---
title: Jena Atomic Calculator (JAC) Framework
url: https://www.emergentmind.com/topics/jena-atomic-calculator-jac
type: topic
---

# Jena Atomic Calculator (JAC) Framework

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. [2512.22574], 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 [2512.22574].

Source: https://www.emergentmind.com/topics/jena-atomic-calculator-jac