Flexible Atomic Code (FAC)

• FAC is a fully relativistic atomic-physics code that calculates atomic structure, radiative, and collisional data across a wide range of ions.
• It employs Dirac equation solutions with Breit interaction and vacuum polarization to model configuration interactions and accurately determine level energies for spectroscopy and plasma simulations.
• FAC serves both as a primary production tool and an independent benchmark, though its distorted-wave approach may underrepresent resonance effects in some collisional processes.

Flexible Atomic Code (FAC) is a fully relativistic atomic-physics code used to generate atomic structure, radiative, and collisional data for ions ranging from light elements to open-$4f$ heavy systems. In the literature represented here, FAC appears both as a primary production code and as an independent benchmark against GRASP, DARC, HULLAC, AUTOSTRUCTURE, CHIANTI-based datasets, and experiment. Its outputs include level energies, wavelengths, radiative transition probabilities, oscillator strengths, collision strengths, effective collision strengths, ionization cross sections, and data products for collisional-radiative, non-local thermodynamic equilibrium, and opacity modeling (Zhang et al., 2023, Lu et al., 2021, Han et al., 2015, Flörs et al., 10 Jul 2025).

1. Formalism and computational character

FAC is described in multiple studies as a fully relativistic code that solves the Dirac equation and includes Breit interaction, vacuum polarization, and electron self-energy with standard procedures (Zhang et al., 2023, Zhang et al., 2024). In the lanthanide opacity study, FAC is presented as an open-source relativistic atomic-structure package based on the Dirac-Coulomb Hamiltonian,

$$H_{DC}=\sum_{i=1}^{N}\left(c \bm{\alpha}_{i}\cdot \bm{p}_{i}+(\beta_i-1)c^2+V_i\right)+\sum_{i<j}^{N}\frac{1}{r_{ij}},$$

with atomic state functions expanded in configuration state functions,

$$\Psi(nJM_JP)=\sum_{i=1}^{N_{\mathrm{CSF}}} c_i^n\,\Phi_i(\gamma_iJM_JP).$$

The same work emphasizes that FAC employs a common local central potential for all orbitals, yielding inherent orbital orthogonality, and that its native output is in $jj$-coupling (Flörs et al., 10 Jul 2025).

For lowly charged tungsten, FAC was used in relativistic configuration-interaction mode rather than in its RMBPT option, because only second-order perturbative corrections are included in FAC’s MBPT implementation and this was judged unsuitable for the strong-correlation regime of $\mathrm{W}^{8+}$ (Lu et al., 2021). In that study, orbital optimization proceeds through a fictitious mean configuration with fractional occupation numbers in the Dirac-Hartree-Fock-Slater approximation, after which Breit interaction, vacuum polarization, and electron self-energy are added (Lu et al., 2021). This suggests that FAC’s numerical architecture is flexible enough to support both compact production runs and deliberately enlarged correlation models, but that the practical quality of a calculation is strongly conditioned by how the mean configuration and configuration set are chosen.

2. Atomic structure calculations and configuration interaction practice

FAC is routinely used to compute level energies, theoretical wavelengths, radiative transition probabilities, and oscillator strengths. In tokamak spectroscopy of aluminum, FAC contributed level energies, wavelengths, radiative rates, and collisional rate coefficients across ions from Al$^{3+}$ through Al$^{12+}$; the configuration expansions documented there range from 24 levels and 92 E1 transitions for Li-like Al$^{10+}$ to 1088 levels and 152,729 E1 transitions for O-like Al$^{5+}$ (Zhang et al., 2023). In the $\mathrm{W}^{8+}$ EBIT study, a large FAC relativistic CI expansion with extensive $$H_{DC}=\sum_{i=1}^{N}\left(c \bm{\alpha}_{i}\cdot \bm{p}_{i}+(\beta_i-1)c^2+V_i\right)+\sum_{i<j}^{N}\frac{1}{r_{ij}},$$0- and $$H_{DC}=\sum_{i=1}^{N}\left(c \bm{\alpha}_{i}\cdot \bm{p}_{i}+(\beta_i-1)c^2+V_i\right)+\sum_{i<j}^{N}\frac{1}{r_{ij}},$$1-correlation produced 537,988 energy levels for the high-accuracy structure calculation of the lowest 30 levels (Lu et al., 2021).

