Atoms doi: 10.3390/atoms14080069

Authors: Liu-Yuan Shen Xin-Peng Wang Hong-Hui Li Hong-Yu Zhu Wei Zuo Qi Yuan

We present an ab initio quadrupole-invariant study of neutron-rich nuclei around N=28. Effective Hamiltonians and consistently evolved E2 operators are obtained from chiral NN+3N interactions using the ab initio valence-space in-medium similarity renormalization group, and the resulting E2 matrix elements are used to extract the deformation parameters. Along the even–even N=28 isotonic chain, the β values increase systematically from 48Ca to 40Mg, providing a direct deformation signature of the weakening of the N=28 shell closure. The extracted β and γ values show a prolate–oblate–triaxial evolution of the 01+ states from 40Mg to 42Si and 44S, while the 02+ states remain mainly in the prolate-to-triaxial region. Extending the analysis to the neighboring N=27 and 29 isotones, we find a systematic evolution from strongly prolate Mg isotopes, to oblate-side Si isotopes, and to prolate-to-triaxial S isotopes. These results provide a unified picture of quadrupole deformation, triaxiality, and configuration coexistence around N=28.

Atoms doi: 10.3390/atoms14080068

Authors: Sanja Tošić Vladimir A. Srećković Veljko Vujčić

We investigate reactive collision dynamics and effective interaction cross-sections using classical trajectory simulations on a reduced-dimensional reactive potential-energy surface containing reactant and product wells separated by an intermediate barrier region. The simulations are performed over a range of collision velocities for which direct scattering, transient trapping, and reactive trajectories coexist within the same interaction landscape. Trajectories are propagated using a velocity-Verlet integration scheme, while reaction probabilities are analyzed as functions of the impact parameter and initial projectile velocity. The calculated probability distributions exhibit strongly localized reactive windows in phase space separated by extended nonreactive regions, indicating pronounced sensitivity of the dynamics to both collision geometry and initial conditions. Probability maps in the (vx,b) plane reveal a fragmented phase-space structure and highly nonuniform accessibility of the interaction region across the investigated parameter range. The simulations further show substantial variations in the relative importance of reactive, trapped, and back-scattering trajectories with increasing collision velocity, together with non-monotonic behavior of the effective reactive cross-sections. Despite the intentionally reduced dimensionality of the model, the trajectory ensembles reproduce several characteristic features of complex reactive scattering dynamics, including transient trapping, competing dynamical pathways, and broad residence-time distributions. The present results demonstrate that reduced-dimensional classical trajectory approaches can already capture important phase-space mechanisms governing dynamical accessibility and channel competition in reactive molecular collisions.

Atoms doi: 10.3390/atoms14080067

Authors: Georgios Tzanoulinos Spiros Makris Vaios Lappas

Space-based navigation systems rely on atomic clocks aboard satellites to provide precise time and positioning information through synchronized radio-frequency signals. At the core are Atomic Frequency References (AFRs), which stabilize a local oscillator using atomic transitions to achieve exceptional accuracy and stability. To maintain 1 m positioning precision, timing uncertainties below 3 ns are required, achievable only with high-quality atomic clocks. This paper surveys the principles, architectures, and performance of AFRs in satellite navigation, including the types deployed in major Global Navigation Satellite System (GNSS) constellations such as GPS, Galileo, GLONASS, and BeiDou, and discusses current practices and emerging trends in satellite timing technologies, including the nascent Low-Earth-Orbit Positioning, Navigation, and Timing (LEO-PNT) paradigm.

Atoms doi: 10.3390/atoms14080066

Authors: Aaron T. Bondy Klaus Bartschat

We outline an intermediate step toward a semirelativistic BSR–RMT interface by using inner-region structure information generated with the B-spline R-matrix (BSR) method as the input to the Seaton/Badnell STGF/STGBF outer-region asymptotic codes used in R-matrix photoionization calculations. The long-term goal is to exploit the compact, nonorthogonal, term-dependent target descriptions available in BSR for time-dependent R-matrix calculations with the R matrix with time dependence (RMT) method, especially for processes sensitive to semirelativistic and spin-orbit effects. To probe these effects, we consider the ground-state photoionization of Ne8+, Ar16+, Fe24+, and Kr34+, focusing on resonance structures and the singlet-triplet separation of the predominantly 2s2p3Po and 2s2p1Po autoionizing states and their spin-orbit mixing. For Fe24+, we also analyze higher resonances and the region between the ionic thresholds, with R-matrix I (RM-I) calculations using Badnell’s version for comparison. The BSR results agree well overall with the available Iron Project data and with the NIST separations between the predominantly 2s2p3Po and 2s2p1Po levels. Since semirelativistic RMT currently uses RM-I input; the successful interfacing of BSR inner-region data with STGF/STGBF codes that likewise use RM-I input represents a direct precursor to semirelativistic BSR–RMT capability.

Atoms doi: 10.3390/atoms14080065

Authors: Luis A. Poveda Bill Poirier Arthur R. B. de Magalhães

Quantum relativistic solutions of a “square-root” version of the Klein–Gordon equation, for a particle in a one-dimensional Morse potential, are presented using methods previously proposed and applied to a particle in a harmonic oscillator. The methods lead to both numerical and analytical solutions, with the latter allowing smooth variation of the system parameters from non-relativistic to ultra-relativistic limits. Analytical expressions for the energy levels and wavefunctions are obtained, as solutions to a Schrödinger-type equation, including relativistic effects through a state-dependent rescaled mass. The eigenstates of the Morse potential exhibit suitable and smooth behavior and approach the corresponding harmonic oscillator solutions as the depth of the Morse potential well increases, as expected. A comparison is also presented between the relativistic harmonic oscillator obtained with this method and the so-called “Klein–Gordon oscillator”.

Atoms doi: 10.3390/atoms14080064

Authors: Manpreet Kaur Sushil Kumar Sukhjeet Singh A. K. Jain

In the present work, configuration assignments to K = 1/2 bandheads of given three-quasiparticle (3qp) quadruplets experimentally observed and yet to be observed in odd-A nuclei lying in the rare-earth mass region are carried out based on an improved version of an earlier empirical formulation. The important contributions of the irrotational component, residual n-p interactions, and rotor–particle coupling terms are explored and highlighted. The magnitude of the RPC term is found to be comparatively smaller (≈5–10 keV) than the irrotational and residual interactions and can affect the 3qp bandheads with small energy separation. We confirm the tentative configuration assignments of three K = 1/2 bandheads having configurations 1/2[411]π⊗7/2[523]π⊗5/2[523]ν and 1/2[411]π⊗9/2[514]π⊗7/2[514]ν in 175Yb and 9/2[514]π⊗7/2[514]ν⊗1/2[521]ν in 177Ta nuclides. The average deviation between experimental data and present model calculations for 175Yb and 177Ta is 37.9 keV and 163.2 keV, as compared to the earlier model estimates, which yielded a deviation of 249.7 keV and 222.3 keV, respectively. We also resolved some issues pertaining to ambiguous configuration assignments of 3qp quadruplets observed in 177Lu, 173Ta, and 179Ta nuclides. Encouraged by the good overall agreement among experimental and calculated bandhead energies, we have extended these calculations to predict the locations of 18 K = 1/2 bandheads, which are members of 3qp quadruplets of 171,175,177Lu, 173,179,181Ta, and 179,183Re nuclides but have not been observed experimentally and will be useful for predicting bandheads associated with various 3qp states.

Atoms doi: 10.3390/atoms14080063

Authors: Jinn Ming Yap Peter Schury Sarah Naimi Gemma Bartram Alison M. Bruce Navodhayan Chandrakumar Jiulong Chen James G. Cubiss Siddharth Doshi Sunil Dutt Jilehu Gada Song Guo Yoshikazu Hirayama George Hudson-Chang Sota Kimura Filip G. Kondev Gregory Lane Jenny Lee Guangshun Li Yury Litvinov Zhong Liu Pengcong Ma Sorin Pascu Zsolt Podolyák Aiko Takamine Michiharu Wada Philip Walker Hiroshi Watanabe Yutaka X. Watanabe

Mass measurements of isotopes around the A=184 region near stability were performed at the KEK Isotope Separation System within the RIKEN Nishina Center, Japan. The isotopes were produced via multi-nucleon transfer reactions with a 136Xe beam of 7.2 MeV/u impinging on a natural tungsten target. We report the first direct mass measurement of 184Ta, showing a 62(27) keV deviation compared to previous indirect measurements.

Atoms doi: 10.3390/atoms14070062

Authors: Zi-Jian Xiang Rui-Yuan Zhang Jing Guo

Tailoring the spatial and angular-momentum structure of high-harmonic radiation is an important route toward controllable extreme-ultraviolet structured light. Here, we introduce an orbital angular momentum (OAM)-resolved photon-channel framework to elucidate spatial-mode formation in high-harmonic generation driven by bichromatic vortex fields. Combined with numerical solutions of the time-dependent Schrödinger equation for argon, this framework shows that the spatial structure of the emitted harmonics is governed by spectral broadening and interference among dominant OAM-carrying photon channels. This mechanism drives a continuous transition from clean vortex beams at harmonic peaks, characterized by high OAM purity, nearly pure circular polarization, and strong transverse coherence, to disordered speckle-like patterns in harmonic valleys, where multi-channel interference induces pronounced phase distortion. These results identify photon-channel interference as a useful mechanism for controlling the OAM, polarization, and spatial coherence of harmonic beams. The proposed analysis provides a field-control and OAM-channel framework that may be extended in future studies to structured-light probes of ultrafast dynamics and topological phases in quantum materials.

Atoms doi: 10.3390/atoms14070061

Authors: Michele A. Quinto Juan M. Monti Roberto D. Rivarola

Single-electron capture in the collisions of carbon ions, Cq+ (q = 1–6), with water molecules is theoretically investigated using the continuum distorted wave-eikonal initial state (CDW-EIS) formalism. The projectile–electron interaction is modeled using three different approaches to account for screening effects: (i) the Green–Sellin–Zachor (GSZ) potential, which combines long-range Coulomb and short-range screening terms; (ii) a modified GSZ-ZPr variant employing an effective radius-dependent nuclear charge; and (iii) the binding energy screening (BES) approximation, which treats the projectile as a rigid Coulomb core. Total cross-sections for single-electron capture are computed over the energy range of 40 keV/u to 10 MeV/u. A detailed analysis is presented as a function of the initial molecular orbital, the final bound state of the captured electron, and the projectile charge state q. The results reveal strong dependencies on orbital binding energies and the degree of projectile ionization. Additionally, of radiobiological interest, the average binding energy for the different projectiles are computed using the presented cross-sections and reported.

Atoms doi: 10.3390/atoms14070060

Authors: André F. F. da Silva Sergio Da Silva Raul Matsushita Giovanni F. Caramori

Earlier work by some of the present authors identified power-law statistical regularities across the periodic table: a near-Zipf relationship W∝Z1.09 between atomic weight and atomic number in neutral atoms, and a fine-structure splitting scaling ΔE∝Z1.78 in the 3d2+ (Cr-like) isoelectronic sequence of highly charged ions (HCIs). The apparent tension between this superlinear empirical exponent and the linear-in-Z scaling derived analytically by Lyu, Keitel, and Harman for relativistic clock-state transitions in nd6 ions is resolved by writing the scaling law in the form ΔE=A(Z−Z∗)γ, which gives γ=1.03 and Z∗=25.83 for the 3d2+ data and restores a near-Zipf interpretation. We extend the thresholded model from the single 3d2+ sequence to fourteen isoelectronic sequences spanning C-like (Z≥6) through Mo-like (Z≥42), drawing on the NIST Atomic Spectra Database, evaluated compilations of strontium, copper, and tungsten ion data, and digitization of GRASP MCDHF Mo-like results. Comparing the fitted threshold Z∗ against the classical Slater shielding σSlater of the valence shell reveals three regimes: (i) Coulomb-LS sequences (C/N/O/Ne-like) with Z∗ fixed at zero by construction; (ii) a mainstream cluster of nine sequences (a mix of spin–orbit fine-structure and, for the d2 Ca-/Sr-like members, Coulomb term separations) for which Z∗ tracks σSlater within ±1.5 units; (iii) a pair of nd6D05 sequences (Cr-like 3d6 and Mo-like 4d6) with Z∗−σSlater=+7.20 and +10.36, scaling linearly with Z in the jj regime (after resolving a low-Z LS–jj crossover for the Mo-like member). The slope ratio AMo/ACr≈0.48 is of the order of the hydrogenoid radial-extent ratio ⟨r2⟩(3d)/⟨r2⟩(4d)≈0.32, exceeding it by ∼50% as expected from relativistic 4d contraction. A falsifiable extrapolation for the W-like 5d6D05 sequence is offered: Δ∈[+7,+11], AW∈[0.12,0.18] eV/Z, with explicit failure modes specified for an independent GRASP MCDHF test.

