2 Target representations

Table 1 lists the 32 electron configurations that are used to represent the target states for the scattering problem.

 

Table 1: Target description for P  III, S  IV and Cl  V: 32 configurations with closed shells 1s²2s²2p⁶

  

Figure 1: The term structure of Cl V. The horizontal dashed line indicates the cutoff of the CC expansion. The number of terms in each electron configuration is given in brackets

A schematic diagram of the term structure of Cl V is given in Fig. 1, where the dashed line indicates the cutoff in the close-coupling expansion for this ion. As the effective charge of the ions in the series decreases, the separation in energy between states of the n=3 complex and the 3s²nl Rydberg states decreases so that for P III the close-coupling expansion needs to be extended to include terms 3s²5s ²S and 3s²5p ²P, which lie among the terms of the configuration 3s3p3d. The cutoff of the close-coupling expansion for the three ions, P III, S IV and Cl V, was such that respectively 22, 21 and 21 target states were included, as shown in Tables 2, 3 and 4.

The program SUPERSTRUCTURE by Eissner et al. (1974) as modified by Nussbaumer & Storey (1978) was used to construct the target wavefunctions. The scaling parameters were optimised in two stages, first, the sum of the energies of all configurations was minimised, and in the second stage, the potential parameters for orbitals with principal quantum numbers n=4 and 5 were kept fixed whilst the others were optimised so as to minimise the energies of all the target terms that were to be retained in the CC expansion. This method ensured that the n=4 and 5 valence electron states were not dragged into the core by the optimisation process. Whilst this optimisation was carried out in LS coupling the contributions of the mass and Darwin relativistic operators were included at all further stages of the calculation.

 

Table 2: Target energies ($E_{\rm target}$) for all terms included in the CC expansion for P  III. Columns 1 to 3 are from Martin et al. (1985) but ($E_{\rm exp}$) are the centres of gravity of the experimental energies. Calculated energies ($E_{\rm calc}$) include the mass and Darwin type relativistic contributions. The abbreviations ¹P$^{\rm o}$and ³P$^{\rm o}$stand for 3s3p(¹P$^{\rm o}$) and 3s3p(³P$^{\rm o}$) respectively

The term systems of P III and S IV are well known (Martin et al. 1985) so that the calculated energies could be checked against experiment. The analysis is less complete for Cl V. The term structure of these ions shows some interesting configuration interaction. Terms of configurations 3p³ and 3s3p3d are strongly mixed. This led to the configuration assignment 3s3p3d to the lowest term of symmetry ²D$^{\rm o}$ and the third term ²P$^{\rm o}$ in P III (Magnusson & Zetterberg 1977). In S IV the pattern is further complicated by terms of configurations 3s²4l appearing among those of the n=3 complex (Reistad & Engström 1989). In Tables 2 to 4 we give the centres of gravity of the experimental term energies ($E_{\rm exp}$) and the energies that were actually used in the scattering calculation ($E_{\rm target}$). The latter are as close as practicable to the experimental values without changing the calculated order of the target terms. $E_{\rm exp}$ and $E_{\rm target}$ differ only for a few of the higher terms. But it is evident from the tables that the term structure of P III is much more difficult to represent than that of the more highly ionised systems. For some terms of Cl V, which we have indicated with an "*", term energies are not available experimentally and the corresponding $E_{\rm target}$ were obtained by interpolation.

 

Table 3: Target energies ($E_{\rm target}$) for all terms included in the CC expansion for S IV. Columns 1 to 3 are from Martin et al. (1990) but ($E_{\rm exp}$) are the centres of gravity of the experimental energies. Calculated energies ($E_{\rm calc}$) include the mass and Darwin type relativistic contributions. The abbreviations ¹P$^{\rm o}$and ³P$^{\rm o}$stand for 3s3p(¹P$^{\rm o}$) and 3s3p(³P$^{\rm o}$) respectively

,Martin et al. (1985, 1990) ask for further calculations of mixing coefficients. They give leading percentages alongside their term table. Therefore we give in the last Cols. of Tables 2 to 4 the theoretical target term energies ($E_{\rm calc}$) including contributions from the mass and Darwin terms, and calculated leading percentages where we adopt the presentation of Martin et al. Aashamar et al. (1984) give mixing coefficients for some terms of Al-like ions. Reistadt & Engström (1989) give leading percentages for all experimentally determined terms of S IV. Martin et al. also quote some unpublished eigenvector components from Froese-Fischer (1981).

 

Table 4: Target energies ($E_{\rm target}$) for all terms included in the CC expansion for Cl V. ($E_{\rm exp}$) are the centres of gravity of the experimental energies taken from Moore (1971) and from Baudinet-Robinet et al. (1982). The identifications (Col. 1) that are marked with an ^* do not agree with those by Baudinet-Robinet et al., see Sect. 2. Calculated energies ($E_{\rm calc}$) include the mass and Darwin type relativistic contributions. The abbreviations ¹P$^{\rm o}$and ³P$^{\rm o}$stand for 3s3p(¹P$^{\rm o}$) and 3s3p(³P$^{\rm o}$) respectively

For most terms we agree very well with other calculated leading percentages. However, where our calculated term energy disagrees significantly with experiment the associated mixing coefficients are unreliable. Tables 2 to 4 would suggest a few changes to the configuration assignments that might be of interest to experimentalists. Particularly the order of terms 3s²5p ²P$^{\rm o}$and 3s3p(³P$^{\rm o}$)4s ²P$^{\rm o}$ in P III should be interchanged. In S IV terms ²P$^{\rm o}$ and ²D$^{\rm o}$ appear more strongly mixed in our calculation than in that of Reistadt & Engström (1989). For Cl V we give, in Table 4, identifications according to our calculated leading percentages that do differ from those given by Baudinet-Robinet et al. (1982) for some terms, we have indicated the affected terms by an *. For a more comprehensive theoretical treatment of the term structure of Al-like ions, Al I, Si II, S IV, Ar VI, Ca VIII, and Fe XIV, see Mendoza et al. (1995).