The configuration-management problem becomes much more severe in open-$$H_{DC}=\sum_{i=1}^{N}\left(c \bm{\alpha}_{i}\cdot \bm{p}_{i}+(\beta_i-1)c^2+V_i\right)+\sum_{i<j}^{N}\frac{1}{r_{ij}},$$2 systems. The lanthanide opacity study used FAC for all singly and doubly ionized lanthanides from La to Yb and reported 146,856 energy levels below the ionization threshold and 28,690,443 E1 transitions among those levels; for ions near half-filled $$H_{DC}=\sum_{i=1}^{N}\left(c \bm{\alpha}_{i}\cdot \bm{p}_{i}+(\beta_i-1)c^2+V_i\right)+\sum_{i<j}^{N}\frac{1}{r_{ij}},$$3 shells, computational limits reduced the usable configuration set to fewer than 10 configurations, whereas ions near the ends of the series admitted hundreds (Flörs et al., 10 Jul 2025). Because experimental compilations are predominantly LS-labeled while FAC outputs are naturally $$H_{DC}=\sum_{i=1}^{N}\left(c \bm{\alpha}_{i}\cdot \bm{p}_{i}+(\beta_i-1)c^2+V_i\right)+\sum_{i<j}^{N}\frac{1}{r_{ij}},$$4-coupled, that work transformed FAC wave functions with JJ2LSJ before manual calibration against measured levels (Flörs et al., 10 Jul 2025).

Several studies use FAC structure calculations to isolate whether disagreements in scattering data arise from the target representation or from the collision treatment. In the Br XXVII comment, level energies and oscillator strengths from FAC and GRASP were described as comparable, and the near identity of representative $$H_{DC}=\sum_{i=1}^{N}\left(c \bm{\alpha}_{i}\cdot \bm{p}_{i}+(\beta_i-1)c^2+V_i\right)+\sum_{i<j}^{N}\frac{1}{r_{ij}},$$5-values was used to argue that anomalous published collision strengths could not be explained by target-structure differences (Aggarwal, 2018). In F-like W LXVI, FAC energies for the 113-level model were used as an independent check on the GRASP target and were reported to differ by less than 0.6 Ryd (Aggarwal, 2016).

3. Radiative, collisional, and rate-data outputs

FAC is not limited to bound-state structure. Across the papers considered here, it is used to produce radiative rates, distorted-wave excitation data, ionization data, autoionization rates, and the rate coefficients required for collisional-radiative models. In the RCF NLTE study, the SFAC interface to FAC supplied $$H_{DC}=\sum_{i=1}^{N}\left(c \bm{\alpha}_{i}\cdot \bm{p}_{i}+(\beta_i-1)c^2+V_i\right)+\sum_{i<j}^{N}\frac{1}{r_{ij}},$$6-coupled energies $$H_{DC}=\sum_{i=1}^{N}\left(c \bm{\alpha}_{i}\cdot \bm{p}_{i}+(\beta_i-1)c^2+V_i\right)+\sum_{i<j}^{N}\frac{1}{r_{ij}},$$7, spontaneous decay rates $$H_{DC}=\sum_{i=1}^{N}\left(c \bm{\alpha}_{i}\cdot \bm{p}_{i}+(\beta_i-1)c^2+V_i\right)+\sum_{i<j}^{N}\frac{1}{r_{ij}},$$8, electron-impact excitation cross sections $$H_{DC}=\sum_{i=1}^{N}\left(c \bm{\alpha}_{i}\cdot \bm{p}_{i}+(\beta_i-1)c^2+V_i\right)+\sum_{i<j}^{N}\frac{1}{r_{ij}},$$9, photoionization cross sections $$\Psi(nJM_JP)=\sum_{i=1}^{N_{\mathrm{CSF}}} c_i^n\,\Phi_i(\gamma_iJM_JP).$$0, electron-impact ionization cross sections $$\Psi(nJM_JP)=\sum_{i=1}^{N_{\mathrm{CSF}}} c_i^n\,\Phi_i(\gamma_iJM_JP).$$1 or $$\Psi(nJM_JP)=\sum_{i=1}^{N_{\mathrm{CSF}}} c_i^n\,\Phi_i(\gamma_iJM_JP).$$2, autoionization rates $$\Psi(nJM_JP)=\sum_{i=1}^{N_{\mathrm{CSF}}} c_i^n\,\Phi_i(\gamma_iJM_JP).$$3, and radiative recombination cross sections $$\Psi(nJM_JP)=\sum_{i=1}^{N_{\mathrm{CSF}}} c_i^n\,\Phi_i(\gamma_iJM_JP).$$4; inverse processes such as collisional de-excitation, three-body recombination, and dielectronic capture were then derived by detailed balance (Han et al., 2015).

