# An Introduction to Relativistic Theory as Implemented in GRASP

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## Abstract

**:**

## 1. Introduction

## 2. Grasp Theory—Wave Functions

#### 2.1. One-Electron Dirac Orbital Functions

#### 2.2. Atomic State Function and Configuration State Functions

#### 2.3. Dirac-Coulomb, Dirac-Coulomb-Breit Hamiltonians and QED Corrections

`rnucleus`and saved in a file

`isodata`(see accompanying manual §2.2 and §8.1).

`rci`code, as implemented in the original grasp program [78], to yield the final Hamiltonian

#### 2.4. Building Configuration State Functions

- By using the well-known vector coupling techniques of angular momentum theory [85] to couple sequentially, from left to right, the subshell angular momenta $\mathbf{J}={\sum}_{i=1}^{m}{\mathbf{J}}_{i}$,$$(\left(({J}_{1},{J}_{2}){J}_{12}\right),{j}_{3}){J}_{123},\dots ,{J}_{12\dots m-1},{J}_{m})J{M}_{J}\phantom{\rule{0.277778em}{0ex}},$$$$|{\left({n}_{i}{\kappa}_{i}\right)}^{{w}_{i}}{\mathit{\alpha}}_{i}{\nu}_{i}{J}_{i}{M}_{i}\rangle \phantom{\rule{0.277778em}{0ex}},$$
- Antisymmetrize the resulting coupled products through the permutations restricted to the exchange of electron coordinates involving different subshells [86].

`rcsf.out`produced by the

`rcsfgenerate`program; see accompanying manual §3.2. However, we deliberately keep the $Nr$ notation, since the spin-angular library [93] allows us to deal with the $j=9/2$ subshell without any occupation restriction.

**Q**$({Q}_{+},{Q}_{-},{Q}_{z})$ is that the ladder operators ${Q}_{\pm}$ connect subshell wave functions with occupation numbers w and $w\pm 2$ having the same seniority $\nu $. The Wigner–Eckart (WE) theorem can be applied in the space of quasi-spin for all individual subshell states much in the same way as for J-space, allowing an efficient reduction and factorization of matrix elements and CFP matrices [96].

#### 2.5. Second Quantization and Composite Tensor Operators

`rangular`code from [96] is based on the quasi-spin constructions, replacing the older formulation following Fano [86] used in earlier versions of the atsp2K [28] and grasp [43] packages.

#### 2.6. Calculation of Matrix Elements

`librang90`library, which are then called by the

`rangular`program for MCDHF approach, the

`rci`code for CI calculations, the

`rbiotransform`and

`rtransition`programs for transition properties, the

`rhfs`and

`hfszeeman95`codes for hyperfine parameters and magnetic interactions, the

`ris4`programs for isotope shifts and the

`rdensity`program for radial electron densities and natural orbitals, see accompanying manual §1.3 and §2.2.

**r**${}_{i}$. The effective interaction strength, ${X}^{k}\left(abcd\right)$, is specific to the nature of the interaction and involves only the active orbitals [73,106]. It can be written in terms of a radial double integral ${\mathcal{R}}^{k}\left(abcd\right)$ and factors involving matrix elements of the spherical spinors of the active orbitals. The spin-angular ${\xi}_{abcd;k}$ coefficients are computed by the routines of the

`librang90`library, which are then called by the

`rangular`program for MCDHF approach, the

`rci`program for CI calculations, and the

`ris4`program for isotope shifts; compare to the accompanying manual §2.2.

#### 2.7. Multiconfiguration Dirac–Hartree–Fock

`rmcdhf`program, which reads the nuclear parameters, the CSFs list, as generated by the

`rcsfgenerate`program and the necessary angular data, produced by

`rangular`, from disk files. The initial estimates of the radial orbitals for the SCF procedure are generated by the

`rwfnestimate`program and can be taken as screened hydrogenic functions, functions from a Thomas–Fermi calculation or converted non-relativistic orbitals; see the accompanying manual §3.3. For an overview of the program and file flows for the MCDHF calculations, see Figures 1 and 2 of the manual.

#### 2.8. Configuration Interaction

`rci`program. Nuclear parameters, the CSFs list, as generated by

`rcsfgenerate`and radial orbitals, as produced by

`rmcdhf`, are read from disk files. Angular data needed to compute the Hamiltonian matrix elements are computed on the fly by calls to the routines of the

`librang90`library. The program and file flows associated with a CI calculation are displayed in Figures 1 and 2 in the accompanying manual. In grasp, the program

`rcsfzerofirst`is used to partition the CSF expansion in zero- and first-order sets. The ZF method is not the default mode but is only used to handle very large expansions and matrices, as discussed in detail in section 14 of the manual [72] that is entirely devoted to strategies for ZF-MCDHF and ZF-CI.

#### 2.9. Transformation to Different Coupling Schemes

`jj2lsj`program that also determines unique labels. Transformations to other coupling schemes, e.g., $JK$ or $LK$, are completed by the

`coupling`program; see the accompanying manual §6.2 and §6.3. Needless to say, the programs to transform the wave functions and assigning unique labels are important parts of the grasp package.

## 3. Grasp Theory—Atomic Properties

#### 3.1. Hyperfine Structures

`rhfs`program. The nuclear parameters, the CSF expansion, the radial orbitals, as obtained by

`rmcdhf`, and the expansion coefficients of the CSFs, as obtained by the

`rmcdhf`or

`rci`programs, are read from files. Specific examples of computations of hyperfine constants are given in §6.1 of the manual. For more recent developments, using several independently optimized radial orbital sets in the computation of the hyperfine constants, see the article by Yan Ting et al. [128] in the present Special Issue.

#### 3.2. External Magnetic Fields

`hfszeeman95`program. The nuclear parameters, the CSF expansion, the radial orbitals, as obtained by

`rmcdhf`, and the expansion coefficients of the CSFs, as obtained by the

`rmcdhf`or

`rci`programs, are read from files. Specific examples are given in §6.9 of the accompanying manual.

#### 3.3. Isotope Shift

#### 3.3.1. Mass Shift

`ris4`program (see §6.1 in the manual for a specific example). The three contributions associated with the three terms of (72) and (73) are reported separately.

#### 3.3.2. Field Shift

`ris4`[68]. For lighter systems, where the electronic density is essentially constant inside the nuclear volume, it is justified to only consider the first electronic factor $\mathsf{\Delta}{F}_{k,0}=\frac{2}{3}\pi Z\mathsf{\Delta}{\rho}_{\mathsf{\Gamma}J}^{e}\left(\mathbf{0}\right)$, and the nuclear quantity $\delta {\langle {r}^{2}\rangle}^{A,{A}^{\prime}}$ can be extracted from observed line shifts along isotope chains [63] and be directly compared with predictions from nuclear theory [133]. The reduction (31) for the field shift operator is relatively simple thanks to the scalar property of the radial electron density. The spin-angular coefficients are identical to the weighing factors ${t}_{ab}^{\mathit{\alpha}\beta}$ of the one-electron radial integrals $I(a,b)$ appearing in Equation (41) for the Hamiltonian, which is another $k=0$ tensorial operator. More details can be found in [68]—see also Section 3.4. In the grasp suite of codes, the electronic factors (81) of the level field shift (80), together with ${F}_{\phantom{\rule{0.277778em}{0ex}}\mathsf{\Gamma}J,0}^{\phantom{\rule{0.277778em}{0ex}}\left(0\right)\mathrm{ved}}$ and ${F}_{\phantom{\rule{0.277778em}{0ex}}\mathsf{\Gamma}J,0}^{\phantom{\rule{0.277778em}{0ex}}\left(1\right)\mathrm{ved}}$ of (86) are computed by the

`ris4`program; see §6.1 and §8.1 in the manual.

#### 3.3.3. Total Shift

`fical`of the grasp suite of codes allows this calculation. Section 12 of the manual provides a specific example of how to use

`ris4`together with

`fical`to compute the effect of nuclear deformation on the frequency isotope shift for the $1{s}^{2}2s{\phantom{\rule{3.33333pt}{0ex}}}^{2}{S}_{1/2}-1{s}^{2}2p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2,3/2}^{o}$ transitions in ${}^{150,142}$Nd${}^{57+}$. Careful attention should be paid to the conventions used to define the relevant ingredients [134].

#### 3.4. Electronic Radial Densities and Natural Orbitals

`rdensity`program. The nuclear parameters, the CSF expansion, the radial orbitals, as obtained by

`rmcdhf`, and the expansion coefficients of the CSFs, as obtained by the

`rmcdhf`or

`rci`programs, are read from files. The radial density function is written to file in the format that makes it easy to print. The NOs are written to file in the same format as the radial wave functions from

`rmcdhf`and can be directly used by the

`rci`program. Examples of how to use the

`rdensity`program to produce the radial density function and the NOs are given in §6.8 in the manual.

#### 3.5. Radiative Transition Properties

- 1.
- Perform MCDHF or CI calculations for the initial and< the final states, where the radial one-electron orbital sets $\left\{({P}_{n\kappa}\phantom{\rule{0.277778em}{0ex}},{Q}_{n\kappa})\right\}$ and $\left\{({P}_{{n}^{\prime}\kappa}^{\prime}\phantom{\rule{0.277778em}{0ex}},{Q}_{{n}^{\prime}\kappa}^{\prime})\right\}$ of the lower $\mathsf{\Gamma}J{M}_{J}$ and upper ${\mathsf{\Gamma}}^{\prime}{J}^{\prime}{M}_{{J}^{\prime}}$ state wave functions are not assumed to be the same.
- 2.
- For each $\kappa $, compute the radial orbital overlap matrix. Transform the two radial one-electron orbital sets$$\left\{({P}_{n\kappa}\phantom{\rule{0.277778em}{0ex}},{Q}_{n\kappa})\right\}\to \left\{({\tilde{P}}_{n\kappa}\phantom{\rule{0.277778em}{0ex}},{\tilde{Q}}_{n\kappa})\right\},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\left\{({P}_{{n}^{\prime}\kappa}^{\prime}\phantom{\rule{0.277778em}{0ex}},{Q}_{n\kappa}^{\prime})\right\}\to \left\{({\tilde{{P}^{\prime}}}_{{n}^{\prime}\kappa}\phantom{\rule{0.277778em}{0ex}},{\tilde{{Q}^{\prime}}}_{{n}^{\prime}\kappa})\right\}$$$$\langle ({\tilde{P}}_{n\kappa}\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.166667em}{0ex}}{\tilde{Q}}_{n\kappa})|({\tilde{{P}^{\prime}}}_{{n}^{\prime}\kappa}\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.166667em}{0ex}}{\tilde{{Q}^{\prime}}}_{{n}^{\prime}\kappa})\rangle ={\delta}_{n,{n}^{\prime}}\phantom{\rule{0.277778em}{0ex}}.$$The orbital transformation in effect changes the CSFs, and we have$$\left\{\mathsf{\Phi}\left({\gamma}_{\mathit{\alpha}}J\right)\right\}\to \left\{\tilde{\mathsf{\Phi}}\left({\gamma}_{\mathit{\alpha}}J\right)\right\},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\left\{\mathsf{\Phi}\left({\gamma}_{\beta}^{\prime}{J}^{\prime}\right)\right\}\to \left\{\tilde{\mathsf{\Phi}}\left({\gamma}_{\beta}^{\prime}{J}^{\prime}\right)\right\}.$$The orbital transformation is followed by a counter transformation of the CI expansion coefficients$$\left\{{C}_{\mathit{\alpha}}^{\phantom{\rule{0.166667em}{0ex}}\mathsf{\Gamma}J}\right\}\to \left\{{\tilde{C}}_{\mathit{\alpha}}^{\phantom{\rule{0.166667em}{0ex}}\mathsf{\Gamma}J}\right\},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\left\{{C}_{\beta}^{\phantom{\rule{0.166667em}{0ex}}{\mathsf{\Gamma}}^{\prime}{J}^{\prime}}\right\}\to \left\{{\tilde{C}}_{\beta}^{\phantom{\rule{0.166667em}{0ex}}{\mathsf{\Gamma}}^{\prime}{J}^{\prime}}\right\}$$$$\sum _{\mathit{\alpha}}{C}_{\mathit{\alpha}}^{\phantom{\rule{0.166667em}{0ex}}\mathsf{\Gamma}J}\mathsf{\Phi}\left({\gamma}_{\mathit{\alpha}}J\right)\equiv \sum _{\mathit{\alpha}}{\tilde{C}}_{\mathit{\alpha}}^{\phantom{\rule{0.166667em}{0ex}}\mathsf{\Gamma}J}\tilde{\mathsf{\Phi}}\left({\gamma}_{\mathit{\alpha}}J\right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\sum _{\beta}{C}_{\beta}^{{\mathsf{\Gamma}}^{\prime}{J}^{\prime}}\mathsf{\Phi}\left({\gamma}_{\beta}^{\prime}{J}^{\prime}\right)\equiv \sum _{\beta}{\tilde{C}}_{\beta}^{{\mathsf{\Gamma}}^{\prime}{J}^{\prime}}\tilde{\mathsf{\Phi}}\left({\gamma}_{\beta}^{\prime}{J}^{\prime}\right)$$
- 3.
- Use standard Racah algebra to compute the transition matrix elements in the new biorthonormal representation.