Atoms doi: 10.3390/atoms14070059

Authors: Espen Gaarder Haug

The Bohr radius is normally presented as a non-relativistic length scale. Less widely discussed is that Bohr’s 1913 work also indicated how the radius formula changes when the orbital velocity is not negligible compared with the speed of light. We revisit this relativistic prescription and show that, for a point nucleus and a one-electron Coulomb field, it gives a0,r(Z)=a0Z1−Z2α2, which is exactly the most-probable radius obtained independently from the Dirac 1s1/2 radial probability density. The two radii are calculated independently and are found to be analytically identical. This equality does not derive the Dirac result from the Bohr model; rather, it shows that Bohr’s relativistic circular-orbit prescription selects the same radial scale as the maximum of the Dirac probability distribution. We also show that the same construction extends to the node-free circular Dirac excited states, for which rmpD(n,κ=−n)=(a0/Z)nn2−Z2α2. We emphasize throughout that most-probable radii are distinct from expectation values and from empirical radii of many-electron atoms.

Atoms doi: 10.3390/atoms14070058

Authors: Radu I. Campeanu Ladislau Nagy

The distorted-wave approach is applied to electron impact single ionization of krypton. We find that our models CPE4 and DW2 give results which are in fair agreement with the experiments. Both models contain a final state representation which includes the electrostatic repulsion between the ejected electron and the scattered electron.

Atoms doi: 10.3390/atoms14070057

Authors: Mehrdad Adibzadeh Constantine E. Theodosiou

We present an extensive set of theoretical results for spin asymmetry in the form of Sherman functions for the elastic scattering of electrons by zinc, cadmium, and mercury. This study extends the application to these three atoms of our earlier method of calculations, previously employed for stable inert gases and alkaline-earth metals. Our predictions are in adequate agreement with experimental values and precise theoretical results and extend the available theoretical data to a large impact energy range.

Atoms doi: 10.3390/atoms14070056

Authors: László Gulyás

Single-electron emission from the coronene (C24H12) molecule induced by the impact of 100 keV H+ projectiles is investigated using the continuum distorted wave–eikonal initial state approximation within a semi-classical impact-parameter framework. Multi-center orbitals in the initial channel are derived using the Gaussian quantum chemistry software package, while wave functions of the emitted electron in the exit channel are evaluated using a spherically averaged static potential of the molecular core. The evaluated total ionization cross sections and probabilities are compared and discussed with results obtained from a simplified description of the molecule in which the molecular geometry is ignored. The ionization cross sections of the molecular orbitals generally increase monotonically with decreasing binding energy; however, several deviations from this overall trend are observed. Both the overall trend and the deviations are interpreted in terms of information-theoretical quantities.

Atoms doi: 10.3390/atoms14070055

Authors: Haoran Lin Sijie Wu Ran Si Per Jönsson Chongyang Chen

In this work, we implement the finite-field (FF) method for isotope shift (IS) calculations within the relativistic multiconfiguration Dirac–Hartree–Fock (MCDHF) framework of the GRASPG program package. The implementation is benchmarked using the ten lowest-lying fine-structure levels of B II by comparing FF results for the IS parameters with first-order perturbation theory (PT) results obtained using the RIS4 program. The relative deviations for the IS parameters are within 0.03%, and the computed transition isotope shifts agree with previous theoretical predictions and experimental measurements within the reported experimental uncertainties. Additional calculations for Ar XV further confirm the consistency between the FF and PT methods for heavier ions, and the resulting isotope shifts are consistent with the experimental values. This implementation therefore provides GRASPG with two equivalent and reliable approaches for isotope shift calculations.

Atoms doi: 10.3390/atoms14070054

Authors: Imam S. Wahyutama

Molecular tunnel ionization sets the stage for the subsequent attosecond processes. Therefore, a comprehensive understanding of its dependence on various molecular properties, such as binding energy, wave function shape, dipole moment, and polarizability, is crucial for tailoring attosecond science concepts for scientific and technological applications. While the first three properties have been relatively well understood, the effects of polarizability have not been explored to the same extent in studies of molecular tunnel ionization. To address this gap, in this work we reveal the peak-shifting effect of permanent polarizability in tunnel ionization rates as the ionizing field varies. By analyzing a specially defined rate gradient, we determine the characteristic directions of the shift from the sign of the local slope of the gradient curve. Furthermore, we show how ionization into higher parabolic channels can transform minima of orientation-dependent rates into maxima. These findings demonstrate how the common assumption that the orientation dependence of tunneling rates resembles orbital shape can break down. We benchmark the method employed, the weak-field asymptotic theory (WFAT), and demonstrate good agreement with ab initio results, underscoring the efficiency and accuracy of WFAT in simulating strong-field ionization at low-to-moderate field strengths.

Atoms doi: 10.3390/atoms14070053

Authors: Thomas A. Gomez Mark C. Zammit Jackson White

Spectral line shapes are used to diagnose parameters of white dwarfs and neutron stars in particular. In magnetized plasmas, the motion of the radiating atom in the plasma needs to be considered in the collision process as the electronic structure of the atom depends on its center-of-mass translational momentum. More broadly, collision models do not explicitly or fully account for the motion of the nucleus, accounting for deflection through conservation of momentum. Traditionally, the correlation between electronic and nuclear motion has been captured through mass-polarization terms involving momenta scalar products between different electrons. We reformulate the collision problem accounting for the motion of the nucleus, taking advantage of unitary transformations. In this new formulation, Coulomb interactions between the atom and projectile/plasma particle become displaced Coulomb interactions, and exchange interactions include corrections of order 1/MA. We demonstrate the resulting impact on the elastic scattering T-matrices of the 1s state of hydrogen, where the lowest-energy electrons increase the real part by 20–30% while leaving the imaginary part practically unaltered. Lastly, we present a formulation so that the atomic motion can be explicitly included in the collision problem for magnetic-field applications.

Atoms doi: 10.3390/atoms14070052

Authors: Veljko Vujčić Bratislav P. Marinković Janina Kopyra Jelena B. Maljković Vladimir A. Srećković Sanja Tošić Nenad Aničić Nigel J. Mason

This work presents an extension of the Belgrade ACol collisional database within the Virtual Atomic and Molecular Data Centre (VAMDC) framework to include dissociative electron attachment (DEA) processes. DEA, a low-energy electron-driven resonant mechanism leading to molecular fragmentation, is relevant in fields such as plasma science, radiation damage, nanofabrication, and EUV lithography. The ACol data model was redesigned to improve semantic clarity and flexibility by separating physical collision data from bibliographic and serialization structures. A new DataSource entity and a redefined TabulatedData–Collision relationship enable a more normalized database while preserving compatibility with the XSAMS schema through dynamic, query-time construction of hierarchical structures. The implementation is demonstrated using DEA to isoflurane, incorporating experimental data from independent studies and cataloging resulting fragment anions and energy-dependent yields. Elastic cross sections of isoflurane are included as they are immanently connected to DEA, providing an essential tool for understanding the complex DEA process. The novelty of the present implementation lies not in the first representation of DEA within VAMDC, but in its integration into a multi-process collision database based on a compact, normalized data model that separates scientific data and bibliographic provenance from the XSAMS serialization structure. The updated architecture enhances interoperability, reduces redundancy, and improves data management while maintaining compliance with VAMDC standards.

Atoms doi: 10.3390/atoms14070051

Authors: Grant B. Bunker

The critical binding of quantum states in Screened Coulomb Potentials such as Yukawa/Debye, Hulthén, and ECSC (Exponential Cosine Screened Coulomb) potentials is of perennial interest and relevance in many fields of science, ranging from nuclear and particle physics; plasma physics, astrophysics, cosmology, and nuclear fusion; physical chemistry, condensed matter, and materials physics; to synthetic nanostructures and nanophotonics. The purpose of this paper is to heuristically explore two related mysteries, one new, the other more than 50 years old. The solutions to these mysteries have implications for a much broader class of potentials, those addressed by Klaus and Simon. In our recent paper we presented numerical calculations using the Phase Method (PM), which is accurate to 60 digits and to screening lengths D≤103 au and l = 0–20 of the critical binding parameters for these potentials and, for Yukawa and ECSC, l = 0–12 to D≤105 au, at 30 digits. In doing so, we discovered an anomalous period-40 sawtooth structure in the critical parameters of the ECSC potential that is not observed for the Yukawa potential. In this second paper, we quantitatively explain the origin and periodicity of this newly discovered structure. To do so, we use two complementary approaches: a “neoclassical” (NC) variant of conventional semiclassical phase-space quantization and the PM for very precise fully quantum calculations. The observed period-40 sawtooth structure is quantitatively explained in terms of a novel “tick-tock” mechanism. The periodicity is calculated in terms of the ratio of phase-space integrals for the primary and secondary potential wells. A quartic double-well potential is used as a simple model to further illustrate the tick-tock mechanism. Using the NC method, an approximate expression is derived to predict the locations of tick-tock glitches from higher-order wells; it is confirmed by a PM calculation up to D≤106 au. The second mystery is a strangely linear dependence of the total number of bound states vs. screening length for both the Yukawa and ECSC potentials. Using the PM, we confirm and extend these empirical relations. We show, using the PM, that an approximate trivariate linear relation between the square root of the critical screening length Dc, state number n, and angular momentum l applies to these potentials. This, plus a geometrical state accumulation argument, solve the second mystery. We show these properties derive from the scaling relation between screening length and coupling constant and, as such, are predicted to be applicable to the whole class of potentials. These results are expected to be of both theoretical interest and experimental relevance when interpreting spectra or calculating thermal properties. The significance of these results, and the applicability of these methods and conclusions to a vast array of related potentials, is briefly discussed.

Atoms doi: 10.3390/atoms14070050

Authors: Mehrdad Adibzadeh Constantine E. Theodosiou

We present an extensive set of theoretical results for differential, integrated, and momentum-transfer cross sections for the elastic scattering of electrons by zinc, cadmium, and mercury. Our approach is a self-consistent relativistic calculation, with a semi-empirically adjustable cutoff radius of the polarization potential. This study further extends the application of our method of calculations, previously employed for stable inert gases and alkaline-earth metals. Based on the satisfactory agreement of our previous investigations with experimental values and other precise theoretical results, we expect to provide a set of accurate data for Zn, Cd and Hg.

Atoms doi: 10.3390/atoms14070049

Authors: María Laura de Sanctis Marie-Françoise Politis Rodolphe Vuilleumier Omar Ariel Fojón

We theoretically study the single ionization of liquid water by impact of fast electrons. A realistic description of the wavefunction for an isolated water molecule in the liquid phase is obtained by means of a Wannier orbital formalism. In this way, we consider ionization from the most external orbitals 1B1, 2A1, 1B2 and 1A1 of a single liquid water molecule. Triple, double, single differential and total cross sections are computed through a first order model with proper Coulomb conditions for the ionized electron. We compare our calculations with measurements and other theoretical results for liquid and gaseous phases. An analysis of the main features of the cross sections is performed. Previous theoretical works found almost no discrepancies between these observables in spite of the physical dissimilarities of both phases. In this work, we compare our results with other theories and with the available experiments for vapor. We report interesting differences between the differential and total cross sections of the mentioned phases.

Atoms doi: 10.3390/atoms14070048

Authors: Kayim Pineda-Urbina

Ionization potentials and electron affinities provide the energetic basis for several conceptual density functional theory descriptors, but their use in donor–acceptor maps requires careful distinction between physically bound anions, weak or borderline electron-affinity cases, and formally computed diagnostic states. In this work, a periodic donor–acceptor descriptor map was constructed for main-group atoms from H to Kr using a ΔSCF-DFT framework. Neutral atoms, monocations, and formally defined monoanionic states were evaluated to obtain ionization potentials, electron affinities, and global reactivity descriptors, including electronegativity, chemical hardness, chemical potential, electrophilicity, electrodonating power, and electroaccepting power. The production dataset was calculated at the ωB97X-D4/def2-QZVPPD level and benchmarked against reference atomic data. This protocol reproduced ionization potentials with a mean absolute error of 0.134 eV and electron affinities with a mean absolute error of 0.116 eV for the reference EA set, including the weak calcium case. A functional and basis-set sensitivity analysis using ωB97X-D4/def2-TZVPPD, PBE0/def2-QZVPPD, and PBE0/def2-TZVPPD showed that ionization potentials are comparatively robust, whereas electron affinities are strongly affected by the quality of the diffuse basis set. The normalized donor–acceptor map reproduces chemically intuitive periodic trends, with alkali metals occupying the strong-donor region and halogens defining the strong-acceptor region. The analysis explicitly separates core validation atoms from weak or borderline electron-affinity cases and diagnostic finite-basis anionic states, emphasizing that formally computed negative electron affinities for unbound anions should not be interpreted as physical bound states. The resulting nonrelativistic dataset provides a reproducible atomic descriptor reference for interpreting donor–acceptor behavior in atoms, clusters, superatoms, doped materials, and charge-transfer systems.

