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Appendix A: DCO⁺–H₂ collisional coefficients

Hyperfine-resolved DCO⁺–He rate coefficients have been published recently by Buffa (2012). It is generally assumed that rate coefficients with He multiplied by a factor of 1.4 can provide an estimate of rate coefficients with H₂(J = 0) (Lique et al. 2008). However, for a molecular ion like DCO⁺, this approximation may be invalid, since the electrostatic interaction of an ion with H₂ differs significantly from that with He. We have thus decided to determine hyperfine DCO⁺–H₂ rate coefficients from close coupling (CC) HCO⁺–H₂ rate coefficients () of Flower (1999) using the Infinite order sudden (IOS) approximation described in Faure & Lique (2012). Here, the isotopic substitution of HCO⁺ has been ignored since it is expected to have a little impact on the magnitude of the rate coefficients (Buffa 2012).

In DCO⁺, the coupling between the nuclear spin (I = 1) of the deuterium atom and the molecular rotation results in a weak splitting (Alexander & Dagdigian 1985) of each rotational level J, into three hyperfine levels (except for the J = 0 level which is split into only 1 level). Each hyperfine level is designated by a quantum number F (F = I + J) varying between |I − J| and I + J.

Within the IOS approximation, inelastic rotational rate coefficients can be calculated from the “fundamental” rates (those out of the lowest J = 0 level) as follows (e.g. Corey & McCourt 1983): (A.1)Similarly, IOS rate coefficients among hyperfine structure levels can be obtained from the rate coefficients using the following formula (e.g. Corey & McCourt 1983): (A.2)

where and {   } are respectively the “3 − j” and “6 − j” Wigner symbols.

The IOS approximation is expected to be inaccurate at low temperature. However, it is also expected to correctly predict the relative rates among hyperfine levels within a rotational J → J′ transition. Propensity rules are indeed properly included through the Wigner coefficients. As a result, Neufeld & Green (1994) have suggested to compute the hyperfine rates as (A.3)using the CC rate coefficients k^CC(0 → L) of Flower (1999) for the IOS “fundamental” rates () in Eqs. (A.1) and (A.2).

In addition, the fundamental excitation rates were replaced by the fundamental de-excitation rates using the detailed balance relation: (A.4)This procedure is found to significantly improve the results at low temperature due to important threshold effects.

Then, from the HCO⁺–H₂ rotational rate coefficients of Flower (1999), we have determined the IOS DCO⁺–H₂ hyperfine rate coefficients using the computational scheme described above. The complete set of (de-)excitation rate coefficients is available online from the LAMDA¹ and BASECOL² websites.The present approach has been shown to be accurate, even at low temperature, and has also been shown to induce almost no consequence on the radiative transfer modeling compared to a more exact calculation of the DCO⁺–H₂ rate coefficients (Faure & Lique 2012). However, we note that with the present approach, some hyperfine rate coefficients (those from the J = 1,F = 0 level to a J = F level) are strictly zero. This selection rule is explained by the “3 − j” and “6 − j” Wigner symbols that vanish for these kinds of transitions. Using a more accurate approach, these rate coefficients will not be strictly zero but will generally be smaller than the other rates. In addition, Faure & Lique (2012) have shown that it should imply almost no consequences for astrophysical modeling.

© ESO, 2012