Physical Sciences

Abstract: Building on our previous Multiconfiguration-Dirac-Hartree-Fock (MCDHF) computational strategies taylored for the hyperfine structure (HFS) of low-lying levels in one-valence electron Sr II and Ba II ions, and in two-valence electrons Ba I atom [Nezosi et al, Atoms 14, 17 (2026). DOI: 10.3390/atoms14030017], we successfully extend these methodologies along the alkaline-earth elements to the lighter Ca II and the heavier Ra I-II ions. MCDHF recommended HFS constants, along with their uncertainty estimates, are reported for the first time and critically discussed. Where applicable, the Bohr-Weisskopf correction is applied to the HFS constants, and its effects are analyzed. This is particularly valuable for cases where no measurement is available such as for the [Rn]\(6d ^2D_{3/2,5/2}\) levels in ^223,225Ra II, for the [Rn]\(7s6d~^3D_{1,2,3}\) and ¹D₂ levels in ²²³Ra I, for the [Rn]\(7s6d ^3D_{2,3}\) and ¹D₂ levels in ²²³Ra I, and for the [Rn]\(7s7p~^3P^o_2\) level in ²²⁵Ra I. In all cases, our MCDHF values agree with the ones found in the literature within our error bars. For the ²D_5/2 levels in Ca II and Ra II, the discrepancies with experiment observed in Sr II and Ba II for the HFS \(A\) constant are not seen.

Abstract: The R-matrix method has long been employed as the most advanced and powerful approach to treat a variety of atomic processes. Its main application has been in spectral analysis and modeling of astrophysical and laboratory sources. We describe the R-matrix methodology, computational codes, and extensions developed for numerical simulations under wide-ranging physical conditions in astronomical objects such as stellar atmospheres in non-local-thermodynamic-equilibrium (NLTE), interstellar medium, black hole environments and active galactic nuclei, and in high-energy-density (HED) environments such as high-intensity lasers, plasma fusion devices and stellar interiors. We present exemplary results for atomic cross sections for photoionization, electron impact excitation and electron-ion recombination, as well as for plasma effects in HED sources. In addition, we illustrate selected applications at different and varying temperatures, densities, and other variables.

Abstract: Calculation of nonbonded interactions has been a bottleneck for molecular simulations (for molecular dynamics (MD) and Monte Carlo (MC) simulations) that are limiting the system size and the time scale in MD up to nanoseconds. To reduce the computation time/load in calculating nonbonded interactions, an efficient sampling techniques were introduced in this study, for calculating total potential acting on individual atoms. Conventional way of calculating nonbonded interaction for a particular atom is by calculating pairwise van Der Waal and Coulomb interactions for all neighboring atoms with in cutoff radius and adding tail correction beyond the cutoff. Here, we introduce Theory of Local Sampling (TLS), where nonbonded interactions will be calculated by using selective local region around the atom (instead of considering all region within cutoff). Basic principle in TLS is that, any effect/change in global region will reflect in selective local region and can be integrated into molecular simulations. This local sampling technique, instead of global sampling, helps in reducing total computation time/load in calculating nonbonded interactions for the system at each step in MD and in MC simulations. Applying TLS in MD simulation for Argon atoms gives reduction in computation time up to 60%. For MC simulation, up to 37% of reduction in computation time was acheived. Energies and radial distribution function (RDF) comparison for Argon liquid system gives good agreement with conventional MC and MD simulations. Another approach presented in reducing computation time in MD simulations is using "Curve Fitting" techniques in predicting force/acceleration acting on individual atoms using previous time steps, giving 24% reduction in computation time. Combining "TLS & Curve Fitting" techniques further reduces computation time up to 67% in MD simulations.