Atoms doi: 10.3390/atoms14070047

Authors: Sebastián D. López Matías L. Ocello Martín Barlari Diego G. Arbó

We present a theoretical study focused on the photoelectron spectrum of near-infrared (NIR) laser-driven ionization of hydrogen atoms by attosecond pulse trains composed of several HHs of the former. We analyze the effects of increasing the intensity of the NIR probe laser to account for the interference of multiple quantum pathways arising from mainbands formed in ionization by the attosecond pulse train within the strong-field approximation (SFA) beyond the commonly used first-order perturbative (in the NIR laser intensity) reconstruction of attosecond beating by interference of two-photon transitions (RABBIT). The structure of the energy bands formed in the photoelectron spectrum is governed by quantum interferences of the photoelectron wave packet released within one optical cycle of the NIR probe laser field—intracycle interference—and by the number of active high harmonic components, leading to higher-order Fourier contributions as a function of the NIR–XUV relative phase delay. We show that Fourier terms can be interpreted in terms of well-defined semiclassical trajectories. Our results demonstrate a significant departure from the standard two-path quantum-interference RABBIT picture, showing that both the phase-dependent oscillations of mainbands and sidebands and the extracted phase delays depend strongly on the probing laser intensity. The predictions of the SFA reveal that the above-threshold ionization bands exhibit systematic splitting and oscillation patterns as a function of the NIR intensity. SFA predictions are compared with results obtained within ab initio solutions of the time-dependent Schrödinger equation (TDSE), showing an excellent agreement, which evidences the minor effect of the Coulomb potential of the remaining ion on the escaping photoelectron for high energy above-threshold ionization. The precise study of the SFA reference phases is essential for the determination of the effect of the Coulomb potential on the escaping photoelectron for what these findings provide new insights into attosecond chronoscopy in the strong-field regime.

Atoms doi: 10.3390/atoms14070046

Authors: Stephan Fritzsche Nishita M. Hosea Houke Huang Tianluo Luo Aloka K. Sahoo

Electron configurations are known to provide valuable insights into the electronic structure and behavior of atoms. They specify which and how the electronic (sub-) shells are occupied, and are thus an essential ingredient for most atomic observables. When combined with the shell model and the successive filling of shells, these configurations help explain the Periodic Table and much of chemical binding. They also establish a qualitative framework for analyzing excitation, ionization and relaxation processes and may facilitate a wide range of astrophysical and plasma simulations. Here, we review the role of electron configurations for understanding atomic behavior in interactions with particles and radiation. In particular, we identify several central requirements for an efficient treatment of configuration lists and define a domain-specific language in order to generate, manipulate and analyze such lists as well as to extract physically relevant information. We also demonstrate the implementation of this language in Jac, the Jena Atomic Calculator. An efficient handling of configurations will refine the coupling of structure codes with the spectral synthesis of plasma radiation, the setup of ionic cascades or even non-LTE plasma simulations. This common framework for dealing with electron configurations therefore improves consistency, reproducibility and scalability of atomic modeling.

Atoms doi: 10.3390/atoms14060045

Authors: Giulio Del Zanna

Ian Philip Grant (see Figure 1) was a monumental figure in relativistic atomic and molecular atomic physics [...]

Atoms doi: 10.3390/atoms14060044

Authors: Paweł Syty Michał P. Piłat Moein Sahraei Józef E. Sienkiewicz

Total, momentum-transfer, and differential cross sections, together with spin-polarization (Sherman) functions, are reported for elastic scattering of low-energy electrons from neutral strontium atoms over the energy range 0.001–15 eV. The calculations are performed within a fully relativistic Dirac framework for the continuum states. The target structure is described using multi-configuration Dirac–Hartree–Fock wavefunctions obtained with the GRASP2018 package, while continuum orbitals are generated using the recently developed GRASPC extension. Long-range target polarization effects are incorporated using a dipole model potential, and exchange interactions are treated explicitly for the large and small components of the continuum wavefunctions. Particular attention is given to the ultralow-energy regime, where reliable cross section data for Sr remain limited. The calculated total cross section exhibits a broad maximum near 1 eV, while the momentum-transfer cross section shows a shallow minimum near 0.05–0.06 eV. The differential cross sections are in good agreement with earlier static-exchange-plus-polarization calculations over much of the 1–5 eV range, whereas at lower energies, visible differences appear, especially at forward angles where the results are most sensitive to the polarization interaction. In the ultralow-energy region, the present differential cross sections remain smooth and show no indication of additional low-lying shape resonances within the adopted model. The calculated Sherman functions follow the general trends of earlier theoretical studies at higher energies and decrease rapidly in the sub-eV range. Overall, the present results provide a consistent relativistic dataset for elastic e–Sr scattering at low energies, with emphasis on the near-threshold region.

Atoms doi: 10.3390/atoms14060043

Authors: Xuechun Li Xuhui Bai Yanpei Zhang Chuanqi Jin Jie Chen Aritra K. Mukhopadhyay Yanting Zhao Zhonghua Ji Yongchang Han Gaoren Wang

We study exotic orbits in the photoassociation process by considering both the vibrational and rotational motions in a classical model. In the presence of rotational motion, the exotic orbits possess regular bound segments even when the total energy exceeds the threshold energy of the interaction potential. Such features of the exotic orbits are interpreted by introducing an effective potential. We employ Lagrangian descriptors and escape time to characterize the phase-space structure and show that exotic orbits are distributed around the stable region in the short-range phase space. We further calculate the photoassociation probability. Our work provides new insights into the dynamical mechanisms of photoassociation processes.

Atoms doi: 10.3390/atoms14060042

Authors: Nicolas Bachi Emiliano Acebal Nelson D. Cariatore Sebastian Otranto

In this work, state-selective electron-capture processes in collisions of Ar16+ with ground-state hydrogen are analyzed in classical terms by means of the classical trajectory Z-CTMC method. Oscillations in the n-state-selective charge-exchange cross-sections are observed in the impact-energy range 1–10 keV/u for n-values greater than the nmax value at which charge exchange maximizes. The oscillations are ascribed to an electron-swap mechanism between centers previously identified in ion–Rydberg and ion–alkali charge-exchange collisions. A detailed analysis of the structures in the perpendicular momentum-transfer distributions and their association with the different numbers of swaps is developed. Their dynamics in terms of the collisional impact parameters are also presented.

Atoms doi: 10.3390/atoms14060041

Authors: Jay Jay Tsui Tom Kirchner

Electron emission from hydrogen atoms induced by antiproton impact at intermediate energies is investigated using the one-centre Basis Generator Method within a semi-classical impact-parameter framework. The formulation employs a single-centre expansion of the time-dependent Schrödinger equation with a pseudostate basis consisting of hydrogenic orbitals acted upon by powers of a Yukawa-regularized potential, providing a compact and effective representation of the electronic continuum. Ionization probabilities are obtained by projecting the time-evolved wavefunction onto Coulomb continuum states, from which energy-differential cross sections (EDCS) are extracted. Exponential piecewise functions are constructed to interpolate between the pseudostate eigenenergies, yielding smooth EDCS profiles for each partial wave. The total EDCS, reconstructed by summing over all partial-wave contributions, exhibits good agreement with results from other pseudostate-based approaches.

Atoms doi: 10.3390/atoms14050040

Authors: Gediminas Gaigalas Pavel Rynkun Laima Kitovienė

A developed method, based on the stationary second-order Rayleigh–Schrödinger many-body perturbation theory in an irreducible tensorial form, allows us to determine the most important core–valence, core, core–core, and valence–valence correlations for any atom or ion with an arbitrary number of valence and core electrons. This paper presents the Feynman diagrams that describe these correlations. Additionally, it provides the rules for obtaining algebraic expressions in an irreducible tensorial form for any Feynman diagram coming from second-order many-body perturbation theory. Whereas some types of the valence–valence and core–valence correlations are described by the three-particle Feynman diagrams, additional developments to calculate the spin-angular parts of these diagrams have been made to the program library librang of the Grasp2018 As an example of the application of the developed method, the atomic calculations of the energy level structure and transition data for Ar II are presented.

Atoms doi: 10.3390/atoms14050039

Authors: Hristina Delibašić-Marković Veljko Vujčić Vladimir A. Srećković Violeta Petrović

Laser-induced breakdown in water-rich biological media results from the interplay between primary photoionization processes and avalanche amplification of free electrons. Understanding this competition is essential for predicting ablation thresholds under ultrashort-pulse irradiation. In this work, we develop an analytical rate-equation model for the buildup of electron density in water-like biological tissues. It combines photoionization and chromophore ionization into a single seed-generation term, while avalanche ionization is described through a cascade gain factor. This formulation provides a framework for describing cascade electron-impact ionization processes in liquid-like media under strong-field excitation. Our approach gives an analytical expression for the temporal evolution of electron density driven by a Gaussian laser pulse and makes it possible to separate the contributions of direct ionization of water and ionization of chromophore centers. The analytical results are compared with numerical simulations that include carrier diffusion, bimolecular recombination and trapping. The comparison clarifies the roles of seed formation and cascade amplification in the growth of the electron population. The predicted dependence of threshold fluence on pulse duration agrees well with experimental data reported for water-like tissues such as the corneal tissues at a wavelength of 800 nm. The model provides a simple analytical picture of ultrafast plasma formation and electron-driven energy deposition in water-like biological media.

Atoms doi: 10.3390/atoms14050038

Authors: Theo J. M. Zouros Emmanouil P. Benis

Resonant transfer and excitation (RTE) is a correlated two-electron ion–atom collision process mediated by the two-center electron–electron interaction: a projectile electron is excited while a target electron is captured, forming doubly excited states. These states decay via X-ray (RTEX) or Auger (RTEA) emission. For sufficiently fast collisions with light targets, RTE becomes analogous to dielectronic capture (DC)—a key plasma process—and is successfully described by the impulse approximation (IA). Early (1983–1992) RTEX and more stringent, state-selective RTEA measurements provided essential indirect DC cross-section information before direct electron–ion measurements became available. A 1992 review by the first author, focusing on zero-degree Auger projectile spectroscopy (ZAPS) of state-selective KLL D states, validated the IA for low-Zp (Zp≤9) projectile ions, yet a puzzling systematic discrepancy remained: IA RTEA cross-sections were consistently larger than experimental, with the disagreement increasing as Zp decreased. The present article reviews RTEA progress since 1992, including new refinements to IA calculations, an exact analytic IA formulation, and instrumental ZAPS improvements. A methodical analysis demonstrates impressive agreement across measurements spanning both pre- and post-1992 eras, including new experimental results, effectively eliminating previous systematic discrepancies. IA validity is confirmed down to boron ions, with He+ and certain Li-like ions remaining the only notable exceptions. Recently, a rigorous quantum mechanical ion–atom collision treatment has emerged: nonperturbative close-coupling calculations of transfer excitation for He-like carbon ions colliding with He confirm the dominance of RTE via two-center electron–electron interactions at large impact parameters, yielding RTEA results in excellent agreement with experiments.

Atoms doi: 10.3390/atoms14050037

Authors: Petar Danev Iavor Boradjiev Hristo Tonchev

We propose an analytical model and perform numerical simulations to study the time distribution of the characteristic muonic oxygen X-ray emission following muon transfer from muonic hydrogen to oxygen in a H2 + O2 gas mixture. The model accounts for all fundamental processes that alter the kinetic energy and spin distribution of muonic hydrogen atoms. The impact of the uncertainties in various experimental parameters on the precision of the computed results is studied in detail by means of the Monte Carlo method. Specifically, we observe the presence of a minimum in the time dependence of the relative standard deviation of X-ray emission for realistic parameter combinations, which can serve as a benchmark for comparing experiments and numerical simulations. Verification against available experimental data reveals the potential of this approach for both description and parameter optimization in the planning and analysis of muonic experiments

Atoms doi: 10.3390/atoms14050036

Authors: Sergey A. Zaytsev Darya S. Zaytseva Alexander S. Zaytsev Lorenzo U. Ancarani Konstantin A. Kouzakov Yury V. Popov

We investigate theoretically the 75 keV proton-impact ionization of atomic helium. The convoluted quasi-Sturmian approach is extended to treat, on an equal footing, both the direct mechanism and the electron capture to the continuum. This is achieved by proposing an ansatz of the Green’s function of the three-body Coulomb system (e−,He+,p+) that is compatible with the well-known 3C correlated continuum wave function. The model that stems from this approximation, named 3C˜, is tested numerically using parabolic Sturmian expansions. Calculations of fully differential cross sections are presented for different regimes of energy losses, namely for ejected electron energies below, nearly equal to, and above the cusp energy. Our results are compared with recent experimental measurements and other theoretical calculations. The proposed 3C˜ model yields very encouraging results and paves the way towards a more advanced Lippmann–Schwinger approach based on the 3C model.

Atoms doi: 10.3390/atoms14050035

Authors: Karol Kozioł

The Rci-Q package is an extension to the Grasp2018 suite, improving the model of estimating the quantum electrodynamics corrections to the energy levels. The Flambaum–Ginges radiative potential method is used to estimate the leading self-energy correction to electron energy in many electron atoms. The new fitting prefactors to parameterize radiative potential are presented. The correction to self-energy originating from finite nucleus size is included. The Wichmann–Kroll part of the vacuum polarization potential is also implemented.

