Astron. Astrophys. 324, 185-195 (1997)

3. New H₂-H₂ CIA opacities

No quantum mechanical models have been available for H₂-H₂ CIA spectra in the first and second hydrogen overtone bands. Below, we summarize our attempts in making such models, which we applied to our present studies. Our models are applicable at temperatures from 1000 to 7000 K.

For stars with effective temperatures between 3000 and 4000 K, the maximum of the black body radiation is emitted between about 10000 and 14000 cm^-1. The centre of the first overtone CIA band of H₂-H₂ falls around the frequency  8000 cm^-1, and that of the second overtone band, at  12000 cm^-1. The second overtone band is extremely weak compared to the first overtone ( for peak intensity). At the highest temperatures (  7000 K), the peaks of the second overtone band are even smaller than the far wings of the first overtone band at the same frequencies. Thus we have extended the wings of the first overtone band up to 25000 cm^-1, and predict that overtones higher than the second are not important. Since our analytical models are not designed to work well in the far wings (roughly at intensities peak intensity), the uncertainties of the extended wings of the first overtone band may be substantial.

3.1. The first overtone band

Ab initio induced dipole functions (Meyer et al. 1993) and an "effective" isotropic H₂-H₂ potential (Ross et al. 1983) were used as input to produce the lowest three semi-classical spectral moments. These moments were then used to model the H₂-H₂ CIA spectra by the model lineshapes, used successfully to model the low (T 500 K) temperature spectra of this band (Zheng & Borysow 1995a).

At low temperatures only the ground vibrational state, , is populated, but at the high temperatures of importance here, the higher vibrational states are also significantly populated. We account for the initial vibrational states of two interacting hydrogen molecules and 2. In this band, there are two different kinds of vibrational transitions present: (i) single transition, , (or , ); and (ii) double transitions: . Both kinds of vibrational transitions fall in the same frequency region centred around 8000 cm^-1.

Dipolar terms (Meyer et al. 1993) 2023, 0223, 2021, 0221 and 2233 have been included for the single vibrational transition, and = 2023, 0223, 0001, 2021, 0221 and 0445 for the double vibrational transitions. The other terms were found to contribute less than 2% of the total intensity.

The induced dipole functions also depend on the rotational states of the two interacting molecules. We account for such dependencies only for the largest, 2023 and 0223, terms (for both single and double transitions), which contribute more than 70% to the total intensity. Since these dependencies are available only for the lowest three J values (Meyer et al. 1993), we have scaled the entire functions according to their long range asymptotic forms, in the same way as described previously (Zheng & Borysow 1995b):

and

with R being the H₂-H₂ intermolecular distance. We have computed the matrix elements of the electronic polarizability and quadrupole moment of the hydrogen molecule, for all needed J values, based on the functions (Kolos & Wolniewicz 1967), (Poll & Wolniewicz 1978), and the H-H potential (Kolos & Wolniewicz 1965, 1968, 1975), with r being the H-H internuclear distance.

It is difficult to predict the accuracy of our model. Over the frequency range where the spectral intensities are larger than 1% of the peak intensity, the accuracy mainly depends on our selection of the "effective" isotropic H₂-H₂ potential (Ross et al. 1983); the word "effective" is being used to indicate that it effectively accounts for the anisotropy, and the v -dependence of the H₂-H₂ interaction and is expected to describe accurately H₂-H₂ interactions at high temperature. According to the discussion in our previous work (Zheng & Borysow 1995b), we expect the accuracy to be better than 50% at this frequency range. However, the uncertainty becomes less predictable in the far wings where the spectral intensities fall below 1% of those of the peak. Among various reasons we name the use of the model lineshapes, which become less predictable in the far wings, the dependence of the far wings on the dipole functions at short range (largely uncertain), and the use of the effective intermolecular potential rather than the real one. We expect the far wings to be accurate within an order of magnitude. Further work on this band is in progress (Zheng, Fu, Borysow 1996, in preparation).

3.2. The second overtone band

The prediction of the second overtone band H₂-H₂ CIA spectra presents an even greater challenge, due to the lack of detailed knowledge of the induced dipole functions. Vibrational transitions with and were included (each of them falling into the same frequency region centred around 12000 cm^-1), and both interacting hydrogen molecules were assumed to be in the initial ground vibrational state ().

The ab initio dipole functions are not available for this band. Since the = 2023 and 0223 terms contribute more than 70% to the total intensity in the first overtone band, we included only 2023 and 0223 in our model of the second overtone band, considering all other (unknown) terms to be small. The asymptotic form of those dipole functions is well known, and is given by Eqs  1 and 2. However, at high temperatures (i.e. at high collisional energies, probing the short intermolecular distances), the importance of the electronic overlap increases. In order to correct for this unknown short-range behaviour, we have scaled the known dipole functions for the and transitions (of the first overtone band) so that we matched the long-term dipole magnitudes given by Eqs  1 and 2 for the second overtone. Next, we proceeded with the same modelling procedure as we have used for the first overtone.

The assumption of introduces additional inaccuracy, which we consider minor, in view of other uncertainties involved, like the choice of the H₂-H₂ potential, or scaling of the dipole functions. We estimate that the model for this band is accurate within an order of magnitude.

3.3. The combined quantum mechanical CIA data

Figs. 1 and 2 show the total CIA opacities of H₂-He and H₂-H₂ plotted at frequencies from 0 to 20000 cm^-1, at temperatures from 1000 to 7000 K. At lower temperatures separate vibrational bands are identifiable, but with increasing temperature the spectral bands broaden and result in one featureless continuum. Both H₂-H₂ and H₂ -He opacities show the maximum of intensity at frequencies around 5000 cm^-1. We observe a very wide range of intensities over the frequency band from 0 to 20000 cm^-1.

Fig. 1. CIA opacities of H2-H2 at temperatures between 1000 and 7000 K.

Fig. 2. CIA opacities of H2 -He at temperatures between 1000 and 7000 K.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

helpdesk.link@springer.de