Abstract: In strong-field ionization, complex-time saddle points govern electron ionization instant and quantum-orbit interference patterns. Neverthless, not all solutions to the saddle-point equation carry physical contributions. And saddle coalescence can also invalidate the isolated-saddle approximation. Within the strong-field-approximation framework, we propose a topology-constrained approach to reconstruct photoelectron momentum spectra. A physics-informed neural network is employed to produce candidate roots, which are subsequently refined via a Newton-type iteration. Picard–Lefschetz upward flows are adopted to pick out physically connected saddles, while local coherent continuation for pole-saddle pairs handles saddle-coalescence regions. A topological selector is defined by the non-zero support of intersection numbers to filter unphysical solutions. We test our method on hydrogen atoms subjected to six-cycle elliptically polarized laser fields and compare reconstructed spectra against direct numerical strong-field-approximation results. For 300 nm and 500 nm driving fields, the normalized relative L₂ errors on the linear probability scale reach 3.793% and 3.520%, respectively.The topological selection eliminates disconnected non-physical contributions, and local continuation recovers quantum cancellation effects near saddle coalescence. Our method improves the physical interpretability of saddle-based spectrum reconstruction under strong-field approximation and is promising for other computational-physics problems involving saddle-point equations.

Abstract: We present a theoretical investigation of the attoclock method in laser-assisted photoemission (LAPE) processes. We analyze atomic photoionization driven by a short XUV pulse assisted by a circularly polarized strong infrared (IR) laser under accessible experimentally conditions. We calculate three-dimensional photoelectron momentum distributions (PMDs) in order to extract time delays from their angular information based on the attoclock principle. Furthermore, we explore different XUV parameters to determine the optimal achievable temporal precision for the laser-assisted attoclock (LAAC). The analyses performed within the the strong-field approximation (SFA) are further supported by improved Coulomb methods, including the Coulomb–Volkov approximation (CVA) and numerical solutions of the time-dependent Schrödinger equation (TDSE). Our work lays a theoretical foundation for understanding time-resolved electronic dynamics in cutting-edge ultrafast experiments of photonic collisions.

Abstract: The electron is modelled as a two-photon bound system rather than a point particle, admitting a discrete set of stable orbital states selected by a single standing-wave closure condition; the nuclear-particle mass spectrum then follows as products of these kernel states with no particle-by-particle fitted parameters. The atomic and nuclear domains appear as the two lowest exact solutions of one closure condition—the closed full-orbit kernel and the open half-orbit kernel—rather than as independently postulated structures. The neutron mass, computed independently of the proton, agrees to 0.268 ppm raw and +0.0013 ppm with a derived one-loop coefficient; the charged pion agrees with PDG 2024 at 1.5σ of its ±1.3 ppm measurement; the top quark agrees to 25 ppm; the muon closes to +0.030 ppm; and the tauon carries a single falsifiable prediction, 3477.2269 electron masses, resolvable by Belle II. The proton mass is reproduced to eleven significant figures through an exact closed form, disclosed as the calibration identity fixing the Anomalous fNuclear Ratio. All results follow from the fine-structure constant and the electron gyromagnetic anomaly alone, presented as evidence that the nuclear mass hierarchy has a geometric origin in electron structure.

Abstract: High-precision two-dimensional intermolecular potential energy surfaces (PESs) for Rg–CuF (Rg = Ar, Kr, Xe) were constructed at the coupled-cluster singles and doubles with non-iterative triples [CCSD(T)] level, by employing aug-cc-pVXZ (X = D, T, Q) basis sets and the energies were extrapolated to the complete basis set (CBS) limit. All three complexes exhibit a consistent topological pattern: the global minimum corresponds to a collinear Rg–Cu–F configuration, and the local minimum corresponds to an anti-linear Rg–F–Cu configuration. As the atomic number of noble gas increases, the Rg–Cu equilibrium distance lengthens while the binding strength remarkably enhance. Bound state calculations were performed based on these PESs to yield rotational levels, which can be used derive the intermolecular vibrational frequencies, molecular structures and spectroscopic parameters for all primary isotopologues. The predicted rotational constants B are in excellent with the experimental observations, attaining a sub-MHz accuracy at the AVTZ level for Kr–CuF and at the CBS limit for Ar–CuF and Xe–CuF. Vibrational wavefunction analysis reveals that the intermolecular vibrational modes of Kr–CuF and Xe–CuF are highly localized, consistent with the pronounced molecular rigidity observed experimentally. Isotopic effect analysis reveals a well-defined linear relationship between the changes in the rotational constant B and the intermolecular vibrational frequency in relation to the reduced mass of the complex, which provides a reliable basis for predicting spectroscopic parameters of unobserved isotopologues.

