# Independently Optimized Orbital Sets in GRASP—The Case of Hyperfine Structure in Li I

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## Abstract

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## 1. Introduction

## 2. Variational Calculations

#### 2.1. The MCDHF Method

#### 2.2. Localization of the Radial Orbitals and Their Dependence on the Energy Functional

## 3. Computed Properties and Their Dependence on Correlation Effects

#### 3.1. Hyperfine Structure

#### 3.2. Polarization Effects

#### 3.3. Localization of the Polarization Orbitals

- Perform a weighted average Dirac–Fock calculation for $1{s}^{2}2s{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{S}_{1/2}$ and $1{s}^{2}2p{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2,\phantom{\rule{0.166667em}{0ex}}3/2}^{\phantom{\rule{0.166667em}{0ex}}o}$;
- Keep $1s,2s,2p\text{-},2p$ frozen and perform weighted average MCDHF calculations for $1{s}^{2}2s{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{S}_{1/2}$ and $1{s}^{2}2p{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2,\phantom{\rule{0.166667em}{0ex}}3/2}^{\phantom{\rule{0.166667em}{0ex}}o}$ based on the CSF expansions formed by allowing S’s substitution from the reference configuration to a set of $s,d\text{-},d$ orbitals;
- Compute the hyperfine interaction constants and monitor the convergence as the set of $s,d\text{-},d$ orbitals is increased;
- Stop when the hyperfine interaction constants are not changing anymore.

## 4. Hyperfine Interaction Constants in Different Orbital Bases

#### 4.1. Orbital Basis from Energy-Driven Calculations

#### 4.2. Polarization Orbitals Augmented to the Orbital Basis from Energy-Driven Calculations

`rwfnrelabel`program in grasp to relabel (change the principal quantum number that serves as the orbital identifier) the polarization orbitals so that they appear after the orbitals from the energy-driven layer-by-layer calculation. The relabeled polarization orbitals are then, using the

`rwfnestimate`program in grasp, orthogonalized against the orbitals from the energy-driven layer-by-layer calculation and finally augmented. Then, we perform CI calculations with the CSFs obtained by allowing SDT substitutions from the {$1{s}^{2}2s,1{s}^{2}2p$} reference configurations to orbitals optimized in the energy-driven calculation. To these CSFs, which mainly describe correlation effects that are important for lowering the energy and, to a lesser extent, the effects that are important for the hyperfine structure, we add the CSFs obtained by S substitutions from the atomic core to the augmented, orthogonalized polarization orbitals. For energy-optimized orbital bases with $n<10$, four layers of orthogonalized polarization orbitals were augmented. For energy-optimized orbital bases with $n\ge 10$, due to linear dependencies in the orbital basis, only two polarization layers were augmented. The number of CSFs in the final even and odd state expansions are, respectively, 92,541 and 258,432, i.e., only negligibly larger than those from the energy-driven calculation. In Figure 4, we display, in blue crosses, the resulting hyperfine interaction constants A and B in MHz, as functions of the increasing energy-driven orbital set (to which two or four layers of polarization orbitals have been added). Now, the oscillations are almost completely damped out, and the final values of the interaction constants can already be accurately established from a limited, $n=9$, energy-driven orbital set merged with the polarization orbitals. The CPU times of the $n=13$ CI calculations are 20m42s in energy-driven calculations and 25m5s for energy-driven orbitals merged with the polarization orbitals’ calculation for the $1{s}^{2}2s{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{S}_{1/2}$ state, using 64 nodes on a cluster with the AMD EPYC 7542 32-Core Processor. The bulk of the CPU time is spent on the calculations for energy-driven orbital sets. The interaction constants, on the other hand, can be accurately established at $n=10$ during the calculation of the energy-driven orbital set merged with the polarization orbitals, for which the CPU time of the CI calculation is 6m30s on the $1{s}^{2}2s{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{S}_{1/2}$ state. That means that the efficiency can be improved by using a smaller orbital set compared with energy-driven calculations. The use of separately optimized polarization orbitals merged with energy-driven orbital sets represents a great improvement in both efficiency (the time-consuming parts are the calculations for large energy-driven orbital sets) and accuracy. To provide the final values of the interaction constants, the effects from the neglected orbitals with a higher l, as well as from the Breit interaction, QED, and nuclear recoil, have to be added [31,36,37]. This is, however, outside the scope of the present paper.