For electron-impact excitation studies, FAC frequently provides $$\Psi(nJM_JP)=\sum_{i=1}^{N_{\mathrm{CSF}}} c_i^n\,\Phi_i(\gamma_iJM_JP).$$5 and, after averaging, $$\Psi(nJM_JP)=\sum_{i=1}^{N_{\mathrm{CSF}}} c_i^n\,\Phi_i(\gamma_iJM_JP).$$6. The effective collision strength is defined in several of these papers as

$$\Psi(nJM_JP)=\sum_{i=1}^{N_{\mathrm{CSF}}} c_i^n\,\Phi_i(\gamma_iJM_JP).$$7

with $$\Psi(nJM_JP)=\sum_{i=1}^{N_{\mathrm{CSF}}} c_i^n\,\Phi_i(\gamma_iJM_JP).$$8 entering excitation and de-excitation rate coefficients (Aggarwal, 2018, Aggarwal et al., 2012). In spectroscopy-oriented collisional-radiative models, FAC-derived rate coefficients populate standard level-balance equations. For the EAST aluminum analysis, the level populations satisfy

$$\Psi(nJM_JP)=\sum_{i=1}^{N_{\mathrm{CSF}}} c_i^n\,\Phi_i(\gamma_iJM_JP).$$9

and line emissivities are then

$jj$0

FAC data were one of the inputs to that model (Zhang et al., 2023).

Quantity class	FAC output or role	Representative use
Atomic structure	Energies, wavelengths, $jj$1-values, $jj$2, level mixing	Al EUV identification; $jj$3 levels; lanthanide line lists
Excitation and ionization	$jj$4, $jj$5, $jj$6, direct ionization, excitation-autoionization, autoionization	Ti XIX and W LXVI benchmarking; W$jj$7 influx modeling
Plasma modeling	Level populations, emissivities, S/XB coefficients, NLTE and LTE inputs	EAST CR models, RCF, kilonova opacity calculations

In the W$jj$8 influx study, FAC was also the sole ionization code: it provided direct electron-impact ionization in the distorted-wave approximation and excitation-autoionization contributions up to $jj$9 manifolds, which were then folded into metastable-resolved S/XB coefficients for EUV influx diagnostics (Zhang et al., 2024).

4. FAC as benchmark, comparator, and source of controversy

A recurrent role of FAC is that of an independent comparator against close-coupling $\mathrm{W}^{8+}$0-matrix calculations. In F-like W LXVI, present FAC collision strengths were reported to agree with present DARC values within 20% for three representative forbidden transitions, and the same paper used FAC to correct earlier published FAC energies that had been wrong by up to $\mathrm{W}^{8+}$1 Ryd for the highest levels (Aggarwal, 2016). In Be-like Ti XIX, FAC1 energies agreed with preferred GRASP2 energies within 0.04 Ryd, and for about 60% of transitions the DARC and FAC collision strengths agreed within 20% at about 700 Ryd; however, the paper also stated that FAC $\mathrm{W}^{8+}$2 values for weak forbidden transitions were not assessed to be accurate and attributed some anomalous high-energy behavior to interpolation and extrapolation procedures in FAC (Aggarwal et al., 2012).

The same general pattern appears in He-like systems. For Kr XXXV, 64% of transitions showed DARC–FAC $\mathrm{W}^{8+}$3 agreement within 20% at about 2800 Ryd, but forbidden transitions and low-energy behavior were more problematic, and FAC-based $\mathrm{W}^{8+}$4 missed resonance enhancement present in DARC (Aggarwal et al., 2012). For Ga XXX, Ge XXXI, As XXXII, Se XXXIII, and Br XXXIV, FAC again served as the non-resonant distorted-wave reference against which DARC resonance effects were judged; strong radiative data agreed well, while differences in $\mathrm{W}^{8+}$5 for forbidden lines were ascribed to resonance physics absent from FAC (Aggarwal et al., 2013).

The Br XXVII comment is a more pointed case. There FAC is not criticized as defective; rather, it is the principal independent code used to expose inconsistencies in published DARC data. Aggarwal argued that the published DARC $\mathrm{W}^{8+}$6 for allowed transitions were nearly constant with energy when they should rise, underestimated by roughly a factor of two, and likely affected by missing or incorrectly applied top-up. Published $\mathrm{W}^{8+}$7 values were called “simply wrong,” with zeros, missing entries, and unphysical temperature dependence, and the coarse resonance mesh $\mathrm{W}^{8+}$8 Ryd was identified as a major source of failure (Aggarwal, 2018). A common misconception is therefore avoided by the literature itself: disagreement between FAC and another code is not, by itself, evidence of a defect in FAC; in several papers FAC is the consistency check used to reveal problems elsewhere (Aggarwal, 2018, Aggarwal, 2016).

5. Spectroscopy, plasma diagnostics, and astrophysical modeling

FAC has been used extensively in experimental spectroscopy. In EAST tokamak aluminum spectra, FAC, HULLAC, and AUTOSTRUCTURE were jointly used to identify EUV lines from Al$\mathrm{W}^{8+}$9 to Al$^{3+}$0; the paper states that the general agreement among the three was very good, while also noting a systematic trend in which FAC tended to predict slightly longer wavelengths than experiment, AUTOSTRUCTURE tended to predict shorter wavelengths, and HULLAC was often closest (Zhang et al., 2023). For $^{3+}$1 in an EBIT, FAC structure calculations and the collisional-radiative model implemented in FAC were central to identifying six visible M1 lines. The average wavelength deviation was about 1.38% when using the smaller FAC CRM structure directly, and about 0.88% when large-scale FAC wavelengths were merged with FAC CRM intensities (Lu et al., 2021).

