Atoms

19 pages, 546 KB

Open AccessArticle

Multichannel Quantum Defect Theory with Numerical Reference Functions: Applications to Cold Atomic Collisions

by Dibyendu Sardar, Arpita Rakshit, Somnath Naskar and Bimalendu Deb

Atoms 2026, 14(3), 26; https://doi.org/10.3390/atoms14030026 - 21 Mar 2026

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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 [...] Read more.

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. Full article

(This article belongs to the Section Quantum Chemistry, Computational Chemistry and Molecular Physics)

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1 pages, 129 KB

Open AccessCorrection

Correction: Pawelkiewicz et al. Introducing Machine Learning in Teaching Quantum Mechanics. Atoms 2025, 13, 66

by M. K. Pawelkiewicz, Filippo Gatti, Didier Clouteau, Viatcheslav Kokoouline and Mehdi Adrien Ayouz

Atoms 2026, 14(3), 25; https://doi.org/10.3390/atoms14030025 - 19 Mar 2026

Viewed by 183

Abstract

Missing Supplementary Materials [...] Full article

(This article belongs to the Special Issue Artificial Intelligence for Quantum Sciences)

8 pages, 480 KB

Open AccessArticle

Ni- and Co-like Xe Ion EUV Spectra Produced by Excitation Around the Ionisation Threshold of Xe XXVII

by Elmar Träbert

Atoms 2026, 14(3), 24; https://doi.org/10.3390/atoms14030024 - 12 Mar 2026

Viewed by 265

Abstract

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 [...] Read more.

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

E 2

and

M 3

transitions from long-lived levels, the present report shows spectra of 3d-4p and 3d-4f

E 1

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

M 1

and

E 2

transitions between these configurations are permitted. Full article

(This article belongs to the Section Atomic, Molecular and Nuclear Spectroscopy and Collisions)

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24 pages, 1742 KB

Open AccessReview

Quantum Encryption in Phase Space

by Randy Kuang

Atoms 2026, 14(3), 23; https://doi.org/10.3390/atoms14030023 - 11 Mar 2026

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Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Quantum Optics and Quantum Information)

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20 pages, 819 KB

Open AccessArticle

Multiplatform Computing of Transition Probabilities in Os V

by Patrick Palmeri, Saturnin Enzonga Yoca, Exaucé Bokamba Motoumba, Alix Niels, Maxime Brasseur and Pascal Quinet

Atoms 2026, 14(3), 22; https://doi.org/10.3390/atoms14030022 - 11 Mar 2026

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Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Section Atomic, Molecular and Nuclear Spectroscopy and Collisions)

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28 pages, 491 KB

Open AccessArticle

Extension of an Efficient Approach for Spin-Angular Integrations in Atomic Structure Calculations

by Gediminas Gaigalas

Atoms 2026, 14(3), 21; https://doi.org/10.3390/atoms14030021 - 9 Mar 2026

Cited by 1 | Viewed by 320

Abstract

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- [...] Read more.

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. Full article

(This article belongs to the Special Issue Future Directions in Atomic Physics Inspired by the Pioneering Work of Charlotte Froese Fischer and Ian Philip Grant)

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12 pages, 324 KB

Open AccessArticle

jj to LSJ Transformation for Configuration State Functions with an Arbitrary Number of Open Shells

by Gediminas Gaigalas

Atoms 2026, 14(3), 20; https://doi.org/10.3390/atoms14030020 - 9 Mar 2026

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Abstract

This paper presents a methodology that allows for calculated energy levels and other atomic characteristics in relativistic atomic theory, i.e., using the

j j

-coupling scheme, to be identified in terms of

L S J

-coupling characteristics. The paper begins with outlining the [...] Read more.

This paper presents a methodology that allows for calculated energy levels and other atomic characteristics in relativistic atomic theory, i.e., using the

j j

-coupling scheme, to be identified in terms of

L S J

-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. Full article

(This article belongs to the Special Issue Future Directions in Atomic Physics Inspired by the Pioneering Work of Charlotte Froese Fischer and Ian Philip Grant)

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12 pages, 527 KB

Open AccessPerspective

Diabatic Potential Energy Matrices at the Interface of Nonadiabatic Dynamics, Machine Learning, and Quantum Computing

by Yuchen Wang

Atoms 2026, 14(3), 19; https://doi.org/10.3390/atoms14030019 - 8 Mar 2026

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Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Section Quantum Chemistry, Computational Chemistry and Molecular Physics)

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44 pages, 1063 KB

Open AccessArticle

Numerical Computation of Critical Binding Parameters of Screened Coulomb Potentials

by Grant B. Bunker

Atoms 2026, 14(3), 18; https://doi.org/10.3390/atoms14030018 - 5 Mar 2026

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Abstract

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, [...] Read more.

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

μ_{c} = 1 / D_{c}

for screening lengths

D \leq 1000

au and angular momenta

l = 0, \dots, 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 \leq 10^{5}

and

l \leq 12

, to 30 digit accuracy, which show interesting (and to our knowledge, not previously reported) periodic structure in

D_{c} (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. Full article

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20 pages, 361 KB

Open AccessArticle

Study of the Hyperfine Structure of Sr II, Ba I and Ba II: An MCDHF Approach for Modeling the Low-Lying Levels

by Lorenzo Nezosi, Lucas Maison, Patrick Palmeri, Per Jönsson and Michel Godefroid

Atoms 2026, 14(3), 17; https://doi.org/10.3390/atoms14030017 - 5 Mar 2026

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Abstract

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 [...] Read more.

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

{}^{2}D_{5 / 2}

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. Full article

(This article belongs to the Special Issue Computational Atomic Physics in Astrophysics)

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20 pages, 1083 KB

Open AccessReview

Application of Atomic Models to Determine Elemental Abundances in Stars in the Non-LTE Approximation: Neutral Potassium and Copper

by Sergei M. Andrievsky and Sergey A. Korotin

Atoms 2026, 14(3), 16; https://doi.org/10.3390/atoms14030016 - 4 Mar 2026

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Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Atomic Processes and Their Role in Astrophysical Phenomena)

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10 pages, 4426 KB

Open AccessArticle

Atomic Ions Ionization Energy Values Assessment: Interpolative Empirical Analysis

by Mariana S. Sendova

Atoms 2026, 14(3), 15; https://doi.org/10.3390/atoms14030015 - 28 Feb 2026

Viewed by 430

Abstract

In this paper, a novel ionization energy,

I E

, set theory-based organizational structure is suggested: (i) iso-protonic sets,

{}_{Z}I E

; (ii) iso-electronic sets,

I E_{s}

; and (iii) iso-ionic sets,

I E^{i +}

. A computational algorithm is [...] Read more.

In this paper, a novel ionization energy,

I E

, set theory-based organizational structure is suggested: (i) iso-protonic sets,

{}_{Z}I E

; (ii) iso-electronic sets,

I E_{s}

; and (iii) iso-ionic sets,

I E^{i +}

. A computational algorithm is proposed which was demonstrated on twenty-five iso-electronic

I E_{s}

-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. Full article

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