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.

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.