Abstract: Accurate atomic data for heavy ions are required for spectroscopic diagnostics and impurity transport modeling in magnetic-confinement fusion devices, particularly for tungsten and its neutron-induced transmutation products. In the present work, we investigate the ytterbium-like ions Ta IV, W V, Re VI, Os VII, Ir VIII, and Pt IX using complementary pseudo-relativistic Hartree–Fock (HFR) and fully relativistic multiconfiguration Dirac–Hartree–Fock (MCDHF) approaches. Semi-empirical calculations were first performed using the HFR method including core polarization corrections (HFR+CPOL). The evolution of correlation effects along the isoelectronic sequence was then analyzed through an alternative HFR(CV) model in which the most important core-excited configurations were introduced explicitly to account for core–valence interactions. Extensive least-squares fits to the available experimental energy levels were carried out for all ions considered. The results reveal that the HFR+CPOL approach provides an adequate description of Ta IV and W V, while explicit treatment of core–valence correlations becomes increasingly important from Re VI onwards owing to the growing interaction between the 5d5f and 5p⁵5d³ configurations. New energy level classifications, level compositions, oscillator strengths, and transition probabilities are reported and critically assessed through comparisons between HFR and MCDHF calculations. The resulting dataset extends previous investigations of rhenium and osmium ions and provides benchmark atomic data for heavy-ion spectroscopy and future fusion-plasma applications.

Abstract: The Bohr radius is normally presented as a non-relativistic length scale. Less widely discussed is that, in the second part of his 1913 trilogy, Bohr briefly indicated how the radius formula could be modified when the orbital velocity is not negligible compared with the speed of light. In this paper, we revisit that overlooked prescription and show that replacing the electron rest mass by the Lorentz-factor-adjusted mass term leads to a relativistic Bohr radius a_(0,r) = a₀ √(1 -α²) = (λ̅_e√(1 -α²))/α for hydrogen, and to a^Z_(0,r) = a₀/Z √(1- Z²α²) for a hydrogen-like one-electron ion. The central result is that this semi-classical expression is not merely an ad hoc relativistic contraction: under the assumptions of a point nucleus, infinite nuclear mass, and a one-electron Coulomb field, it is exactly identical to the most probable radius obtained from the Dirac 1s_1/2 radial probability density. This identification appears to be the first explicit demonstration that Bohr’s historical relativistic prescription selects the Dirac most-probable radius, rather than the Dirac expectation value ⟨r⟩D. We emphasize the important distinction between a most-probable radius and a mean radius: even in the non-relativistic Schrödinger 1s state, the most-probable radius is a₀/Z, whereas the expectation value is 3a₀/(2Z) and is therefore larger by a factor of 3/2. For hydrogen the relativistic correction to the most-probable radius is only 1.40898891811 × 10⁻¹⁵ m, or 0.0026626%, but the result is conceptually significant because it links Bohr’s semi-classical radius, the relativistically contracted reduced Compton wavelength, and the Dirac-Coulomb ground-state probability maximum in one compact formula. We also provide numerical values for hydrogen-like ions and explain why these smooth one-electron Dirac radii should not be confused with expectation values or with empirical radii of neutral many-electron atoms, which can vary non-monotonically across the periodic table.

