Ro-vibrational excitation of H_2 by H extended to high temperatures

Collisions between H2 and H are crucial to the thermal balance of astrophysical media, especially that of the early Universe. We report quasi-classical trajectory (QCT) calculations of rate coe cients for the collisional (de-)excitation of H2 by H—including the three-body collisional dissociation— on a highly accurate H3 potential energy surface (PES). Translational temperatures ranging from 100 to 15 000 K, and ro-vibrational energy levels of H2 up to ⇡ 50 000 K are considered. Our QCT rate coe cients agree well with those obtained from previous quantum calculations on the same PES. This good agreement supports our claim that, in order to study the 5000-to-15 000 K temperature range, typical of the early Universe, inexpensive QCT calculations should be preferred to extrapolation of costly quantum calculations run at lower temperatures. The new collisional rate coe cients should enhance our ability to model the cooling and heating budget of the primordial gas.


INTRODUCTION
In astrophysical media, collisional excitation processes followed by radiative decay lead to the conversion of thermal energy into photons, which can escape from the medium. Hence, they contribute significantly to the thermal balance of astrophysical media. Since atomic hydrogen and helium, the two most abundant species in space, have no fine structure in their ground electronic state, the cooling in the interstellar gas takes place mostly through collision with heavier atoms. Hence, cooling in the interstellar medium is dominated by collisions with the next most abundant elements, carbon and oxygen (Wolfire et al. 1995).
In the early Universe, the situation is di↵erent since such atoms are absent. Such kind of cooling takes place thanks to the first molecules that have been formed. Even if H 2 is on small abundance (Galli & Palla 2013), the cooling by infrared emission of H 2 following its collisional excitation is the dominant cooling mechanism. Indeed, collisions between H 2 and H provide the cooling essential to the formation of protogalaxies and of the first stars (Flower & Pineau des Forêts 2001). Modeling of collisions between H 2 and H are then crucial in order to qualitatively and quantitatively model the thermal balance of the early Universe.
Because of their astrophysical interest, and especially ? E-mail: yohann.scribano@umontpellier.fr because of their experimental and theoretical accessibility, collisions between H 2 and H have been the object of constant and detailed investigations (See Aoiz et al. 2005, for a review). Collisional rate coe cients computed using either quasiclassical trajectory (QCT) (see Mandy & Martin 1993, and references therein) or time independent quantum mechanical (TIQM) , and references therein) approaches have been provided to the astrophysical community during the last three decades. Current models of the early Universe Glover & Abel 2008) generally rely on the later TIQM collisional data. Indeed, the data of  were obtained using the state-of-the-art potential energy surface (PES) of Mielke et al. (2002) that has been shown to reproduce all the main experimental features (Mielke et al. 2003). However, these TIQM rate coe cients were obtained using pure inelastic approaches assuming that the reactive cross sections are negligible and hence neglecting the possible ortho-to-para-H 2 1 nuclear spin conversion process during hydrogen exchange reactive collisions. Whereas such an approximation may be valid at low temperature, it is much more questionable at temperatures above 1000 K and/or when dealing with ro-vibrational (de-)excitation. Hence, in order to provide the astrophysical community with highly accurate data including the possible ortho-topara-H 2 conversion process, one of the authors contributed new TIQM results (Lique et al. 2012(Lique et al. , 2014Lique 2015) for the ro-vibrational excitation of H 2 by H at low to moderate temperatures (T < 5000 K), using the PES of Mielke et al. (2002). Ro-vibrational energy levels of H 2 with internal energies up to ' 22 000 K were considered. The computed data are in very good agreement with the available experimental data. Significant di↵erences were found with previous TIQM calculations by , mainly due to the neglect of reactive channels in the latter. These new data were used to derive an updated temperature and densitydependent cooling function for H 2 colliding with atomic hydrogen (Coppola et al. 2018). The inclusion of reactive channels in the calculation results in a larger ortho-to-para ratio, which in turn gives rise to a stronger cooling e↵ect at low temperature.
For high temperature applications (T > 5000 K), the only available data (Mandy & Martin 1993) are based on QCT calculations on the Liu-Siegbahn-Truhlar-Horowitz (LSTH) PES of Truhlar & Horowitz (1978). Clearly, rate coe cients for the H + H 2 collision on the recent PES of Mielke et al. (2002) are desirable and would be very useful to the astrophysical community. In this paper, we present QCT calculations of rate coe cients for the ro-vibrational (de-)excitation of H 2 by H-including the ortho-para conversion and the three-body collisional dissociation-for temperatures up to 15 000 K, and for ro-vibrational energy levels of H 2 up to ' 50 000 K. Our QCT data are benchmarked against the recent TIQM calculations of Lique (2015) in order to assess the accuracy of our QCT calculations, normally expected to improve with increasing temperature. The paper is organized as follows. Section 2 provides a brief description of the scattering calculations. In Section 3, we present the results. Possible astrophysical consequences are drawn in Section 4.