`rbiotransform`. The CSF expansions, required to be closed under de-excitation (see accompanying manual §3.8), the radial orbitals, as obtained by

`rmcdhf`, and expansion coefficients obtained by

`rmcdhf`or

`rci`are read from file. The transformed radial orbitals and expansion coefficients are written back to file. The computation of transition rates and weighted oscillator strengths, in both Babushkin and Coulomb gauges for electric multipoles, are completed with

`rtransition`. The CSF expansions, the transformed radial orbitals and expansion coefficients, along with relevant energies needed to compute the transition frequencies $\omega $, are read from file. The program and file flow leading up to the transition calculations are displayed in Figures 1 and 2 in the manual.

`rtransition`program along with the transition rates and weighted oscillator strengths in both Coulomb and Babushkin gauges. Specific examples of transition calculations are given in Sections 6.1, 6.3, 6.4, 9, 10, and 11 of the manual. In §11.3, the programs are used to study the change of transition rates along an iso-electronic sequence.

#### 3.6. Unexpected Transitions

`hfszeeman95`program [67]. The nuclear parameters, the CSF expansion as produced by the

`rcsfgenerate`, the radial wave functions as produced by

`rmcdhf`, and the expansion coefficients of the CSFs as obtained by the

`rmcdhf`or

`rci`programs are read from file. The

`hfszeeman95`program diagonalizes the interaction matrix to give the energies and expansion coefficients in Equation (109). In external magnetic fields, the latter quantities are dependent on B and, given the

`mithit`tool of

`hfszeeman95`, it is possible to map out the energy structure as a function of B beyond the weak Zeeman– or Paschen–Back limits [149].

`mithit`program. The program reads matrix elements produced by

`hfszeeman95`, and then, it constructs and diagonalizes the interaction matrix for a specified B value to determine the expansion coefficients in (109). The needed transition matrix elements are computed from the output of the

`rtransition_phase`, which is a modification of the

`rtransition`program, that gives also phase information. In Section §6.9 of the manual, there is an example of how to use the above programs to compute hyperfine and magnetically induced transitions $2s2p{\phantom{\rule{3.33333pt}{0ex}}}^{3}{P}_{0}^{o}-2{s}^{2}{\phantom{\rule{3.33333pt}{0ex}}}^{1}{S}_{0}$ in Ni XXV.

## 4. Selection of CSFs

#### 4.1. Electron Correlation

#### 4.2. Z-Dependent Perturbation Theory

#### 4.3. Classification of Correlation Effects

- 1.
- Those that differ from $\left|\right\{nl\}\gamma LS\rangle $ by one principal quantum number, but retain the same spin and orbital angular coupling. These configuration states are part of radial correlation.
- 2.
- Those that differ by one principal quantum number and also differ in their coupling. Often the only change is the coupling of the spins, in which case the configuration states are part of spin-polarization.
- 3.
- Those that differ in the angular momentum of exactly one electron and are accompanied by a change in orbital angular coupling of the configuration state and possibly also the spin coupling. These represent orbital polarization.

- 1.
- If $ab$ are orbitals for outer electrons, the replacement represents outer or valence correlation.
- 2.
- If a is a core orbital but b is an outer orbital, the effect represents the polarization of the core and is referred to as core–valence correlation.
- 3.
- If both orbitals are from the core, the replacement represents core–core correlation.

#### 4.4. The Active Set Approach

#### 4.5. CSF Expansions for Energy Differences

`rwfnplot`program is used for plotting, and examples are given in §7.3 of the manual.

#### 4.6. The Active Set Approach as Implemented in GRASP

`rcsfgenerate`program. The user is required to specify the configurations (non-relativistic notation) in the MR along with information whether the orbitals in the configurations are inactive (

`i`), i.e., no substitutions are allowed, active (

`*`), i.e., unrestricted substitutions are allowed, or have a minimal occupation (

`m`). To generate a valence expansion for the ground state in Mg, the specification would be

`1s(2,i)2s(2,i)2p(6,i)3s(2,*)`

`1s(2,i)2s(2,i)2p(6,5)3s(2,*)`

`1s(2,i)2s(2,1)2p(6,i)3s(2,*)`

`1s(2,i)2s(2,i)2p(6,5)3p(2,*)`

`1s(2,i)2s(2,1)2p(6,i)3p(2,*)`

`1s(2,i)2s(2,i)2p(6,5)3d(2,*)`

`1s(2,i)2s(2,1)2p(6,i)3d(2,*)`

`rcsfgenerate`program is given in Sections 4.3 and 4.4, as well as in Sections 5.1–5.10 of the manual.

`rcsfgenerate`program gives a number of configurations as determined by the substitution rules, the active set of orbitals and the specified J values. Given these configurations, a set of CSFs follows by applying the angular couplings described in Section 2.4. Not all of these CSFs interact with the CSFs formed by the configurations in the MR. To include only CSFs that interact, one has to run the

`rcsfinteract`program. It should be remembered that the Z-dependent perturbation theory underlying the active set approach is applicable mainly for ionized systems. For neutral and near neutral systems, it may be necessary to include all generated CSFs and not only the ones interacting with the CSFs formed by the configurations in the MR; see [152] for a discussion. Examples of how to use the

`rcsfinteract`program are given in the manual §5.5.

## 5. Examples of Applications

`rci`program. For each considered case, the details of the computational strategies can be found in the original publications.

#### 5.1. Determination of the Nuclear Quadrupole Moment $Q{(}^{67}$Zn)

#### 5.2. Determination of Changes in Nuclear Radii

#### 5.3. Spectroscopic Data for Astrophysics—Al-like Ions

#### 5.4. Impact of External Magnetic Fields on Hyperfine Spectra: The 4205 Å Line in Eu II

`hfszeeman95`program [67] allows the computation of complete hyperfine-Zeeman interaction matrices in a representation of the unperturbed ASFs, which were pre-calculated with grasp. From these interaction matrices, new perturbed eigenstates, labeled by ${M}_{J}$, F, or ${M}_{F}$ depending on the specified magnetic field and nuclear parameters (see Equation (109)) can be obtained through standard perturbation theory or from smaller configuration–interaction calculations, which in turns allows for determining radiative properties between these new eigenstates [148]. The perturbed eigenstates and associated radiative properties are conveniently computed with the

`mithit`tool associated with the

`hfszeeman95`program.

`hfszeeman95`calculation takes all off-diagonal effects into account, most notably, in this case, the mixing between different hyperfine states due to the external magnetic field. It is striking how complex a single fine-structure transition may become if these effects are taken into consideration.

## 6. Summary and Conclusions

- (i)
- (ii)
- Designing efficient CSF generators that drastically reduce the computational load of MCDHF and RCI calculations [152];
- (iii)
- (iv)
- (v)
- Lifting current restrictions on maximum occupation numbers of two for orbitals $j=9/2$ in the CSF list generation to fully exploit the available spin-angular library;
- (vi)
- Searching for original methods that open promising perspectives for performing rigorous QED calculations within the grasp framework [180];
- (vii)
- (viii)

