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2 The model

The basic atomic theory, the approximations and the computer codes employed in the IRON Project are described in Paper I of the A&A series (Hummer et al. 1993). 80 terms are included in the target expansion, arising from the 3d2, 3d4s, 3d4p, 3d4d, 3d4f and 3p53d3 configurations. Figure 1 shows the calculated energy level diagram as an illustration of our model.

\put(10,0){\framebox (160,280)}
\put(10, 267){\line(1,0){80}}
\end{picture}\par\end{figure} Figure 1: The model atom: calculated energies (Ryds) of the 80 terms included for Fe VII

The radial orbitals for the Ca-like Fe ion target were taken from Clementi & Roetti (1974) for 1s, 2s, 2p, 3s, 3p, 3d. Also included were n=4 orbitals, optimised using Hibbert's (1975) variational program CIV3 in the following way: 4s optimised on the 3d4s3,1 D$^{\rm e}$; 4p optimised on the 3d4p3,1 D$^{\rm o}$; 4d optimised on the 3d4d3,1 F$^{\rm e}$; 4f optimised on the 3d4f3,1 G$^{\rm o}$. The exponents of the n=4 orbitals are summarised in Table 1.

A configuration interaction wavefunction is used to describe the target terms. The following configurations are included in the target description: 3d2, 3d4l, 4l4l', 3p53d3, 3p53d24s, 3p53d4s2, 3p43d4, 3p43d34s, 3p43d34p and 3s3p63d3. This includes configurations to improve correlation in the target energies; similar configurations are included in the scattering "N+1 electron'' system.


Table 1: Ca-like Fe target n=4 Slater-type orbital exponents for each power of r; the orbital coefficients are fixed by orthonormality conditions

$P_{{\rm nl}}$
r1 r2 r3 r4
4s 3.05595 9.14246 1.28822 1.48174
4p   11.15581 4.58536 2.46411
4d     5.16564 1.97463
4f       1.89000

In order to assess our model target energies, we calculated level energies using the Breit-Pauli Hamiltonian with our wavefunction. These energies are shown in Table 2, where they are compared with those from the multiconfigurational Dirac-Fock (MCDF) calculation including full transverse Breit and QED contributions (Norrington & Grant 1987) and with experiment (Ekberg 1981). Also included in Table 2 are energies which we calculated from our orbitals in a "frozen core'' model (i.e. with no open p-shell configuration interaction), which agrees remarkably well with the "MCDF'' results (which was also a "frozen core'' model), indicating that core correlation is more significant than relativistic effects and is responsible for most of the improvement of our present energies relative to experiment. In general, our calculated energy levels are now reasonably accurate for a scattering calculation, though we note that the only level which fails to improve in any calculation is 3d2 1G.


Table 2: 3d2 energy levels (Ryd) of Fe VII. The present calculation (together with a frozen-18-electron approximation) is compared with the multiconfigurational Dirac-Fock (MCDF) calculation of Norrington & Grant (1987) and with experiment (Ekberg 1981)

3d2 term MCDF frozen core Present Experiment
1 $^{3}{\rm F}_{2}$ 0.0 0.0 0.0 0.0
2 $^{3}{\rm F}_3$ 0.009 0.010 0.011 0.010
3 $^{3}{\rm F}_4$ 0.021 0.023 0.024 0.021
4 $^{1}{\rm D}_{2}$ 0.193 0.194 0.167 0.159
5 $^{3}{\rm P}_0$ 0.223 0.224 0.196 0.183
6 $^{3}{\rm P}_{1}$ 0.226 0.227 0.200 0.186
7 $^{3}{\rm P}_2$ 0.233 0.236 0.209 0.194
8 $^{1}{\rm G}_{4}$ 0.294 0.298 0.298 0.264
9 $^{1}{\rm S}_{0}$ 0.715 0.711 0.634 0.611

\includegraphics[width=6cm]{}\end{tabular}\end{figure} Figure 2: Collision strength for electron excitation from the ground state $^{3}{\rm F}_{2}$ to $^{1}{\rm D}_{2}$ (upper plot) and $^{3}{\rm P}_{1}$ (lower) of Fe VII: -- present 80-term R-matrix calculation; - - - 9-level Dirac R-matrix (Norrington & Grant 1987); - - - - - - distorted-wave (Nussbaumer & Storey 1982)

The present 80-term calculation is in LS coupling, using the R-matrix programs of Berrington et al. (1995). The R-matrix boundary is at 9.6 a.u., we include 16 continuum terms per channel and mass correction and Darwin non-finestructure relativistic terms. A transformation to intermediate coupling is applied to the T-matrix using the JAJOM procedure of Saraph (1978), with term coupling included amongst the 3d2 terms (though the coupling coefficients are not very significant: the largest is between $^{1}{\rm D}_{2}$ and $^{3}{\rm P}_2$ where it is 5%). This model is used in the resonance region between the 3d2 states and the higher excited states to a maximum of 16 Ryds. A "top-up'' in angular momentum is applied, and an explicit calculation to J=20.5 at the higher energies confirms convergence of these forbidden transitions.

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