odd-parity (5f 6d + 5f 7d + 5f 7s + 5f 8s + 6d 6f + 6d 7p)
grid = Radial.Grid(Radial.Grid(false), rnt = 4.0e-6, h = 5.0e-2, hp = 0.6e-2, rbox = 10.0)
startOrbs = multiplet.levels[1].basis.orbitals
subshells = multiplet.levels[1].basis.subshells
asfSettings = AsfSettings(AsfSettings(), eeInteractionCI=CoulombInteraction(), jjLS=LSjjSettings(true),
                         startScfFrom=StartFromPrevious(startOrbs), frozenSubshells=subshells)
oddConfigs = [Configuration("[Rn] 5f 6d"), Configuration("[Rn] 5f 7d"), Configuration("[Rn] 5f 7s"),
              Configuration("[Rn] 5f 8s"), Configuration("[Rn] 6d 6f"), Configuration("[Rn] 6d 7p")]
wa = Atomic.Computation(Atomic.Computation(), name="Th^2+ odd-partiy level energies", grid=grid,
                       nuclearModel = Nuclear.Model(90.),
                       configs = oddConfigs, asfSettings = asfSettings)
wb = perform(wa, output=true)
pSettings = PhotoEmission.Settings(PhotoEmission.Settings(), multipoles=[E1])
evenConfigs = [Configuration("[Rn] 6d^2"), Configuration("[Rn] 5f^2"), Configuration("[Rn] 7s^2"),
              Configuration("[Rn] 5f 7p"), Configuration("[Rn] 5f 6f"), Configuration("[Rn] 6d 7s")]
wa = Atomic.Computation(Atomic.Computation(), name="Th^2+: Lifetimes of even-partiy level energies",
                       grid=grid, nuclearModel=Nuclear.Model(90.),
                       initialConfigs = evenConfigs, initialAsfSettings = asfSettings,
                       finalConfigs = oddConfigs, finalAsfSettings = asfSettings,
                       processSettings = pSettings)
```
**Figure 6.** Input for the Atomic.Computation of the low-lying levels of Th2+. In these calculations (**upper panel**), the orbitals of the [Rn] 5 *f* 6*d* configuration have first been optimized independently and then be kept frozen in the computation above. In the (**lower panel**), in addition, the transition probabilities and lifetimes are obtained by just specifying the even-parity configurations as well as the Settings for the photo emission.

While Figure 6 shows perhaps a surprisingly simple input to calculate and analyze the fine-structure of Th2<sup>+</sup> ions, this "simplicity" becomes relevant especially if other open *f*-shell ions or their properties and processes need to be considered. Apart from the *standard* input, the user has extensive control about the interatomic interactions and the amount of correlations, if the defaults are carefully overwritten.

#### *3.2. Transition Probabilities. Lifetimes and Branching Fractions*

For open *f*-shell elements, the transition probabilities and lifetimes need often to be estimated in order to identify and characterize the low-lying level structure from the intensities of the observed line spectra. In JAC, such estimates can readily be done by just modifying a few lines in the input as shown in the lower panel of Figure 6.

Here, the (value of the) processSettings tells JAC to calculate the Einstein A and B coefficients as well as the oscillator strength for the photon emission from the evento odd-parity levels, and as associated with the initial- and final-state configurations. In these computations, we just consider—in line with the defaults of the JAC toolbox—the electric-dipole transitions, although these *defaults* can also be readily modified within the code. Once the Atomic.Computation has been performed, all results are usually tabulated in a neat format, both at screen and within a summary file. In these tables, the atomic levels and transitions are then listed in terms of the level numbers as they arise from the diagonalization of the associated Hamiltonian matrix within the JAC program [32,40]. Moreover, the (full) representation of the initial and final-state multiplets as well as all the computed transition data can be obtained eventually if the optional argument output=true is given to the function perform(). Since the initial and final-state multiplets are determined

independently in the computation of all atomic processes, the rearrangement of the electron density is partly taken into account, though no attempt has been made so far to deal with this non-orthogonality in the evaluation of the angular coefficients.

**Table 3.** Excitation energies [eV] and lifetimes [s] of Th2+. Data from the JAC toolbox are compared with the NIST database ([58]), experiments and previous computations. Results are shown for 16 low-lying levels. These energetically low-lying levels can be identified uniquely using their energy and total symmetry despite of certain level crossings. In general, strong admixtures of other *LSJ* symmetries are typically found for these levels, which increase further as the valence-shell structure is "opened" further. See text for discussion.


We shall not display and compare here explicitly the transition probabilities with previous computations, and with typically a better agreement for the strong than the weak transitions. However, Table 3 compares the lifetime estimates from the present computations with the measurements by Biemont et al. [57] and recent calculations [60]. For these even parity levels with energies 32,000 cm−1, both the energies and lifetimes still exhibit rather large uncertainties due to the limited configuration basis of the present computations. Again, the identification of these levels is possible by means of a *jjJ* − *LSJ* transformation and the analysis of the leading *LSJ* terms. The lifetimes are shown here in velocity gauge and were found to differ by up to a factor of 3 from the corresponding lengthgauge computations. Since we wish to demonstrated the simple use of the JAC toolbox, no further enlargement is shown for the wave-function expansion nor the derived properties. Our two examples however manifest how JAC can be employed to generate much larger surveys of fine-structure levels as well as the—radiative and nonradiative—decay branches of the resonantly excited ion.

#### **4. Summary and Conclusions**

While the difficulties with open *d*- and *f*-shell elements can hardly be overrated, we have shown how the JAC toolbox *is* utilized to perform reasonably accurate computations for these shell structures. In particular, we explain how these tools help estimate the energies and properties for complex fine-structures. Apart from such simple estimates, JAC can also be applied to approximate and systematically improve relativistic ASF by including different classes (schemes) of virtual excitations with regard to a given set of reference configurations. In addition, the JAC toolbox also facilitates the computation of atomic processes, cascades or even the symbolic simplification of expressions from Racah's algebra [61].

Numerical results are shown above for the low-lying level structure of Th2<sup>+</sup> ions. These and similar computations for other actinide ions will be useful for developing new excitation schemes for heavy elements and for applications in medicine, radiation safety or elsewhere. Although, at present, the JAC program can often not immediately compete with its (numerical) accuracy with many-body perturbation or all-order techniques; these tools will help to go *beyond* the currently available applications of relativistic atomic structure theory.

Since JAC's very first design in 2017, the number of atomic properties and processes that can be handled by this code has grown steadily and it now supports the generation of (atomic) data for astro and plasma physics [62]. In fact, there are at present various demands to further advance the JAC toolbox: For open *d*- and *f*-shell elements, these requests mainly refer to efficiency and memory issues, the re-use of angular coefficients or the coupling of free electrons in ionization or capture processes. With the present version, however, a major step has already been made to obtain useful estimates and data for a large class of heavy and super-heavy elements.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**

