3 Computations

3.1 The BPRM calculations

The Fe V wavefunctions are computed with eigenfunction expansions over the "target'' ion Fe VI. Present work employs a 19-level eigenfunction expansion of Fe VI corresponding to the 8-term LS basis set of 3d³ (⁴F, ⁴P, ²G, ²P, ²D2, ²H, ²F, ²D1), as used in Nahar & Pradhan (2000). The target wavefunctions were obtained by Chen & Pradhan (1999) using the Breit-Pauli version of the atomic structure code, SUPERSTRUCTURE (Eissner et al. 1974). The bound channel set of functions $\Phi_j$ in Eq. (1), representing additional (N+1)-electron correlation includes a number of Fe V configurations, particularly from the important n = 3 complex, i.e. 3s²3p⁶3d⁴, 3p⁶3d⁶, 3s²3p⁵3d⁵, 3s²3p⁴3d⁶; the complete list of $\Phi_j$ for the n = 3 and 4 configurations is given in Chen & Pradhan (1999).

The Breit-Pauli calculations consider all possible fine-structure bound levels of Fe V with (2S + 1) = 1, 3, 5 and L = 0 - 10, $n\leq 10, \ \ell \leq n-1$, and $J \leq$ 8, and the transitions among these levels. In the R-matrix computations, the calculated energies of the target levels were replaced by the observed ones. The calculations are carried out using the BPRM codes (Berrington et al. 1995) extended from the Opacity Project codes (Berrington et al. 1987).

STG1 of the BPRM codes computes the one- and two-electron radial integrals using the one-electron target orbitals generated by SUPERSTRUCTURE. The number of continuum R-matrix basis functions is chosen to be 12. The intermediate coupling calculations are carried out on recoupling these LS symmetries in a pair-coupling representation in stage RECUPD. The computer memory requirement for this stage has been the maximum as it carries out angular algebra of dipole matrix elements of a large number of levels due to fine-structure splitting. The (e + Fe VI) Hamiltonian is diagonalized for each resulting $J\pi$ in STGH.

3.1.1 Energy levels and identification

The negative eigenvalues of the (e + Fe VI) Hamiltonian correspond to the bound levels of Fe V, determined using the code STGB. Splitting of each target LS term into its fine-structure components also increases the number of Rydberg series of levels converging on to them. These result in a large number of fine-structure levels in comparatively narrow energy bands. An order of magnitude finer mesh of effective quantum number ( $\Delta \nu$ = 0.001), compared to that needed for the locating the bound LS states, was needed to search for the BP Hamitonian eigenvalues in order to avoid missing energy levels. The computational requirements were, therefore, increased considerably for the intermediate coupling calculations of bound levels over the LS coupling case by several orders of magnitude. The calculations take up to several CPU hours per $J\pi$ in order to determine the corresponding eigenvalues in the asymptotic program STGB.

The identification of the fine-structure bound levels computed in intermediate coupling using the collision theory BPRM method is rather involved, since they are labeled with quantum numbers related to electron-ion scattering channels. The levels are associated with collision complexes of the (e + ion) system which, in turn, are initially identified only with their total angular momenta and parity, $J\pi$. A scheme has been developed (Nahar & Pradhan 2000) to identify the levels with complete spectroscopic information giving

$$C_{\rm t} (\ S_{\rm t} \ L_{\rm t})\ J_{\rm t}~\pi_{\rm t} n\ell \ [{\rm K}] {\rm s}\ \ J \ \pi,$$ (10)

and also to designate the levels with a possible $SL\pi$ symmetry ($C_{\rm t}$ is the target configuration).

Most of the spectroscopic information of a computed level is extracted from the few bound channels that dominate the wavefunction of that level. A new code PRCBPID has been developed to carry out the identification, including quantum defect analysis and angular momentum algebra of the dominant channels. Two additional problems are addressed in the identification work: (A) correspondence of the computed fine-structure levels to the standard LS coupling designation, $SL\pi$, and (B) completeness checks for the set of all fine-structure components within all computed LS multiplets. A correspondence between the sets of $SL\pi$ and $J\pi$ of the same configuration are established from the set of $SL\pi$ symmetries, formed from the target term, $S_{\rm t}L_{\rm t}\pi_{\rm t}$, nl quantum numbers of the valence electron, and $J\pi$ of the fine-structure level belonging to the LS term. The identification procedure is described in detail in Nahar & Pradhan (2000).