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Hartree, D.R. Wave Mechanics of an Atom with a Non-Coulomb Central Field: Part I. Theory and Methods. Proc. Camb. Philos. Soc.
**1928**, 24, 89. [Google Scholar] [CrossRef] - Hartree, D.R. Wave Mechanics of an Atom with a Non-Coulomb Central Field: Part II. Some Results and Discussions. Proc. Camb. Philos. Soc.
**1928**, 24, 111. [Google Scholar] [CrossRef] - Hartree, D.R. Wave Mechanics of an Atom with a Non-Coulomb Central Field:Part III. Term Values and Series in Optical Spectra. Proc. Camb. Philos. Soc.
**1928**, 24, 426. [Google Scholar] - Hartree, D.R. Wave Mechanics of an Atom with a Non-Coulomb Central Field: Part IV. Further Results relating to Terms of the Optical Spectrum. Proc. Camb. Philos. Soc.
**1929**, 25, 310. [Google Scholar] [CrossRef] - Slater, J.C. Note on Hartree’s method. Proc. Camb. Philos. Soc.
**1930**, 35, 210. [Google Scholar] [CrossRef] - Fock, V.A. Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Z. Phys.
**1930**, 35, 210. [Google Scholar] - Hartree, D.R.; Hartree, W. Self-consistent field, with exchange, for beryllium. Proc. R. Soc. A
**1935**, 150, 9. [Google Scholar] - Dirac, P.A.M. The Quantum Theory of the Electron. Proc. R. Soc. A
**1928**, 117, 610. [Google Scholar] - Swirles, B. Relatvistic self-consistent fields. Proc. R. Soc. A
**1935**, 152, 625. [Google Scholar] - Löwdin, P.O. Quantum Theory of Many-Particle Systems. III. Extension of the HF Scheme to Include Degenerate Systems and Correlation Effects. Phys. Rev.
**1955**, 97, 1509. [Google Scholar] [CrossRef] - Bacher, R.F. The Interaction of Configurations: sd-p
^{2}. Phys. Rev.**1933**, 43, 264. [Google Scholar] [CrossRef] [Green Version] - Ufford, C.W. Configuration Interaction in Complex Spectra. Phys. Rev.
**1933**, 44, 732. [Google Scholar] [CrossRef] - Slater, J.C. The Theory of Complex Spectra. Phys. Rev.
**1929**, 34, 1293. [Google Scholar] [CrossRef] - Condon, E.U. The Theory of Complex Spectra. Phys. Rev.
**1930**, 36, 1121. [Google Scholar] [CrossRef] - Hartree, D.R.; Hartree, W.; Swirles, B. Self-consistent field, including exchange and superposition of configurations, with some results for oxygen. Philos. Trans. R. Soc. Lond. Ser. Math. Phys. Sci.
**1939**, 238, 229. [Google Scholar] - Machine Solves Mathematical Problems—A Wonderful Meccano Mechanism. Meccano Mag.
**1934**, XIX, 442. - Hartree, D.R. The Mechanical Integration of Differential Equations. Math. Gaz.
**1938**, 22, 342. [Google Scholar] [CrossRef] - Hartree, D.R. The calculation of atomic structures. Rep. Prog. Phys.
**1947**, 11, 113. [Google Scholar] [CrossRef] - Hartree, D.R. The Calculation of Atomic Structures; John Wiley and Sons: New York, NY, USA, 1957. [Google Scholar]
- Shavitt, I. The history and evolution of configuration interaction. Mol. Phys.
**1998**, 94, 3. [Google Scholar] [CrossRef] - Mayers, D.F. Relativistic Self-Consistent Field Calculations for Mercury. Philos. Trans. R. Soc. Lond. Ser. Math. Phys. Sci.
**1957**, 241, 93. [Google Scholar] - Froese, C. The self-consistent field with exchange for some 10 and 12 electron systems. Math. Proc. Camb. Philos. Soc.
**1957**, 53, 206. [Google Scholar] [CrossRef] - Froese Fischer, C. Douglas Rayner Hartree: His Life in Science and Computing; World Scientific Publishing Co Pte Ltd.: Singapore, 2003. [Google Scholar]
- Froese, C. Numerical solution of the Hartree-Fock equations. Can. J. Phys.
**1963**, 41, 1895. [Google Scholar] [CrossRef] - Froese Fischer, C. Self-consistent-field (SCF) and multiconfiguration (MC) Hartree-Fock (HF) methods in atomic calculations: Numerical integration approaches. Comp. Phys. Rep.
**1986**, 3, 274. [Google Scholar] [CrossRef] - Froese Fischer, C. The Hartree-Fock Method for Atoms. A Numerical Approach; John Wiley and Sons: New York, NY, USA, 1977. [Google Scholar]
- Froese Fischer, C. A general multi-configuration Hartree-Fock program. Comp. Phys. Commun.
**1978**, 14, 145. [Google Scholar] [CrossRef] - Froese Fischer, C.; Tachiev, G.; Gaigalas, G.; Godefroid, M. An MCHF atomic-structure package for large-scale calculations. Comput. Phys. Commun.
**2007**, 176, 559. [Google Scholar] [CrossRef] [Green Version] - Froese Fischer, C.; Brage, T.; Jönsson, P. Computational Atomic Structure—An MCHF Approach; Institute of Physics Publishing (IoP): Bristol, UK, 1997. [Google Scholar]
- Froese Fischer, C.; Godefroid, M.; Brage, T.; Jönsson, P.; Gaigalas, G. Advanced multiconfiguration methods for complex atoms: I. Energies and wave functions. J. Phys. B At. Mol. Opt. Phys.
**2016**, 49, 182004. [Google Scholar] [CrossRef] [Green Version] - Grant, I.P. Relativistic self-consistent fields. Proc. R. Soc. Lond. A
**1961**, 262, 555. [Google Scholar] [CrossRef] - Grant, I.P. Relativistic self-consistent fields. Proc. Phys. Soc.
**1965**, 86, 523. [Google Scholar] [CrossRef] - Grant, I.P. Relativistic calculation of atomic structures. Adv. Phys.
**1970**, 19, 747. [Google Scholar] [CrossRef] - Grant, I.P.; CECAM Workshop organized by Carl Moser, Paris, France. Private communication, September–December 1970.
- Desclaux, J.-P.; Mayers, D.F.; O’Brien, F. Relativistic atomic wave functions. J. Phys. B At. Mol. Opt. Phys.
**1971**, 4, 631. [Google Scholar] [CrossRef] - Desclaux, J.P. A multiconfiguration relativistic Dirac-Fock program. Comput. Phys. Commun.
**1975**, 9, 31. [Google Scholar] [CrossRef] - Grant, I.P.; McKenzie, B.J.; Norrington, P.H.; Mayers, D.F.; Pyper, N.C. An atomic multiconfigurational Dirac-Fock package. Comput. Phys. Commun.
**1980**, 21, 207. [Google Scholar] [CrossRef] - Grant, I.P. A general program to calculate angular momentum coefficients in relativistic atomic structure. Comput. Phys. Commun.
**1973**, 5, 263. [Google Scholar] [CrossRef] - Grant, I.P. A program to calculate angular momentum coefficients in relativistic atomic structure - revised version. Comput. Phys. Commun.
**1976**, 11, 397. [Google Scholar] [CrossRef] - Dyall, K.G.; Grant, I.P.; Johnson, T.; Parpia, F.A.; Plummer, E.P. GRASP: A general-purpose relativistic atomic structure program. Comput. Phys. Commun.
**1989**, 55, 425. [Google Scholar] [CrossRef] - Parpia, F.A.; Froese Fischer, C.; Grant, I.P. GRASP92: A package for large-scale relativistic atomic structure calculations. Comput. Phys. Commun.
**1996**, 94, 249. [Google Scholar] [CrossRef] - Jönsson, P.; Gaigalas, G.; Bieroń, J.; Froese Fischer, C.; Grant, I.P. New version: Grasp2K relativistic atomic structure package. Comput. Phys. Commun.
**2013**, 184, 2197. [Google Scholar] [CrossRef] [Green Version] - Froese Fischer, C.; Gaigalas, G.; Jönsson, P.; Bieroń, J. GRASP2018—A Fortran 95 version of the General Relativistic Atomic Structure Package. Comput. Phys. Commun.
**2019**, 237, 184. [Google Scholar] [CrossRef] - Computational Atomic Structure Group (CompAS). Available online: https://compas.github.io/ (accessed on 30 October 2022).
- Grumer, J.; Zhao, R.; Brage, T.; Li, W.; Huldt, S.; Hutton, R.; Zou, Y. Coronal lines and the importance of deep-core-valence correlation in Ag-like ions. Phys. Rev. A
**2014**, 89, 062511. [Google Scholar] [CrossRef] [Green Version] - Jönsson, P.; Gaigalas, G.; Rynkun, P.; Radžiūtė, L.; Ekman, J.; Gustafsson, S.; Hartman, H.; Wang, K.; Godefroid, M.; Froese Fischer, C.; et al. Multiconfiguration Dirac-Hartree-Fock Calculations with Spectroscopic Accuracy: Applications to Astrophysics. Atoms
**2017**, 5, 16. [Google Scholar] [CrossRef] - Lu, Q.; He, J.; Tian, H.; Li, M.; Yang, Y.; Yao, K.; Chen, C.; Xiao, J.; Li, J.G.; Tu, B.; et al. Observation of indirect ionization of W
^{7+}in an electron-beam ion-trap plasma. Phys. Rev. A**2019**, 99, 042510. [Google Scholar] [CrossRef] [Green Version] - Lu, Q.; Yan, C.L.; Meng, J.; Xu, G.Q.; Yang, Y.; Chen, C.Y.; Xiao, J.; Li, J.G.; Wang, J.G.; Zou, Y. Visible spectra of W
^{8+}in an electron-beam ion trap. Phys. Rev. A**2021**, 103, 022808. [Google Scholar] [CrossRef] - Zhang, X.H.; Del Zanna, G.; Wang, K.; Rynkun, P.; Jönsson, P.; Godefroid, M.; Gaigalas, G.; Radžiūtė, L.; Ma, L.H.; Si, R.; et al. Benchmarking Multiconfiguration Dirac–Hartree–Fock Calculations for Astrophysics: Si-like Ions from Cr XI to Zn XII. Astrophys. J. Suppl. Ser.
**2021**, 257, 56. [Google Scholar] [CrossRef] - Tanaka, M.; Kato, D.; Gaigalas, G.; Rynkun, P.; Radžiūtė, L.; Wanajo, S.; Sekiguchi, Y.; Nakamura, N.; Tanuma, H.; Murakami, I.; et al. Properties of Kilonovae from Dynamical and Post-merger Ejecta of Neutron Star Mergers. Astrophys. J.
**2018**, 852, 109. [Google Scholar] [CrossRef] - Tanaka, M.; Kato, D.; Gaigalas, G.; Kawaguchi, K. Systematic opacity calculations for kilonovae. Mon. Not. R. Astron. Soc.
**2020**, 496, 1369. [Google Scholar] [CrossRef] - Radžiūtė, L.; Gaigalas, G.; Kato, D.; Rynkun, P.; Tanaka, M. Extended Calculations of Energy Levels and Transition Rates for Singly Ionized Lanthanide Elements. II. Tb-Yb. Astrophys. J. Suppl. Ser.
**2021**, 257, 29. [Google Scholar] [CrossRef] - Brage, T.; Judge, P.G.; Proffitt, C. Determination of hyperfine-induced transition rates from observations of a planetary nebula. Phys. Rev. Lett.
**2002**, 77, 281101. [Google Scholar] [CrossRef] - Si, R.; Brage, T.; Li, W.; Grumer, J.; Li, M.; Hutton, R. A first spectroscopic measurement of the magnetic-field strength for an active region of the solar corona. Astrophys. J. Lett.
**2020**, 898, L34. [Google Scholar] [CrossRef] - Li, W.; Grumer, J.; Yang, Y.; Brage, T.; Yao, K.; Chen, C.; Watanabe, T.; Jönsson, P.; Lundstedtr, H.; Hutton, R.; et al. A Novel Method to Determine Magnetic Fields in Low-density Plasma Facilitated through Accidental Degeneracy of Quantum States in Fe
^{9+}. Astrophys. J.**2015**, 807, 69. [Google Scholar] [CrossRef] [Green Version] - Li, W.; Yang, Y.; Tu, B.; Xiao, J.; Grumer, J.; Brage, T.; Watanabe, T.; Hutton, R.; Zou, Z. Atomic-level Pseudo-degeneracy of Atomic Levels Giving Transitions Induced by Magnetic Fields, of Importance for Determining the Field Strengths in the Solar Corona. Astrophys. J.
**2016**, 826, 219. [Google Scholar] [CrossRef] - Tang, R.; Si, R.; Fei, Z.; Fu, X.; Lu, Y.; Brage, T.; Liu, H.; Chen, C.; Ning, C. Candidate for Laser Cooling of a Negative Ion: High-Resolution Photoelectron Imaging of Th
^{-}. Phys. Rev. Lett.**2019**, 123, 203002. [Google Scholar] [CrossRef] [PubMed] - Si, R.; Guo, X.L.; Brage, T.; Chen, C.Y.; Hutton, R.; Froese Fischer, C. Breit and QED effects on the 3d
^{92}D_{3/2}→^{2}D_{5/2}in Co-like ions. Phys. Rev. A**2018**, 98, 012504. [Google Scholar] [CrossRef] [Green Version] - Li, M.C.; Si, R.; Brage, T.; Hutton, R.; Zou, Y. Proposal of highly accurate tests of Breit and QED effects in the ground state 2p
^{5}of the F-like isoelectronic sequence. Phys. Rev. A**2018**, 98, 020502. [Google Scholar] [CrossRef] - Zhang, C.Y.; Li, J.Q.; Wang, K.; Si, R.; Godefroid, M.; Jönsson, P.; Xiao, J.; Gu, M.F.; Chen, C.Y. Benchmarking calculations of wavelengths and transition rates with spectroscopic accuracy for W XLVVIII through W LVI tungsten ions. Phys. Rev. A
**2022**, 105, 022817. [Google Scholar] - Filippin, L.; Bieroń, J.; Gaigalas, G.; Godefroid, M.; Jönsson, P. Multiconfiguration calculations of electronic isotope-shift factors in Zn I. Phys. Rev. A
**2017**, 96, 042502. [Google Scholar] [CrossRef] [Green Version] - Papoulia, A.; Schiffmann, S.; Bieroń, J.; Gaigalas, G.; Godefroid, M.; Harman, Z.; Jönsson, P.; Oreshkina, N.S.; Pyykkö, P.; Tupitsyn, I.I. Ab initio electronic factors of the A and B hyperfine structure constants for the 5s
^{2}5p6s^{1,3}P_{1}^{o}states in Sn I. Phys. Rev. A**2021**, 103, 022815. [Google Scholar] [CrossRef] - Barzakh, A.; Andreyev, A.N.; Raison, C.; Cubiss, J.G.; Van Duppen, P.; Péru, S.; Hilaire, S.; Goriely, S.; Andel, B.; Antalic, S.; et al. Large Shape Staggering in Neutron-Deficient Bi Isotopes. Phys. Rev. Lett.
**2021**, 127, 192501. [Google Scholar] [CrossRef] - Brage, T.; Grumer, J. Resolving a discrepancy between experimental and theoretical lifetimes in atomic negative ions. J. Phys. B At. Mol. Opt. Phys.
**2016**, 50, 025001. [Google Scholar] [CrossRef] [Green Version] - Si, R.; Schiffmann, S.; Wang, K.; Chen, C.Y.; Godefroid, M. Ab initio multiconfiguration Dirac-Hartree-Fock calculations of the In and Tl electron affinities and their isotope shifts. Phys. Rev. A
**2021**, 104, 012802. [Google Scholar] [CrossRef] - Jönsson, P.; Parpia, F.A.; Froese Fischer, C. hfs92: A program for relativistic atomic hyperfine structure calculations. Comput. Phys. Commun.
**1996**, 96, 301. [Google Scholar] [CrossRef] - Li, W.; Grumer, J.; Brage, T.; Jönsson, P. Hfszeeman95: A program for computing weak and intermediate magnetic-field- and hyperfine-induced transition rates. Comput. Phys. Commun.
**2020**, 253, 107211. [Google Scholar] [CrossRef] - Ekman, J.; Jönsson, P.; Godefroid, M.; Nazé, C.; Gaigalas, G.; Bieroń, J. ris4: A program for relativistic isotope shift calculations. Comput. Phys. Commun.
**2019**, 235, 433. [Google Scholar] [CrossRef] - Schiffmann, S.; Li, J.G.; Ekman, J.; Gaigalas, G.; Godefroid, M.; Jönsson, P.; Bieroń, J. Relativistic radial electron density functions and natural orbitals from GRASP2018. Comput. Phys. Commun.
**2022**, 278, 108403. [Google Scholar] [CrossRef] - Gaigalas, G.; Froese Fischer, C.; Rynkun, P.; Jönsson, P. JJ2LSJ Transformation and Unique Labeling for Energy Levels. Atoms
**2017**, 5, 6. [Google Scholar] [CrossRef] - Gaigalas, G. Coupling: The program for searching optimal coupling scheme in atomic theory. Comput. Phys. Commun.
**2020**, 247, 106960. [Google Scholar] [CrossRef] - Jönsson, P.; Gaigalas, G.; Froese Fischer, C.; Bieroń, J.; Grant, I.P.; Brage, T.; Ekman, J.; Godefroid, M.; Grumer, J.; Li, J.; et al. GRASP Manual for Users. Atoms
**2023**. accepted. [Google Scholar] - Grant, I.P. Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation; Springer Science and Business Media, LLC: New York, NY, USA, 2007. [Google Scholar]
- Tiesinga, E.; Mohr, P.J.; Newell, D.B.; Taylor, B.N. CODATA recommended values of the fundamental physical constants: 2018. Rev. Mod. Phys.
**2021**, 93, 025010. [Google Scholar] [CrossRef] - Furry, W.H. On Bound States and Scattering in Positron Theory. Phys. Rev.
**1981**, 81, 115. [Google Scholar] [CrossRef] - Parpia, F.A.; Mohanty, A.K. Relativistic basis-set calculations for atoms with Fermi nuclei. Phys. Rev. A
**1992**, 46, 3735. [Google Scholar] [CrossRef] - Bethe, H.; Salpeter, E. Quantum Mechanics of One- and Two-Electron Atoms; Springer: Berlin, Germany; New York, NY, USA, 1957. [Google Scholar]
- McKenzie, B.; Grant, I.P.; Norrington, P. A program to calculate transverse Breit and QED corrections to energy levels in a multiconfiguration Dirac-Fock environment. Comput. Phys. Commun.
**1980**, 21, 233. [Google Scholar] [CrossRef] - Johnson, W. Atomic Structure Theory: Lectures on Atomic Physics; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Uehling, E.A. Polarization Effects in the Positron Theory. Phys. Rev.
**1935**, 48, 55. [Google Scholar] [CrossRef] - Fullerton, L.W.; Rinker Jr, G.A. Accurate and efficient methods for the evaluation of vacuum-polarization potentials of order Zα and Zα
^{2}. Phys. Rev. A**1976**, 13, 1283. [Google Scholar] [CrossRef] - Mohr, P.J. Energy levels of hydrogen-like atoms predicted by quantum electrodynamics, 10≤Z≤40. At. Data Nucl. Data Tables
**1983**, 29, 453. [Google Scholar] [CrossRef] - Mohr, P.J.; Kim, Y.K. Self-Energy of Excited States in a Strong Coulomb Field. Phys. Rev. A
**1992**, 45, 2727. [Google Scholar] [CrossRef] [PubMed] - Andersson, M.; Jönsson, P. HFSZEEMAN. A program for computing weak and intermediate field fine and hyperfine structure Zeeman splittings from MCDHF wave functions. Comput. Phys. Commun.
**2008**, 178, 156. [Google Scholar] [CrossRef] - Edmonds, A.R. Angular Momentum in Quantum Mechanics; Princeton University Press: Hoboken, NJ, USA, 1957. [Google Scholar]
- Fano, U. Interaction between configurations with several open shells. Phys. Rev. A
**1965**, 67, 140. [Google Scholar] [CrossRef] - Racah, G. Theory of Complex Spectra. III. Phys. Rev.
**1943**, 63, 367. [Google Scholar] [CrossRef] - Flowers, B.H. Studies in jj-Coupling. I. Classification of Nuclear and Atomic States. Proc. R. Soc. Lond. Math. Phys. Eng. Sci.
**1952**, 212, 248. [Google Scholar] - Gaigalas, G.; Fritzsche, S.; Rudzikas, Z. Reduced Coefficients of Fractional Parentage and Matrix Elements of the Tensor W
^{(kqkj)}in jj-Coupling. At. Data Nucl. Data Tables**2000**, 76, 235. [Google Scholar] [CrossRef] - Racah, G. Theory of Complex Spectra. IV. Phys. Rev.
**1949**, 76, 1352. [Google Scholar] [CrossRef] - Rudzikas, Z.B. Theoretical Atomic Spectroscopy; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Gaigalas, G.; Froese Fischer, C. Extension of the HF program to partially filled f-subshells. Comput. Phys. Commun.
**1996**, 98, 255. [Google Scholar] [CrossRef] - Gaigalas, G. A Program Library for Computing Pure Spin-Angular Coefficients for One- and Two-Particle Operators in Relativistic Atomic Theory. Atoms
**2022**, 10, 129. [Google Scholar] [CrossRef] - Judd, B.R. Second Quantization and Atomic Spectroscopy; The Johns Hopkins Press: Baltimore, MD, USA, 1967. [Google Scholar]
- Flowers, B.H.; Szpikowski, S. A generalized quasi-spin formalism. Proc. Phys. Soc.
**1964**, 84, 193. [Google Scholar] [CrossRef] - Gaigalas, G.; Fritzsche, S.; Grant, I.P. Program to calculate pure angular momentum coefficients in jj-coupling. Comput. Phys. Commun.
**2001**, 139, 263. [Google Scholar] [CrossRef] [Green Version] - Brink, D.M.; Satchler, G.R. Angular Momentum; Clarendon Press: Oxford, UK, 1968. [Google Scholar]
- Fano, U.; Racah, G. Irreducible Tensorial Sets; Academic Press: New York, NY, USA, 1959. [Google Scholar]
- Racah, G. Theory of Complex Spectra. II. Phys. Rev.
**1942**, 62, 438. [Google Scholar] [CrossRef] - Judd, B.R. Operator Techniques in Atomic Spectroscopy; McGraw-Hill Book Company, Inc.: New York, NY, USA, 1963. [Google Scholar]
- Cowan, R.D. The Theory of Atomic Structure and Spectra; University of California Press: Berkeley, CA, USA, 1981. [Google Scholar]
- Rose, M.E. Elementary Theory of Angular Momentum; John Wiley and Sons: New York, NY, USA, 1957. [Google Scholar]
- Judd, B.R. Lie groups for atomic shells. Phys. Rep.