Abstract: We consider quasi-one-dimensional (Q1D) continuous waves (CWs) in the two-dimensional (2D) optical system with the cubic-quintic nonlinearity and a Q1D potential trough. In the case of a smooth trough profile, we confirm the known modulational instability (MI) of Q1D CWs with the transverse structure corresponding to the 1D ground state (GS) in the potential trough, and demonstrate the MI of CWs with the dipole-mode (DM) transverse structure, corresponding to the lowest 1D excited state in the potential trough. The CWs of both GS and DM types remain nearly stable close to edges of their existence regions. Stable stationary states in the form of periodic chains of 2D solitons, trapped in the potential trough, are produced in a numerical form. The dynamics of the soliton chains excited by a localized kick is studied too. For the potential trough with the singular delta-functional profile, we find two species of exact analytical solutions for CWs, one of which is completely stable.

Abstract: The new approach is develop to determine the hyperfine structure of the rotationally excited 1s_µnℓ_e-states in the three-body helium-muonic ³He µ e and ⁴He µ e atoms. The formulas derived in this study allow one to determine the hyperfine structure splittings for an arbitrary rotationally excited 1s_µnℓ_e-state in the three-body helium-muonic atoms. We also created the new approximate method which can be used for fast numerical evaluations of the hyperfine structure splittings in the both 1s_µnp_e and 1s_µnd_e-states of the 3He µ e and 4He µ e atoms. Similar formulas were never produced in earlier studies.

Abstract: We reformulate matrix mechanics by incrementally heating a single hydrogen atom in order to derive the complete, diagonalized Hamiltonian matrix. The spectral lines and transition probabilities cannot be generated by reversing time thereby demonstrating asymmetry. Experimental evidence in support of the theoretical model is obtained from experiments performed with the simplest quantum system, an electron cyclotron. Wave mechanics is also reformulated by altering the original Schrödinger equation describing a time symmetric, conservative system to a system of two independent equations describing a time asymmetric, non-conservative system.

Abstract: 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 [1], we presented numerical calculations using the Phase Method (PM), accurate to 60 digits and to screening lengths D ≤ 10³ au, l = 0–20, of the critical binding parameters for these potentials; an for Yukawa and ECSC, l = 0–12, to D ≤ 10⁵ au, at 30 digits. In doing so, we discovered 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 fullyquantum 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 NC, 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 √D_c, 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. Tables of critical screening parameters for Yukawa, Hulthén and ECSC D ≤ 10⁵ and l = 0 − 12 are posted as supplementary data.

Abstract: We present new and improved calculations of Stark-broadened profiles for ionized helium, a key ingredient in the spectroscopic analysis of helium-atmosphere DO white dwarfs. Our approach builds upon the computer simulation framework previously developed for neutral helium, which fully accounts for the dynamical interactions of both ions and electrons with the emitting helium atom. We extend this theoretical formalism by relaxing the assumption of straight-line trajectories for the perturbing particles (electrons and ionized helium) and adopting the hyperbolic trajectories appropriate for their interaction with a charged emitter, thereby accounting for their dynamical influence on the line-broadening process. In this exploratory study, we focus on the He II λ4686 line, the strongest absorption feature observed in the spectra of DO white dwarfs. We present the resulting Stark profiles and perform a detailed comparison with those available in the literature.

Abstract: 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 is 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.

Abstract: 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 high-order harmonics 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. 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 that the negligible effect of the Coulomb potential of the remaining ion on the escaping photoelectron for high energy above-threshold ionization. These findings provide new insights into attosecond chronoscopy in the strong-field regime.

Abstract: Classical subatomic particles simulations were applied to protons, neutrons and electrons to simulate a complete atom using pseudo potentials to simulate nucleus and electrons distributed around nucleus. Molecular dynamics algorithms were applied to subatomic particles to simulate a complete atom for Hydrogen, Carbon and Uranium atoms that were reported in previous studies. Further analysis of energies of Carbon atom reveals that atom has quantized discrete energy values rather than continuous energies as expected in classical simulations. Changes in atom's total energy were observed to be in step changes going to higher/lower energy states and energy values are observed to be in quantized rather than continuous. Electrons energy profiles was observed during the atom's quantum energy shifts and only one electron got affected that has step change in its energies, for short duration, leaving other electrons unaffected. To observe similar phenomenon in other atoms, Fluorine, Magnesium and Chlorine atom were considered in this study and found to have similar quantized discrete energy values properties. This study implements subatomic particles simulations using classical mechanics that explains why atom's energies are quantized in nature as observed in discrete lines in atomic emission/absorption spectrum for atoms.