Atoms doi: 10.3390/atoms14050034

Authors: Gregory J. Boyle Dale L. Muccignat Joshua R. Machacek Robert P. McEachran

We present fully relativistic calculations of integral cross sections and swarm transport properties for positron–radon scattering over a wide energy range (0–1000 eV) and reduced electric field range (0.01–1000 Td). Elastic (total, momentum-transfer and viscosity-transfer), discrete excitation, direct annihilation, positronium formation and positron-impact ionization cross sections are obtained using a complex relativistic optical potential method. Owing to the large atomic number of radon and the absence of experimental scattering data, a consistent relativistic treatment is essential. The present work provides the first fully relativistic, internally consistent cross-section dataset for positron swarms in radon gas. Using a multi-term solution of Boltzmann’s equation, steady-state transport coefficients are calculated and found to be strongly influenced by energy-dependent reactive loss, particularly positronium formation. Significant divergence between bulk and flux transport coefficients is observed, including non-monotonic bulk drift velocities and pronounced suppression of longitudinal bulk diffusion at intermediate fields (0.3–1000 Td). Time-dependent field-free calculations further quantify thermalization and annihilation dynamics through the evolution of the mean energy and ⟨Zeff⟩(t). These results provide a robust theoretical foundation for modelling positron transport and annihilation in radon and other heavy noble gases where relativistic and reactive effects are crucial.

Atoms doi: 10.3390/atoms14040033

Authors: James A. Perez Josh A. Muller

A three-body classical trajectory Monte Carlo method is used to investigate state-specific electron capture from H2O by highly charged ions. The radial and momentum distributions of the target electron are modeled using a one-center molecular orbital wave function. Total single-electron capture cross sections, as well as cross sections for capture into specific nl-states, are calculated for the highly charged ion projectiles, C6+, N7+, Ne10+, and Ar18+, at relative collision energies ranging from 0.01 keV/amu to 50 keV/amu. Comparisons of relative n-state capture populations and total single-electron capture cross sections are made with experimental results. The results show a marked improvement in the prediction of relative n-states populated, with the overall single-electron single capture cross sections being slightly low compared with experimental values. Overall, this method of calculating nl-states of the captured electron appears to be a promising approach for those wishing to model X-ray and Extreme Ultraviolet (EUV) emissions from comets bombarded by solar wind ions, and fusion researchers trying to determine the effects of impurities in Tokomak reactors.

Atoms doi: 10.3390/atoms14040032

Authors: Elmar Träbert

In Na- through Ca-like ions of Fe, rather low-lying 3d levels feature level lifetimes in the range from picoseconds to many seconds. This lifetime range is somewhat wider than that of the 3p resonance levels of the same ions. When trying to measure such level lifetimes, the width of the range exceeds the capabilities of a single measurement scheme. However, there also is the fundamental problem of multi-exponential decay curves and the reliability of their analysis. This problem has arisen afresh for the analysis of long-lived 3p levels in the ground configurations of many isoelectronic sequences that are replenished by cascades from long-lived 3d levels that have no E1 decay channels, but consequently feature lifetimes of the same order of magnitude as those in the ground configuration. This tutorial addresses the measurement situation for lifetimes of 3d levels in a number of ions of Fe and nearby elements.

Atoms doi: 10.3390/atoms14040031

Authors: Sh. U. Alladustov K. H. Spicer N. W. Antonio A. M. Kotian A. S. Kadyrov

Differential ionization in C6++He collisions is investigated using the single- and two-center wave-packet convergent close-coupling (WP-CCC) method for projectile energies of 1–6 MeV/u. We present three types of singly differential cross-sections (SDCSs) as functions of the ejection angle, ejection energy, and projectile scattering angle. The two-center framework incorporates couplings across all channels as well as electron correlations. Overall, both the single- and two-center WP-CCC results agree well with existing experimental and theoretical data (apart from the first Born ones) for the SDCS as a function of electron energy and the SDCS as a function of ejection angle, laying a foundation for investigation of doubly and fully differential ionization cross-sections. The cross-sections differential in the projectile scattering angle are presented for the first time.

Atoms doi: 10.3390/atoms14040030

Authors: Yijiao Wu Hao Yin Bingsheng Tu Tianming Meng Pufang Ma Xu Tan Ke Yao Jun Xiao Yaming Zou Baoren Wei

The energy-dependent population of fine quantum states in single electron capture (SEC) reflects the intrinsic collision dynamics. Here we report experimental studies of Ar7+ ions colliding with He in the energy range of 1.05–17.5 keV/u. Owing to the high resolution of a recoil-ion momentum spectrometer, the n-, l-, and spin-state electron capture populations are well resolved, and a strong energy dependence of the SEC cross sections is observed. Most importantly, a clear inversion of the cross-section ratio between the spin-resolved triplet and singlet 3s3d configurations is found, demonstrating a breakdown of spin statistics. Together with recent spin-resolved studies of C3+-He collisions (PRL 133, 173002 (2024)), these results suggest that the breakdown of spin statistics is likely a general feature of charge exchange in open-shell highly charged ion systems.

Atoms doi: 10.3390/atoms14040029

Authors: Gediminas Gaigalas

A program library, libang77, for computing pure spin-angular coefficients for any one- and scalar two-particle operator is presented. The method is based on the combination of the second quantization and quasi-spin techniques with angular momentum theory and the method of irreducible tensorial sets. A non-relativistic approach is used, in which relativistic corrections may be included in the Breit–Pauli approximation. This program library, libang77, is integrated into the Atomic Structure Package ATSP2K [ATSP2K, C. Froese Fischer, G. Tachiev, G. Gaigalas, and M.R. Godefroid, Comput. Phys. Commun. (2007). DOI: 10.1016/j.cpc.2007.01.006], but it can be implemented in other program packages too.

Atoms doi: 10.3390/atoms14040028

Authors: Hanfeng Yu Dalong Guo Xiaolong Zhu Xuexia Pang Yong Gao Dongmei Zhao Kaizhao Lin Jinjian Yu Shaofeng Zhang Xinwen Ma

State-selective single-electron capture in He+ + Ne collisions was studied at laboratory He+ projectile kinetic energies of 30–100 keV (corresponding to 7.5–25 keV/u) using a reaction microscope. Q-value spectra were obtained through recoil-ion momentum reconstruction, enabling the decomposition of the capture yield into three distinct contributions: (i) capture into excited states of the projectile without target excitation, (ii) capture into the projectile ground state accompanied by excitation of the residual Ne+ ion, and (iii) capture involving simultaneous excitation of both He and Ne+. Across the studied energy range, capture into the projectile ground state accompanied by target excitation is the dominant pathway. With increasing impact energy, the fraction of joint projectile–target excitation increases markedly, whereas the target-excitation-only contribution decreases; the projectile-excitation-only fraction remains at a low, nearly constant level. These findings underscore the significance of multi-electron dynamics in intermediate-energy collisions involving many-electron targets.

Atoms doi: 10.3390/atoms14040027

Authors: Anand K. Bhatia

The ionizations of the 1S state of hydrogenic systems with a nuclear charge of Z = 2 and 3 have been carried out using the hybrid theory. This is a continuation of the work started earlier. The present results are compared with the published cross-sections for Z = 1 [Bhatia, A.K. 2025]. The distortion of the orbit is considered irrespective of the position of the incident particle, whether it is outside or inside the orbit. Only the distortion of the target orbit in the initial state is considered, but the distortion in the final state is not considered. Cross-sections decrease as the nuclear charge increases.

Atoms doi: 10.3390/atoms14030026

Authors: Dibyendu Sardar Arpita Rakshit Somnath Naskar Bimalendu Deb

We develop a method for calculating multichannel wavefunctions in the spirit of quantum defect theory, based on numerically calculated reference functions. We benchmark the method by calculating cold collisional properties of 85Rb and 6Li in the presence of external magnetic fields tuned across specific s-wave Feshbach resonances and thereby reproducing known results. We then apply the method to calculate experimentally observed d-wave Feshbach resonance in 87Rb-85Rb cold collisions. Our numerical results for this d-wave resonance show good agreement with the experimental observations. The method is applicable to arbitrary interaction potentials and to any energy range near the scattering threshold. The implementation of our method to any multichannel two-body scattering problem is straightforward.

Atoms doi: 10.3390/atoms14030025

Authors: M. K. Pawelkiewicz Filippo Gatti Didier Clouteau Viatcheslav Kokoouline Mehdi Adrien Ayouz

Missing Supplementary Materials [...]

Atoms doi: 10.3390/atoms14030024

Authors: Elmar Träbert

A high-resolution flat-field grating spectrometer has been employed at the Livermore EBIT-I electron beam ion trap for observations of extreme-uv spectra of Ni-like ions Xe26+ and Co-like ions Xe27+. Multistep ionisation involving the long-lived 3d94s 3D3 level in the Ni-like ion as a stepping stone has a significant influence on the charge state distribution at a given electron beam energy, as has been reported elsewhere. Complementing those observations of 3d-4s E2 and M3 transitions from long-lived levels, the present report shows spectra of 3d-4p and 3d-4f E1 transitions that arise from the decays of short-lived levels in both ions and their neighbouring ions of higher charge states and provide bright reference signals for the changes in the charge state distribution. Their observation is serendipitously furthered by the visual absence of 3d-4d transitions from the observed spectra, although M1 and E2 transitions between these configurations are permitted.

Atoms doi: 10.3390/atoms14030023

Authors: Randy Kuang

Quantum Encryption in Phase Space (QEPS) is a physical-layer encryption framework that harnesses the quantum-mechanical properties of coherent states to secure optical communications against both classical and quantum computational threats. By applying randomized phase shifts, displacements, or their dynamic combinations—implemented as unitary transformations in phase space—QEPS disrupts the phase reference essential for coherent detection, establishing aphase synchronization barrier. This review synthesizes the theoretical foundations, security mechanisms, and experimental progress of the QEPS framework, encompassing its three principal variants: the round-trip Quantum Public Key Envelope (QPKE) protocol—a public-key-like scheme built upon phase randomization (QEPS-p), the symmetric phase-only QEPS-p, and the displacement-based QEPS-d. Experimental validations demonstrate that authorized users achieve bit-error rates (BERs) below the forward-error-correction threshold, whereas eavesdroppers are confined to BERs near 50%, equivalent to random guessing—all while utilizing standard coherent optical transceivers at data rates up to 200 Gb/s over 80 km of fiber. We further examine QEPS’s robustness to channel impairments, its seamless compatibility with existing digital signal processing (DSP) pipelines, and its distinctive position within the post-quantum cryptography landscape. Finally, we outline key challenges and future research directions toward deploying QEPS as a practical, quantum-resistant security layer for next-generation optical networks.

Atoms doi: 10.3390/atoms14030022

Authors: Patrick Palmeri Saturnin Enzonga Yoca Exaucé Bokamba Motoumba Alix Niels Maxime Brasseur Pascal Quinet

Osmium is an element of the Periodic Table with an atomic number Z equal to 76. In Tokamaks with divertors made of tungsten (Z=74), it is produced in the neutron-induced transmutation of the latter. Therefore one can expect that their sputtering may generate ionic impurities of all possible charge states in the fusion plasma. As a consequence, these could contribute to radiation losses in these controlled nuclear devices. The knowledge of radiative rates in all the spectra of osmium is thus important in this field. In this framework, a multiplatform approach has been used to determine the Os V radiative properties and estimate their accuracy. The transition probabilities have been computed for the 2677 electric dipole (E1) transitions falling in the spectral range from 400 Å to 12,000 Å. Three independent atomic structure models have been considered; one based on the fully relativistic ab initio multiconfiguration Dirac–Hartree–Fock (MCDHF) method and two based on the semi-empirical pseudo-relativistic Hartree–Fock (HFR) method.

Atoms doi: 10.3390/atoms14030021

Authors: Gediminas Gaigalas

In this study, an extension of the general method [G. Gaigalas, Z. Rudzikas, C. Froese Fischer, J. Phys. B, At. Mol. Phys. (1997). DOI: 10.1088/0953-4075/30/17/006] is described for finding algebraic expressions of the spin-angular parts of the reduced matrix elements of any one- and two-particle operator for an arbitrary number of shells in an atomic configuration. This extension is related, at first, to a change in the definition of tensor structure, where a non-scalar space with respect to l and s for any two-particle operator acts on four different shells. This leads to more efficient expressions for recoupling matrices and amplitudes, which are presented in the paper. In addition, the paper presents new expressions for some of the recoupling matrices, in which 6j- and 9j-coefficients are summed up algebraically. All this leads to a significantly simpler and faster calculation of the spin-angular parts of any non-scalar two-particle operator.

Atoms doi: 10.3390/atoms14030020

Authors: Gediminas Gaigalas

This paper presents a methodology that allows for calculated energy levels and other atomic characteristics in relativistic atomic theory, i.e., using the jj-coupling scheme, to be identified in terms of LSJ-coupling characteristics. The paper begins with outlining the general principles for effectively addressing this problem. Furthermore, it provides a general expression that enables such identification when the atomic state function consists of any number of configuration state functions, each with any number of open shells, and explains how this expression was obtained. The methodology developed in this paper has been successfully implemented in the General Relativistic Atomic Structure Package and can be applied to other similar packages.

