2 Method

The calculations of 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 functions. The method has been described in previous IP reports (Galavís et al. 1997, 1998). As discussed by Eissner (1991) the LS terms are represented by CI wavefunctions of the type

$$\Psi=\sum_i\phi_ic_i \ ,$$ (2)

where the configuration basis functions $\phi_i$ are constructed from one-electron orbitals generated in two types of potentials $V(\lambda_{nl})$:the spectroscopic orbitals P(nl) are calculated in a statistical Thomas-Fermi-Dirac model potential (Eissner & Nussbaumer 1969) whereas the correlation orbitals $P(\bar{nl})$ are obtained in a Coulomb potential (Nussbaumer & Storey 1978). With regards to configuration bases, four approximations are considered as shown in Table 1. Approximation A is our main standard framework containing only configurations with $n \leq 3$ orbitals. The representations for the lower members of the sequence are progressively refined with approximations B, C and D which include configurations with n=4 orbitals of increasing complexity. The adopted variational procedure minimises with equal weights the sum of energies of all the terms in specific configurations, that is

$${\cal F}=\sum_{i=1}^N E(S_i,L_i) \ .$$ (3)

The 1s, 2s and 2p orbitals are chosen to minimise the sum of energies of the three terms in the 2s$\sp2$2p$\sp2$ ground configuration. The $\bar{3}$s, $\bar{3}$p and $\bar{3}$d correlation orbitals are subsequently chosen to minimise the sum of the energies of the twelve terms of the n=2 complex. Finally, the $\bar{4}$s, $\bar{4}$p, $\bar{4}$d and $\bar{4}$f correlation orbitals in approximations B, C and D are optimised on the sum of the energies of the three terms within the ground configuration. The final scaling parameters are listed in Table 2.

 

Table 1: Configuration bases used in approximations A, B, C and D

 

Table 2: Scaling parameters $\lambda_{\rm nl}$ used to generate the orbitals for the four approximations (A, B, C and D) as a function of nuclear charge number Z. The negative scaling parameters denote Coulombic correlation orbitals

In SUPERSTRUCTURE the Hamiltonian is taken to be of the form

$$H=H_{\rm nr}+H_{\rm rc}$$ (4)

where $H_{\rm nr}$ is the usual non-relativistic Hamiltonian. The relativistic corrections $H_{\rm rc}$ are taken into account through the Breit-Pauli (BP) approximation (Jones 1970, 1971)

$$\begin{eqnarray} H_{\rm rc}= & \sum_{i=1}\sp N [f_i({\rm mass})+f_i({\rm d})+f_i... ... \\ & \sum_{i\gt j}[g_{ij}({\rm so}+{\rm so}')+g_{ij}({\rm ss}')]\end{eqnarray}$$ (5)

where the one-body terms $f_i({\rm mass})$, $f_i({\rm d})$ and $f_i({\rm so})$correspond respectively to the mass-variation correction, the Darwin term and the spin-orbit interaction; the two-body Breit terms $g_{ij}({\rm so}+{\rm so}')$and $g_{ij}({\rm ss}')$ are the spin-other-orbit plus mutual spin-orbit and spin-spin interactions. In the present calculations, the spin-spin interaction, when included, is fully taken into account for the first three configurations in Table 1, i.e. the spectroscopic configurations. To estimate the spin-spin contribution for all configurations in the largest basis set considered here would be very costly, if not impossible. Fortunately, small-scale tests showed that the numbers reported here are not affected by more than a few % at low Z by this approximation. When Z increases, this effect decreases rapidly. Two-body non fine-structure terms are currently neglected in SUPERSTRUCTURE.

From perturbation theory, the relativistic wavefunction $\psi_i^{\rm r}$ can be expanded in terms of the non-relativistic functions $\psi_j^{\rm nr}$:

$$\psi_i^{\rm r}= \psi_i^{\rm nr}+ \sum_{j \not=i}\psi_j^{\rm... ...}\vert\psi_i^{\rm nr}\gt} {E_i^{\rm nr}-E_j^{\rm nr}} + \ldots$$ (6)

This expansion demonstrates the importance of accurate term energy separations when constructing the relativistic wave functions $\psi_i^{\rm r}$.Using accurate experimental level energies (Edlén 1985), $H_{\rm nr}$ is adjusted in order to obtain term energies (calculated from the weighted fine-structure level energies) which match experiment. This semi-empirical term energy correction (TEC) procedure was originally implemented in SUPERSTRUCTURE by Zeippen et al. (1977). Galavís et al. (1997, 1998) have shown that, in the treatment of the forbidden transitions in the carbon and oxygen sequences and of the intercombination transitions in the boron sequence, the inclusion of TECs can lead to a high degree of accuracy with a much reduced configuration basis than in a comparable purely ab initio treatment. This previous experience led to the choice of our standard representation (approximation A). Of course, the validity of this semi-empirical procedure must be checked in the case under consideration here.

The radiative rate for an electric dipole (E1) transition is given by the expression  

$$A_{ij}({E1})= 2.6774\ 10^9(E_i-E_j)^3\frac{1}{g_i} S_{ij}^{E1} \quad\quad\mbox{(\rm s$^{-1}$)} \ ,$$ (7)

where 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^E1, but even relatively small errors in the energy differences (E_i-E_j) can reduce it because of the exponent 3 in (7). Therefore, the transition probabilities are computed with an accurate and consistent dataset of experimental energy levels (Edlén 1985).