We report fully-quantum time-independent calculations of cross sections and rate coefficients for the collisional excitation and dissociation of D2 by H, two astrophysically relevant processes. Our calculations are based on the recent H3 global potential energy surface of Mielke et al [J. Chem. Phys.116, 4142 (2002)

]. Results of exact three-dimensional calculations, i.e., including the reactive channels, are compared to pure inelastic two-dimensional calculations based on the rigid rotor approximation. A reasonable agreement is found between the two sets of inelastic cross sections over the whole energy range 10–9000 cm−1. At the highest collisional energies, where the reactive channels are significant, the rigid rotor approach slightly overestimates the cross sections, as expected. At moderate collisional energies, however, the opposite behaviour is observed. The rigid rotor approach is found to be reliable at temperatures below ∼500 K, with a significant but moderate contribution from reactive channels

Along with the F+H2 reaction, the H+H2 reaction (and its isotopic analogs) has became a prototype for triatomic reactions. Because of its theoretical and experimental accessibility, the reaction has been the object of many detailed theoretical and experimental studies, allowing a very accurate understanding of the dynamics (see Ref. 1 for a review). In contrast, the corresponding inelastic processes have received less attention. It is even more so for the H–D2 inelastic collisions that have been the object of very few studies. Indeed, the vibrational excitation of D2 by H has been the object of little experimental and theoretical works2–4 and no recent studies on the pure rotational excitation of D2 by H has been performed.

The rotationally inelastic scattering of D2 by H has been actually studied almost half a century ago by Dalgarno et al.5 and by McGuire and Krüger.6 These studies, as most of the work on the rotational excitation of HD by H (Ref. 7) were done using the rigid rotor approximation and neglecting the reactive channels. The goal of these studies was the calculation of rotationally inelastic rate coefficients for astrophysical applications at temperatures below 1500 K. They used the rigid rotor approach since the reaction between H and HD/D2 is inhibited by a large barrier (∼5000 K), and the reactive rate constants are expected to be very small at low temperatures.

The determination of molecular abundances in the interstellar medium from spectral line data requires collisional rate coefficients with the most abundant species, i.e., H, He, or H2 because collisions compete with radiative processes in altering populations in molecular ro-vibrational levels. Collisions between closed-shell interstellar species and He or H2 are generally non-reactive collisions but most of the collisions with H atoms or between open-shell molecules and H2 molecules are reactive collisions. In some cases, the reaction proceeds through a large barrier and the reactive pathways may be neglected. However, to the best of our knowledge, this common approximation has never been tested in detail, i.e., at the state-to-state level.

In this paper, we study the rotational excitation of D2 by H using an exact quantum three-dimensional (3D) approach including the reactive channels, and the results are compared to rigid rotor calculations. The choice of the H–D2 collisional system is governed by the relative simplicity of the system that will allow detailed comparisons, and also by the possibility to easily distinguish the reactive and inelastic processes, unlike the H–H2 collisional system. The H–D2 system is also astrophysically relevant since the deuterated isotopologues of H2 play an important role both in the interstellar medium8 and in the early Universe.9 We will try below to determine the validity of neglecting the reactive channels in the study of inelastic collisions for a reactive (through a barrier) system. The paper is organized as follows: Section II provides a brief description of the employed potential energy surface and of the scattering calculations. In Sec. III, we present the results. Concluding remarks are drawn in Sec. IV.

In our investigation of the scattering dynamics, we use the H3 global potential energy surface (PES) of Mielke et al.10 The ab initio H3 PES has been calculated at the full configuration interaction level using a complete basis set extrapolation. This PES proved to be very successful in the calculation of H+H2 reactive rate coefficients.11 

We used the ABC reactive scattering code12 to carry out inelastic and reactive calculations for all values of the total angular momentum that made a non-zero contribution to elastic, inelastic, and reactive processes on a grid of energies (N = 170), ranging from 10 to ≃ 9000 cm−1. In each arrangement (H+D2 or HD+D) all rotational and vibrational levels of the diatomic moiety were included subject to the exclusion of (a) levels with internal energy greater than ≃ 16 000 cm−1 and (b) levels in which the angular momentum j of the diatomic moiety (either D2 or HD) is greater than 15. Converged integral reactive cross sections could be obtained by restricting the projection quantum number of the total angular momentum to K ⩽ 6. In all calculations, the integration range was divided into 150 sectors spanning a range in the hyperradius ρ from 0.8 to 15 a0. We refer hereafter to these calculations as 3D calculations. The calculations of reactive cross sections and rate coefficients for the H+D2 reaction have been already performed in many studies. An excellent agreement between theoretical and experimental results was generally found in Refs. 11 and 13 and we do not further discuss these results in the present paper.