**1997**, 285, 1. [Google Scholar] [CrossRef] - Robb, W.D. A Program to Evaluate Reduced Matrix-Elements of Summations of One-Particle Tensor Operators. Comput. Phys. Commun.
**1973**, 6, 132. [Google Scholar] [CrossRef] - Gaigalas, G.; Fritzsche, S. Pure spin-angular momentum coefficients for non-scalar one-particle operators in jj-coupling. Comput. Phys. Commun.
**2002**, 148, 349. [Google Scholar] [CrossRef] [Green Version] - Gaigalas, G.; Rudzikas, Z.B.; Froese Fischer, C. An efficient approach for spin-angular integrations in atomic structure calculations. J. Phys. B At. Mol. Phys.
**1997**, 30, 3747. [Google Scholar] [CrossRef] - Gaigalas, G. Integration over spin-angular variables in atomic physics. Lith. J. Phys.
**1999**, 39, 80. [Google Scholar] - Kaniauskas, J.M.; Rudzikas, Z.B. Quasi-spin method for jj coupling in the theory of many-electron atoms. J. Phys. B At. Mol. Phys.
**1980**, 13, 3521. [Google Scholar] [CrossRef] - Gaigalas, G.; Rudzikas, Z.B. On the secondly quantized theory of the many-electron atom. J. Phys. B: At. Mol. Phys.
**1996**, 29, 3303. [Google Scholar] [CrossRef] - Froese Fischer, C.; Senchuk, A. Numerical Procedures for Relativistic Atomic Structure Calculations. Atoms
**2020**, 8, 85. [Google Scholar] [CrossRef] - Froese Fischer, C. A B-spline Hartree-Fock program. Comput. Phys. Commun.
**2011**, 182, 1315. [Google Scholar] [CrossRef] - Grant, I.P. Relativistic Atomic Structure Calculations. In Methods in Computational Chemistry; Wilson, S., Ed.; Plenum Press: New York, NY, USA, 1988; Volume 2, Chapter 2; pp. 1–71. [Google Scholar]
- Froese Fischer, C.; Godefroid, M. Electron correlation in the lanthanides: 4f
^{2}spectrum of Ce^{2+}. Phys. Rev. A**2019**, 99, 032511. [Google Scholar] [CrossRef] [Green Version] - Papoulia, A.; Ekman, J.; Jönsson, P. Extended transition rates and lifetimes in Al I and Al II from systematic multiconfiguration calculations. Astron. Astrophys.
**2019**, 621, A16. [Google Scholar] [CrossRef] [Green Version] - Burke, P.G. R-Matrix Theory of Atomic Collisions: Application to Atomic, Molecular and Optical Processes; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Kato, D.; Tong, X.-M.; Watanabe, H.; Fukami, T.; Kinugawa, T.; Yamada, C.; Ohtani, S.; Watanabe, T. Fine-structure in 3d
^{4}States of Highly Charged Ti-like Ions. J. Chin. Chem. Soc.**2001**, 48, 525. [Google Scholar] [CrossRef] - Kotochigova, S.; Kirby, K.P.; Tupitsyn, I. Ab initio fully relativistic calculations of x-ray spectra of highly charged ions. Phys. Rev. A
**2007**, 76, 052513. [Google Scholar] [CrossRef] - Gustafsson, S.; Jönsson, P.; Froese Fischer, C.; Grant, I.P. Combining Multiconfiguration and Perturbation Methods: Perturbative Estimates of Core-Core Electron Correlation Contributions to Excitation Energies in Mg-Like Iron. Atoms
**2017**, 5, 3. [Google Scholar] [CrossRef] [Green Version] - Froese Fischer, C. The MCHF atomic-structure package. Comput. Phys. Commun.
**1991**, 64, 369. [Google Scholar] [CrossRef] - Gaigalas, G.; Rynkun, P.; Radžiūtė, L.; Kato, D.; Tanaka, M.; Jönsson, P. Energy Level Structure and Transition Data of Er
^{2+}. Astrophys. J. Suppl. Ser.**2020**, 248, 13. [Google Scholar] [CrossRef] - Stathopoulos, A.; Froese Fischer, C. A Davidson program for finding a few selected extreme eigenpairs of a large, sparse, real, symmetric matrix. Comput. Phys. Commun.
**1994**, 79, 268. [Google Scholar] [CrossRef] - Gaigalas, G.; Zalandauskas, T.; Rudzikas, Z. LS-jj transformation matrices for a shell of equivalent electrons. At. Data Nucl. Data Tables
**2003**, 84, 99. [Google Scholar] [CrossRef] - Froese Fischer, C.; Gaigalas, G. Multiconfiguration Dirac-Hartree-Fock energy levels and transition probabilities for W XXXVIII. Phys. Rev. A
**2012**, 85, 042501. [Google Scholar] [CrossRef] - Pyykkö, P. Spectroscopic nuclear quadrupole moments. Mol. Phys.
**2001**, 99, 1617. [Google Scholar] [CrossRef] - Lindgren, I.; Rosén, A. Case Stud. At. Phys.
**1974**, 3, 197. - Stone, N.J. Table of Nuclear Magnetic Dipole and Electric Quadrupole Moments; Report INDC(NDS)–0658; International Atomic Energy Agency (IAEA): Vienna, Austria, 2014. [Google Scholar]
- Stone, N.J. Table of nuclear electric quadrupole moments. At. Data Nucl. Data Tables
**2016**, 111, 1. [Google Scholar] [CrossRef] - Yan Ting, L.; Jönsson, P.; Godefroid, M.; Gaigalas, G.; Bieroń, J.; Marques, J.P.; Indelicato, P.; Chen, C. Independently Optimized Orbital Sets in GRASP—The Case of Hyperfine Structure in Li I. Atoms
**2023**, 111, 4. [Google Scholar] [CrossRef] - Cheng, K.T.; Childs, W.J. Ab initio calculation of 4f
^{N}6s^{2}hyperfine structure in neutral rare-earth atoms. Phys. Rev. A**1985**, 31, 2775. [Google Scholar] [CrossRef] - Palmer, C.W.P. Reformulation of the theory of the mass shift. J. Phys. B At. Mol. Opt. Phys.
**1987**, 20, 5987. [Google Scholar] [CrossRef] - Shabaev, V.M. Nuclear recoil effect in the relativistic theory of multiply charged ions. Sov. J. Nucl. Phys.
**1988**, 47, 69. [Google Scholar] - Gaidamauskas, E.; Rynkun, P.; Nazé, C.; Gaigalas, G.; Jönsson, P.; Godefroid, M. Tensorial form and matrix elements of the relativistic nuclear recoil operator. J. Phys. B At. Mol. Opt.
**2011**, 44, 175003. [Google Scholar] [CrossRef] [Green Version] - Reinhard, P.-F.; Nazarewicz, W. Nuclear charge densities in spherical and deformed nuclei: Toward precise calculations of charge radii. Phys. Rev. C
**2021**, 103, 054310. [Google Scholar] [CrossRef] - Godefroid, M.; Ekman, J.; Jönsson, P. Signs in isotope shifts: A perennial headache. arXiv
**2022**, arXiv:2211.00798. [Google Scholar] - Borgoo, A.; Scharf, O.; Gaigalas, G.; Godefroid, M. Multiconfiguration electron density function for the ATSP2K-package. Comput. Phys. Commun.
**2010**, 181, 426. [Google Scholar] [CrossRef] - Carette, T.; Godefroid, M. Isotope shift on the chlorine electron affinity revisited by an MCHF/CI approach. J. Phys. B At. Mol. Opt. Phys.
**2013**, 46, 095003. [Google Scholar] [CrossRef] [Green Version] - Layzer, D. On a Screening Theory of Atomic Spectra. Ann. Phys.
**1959**, 8, 271. [Google Scholar] [CrossRef] - Schiffmann, S.; Godefroid, M.; Ekman, J.; Jönsson, P.; Froese Fischer, C. Natural orbitals in multiconfiguration calculations of hyperfine-structure parameters. Phys. Rev. A
**2020**, 101, 062510. [Google Scholar] [CrossRef] - Froese Fischer, C. Towards B-Spline Atomic Structure Calculations. Atoms
**2021**, 9, 50. [Google Scholar] [CrossRef] - Grant, I.P. Gauge invariance and relativistic radiative transitions. J. Phys. B: At. Mol. Opt.
**1974**, 7, 1458. [Google Scholar] [CrossRef] - Kaniauskas, J.; Kičkin, I.; Rudzikas, Z. J. Lit. Fiz. Sb.
**1974**, 14, 463. - Olsen, J.; Godefroid, M.; Jönsson, P.; Malmqvist, P.-Å.; Froese Fischer, C. Transition probability calculations for atoms using non-orthogonal orbitals. Phys. Rev. E
**1995**, 52, 4499. [Google Scholar] [CrossRef] - Papoulia, A.; Ekman, J.; Gaigalas, G.; Godefroid, M.; Gustafsson, S.; Hartman, H.; Li, W.; Radžiūtė, L.; Rynkun, P.; Schiffmann, S.; et al. Coulomb (Velocity) Gauge Recommended in Multiconfiguration Calculations of Transition Data Involving Rydberg Series. Atoms
**2019**, 7, 106. [Google Scholar] [CrossRef] [Green Version] - Ekman, J.; Godefroid, M.; Hartman, H. Validation and Implementation of Uncertainty Estimates of Calculated Transition Rates. Atoms
**2014**, 2, 215. [Google Scholar] [CrossRef] - Togawa, M.; Kühn, S.; Shah, C.; Amaro, P.; Steinbrügge, R.; Stierhof, J.; Hell, N.; Rosner, M.; Fujii, K.; Bissinger, M.; et al. Observation of strong two-electron–one-photon transitions in few-electron ions. Phys. Rev. A
**2022**, 102, 052831. [Google Scholar] [CrossRef] - Indelicato, P. Radiative de-excitation of the 1s
^{2}2s3p^{3}P_{0}level in beryllium-like ions: A competition between an E2 and a two-electron one-photon E1 transition. Hyperfine Interact.**1997**, 108, 39. [Google Scholar] [CrossRef] - Li, J.; Jönsson, P.; Dong, C.; Gaigalas, G. Two-electron-one-photon M1 and E2 transitions between the states of the 2p
^{3}and 2s^{2}2p odd configurations for B-like ions with 18 ≤ Z ≤ 92. J. Phys. B At. Mol. Opt. Phys.**2010**, 43, 035005. [Google Scholar] [CrossRef] [Green Version] - Grumer, J.; Brage, T.; Andersson, M.; Li, J.; Jönsson, P.; Li, W.; Yang, Y.; Hutton, R.; Zou, Y. Unexpected transitions induced by spin-dependent, hyperfine and external magnetic-field interactions. Phys. Scr.
**2014**, 89, 114002. [Google Scholar] [CrossRef] [Green Version] - Bransden, B.H.; Joachain, C.J. Physics of Atoms and Molecules; Prentice Hall: Harlow, UK, 2003. [Google Scholar]
- Kato, T. On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math.
**1957**, 10, 151. [Google Scholar] [CrossRef] - Layzer, D.; Bahcall, J. Relativistic Z-Dependent Theory of Many-Electron Atoms. Ann. Phys.
**1962**, 17, 177. [Google Scholar] [CrossRef] - Li, Y.T.; Wang, K.; Si, R.; Godefroid, M.; Gaigalas, G.; Chen, C.Y.; Jönsson, P. Reducing the computational load—Atomic multiconfiguration calculations based on configuration state function generators. Comput. Phys. Commun.
**2023**, 283, 108562. [Google Scholar] [CrossRef] - Laulainen, N.S.; McDermott, M.N. Spin and Nuclear Moments of the Zn
^{63}Ground State. Phys. Rev.**1969**, 177, 1606. [Google Scholar] [CrossRef] - Byron, J.F.W.; McDermott, M.N.; Novick, R.; Perry, B.W.; Saloman, E.B. Spin and Nuclear Moments of 245-Day Zn
^{65}; Redetermination of the hfs of Zn^{67}and τ(^{3}P_{1}) of Zinc. Phys. Rev.**1964**, 134, A47. [Google Scholar] [CrossRef] - Lurio, A. Hyperfine Structure of the
^{3}P States of Zn^{67}and Mg^{25}. Phys. Rev.**1962**, 126, 1768. [Google Scholar] [CrossRef] - Bieroń, J.; Filippin, L.; Gaigalas, G.; Godefroid, M.; Jönsson, P.; Pyykkö, P. Ab initio calculations of the hyperfine structure of zinc and evaluation of the nuclear quadrupole moment Q(
^{67}Zn). Phys. Rev. A**2018**, 97, 062505. [Google Scholar] [CrossRef] [Green Version] - Liu, Y.; Hutton, R.; Zou, Y.; Andersson, M.; Brage, T. MCDF calculations for the lowest excited states in the Zn-like sequence. J. Phys. B At. Mol. Opt.
**2006**, 39, 3147. [Google Scholar] [CrossRef] - Palffy, A. Nuclear effects in atomic transitions. Contemp. Phys.
**2010**, 51, 471. [Google Scholar] [CrossRef] [Green Version] - Angeli, I. A consistent set of nuclear rms charge radii: Properties of the radius surface R(N,Z). At. Data Nucl. Data Tables
**2004**, 87, 185. [Google Scholar] [CrossRef] - Nörtershäuser, W.; Tiedemann, D.; Žáková, M.; Andjelkovic, Z.; Blaum, K.; Bissell, M.L.; Cazan, R.; Drake, G.W.F.; Geppert, C.; Kowalska, M.; et al. Nuclear Charge Radii of
^{7,9,10}Be and the One-Neutron Halo Nucleus^{11}Be. Phys. Rev. Lett.**2009**, 102, 062503. [Google Scholar] [CrossRef] [Green Version] - Kluge, H.-J. Atomic physics techniques for studying nuclear ground state properties, fundamental interactions and symmetries: Status and perspectives. Hyperfine Interact.
**2010**, 196, 295. [Google Scholar] [CrossRef] - Filippin, L.; Godefroid, M.; Ekman, J.; Jönsson, P. Core correlation effects in multiconfiguration calculations of isotope shifts in Mg I. Phys. Rev. A
**2016**, 93, 062512. [Google Scholar] [CrossRef] [Green Version] - Nazé, C.; Verdebout, S.; Rynkun, P.; Gaigalas, G.; Godefroid, M.; Jönsson, P. Isotope shifts in beryllium-, boron-, carbon-, and nitrogen-like ions from relativistic configuration interaction calculations. At. Data Nucl. Data Tables
**2014**, 100, 1197. [Google Scholar] [CrossRef] [Green Version] - Li, J.; Nazé, C.; Godefroid, M.; Fritzsche, S.; Gaigalas, G.; Indelicato, P.; Jönsson, P. Mass- and field-shift isotope parameters for the 2s-2p resonance doublet of lithiumlike ions. Phys. Rev. A
**2012**, 86, 022518. [Google Scholar] [CrossRef] [Green Version] - Kozhedub, Y.S.; Volotka, A.V.; Artemyev, A.N.; Glazov, D.A.; Plunien, G.; Shabaev, V.M.; Tupitsyn, I.I.; Stohlker, T. Relativistic recoil, electron-correlation, and QED effects on the 2p
_{j}-2s transition energies in Li-like ions. Phys. Rev. A**2010**, 81, 042513. [Google Scholar] [CrossRef] [Green Version] - Brandau, C.; Kozhuharov, C.; Harman, Z.; Müller, A.; Schippers, S.; Kozhedub, Y.S.; Bernhardt, D.; Böhm, S.; Jacobi, J.; Schmidt, E.W.; et al. Isotope Shift in the Dielectronic Recombination of Three-Electron
^{A}Nd^{57+}. Phys. Rev. Lett.**2008**, 100, 073201. [Google Scholar] [CrossRef] [Green Version] - Ekman, J.; Jönsson, P.; Radžiūtė, L.; Gaigalas, G.; Del Zanna, G.; Grant, I.P. Large-scale calculations of atomic level and transition properties in the aluminum isoelectronic sequence from Ti X through Kr XXIV, Xe XLII, and W LXII. At. Data Nucl. Data Tables
**2018**, 120, 152. [Google Scholar] [CrossRef] - Kramida, A.; Ralchenko, Y.; Reader, J. NIST ASD Team, NIST Atomic Spectra Database (Ver. 5.4); National Institute of Standards and Technology: Gaithersburg, MD, USA, 2021. Available online: https://physics.nist.gov/asd (accessed on 18 October 2016).
- Del Zanna, G. Benchmarking atomic data for astrophysics: A first look at the soft X-ray lines. Astron. Astrophys.
**2012**, 546, A97. [Google Scholar] [CrossRef] - Froese Fischer, C.; Tachiev, T.; Irimia, A. Relativistic energy levels, lifetimes, and transition probabilities for the sodium-like to argon-like sequences. At. Data Nucl. Data Tables
**2006**, 92, 607. [Google Scholar] [CrossRef] - Santana, J.A.; Ishikawa, Y.; Träbert, E. Multireference Møller-Plesset perturbation theory results on levels and transition rates in Al-like ions of iron group elements. Phys. Scr.
**2009**, 79, 065301. [Google Scholar] [CrossRef] - Degl’Innocenti, E.L.; Landolfi, M. Polarization in Spectral Lines; Kluwer Academic Publishers: New York, NY, USA, 2004. [Google Scholar]
- Verdebout, S.; Rynkun, P.; Jönsson, P.; Gaigalas, G.; Froese Fischer, C.; Godefroid, M. A partitioned correlation function interaction approach for describing electron correlation in atoms. J. Phys. B At. Mol. Opt.
**2013**, 46, 085003. [Google Scholar] [CrossRef] [Green Version] - Froese Fischer, C.; Godefroid, M. Variational Methods for Atoms and the Virial Theorem. Atoms
**2022**, 10, 110. [Google Scholar] [CrossRef] - Froese Fischer, C.; Verdebout, S.; Godefroid, M.; Rynkun, P.; Jönsson, P.; Gaigalas, G. Doublet-quartet energy separation in boron: A partitioned-correlation-function-interaction method. Phys. Rev. A
**2013**, 88, 062506. [Google Scholar] [CrossRef] [Green Version] - Li, J.G.; Gaigalas, G.; Bieroń, J.; Ekman, J.; Jönsson, P.; Godefroid, M.; Froese Fischer, C. Re-evaluation of the nuclear magnetic octupole moment of
^{209}Bi. Atoms**2022**, 10, 132. [Google Scholar] [CrossRef] - Li, J.G.; Ekman, J.; Gaigalas, G.; Bieroń, J.; Jönsson, P.; Godefroid, M.; Froese Fischer, C. New Version of RHFS code. Comput. Phys. Commun.
**2023**. in preparation. [Google Scholar] - Xiao, D.; Li, J.; Campbell, W.C.; Dellaert, T.; McMillin, P.; Ransford, A.; Roman, C.; Derevianko, A. Hyperfine structure of
^{173}Yb^{+}: Toward resolving the^{173}Yb nuclear-octupole-moment puzzle. Phys. Rev. A**2020**, 102, 022810. [Google Scholar] [CrossRef] - De Groote, R.P.; Kujanpää, S.; Koszorús, Á.; Li, J.G.; Moore, I.D. Magnetic octupole moment of
^{173}Yb using collinear laser spectroscopy. Phys. Rev. A**2021**, 103, 032826. [Google Scholar] [CrossRef] - Grant, I.P.; Quiney, H. GRASP: The future? Atoms
**2022**, 10, 108. [Google Scholar] [CrossRef] - Shabaev, V.M.; Tupitsyn, I.I.; Yerokhin, V.A. Model operator approach to the Lamb shift calculations in relativistic many-electron atoms. Phys. Rev. A
**2013**, 88, 012513. [Google Scholar] [CrossRef] [Green Version] - Shabaev, V.M.; Tupitsyn, I.I.; Yerokhin, V.A. QEDMOD: Fortran program for calculating the model Lamb-shift operator. Comput. Phys. Commun.
**2015**, 189, 175. Available online: https://www.sciencedirect.com/science/article/abs/pii/S0010465514004081 (accessed on 31 October 2022). [CrossRef] - Shabaev, V.M.; Tupitsyn, I.I.; Yerokhin, V.A. QEDMOD: Fortran program for calculating the model Lamb-shift operator. Comput. Phys. Commun.
**2018**, 223, 69. Available online: https://www.sciencedirect.com/science/article/abs/pii/S0010465517303478 (accessed on 31 October 2022). [CrossRef] - Malyshev, A.V.; Glazov, D.A.; Shabaev, V.M.; Tupitsyn, I.I.; Yerokhin, V.A.; Zaytsev, V.A. Model-QED operator for superheavy elements. Phys. Rev. A
**2022**, 106, 012806. [Google Scholar] [CrossRef] - Zhang, C.Y.; Wang, K.; Si, R.; Godefroid, M.; Jönsson, P.; Xiao, J.; Gu, M.F.; Chen, C.Y. Benchmarking calculations with spectroscopic accuracy of level energies and wavelengths in W LVII-W LXII tungsten ions. J. Quant. Spectrosc. Rad. Transf.
**2021**, 269, 107650. [Google Scholar] [CrossRef] - Welton, T.A. Some Observable Effects of the Quantum-Mechanical Fluctuations of the Electromagnetic Field. Phys. Rev.
**1948**, 74, 1157. [Google Scholar] [CrossRef] - Pyykkö, P.; Zhao, L.-B. Search for Effective Local Model Potentials for Simulation of Quantum Electrodynamic Effects in Relativistic Calculations. J. Phys. B At. Mol. Phys.
**2003**, 36, 1469. [Google Scholar] [CrossRef] - Flambaum, V.V.; Ginges, J.S.M. Radiative Potential and Calculations of QED Radiative Corrections to Energy Levels and Electromagnetic Amplitudes in Many-Electron Atoms. Phys. Rev. A
**2005**, 72, 1094. [Google Scholar] [CrossRef] [Green Version] - Lowe, J.A.; Chantler, C.T.; Grant, I.P. Self-Energy Screening Approximations in Multi-Electron Atoms. Radiat. Phys. Chem.
**2013**, 85, 118. [Google Scholar] [CrossRef] - Piibeleht, M. Numerical Investigations of the Dirac Equation and Bound State Quantum Electrodynamics in Atoms. Ph.D. Thesis, Massey University, Albany, New Zealand, 2022. [Google Scholar]
- Bieroń, J.; Indelicato, P.; Jönsson, P. Multiconfiguration Dirac-Hartree-Fock calculations of transition rates and lifetimes of the eight lowest excited levels of radium. Eur. Phys. J. Spec. Top.
**2007**, 144, 75. [Google Scholar] [CrossRef] - Bieroń, J.; Froese Fischer, C.; Indelicato, P.; Jönsson, P.; Pyykkö, P. Complete Active Space multiconfiguration Dirac-Hartree-Fock calculations of hyperfine structure constants of the gold atom. Phys. Rev. A
**2009**, 79, 052502. [Google Scholar] [CrossRef] [Green Version] - Bieroń, J.; Gaigalas, G.; Gaidamauskas, E.; Indelicato, P.; Fritzsche, S.; Jönsson, P. MCDHF calculations of the electric dipole moment of radium induced by the nuclear Schiff moment. Phys. Rev. A
**2009**, 80, 012513. [Google Scholar] [CrossRef] [Green Version] - Indelicato, P.; Bieroń, J.; Jönsson, P. Are MCDF calculations 101% correct in the superheavy elements range? Theor. Chem. Acc.
**2011**, 129, 495. [Google Scholar] - Sampaio, J.M.; Parente, F.; Nazé, C.; Godefroid, M.; Indelicato, P.; Marques, J.P. Relativistic calculations of 1s
^{2}2s2p level splitting in Be-like Kr. Phys. Scr.**2013**, T156, 014015. [Google Scholar] [CrossRef] - Bieroń, J.; Froese Fischer, C.; Fritzsche, S.; Gaigalas, G.; Grant, I.P.; Indelicato, P.; Jönsson, P.; Pyykkö, P. Ab initio MCDHF calculations of electron-nucleus interactions. Phys. Scr.
**2015**, 90, 054011. [Google Scholar] [CrossRef] [Green Version] - Sampaio, J.M.; Ekman, J.; Tee, B.P.E.; du Rietz, R.; Lee, B.Q.; Pires, M.S.; Jönsson, P.; Kibédi, T.; Vos, M.; Stuchbery, A.E.; et al. Simulation of
^{125}I Auger emission spectrum with new atomic parameters from MCDHF calculations. J. Quant. Spectrosc. Rad. Transf.**2022**, 277, 107964. [Google Scholar] [CrossRef] - Desclaux, J.-P.; Indelicato, P. A General Multiconfiguration Dirac-Fock Code. Available online: http://www.lkb.upmc.fr/metrologysimplesystems/mdfgme-a-general-purpose-multiconfiguration-dirac-foc-program (accessed on 31 October 2022).

