Jena Atomic Calculator (JAC) Framework

• JAC is a fully relativistic, configuration-based computational framework designed to predict atomic observables, including photoabsorption cross sections and charge-state distributions in heavy ions.
• It couples Dirac single-electron orbitals into configuration state functions (CSFs) and employs advanced algorithms to automate radiative and autoionizing cascade tracking.
• Validation against experimental benchmarks shows JAC achieves accuracies within 25% for photoabsorption and effectively reproduces charge distributions in La+ ions.

The Jena Atomic Calculator (JAC) is a fully relativistic, configuration-based computational framework designed for the quantum-atomic many-body problem, with particular emphasis on the calculation of bound and continuum states, transitions, and deexcitation cascades in heavy ions. As detailed in Looshorn et al. (Looshorn et al., 27 Dec 2025), JAC’s methodological architecture couples Dirac single-electron orbitals into configuration state functions (CSFs), enabling large-scale, tractable theoretical predictions for atomic observables such as photoabsorption cross sections and charge-state distributions pertinent to nonequilibrium plasma environments and kilonova opacity modeling.

1. Theoretical and Computational Architecture

JAC adopts a fully relativistic approach for constructing atomic wavefunctions $\Psi$, employing the coupling of single-electron Dirac orbitals into CSFs. For smaller mechanisms, it supports multiconfiguration interaction (CI); however, for production-scale runs involving lanthanum-like ions ($Z=57$), a single-configuration description is predominantly used to manage the size of Hamiltonian matrices and the ensuing cascade networks. Photoionization cross sections are formulated in the dipole approximation. The length-gauge cross section expression is

$$\sigma(\omega) = 4\pi^2 \alpha a_0^2 \sum_{i,f} \left| \langle \Psi_f \mid \mathbf{r} \cdot \hat{\epsilon} \mid \Psi_i \rangle \right|^2 \delta(E_f - E_i - \hbar\omega)$$

with $\alpha$ the fine-structure constant, $a_0$ the Bohr radius, and $\hat{\epsilon}$ the photon polarization. JAC numerically evaluates reduced dipole matrix elements using a projection method, with continuum orbitals discretized on a positive-energy Dirac mesh.

Bound discrete states are solved via a self-consistent-field Dirac–Hartree–Fock procedure, optionally extended by CI among selected configurations. Continuum orbitals are generated using the same central-field and orthonormalized to the bound set. In the application to La$^+$, the continuum is discretized with 0.1 eV energy steps, and the delta function in the cross section is represented by a Voigt profile (Gaussian 1 eV plus Lorentzian 0.6 eV) to match experimental linewidths.

2. Innovations and Algorithmic Upgrades for Heavy Ion Cascades

To extend JAC’s applicability to the M-shell region of La$^+$, several upgrades were necessary. A notable innovation is the algorithmic enumeration of all reachable ionic configurations and transitions following the creation of a $3d$ hole, which initiates cascades involving radiative (R) and autoionizing (A) steps. The advanced JAC implementation tracks final charge states $q=2\dots10$, automating the generation of all $1\,025$ configurations and $141\,245$ cascade links ($54\,738$ autoionization, $86\,507$ radiative) relevant to these steps. Computational tractability is achieved by pruning transitions with rates below $10^7$ s$^{-1}$ and truncating cascade paths for $q>10$.

A branching-ratio formalism quantifies the transition probability at each cascade step $k$ as

$$B_k(E) = \frac{A_k(E)}{\sum_{R}A_R(E) + \sum_{A}A_A(E)}$$

(over all open radiative and autoionization channels). The probability to reach a charge state $q$ after photoionization with photon energy $E$ is

$$P_q(E) = \sum_{\text{paths}\rightarrow q} \prod_{k\in\text{path}} B_k(E)$$

yielding the partial cross section for net $(q-1)$-fold ionization:

$$\sigma_q(E) = \sigma_\text{abs}^\text{dir}(E) \cdot P_q(E)$$

where only direct continuum absorption (excluding cascade resonances) is considered in the kernel.

