scaled He rate coefficients of (Green et al. 1993) for deexcitation transitions from the first 16 levels of paraH_{2}O at 200 K. These last two sets coincide for some transitions since the Faure et al. (2007) set includes both QCT calculations and scaled He rate coefficients for the weakest transitions. Overall we find that the QCT calculations give better absolute values than scaled He calculations.
Figure 4: CC averaged deexcitation rate coefficients (Eq. (4)) of paraH_{2}O with orthoH_{2}, set of rate coefficients published by Faure et al. (2007) (black line) and scaled He rate coefficients of (Green et al. 1993) (red line) at 200 K for the 1st to the 28th deexcitation transitions. The abscissae indicate the labeling of the deexcitation transitions as indicated in Table 4. 

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Figure 5: CC averaged deexcitation rate coefficients (Eq. (4)) of paraH_{2}O with orthoH_{2}, set of rate coefficients published by Faure et al. (2007) (black line) and scaled He rate coefficients of (Green et al. 1993) (red line) at 200 K for the 29th to the 66th deexcitation transitions. The abscissae indicate the labeling of the deexcitation transitions as indicated in Table 4. 

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Figure 6: CC averaged deexcitation rate coefficients (Eq. (4)) of paraH_{2}O with orthoH_{2}, set of rate coefficients published by Faure et al. (2007) (black line) and scaled He rate coefficients of (Green et al. 1993) (red line) at 200 K for the 67th to the 190th deexcitation transitions. The abscissae indicate the labeling of the deexcitation transitions as indicated in Table 4. 

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3.5 Comparison with the H O + H effective rate coefficients of Phillips et al. (1996)
Comparison with effective rate coefficients of Phillips et al. (1996) can only be performed for the first 10 levels of paraH_{2}O and for temperatures in the range 20 K to 140 K. It is recalled that between 20 K and 140 K, effective rate coefficients (Eq. (3)) for j_{2}=1 are equal to averaged rate coefficients (Eq. (4)) since j_{3} is scarcely populated. The ratios of effective rate coefficients given in Table 5 are very close to 1 for the strongest transitions and do not have a strong temperature dependence. Overall the new PES of Faure et al. (2005) does not induce a significant change in rate coefficients for collision with orthoH_{2}.
Table 5: Ratios of the 5 effective deexcitation rate coefficients of Phillips et al. (1996) over our effective rate coefficients (Eq. (3)) for j_{2}=1 (equal to our averaged rate coefficients).
3.6 Fitted rate coefficients
The statetostate rate coefficients for the deexcitation of paraH_{2}O with paraH_{2} (j_{2} = 1, 3) and are fitted to an analytical form very similar to the one used by Mandy & Martin (1993):The fits were performed using numerical rate coefficients calculated at 100 temperatures ranging from to , which are indicated in Table 3. The fitted coefficients are such that the maximum error between initial data points and fitted values is minimal. A maximum value of N=14 is needed for good accuracy over the whole range of temperature. The fitted rate coefficients were subsequently compared to numerical rate coefficients calculated with a step of T=1 K from to , and the maximum error found is less than 0.5%. We emphasize that these fits have no physical meaning; they are only valid in the temperature range of the relevant , and should not be used to perform extrapolations. The complete fitting coefficients sets corresponding to sets (1), (2a,b), (3), and (4) of Table 3 will be available in the BASECOL database (Dubernet et al. 2006). The quality of the fits can be checked online through the graphic interface.
4 Concluding remarks
We provide statetostate rate coefficients among the 20 lowest levels of paraH_{2}O with H_{2}(j_{2}=1) and and amongthe 10th levels of paraH_{2}O with orthoH_{2}(j_{2}=3) and . We predict the effective rate coefficients for j_{2} = 5,7.For the given PES, the accuracy of quantum rate coefficients, explicitly given for different temperatures and transitions, is rather homogeneous and lies between 5% and 40% for the first 20 levels of paraH_{2}O. For the available transitions and temperature, we strongly recommend using the present sets of effective rate coefficients instead of either the scaled H_{2}OHe data of Green et al. (1993) or the set published in Faure et al. (2007). For the uncalculated effective rate coefficients with j_{2}=3, 5,7, the user might use guesses as explained in Sect. 3.3. Alternatively, the user may use the sets of thermalized rate coefficients published in Faure et al. (2007), being aware that the weakest transitions are given by scaled H_{2}OHe rate coefficients that are sometimes wrong by large factors. We find that the scaled He rate coefficients (Green et al. 1993) are representative neither of the effective rate coefficients nor of the averaged CC rate coefficients in absolute values.
Collisions with orthoH_{2} (j_{2}=1) are relevant whenever the ortho/para ratio of H_{2} is high. It would be useful to identify the astrophysical cases where rotationally excited orthoH_{2} is relevant for observations of paraH_{2}O and where more extensive calculations should be carried out to complete the collisional sets provided here.
We are currently carrying out similar calculations for orthoH_{2}O with orthoH_{2} (j_{2}=1,3) and for paraH_{2}O with paraH_{2} (j_{2}=0,2).
AcknowledgementsMost scattering calculations were performed at the IDRISCNRS and CINES under project 2006070809 04 1472. 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: MRTNCT2004512302.
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All Tables
Table 1: Energy levels of paraH_{2}O.^{a}
Table 2: Contribution of the statetostate rate coefficients with j_{2}=1 to j'_{2}=3 to the effective rate coefficients for the four largest transitions with .
Table 3: Summary of the sets of statetostate rate coefficients (STSR) available in BASECOL (sets (1), (2), (3), (4) below 10th level), and effective rate coefficients (ER) .
Table 4: Labels of transitions.
Table 5: Ratios of the 5 effective deexcitation rate coefficients of Phillips et al. (1996) over our effective rate coefficients (Eq. (3)) for j_{2}=1 (equal to our averaged rate coefficients).
All Figures
Figure 1: Statetostate rate coefficients (cm^{3}s^{1}) of the 6_{24} to 4_{22} and the 6_{15} to 4_{13} transition of paraH_{2}O as a function of temperature (Kelvin).The full line indicates the statetostate rate coefficients for , and the broken line corresponds to . 