**Figure 1.**Left panel/in black: radial electron density ${D}^{SD}\left(r\right)$ for the ground state of beryllium using the $MR(1{s}^{2}2{s}^{2}+1{s}^{2}2{p}^{2})-SD(n=3)$ correlation model. In blue, difference between the radial electron densities calculated using, respectively, this correlation model (${D}^{SD}\left(r\right)$) and correlation limited to the Layzer complex (${D}^{LC}\left(r\right)$). Right panel: analysis of diagonal (in red) and off-diagonal contributions (in green) to the total ${D}^{SD}\left(r\right)-{D}^{LC}\left(r\right)$ difference (in blue). See text for discussion.

**Figure 2.**Relative line strengths of the magnetic-field split hyperfine components within the 4205 Å fine-structure transition of ${}^{151,153}$Eu${}^{+1}$ in a natural abundance mix under influence of a uniform external magnetic field of 6 kG. The $\pi $ ($\delta {M}_{F}=0$) components and ${\sigma}_{\mathrm{r}}$ ($\delta {M}_{F}=+1$; red-shifted) components are shown in the upper and lower panels, respectively. The relative line strengths are presented in log${}_{10}$ scale, and transitions with a relative strength smaller than ${10}^{-8}$ are excluded. This figure is inspired by Figure 3.12 in the book Polarization in Spectral Lines by Landi Deg’Innocenti and Landolfi [172].