QUASI-CLASSICAL TRAJECTORY METHOD
Standard methods for performing quasi-classical trajectory calculations have been reviewed by Karplus et al. (1965) and Truhlar & Muckerman (1979). Here we simply give a short overview of the QCT method with, however, some details relative to the trajectory analysis. The following processes are considered: where v and j are the vibrational and rotational quantum numbers, respectively, of the H 2 target molecule. We assume that the Born-Oppenheimer approximation (Messiah (1995)) is valid, so that only the ground state PES of H 3 is considered in the present calculations. The twelve coupled Hamilton equations of motion in Jacobi coordinates are propagated by means of a standard Runge-Kutta integrator with adaptive time stepsize (Press et al. (1992)). Space derivatives are computed numerically by means of a finite di↵erence formula of order three. The local truncation error on the coordinates and momenta is monitored to achieve a relative precision better than 10 6 in the conservation of the total energy, and the total angular momentum. Preliminary to the trajectory propagation, the quantum bound vibrational states of the free H 2 molecule are computed by means of the Fourier-Grid-Hamiltonian (FGH) method of Balint-Kurti (Marston & Balint-Kurti (1989)). Note that this has to be done for each rotational state j of H 2 , since the e↵ective potential interaction of H 2 , V e↵ , includes a centrifugal term due to the rotational motion of the molecule: where r is the H-H internuclear distance, V (r) is the potential energy curve of H 2 , and m is its reduced mass.
In the trajectory calculation, the initial ro-vibrational energy ✏ v, j obtained from the FGH calculation, as well as the initial collision energy, are fixed. All other collision parameters are selected at random according to their probability distributions (Truhlar & Muckerman (1979)). In particular, the impact parameter b is ascribed the usual b 2 distribution between zero and a maximum value, which is su ciently large that no reaction takes place beyond it. Moreover, the maximum value of b was found to be strongly dependent on the collision energy E c , so that the maximum impact parameter function b max (E c ) had to be determined, prior to the trajectory calculation, by running small batches of trajectories at selected collision energies. Values of b max for a particular value of E c are then obtained by linear interpolation of b max (E c ).
We ran batches of 80 000 to 110 000 trajectories per energy for 20 collision energies between E thr esh = 0.1 eV (equivalent to 1160 K) and E max = 2.7 eV (equivalent to 31 330 K) for each initial ro-vibrational state (v, j) with internal energies up to 4.3 eV (equivalent to 50 000 K). The highest internal energy levels (v, j) are: (0,30), (1,29), (2,26), (3,25), (4,24), (5,22), (6,20), (7,18), (8,16), (9,14), (10,11), (11,8), (12,4), with internal energies less than or equal to 50 000K. A total of 260 initial states were taken into account in the calculations. Each individual trajectory is propagated in time until any two of the three internuclear distances get larger than a "collision shell" distance ⇢ 0 = 25a 0 . The remaining H-H fragment with the smallest internuclear separation may be a stable molecule (resulting from an inelastic or reactive collision), a quasibound molecule (trapped behind its centrifugal barrier), or an unbound molecule still evolving toward dissociation; analysis of this fragment's internal energy and angular momentum allows a definitive assignment. In the present calculations, quasibound H 2 molecules are counted as pertaining to the three-body dissociative channel. Around 5000 K they contribute up to 50% of the dissociative channel, and their contribution decreases with temperature to reach 30% at 15 000 K. The above assumption remains valid as long as the quasibound molecules have a life-time lower than the typical collision time. In high density media, collisions are more frequent and a part of the quasibound molecules may interact before dissociating.
Once a trajectory has met the above end-of-trajectory criterion that corresponds to a bound H-H fragment, continuous "pseudo" quantum numbers for this diatom, which will be marked with a tilde sign, are derived from the trajectory's final positions and momenta. The rotational quantum numberj 0 is taken as the root of the equation: where J 0 int is the internal angular momentum of the H 2 product. The vibrational quantum numberṽ 0 is obtained through the semiclassical Wentzel-Kramers-Brillouin (WKB) quantization condition (Berry & Mount (1972)): 2mr 2 3 7 7 7 7 5 9 > = > ; where r and r + are the left and right turning points, respectively, relative to the classical internal energy of the relevant H 2 fragment, noted ✏ 0 int (Truhlar & Muckerman (1979)). Once the above continuous pseudo-quantum numbers (ṽ 0 ,j 0 ) have been determined, they are rounded to the nearest integers through a "binning" procedure (histogram method) (Truhlar & Muckerman (1979)), and the corresponding trajectory is attributed to the bin labeled by the couple of discrete quantum numbers (v 0 , j 0 ). Notice that this method has to be adapted to account for the fact that orthopara conversion of H 2 is forbidden in purely inelastic collisions. Then, a j = 2 parity rule is enforced through the binning procedure, i.e., the continuousj 0 is rounded to the nearest integer j 0 of same parity as j. In H + H 2 reactive collisions, ortho-para conversion is allowed, and the regular binning process described above is applied.
Finally, after being analysed as described above, all trajectories are accumulated in the histogram. The trajectory count in the histogram bin labeled (v 0 , j 0 ), denoted N r (v 0 , j 0 ; E c ), allows us to determine the state-to-state cross section for the reaction: where N tot (v, j; E c ) is the total number of trajectories started in the ro-vibrational state (v, j) and propagated with the collision energy E c . The estimated statistical error on this cross section is given by (Truhlar & Muckerman (1979) In this study, the average error on the cross section is about 10% for the less populated states and it is below 3% for the most populated states.
The state-to-state rate constant is given by: where k B is Boltzmann's constant, µ is the reduced mass for the H-H 2 motion, and where the integration was carried out numerically.

