next previous
Up: Rovibrational excitation of HD


2 Numerical methods

The computer code MOLSCAT developed by Hutson & Green ([1995]) has been used in this work. Close-coupling quantal collision equations are solved for a rotating harmonic oscillator perturbed by a structureless atom, in this case helium. Different numerical methods are implemented in the code to solve the coupled, second-order differential equations. Among the various possibilities, the hybrid modified LOG-DERIVATIVE/AIRY propagator (Alexander & Manolopoulos [1987]) and the R-MATRIX propagator (Stechel et al. [1978]) have been tried, with mutually consistent results. The potential surface of Muchnik & Russek ([1994]), relative to the H2-He system, is expressed as a function of the distances between the three atoms in the system including a range of H2 internuclear distances adequate to probe the vibrationally excited wave functions, as shown in Flower et al. ([1998]). The potential surfaces of the H2-He and HD-He systems are identical from the adiabatic point of view where nuclei are fixed. As in the previous related studies (Flower et al. [1998]; Flower & Roueff [1999]; Roueff & Zeippen [1999]), the facility provided by the MOLSCAT program to expand the potential in terms of Legendre polynomials was used. Now, contrary to the H2-He case, the Legendre expansion for the HD-He interaction contains odd as well as even contributions of $\lambda$ since the system is not symmetric anymore with respect to the exchange of nuclei within the molecule. In the present collision calculations, terms up to $\lambda$ = 15 have been retained in the potential expansion and vibrational levels up to v = 3 have been included.
  \begin{figure}\par\includegraphics[width=8.8cm,clip]{fig1.eps}\end{figure} Figure 1: Comparison between present (filled points) and rigid rotor results (Roueff & Zeippen [1999]) (open points) for the de-excitation collisional cross-sections as a function of the relative center-of-mass velocity: a) de-excitation from the J=1 level, b) de-excitation from the J = 2 level, c) de-excitation from the J = 3 level


 

 
Table 1: Labels and energies of HD rovibrational levels used in the present expansion basis
J v E(1) E(K)  
0 0 0.00 0.00  
1 0 89.23 128.4  
2 0 267.12 384.3  
3 0 532.32 765.9  
4 0 883.30 1270.7  
5 0 1317.45 1895.4  
6 0 1832.55 2635.8  
7 0 2424.14 3487.5  
8 0 3089.46 4445.3  
0 1 3632.568 5226.7  
1 1 3717.938 5349.6  
9 0 3824.924 5503.5  
2 1 3888.082 5594.4  
3 1 4141.820 5959.5  
4 1 4477.420 6442.3  
10 0 4626.133 6656.3  
5 1 4892.634 7039.8  
6 1 5384.750 7747.8  
11 0 5488.828 7897.6  
7 1 5950.651 8562.1  
12 0 6408.644 9221.1  
8 1 6586.873 9477.5  
0 2 7087.660 10198.1  
1 2 7169.249 10315.5  
9 1 7289.669 10488.7  
2 2 7331.849 10549.4  
13 0 7381.163 10620.4  
3 2 7574.313 10898.3  
4 2 7894.961 11359.7  
10 1 8055.076 11590.0  
5 2 8291.609 11930.4  
14 0 8401.964 12089.2  
6 2 8761.628 12606.7  
11 1 8878.973 12775.5  
7 2 9301.995 13384.2  
15 0 9466.661 13621.1  
12 1 9757.136 14039.0  
8 2 9909.350 14258.1  
0 3 10368.877 14919.3  
1 3 10446.746 15031.3  
16 0 10570.939 15210.0  
9 2 10580.064 15223.1  
2 3 10601.922 15254.6  
13 1 10685.293 15374.5  
3 3 10833.290 15587.5  


The integral cross-sections are obtained by summing the partial cross-sections $\sigma_{\rm J}$ until convergence is reached. A step of 1 is taken for collision energies smaller than 5000 1. Larger step values may be used for higher energies since the partial cross-sections vary smoothly with J. This is fortunate because, as expected, the number of partial cross-sections becomes very large as collision energy increases. The expansion basis is made of the first 45 rovibrational levels in HD whose energies, displayed in Table 1, are taken from Dabrowski & Herzberg ([1976]) and Abgrall et al. ([1982]). Cross-sections for rovibrational transitions were calculated on a grid of barycentric collision energies extending from the threshold of the first rotational level at 89.23 cm-1 up to 100 000 cm-1.


next previous
Up: Rovibrational excitation of HD

Copyright The European Southern Observatory (ESO)