${\mathit{s}}_{1/2}$ | ${\mathit{p}}_{1/2}$ | ${\mathit{p}}_{3/2}$ | ${\mathit{d}}_{3/2}$ | ${\mathit{d}}_{5/2}$ | ${\mathit{f}}_{5/2}$ | ${\mathit{f}}_{7/2}$ | ${\mathit{g}}_{7/2}$ | ${\mathit{g}}_{9/2}$ | |
---|---|---|---|---|---|---|---|---|---|

s | p- | p | d- | d | f- | f | g- | g | |

l | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 |

j | $1/2$ | $1/2$ | $3/2$ | $3/2$ | $5/2$ | $5/2$ | $7/2$ | $7/2$ | $9/2$ |

$\kappa $ | $-1$ | $+1$ | $-2$ | $+2$ | $-3$ | $+3$ | $-4$ | $+4$ | $-5$ |

**Table 2.**Subshell states ${\left[j\right]}^{w}$ are listed for $j\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1/2,\phantom{\rule{0.222222em}{0ex}}\dots ,\phantom{\rule{0.222222em}{0ex}}9/2$, in both seniority $\left(\nu \right)$ and quasi-spin $\left(2Q\right)$ representations. An extra number $Nr$ is introduced for $j=9/2$.

$\mathit{subshell}$ | $\mathit{\nu}$ | J | $2\mathit{Q}$ | $\mathit{Nr}$ | $\mathit{subshell}$ | $\mathit{\nu}$ | J | $2\mathit{Q}$ | $\mathit{Nr}$ |
---|---|---|---|---|---|---|---|---|---|

${\left(\right)}^{1}$ or ${\left(\right)}^{1}$ | 0 | 0 | 1 | 3 | $5/2$ | 2 | |||

${\left(\right)}^{1}$ | 1 | $1/2$ | 0 | 3 | $7/2$ | 2 | |||

3 | $9/2$ | 2 | |||||||

${\left(\right)}^{3}$ or ${\left(\right)}^{3}$ | 0 | 0 | 2 | 3 | $11/2$ | 2 | |||

${\left(\right)}^{3}$ or ${\left(\right)}^{3}$ | 1 | $3/2$ | 1 | 3 | $13/2$ | 2 | |||

${\left(\right)}^{3}$ | 0 | 0 | 2 | 3 | $15/2$ | 2 | |||

2 | 2 | 0 | 3 | $17/2$ | 2 | ||||

3 | $21/2$ | 2 | |||||||

${\left(\right)}^{5}$ or ${\left(\right)}^{5}$ | 0 | 0 | 3 | ${\left(\right)}^{9}$ or ${\left(\right)}^{9}$ | 0 | 0 | 5 | ||

${\left(\right)}^{5}$ or ${\left(\right)}^{5}$ | 1 | $5/2$ | 2 | 2 | 2 | 3 | |||

${\left(\right)}^{5}$ or ${\left(\right)}^{5}$ | 0 | 0 | 3 | 2 | 4 | 3 | |||

2 | 2 | 1 | 2 | 6 | 3 | ||||

2 | 4 | 1 | 2 | 8 | 3 | ||||

${\left(\right)}^{5}$ | 1 | $5/2$ | 2 | 4 | 0 | 1 | |||

3 | $3/2$ | 0 | 4 | 2 | 1 | ||||

3 | $9/2$ | 0 | 4 | 3 | 1 | ||||

4 | 4 | 1 | 1 | ||||||

${\left(\right)}^{7}$ or ${\left(\right)}^{7}$ | 0 | 0 | 4 | 4 | 4 | 1 | 2 | ||

${\left(\right)}^{7}$ or ${\left(\right)}^{7}$ | 1 | $7/2$ | 3 | 4 | 5 | 1 | |||

${\left(\right)}^{7}$ or ${\left(\right)}^{7}$ | 0 | 0 | 4 | 4 | 6 | 1 | 1 | ||

2 | 2 | 2 | 4 | 6 | 1 | 2 | |||

2 | 4 | 2 | 4 | 7 | 1 | ||||

2 | 6 | 2 | 4 | 8 | 1 | ||||

${\left(\right)}^{7}$ or ${\left(\right)}^{7}$ | 1 | $7/2$ | 3 | 4 | 9 | 1 | |||

3 | $3/2$ | 1 | 4 | 10 | 1 | ||||

3 | $5/2$ | 1 | 4 | 12 | 1 | ||||

3 | $9/2$ | 1 | ${\left(\right)}^{9}$ | 1 | $9/2$ | 4 | |||

3 | $11/2$ | 1 | 3 | $3/2$ | 2 | ||||

3 | $15/2$ | 1 | 3 | $5/2$ | 2 | ||||

${\left(\right)}^{7}$ | 0 | 0 | 4 | 3 | $7/2$ | 2 | |||

2 | 2 | 2 | 3 | $9/2$ | 2 | ||||

2 | 4 | 2 | 3 | $11/2$ | 2 | ||||

2 | 6 | 2 | 3 | $13/2$ | 2 | ||||

4 | 2 | 0 | 3 | $15/2$ | 2 | ||||

4 | 4 | 0 | 3 | $17/2$ | 2 | ||||

4 | 5 | 0 | 3 | $21/2$ | 2 | ||||

4 | 8 | 0 | 5 | $1/2$ | 0 | ||||

5 | $5/2$ | 0 | |||||||

${\left(\right)}^{9}$ or ${\left(\right)}^{9}$ | 0 | 0 | 5 | 5 | $7/2$ | 0 | |||

${\left(\right)}^{9}$ or ${\left(\right)}^{9}$ | 1 | $9/2$ | 4 | 5 | $9/2$ | 0 | |||

${\left(\right)}^{9}$ or ${\left(\right)}^{9}$ | 0 | 0 | 5 | 5 | $11/2$ | 0 | |||

2 | 2 | 3 | 5 | $13/2$ | 0 | ||||

2 | 4 | 3 | 5 | $15/2$ | 0 | ||||

2 | 6 | 3 | 5 | $17/2$ | 0 | ||||

2 | 8 | 3 | 5 | $19/2$ | 0 | ||||

${\left(\right)}^{9}$ or ${\left(\right)}^{9}$ | 1 | $9/2$ | 4 | 5 | $25/2$ | 0 | |||

3 | $3/2$ | 2 |

**Table 3.**Mixing coefficients for the $n=4$ active space valence correlation expansion of the $1{s}^{2}2{s}^{2}{\phantom{\rule{3.33333pt}{0ex}}}^{1}{S}_{0}$ ground state of Be, using two different orbital bases: the MCDHF-optimized orbitals and their corresponding natural orbitals (MCDHF/NO).