Abstract: A phenomenological geometrical model is presented for the hyperfine structure of hydrogenic systems. The approach extends a previously published fine-structure analysis to hyperfine splittings by introducing a compact set of empirically constrained geometric corrections. Unlike conventional quantum electrodynamic treatments which reference the Lamb shift to the hyperfine centroid, the present framework targets the hyperfine mid-point, corresponding to a substantially larger reference interval. Despite this difference in reference scale, agreement with experimental hyperfine and Lamb-shift-related data at the 0.01 MHz level is obtained across multiple hydrogen states. The model is further extended to deuterium, tritium, 3He⁺, and 7Li²⁺, revealing systematic cross-nuclear behaviour and a universal scaling relation involving nuclear mass number and charge. The calculations employ a simplified geometric scheme that does not rely on perturbative quantum electrodynamics but is intended as a complementary phenomenological description. Possible physical interpretations in terms of structured internal dynamics are discussed, together with limitations and directions for further development.

Abstract: String theory has long pursued a mechanism to compactify its extra dimensions into the observable physical constants of our universe, yet the vast landscape of ~10⁵⁰⁰ possible vacua remains unresolved. Our 4G Model Solution: The 4G Model introduces four interaction-dependent scalar gravitational constants (GN, Ge, Gn, Gw) as the practical bridge, deriving a fundamental 33 pm interaction length—the geometric mean of nuclear and electromagnetic gravity. This scale constrains atomic structure and nuclear radii via the empirical A¹/³ scaling law. Validation: The 4G Model’s fundamental scaling law (A¹/³ × 33 pm) predicts Carbon’s covalent radius at 75.6 pm-matching the experimental 75 pm value to under 1% error without the proposed common correction term. This precise agreement without corrections suggests that the geometric mean of nuclear and electromagnetic gravity [√(Gn*Ge)] may play a key role in atomic structure. Broader Deviations Contextualized: Secondary deviations in other groups stem from Z-dependent quantum screening of Ge and Gn, not flaws in the underlying scale- paralleling Bohr model successes for hydrogen before Sommerfeld’s fine-structure refinements addressed relativistic effects. Theoretical Confirmation: This selective precision affirms 4G’s unification: atomic radii emerge directly from string-like compactification geometry, with screening as tunable perturbations. Carbon’s validation anchors the model as a working hypothesis, indicating that gravitational constants could play a significant role in constraining chemistry at the 33 pm scale. Extension: Finally, by applying the proton’s charge‑mass “dual discreteness formalism,” we propose that atoms can be interpreted as quantum gravitational compact objects within this framework. These are structured into a hierarchy of 7 fundamental shells, dictated by the stability condition , n=1,2,3.., Z/Root(A) . Light magic numbers emerge from the integer values of Z/Root(A_stable) , while heavy magic numbers correspond to the half-integer form, [Z/Root(A_stable)+0.5].

Abstract: Electron collision broadening parameters calculations were performed for 33 singly ionized yttrium spectral lines, 27 with special relevance in astrophysical applications (can be found in the Atmospheres of FGK Stars). Calculations were made in the semi-empirical Griem approach using the Gaunt factors proposed by Van Regemorter and those proposed by Douglas H. Sampson. Furthermore, to test our calculations, the electron collision broadening of 4 well isolated spectral lines of Y II was experimentally measured in laser- induced breakdown experiments using a Q-switched Nd:YAG laser focused on Pb-Y alloy samples.