Comparison with previous quantum results
It is first important to estimate the accuracy and validity domain of the QCT results. To this aim, we compare the new QCT rate coe cients with the TIQM one of Lique (2015). We compare QCT and TIQM results for collisional (de-)excitation of H 2 by H. The upper panels of Fig. 1 shows the QCT and TIQM results of the pure rotational de-excitation of H 2 (v, j) induced by collisions with H. The ro-vibrational (de-)excitation is studied on the lower panels of Fig. 1. As anticipated, significant deviations occur at low temperatures because of the inability of QCT treatments to conserve the vibrational zero-point energy that renders this method unreliable near reaction thresholds. However, at higher temperatures, above a few hundreds of Kelvins, the QCT approach is relatively accurate for both rotational and ro-vibrational transitions, the mean deviations being of the order of a factor 3 and tend to decrease with increasing temperatures, as expected. Fig. 2 presents a comparison between the QCT and TIQM results of Lique (2015) at 5000 K for all (de-)excitation transitions between levels with internal energy lower than 22 000 K. One can observe a global agreement within a factor of 3 between the two sets of data. It is interesting to note that the agreement improves with increasing magnitude of the data. It means that the accuracy of the rate coe cients for the dominant transitions will be relatively high. The QCT rate coe cients are then closed to the values computed by Lique (2015), and with a better agreement for the inelastic channel than for the reactive one where quantum e↵ects are not negligible, especially at low and intermediate collisional energy.
The good agreement between the previous quantum and the new QCT rate coe cients allows us to be confident in the accuracy of the new data that will be provided for high temperatures and highly excited ro-vibrational levels.