We also performed calculations for pure rotational excitation of D2 by H using the rigid rotor approximation. We have chosen for the D2 internuclear separation

$r_{D_2}= 1.435$
rD2=1.435 a0, the ground state vibrationally averaged value. The standard time-independent coupled scattering equations were solved using the MOLSCAT code.14 Calculations were carried out at values of the total energy ranging from 10 to 9000 cm−1. The integration parameters were chosen to ensure convergence of the cross sections. The D2 rotational basis was extended to j = 16 to ensure convergence of the inelastic cross sections for j, j ⩽ 10. The maximum value of the total angular momentum J used in the calculations was set large enough that the elastic and inelastic cross sections were converged to within 0.1 Å2 and 0.005 Å2, respectively. We refer hereafter to these calculations as two-dimensional (2D) calculations

For both calculations, the coupled equations have been solved using the close-coupling approach without any approximations. From the calculated inelastic cross sections

$\sigma _{j \rightarrow j^{\prime }} (E_{c})$
σjj(Ec)⁠, one can obtain the corresponding thermal rate coefficients at temperature TK by an average over the collision energy (Ec),

\begin{eqnarray}k_{j \rightarrow j^{\prime }}(T_K) & = & \left(\frac{8}{\pi \mu k_B^3 T_K^3}\right)^{{1}/{2}} \nonumber \\ [6pt]& & \times \int _{0}^{\infty } \sigma _{j \rightarrow j^{\prime }}(E_{c})\, E_{c}\, e^{{-E_{c}}/{k_BT_K}} dE_{c},\end{eqnarray}
kjj(TK)=8πμkB3TK31/2×0σjj(Ec)EceEc/kBTKdEc,
(1)

where μ is the reduced mass and kB is Boltzmann's constant. Calculations up to total energies of 9000 cm−1 allow us to determine rates up to 1500 K.

Figures 1 and 2 display the energy dependence of the calculated integral cross sections for rotational excitation of ortho- (j even) and para-D2 (j odd) by H, respectively.

FIG. 1.

Collision energy dependence of the integral cross section for the rotational excitation of ortho-D2(j = 0) (upper panel) and ortho-D2(j = 2) (lower panel) by H. The solid lines correspond to exact 3D results. The dashed lines correspond to 2D results. The open circles indicate the reactive cross section and the crosses indicate the vibrational excitation [H+D2(v = 0, j) →

$\sum _{j^{\prime }}$
j H+D2(v ⩾ 1, j)] cross section.

FIG. 1.

Collision energy dependence of the integral cross section for the rotational excitation of ortho-D2(j = 0) (upper panel) and ortho-D2(j = 2) (lower panel) by H. The solid lines correspond to exact 3D results. The dashed lines correspond to 2D results. The open circles indicate the reactive cross section and the crosses indicate the vibrational excitation [H+D2(v = 0, j) →

$\sum _{j^{\prime }}$
j H+D2(v ⩾ 1, j)] cross section.

Close modal
FIG. 2.

Same as Fig. 1 for para-D2.

FIG. 2.

Same as Fig. 1 for para-D2.

Close modal

One can see that the 2D approach leads to results in reasonable agreement with those of the 3D approach, especially for the largest cross sections corresponding to Δj = 2. However, some differences exist between the two sets of results. At low collisional energies, close to the thresholds, the 2D and 3D results are in very good agreement (especially for Δj = 2 transitions). At intermediate energies, 2D cross sections are smaller than the 3D ones, whereas at high kinetic energies, the 2D cross sections slightly overestimates the 3D cross sections.

The following two obvious reasons can explain these differences:

  1. The 2D approach does not take into account the vibration of D2 in the scattering calculations.

  2. The 2D approach neglects the reactive channels.

Hence, we plot on Figs. 1 and 2 the reactive and vibrational excitation cross sections. One can clearly see that the reaction competes with the inelastic processes at kinetic energies larger than 3000–5000 cm−1, depending on the initial rotational levels of D2. At these collisional energies, the H and D2 reactants have enough energies to pass the barrier and this simply explains that the 2D approach, that neglect the reactive pathway, overestimates the inelastic process. Interestingly, we note that at intermediate energies the reaction cross section increases with increasing j as already mentioned in Ref. 1. Hence, the validity of the rigid rotor approximation decreases with increasing j.

