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23 pages, 425 KB

Open AccessArticle

Efficient Methods for Dynamic Correlation in Atoms

by Kenneth G. Dyall

Atoms 2026, 14(8), 70; https://doi.org/10.3390/atoms14080070 - 15 Aug 2026

Viewed by 111

Abstract

An algorithm for large-scale correlated calculations on atoms is presented that significantly reduces the scaling of these calculations with the number of single-particle functions used to construct the N-particle states. The reduction is provided in the stage in which the Hamiltonian matrix is [...] Read more.

An algorithm for large-scale correlated calculations on atoms is presented that significantly reduces the scaling of these calculations with the number of single-particle functions used to construct the N-particle states. The reduction is provided in the stage in which the Hamiltonian matrix is contracted with the coefficients or amplitudes of the basis states in an iterative procedure such as the Davidson method. The algorithm relies on the representation of the radial one-particle functions on a grid, and it makes use of the multipole expansion of the electron–electron interaction in a sequence of transformations on the spinors, coefficients, and potentials. It also uses prototyping for the angular integrals and evaluation of recoupling coefficients for the configuration state functions (CSFs) separately rather than for pairs of CSFs. The scaling is verified with calculations on two-electron atoms and analyzed in terms of the number of operations required for each stage of both the proposed algorithm and the conventional methods. 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, 1054 KB

Open AccessArticle

Semirelativistic BSR–RMT Interface: Photoionization of Highly Charged Two-Electron Ions

by Aaron T. Bondy and Klaus Bartschat

Atoms 2026, 14(8), 66; https://doi.org/10.3390/atoms14080066 - 1 Aug 2026

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Abstract

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 [...] Read more.

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

{2 s 2 p}^{3} P^{o}

and

{2 s 2 p}^{1} P^{o}

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

{2 s 2 p}^{3} P^{o}

and

{2 s 2 p}^{1} P^{o}

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. 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, 472 KB

Open AccessArticle

MCDHF Calculations of Isotope Shifts for Be-like Ions Using Perturbation Theory and Finite-Field Method

by Haoran Lin, Sijie Wu, Ran Si, Per Jönsson and Chongyang Chen

Atoms 2026, 14(7), 55; https://doi.org/10.3390/atoms14070055 - 13 Jul 2026

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Abstract

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 [...] Read more.

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. 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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16 pages, 3097 KB

Open AccessArticle

Total, Momentum-Transfer, Differential and Spin-Polarization Cross Sections for Elastic Electron–Strontium Scattering at Low Energies

by Paweł Syty, Michał P. Piłat, Moein Sahraei and Józef E. Sienkiewicz

Atoms 2026, 14(6), 44; https://doi.org/10.3390/atoms14060044 - 31 May 2026

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Abstract

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. [...] Read more.

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. 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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40 pages, 755 KB

Open AccessArticle

Second-Order Rayleigh–Schrödinger Perturbation Theory for the Grasp2018 Package

by Gediminas Gaigalas, Pavel Rynkun and Laima Kitovienė

Atoms 2026, 14(5), 40; https://doi.org/10.3390/atoms14050040 - 21 May 2026

Cited by 1 | Viewed by 652

Abstract

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 [...] Read more.

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. 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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15 pages, 441 KB

Open AccessArticle

Rci-q: An Improved QED Correction Model for the Grasp2018 Package

by Karol Kozioł

Atoms 2026, 14(5), 35; https://doi.org/10.3390/atoms14050035 - 24 Apr 2026

Viewed by 553

Abstract

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 [...] Read more.

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. 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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97 pages, 1163 KB

Open AccessArticle

A Program Library for Computing Pure Spin-Angular Coefficients for One- and Two-Particle Operators in Non-Relativistic Atomic Theory

by Gediminas Gaigalas

Atoms 2026, 14(4), 29; https://doi.org/10.3390/atoms14040029 - 1 Apr 2026

Cited by 2 | Viewed by 891

Abstract

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. [...] Read more.

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. 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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28 pages, 491 KB

Open AccessEditor’s ChoiceArticle

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 2 | Viewed by 780

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 AccessEditor’s ChoiceArticle

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

Cited by 1 | Viewed by 991

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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