## 5. Summary and Conclusions

`rwfnrelabel`, and

`rwfnestimate`[22], the present procedure has the drawback that it requires CSFs that are important for the property, or, in the case of energy separations, core–valence, and valence–valence correlation, to be built on merged orbital sets, which implies a certain overhead. An alternate and more general solution circumventing the problems above would be to employ different and mutually non-orthogonal orbital sets and deal with the non-orthogonalities by bi-orthogonal orbital transformations, as in [35,38]. This work is in progress based on the new concept of configuration state function generators (CSFGs) [39].

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- 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, 184–187. [Google Scholar] [CrossRef] - 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] - Zhang, C.Y.; Wang, K.; Godefroid, M.; Jönsson, P.; Si, R.; Chen, C.Y. Benchmarking calculations with spectroscopic accuracy of excitation energies and wavelengths in sulfur-like tungsten. Phys. Rev. A
**2020**, 101, 032509. [Google Scholar] [CrossRef] [Green Version] - 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. Radiat. Transf.
**2021**, 269, 107650. [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 xlviii through W lvi tungsten ions. Phys. Rev. A
**2022**, 105, 022817. [Google Scholar] [CrossRef] - Andersson, M.; Grumer, J.; Ryde, N.; Blackwell-Whitehead, R.; Hutton, R.; Zou, Y.; Jönsson, P.; Brage, T. Hyperfine-dependent gf values of Mn I lines in the 1.49–180 μm H Band. Astrophys. J. Suppl.
**2015**, 216, 21. [Google Scholar] [CrossRef] [Green Version] - 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] - 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] - 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, 433–446. [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] - 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] - Li, J.; 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] - 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] [PubMed] - Wraith, C.; Yang, X.; Xie, L.; Babcock, C.; Bieroń, J.; Billowes, J.; Bissell, M.; Blaum, K.; Cheal, B.; Filippin, L.; et al. Evolution of nuclear structure in neutron-rich odd-Zn isotopes and isomers. Phys. Lett. B
**2017**, 771, 385–391. [Google Scholar] [CrossRef] - Barzakh, A.; Cubiss, J.; Andreyev, A.; Seliverstov, M.; Andel, B.; Antalic, S.; Ascher, P.; Atanasov, D.; Beck, D.; Bieroń, J.; et al. Inverse odd-even staggering in nuclear charge radii and possible octupole collectivity in
^{217,218,219}At revealed by in-source laser spectroscopy. Phys. Rev. C**2019**, 99, 054317. [Google Scholar] [CrossRef] [Green Version] - 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
**2022**, in press. [Google Scholar] - Froese Fischer, C.; Brage, T.; Jönsson, P. Computational Atomic Structure; Institute of Physics Publishing (IoP): Bristol, UK, 1997. [Google Scholar]
- Godefroid, M.R.; Van Meulebeke, G.; Jönsson, P.; Froese Fischer, C. Large-scale MCHF calculations of hyperfine structures in nitrogen and oxygen. Z. Phys. D—Atoms Mol. Clust.
**1997**, 42, 193–201. [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] - 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]
- 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] - Jönsson, P.; Godefroid, M.; Gaigalas, G.; Ekman, J.; Grumer, J.; Li, W.; Li, J.; Brage, T.; Grant, I.P.; Bieroń, J.; et al. GRASP Manual for Users. Atoms