Considerable effort has been devoted to a precise and unique identification of levels. However, a complex ion such as Fe V involves many highly mixed levels and it becomes difficult to assign a definite configuration and parentage to all bound states. Nonetheless, most of the levels have been uniquely identified. In particular all calculated levels corresponding to the experimentally observed ones are correctly (and independently) assigned to their proper spectroscopic designation by the identification procedure employed.

3.1.2 E1 oscillator strengths

The oscillator strengths and transition probabilities were obtained using STGBB of the BPRM codes. STGBB computed the transition matrix elements using the bound wavefunctions created by STGB, and the dipole operators computed by STGH. The fine structure of the core and the (N+1) electron system increased the computer memory and CPU time requirements considerably over the LS coupling calculations. About 31 MW of memory, and about one CPU hour on the Cray T94, was required to compute the oscillator strengths for transitions among the levels of a pair of $J\pi$ symmetries. These are over an order of magnitude larger than those needed for f-values in LS coupling. The number of f-values obtained from the BPRM calculations ranges from over 5000, among J = 8 levels, to over 123000, among J = 3 levels, for a pair of symmetries.

These computations required over 120 CPU hours on the Cray T94. Total memory size needed was over 42 MW to diagonalise the BP Hamilitonian. Largest computations involved a single $J\pi$ Hamiltonian of matrix size 3555, 120 channels, and 2010 configurations.

We have included extensive tables of all computed bound levels, and associated E1 A-values, with full spectroscopic identifications, as standardized by the U.S. National Institute for Standards and Technology (NIST). In addition, rather elaborate (though rather tedious) procedures are implemented to check and ensure completeness of fine-structure components within all computed LS multiplets. The complete data tables are available in electronic format. A sample of the datasets is described in the next section.

3.2 SUPERSTRUCTURE calculations for the forbidden E2, M1 transitions

The only available dataset by Garstang (1957) comprises of the E2, M1 A-values for transitions within the ground 3d⁴ levels. The CI expansion for Fe V consists of the configurations (1s²2s²2p⁶3s²3p⁶) 3d⁴, 3d³4s, 3d³4p as the spectroscopic configurations and 3d³4d, 3d³5s, 3d³5p, 3d³5d, 3d²4s², 3d²4p², 3d²4d², 3d²4s4p, 3d²4s4d as correlation configurations. The eigenenergies of levels dominated by the spectroscopic configurations are minimised with scaling parameters $\lambda_{n\ell}$ in the Thomas-Fermi-Dirac potential used to calculate the one-electron orbitals in SUPERSTRUCTURE (see Nussbaumer & Storey 1978): ${ \lambda_{[1{\rm s}-5{\rm d}]}}$ = 1.42912, 1.13633, 1.08043, 1.09387, 1.07756, 0.99000, 1.09616, 1.08171, -0.5800, -0.6944, -1.0712, -3.0000.

There are 182 fine-structure levels dominated by the configurations 3d⁴, 3d³4s and 3d³4p, and the respective number of LS terms is 16, 32 and 80. The $\lambda_{1{\rm s}-3{\rm d}}$ are minimised over the first 16 terms of 3d⁴, $\lambda_{4{\rm s}}$ over 32 terms including 3d³4s, and $\lambda_{4{\rm p}}$, $\lambda_{5{\rm p}}$ over all 80 terms. The $\lambda_{5{\rm s}}$ and $\lambda_{5{\rm d}}$ are optimised over the 3d⁴ terms to further improve the corresponding eigenfunctions.

The numerical experimentation entailed a number of minimisation trials, with the goal of optimisation over most levels. The final set of calculated energies agree with experiment to within 10%, although more selective optimisation can lead to much better agreement for many (but not all) levels. Finally, semi-empirical term energy corrections (TEC) (Zeippen et al.1977) were applied to obtain the transition probabilities. This procedure has been successfully applied in a large number of studies (see e.g. Biémont et al.1994). The electric quadrupole (E2) and the magnetic dipole (M1) transition probabilites, $A^{\rm q}$ and $A^{\rm m}$, are obtained using observed energies according to the expressions:

$$A_{j,i}^{\rm q}({\rm E}2) = 2.6733~10^3 (E_j - E_i)^5 {\cal S}^{\rm q}(i,j) {\rm s}^{-1},$$ (11)

and

$$A_{j,i}^{\rm m}({\rm M}1) = 3.5644~10^4 (E_j - E_i)^3 {\cal S}^{\rm m}(i,j) {\rm s}^{-1},$$ (12)

where E_j > E_i (the energies are in Rydbergs), and ${\cal S}$ is the line strength for the corresponding transition.