# 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

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**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 | $2\mathit{Q}$ | $\mathit{Nr}$ |
---|---|---|---|---|---|---|---|---|---|

${\left[1/2\right]}^{0}$ or ${\left[1/2\right]}^{2}$ | 0 | 0 | 1 | 3 | $5/2$ | 2 | |||

${\left[1/2\right]}^{1}$ | 1 | $1/2$ | 0 | 3 | $7/2$ | 2 | |||

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

${\left[3/2\right]}^{0}$ or ${\left[3/2\right]}^{4}$ | 0 | 0 | 2 | 3 | $11/2$ | 2 | |||

${\left[3/2\right]}^{1}$ or ${\left[3/2\right]}^{3}$ | 1 | $3/2$ | 1 | 3 | $13/2$ | 2 | |||

${\left[3/2\right]}^{2}$ | 0 | 0 | 2 | 3 | $15/2$ | 2 | |||

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

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

${\left[5/2\right]}^{0}$ or ${\left[5/2\right]}^{6}$ | 0 | 0 | 3 | ${\left[9/2\right]}^{4}$ or ${\left[9/2\right]}^{6}$ | 0 | 0 | 5 | ||

${\left[5/2\right]}^{1}$ or ${\left[5/2\right]}^{5}$ | 1 | $5/2$ | 2 | 2 | 2 | 3 | |||

${\left[5/2\right]}^{2}$ or ${\left[5/2\right]}^{4}$ | 0 | 0 | 3 | 2 | 4 | 3 | |||

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

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

${\left[5/2\right]}^{3}$ | 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 | ||||||

${\left[7/2\right]}^{0}$ or ${\left[7/2\right]}^{8}$ | 0 | 0 | 4 | 4 | 4 | 1 | 2 | ||

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

${\left[7/2\right]}^{2}$ or ${\left[7/2\right]}^{6}$ | 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 | ||||

${\left[7/2\right]}^{3}$ or ${\left[7/2\right]}^{5}$ | 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 | ${\left[9/2\right]}^{5}$ | 1 | $9/2$ | 4 | |||

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

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

${\left[7/2\right]}^{4}$ | 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 | |||||||

${\left[9/2\right]}^{0}$ or ${\left[9/2\right]}^{10}$ | 0 | 0 | 5 | 5 | $7/2$ | 0 | |||

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

${\left[9/2\right]}^{2}$ or ${\left[9/2\right]}^{8}$ | 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 | ||||

${\left[9/2\right]}^{3}$ or ${\left[9/2\right]}^{7}$ | 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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**MDPI and ACS Style**

Jönsson, P.; Godefroid, M.; Gaigalas, G.; Ekman, J.; Grumer, J.; Li, W.; Li , J.; Brage, T.; Grant, I.P.; Bieroń, J.; Fischer, C.F. 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, Fischer CF. 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 Charlotte Froese Fischer. 2023. "An Introduction to Relativistic Theory as Implemented in GRASP" *Atoms* 11, no. 1: 7.
https://doi.org/10.3390/atoms11010007