Open with DEXTER  
In the text 
Figure 2: Ratios of effective deexcitation rate coefficients / (Eq. (3)) for the first 10 levels of paraH_{2}O and for temperatures ranging from 20 K to 1600 K. The abscissae indicate the labeling of the deexcitation transitions as indicated in Table 4. 

Open with DEXTER  
In the text 
Figure 3: Ratios of CC averaged deexcitation rate coefficients (Eq. (4)) of paraH_{2}O with orthoH_{2} over data sets published by Faure et al. (2007) as a function of the CC averaged deexcitation rate coefficients of paraH_{2}O with orthoH_{2} (in cm^{3}s^{1}). These data sets include both the quasiclassical calculations (crosses in squares) and scaled waterHe rate coefficients (crosses without squares) published in Faure et al. (2007). 

Open with DEXTER  
In the text 
Figure 4: CC averaged deexcitation rate coefficients (Eq. (4)) of paraH_{2}O with orthoH_{2}, set of rate coefficients published by Faure et al. (2007) (black line) and scaled He rate coefficients of (Green et al. 1993) (red line) at 200 K for the 1st to the 28th deexcitation transitions. The abscissae indicate the labeling of the deexcitation transitions as indicated in Table 4. 

Open with DEXTER  
In the text 
Figure 5: CC averaged deexcitation rate coefficients (Eq. (4)) of paraH_{2}O with orthoH_{2}, set of rate coefficients published by Faure et al. (2007) (black line) and scaled He rate coefficients of (Green et al. 1993) (red line) at 200 K for the 29th to the 66th deexcitation transitions. The abscissae indicate the labeling of the deexcitation transitions as indicated in Table 4. 

Open with DEXTER  
In the text 
Figure 6: CC averaged deexcitation rate coefficients (Eq. (4)) of paraH_{2}O with orthoH_{2}, set of rate coefficients published by Faure et al. (2007) (black line) and scaled He rate coefficients of (Green et al. 1993) (red line) at 200 K for the 67th to the 190th deexcitation transitions. The abscissae indicate the labeling of the deexcitation transitions as indicated in Table 4. 

Open with DEXTER  
In the text 
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