Atoms doi: 10.3390/atoms14030019

Authors: Yuchen Wang

The accurate description of nonadiabatic quantum molecular dynamics represents one of the most significant challenges in modern computational chemistry, serving as a gateway to understanding complex phenomena ranging from photochemistry and electron transfer to surface scattering and biological exciton transport. A key difficulty lies in bridging high-level electronic structure theory for ground and excited states with accurate quantum dynamics theory. Although on-the-fly semiclassical approaches are increasingly viable, most quantum dynamics simulations still rely on pre-constructed potential energy surfaces, or in the nonadiabatic context, diabatic potential energy matrices (DPEMs). This perspective paper addresses the theoretical foundations, construction methodologies, and emerging frontiers of DPEMs. We examine the mathematical framework of the adiabatic-to-diabatic transformation, addressing the inherent topological challenges imposed by the geometric phase and the curl condition. We further analyze the transformative impact of machine learning, detailing how machine learning algorithms, such as permutation invariant polynomial neural networks and deep learning architectures, are reshaping the construction of global, high-dimensional DPEMs. Finally, we explore the disruptive potential of quantum computing, discussing how quantum algorithms are automating the direct simulation of nonadiabatic dynamics. In emerging quantum-centric workflows, DPEMs will continue to provide the critical bridge which enables the mapping of realistic, time-dependent molecular Hamiltonians onto quantum hardware.

Atoms doi: 10.3390/atoms14030018

Authors: Grant B. Bunker

For nearly a century, screened Coulomb potentials have been of recognized importance in diverse areas of physics and chemistry. A key feature of interest in these potentials is the phenomenon of critical screening. This paper has three main purposes: to present an extensive, open-access, high accuracy (60 digit) benchmark reference dataset of critical screening parameters, with validation; to confirm excellent past work in the field (to 30 digits), and to correct an historical oversight in its literature; and to present the essentials of our new approach, the “Phase Method” (PM), for computing them. Using the PM, we calculate critical screening parameters, accurate to 60 decimal digits, for the Yukawa/Debye, Hulthén, Pseudo-Hulthén, and Exponential Cosine Screened Coulomb (ECSC)) potentials. The practical feasibility of such calculations on inexpensive hardware opens up new possibilities in research and education. We highlight an apparently overlooked 1989 paper of Demiralp on critical screening parameters of the Yukawa potential, which accurately calculated them to 30 decimal digits. Our main results are computations of the critical screening parameters μc=1/Dc for screening lengths D≤1000 au and angular momenta l=0,…,20. The claimed accuracy of our results is supported by several independent lines of evidence: comparison with the most accurate (30 digit) values available in the print literature for the Yukawa, Hulthén, and ECSC potentials; comparison to 60 decimal digits accuracy with exactly known eigenvalues and critical binding parameters of the Pseudo-Hulthén potential; consistency tests between computed critical parameters, for various l-values for the Pseudo-Hulthén Potential, and known exact relations between eigenvalues; and application of a novel consistency test between results with different potential parameters, that exploits an exact scaling symmetry of this entire class of potentials. Similar calculations were done for ECSC and Yukawa potentials for screening lengths up to D≤105 and l≤12, to 30 digit accuracy, which show interesting (and to our knowledge, not previously reported) periodic structure in Dc(n,l) for the ECSC potential that is not observed for the Yukawa potential. The asymptotic scaling behavior of critical parameters for the Yukawa and Hulthén potentials is explained quantitatively by simple semiclassical calculations, as is the scaling of circular states for those and other potentials.

Atoms doi: 10.3390/atoms14030017

Authors: Lorenzo Nezosi Lucas Maison Patrick Palmeri Per Jönsson Michel Godefroid

Using the Multiconfiguration Dirac–Hartree–Fock method as implemented in the General Relativistic Atomic Structure Package, the magnetic dipole and electric quadrupole hyperfine structure constants were determined for the ground and first excited levels of 135,137Ba II isotopes, as well as for 137Ba I and 87Sr II, to assess the robustness of the developed model. This study builds upon and extends previous investigations by examining the levels involved in resonance lines, with the aim of resolving persistent discrepancies in the hyperfine structure of 137Ba II and 87Sr II. New code developments such as the use of natural orbitals, as well as the addition of polarization effects and Configuration State Function Generators, as implemented in GRASPG, were tested for these heavy elements. The developed strategy allowed us to achieve encouraging results that satisfactorily agree with experiments for all studied levels but D5/22 in the 137Ba II isotope. This disagreement was also observed in 135Ba II isotope as well as in 87Sr II. With two valence electrons, 137Ba I is definitely more complex, requiring a multireference approach. Even with the latter, the theory–observation disagreement observed for the hyperfine structure of the low-lying levels remains large in comparison with the alkali-like systems. Possible ongoing developments to remediate this issue are discussed in the conclusions.

Atoms doi: 10.3390/atoms14030016

Authors: Sergei M. Andrievsky Sergey A. Korotin

In this paper, we discuss the atomic models developed for the non-local thermodynamic equilibrium (LTE) analysis of the spectra of two odd-Z chemical elements, the little-studied potassium and copper, whose nuclei are often thought to form in Cosmos through different astrophysical processes. The K I and Cu I atomic models have been developed and updated over the past decade and applied to determine non-LTE abundances of these elements in the hot and cool dwarfs, giants, and supergiants of different metallicities, from solar to extremely low metallicity. The abundances of potassium and copper in old metal-poor halo stars are of considerable interest because these objects bear the imprints of nucleosynthesis in Type II supernovae and hypernovae in the early Galaxy. The vast majority of the studies of the spectra of these atoms have been based on the assumption of LTE. In some cases, this approach has led to incorrect results, which have sometimes affected our understanding of evolutionary processes in stars and stellar systems. The main objective of this article is to highlight the importance of using the non-LTE stellar abundance data to improve or modify existing theoretical models of cosmic chemical evolution. In particular, significantly different results for the copper abundance in old Galactic stars were obtained compared to LTE data. This finding could inspire specialists working in the field of chemodynamic models to search for realistic pathways for the formation of this element in massive stars. Despite this, since the first non-LTE results on the copper abundance in the oldest Galactic stars, LTE data remained in use for several years. This situation seriously hinders progress in research into some certain aspects of cosmic nucleosynthesis.

Atoms doi: 10.3390/atoms14030015

Authors: Mariana S. Sendova

In this paper, a novel ionization energy, IE, set theory-based organizational structure is suggested: (i) iso-protonic sets, IZE; (ii) iso-electronic sets, IEs; and (iii) iso-ionic sets, IEi+. A computational algorithm is proposed which was demonstrated on twenty-five iso-electronic IEs-sets plotted vs. the nuclear charge, Z. The algorithm allows for: (i) the interpolative assessment of 162 new (not measured) IE values, with their uncertainties estimated by the Lagrange method, for ions from 30Zn to 41Nb; and (ii) effective atomic nuclear charge assessment. It is shown that the IE effective atomic nuclear charge assessment is strongly correlated with the Slater’s effective charge and Pauling electronegativity scale.

Atoms doi: 10.3390/atoms14020014

Authors: Hirokazu Maruyama

We propose a novel theoretical framework for describing photon–electron interactions and electron collision processes in a unified manner within quantum electrodynamics. Specifically, we develop a method to construct the Dirac operator in curved spacetime using only matrix representations rooted in the basis structure of four-dimensional gamma matrix algebra, without introducing vierbeins (tetrads) or independent spin connections. We realize 16 gamma matrices with two indices as 256×256 matrices and embed the spacetime metric directly into the matrix elements. This reduces geometric operations such as covariantization, connection-like operations, and basis transformations to matrix products and trace calculations, yielding a unified and transparent computational scheme. The spacetime dimension remains as four, and the number “16” represents the number of basis elements of four-dimensional gamma matrix algebra (24=16). Based on the extended QED Lagrangian, vertex rules, propagators, spin sums, and traces can be handled uniformly, making it suitable for automation. As validation of this method, we analyzed four fundamental scattering processes in atomic and particle physics: (i) Compton scattering (photon–electron scattering), (ii) muon pair production (e+e−→μ+μ−), (iii) Møller scattering (electron–electron collision), and (iv) Bhabha scattering (electron–positron collision). In the flat spacetime limit, we confirmed the exact reproduction of standard quantum electrodynamics (QED) results including the Klein–Nishina formula. Furthermore, trial calculations using a metric with off-diagonal components show systematic deviations from flat results near scattering angle θ≈90∘, suggesting that metric-induced angular dependence could in principle serve as an observable signature. The matrix representation developed in this work enables unified pipeline execution of theoretical calculations for photon interactions and charged particle collision processes, with expected applications to precision calculations in atomic and particle physics.

Atoms doi: 10.3390/atoms14020013

Authors: Houke Huang Zhongkui Huang Yang Yuan Hanbing Wang Zeshan Muhammad Chang Liu Weiqiang Wen Linfan Zhu Xinwen Ma Stephan Fritzsche

Dielectronic recombination (DR) is widely recognized as a fundamental atomic process in many astrophysical and laboratory plasmas, where it plays a crucial role in determining ionization balance and level populations over a broad temperature range. Reliable DR resonance strengths and plasma rate coefficients for such plasma modeling can be computed using the Jena Atomic Calculator (JAC)—a relativistic code based on the multiconfiguration Dirac–Hartree–Fock (MCDHF) method. In this work, we investigate the DR of Li-like Ar15+ ions in their ground state (2s), focusing on resonances associated with the fine-structure core excitations 2s1/2→2p1/2,3/2. The resulting fine-structure-resolved DR resonance strengths and plasma rate coefficients are in good agreement with recent high-resolution DR measurements of Ar15+ ions performed at the Main Cooler Storage Ring (CSRm) in Lanzhou, China. These results provide a stringent benchmark for JAC calculations and support their applicability in plasma modeling.

Atoms doi: 10.3390/atoms14020012

Authors: Paul R. Berman

The radiation pattern emitted by two atoms, interacting with each other via the vacuum radiation field, has been calculated, including effects of magnetic state degeneracy for atoms with a ground state having G=0 angular momentum and an excited state having H=1 angular momentum. For an initial condition in which both atoms are inverted, the time-integrated radiation pattern is identical to that for non-interacting atoms if the atoms lie on the z-axis, but differs if the atoms lie on the x-axis. The underlying dynamics giving rise to this behavior are examined.

Atoms doi: 10.3390/atoms14020011

Authors: Miguel E. Gómez Quintanar Adrian M. Escobar-Ruiz

We study the planar repulsive two-center Coulomb system in the presence of a uniform magnetic field perpendicular to the plane, taking the inter-center separation a and the magnetic field strength B as independent control parameters. The free-field system B=0 is Liouville integrable and the motion is unbounded. The magnetic confinement introduces nonlinear coupling that breaks integrability and gives rise to chaotic bounded dynamics. Using Poincaré sections and maximal Lyapunov exponents, we characterize the transition from regular motion at aB=0 to mixed regular–chaotic dynamics for aB≠0. To probe the recoverability of the dynamics, we apply sparse regression techniques to numerical trajectories and assess their ability to capture the equations of motion across mixed dynamical regimes. Our results clarify how magnetic confinement competes with two-center repulsive interactions in governing the emergence of chaos and delineate fundamental limitations of data-driven model discovery in nonintegrable Hamiltonian systems. We further identify an organizing mechanism whereby the repulsive two-center system exhibits locally one-center-like dynamics in the absence of any static confining barrier.

Atoms doi: 10.3390/atoms14020010

Authors: Lin Li Zi Li

Through finite-element simulation, the axial potential distribution of the ion trap is analyzed. The effects of the central hole diameter of the end cap and the spacing between the two end caps on the axial geometric parameters of the ion trap are investigated. Based on these findings, a set of linear Paul traps is designed by selecting suitable end caps and quadrupoles. Stable trapping of calcium ions (Ca+) is successfully achieved, and these ions are subsequently laser-cooled into ionic Coulomb crystals. In the experiment, secular motion excitation of the Ca+ ion Coulomb crystal is performed, yielding an axial geometric parameter of 0.115(1) for the ion trap. This value aligns well with the simulation result of 0.114(2). The precise determination of the axial geometric parameter provides a solid foundation for subsequent high-precision optical or mass spectrometry measurements.

Atoms doi: 10.3390/atoms14020009

Authors: Bratislav P. Marinković Lorenzo Avaldi Jozo J. Jureta

In this work, the He and Ar triplet autoionizing states have been studied using a non-monochromatic electron beam and a high-resolution electrostatic analyzer at low incident electron energies and three ejection angles: 40°, 90°, and 130°. Low-energy electrons have been used because they have a high probability of exciting triplet states regardless of whether they are discrete isolate states or are embedded in the ionization continuum. Additionally, the He ejected electron spectra have been measured at several ejection angles between 20° and 130° and two incident energies, namely 60.5 eV and 101 eV. The anisotropic angular distributions indicate that orbital angular momentum exchange between the ejected and scattered electrons occurred. The energies of the first triplets 3s3p64s(3S) and 3s3p64p(3P) states of argon are found to be (24.985 ± 0.020) eV and (26.52 ± 0.02) eV, respectively.

Atoms doi: 10.3390/atoms14020008

Authors: Mbuso K. Matfunjwa Harindranath B. Ambalampitiya Ilya I. Fabrikant

A program package for calculating regularized classical trajectories for Coulomb n–body problem is developed. The Coulomb singularities from the equations of motion are removed by transformations of variables including the time. This effectively conserves the energy of the time-independent systems to a high accuracy for long time propagation. Sample calculations are shown for the cases of 2, 3, 4, and 5 particle systems giving comparisons with the un-regularized trajectories. The program can be used for general purposes including the classical-trajectory Monte-Carlo simulations for charged-particle collisions in free or laser environments.