CSF | MCDHF | MCDHF/NO |
---|---|---|

2s (2) | 0.953738 | 0.953740 |

2s (1) 3s (1) | −0.001117 | 0.000000 |

2s (1) 4s (1) | −0.001846 | 0.000000 |

2p (2) | 0.242750 | 0.242750 |

2p-(2) | 0.171674 | 0.171674 |

2p (1) 3p (1) | 0.000254 | 0.000000 |

2p (1) 4p (1) | 0.000302 | 0.000000 |

2p-(1) 3p-(1) | 0.000178 | 0.000000 |

2p-(1) 4p-(1) | 0.000214 | 0.000000 |

3s (2) | −0.039770 | −0.039787 |

3s (1) 4s (1) | −0.001052 | 0.000000 |

3p (2) | 0.004905 | 0.004922 |

3p-(2) | 0.003467 | 0.003479 |

3p (1) 4p (1) | −0.000333 | 0.000000 |

3p-(1) 4p-(1) | −0.000237 | 0.000000 |

3d (2) | −0.013120 | −0.013134 |

3d-(2) | −0.010712 | −0.010723 |

3d (1) 4d (1) | 0.000530 | 0.000000 |

3d-(1) 4d-(1) | 0.000432 | 0.000000 |

4s (2) | −0.004103 | −0.004089 |

4p (2) | 0.001628 | 0.001611 |

4p-(2) | 0.001150 | 0.001138 |

4d (2) | −0.002808 | −0.002794 |

4d-(2) | −0.002291 | −0.002280 |

4f (2) | 0.004766 | 0.004766 |

4f-(2) | 0.004127 | 0.004127 |

**Table 4.**CSFs and corresponding expansion coefficients for the MR of the $1{s}^{2}2{s}^{2}2{p}^{6}3{s}^{2}{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}{S}_{0}$ ground state and the $1{s}^{2}2{s}^{2}2{p}^{6}3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}{P}_{1}^{o}$, ${}^{1}\phantom{\rule{-0.166667em}{0ex}}{P}_{1}^{o}$ excited states in Mg I.

CSFs | $3{s}^{2}{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}{S}_{0}$ | |

1s (2) 2s (2) 2p-(2) 2p (4) 3s (2) | 0.964240 | |

1s (2) 2s (2) 2p-(2) 2p (4) 3p (2) | 0.214715 | |

1s (2) 2s (2) 2p-(2) 2p (4) 3p-(2) | 0.152334 | |

1s (2) 2s (2) 2p-(2) 2p (4) 3d (2) | −0.023696 | |

1s (2) 2s (2) 2p-(2) 2p (4) 3d-(2) | −0.019299 | |

CSFs | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}{P}_{1}^{o}$ | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}{P}_{1}^{o}$ |

1s (2) 2s (2) 2p-(2) 2p (4) 3s (1) 3p-(1) | 0.811480 | 0.756818 |

1s (2) 2s (2) 2p-(2) 2p (4) 3s (1) 3p (1) | −0.571905 | 0.532680 |

1s (2) 2s (2) 2p-(2) 2p (4) 3p (1) 3d-(1) | 0.082429 | −0.233472 |

1s (2) 2s (2) 2p-(2) 2p (4) 3p (1) 3d (1) | −0.061506 | 0.191522 |

1s (2) 2s (2) 2p-(2) 2p (4) 3p-(1) 3d-(1) | 0.046401 | −0.166818 |

1s (2) 2s (2) 2p-(2) 2p (4) 3s (1) 4p-(1) | 0.032508 | 0.142870 |

1s (2) 2s (2) 2p-(2) 2p (4) 3s (1) 4p (1) | −0.025260 | 0.063628 |

**Table 5.**A (MHz), $B/Q$ (MHz/b), and Q (in barn) values as functions of the increasing active set of orbitals for the $4s4p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}{P}_{1}^{o}$ and $4s4p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}{P}_{2}^{o}$ states in ${}^{67}$Zn i. ${I}^{\pi}=5/{2}^{-}$ and ${\mu}_{\mathrm{expt}}=0.875479\left(9\right)\phantom{\rule{0.166667em}{0ex}}{\mu}_{N}$. The Q-values are extracted from the relation $Q={B}_{\mathrm{expt}}/(B/Q)$, where the experimental values are ${B}_{\mathrm{expt}}{(}^{3}\phantom{\rule{-0.166667em}{0ex}}{P}_{1}^{o})=-18.782\left(8\right)$

^{1}MHz and ${B}_{\mathrm{expt}}{(}^{3}\phantom{\rule{-0.166667em}{0ex}}{P}_{2}^{o})=35.806\left(5\right)$

^{2}MHz.

Active Set | $4\mathit{s}4\mathit{p}{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{1}^{\mathit{o}}$ | $4\mathit{s}4\mathit{p}{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{2}^{\mathit{o}}$ | ||||||
---|---|---|---|---|---|---|---|---|

${\mathit{N}}_{\mathbf{CSFs}}$ | A (MHz) | $\mathit{B}/\mathit{Q}$ (MHz/b) | Q (barn) | ${\mathit{N}}_{\mathbf{CSFs}}$ | A (MHz) | $\mathit{B}/\mathit{Q}$ (MHz/b) | Q (barn) | |

MCDHF-SrDT-SP | ||||||||

DHF | 2 | 473.40 | $-100.373$ | 0.1098 | 1 | $419.93$ | 192.924 | 0.1159 |

VV+CV | ||||||||

$5s5p4d4f$ | $1\phantom{\rule{0.166667em}{0ex}}592$ | $558.02$ | $-131.036$ | $0.1433$ | $2\phantom{\rule{0.166667em}{0ex}}122$ | $483.71$ | $254.975$ | $0.1404$ |

$6s6p5d5f5g$ | $11\phantom{\rule{0.166667em}{0ex}}932$ | $590.45$ | $-146.084$ | $0.1286$ | $16\phantom{\rule{0.166667em}{0ex}}961$ | $507.74$ | $280.708$ | $0.1276$ |

$7s7p6d6f6g6h$ | $48\phantom{\rule{0.166667em}{0ex}}574$ | $610.80$ | $-150.997$ | $0.1244$ | $71\phantom{\rule{0.166667em}{0ex}}610$ | $529.87$ | $290.233$ | $0.1234$ |

$8s8p7d7f7g7h$ | $128\phantom{\rule{0.166667em}{0ex}}264$ | $613.17$ | $-152.617$ | $0.1231$ | $191\phantom{\rule{0.166667em}{0ex}}495$ | $532.46$ | $292.535$ | $0.1220$ |

$9s9p8d8f8g8h$ | $267\phantom{\rule{0.166667em}{0ex}}998$ | $617.02$ | $-154.391$ | $0.1217$ | $402\phantom{\rule{0.166667em}{0ex}}586$ | $536.97$ | $296.441$ | $0.1208$ |

Liu et al. [157] | $605.9$ | $-150.7$ | $0.1247$ | |||||

Expt. | $609.086\left(2\right)$^{1} | $531.987\left(5\right)$^{2} |

**Table 6.**Relativistic mass shift $\mathsf{\Delta}{K}_{MS}$ (in GHz u) and field shift F (in MHz/fm${}^{2}$) factors for the $2s{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{S}_{1/2}-2p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}^{o}$ and $2s{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{S}_{1/2}-2p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}^{o}$ transitions in Nd${}^{57+}$ from Li et al. [164]. For comparison, individual relativistic normal mass shift $\mathsf{\Delta}{K}_{NMS}$ and specific mass shift $\mathsf{\Delta}{K}_{SMS}$ coefficients (in GHz u) are also included. The results in the second row, labeled MCDHF, were obtained in the MCDHF model, with the largest size of the active set ($n=5$). The numbers in square brackets represent powers of 10.

Model | $2\mathit{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{\mathit{S}}_{1/2}-2\mathit{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{1/2}^{\mathit{o}}$ | $2\mathit{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{\mathit{S}}_{1/2}-2\mathit{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{3/2}^{\mathit{o}}$ | ||||||
---|---|---|---|---|---|---|---|---|

$\mathsf{\Delta}{\mathit{K}}_{\mathit{NMS}}$ | $\mathsf{\Delta}{\mathit{K}}_{\mathit{SMS}}$ | $\mathsf{\Delta}{\mathit{K}}_{\mathit{MS}}$ | $\mathit{F}$ | $\mathsf{\Delta}{\mathit{K}}_{\mathit{NMS}}$ | ${\mathit{K}}_{\mathit{SMS}}$ | $\mathsf{\Delta}{\mathit{K}}_{\mathit{MS}}$ | $\mathit{F}$ | |

DHF | −1083[1] | −8227[2] | −8336[2] | −7903[3] | −8721[1] | −8768[2] | −9640[2] | −8215[3] |

MCDHF ($n=5$) | −1342[1] | −8196[2] | −8331[2] | −7929[3] | −8589[1] | −8761[2] | −9620[2] | −8203[3] |

CI + Breit | −1449[1] | −8196[2] | −8341[2] | −7885[3] | −8577[1] | −8775[2] | −9632[2] | −8157[3] |

Kozhedub et al. [165] | −1641.8[1] | −8180.90[2] | −8345.08(25)[2] | −8573.3[1] | −8769.29[2] | −9626.62(25)[2] | ||

Brandau et al. [166] | −7520[3] | −7810[3] |

**Table 7.**Energies in cm${}^{-1}$ and $LS$-compositions for the first 40 levels in Fe XIV. ${E}_{CI}$ Ekman et al. [167], ${E}_{NIST}$ NIST Atomic Spectra Database (2013) [168] and $\mathsf{\Delta}E$ difference between ${E}_{CI}$ and ${E}_{NIST}$. Indices “a” and “b” are used to differentiate between identical configurations which share the same coupling and leading $LS$-percentage composition. The first number in the $LS$-compositions is the expansion coefficient for the leading configuration and $LSJ$ term in column 3.

No. | Level | $\mathbf{LS}$-Composition | ${\mathit{E}}_{\mathbf{CI}}$ | ${\mathit{E}}_{\mathbf{NIST}}$ | $\mathbf{\Delta}\mathit{E}$ |
---|---|---|---|---|---|

1 | $3{s}^{2}\phantom{\rule{0.166667em}{0ex}}3p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}^{\circ}$ | 0.97 | 0 | 0 | 0 |

2 | $3{s}^{2}\phantom{\rule{0.166667em}{0ex}}3p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}^{\circ}$ | 0.97 | 18 855 | 18 853 | 2 |

3 | $3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}$ | 0.98 | 225 086 | 225 114 | −28 |

4 | $3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}$ | 0.99 | 232 777 | 232 789 | −12 |

5 | $3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{5/2}$ | 0.97 | 242 372 | 242 387 | −15 |

6 | $3s3{p}^{2}{(}_{2}^{1}D){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}_{3/2}$ | 0.86 + 0.11 $3{s}^{2}\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}D$ | 299 402 | 299 242 | 160 |

7 | $3s3{p}^{2}{(}_{2}^{1}D){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}_{5/2}$ | 0.85 + 0.11 $3{s}^{2}\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}D$ + 0.02 $3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}P$ | 301 627 | 301 469 | 158 |

8 | $3s3{p}^{2}{(}_{0}^{1}S){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{S}_{1/2}$ | 0.75 + 0.21 $3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}P$ | 364 945 | 364 693 | 252 |

9 | $3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}$ | 0.75 + 0.22 $3s3{p}^{2}{(}_{0}^{1}S){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}S$ | 388 711 | 388 510 | 201 |

10 | $3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}$ | 0.95 + 0.02 $3{p}^{2}{(}_{2}^{1}D){\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}D\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}P$ | 396 687 | 396 512 | 175 |

11 | $3{s}^{2}\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}_{3/2}$ | 0.86 + 0.11 $3s3{p}^{2}{(}_{2}^{1}D){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}D$ + 0.02 $3{p}^{2}{(}_{0}^{1}S){\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}S\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}D$ | 473 231 | 473 223 | 8 |

12 | $3{s}^{2}\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}_{5/2}$ | 0.86 + 0.11 $3s3{p}^{2}{(}_{2}^{1}D){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}D$ + 0.02 $3{p}^{2}{(}_{0}^{1}S){\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}S\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}D$ | 475 215 | 475 202 | 13 |