Laboratory benchmarking has also shown where FAC-based diagnostics succeed and fail. In EBIT measurements of Fe XII–XIV density diagnostics, FAC theory was found reliable for Fe XIII 196.53/202.04 and Fe XIV 264.79/274.21 and 270.52/274.21, but showed a large discrepancy for the widely used Fe XII $^{3+}$2 ratio, implying that the associated atomic calculations are problematic for that ion (Arthanayaka et al., 2019). In carbon ionization modeling, FAC-based cross sections from the Dere (2007)/CHIANTI compilation behaved overall similarly to GIPPER and MRBEB results, with the largest deviations near the peak maximum and with plasma consequences that decreased as temperature and ionization degree increased (Avillez et al., 2019).

In astrophysical X-ray spectroscopy, FAC underpins much larger atomic datasets. The Fe-L complex study recalculated direct excitation, resonant excitation, dielectronic recombination, and associated cascade networks with FAC for Fe XVII–Fe XXIV. The resulting main Fe-L lines were found consistent within 20% with recent $^{3+}$3-matrix calculations, and when implemented in SPEX the new FAC-based model yielded systematically better fits than SPEX v3.04, with the mean Fe abundance decreasing by 12% and the O/Fe ratio increasing by 16% in two-temperature fits (Gu et al., 2019). FAC-derived atomic data have likewise been used for radiative-transfer opacities in kilonova modeling, where low-lying levels were calibrated against experiment and 66,591 transitions acquired experimentally calibrated wavelength information (Flörs et al., 10 Jul 2025).

Fusion-edge impurity diagnostics provide another specialized use. For W$^{3+}$4 in EAST, FAC generated the ionization data, radiative rates, excitation rates, and collisional-radiative coefficients needed for S/XB analysis of the 382.13 Å and 394.07 Å lines. In the edge-relevant density regime, the resulting S/XB ratios were reported to be almost independent of electron density but strongly temperature dependent; at $^{3+}$5 eV and $^{3+}$6, the S/XB values were of order 3 (Zhang et al., 2024).

6. Limitations, calibration strategies, and methodological lessons

The literature portrays FAC as powerful but not universal. Its distorted-wave treatment is repeatedly identified as the main limitation in excitation and ionization work requiring near-threshold resonances or strong channel coupling. For Ti XIX, Kr XXXV, and the He-like sequence from Ga to Br, FAC systematically underestimates many forbidden-transition effective collision strengths relative to DARC because resonances are absent, while some anomalously large FAC values are instead linked to interpolation or extrapolation artifacts in $^{3+}$7 (Aggarwal et al., 2012, Aggarwal et al., 2012, Aggarwal et al., 2013). The W$^{3+}$8 influx paper similarly notes that distorted waves are known for generally overestimating ionization cross sections, even though the FAC results agreed well with prior calculations and experiment there (Zhang et al., 2024).

At the same time, several studies show that FAC’s limitations are often dominated not by the relativistic formalism itself but by the completeness and calibration of the model space. In $^{3+}$9, $^{12+}$0-correlation contributed about 20% to excitation energies and single $^{12+}$1-electron excitations into $^{12+}$2–$^{12+}$3 about 10%, making large CI indispensable for accurate low-lying structure (Lu et al., 2021). In the lanthanide dataset, strong lines with $^{12+}$4 were found to agree well with experiment and semi-empirical calculations, but raw FAC wavelengths were not sufficient for spectroscopic identification; manual matching and energy correction against experimental levels produced 66,591 calibrated transitions, while the paper also concluded that calibration has only a marginal effect on opacities compared with level-density completeness (Flörs et al., 10 Jul 2025).

FAC’s breadth becomes a practical advantage when completeness and consistency across many processes are required. The RCF paper is explicit on this point: by generating essentially all atomic data from FAC, the NLTE model avoids patching together heterogeneous databases, and in the Fe photoionization benchmark the resulting level energies agreed with NIST within about 0.4%, more than 80% of radiative decay rates agreed within 20%, and collisional excitation cross sections agreed with ICFT and Dirac $^{12+}$5-matrix calculations (Han et al., 2015). A plausible implication is that FAC is most robust when used in one of two ways: either as a consistent single-source atomic-data engine for broad plasma modeling, or as an independently cross-checked benchmark within multi-code workflows. The papers surveyed here support both modes.