Rovibrational excitation of HD in collisions with atomic and molecular hydrogen

Abstract

We have computed cross-sections and rate coefficients for rovibrational transitions in HD, induced by collisions with atomic and molecular hydrogen. We employed fully quantum-mechanical methods and the potential of Boothroyd et al. for H—HD, and that of Schwenke for H₂-HD. The rate coefficients for vibrational relaxation v=1→0 of HD are compared with the corresponding values for H₂. The influence of vibrationally excited channels on the rate coefficients for rotational transitions within the v=0 vibrational ground state of HD is shown to be small at T=500 K, where T is the kinetic temperature. The rate coefficients, for 100T2000 K, are available from http://ccp7.dur.ac.uk/.

molecular processes, ISM: clouds, early Universe

1 Introduction

In dark interstellar clouds, deuterium, like hydrogen, is mainly in molecular form, and the ratio of densities n(HD)/n(H₂)≈2n_D/n_H (Millar, Bennett & Herbst 1989; Pineau des Forêts, Roueff & Flower 1989), where n_D/n_H≈10⁻⁵ is the ‘cosmic’ value of the elemental abundance ratio (Vidal-Madjar 1983; Bertoldi et al. 1999). However, in diffuse clouds, where H and D are predominantly atomic, significant chemical fractionation can occur, owing to the reaction D⁺(H₂, HD)H⁺, which is endoergic (by 464 K; energies are expressed in kelvins through division by Boltzmann's constant) in the reverse direction. Thus, the HD/H₂ ratio can be enhanced in diffuse clouds by factors of 100 or more (see Dalgarno 1975; Watson 1975). Similar chemical enhancements of the HD/H₂ ratio are believed to have occurred in the primordial gas, in which non-negligible amounts of H₂ and HD were present (cf. Galli & Palla 1998). The smaller rotational constant of HD, relative to H₂, and its finite dipole moment then lead to HD being a significant coolant, at kinetic temperatures T≈100 K, both of diffuse interstellar gas (Dalgarno & McCray 1972) and of the primordial gas (cf. Galli & Palla 1998). Rate coefficients for the rotational excitation of HD by He (Roueff & Zeippen 1999), by H (Roueff & Flower 1999) and by H₂ (Flower 1999) have been computed recently. The first and last of these studies extend and complement earlier work by Schaefer (1990).

Whilst vibrationally excited H₂ has been extensively observed, notably in the K band at 2.12 μm, vibrationally excited HD has yet to be detected. Timmermann (1996) calculated the intensities of transitions in the 1-0 fundamental band of HD from C-type shock waves propagating in the interstellar medium. Owing to the incompleteness of both the experimental and theoretical data pertaining to the vibrational excitation of HD, Timmermann scaled (by a factor of 2) the rate coefficients for the vibrational excitation of H₂, taken from the compilation of Draine, Roberge & Dalgarno (1983). The assumed scaling factor remains to be confirmed, but, in any case, the data compiled by Draine et al. (1983), relating to the excitation of H₂, have been superseded by more recent calculations (see Le Bourlot, Pineau des Forêts & Flower 1999). Clearly, if reliable predictions of the intensities of the transitions of HD are to be made, the requisite collisional data need to be properly calculated.

In the present paper, we give the results of what we believe to be the first theoretical study of the rovibrational excitation of HD by atomic and molecular hydrogen. The equivalent calculations for the excitation of HD by He will be presented elsewhere (Roueff & Zeippen, in preparation). Section 2 summarizes the theoretical methods that have been used, and the results are given in Section 3.

2 Theoretical Methods

Our theoretical approach to the problems of rovibrationally inelastic, non-reactive scattering of H and H₂ on HD follows closely that in our recent studies of H—H₂ (Flower & Roueff 1998a) and H₂-H₂ (Flower & Roueff 1998b, 1999) collisions. In particular, the same interaction potentials (Schwenke 1988; Boothroyd et al. 1996) have been employed, together with a fully quantum-mechanical treatment of the collision processes, using the molcol program of Flower & Launay (in preparation). Important differences arise, in practice, from the displacement of the centre of mass of the HD molecule from the mid-point of the internuclear line, which results in terms of both even and odd symmetry being present in the Legendre polynomial/spherical harmonic expansions of the interaction potentials (cf. Roueff & Flower 1999; Flower 1999). Furthermore, as HD does not possess the ortho/para symmetry of (homonuclear) H₂, rotational states of both even and odd J have to be considered simultaneously. This fact, together with the smaller rotational (and vibrational) constant of HD, leads to a large increase in the number of open channels at a given collision energy and hence in the time required to perform the quantal coupled channels calculations. Partly for this reason, we have restricted the HD basis set to the rovibrational states v2, J9, i.e. to a total of 30 levels. Their eigenenergies were taken from the experimental work of Dabrowski & Herzberg (1976), complemented by the theoretical values of Abgrall, Roueff & Viala (1982). In the case of excitation of HD by H₂, only the ground rotational states of para-H₂ (J=0) and of ortho-H₂ (J=1) were included in the bases. Previous work on the rotational excitation of HD by H₂ (Schaefer 1990; Flower 1999) suggests that these restrictions do not significantly compromise the results. Rate coefficients q(T)=〈σv〉 for collisional de-excitation were obtained by numerically averaging the product of the cross-section σ and the barycentric collision velocity v over a Maxwellian velocity distribution at kinetic temperature T. The rate coefficients for excitation were obtained from detailed balance.