Collisional dissociation process vs. inelastic and reactive processes
As mentioned above, the three body collisional dissociation process has been included in the calculations. Such process is usually neglected in TIQM calculations and in some of the previous QCT calculations whereas it is expected to be the dominant channel at high temperature and for ro-vibrational levels highly excited. The respective importance of inelastic and reactive collisions and of the dissociation process is investigated on Fig. 3. As can be seen, pure rotational excitations 2 are principally induced by the purely inelastic process, the e↵ect being even enhanced with increasing temperature. For rovibrational excitation, the behavior is relatively similar. However, the inelastic process is less dominant than for pure rotational transition. As expected while temperature increases above a thousand of Kelvins, the excitation process due to reactive collisions decreases because of the dissociative process that becomes dominant. For temperature above ⇠ 3000 K, there is a clear competition between collisional excitation through the reactive process and dissociation and for temperature above ⇠ 5000 K, the collisional dissociation becomes the dominant process. Hence, collisional data obtained neglecting such process may be doubtful for high temperature application. It is worth mentioning that the importance of dissociation process increases with increasing energy of the H 2 ro-vibrational level, with di↵erences up to 4 orders of magnitude for the rate coe cients between the lowest and the highest ro-vibrational state. For the lowest vibrational level v = 0, the rate coecients for both nuclear spin conversion and conversion transitions increase rapidly with increasing temperatures up to 5000 K and then reach a plateau up to 15 000 K. As no-ticed by Lique (2015), nuclear spin conversion transitions are slower than those corresponding to nuclear spin conservation because of a significant activation barrier in the PES that inhibit the hydrogen exchange process that lead to nuclear spin conversion process.

Pure rotational (de-)excitation
While increasing the vibrational level, there is an increase of the pure rotational rate coe cients (for both nuclear spin conservation and conversion transitions) as already found in previous TIQM calculations Lique 2015) and can be explained by the anisotropy of the PES that is is strongly dependent on the vibration of the diatomic molecule (Lique et al. 2014). However, for a given rotational transition, we note that the rotational rate coe cients for the di↵erent vibrational levels tend to converge at high temperatures. The rate coe cients for nuclear spin conservation transitions reach a plateau at lower temperatures than for the vibrational level v = 0 as can be easily seen in Fig. 4.
The noticeable point is the decrease of the rate coefficients for p-o-H 2 and o-p-H 2 conversion process for high  temperatures. Such e↵ect can be explained by the fact that the only reactive collisions contribute at such process and that at high temperature, and the kinetic energy of the reaction starts to be exchanged for internal rotational energy leading to formation of H 2 in various highly excited states. Such e↵ect is not or weakly seen in the temperature variation of the rate coe cients for spin conservation process as the dominant contribution come from non-reactive process that favor a weak change in internal energy. As noticed by Lique et al. (2012) the o-p-H 2 conversion process is slower than the inverse conversion process.

Ro-vibrational (de-)excitation
Fig. 5 presents the temperature dependence of the rate coe cients for a given v transition and rotational (de-)excitations. For both the spin conservation and conversion processes, the rate coe cients increase with increasing final rotational state, and rotational excitation is faster than rotational de-excitation. This can be explained by a propensity rules in conserving, as much as possible, the internal energy.
For the spin conversion process corresponding to reactive collisions there is a steep increase at low temperature and a slow regular decrease over temperature as seen and explained on Fig. 4. For the spin conservation process, where both inelastic and reactive collisions are involved, the rate (7,7)! (2, 5) (7,7)! (3, 5) (7,7)! (4, 5) (7,7)! (5, 5) (7,7)! (6, 5) (7,7)! (7, 5) coe cients tend to be quite constant above few hundreds of Kelvins. Fig. 6 shows, for a fixed j, the temperature dependence of the rate coe cients for ro-vibrational (de-)excitations conserving the rotational momentum of H 2 . The temperature variation of these rate coe cients is similar to that seen in Fig. 5, with an increase of the rate coe cient with decreasing v transitions. It is interesting to note that the ro-vibrational rate coe cients are smaller in magnitude than that of the pure rotational excitation ones. However, the differences are less than or about an order of magnitude even for large v > 1 transitions and decrease with increasing v. As discussed by Lique (2015), the strong coupling between H 2 vibrational states induced by hydrogen collisions explains this behavior.  Fig. 7 shows the rate coe cients for nuclear spin conservation transitions with di↵erent v = 2. At temperatures below 1000-1500 K, the rate coe cients increase with the initial vibrational level. At the opposite, at higher temperatures, the e↵ect is totally reversed, and the lowest initial vibrational level is having the faster rate coe cients. At high temperatures, the kinetic energy starts to be exchanged for internal energy leading to formation of H 2 in various excited states. As the number of open final ro-vibrational levels increases with increasing vibrational levels, there is a decrease of the magnitude of the rate coe cients with increasing initial vibrational levels.
The unusual behavior of the rate coe cients (i.e. a decrease and then an increase of the magnitude of the rate coefficients) for the (4, 0) ! (2, 0) (also seen for other transitions) can be understood on the basis of the respective contribution of the inelastic and reactive processes. At low temperature, as discussed in Lique (2015), the ro-vibrational transitions mainly proceed through the reactive exchange process leading to a decrease of the rate coe cients with increasing temperature. At higher temperature above ⇠ 5000 K, the rovibrational relaxation process (for nuclear spin conservation transitions) is again dominated by the non reactive collisions and increase again with increasing temperatures.
We also found that, at high temperatures, the rate coe cients for ro-vibrational de-excitation cross the pure rotational de-excitation rate coe cients which is visible on Fig. 8. It happens above 6000 K for an initial vibrational level v = 1, and the crossing moves toward higher temperatures while increasing the initial vibrational level. Contrarily to the o-p-H 2 and p-o-H 2 conversion processes which decrease while temperature increases because of su↵ering of the increase of channels opening, the rate coe cients of the vibrational de-excitation do not show any evolution going in that sense and there is still a smooth slope while temperature increases. Table 1. Room temperature thermal rate coe cients for H 2 (v = 1)+H! H 2 (v 0 = 0)+H process from experimental and theoretical calculations. Thermal rate coe cients are in units of 10 13 cm 3 s 1 .

