2 Method

The calculations of the transition probabilities have been carried out with the computer program SUPERSTRUCTURE, originally developed by Eissner et al. (1974) and later modified by Nussbaumer & Storey (1978) to ensure greater flexibility in the radial orbital functions. The method has been summarized in our previous article (Galavís et al. 1997). As described by Eissner (1991) the LS terms are represented by CI wavefunctions of the type

$$\Psi=\sum_i\phi_ic_i \ ,$$ (1)

where the configuration basis functions $\phi_i$ are constructed from one-electron spectroscopic orbitals P(nl), generated within a statistical Thomas-Fermi-Dirac model potential $V(\lambda_{nl})$ (Eissner & Nussbaumer 1969). For the boron sequence we have adopted the 11-configuration basis set used by Biémont et al. (1994) for C II and N III, namely 2s²2p, 2s2p², 2p³, 2s²3s, 2s²3p, 2s²3d, 2p²3d, 2s3d², 2s2p3s, 2s2p3p, 2s2p3d. To determine the scaling parameters $\lambda_{nl}$, a special optimization procedure described and discussed in detail by these authors is employed. It is found that the accuracy of the energy levels, and consequently that of the transition probabilities, depends strongly on how the optimization is performed. The variational procedure adopted here minimises with equal weights the sum of energies of all the terms in specific configurations, that is

$${\cal F}=\sum_{i=1}^N E(S_i,L_i) \ .$$ (2)

More precisely, the 1s, 2s, 2p and 3s orbitals are first optimized with a functional ${\cal F}$ containing the energies of all terms in the 2s$\sp2$2p, 2s2p$\sp2$, 2p$\sp3$ and 2s$\sp2$3s configurations. Then the 3p and the 3d orbitals are determined by including in ${\cal F}$ all the terms in the 2s$\sp2$2p, 2s2p$\sp2$,2p$\sp3$, 2s$\sp2$3s, 2s$\sp2$3p and 2s$\sp2$3d configurations. Finally, the 3s function is reoptimized with the states generated by the 2s2p3s configuration so as to obtain a slight improvement in the term energies. The final scaling parameters are listed in Table 1.

  

Table 1: Scaling parameters $\lambda_{nl}$ used to generate the orbitals for the B-like ions

In SUPERSTRUCTURE the Hamiltonian is taken to be of the form

$$H=H_{\rm nr}+H_{\rm bp}$$ (3)

where $H_{\rm nr}$ is the usual non-relativistic Hamiltonian. The relativistic corrections $H_{\rm bp}$ are taken into account through the Breit-Pauli (BP) aproximation (Jones 1970, 1971): the one-body terms which include the mass-variation correction, the Darwin term and the spin-orbit interaction; and the two-body fine-structure terms, namely the spin-spin and spin-other-orbit interactions. Using perturbation theory, the relativistic wavefunction $\psi_i^{\rm r}$ can be expanded in terms of the non-relativistic functions $\psi_j^{\rm nr}$:

$$\psi_i^{\rm r}= \psi_i^{\rm nr}+ \sum_{j \not=i}\psi_j^{\rm... ...}\vert\psi_i^{\rm nr}\gt} {E_i^{\rm nr}-E_j^{\rm nr}} + \ldots$$ (4)

Small fractional errors in the non-relativistic energies $E_i^{\rm nr}$and $E_j^{\rm nr}$ can lead to much larger errors in the differences $E_i^{\rm nr}-E_j^{\rm nr}$ for $j\not=i$. Therefore, by using accurate experimental level energies by Moore (1970, 1975) and Edlén (1983), improved estimates of the non-relativistic energies are obtained and a modified $H_{\rm nr}$ is constructed. This semi-empirical term energy correction (TEC) procedure was originally implemented in SUPERSTRUCTURE by Zeippen et al. (1977).

In the present case, for ions with $Z\geq 8$, corrections are made so as to shift the computed excitation energy of the lowest level in each of the 8 terms in the n=2 complex to its observed value. For C II and N III, due to the mixture with low n=3 states, additional terms arising from configurations of the type 2s²3l and 2s2p3l are also corrected.

The radiative rates for electric dipole (E1), electric quadrupole (E2) and magnetic dipole (M1) transitions are respectively given by the expressions  

$$A_{ij}({\rm E1})= 2.6774\ 10^9(E_i-E_j)^3\frac{1}{g_i} S_{ij}^{E1} \quad\quad\mbox{(\rm s$^{-1}$)} \ ,$$ (5)

 

$$A_{ij}({\rm E2})= 2.6733\ 10^3(E_i-E_j)^5\frac{1}{g_i} S_{ij}^{E2} \quad\quad\mbox{(\rm s$^{-1}$)} \ ,$$ (6)

and  

$$A_{ij}({\rm M1})= 3.5644\ 10^4(E_i-E_j)^3\frac{1}{g_i} S_{ij}^{M1} \quad\quad\mbox{(\rm s$^{-1}$)} \ .$$ (7)

Here g_i is the statistical weight of the upper initial level i and energies E are expressed in Rydbergs. It is clear that the accuracy of the calculated A-values depends primarily on the quality of the wavefunctions used in evaluating the line strengths S_ij. However, relatively small errors in the energy difference (E_i-E_j) are amplified by the large exponents in Eqs. ((5)-(7)). As shown, for example, by Galavís et al. (1997), in order to ensure an acceptable degree of accuracy the transition probabilities must be computed with an accurate and consistent dataset of experimental energy levels. For the boron sequence, the work of Moore (1970, 1975) and Edlén (1983) is fully used.