Atomic spectroscopic properties of highly-charged ions (HCIs) are of great importance in atomic physics, astronomical physics, and fusion plasma research. ^{[ 1 – 4 ]} Among them the electric-dipole forbidden transition lines are of particular importance in plasma diagnostics due to their sensitive intensities to electron temperature and density and other plasma parameters. ^{[ 5 – 11 ]} The currently famous elements (for example, tungsten and gold) have inspired great research interest since their favorable physical properties, and many spectra lines have been measured with high accuracy. ^{[ 12 – 22 ]} Meanwhile, the accurately measured properties, such as wavelength, can provide crucial tests for atomic structure theory of both the electron correlation and relativistic corrections, ^{[ 23 , 24 ]} which are the major questions in theoretical calculations.

Ralchenko et al. ^{[ 24 ]} and Osin et al. ^{[ 25 ]} measured many forbidden lines from the HCIs of hafnium (Hf, Z = 72), tantalum (Ta, Z = 73), tungsten (W, Z = 74), and gold (Au, Z = 79) with an open 3d shell, with a wavelength uncertainty of 0.03%. Unfortunately, the calculated wavelengths, obtained by Ralchenko and Osin  et al. ^{[ 24 , 25 ]} using the relativistic configuration-interaction (RCI) method implemented in the flexible atomic code (FAC), ^{[ 26 , 27 ]} can only agree with the experimental ones in 0.8%. Quinet ^{[ 28 ]} later obtained the energy levels, wavelengths, and transition probabilities for tungsten ions between W ⁴⁷⁺ and W ⁶¹⁺ by using the fully relativistic multi-configuration Dirac–Fock (MCDF) method with the latest version of GRASP, i.e., the general purpose relativistic atomic structure package by Norrington et al. ^{[ 29 ]} However, due to the limited configuration interaction considered, the calculated wavelengths differ from the measured ones by up to 0.7%. Fournier ^{[ 30 ]} also performed calculations for W ⁴⁷⁺ by using the fully relativistic parametric potential code (RELAC), ^{[ 31 ]} but the quantum electrodynamic (QED) corrections were not included. ^{[ 32 ]} It is obvious that the differences between these calculated wavelengths and measured ones are much larger than the stated experimental uncertainties, which may mean that there is a long way to go on the theoretical side.

Recently, by means of the second-order relativistic many-body perturbation theory (RMBPT) approach, we have updated the energy levels and the transition data for these forbidden transitions among the levels within the ground configurations [Ar] 3d ^k ( k = 1,...,9) in Co-like through K-like ions of Hf, Ta, W, and Au elements. ^{[ 33 ]} This RMBPT approach is based on the work of Lindgren on Rayleigh–Schrödinger perturbation theory extended to a multi-configurational zero-order wave function, ^{[ 34 ]} which is also implemented in the FAC code. ^{[ 26 , 27 ]} The RMBPT calculation dramatically removes the existing systematic deviations for wavelengths in previous theoretical calculations and reproduces these measured wavelengths to within 0.2%, which greatly improves the existing theoretical accuracy mentioned above, though still fall outside the experimental uncertainties, 0.03%, remotely.

To date, another state-of-the-art method with high accuracy is the multi-configuration Dirac–Hartree–Fock (MCDHF) approach implemented in the GRASP2K-V3 package, ^{[ 35 ]} which is a revised version of GRASP2K by Jönsson et al. ^{[ 37 ]} This approach is suitable to study different physical effects step by step. ^{[ 37 ]} In this work, with the purpose of analyzing the correlation effects clearly and getting the calculated transition energies close to the experimental data, by means of the MCDHF approach, we focus on the forbidden transition within the relatively simple multi-valence cobalt-like ions, which have the ground configuration 3d ⁹ , with nine valence electrons, i.e., one-vacancy-electron outside the closed core subshell. The one-vacancy-electron system is similar to the one-valence-electron system, ^{[ 38 , 39 ]} but should involve more complex correlations.