Atoms doi: 10.3390/atoms14010007

Authors: Andrey I. Bondarev

The first measurements of the magnetic dipole hyperfine structure constants A in singly ionized thulium revealed substantial discrepancies with the corresponding theoretical calculations. Subsequent measurements expanded the very limited available dataset and demonstrated that two of the previously reported experimental A values were incorrect, thereby motivating new theoretical calculations. In this work, we employ the configuration interaction method to calculate the A constants for several low-lying levels in Tm ii, with the random-phase-approximation corrections also taken into account. Our results show good agreement with the new experimental data and provide reliable predictions for additional states where measurements are not yet available.

Atoms doi: 10.3390/atoms14010006

Authors: Linkai Li Xiaohuang Hu Yuheng Xing Xinxing Wu Ning Xu Yuanrun Zhu Yue Tan Yuheng Wu

Inspired by the well-known experimental connections between X(3872), Zcs(4220), and Y(4620), we systematically study the recently reported strange partner of Tcc, the 1+ccq¯s¯ system, and its orbital excitation state 1−ccq¯s¯. A chiral quark model incorporating SU(3) symmetry is considered to study these two systems. To better investigate their spatial structure, we introduce a precise few-body calculation method, the Gaussian Expansion Method (GEM). In our calculations, we include all possible physical channels, including molecular states and diquark structures, and consider channel coupling effects. To identify the stable structures in the system (bound states and resonance states) we employ a powerful resonance search method, the Real-Scaling Method (RSM). According to our results, in the 1+ccq¯s¯ system, we obtain two bound states with energies of 3890 MeV and 3940 MeV, as well as two resonance states with energies of 3975 MeV and 4090 MeV. The decay channels of these two resonance states are DDs∗ and D∗Ds, respectively. In the 1−ccq¯s¯ system, we obtain only one resonance state, with an energy of 4570 MeV, and two main decay channels: DDs1∗ and D∗Ds1′. We strongly suggest that experimental groups use our predictions to search for these stable structures.

Atoms doi: 10.3390/atoms14010005

Authors: Wissem Zrafi Mohamed Bejaoui Hela Ladjimi Jamila Dhiflaoui Hamid Berriche

The potential energy curves and spectroscopic constants of the ground and several low-lying excited states of the Caq+-Kr (q = 0, 1, 2) van der Waals complexes were investigated using one- and two-electron pseudopotential approaches. This treatment effectively reduces the number of active electrons in Caq+-Kr to a single valence electron for q = 1 and two valence electrons for q = 0, allowing the use of large and flexible basis sets for both Ca and Kr atoms. Within this work, potential energy curves (PECs) were calculated at the SCF level for the Ca+-Kr system, while both SCF and full configuration interaction (FCI) calculations were performed for the neutral Ca-Kr. Spin–orbit coupling effects were explicitly included in all calculations to accurately describe the fine-structure splitting of the asymptotic atomic states. The short-range core–core interaction for Ca2+-Kr was obtained using high-level CCSD(T) calculations. Spectroscopic constants were derived from the computed PECs and compared with available theoretical and experimental results, showing consistent trends. Furthermore, the transition dipole moments (TDM) were evaluated as a function of internuclear distances, including spin–orbit effects, to provide a comprehensive description of the electronic structure and radiative properties of these weakly bound systems.

Atoms doi: 10.3390/atoms14010004

Authors: Marco A. García-Márquez Irán Ramos-Prieto Héctor M. Moya-Cessa

We present a general procedure to describe the dynamics of N degenerate quantized fields interacting resonantly with a two–level atom, all coupled with the same strength, within the rotating–wave approximation. Starting from the analysis of the two and three field cases, we generalize the method by identifying dynamical invariants that lead to a factorized form of the time–evolution operator. A unitary transformation reduces the problem to an effective Jaynes–Cummings Hamiltonian, where only one field interacts with the atom and the remaining modes contribute as free fields. Assuming initially coherent fields and an atomic superposition, we compute the atomic inversion and the mean photon number, revealing vacuum Rabi oscillations with a frequency determined by an effective coupling constant that exceeds the individual atom–field coupling, as well as the characteristic collapse–revival behavior.

Atoms doi: 10.3390/atoms14010003

Authors: Jean-Paul Mosnier Eugene T. Kennedy Denis Cubaynes Ségolène Guilbaud Jean-Marc Bizau

We report experimental measurements of the absolute photoionization cross sections for chlorine ions in different stages of ionization, over photon energy ranges corresponding to the L-shell (2s and 2p subshells) excitations. Single, double and triple photoionization channels were investigated for the ions Cl+, Cl2+, Cl3+ and Cl4+. The measurements were performed on the PLéIADES beamline at the SOLEIL radiation storage ring facility, using the Multi-Analysis Ion Apparatus (MAIA). Resonance energies and line strengths are provided for the isonuclear sequence and the evolution of the inner shell photoionization behaviour is demonstrated for the chlorine ions as the degree of ionization is increased. While dominated by photoionization from the corresponding ground state ions, the photoion yields may also contain contributions from low-lying metastable states. The results provide useful data on these ions for plasma modelling and can serve as benchmarking experimental data for future atomic theoretical calculations.

Atoms doi: 10.3390/atoms14010002

Authors: Nhat A. Nghiem Trung V. Phan

We explore the physical quantum properties of atoms in fractal spaces, both as a theoretical generalization of normal integer-dimensional Euclidean spaces and as an experimentally realizable setting. We identify the threshold of fractality at which Ehrenfest atomic instability emerges, where the Schrödinger equation describing the wavefunction of a single electron orbiting around an atom becomes scale-free, and discuss the potential of observing this phenomena in laboratory settings. We then study the Rydberg states of stable atoms using the Wentzel–Kramers–Brillouin approximation, along with a proposed extension for the Langer modification, in general fractal dimensionalities. We show that fractal space atoms near instability explode in size even at low-number excited state, making them highly suitable to induce strong entanglements and foster long-range many-body interactions. We argue that atomic physics in fractal spaces—“fractatomic physics”—is a rich research avenue deserving of further theoretical and experimental investigations.

Atoms doi: 10.3390/atoms14010001

Authors: Veronica Giardini Luca Guariento Andrea Fantini Shawn Storm Massimo Inguscio Jacopo Catani Giacomo Cappellini Vladislav Gavryusev Leonardo Fallani

We report on the realization of a platform for trapping and manipulating individual 88Sr atoms in optical tweezers. A first cooling stage based on a blue shielded magneto-optical trap (MOT) operating on the |1S0⟩→|1P1⟩ transition at 461 nm enables us to trap approximately 4 × 106 atoms at a temperature of 6.8 mK. Further cooling is achieved in a narrow-line red MOT using the |1S0⟩→|3P1⟩ intercombination transition at 689 nm, bringing 5 × 105 atoms down to 5μK and reaching a density of 4 × 1010 cm3. Atoms are then loaded into 813 nm tweezer arrays generated by crossed acousto-optic deflectors and tightly focused onto the atoms with a high-numerical-aperture objective. Through light-assisted collision processes we achieve the collisional blockade, which leads to single-atom occupancy with a probability of about 50%. The trapped atoms are detected via fluorescence imaging with a fidelity of 99.986(6)%, while maintaining a survival probability of 97(2)%. The release-and-recapture measurement provides a temperature of 12.92(5)μK for the atoms in the tweezers, and the ultra-high-vacuum environment ensures a vacuum lifetime higher than 7 min. These results demonstrate a robust alkaline-earth tweezer platform that combines efficient loading, cooling, and high-fidelity detection, providing the essential building blocks for scalable quantum simulation and quantum information processing with Sr atoms.

Atoms doi: 10.3390/atoms13120099

Authors: Matías L. Ocello Sebastián D. López Martín Barlari Diego G. Arbó

We present a time-dependent nonperturbative theory of the reconstruction of attosecond beating by interference of multiphoton transitions (RABBIT) for photoelectron emission from hydrogen atoms in the transverse direction relative to the laser polarization axis. Extending our recent semiclassical strong-field approximation (SFA) model developed for parallel emission, we deduce analytical expressions for the transition amplitudes and demonstrate that the photoelectron probability distribution can be factorized into interhalf- and intrahalfcycle interference contributions, the latter modulating the intercycle pattern responsible for sideband formation. We identify the intrahalfcycle interference arising from trajectories released within the same half cycle as the mechanism governing attosecond phase delays in the perpendicular geometry. Our results reveal the suppression of even-order sidebands due to destructive interhalfcycle interference, leading to a characteristic spacing between adjacent peaks that doubles the standard spacing observed along the polarization axis. Comparisons with numerical calculations of the SFA and the ab initio solution of the time-dependent Schrödinger equation confirm the accuracy of the semiclassical description. This work provides a unified framework for understanding quantum interferences in attosecond chronoscopy, bridging the cases of parallel and perpendicular electron emission in RABBIT-like protocols.

Atoms doi: 10.3390/atoms13120098

Authors: Jiaolong Zeng Guoqing Wang Aihua Deng Cheng Gao Jianmin Yuan

Auger decay of all levels of the double core-hole states 4d−2 of Xe2+, including collective Auger decay (CAD) pathways, is investigated using the relativistic distorted-wave approximation. Large-scale configuration interaction calculations were performed to obtain level-to-level Auger decay rates. In addition to the typical Auger decay final levels associated with the configurations of 4d−15s25p4, 4d−15s15p5, and 4d−15s05p6, evident contributions are identified from excited channels, leading to configurations such as 4d94f15s25p3, 4d95s25p35d1, 4d95s25p36s1, and 4d95s25p36p1. These contributions arise from strong electron correlation between the valence electronic orbitals and the 4d inner-shell orbital. The CAD rates and branching ratios (BRs) are determined for each double core-hole level with a minimum CAD BR of 1.28% and a maximum of 4.08% among all CAD channels. The configuration-averaged CAD BR is predicted to be 1.93%, which helps explain recent unexplained experimental findings. The inclusion of CAD processes enriches Auger electron spectroscopy, thereby extending potential applications of this important experimental tool in both fundamental and applied research.

Atoms doi: 10.3390/atoms13120097

Authors: George-Tony Constantin Cristian Iorga

This paper investigates the validity of the long wavelength approximation in the calculation of two-photon decay of 2s1/2 level in hydrogen-like ions with nuclear charge Z=1−100 based on time-dependent second-order perturbation theory and angular momentum algebra. While the relativistic structure effects on the two-photon decay rates are highlighted in the literature, the role of slowing effects in the photon electric dipole operators are not discussed extensively. The rate is computed by the sum-over-states method, with bound-bound and bound-free electric dipole matrix elements obtained in the Babushkin and Coulomb gauges, which satisfy the Lorenz gauge condition, as well as their non-relativistic limits in the long-wavelength approximation (Length and Velocity forms, respectively). The present results explicitly show how this approximation breaks gauge invariance by overestimating the Babushkin values by ∼24%(αZ)2 while underestimating the Coulomb rates by ∼31%(αZ)2. Using analytical eigenfunctions of the Dirac equation, we found that the contributions of the negative continuum states to the rate scale are ∼0.0134(αZ)4 in the Babushkin gauge and ∼1.46(αZ)4 in the Coulomb gauge, making the latter gauge more susceptible to errors when attempting to achieve basis completeness in multiphoton calculations. The present results are useful in assessing the complexity requirements of radiative transition rates for atomic systems of interest.

Atoms doi: 10.3390/atoms13120096

Authors: Cherie Sisay Mekonen Ayyagari Venkata Mohan Rao

A comparative study of alpha-induced reactions on cobalt isotope with the predictions by COMPLET code is presented for nine excitation functions, 59Co(α,p5n)57Ni, 59Co (α,p6n)56Ni, 59Co(α,2pn)60Co, 59Co(α,3pn)59Fe, 59Co(α,αn)58Co, 59Co(α,α2n)57Co, 59Co(α,α3n)56Co, 59Co(α,2αn)54Mn, and 59Co(α,2α3n)52Mn. The experimental values were taken from the EXFOR data base. Theoretical cross-sections were calculated using initial exciton number n0 = 4 (4p0h) and level density parameter a (=ACN/10) globally. While several reactions showed excellent agreement with experimental data, others displayed a notable discrepancy. This is because of the limitations of the COMPLET code to take the alpha emission in a pre-equilibrium phase.

