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Atoms, Volume 13, Issue 8 (August 2025) – 6 articles

Cover Story (view full-size image): Helium Rydberg states provide critical benchmarks for quantum electrodynamics, yet their precise energy calculations remain challenging owing to multiscale electron correlations. This work develops a unified method employing correlated B-spline basis functions that computes nonrelativistic energies for helium 1snp^{1,3}P states (n=2-27) with 14-digit precision, providing consistent descriptions from low-energy to Rydberg states. The approach simultaneously captures short-range interactions and long-range asymptotic behavior, demonstrating superior performance to conventional methods. This framework achieves consistent precision for both low- and high-lying Rydberg states, enabling future relativistic and QED investigations while providing crucial theoretical pathways with which to resolve existing theory–experiment discrepancies. View this paper

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12 pages, 320 KB

Open AccessArticle

Inner Products of Spherical Tensor Operators: A Late Chapter of Racah Algebra

by Peter Uylings

Atoms 2025, 13(8), 73; https://doi.org/10.3390/atoms13080073 - 19 Aug 2025

Viewed by 212

Abstract

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. Full article

15 pages, 628 KB

Open AccessArticle

Accurate Nonrelativistic Energy Calculations for Helium 1snp^1,3P (n = 2 to 27) States via Correlated B-Spline Basis Functions

by Jing Chi, Hao Fang, Yong-Hui Zhang, Xiao-Qiu Qi, Li-Yan Tang and Ting-Yun Shi

Atoms 2025, 13(8), 72; https://doi.org/10.3390/atoms13080072 - 4 Aug 2025

Viewed by 452

Abstract

Rydberg atoms play a crucial role in testing atomic structure theory, quantum computing and simulation. Measurements of transition frequencies from the

2^{1, 3} S

states to Rydberg

{}^{1, 3}P

states have reached a precision of several kHz, which poses [...] Read more.

Rydberg atoms play a crucial role in testing atomic structure theory, quantum computing and simulation. Measurements of transition frequencies from the

2^{1, 3} S

states to Rydberg

{}^{1, 3}P

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

24 {}^{1}P

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

1 s n p

{}^{1, 3}P

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. Full article

(This article belongs to the Special Issue Atom and Plasma Spectroscopy)

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7 pages, 1017 KB

Open AccessCommunication

Observing the Ionization of Metastable States of Sn¹⁴⁺ in an Electron Beam Ion Trap

by Qi Guo, Zhaoying Chen, Fangshi Jia, Wenhao Xia, Xiaobin Ding, Jun Xiao, Yaming Zou and Ke Yao

Atoms 2025, 13(8), 71; https://doi.org/10.3390/atoms13080071 - 1 Aug 2025

Viewed by 351

Abstract

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. Full article

(This article belongs to the Special Issue 21st International Conference on the Physics of Highly Charged Ions)

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13 pages, 359 KB

Open AccessReview

Numerical Methods for the Time-Dependent Schrödinger Equation: Beyond Short-Time Propagators

by Ryan Schneider and Heman Gharibnejad

Atoms 2025, 13(8), 70; https://doi.org/10.3390/atoms13080070 - 28 Jul 2025

Viewed by 618

Abstract

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. Full article

(This article belongs to the Special Issue Ab Initio Calculations in Atomic, Molecular, and Optical Physics: A Tribute to Barry Irwin Schneider)

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19 pages, 2243 KB

Open AccessArticle

Theoretical Calculation of Ground and Electronically Excited States of MgRb⁺ and SrRb⁺ Molecular Ions: Electronic Structure and Prospects of Photo-Association

by Mohamed Farjallah, Hela Ladjimi, Wissem Zrafi and Hamid Berriche

Atoms 2025, 13(8), 69; https://doi.org/10.3390/atoms13080069 - 25 Jul 2025

Viewed by 484

Abstract

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. Full article

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

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16 pages, 1681 KB

Open AccessArticle

Thermal–Condensate Collisional Effects on Atomic Josephson Junction Dynamics

by Klejdja Xhani and Nick P. Proukakis

Atoms 2025, 13(8), 68; https://doi.org/10.3390/atoms13080068 - 22 Jul 2025

Viewed by 603

Abstract

We investigate how collisional interactions between the condensate and the thermal cloud influence the distinct dynamical regimes (Josephson plasma, phase-slip-induced dissipative regime, and macroscopic quantum self-trapping) emerging in ultracold atomic Josephson junctions at non-zero subcritical temperatures. Specifically, we discuss how the self-consistent dynamical inclusion of collisional processes facilitating the exchange of particles between the condensate and the thermal cloud impacts both the condensate and the thermal currents, demonstrating that their relative importance depends on the system’s dynamical regime. Our study is performed within the full context of the Zaremba–Nikuni–Griffin (ZNG) formalism, which couples a dissipative Gross–Pitaevskii equation for the condensate dynamics to a quantum Boltzmann equation with collisional terms for the thermal cloud. In the Josephson plasma oscillation and vortex-induced dissipative regimes, collisions markedly alter dynamics at intermediate-to-high temperatures, amplifying damping in the condensate imbalance mode and inducing measurable frequency shifts. In the self-trapping regime, collisions destabilize the system even at low temperatures, prompting a transition to Josephson-like dynamics on a temperature-dependent timescale. Our results show the interplay between coherence, dissipation, and thermal effects in a Bose–Einstein condensate at a finite temperature, providing a framework for tailoring Josephson junction dynamics in experimentally accessible regimes. Full article

(This article belongs to the Special Issue Quantum Technologies with Ultracold Atoms)

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