2 Target expansion

In the close coupling (CC) approximation the total wave function of the electron-ion system is represented as

$$\begin{eqnarray} \Psi(E;SL\pi) = A \sum_{i} \chi_{i}\theta_{i} + \sum_{j} c_{j} \Phi_{j},\end{eqnarray}$$ (1)

where $\chi_{i}$ is the target ion wave function in a specific state S_i L_i, $\theta_{i}$ is the wave function for the free electron, and ${\Phi_j}$ are short range correlation functions for the bound $\rm (e+ion)$ system. Accurate CC calculations of atomic processes require, first of all, a good representation of the target ion. For complex ions an accurate representation must include a large number of correlation configurations. However, in order for the computations to be computationally tractable, the configuration expansion must be carefully and economically chosen.

The CC expansion for this calculation includes 49 LS terms of the target ion Ni III. The atomic structure code SUPERSTRUCTURE (Eissner et al. 1974; Eissner 1991) was used to compute eigenfunctions for the target states dominated by the configurations $\rm 3d^8$, $\rm 3d^7\ 4s$, and $\rm 3d^7\ 4p$. Table 1 presents the complete list of states included in the target, as well as a comparison between the calculated target term energies and the observed energies, averaged over the fine structure, taken from Sugar & Corliss (1985). The agreement between the energies is good; in all but the lowest five terms the agreement with the experimental values is better than 9%. The overall agreement is approximately 5%. Another indicator of the accuracy of the target representation is the good agreement between the length and the velocity oscillator strengths (f-values), which for the present case is typically $10-20\%$.

 

Table 1: Calculated and observed term energies (Rydbergs) for Ni III relative to the $\rm 3d^8 (^3F)$ ground state. The spectroscopic and correlation configurations for Ni III, and the values of the scaling parameters $\lambda_{nl}$ for each orbital in the Thomas-Fermi-Dirac potential used in Superstructure, are also given

Ni III configurations.

Spectroscopic: $\rm 3p^6\ 3d^8$, $\rm 3p^6\ 3d^7\ 4s$, $\rm 3p^6\ 3d^7\ 4p$.

Correlation: $\rm 3p^6\ 3d^6\ 4s^2$, $\rm 3p^6\ 3d^6\ 4p^2$, $\rm 3p^4\ 3d^8\ 4s^2$,$\rm 3p^5\ 3d^7\ 4s^2$, $\rm 3p^6\ 3d^7\ 4d$, $\rm 3p^5\ 3d^8\ 4d^1$,$\rm 3p^6\ 3d^6\ 4s\ 4p^1$, $\rm 3p^4\ 3d^8\ 4p^2$, $\rm 3p^5\ 3d^8\ 4p^1$,$\rm 3p^5\ 3d^7\ 4s$.

$\lambda_{nl}$: 1.43121(1s), 0.93950(2s), 1.28641(2p), 1.15714(3s), 1.02417(3p), 1.01626(3d), 0.97488(4s), 0.96711(4p), 1.26086(4d).