Atoms doi: 10.3390/atoms13120095

Authors: Luca Salasnich

We present a detailed derivation of the quantum and quantum–thermal effective action for non-relativistic systems, starting from the single-particle case and extending to the Gross–Pitaevskii (GP) field theory for weakly interacting bosons. In the single-particle framework, we introduce the one-particle-irreducible (1PI) effective action formalism, taking explicitly into account the choice of the initial quantum state, its saddle-point plus Gaussian-fluctuation approximation, and its finite-temperature extension via Matsubara summation, yielding a clear physical interpretation in terms of zero-point and thermal contributions to the Helmholtz free energy. The formalism is then applied to the GP action, producing the 1PI effective potential at zero and finite temperature, including beyond-mean-field Lee–Huang–Yang and thermal corrections. We discuss the gapless and gapped Bogoliubov spectra, their relevance to equilibrium and non-equilibrium regimes, and the role of regularization. Applications include the inclusion of an external potential within the local density approximation, the derivation of finite-temperature Josephson equations, and the extension to D-dimensional systems, with particular attention to the zero-dimensional limit. This unified approach provides a transparent connection between microscopic quantum fluctuations and effective macroscopic equations of motion for Bose–Einstein condensates.

Atoms doi: 10.3390/atoms13110094

Authors: Shruti Sarswat Saumyashree Baral Jobin Jose

Superheavy elements are an ideal testbed for studying relativistic, exchange, and correlation effects in scattering phenomena. In this work, we investigate electron scattering from copernicium (Z=112) and oganesson (Z=118) atoms. Both the relativistic Dirac and non-relativistic partial wave methods are employed to analyze the scattering dynamics, with the interaction between the projectile and target atom modeled within the framework of the optical potential approach. Our results demonstrate that relativistic, exchange, and correlation effects play a significant role in modifying the scattering cross-sections and scattering length, highlighting the influence of these interactions on the scattering processes from superheavy atomic systems. The work also attempts to identify common features of the scattering cross-section by comparing those of lighter elements in the same group.

Atoms doi: 10.3390/atoms13110093

Authors: Lidia Lappo Jakub Zieliński Fredrik P. Gustafsson Malgorzata Grosbart Georgy Kornakov Michael Doser

The present study investigates the production of highly charged ions (HCIs) through the novel application of antiprotonic atoms and explores their potential for studying atomic and nuclear structures. Utilizing the Geant4 simulation toolkit, comprehensive simulations were conducted for all known isotopes with atomic numbers below 100. These simulations recorded key parameters of the resulting nuclear fragments, including mass, momentum, charge, and yield. The results obtained from this study offer valuable insights into the mechanisms of HCI production and provide a foundation for planning and analyzing future experimental investigations. This work demonstrates the feasibility of using antiprotonic atoms to advance nuclear and atomic physics research.

Atoms doi: 10.3390/atoms13110092

Authors: Kyle A. Hamer Heman Gharibnejad Luca Argenti Nicolas Douguet

We present a time-dependent framework that combines a hybrid basis, consisting of Gaussian-type orbitals (GTOs) and finite-element discrete-variable representation (FEDVR) functions, with a multicenter grid to simulate strong-field and attosecond dynamics in atoms and molecules. The method incorporates the construction of the orthonormal hybrid basis, the evaluation of electronic integrals, a unitary time-propagation scheme, and the extraction of optical and photoelectron observables. Its accuracy and robustness are benchmarked on one-electron systems such as atomic hydrogen and the dihydrogen cation (H2+) through comparisons with essentially-exact reference results for bound-state energies, high-harmonic generation spectra, photoionization cross sections, and photoelectron momentum distributions. This work establishes the groundwork for its integration with quantum-chemistry methods, which is already operational but will be detailed in future work, thereby enabling ab initio simulations of correlated polyatomic systems in intense ultrafast laser fields.

Atoms doi: 10.3390/atoms13110091

Authors: Pravi Mishra Parmanand Pandey Rachana Singh Manisha Yadav Shivani Aftab Ahamad Alka Misra Amritanshu Shukla Poonam Tandon

A high-level ab initio characterization and formation of diaminomethane (DAM), the simplest geminal diamine, is presented to support its spectroscopic detection and astrochemical relevance in the interstellar medium. The C2v DAM conformer is identified as the global minimum, while C1 DAM and C2 DAM represent higher-energy local minima. The proposed reaction pathways are exothermic and proceed without activation barriers. Simulated infrared spectrum reproduces accurate key spectral signatures with several vibrational modes exhibiting strong IR intensities (>80 km mol−1), particularly in the 800–3000 cm−1 range and band shapes. Dipole moments and accurate rovibrational spectroscopic parameters, including rotational constants, anharmonic vibrational frequencies, quartic and sextic distortion constants, and nuclear quadrupole coupling constants are reported to assist with high-resolution spectroscopic identification. This study provides significant theoretical benchmarks for its formation and offers guidance for future laboratory spectroscopy and molecular searches in interstellar environments.

Atoms doi: 10.3390/atoms13110090

Authors: Juan Molina M. Carlos Aragón José Aguilera César Costa-Vera Diego Díaz Pace

In this work, a quantitative analysis of Li in natural brines was carried out by laser-induced breakdown spectroscopy (LIBS) assisted by the τ–algorithm for detailed analysis of the experimental line shapes (τLIBS). Brine samples were collected from different salars located in the Puna plateau (Northwest Argentina) and analyzed by LIBS in the form of solid pressed pellets. The emission intensities of Li I, Hα, and Mg I–II lines were measured and spatially integrated along the line of sight with temporal resolution by using a high-spectral-resolution spectrometer equipped with an intensified charge-coupled device (iCCD) detector. The plasma was characterized through the determination of the electron density and the temperature. The τ–algorithm calculated the optical thicknesses of the Li I lines to generate synthetic intensity profiles that were subsequently fitted to the experimental spectra. By applying the developed τLIBS approach, valuable spectroscopic insight was recovered about the physical processes occurring in the plasma, such as self-absorption. The analytical process involved an univariate external calibration process using the resonant Li I line at 6707.7 Å measured from a series of Li standard samples. Self-absorption effects were evaluated and subsequently compensated. The final LIBS results, with an enhanced accuracy of 15%, were validated by crosschecking them against those obtained with the standard AAS method.

Atoms doi: 10.3390/atoms13110089

Authors: Kaoutar El Haloui Nicolas Thome Nicolas Sisourat

Constructing accurate Potential Energy Surfaces (PES) is a central task in molecular modeling, as it determines the forces governing nuclear motion and enables reliable quantum dynamics simulations. While ab initio methods can provide accurate PES, they are computationally prohibitive for extensive applications. Alternatively, analytical physics-based models such as the Morse potential offer efficient solutions but are limited by their rigidity and poor generalization to excited states. In recent years, neural networks have emerged as powerful tools for determining PES, due to their universal function approximation capabilities, but they require large training datasets. In this work, we investigate hybrid-residual modeling approaches that combine physics-based potentials with neural network corrections, aiming to leverage both physical priors and data adaptability. Specifically, we compare three hybrid models—APHYNITY, Sequential Phy-ML, and PhysiNet—in their ability to reconstruct the potential energy curve of the ground and first excited states of the hydrogen molecule. Each model integrates a simplified physical representation with a neural component that learns the discrepancies from accurate reference data. Our findings reveal that hybrid models significantly outperform both standalone neural networks and pure physics-based models, especially in low-data regimes. Notably, APHYNITY and Sequential Phy-ML exhibit better generalization and maintain accurate estimation of physical parameters, underscoring the benefits of explicit physics incorporation.

Atoms doi: 10.3390/atoms13110088

Authors: Nicolás J. Esponda Michele A. Quinto Roberto D. Rivarola Juan M. Monti

In this work, we present a theoretical model to investigate electron emission in collisions between dressed ions with He atoms and H2 molecules. The projectile potential is described as the sum of a long- and short-range terms. The last term includes a screening function that has its maximum at short distances. The present model is based on the Continuum Distorted Wave Eikonal Initial State (CDW-EIS) theory, but the Eikonal approximation is only made within the long-range transition amplitude. This now leads to physically correct predictions, whenever dressed projectiles are involved, in the binary-encounter peak. Indeed, double-differential cross-sections spectra is calculated and compared with existing experimental data, finding that this model is capable of reproducing some well-known phenomena depending on the projectile charge state. Namely, the dependence of the binary-encounter peak magnitude with the projectile charge state.

Atoms doi: 10.3390/atoms13110087

Authors: Fabrizio Pinto

The presence of dominant interatomic dispersion forces on the nanoscale holds the promise for breakthrough applications in key areas of quantum sensing, such as accelerometry, as well as nano-manipulation and energy storage. In order to do work, nano-machines enabled by dispersion forces must exchange energy with the surrounding environment. Such processes can be described in terms of thermodynamical engine cycles involving individual atoms or material boundaries, separated by possibly empty gaps and interacting via time-dependent dispersion forces. The fundamental strategy indispensable to achieve dispersion force time-modulation, demonstrated experimentally by independent groups on different scales, is based on the illumination of interacting, semiconducting elements by appropriate radiation beams. Here we analyze the operation of ideal nano-engines in the quasi-static regime by means of the Lifshitz theory of dispersion forces involving semiconducting boundary or atom irradiation. Firstly, we verify that the First Law of Thermodynamics is satisfied so that the total energy of the system is rigorously conserved. Secondly, we show that, within this first approximate treatment, the Second Law of Thermodynamics may be violated for extremely small interboundary gap widths. We identify important limitations to be addressed to determine whether this is a reliable conclusion. The technological and historic backdrops are presented, and important topics for future research are identified.

Atoms doi: 10.3390/atoms13110086

Authors: Jean-Patrick Connerade

In Classical Mechanics, time is reversible, i.e., it implies no particular choice: only the observer knows in which direction it flows. The present article re-examines whether this remains true in Quantum Mechanics. In the context of Atomic Physics, it is concluded that the existence of an arrow of time depends on the manner in which the radiation field is introduced, which must be non-perturbative.

Atoms doi: 10.3390/atoms13100085

Authors: Anil K. Pradhan Sultana N. Nahar

Accurate determination of opacity is critical for understanding radiation transport in both astrophysical and laboratory plasmas. We employ atomic data from R-Matrix calculations to investigate radiative properties in high-energy-density (HED) plasma sources, focusing on opacity variations under extreme plasma conditions. Specifically, we analyze environments such as the base of the convective zone (BCZ) of the Sun (2×106 K, Ne=1023/cc), and radiative opacity data collected using the inertial confinement fusion (ICF) devices at the Sandia Z facility (2.11×106 K, Ne=3.16×1022/cc) and the Lawrence Livermore National Laboratory National Ignition Facility. We calculate Rosseland Mean Opacities (RMO) within a range of temperatures and densities and analyze how they vary under different plasma conditions. A significant factor influencing opacity in these environments is line and resonance broadening due to plasma effects. Both radiative and collisional broadening modify line shapes, impacting the absorption and emission profiles that determine the RMO. In this study, we specifically focus on electron collisional and Stark ion microfield broadening effects, which play a dominant role in HED plasmas. We assume a Lorentzian profile factor to model combined broadening and investigate its impact on spectral line shapes, resonance behavior, and overall opacity values. Our results are relevant to astrophysical models, particularly in the context of the solar opacity problem, and provide insights into discrepancies between theoretical calculations and experimental measurements. In addition, we investigate the equation-of-state (EOS) and its impact on opacities. In particular, we examine the “chemical picture” Mihalas–Hummer–Däppen EOS with respect to level populations of excited levels included in the extensive R-matrix calculations. This study should contribute to improving opacity models of HED sources such as stellar interiors and laboratory plasma experiments.

Atoms doi: 10.3390/atoms13100084

Authors: Michele Arcangelo Quinto Juan Manuel Monti Roberto Daniel Rivarola

Single electron capture in collisions involving neutral hydrogen atoms impacted by highly charged dressed projectiles is theoretically investigated using the distorted wave formalism. A series of continuum distorted wave approximations is employed to investigate the electron capture from neutral hydrogen atom impact by boron and carbon projectiles. The projectile potential is described using a two-parameter analytical Green–Sellin–Zachor (GSZ) model potential. The theoretical prediction of total cross sections are compared against other theories and experiments. We looked at a very broad range of collision energies, from 10 keV/u up to 10 MeV/u. In addition, the state-selective cross sections for boron ions are presented.

Atoms doi: 10.3390/atoms13100083

Authors: Camila Strubbia Mangiarelli Verónica B. Tessaro Michaël Beuve Mariel E. Galassi

Proton therapy enables precise dose delivery to tumors while sparing healthy tissues, offering significant advantages over conventional radiotherapy. Accurate prediction of biological doses requires detailed knowledge of radiation interactions with biological targets, especially DNA, a key site of radiation-induced damage. While most biophysical models (LEM, mMKM, NanOx) rely on water as a surrogate, this simplification neglects the complexity of real biomolecules. In this work, we calculate the stopping power and range of protons in liquid water, dry DNA, and hydrated DNA using semi-empirical cross sections for ionization, electronic excitation, electron capture, and electron loss by protons and neutral hydrogen in the 10 keV–100 MeV energy range. Additionally, ionization cross sections for uracil are computed to explore potential differences between DNA and RNA damage. Our results show excellent agreement with experimental and ab initio data, highlighting significant deviations in stopping power and range between water and DNA. Notably, the stopping power of DNA exceeds that of water at most energies, reducing proton ranges in dry and hydrated DNA by up to 20% and 26%, respectively. These findings provide improved input for Monte Carlo simulations and biophysical models, enhancing RBE predictions and dose accuracy in hadrontherapy.