13 | $3{p}^{3}{(}_{3}^{2}D){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}_{3/2}^{\circ}$ | 0.64 + 0.27 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ + 0.04 $3{p}^{3}{(}_{1}^{2}P){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}^{\circ}$ | 576 599 | 576 383 | 216 |

14 | $3{p}^{3}{(}_{3}^{2}D){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}_{5/2}^{\circ}$ | 0.69 + 0.29 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ | 580 450 | 580 233 | 217 |

15 | $3{p}^{3}{(}_{3}^{4}S){\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{S}_{3/2}^{\circ}$ | 0.92 + 0.03 $3{p}^{3}{(}_{3}^{2}D){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ + 0.02 $3{p}^{3}{(}_{1}^{2}P){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}^{\circ}$ | 589 023 | 589 002 | 21 |

16 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{F}_{3/2}^{\circ}$ | 0.96 | 641 955 | ||

17 | $3{p}^{3}{(}_{1}^{2}P){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}^{\circ}$ | 0.80 + 0.13 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}^{\circ}$ + 0.05 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}^{\circ}$ | 642 591 | 642 310 | 281 |

18 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{F}_{5/2}^{\circ}$ | 0.98 | 646 042 | 645 988 | 54 |

19 | $3{p}^{3}{(}_{1}^{2}P){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}^{\circ}$ | 0.71 + 0.13 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}^{\circ}$ + 0.04 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}^{\circ}$ | 646 119 | 645 409 | 710 |

20 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{F}_{7/2}^{\circ}$ | 0.98 | 651 972 | 651 946 | 26 |

21 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{F}_{9/2}^{\circ}$ | 1.00 | 660 304 | 660 263 | 41 |

22 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{5/2}^{\circ}$ | 0.65 + 0.28 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ + 0.02 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ | 690 311 | 690 304 | 7 |

23 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{D}_{3/2}^{\circ}$ | 0.60 + 0.38 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}^{\circ}$ | 692 653 | 692 662 | −9 |

24 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{D}_{1/2}^{\circ}$ | 0.87 + 0.12 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}^{\circ}$ | 694 140 | 694 168 | −28 |

25 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{D}_{7/2}^{\circ}$ | 0.98 | 703 341 | 703 393 | −52 |

26 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}^{\circ}$ | 0.87 + 0.12 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ | 703 826 | 703 750 | 76 |

27 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{D}_{5/2}^{\circ}$ | 0.70 + 0.27 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}^{\circ}$ | 704 114 | 704 114 | 0 |

28 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}^{\circ}$ | 0.60 + 0.39 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ | 704 202 | 704 209 | −7 |

29 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}_{3/2}^{\circ}$ | 0.51 + 0.27 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ + 0.17 $3{p}^{3}{(}_{3}^{2}D){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ | 717 296 | 717 195 | 101 |

30 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}_{5/2}^{\circ}$ | 0.48 + 0.25 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ + 0.16 $3{p}^{3}{(}_{3}^{2}D){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ | 717 937 | 717 861 | 76 |

31 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{F}_{5/2}^{\circ}$ | 0.65 + 0.32 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{F}^{\circ}$ | 745 214 | 744 965 | 249 |

32 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{F}_{7/2}^{\circ}$ | 0.65 + 0.33 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{F}^{\circ}$ | 760 089 | 759 814 | 275 |

33 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}^{\circ}$ | 0.77 + 0.14 $3{p}^{3}{(}_{1}^{2}P){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}^{\circ}$ + 0.06 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ | 807 347 | 807 113 | 234 |

34 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}^{\circ}$ | 0.86 + 0.12 $3{p}^{3}{(}_{1}^{2}P){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}^{\circ}$ | 815 394 | ||

35 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{F}_{7/2}^{\circ}$ | 0.64 + 0.33 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{F}^{\circ}$ | 817 790 | 817 593 | 197 |

36 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{F}_{5/2}^{\circ}$ | 0.64 + 0.32 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{F}^{\circ}$ | 820 795 | 820 601 | 194 |

37 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}^{\circ}$ | 0.90 + 0.04 $3{p}^{3}{(}_{1}^{2}P){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}^{\circ}$ | 839 715 | 839 492 | 223 |

38 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}^{\circ}$${}_{a}$ | 0.38 + 0.37 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ + 0.08 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ | 840 967 | 840 775 | 192 |

39 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}^{\circ}$${}_{b}$ | 0.52 + 0.26 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ + 0.08 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ | 843 862 | 843 656 | 206 |

40 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}_{5/2}^{\circ}$ | 0.67 + 0.17 $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}3d{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ + 0.12 $3{p}^{3}{(}_{3}^{2}D){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}^{\circ}$ | 844 618 | 844 477 | 141 |

Pos | Configuration $\mathit{LS}$. | J | $\mathit{\pi}$ | ${\mathit{E}}_{\mathit{CI}}$ | ${\mathit{E}}_{\mathit{NIST}}$(%) | ${\mathit{E}}_{\mathit{obs}}$(%) |
---|---|---|---|---|---|---|

101 | $3{s}^{2}\phantom{\rule{0.166667em}{0ex}}4s{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{S}_{}$ | 1/2 | + | 1 427 550 | 1 435 020 (0.5206) | 1 426 965 (0.04) T |

125 | $3{s}^{2}\phantom{\rule{0.166667em}{0ex}}4p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{}$ | 1/2 | − | 1 541 937 | 1 568 840 (1.715) | 1 541 394 (0.03) |

128 | $3{s}^{2}\phantom{\rule{0.166667em}{0ex}}4p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{}$ | 3/2 | − | 1 548 618 | 1 574 010 (1.613) | 1 548 258 (0.02) |

136 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}4s{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{}$ | 1/2 | − | 1 690 299 | - | 1 689 695 (0.0004) |

150 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}4p{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{}$ | 3/2 | + | 1 795 164 | - | 1 795 032 (0.007) T |

152 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}4p{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{}$ | 5/2 | + | 1 802 686 | - | 1 802 292 (0.02) T |

181 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}4d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{D}_{}$ | 5/2 | − | 1 930 871 | - | 1 933 758 (0.15) |

184 | $3s3p{\phantom{\rule{3.33333pt}{0ex}}}^{3}\phantom{\rule{-0.166667em}{0ex}}P\phantom{\rule{0.166667em}{0ex}}4d{\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{D}_{}$ | 7/2 | − | 1 935 340 | - | 1 938 452 (0.16) |

**Table 9.**Multipoles (MT), transition energies $\mathsf{\Delta}E$, wavelengths $\lambda $ and transition probabilities A in s${}^{-1}$ in Fe XIV for selected transitions. ${A}_{CI}$ Ekman et al. [167], ${A}_{MCHF}$ Froese Fischer and Tachiev [170], ${A}_{MR-MP}$ Santana et al. [171], and ${A}_{NIST}$ NIST Atomic Spectra Database (2013) [168]. Accuracy estimates $dT$ have been computed based on transition probabilities in Babushkin and Coulomb gauges. The numbers in square brackets are powers of 10.

Upper Level | Lower Level | MT | $\mathbf{\Delta}\mathit{E}$ (cm${}^{-1}$) | $\mathit{\lambda}$ (Å) | ${\mathit{A}}_{\mathit{CI}}$ | $\mathit{dT}$ | ${\mathit{A}}_{\mathit{MCHF}}$ | ${\mathit{A}}_{\mathit{MR}-\mathit{MP}}$ | ${\mathit{A}}_{\mathit{NIST}}$ |
---|---|---|---|---|---|---|---|---|---|

$3{s}^{2}\phantom{\rule{0.222222em}{0ex}}3p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}^{o}$ | $3{s}^{2}\phantom{\rule{0.222222em}{0ex}}3p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}^{o}$ | M1 | 18 854 | 5303.740 | 6.019[1] | 6.016[1] | |||

E2 | 18 854 | 5303.740 | 1.474[−2] | 0.004 | 1.466[−2] | ||||

$3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}$ | $3{s}^{2}\phantom{\rule{0.222222em}{0ex}}3p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}^{o}$ | E1 | 225 086 | 444.274 | 2.657[7] | 0.062 | 2.620[7] | 2.230[7] | |

$3{s}^{2}\phantom{\rule{0.222222em}{0ex}}3p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}^{o}$ | E1 | 206 231 | 484.891 | 9.777[6] | 0.059 | 1.013[7] | 8.693[6] | ||

M2 | 206 231 | 484.891 | 2.801[−1] | 2.892[−1] | |||||

$3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}$ | $3{s}^{2}\phantom{\rule{0.222222em}{0ex}}3p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}^{o}$ | E1 | 232 776 | 429.596 | 5.851[5] | 0.024 | 5.187[5] | 4.833[5] | |

M2 | 232 776 | 429.596 | 2.169[0] | 2.193[0] | |||||

$3{s}^{2}\phantom{\rule{0.222222em}{0ex}}3p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}^{o}$ | E1 | 213 922 | 467.459 | 6.323[6] | 0.083 | 5.908[6] | 5.458[6] | ||

M2 | 213 922 | 467.459 | 6.644[−2] | 6.838[−2] | |||||

$3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}$ | M1 | 7 690 | 13 002.904 | 1.007[1] | 1.014[1] | ||||

E2 | 7 690 | 13 002.904 | 2.050[−5] | 0.001 | 1.003[−5] | ||||

$3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{5/2}$ | E1 | 223 517 | 447.392 | 2.714[7] | 0.061 | 2.491[7] | 2.256[7] | 2.5[7] ${}^{c}$ | |

M2 | 223 517 | 447.392 | 1.633[0] | 1.685[0] | |||||

M2 | 242 372 | 412.589 | 1.433[0] | 1.423[0] | |||||

E2 | 17 285 | 5 785.065 | 6.979[−3] | 0.000 | 3.274[−3] | ||||

$3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{4}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}$ | M1 | 9 595 | 10 421.769 | 1.411[1] | 1.411[1] | ||||

E2 | 9 595 | 10 421.769 | 4.907[−4] | 0.000 | 4.976[−4] | ||||

$3s3{p}^{2}{(}_{2}^{1}D){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}_{3/2}$ | E1 | 299 401 | 333.999 | 2.426[9] | 0.019 | 2.460[9] | 2.3[9] ${}^{b}$ | ||

E1 | 280 547 | 356.446 | 7.560[7] | 0.003 | 8.669[7] | 7.5[7] ${}^{c}$ | |||

$3s3{p}^{2}{(}_{2}^{1}D){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{D}_{5/2}$ | E1 | 282 772 | 353.642 | 1.954[9] | 0.027 | 1.998[9] | 1.9[9] ${}^{b}$ | ||

$3s3{p}^{2}{(}_{0}^{1}S){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{S}_{1/2}$ | E1 | 364 944 | 274.014 | 1.782[10] | 0.011 | 1.716[10] | 1.8[10] ${}^{b}$ | ||

E1 | 346 090 | 288.942 | 1.082[9] | 0.002 | 1.631[9] | 1.2[9] ${}^{c}$ | |||

$3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}$ | E1 | 388 711 | 257.261 | 1.279[10] | 0.012 | 1.511[10] | 1.4[10] ${}^{b}$ | ||

E1 | 369 856 | 270.375 | 2.090[10] | 0.012 | 2.144[10] | 2.1[10] ${}^{b}$ | |||

$3s3{p}^{2}{(}_{2}^{3}P){\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}$ | E1 | 396 687 | 252.088 | 7.427[9] | 0.009 | 7.902[9] | 7.6[9] ${}^{a}$ | ||

E1 | 377 832 | 264.667 | 3.254[10] | 0.012 | 3.429[10] | 3.38[10] ${}^{a}$ |

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Jönsson, P.; Godefroid, M.; Gaigalas, G.; Ekman, J.; Grumer, J.; Li, W.; Li , J.; Brage, T.; Grant, I.P.; Bieroń, J.;
et al. An Introduction to Relativistic Theory as Implemented in GRASP. *Atoms* **2023**, *11*, 7.
https://doi.org/10.3390/atoms11010007

**AMA Style**

Jönsson P, Godefroid M, Gaigalas G, Ekman J, Grumer J, Li W, Li J, Brage T, Grant IP, Bieroń J,
et al. An Introduction to Relativistic Theory as Implemented in GRASP. *Atoms*. 2023; 11(1):7.
https://doi.org/10.3390/atoms11010007

**Chicago/Turabian Style**

Jönsson, Per, Michel Godefroid, Gediminas Gaigalas, Jörgen Ekman, Jon Grumer, Wenxian Li, Jiguang Li , Tomas Brage, Ian P. Grant, Jacek Bieroń,
and et al. 2023. "An Introduction to Relativistic Theory as Implemented in GRASP" *Atoms* 11, no. 1: 7.
https://doi.org/10.3390/atoms11010007