4 Results and discussion

We have obtained energies and wavefunctions of all discrete levels under the dissociation limit H(1s)+H(2s) by solving the Schrödinger coupled equations. We have then calculated all the transition emission probabilities towards the levels of X state, and the emission profile towards the X continuum for values of J up to 25. The amount of data is too considerable to be displayed on a paper form but can be sent on electronic file form if requested. Present data are shown up to J=10.

Tables 1 - 6 display our calculated rovibrational energy level positions measured from the X rovibrational ground state, the total emission probabilities towards X rovibrational states, the total dissociating emission and the mean kinetic enegy $\bar{E}_{k}$ of the emergent H atoms. These quantities are obtained by summation of individual lines and profiles as indicated in Eqs. (3)-(5). In order to compare easily with the calculations without rotation and coupling of Stephens & Dalgarno ([1973]), $\bar{E}_{k}$ is in unit of electron-volt.

As the wavefunctions result from a superposition of B.O. states, we have indicated the fractions $\rho({\rm B})$, $\rho({\rm C})$, $\rho({\rm B'})$, $\rho({\rm D})$ of the B.O. states for each rovibrational level, as defined in Sect. 2. These fractions are significantly linked to the physical properties of the included B.O. states. In most cases one $\rho$is dominant and it is justified to keep the classification B, C, B'D, and to label with the B.O. state of greatest weight. However for some very few rovibrational states the fraction is close to 0.5 and this classification is arbitrary, (see for example B(v=21, J=4) in Table 1).

The C and D states have $\Pi$ symmetry, and the energy degeneracy is removed by rotational coupling. We give the results for $\Pi ^{+}$ and $\Pi ^{-}$ into separate tables.

For the + manifold with J > 0 , B, C⁺, B' and D⁺ states are coupled through rotational coupling and results are obtained by solving a system of 4 equations coupled by a strong rotational coupling (B with C⁺, D⁺ and B' with C⁺, D⁺) and a weak radial coupling (B with B' and C⁺ with D⁺). In the case of J=0, only the radial coupling (B with B') remains, and the system involves 2 equations.

A weak radial coupling (C^- with D^-) is involved in the - manifold and the system involves then 2 coupled equations. We included the different values of $\rho({\rm C})$ and $\rho({\rm D})$ in the corresponding table. Stephens & Dalgarno ([1972], [1973]) calculated the total dissociation probabilities and the kinetic energy released in the dissociation for B and C but neglected all rotational effects and radial couplings. Significant differences are obtained for the lower vibrational levels with J=1 of the C⁺ state, due to the fact that the dissociating branch of the C B.O. state is negligible and the contamination with B is large enough to increase the dissociating branch, giving about the same energy profile as the closer B level. Interference effects which are important for individual spectral lines do not appear in the summed quantities of our tables.

4.1 The case of D state

The D B.O. state dissociates into H(1s)+H(3s). Then all the rovibrational states above the H(1s) + H(2s) limit are predissociated by the B, C and B' continua and cannot be calculated by our method. D⁺ is significantly predissociated by rotational coupling with B' and the corresponding levels are not observed in emission.

D^- undergoes only a weak radial coupling with C^- and the predissociation lifetime is sufficiently large to allow an emission line towards the ground state. It is a good approximation to neglect this coupling. Therefore we have solved the Schrödinger equation with the diagonal term only i.e. the D^- potential as in Abgrall et al. ([1994]). The effect of the different values of the $M_{{\rm DX}}$ transition moment is shown in Table 7 for the line emitted from the D^- (v'=3, J'=1) level. Apart from the very small values of the emission probabilities towards upper v'' X vibrational levels, the results stay within 10%. The weak values are obtained by destructive integration of oscillating functions and any small difference in the shapes or inaccuracy become significantly amplified. When the wavefunctions overlap constructively, the accuracy is expected to be better than 1%. Branchett & Tennyson ([1992]) have calculated the electronic transition moment with the R-Matrix method but used less configurations than Drira ([1999]). Glass-Maujean ([1984]) has used a dipole moment taken from Rothenberg & Davidson ([1967]), with an estimated uncertainty of 10%, and an adiabatic potential built from Kolos & Rychlewski ([1981]) and Lewis-Ford et al. ([1977]). The comparison with the measured lifetime of the D^- levels for J'=1by Glass-Maujean et al. ([1984]) is however very satisfactory.

We compare in Table 8 the total emission probabilities, the total dissociating emission probabilities and the mean kinetic energy $\bar{E}_{k}$ released in the dissociated products obtained with the recent accurate values of Drira ([1999]) and those of Rothenberg & Davidson ([1967]). The two sets of results differ by about 10%. Nevertheless when the dissociative branch is very weak, the two calculations display important differences for the same reason as explained above. However we must note that it is not very important to know $\bar{E}_{k}$ when the dissociative branch is insignificant. In Table 8 we have also displayed the energy levels, they differ from Table 6 by a weak shift due to the small radial coupling.