3 Results

We shall not attempt to present the results comprehensively in this paper. The rate coefficients for rovibrational transitions in HD, induced by H and by para-H₂ (J=0), are available from the CCP7 server (http://ccp7.dur.ac.uk/), or directly from [email protected], for temperatures 100T2000 K, in steps of 20 K. Cross-sections for collisions with ortho-H₂ (J=1) were found to be very similar (generally within 10 per cent) to those for para-H₂ (J=0); this similarity has already been noted by Flower (1999), in the context of rotational transitions within the vibrational ground state, v=0.

First, we compare the rate coefficients for vibrational relaxation v=1→0 of HD and H₂. In Table 1 are listed the relaxation rate coefficients for ortho- and para-H₂ in H (Flower & Roueff 1998a), together with the present results for HD in H. In Table 2 are the results of Flower & Roueff (1998b) for ortho- and para-H₂ in H₂ (J=0) and the present results for HD in H₂ (J=0). Representative kinetic temperatures in the range 100T2000 K are considered. When calculating the rate coefficients, it is assumed that the rotational levels within the v=1 manifold are populated thermally, i.e. that their populations, relative to the lowest level of the manifold, are given by a Boltzmann distribution. Summations are then performed over the initial rotational levels, of the v=1 manifold, and the final rotational levels, of v=0.

Table 1.

Values of the rate coefficients for v=1→0 vibrational relaxation of HD (present work) and of H₂ (Flower & Roueff 1998a) in H. The same interaction potential (Boothroyd et al. 1996) was used in both calculations. Numbers in parentheses are powers of 10.

Table 2.

Values of the rate coefficients for v=1→0 vibrational relaxation of HD (present work) and of H₂ (Flower & Roueff 1998b) in para-H₂ (J=0). The same interaction potential (Schwenke 1988) was used in both calculations. Numbers in parentheses are powers of 10.

Tables 1 and 2 show that the rate coefficients for vibrational relaxation of HD and of H₂, calculated at a given temperature, are similar in magnitude; this is especially true of relaxation in collisions with H (Table 1). In the case of collisions with H₂ (J=0), the scaling factor (of 2) adopted by Timmermann (1996), when deriving rate coefficients for HD from those for H₂, is seen to be approximately correct. Of course, the vibrational relaxation rate coefficient is a summed and averaged quantity, and the use of the same scaling factor for all rovibrational transitions remains a crude, though at that time unavoidable, approximation. Comparing results at a given value of T in Tables 1 and 2, we see that the rate coefficients for vibrational relaxation in collisions with H are much larger than for collisions with H₂. Of course, the relative importance of these two processes also depends on the H/H₂ density ratio. Owing to its permanent dipole moment, transitions |J−J′|=1 dominate collisional population transfer in HD. In Table 3, we compare our present and previous calculations of the rate coefficients for collision-induced dipole transitions within the vibrational ground state of HD, at T=500 K. In the case of H—HD scattering, the present results differ from those of Roueff & Flower (1999) only by the inclusion of vibrationally excited states, v=1 and 2, in the HD basis. On the other hand, in the case of para-H₂-HD scattering, the present results differ from those of Flower (1999) not only by the inclusion of vibrationally excited states of HD but also by the neglect of the first rotationally excited level, J=2, of para-H₂, which was included in the calculations of Flower (1999). The results in Table 3 confirm that the effects of the vibrationally excited states on the rate coefficients at T=500 K are small, even for rotational levels as high as J=8. Differences between the present and previous calculations are larger for para-H₂-HD scattering, where they attain about 5 per cent. We attribute these differences mainly to the influence of the J=2 state of para-H₂, rather than the vibrationally excited states of HD.

Table 3.

Rate coefficients for rotational de-excitation, J−J′, in HD, calculated at T=500 K. Results are given for de-excitation in collisions with H and with para-H₂. For H—HD, the first entry is the result obtained by Roueff & Flower (1999) and the second entry is the present result. For para-H₂-HD, the first entry is from Flower (1999) and the second entry is the present result. Numbers in parentheses are powers of 10.

Acknowledgments

One of the authors (DRF) wishes to thank the University of Durham for the award of a research fellowship.