Furthermore, to investigate the core–valence (CV) correlation effects clearly, two independent approaches, the separate core–valence approach (SCV) and the full core–valence approach (FCV), are adopted. By comparison and analysis, we prove that the SCV is beneficial to give reliable information about important correlation contribution from separate subshell quickly. It is found that the electron correlation from the inner core 2p makes significant contribution to the transition energy, even larger than from the outer electron 3p. At the same time, the contribution from deeper core electron 2s is larger than from the 3s, which also should be considered. The final calculated fine-structure separation are in excellent agreement with the experimental results to within 0.01%.

In the MCDHF method, the ion is represented by the atomic state functions (ASF) Ψ ( γJ ), which is a linear combination of a number of configuration state functions (CSF)

Once we obtain a set of radial orbitals, relativistic configuration interaction calculations (RCI) ^{[ 43 ]} can be performed. In the RCI calculations we include the transverse photon interaction in the Hamiltonian ^{[ 44 ]}

The photon frequency ω _{i j} used by the RCI program in calculating the matrix elements of the transverse photon interaction is taken to be the difference in the diagonal Lagrange multipliers and associated with orbitals. When using the low frequency limit ω _{i j} → 0, the interaction is usually referred to as the Breit interaction. The calculation of the quantum electrodynamic corrections (QED) due to the other leading part of self-energy (SE) and vacuum polarization (VP) are also included in the RCI calculation. ^{[ 45 ]}

One main advantage of the multiconfigurational method is the fact that correlation can be included in a systematic way, by using the notion of an active set (AS) of orbitals, that are in turn used to generate CSFs. ^{[ 46 ]} The foundation of this approach is a restricted active space (RAS) method, ^{[ 47 ]} where the active set of orbits is systematically increased. Considering the relative importance of different excitations from the view point of order-by-order expansion of energy in perturbation theory, we allow single (S) and double (D) replacement from the reference configurations to an active set. According to this approach, the atomic electrons are divided into two types: valence electrons and core electrons. The correlations are then classified into three types, i.e., the valence–valence (VV), core–valence (CV), and core–core (CC) correlations. It should be mentioned that the CC correlation can be further classified according to (a) intra-CC, i.e., correlation within one core subshell; and (b) inter-CC, i.e., correlation between two core subshells. The details of this approach can be found elsewhere. ^{[ 37 , 48 ]}

With the revised GRASP2K-V3 code, we have made a detailed treatment of the electron correlation for the multi-valence-electron systems of Co-like (3d ⁹ ) ions using the MCDHF approach. Highly accurate fine-structure energies between ² D _3/2 and ² D _5/2 are obtained for Hf ⁴⁵⁺ , Ta ⁴⁶⁺ , W ⁴⁷⁺ , and Au ⁵²⁺ , being within the experimental uncertainties. ^{[ 24 , 25 ]} It shows that it is critical to include the deep core-valence correlation down to the n = 2 subshell, to achieve spectroscopic accuracy. In particular, the CV correlation from the deep core 2p is larger than 3p, 2s is larger than 3s, and CV from 1s could be ignored. The CC correction is so small that it could be ignored in the latter large-scale calculations especially for other higher-Z Co-like ions.

Among the SCV and FCV methods used to study CV correlation, the SCV method is reliable enough to get similar accurate results to the FCV model, but with much smaller amount of CSFs and less computation efforts. The SCV method provides a simple and wise way to investigate the core–valence correlation from the individual subshell. The strategies applied here have been successfully employed to investigate the fine-structure splitting within the ground state along with the whole Co I isoelectronic sequence, as well as for the other one-valence sequence (K I and Rb I) and one-vacancy sequence (Rh I). These results are being prepared to be published elsewhere. We believe that the present strategies can also be potentially useful in even more complicated sequences, such as the n d ^k ( k = 2,...,8) ones.