Parallelization is implemented across initial fine-structure levels and continuum energy bins. Employing a sparse-matrix representation reduces memory usage by approximately $60\%$.

3. Validation: Benchmark Comparisons and Computational Metrics

JAC’s direct (non-resonant) photoabsorption cross sections agree to within $25\%$ of NIST-recommended values for neutral La above the $3d$ ($\sim830$ eV) but below the $3p$ ($\sim1120$ eV) threshold. Resonant structures associated with $3d\rightarrow4f$ excitation are accurately predicted when a uniform $-1$ eV empirical shift is applied, resulting in resonance energies matching experiment to $\lesssim0.1$ eV and peak spectral areas aligning to $30$–$50\%$.

Charge state distributions are characterized by partial cross sections $\sigma_q$ for $q=3\dots11$, measured experimentally via a La$^+$ beam prepared with 12 [Xe] $5d^2$, $5d6s$, $6s^2$ levels. The theoretical mean charge $\bar{q}(E)=\sum qf_q(E)$, with $f_q=\sigma_q/\sum\sigma_q$, satisfactorily reproduces experiment: at $E=1000$ eV, models A and B yield $\bar{q}=5.48$ and $5.73$ versus the measured $5.66\pm0.14$. Computed charge distributions are narrower than experiment (FWHM $\sim2.2$ versus $\sim2.5$ charge units), attributable to omitted double-Auger and shake-off processes and the single-configuration method.

A full simulation (photoabsorption plus cascades over 280 energy bins) required approximately $210$ CPU-hours on a 32-core node and peaked at $48$ GB of RAM. Bin-wise parallelization scaled linearly up to 64 cores, above which I/O overhead became limiting.

Benchmark	JAC (Unshifted)	JAC (Shifted)	Experiment/NIST
Mean charge ($\bar{q}$, $E=1000$ eV)	5.48	5.73	5.66$\pm$0.14
FWHM of $f_q$	2.2	2.2	2.5
Photoabsorption cross section accuracy	$\pm25\%$	matches peak positions ($<0.1$ eV)	reference

4. Limitations and Methodological Trade-offs

JAC’s current production architecture for heavy, open-f-shell ions does not include double-Auger, shake-off, or explicit metastable initial-state contributions within the cascade kernel, resulting in predicted charge distributions that are somewhat narrower than experiment. Energy scales for resonances necessitate an empirical $-1$ eV shift, attributed to incomplete inner-shell correlation treatment. The restriction to single-configuration descriptions for each ionic level constrains absolute transition rate accuracy, while simplifying calculations for large-scale networks.

5. Prospective Improvements

Planned enhancements include expansion of CI for inner-shell hole and $4f^2$ dominant configurations, aiming to eliminate empirical energy shifts. Density-matrix or perturbative correction schemes are proposed for the inclusion of double-Auger and shake-off processes. Initial populations could be sampled from multi-level (ground plus metastable) distributions, providing non-LTE realism. Parallelization strategies such as hybrid MPI/OpenMP are identified as necessary to advance scaling to $\gtrsim128$ cores for faster runs and larger systems. Integration of collisional–radiative modules is anticipated, with the explicit goal of direct coupling to non-equilibrium plasma codes and kilonova-opacity solvers.

6. Significance and Impact

JAC now provides a robust, integrated framework for modeling bound and continuum structure, resonant and direct photoabsorption, and radiative/autoionization cascades in systems with thousands of fine-structure levels. Agreement with experimental measurements of La$^+$ M-shell photoionization lies at or near current uncertainty margins. This suggests JAC is poised to be a central computational resource for atomic data generation in advanced plasma modeling, particularly in the astrophysical context of kilonovae and other extreme nonequilibrium plasma phenomena (Looshorn et al., 27 Dec 2025).