Comparison with experiments
Experimental rate coe cients at room temperature for the relaxation of H 2 (v = 1) ! H 2 (v = 0) due to H collisions were provided by Heidner & Kasper (1972). Hence, we compared the thermal rate coe cients at room temperature of our QCT calculations with these data. The thermal rate coe cient for the v = 1 ! v 0 = 0 is computed as follow: where ✏ 1, j is the energy of the v = 1, j rotational levels and w j is a degeneracy weight which is equal to 1 and 3 for paraand ortho-H 2 rotational states, respectively. Results are shown in Tab. 1 together with the rate coe cients computed previously by Lique (2015) using TIQM approach and by Martin & Mandy (1995) and Garcia & Laganà (1986) that both used QCT method.
As could have been anticipated, the present rate coe cients for the relaxation of H 2 (v = 1) to H 2 (v = 0) fail at reproducing the experimental data. The temperature of the experiments (300 K) is out of the accuracy domain of the QCT calculations. However, the agreement between the new QCT results using the state of the art H 3 PES of Mielke et al. (2002) and both the experimental and TIQM data is much better than what was found with previous QCT calculations of Martin & Mandy (1995) and Garcia & Laganà (1986).

DISCUSSIONS AND CONCLUSIONS
We have presented the calculations of new high temperature state specific rate coe cients for the collisional excitation of H 2 by H using QCT method. The accuracy of the present results have been validated through a detailed comparison with TIQM calculations and experiments. In particular, the new collisional data are closer to experimental data than the previous QCT calculations due to the high accuracy of the PES used in this work.
We have found that, because of the competition between inelastic, reaction and dissociative processes, the temperature variation of the rate coe cients is not smooth and strongly dependent of the ro-vibrational level of H 2 . Such findings show the evidence that actual calculations at high temperatures are preferable than extrapolations of low to moderate temperatures data for astrophysical applications where the temperature domain is around 5000 K to 15 000 K as in the early Universe.
As discussed in the introduction, in the early universe, the H 2 cooling function is computed from H 2 radiative and collisional data, the main colliding partner being generally H atoms (Galli & Palla 1998). Hence, H 2 -H collisions dominate the cooling so that the cooling function is almost proportional to the H 2 -H rate coe cients. The new rate coefficients, the first one computed for such high temperatures, will allow accurate modeling the cooling of the warm primordial gas and allow a significant revision of the cooling of the gas at temperature between 5000 and 15 000 K. As the very first stars are believed to have formed thanks to H 2 cooling, our knowledge of star formation should be enhanced by the use of these new data.
Apart from in the early Universe, H 2 is also an important coolant in shocks and photon dominated regions of the interstellar medium because of its high abundance. In photon dominated regions, evidence of highly excited H 2 have been found and the new data will be crucial for modelling the kinetic and thermal evolution of these regions.
Finally, this high temperature study showed some interesting results, making obvious that a collisional excitation study induced by hydrogen atoms in this temperature range for the other dominant molecules in the early Universe such as HD or H + 2 is required in order to update the present data used.