2 Computational procedures

2.1 Relativistic quantum defect orbital method

The RQDO method has been described in detail (Martín & Karwowski 1991). We shall thus only mention here those aspects of the formalism which are relevant to this study.

The relativistic quantum defect orbitals are the analytical solutions of a quasi relativistic second-order Dirac-like equation with a model Hamiltonian that contains the quantum defect as a parameter. This model Hamiltonian allows for an effective variation of the screening effects with the radial distance and, as a consequence, the radial solutions behave approximately correctly in the core region of space, and display a correct behaviour at large radial distances. These have been found to be, in many cases, the most relevant regions contributing to the transition integral. The RQDO's lead to closed-form analytical expressions for the transition integrals, which allows us to calculate transition probabilities with simple algebra and little computational effort.

The RQDO Hamiltonian accounts for core-valence polarization only implicitly, through the inclusion of the quantum defect. However, it is our experience that the explicit inclusion of the above effect is often important. As in some previous works (see, e.g., Charro et al. 1996, 1997) we have performed two types of RQDO f-value calculations: one with the standard dipole-length transition operator, Q(r) = r, and other with a core-polarization corrected form of the former (Bielinska-Waz 1992):  

$$Q(r)= r \left[ 1-\frac{\alpha}{r^3}\, [1-\exp(-r/r_{\rm c})]^3 \right],$$ (1)

where $\alpha$ is the core polarizability and $r_{\rm c}$ is a cut-off radius, chosen to be in these calculations the mean radius of the outermost orbital of the core, calculated in accord with an expression given by Chichkov & Shevelko (1981). The core has been considered in all the cases to be Ar $\rm 3d^{10}\ 4s^2$. For the core polarizabilities we have taken the values supplied by Fraga et al. (1976), which do not go beyond Nb IX (Z =41). For the higher ions in the sequence we have found the core polarizabilities in atomic units from the following expression

$$\ln\alpha=38.838-10.615\, \ln Z,$$ (2)

which has been obtained through a fitting of the $\alpha$-values of Fraga et al. (1976) from As I (Z= 33) to Nb IX (Z = 41), with a correlation factor of 0.9967. Z is the atomic number. The values of the core polarizabilities and cut-off radii, in atomic units, are shown in graph form in Fig. 1.

  

Figure 1: Core dipole polarizabilities and cut-off radii, in atomic units, for As-like ions

In the RQDO context, energy level data are required in order to obtain the quantum defects. For BrIII we have employed the data provided by a compilation by Kelly (1987). For the quadruplet levels of the $\rm 4p^2\ 5s$ configuration we have taken the observed energy values of O'Sullivan (1989) for Rb V and O'Sullivan & Maher (1989) for Sr VI. In both cases, the corresponding spectra of laser- produced plasmas were recorded, and the position of the lines was identified from MCDF calculations, which were also employed to determine f-values. These authors (O'Sullivan 1989; O'Sullivan & Maher 1989) report that the agreement between the calculated and observed energies is generally quite good. For the ions Y VII to Mo X we have used experimental energies by Reader & Acquista (1981). Early energy data measured by Rahimullah et al. (1976) for these ions are in good agreement with those of Reader & Acquista (1981). The energies of the levels of the ground and excited configurations up to Mo X that we have used in our RQDO calculations, for which some experimental data are available, are collected in Table 1.

  

Table 1: Experimental energy levels for some As-like ions, in $\rm cm^{-1}$

All the above data have relied on theoretical predictions and observed intensities for an identification of the lines, as well as an analysis of the systematic trends complied by the level energies along the isoelectronic sequence (O'Sullivan 1989; O'Sullivan & Maher 1989; Reader & Acquista 1981; Rahimullah et al. 1976). Reader & Acquista (1981) employ Hartree-Fock (HF) calculations. These authors estimate uncertainties to range from $\rm 0.6~cm^{-1}$ to $15~{\rm cm}^{-1}$ for the levels of the $\rm 4p^3$ and $\rm 4p^2\ 5s$ configurations, the uncertainties increasing from the ground to the excited configurations, and also from Y VII to Mo X.

  

Table 2: MCDF energy levels for some As-like ions, in $\rm cm^{-1}$

  

Table 3: Doublet MCDF energy levels for the $\rm 4p^3$ configuration, in $\rm cm^{-1}$

  

Table 4: MCDF energy levels for the $\rm 4p^2\ 5s$ configuration, in $\rm cm^{-1}$

 

Table 4: continued

In Tables 2 to 4 we display the theoretical energies obtained in the present work with the MCDF approach and employed in our MCDF calculations of oscillator strengths. In Tables 1 to 3 the energy of the lowest level of the ground configuration, $\rm {}^4S^o_{3/2}$, which is equal to zero in $\rm cm^{-1}$, has been omitted. The comparison of our MCDF energies, as reported in Table 2, with the data by other authors (Table 1) makes us confident as to their accuracy.

In order to have a basis to test the quality of the energies employed, we have plotted the energies of the ground and excited configurations in Figs. 2 and 3, respectively. They present a regular behaviour, in all cases, along the isoelectronic sequence, and the connection between the present MCDF and the experimental energies (Reader & Acquista 1981) is smooth.

  

Figure 2: Energies for the doublet levels of the ground configuration, $\rm 4p^3$, taken from Tables 1 and 2, in As-like ions versus the atomic number

Other input data that are also needed in the RQDO calculations were the ionization energies of the atomic systems. The corresponding values from As I to Kr IV are supplied by Kelly (1987) and a few more by Fraga et al. (1976). For the few remaining ions for which f-values are reported we have employed an extrapolation formula obtained by fitting the ionization energies in $\rm cm^{-1}$of the previous ions. This formula is the following

$$E_i=86.96\, Z^3-2806.66\, Z^2,$$ (3)

where, again, Z is the atomic number. The correlation factor was 0.9998.

2.2 Multiconfiguration Dirac-Fock method

The MCDF method, as implemented in the GRASP code (Grant 1989), is a generalization of the Dirac-Fock formalism. It employs a multiconfigurational trial function in the variation of the total energy. Within this model, an atomic state function is represented as a linear combination of configuration state functions built from antisymetrized products of standard Dirac orbitals. These orbitals are eigenfunctions of the one-electron total angular momentum operator, rather than of the spin and orbital momentum operators independently (as it is the case in the non-relativistic multiconfigurational Hartree-Fock method). For details of the atomic MCDF model we refer to a review by Grant (Grant 1988).

The extended average level (EAL) mode with the Breit and QED corrections (Grant 1988) has been employed. The following configurations have been introduced:

$\rm 4p^3 + 4p^5$ to describe the ground configuration.

  

Figure 3: MCDF energies for the levels of the excited configuration, $\rm 4p^2\ 5s$, in As-like ions versus the atomic number

$\rm 4p^2\ 5s + 4s 4p^4 + 4p^2 4d$ to describe the excited configuration.

The choice of these configurations has been made on the grounds of the comments by Bieron & Migdalek (1980) and O'Sullivan (1989) and by observing that this configuration mixing leads to a good accord between dipole-length (Babushkin gauge) and dipole velocity (Coulomb gauge) MCDF oscillator strengths for most of the transitions studied. Other configurations such as $\rm 4s\ 4p^3\ 4d$ and $\rm 4p^2\ 5p + 4p^2\ 5d$ for the ground and excited states, respectively, have been included in a few ions as a test. However, the final energies and oscillator strengths were practically unaffected by these inclusions whilst the computational time was appreciably increased. Therefore, the data we report in the Tables correspond to the inclusion of the specified configurations solely.