Atoms doi: 10.3390/atoms13100082

Authors: Gregory J. Boyle Dale L. Muccignat Joshua R. Machacek Robert P. McEachran

We report calculations for electron–radon scattering using a complex relativistic optical potential method. The energy range of this study is 0–1000 eV, with results for the elastic (total, momentum-transfer and viscosity-transfer) cross section, summed discrete electronic-state integral excitation cross sections and electron-impact ionization cross sections presented. Here, we obtain our cross sections from a single theoretical relativistic calculation. Since radon is a heavy element, a relativistic treatment is very desirable. The electron transport coefficients are subsequently calculated for reduced electric fields ranging from 0.01 to 1000 Td, using a multi-term solution of Boltzmann’s equation.

Atoms doi: 10.3390/atoms13090081

Authors: Gregory S. Adkins Ulrich D. Jentschura

In exotic atomic systems with hadronic constituent particles, it is notoriously difficult to estimate the strong-interaction correction to energy levels. It is well known that, due to the strength of the nuclear interaction, the problem cannot be solved using Wigner–Brillouin perturbation theory alone. Recently, high-angular-momentum Rydberg states of exotic atomic systems with hadronic constituents have been identified as promising candidates in the search for new physics in the low-energy sector of the Standard Model. We thus derive a generalized Deser–Trueman formula for the induced energy shift for a general hydrogenic bound state with principal quantum number n and orbital angular momentum quantum number ℓ, and we find that the energy shift is given by the formula δE=2αn,ℓβℓ(ah/a0)2ℓ+1Eh/n3, where αn,0=1, αn,ℓ=∏s=1ℓ(s−2−n−2), βℓ=(2ℓ+1)/[(2ℓ+1)!!]2, Eh is the Hartree energy, ah is the hadronic radius and a0 is the generalized Bohr radius. The square of the double factorial, [(2ℓ+1)!!]2, in the denominator implies a drastic suppression of the effect for higher angular momenta.

Atoms doi: 10.3390/atoms13090080

Authors: Darío M. Mitnik Claudia C. Montanari Silvina Segui Silvina P. Limandri Judith A. Guzmán Alejo C. Carreras Jorge C. Trincavelli

A fully connected neural network was trained to model the K-shell ionization cross sections based on two input features: the atomic number and the incoming electron overvoltage. The training utilized a recent, updated compilation of experimental data covering elements from H to U, and incident electron energies ranging from the threshold to relativistic values. The neural network demonstrated excellent predictive performance, compared with the experimental data, when available, and with full theoretical predictions. The developed model is provided in the ikebana code, which is openly available and requires only the user-selected atomic number and electron energy range as inputs.

Atoms doi: 10.3390/atoms13090079

Authors: Luiz C. de Miranda Nelson H. Morgon

Molecular geometry, infrared (IR) vibrational frequencies, and ultraviolet–visible (UV-Vis) electronic absorption spectra of the trivalent iron tris(acetylacetonate) complex, Fe(III)(acac)3, were computed using hybrid meta-generalized gradient approximation (meta-GGA) density functional theory (DFT). Calculations employed the Jorge double-ζ valence plus polarization basis sets (standard DZP and relativistic DZP + DKH). Solvent effects were modeled using the SMD continuum solvation framework with acetonitrile as the dielectric medium. This charge-neutral complex exhibits predominantly ionic metal–ligand bonding character, which simplifies the computational treatment. Despite extensive DFT applications to coordination compounds, systematic benchmarks for this bidentate ligand system remain limited. The computed harmonic frequencies (ν) and electronic excitation energies (λmax) demonstrate excellent agreement with available experimental measurements. These results enable comparative analysis of IR and UV-Vis spectral features, both with and without all-electron scalar relativistic effects with the second-order Douglas–Kroll–Hess approach.

Atoms doi: 10.3390/atoms13090078

Authors: Andrezza O. Arêas Maikel Y. Ballester

Fast-charged particles have been used in diagnosis and treatment since the 19th century. Positrons are widely used in medical imaging through positron emission tomography, but their therapeutic potential remains underexplored due to technology limitations associated with the lack of research on their effectiveness against cancer. One way to understand their behavior is by calculating absorbed dose distributions in tissue, which can be safely and realistically done using computational simulations such as the Monte Carlo Method. This study investigates the interaction of a positron beam with brain tissue and a tumor through simulations using the TOPAS software. Depth dose profiles and absolute absorbed dose values were obtained in the range of 6–24 MeV. Validation was performed using data from the water phantom with electron beams. The results showed that, at certain depths in brain tissue, the absorbed dose by positrons was higher than that of electrons under the same conditions, ranging from 57% to 463% more. These findings suggest that positrons may offer advantages over conventional electron therapy and contribute to the development of novel therapeutic approaches.

Atoms doi: 10.3390/atoms13090077

Authors: Stephan Fritzsche Aloka Kumar Sahoo Lalita Sharma Stefan Schippers

Simulated (or measured) photoabsorption spectra often provide the first indication of how matter interacts with light when irradiated by some radiation source. In addition to the direct, often slowly varying photoabsorption cross-section as a function of the incident photon frequency, such spectra typically exhibit numerous resonances and edges arising from the interaction of the radiation field with the subvalence or even inner-shell electrons. Broadly speaking, these resonances reflect photoexcitation, with its subsequent fluorescence, or the autoionization of bound electrons. Here, a (relativistic) cascade model is developed for estimating the photoabsorption of (many) atoms and multiply charged ions with a complex shell structure across the periodic table. This model helps distinguish between level- and shell-resolved, as well as total photoabsorption, cross-sections, starting from admixtures of selected initial-level populations. Examples are shown for the photoabsorption of C+ ions near the 1s − 2p excitation threshold and for Xe2+ ions in the photon energy range from 10 to 200 eV. While the accuracy and resolution of the predicted photoabsortion spectra remain limited due to the additive treatment of resonances and because of missing electronic correlations in the representation of the levels involved, the present implementation is suitable for ions with quite different open-shell structures and may support smart surveys of resonances along different isoelectronic sequences.

Atoms doi: 10.3390/atoms13090076

Authors: Tengfei Li Zhi Xiao

In this paper, we explore the Larmor time for three types of trapezoidal barriers, and we find consistent results between the traditionally defined Larmor time and a newly defined one. We confirm that the transmission Larmor time for the trapezoidal barriers also satisfies certain properties of mirror-symmetric barriers. Consistent with our expectations, we also find that: 1. as the barrier height increases, the peak of the Larmor time shifts to the right (higher energy); and 2. as the barrier width increases, the peak becomes larger in coincidence with the classical expectation that a particle needs more time to cross a longer path of the same height/inclination.

Atoms doi: 10.3390/atoms13090075

Authors: Juan C. Del Valle Aaron T. Bondy Soumyajit Saha Kathryn R. Hamilton Klaus Bartschat

As a continuation of Schneider et al., Atoms 2022 10, 26, we report recent progress in the development and deployment of the interface between the computational codes B-Spline R-matrix (BSR) and R-Matrix with Time dependence (RMT). These advances have been achieved within the context of the LS-coupling scheme. In its current state, the interface handles atomic target states described by single configurations and supports the Fano–Racah phase convention, as required by RMT. As first example of an application, we use the interface to investigate multiphoton single ionization of helium exposed to a linearly polarized laser field with wavelengths between 280 and 316 nm and a peak intensity of 3×1014 W/cm2. As a second example, we consider high-order harmonic generation (HHG) in carbon, driven by an intense 30-cycle laser field at 800 nm and a peak intensity of 1×1012 W/cm2.

Atoms doi: 10.3390/atoms13090074

Authors: Shucheng Cui Kaizhao Lin Dadi Xing Ling Liu Dongmei Zhao Dalong Guo Yong Gao Shaofeng Zhang Yong Wu Chenzhong Dong Xiaolong Zhu Xinwen Ma

A combined experimental and theoretical study is carried out on the single-electron capture process in He+–He collisions at energies ranging from 0.5 keV/u to 5 keV/u. Using cold target recoil ion momentum spectroscopy, we obtain state-selective cross sections and angular differential cross sections. Within the entire studied energy range, the dominant channel is the electron captured into the ground-state, and the relative contribution of the dominant channel shows a decreasing trend with increasing energy. The angular differential cross sections of ground-state capture exhibit obvious oscillatory structures. To understand the oscillatory structures of the differential cross sections, we also performed theoretical calculations using the two-center atomic orbital close-coupling method, which well reproduced the oscillatory structures. The results indicate that these structures are strongly correlated to the oscillatory structures of the impact parameter dependence of electron probability.

Atoms doi: 10.3390/atoms13080073

Authors: Peter Uylings

Inner products of spherical tensor operators have been used since the early eighties to define orthogonal operators. However, the basic theory and properties are largely missing in the literature. An inner product in any configuration is directly proportional to the inner product taken in the most basic configuration in which it can occur. The formula for the proportionality factor in question is presented for the first time. This allows the inner products in and between arbitrary configurations to be calculated in advance. In addition, inner products are shown to be independent of the coupling scheme used to construct the state functions. Applications such as the orthogonal operator method and projections of ab initio calculations check for the completeness of the used basis of operators and, importantly, check the matrix elements in any arbitrary configuration, as discussed and illustrated with examples. Closed formulae for the inner products of the well-known Slater and spin–orbit operators are given.

Atoms doi: 10.3390/atoms13080072

Authors: Jing Chi Hao Fang Yong-Hui Zhang Xiao-Qiu Qi Li-Yan Tang Ting-Yun Shi

Rydberg atoms play a crucial role in testing atomic structure theory, quantum computing and simulation. Measurements of transition frequencies from the 21,3S states to Rydberg P1,3 states have reached a precision of several kHz, which poses significant challenges for theoretical calculations, since the accuracy of variational energy calculations decreases rapidly with increasing principal quantum number n. Recently the complex “triple” Hylleraas basis was employed to attain the ionization energy of helium 24P1 state with high accuracy. Different from it, we extended the correlated B-spline basis functions (C-BSBFs) to calculate the Rydberg states of helium. The nonrelativistic energies of 1snpP1,3 states up to n=27 achieve at least 14 significant digits using a unified basis set, thereby greatly reducing the complexity of the optimization process. Results of geometric structure parameters and cusp conditions were presented as well. Both the global operator and direct calculation methods are employed and cross-checked for contact potentials. This C-BSBF method not only obtains high-accuracy energies across all studied levels but also confirms the effectiveness of the C-BSBFs in depicting long-range and short-range correlation effects, laying a solid foundation for future high-accuracy Rydberg-state calculations with relativistic and QED corrections included in helium atom and low-Z helium-like ions.

Atoms doi: 10.3390/atoms13080071

Authors: Qi Guo Zhaoying Chen Fangshi Jia Wenhao Xia Xiaobin Ding Jun Xiao Yaming Zou Ke Yao

This study investigates the ionization balance of Sn ions in an electron beam ion trap (EBIT). Highly charged Sn ions are produced via collisions with a quasi-monochromatic electron beam, and the charge state distribution is analyzed using a Wien filter. Significant Sn15+ production occurs at electron energies below the ionization potential of Sn14+ (379 eV). Calculations attribute this to electron-impact ionization from metastable Sn14+ states.

Atoms doi: 10.3390/atoms13080070

Authors: Ryan Schneider Heman Gharibnejad

This article reviews several numerical methods for the time-dependent Schrödinger Equation (TDSE). We consider both the most commonly used approach—short-time propagation, which solves the TDSE by assuming that the Hamiltonian is time-independent over sufficiently small (time) intervals—as well as a number of higher-order alternatives. Our goal is to dispel the notion that the latter are too computationally demanding for practical use. To that end, we cover methods whose numerical building blocks are shared by short-time propagators or can be handled by standard libraries. Moreover, we make the case that these methods are best positioned to take advantage of parallel computing environments. One of the alternatives considered is a “double DVR” solver, which applies an expansion in a product basis of functions in space and time to obtain a solution (over all space and at multiple time points simultaneously) with a single linear system solve. To our knowledge, and despite its simplicity, this approach has not previously been applied to the TDSE.

Atoms doi: 10.3390/atoms13080069

Authors: Mohamed Farjallah Hela Ladjimi Wissem Zrafi Hamid Berriche

In this work, a comprehensive theoretical investigation is carried out to explore the electronic and spectroscopic properties of selected diatomic molecular ions MgRb+ and SrRb+. Using high-level ab initio calculations based on a pseudopotential approach, along with large Gaussian basis sets and full valence configuration interaction (FCI), we accurately determine adiabatic potential energy curves, spectroscopic constants, transition dipole moments (TDMs), and permanent electric dipole moments (PDMs). To deepen our understanding of these systems, we calculate radiative lifetimes for vibrational levels in both ground and low-lying excited electronic states. This includes evaluating spontaneous and stimulated emission rates, as well as the effects of blackbody radiation. We also compute Franck–Condon factors and analyze photoassociation processes for both ions. Furthermore, to explore low-energy collisional dynamics, we investigate elastic scattering in the first excited states (21Σ+) describing the collision between the Ra atom and Mg+ or Sr+ ions. Our findings provide detailed insights into the theoretical electronic structure of these molecular ions, paving the way for future experimental studies in the field of cold and ultracold molecular ion physics.