The ro-vibrational excitation cross sections are significantly lower than that of the rotational excitation and reactive ones over all the energy range considered here. One has to considered very high collisional energies (>10 000 cm−1) to expect a significant influence of this process on the rotational excitation. This confirms results obtained earlier for pure ro-vibrational calculations like the He–CS collisional system.15 

Surprisingly, the largest deviation between the two sets of rotationally inelastic cross sections occurs at moderate energies where the reactive cross sections are negligible. The behaviour with collision energy and the magnitudes of the inelastic cross sections can be related to the radial dependence of the potential expansion coefficients vi(R). Wrathmall and Flower16 found in their study on the rotational excitation of H2 by H, using the PES of Mielke et al.,10 that the cross section for the j = 0 → j = 2 transition significantly decreases at moderate energies due to the shape of the v2(R) expansion coefficient. The present cross sections follow exactly the same behaviour. Wrathmall and Flower16 also found that the minima in the cross sections were lowered and shifted to lower energies when the expansion coefficients were averaged over the vibrational ground state wavefunction (instead of fixing the internuclear separation at its equilibrium value). Hence, the difference between the present 2D and 3D results at moderate energies certainly reflects these intramolecular effects and shows the importance of using 3D potential energy surfaces.

Figure 3 displays the temperature dependence of the rate coefficients for the rotational excitation of ortho- and para-D2 by H, respectively.

FIG. 3.

Temperature dependence of the rate coefficients for the rotational excitation of ortho-D2 (upper panel) and para-D2 (lower panel) by H. The solid lines correspond to exact 3D results, whereas the dashed lines correspond to 2D rigid rotor results.

FIG. 3.

Temperature dependence of the rate coefficients for the rotational excitation of ortho-D2 (upper panel) and para-D2 (lower panel) by H. The solid lines correspond to exact 3D results, whereas the dashed lines correspond to 2D rigid rotor results.

Close modal

One can see that no significant differences exist between the 2D and 3D results at low temperatures (<500 K). At higher temperatures, an agreement within a factor of ∼ 2–3 is obtained for the calculated rate coefficients. This result is not really surprising since the main difference between the 2D and 3D cross sections occurs at moderate kinetic energies. We conclude that the calculation of rotationally inelastic rate coefficients using the 2D approach leads to reasonably accurate results at low temperatures (< 500 K), but at higher temperatures the 3D approach is necessary for an accuracy at the state-to-state level better than a factor of ∼ 3.

We have presented quantum mechanical calculations of cross sections and rate coefficients for the rotational excitation of D2 by H, including the reactive channels (HD+D). The calculations were performed with the recent and accurate H3 potential energy surface of Mielke et al.10 Additional calculations of the rotational excitation of D2 by H within the rigid rotor approximation were performed and the two sets of inelastic results were compared. It was found that a reasonable description of the rotational excitation of D2 by H is obtained using the rigid rotor approach at low temperatures (T <500 K) but at higher temperatures, 3D effects become significant.

From this work, it seems that studies of rotational excitation of interstellar species with H or H2 can be treated using the rigid rotor approach even if the system is reactive, provided that the reaction proceeds through a barrier and that we consider low kinetic energies collisions. The present results have to be confirmed with other systematical studies on reactive systems with barriers and heavy elements such as the rotational excitation of HF or HCl by H.17,18 If the present results are confirmed, they will open the way to many studies of the rotational excitation of reactive interstellar radicals, such as OH and CN by H2, that are urgently needed by the astronomers in order to interpret the recent submillimeter observations of the Herschel Space Observatory.

We acknowledge Mohamed Jorfi for preliminary calculations. F.L. and A.F. acknowledge the CNRS national program “Physique et Chimie du Milieu Interstellaire.” F.L. is grateful for the financial support of the French Commissariat à l’Énergie Atomique and EURATOM, by the Contract No. V3720.001 Reactive collisions in the fusion plasma edge in the frame of Fédération de recherche Fusion Contrôlée Magnétique.

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