**2022**, in press. [Google Scholar] - Godefroid, M.R.; Jönsson, P.; Froese Fischer, C. Atomic structure variational calculations in spectroscopy. Phys. Scr.
**1998**, 1998, 33. [Google Scholar] [CrossRef] [Green Version] - Schwartz, C. Theory of hyperfine structure. Phys. Rev.
**1955**, 97, 380. [Google Scholar] [CrossRef] - Lindgren, I.; Rosén, A. Relativistic self-consistent-field calculations with application to atomic hyperfine interaction. Case Stud. At. Phys.
**1974**, 3, 93–196. [Google Scholar] - Lindgren, I. Effective operators in the atomic hyperfine interaction. Rep. Prog. Phys.
**1984**, 47, 345. [Google Scholar] [CrossRef] - Beckmann, A.; Böklen, K.; Elke, D. Precision measurements of the nuclear magnetic dipole moments of
^{6}Li,^{7}Li,^{23}Na,^{39}K and^{41}K. Z. Phys.**1974**, 270, 173–186. [Google Scholar] [CrossRef] - Orth, H.; Ackermann, H.; Otten, E. Fine and hyperfine structure of the 2
^{2}P term of^{7}Li; determination of the nuclear quadrupole moment. Z. Phys. A Atoms Nucl.**1975**, 273, 221–232. [Google Scholar] [CrossRef] - Desclaux, J. A multiconfiguration relativistic DIRAC-FOCK program. Comput. Phys. Commun.
**1975**, 9, 31–45. [Google Scholar] [CrossRef] - Indelicato, P. Projection operators in multiconfiguration Dirac-Fock calculations: Application to the ground state of heliumlike ions. Phys. Rev. A
**1995**, 51, 1132–1145. [Google Scholar] [CrossRef] - Boucard, S.; Indelicato, P. Relativistic Many-Body and Qed Effects on the Hyperfine Structure of Lithium-Like Ions. Eur. Phys. J. A
**2000**, 8, 59–73. [Google Scholar] [CrossRef] - Indelicato, P.; Lindroth, E.; Desclaux, J. Nonrelativistic Limit of Dirac-Fock Codes: The Role of Brillouin Configurations. Phys. Rev. Lett.
**2005**, 94, 013002. [Google Scholar] [CrossRef] [Green Version] - 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, 207–231. [Google Scholar] [CrossRef] - Dyall, K.G.; Grant, I.P.; Johnson, C.T.; Parpia, F.A.; Plummer, E.P. GRASP: A general-purpose relativistic atomic structure program. Comput. Phys. Commun.
**1989**, 55, 425–456. [Google Scholar] [CrossRef] - 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. Phys.
**2013**, 46, 085003. [Google Scholar] [CrossRef] - Bieroń, J.; Jönsson, P.; Froese Fischer, C. Large-scale multiconfiguration Dirac-Fock calculations of the hyperfine-structure constants of the 2s
^{2}S_{1/2}, 2p^{2}P_{1/2}, and 2p^{2}P_{3/2}states of lithium. Phys. Rev. A**1996**, 53, 2181–2188. [Google Scholar] [CrossRef] [PubMed] - Puchalski, M.; Pachucki, K. Ground State Hyperfine Splitting in
^{6,7}Li Atoms and the Nuclear Structure. Phys. Rev. Lett.**2013**, 111, 243001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - 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, 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]

**Figure 1.**Contraction of the correlation orbitals when going from valence–valence through core–valence to core–core correlation MCDHF calculations of $1{s}^{2}2{s}^{2}{\phantom{\rule{3.33333pt}{0ex}}}^{1}\phantom{\rule{-0.166667em}{0ex}}{S}_{0}$ state of Be. The two thick lines correspond to the $1s$ (red, no node) and $2s$ (blue, one node) orbitals. Other lines represent the radial distributions of the correlation orbitals with n up to 4.

**Figure 2.**Spin polarization of a closed subshell in the core due to the Coulomb interaction with an open valence subshell (cf. Figure 7 in Lindgren’s paper [26]).

**Figure 3.**

**Left**: orbitals from a calculation including CSFs describing polarization effects.

**Right**: orbitals from an energy-driven calculation, including CSFs obtained from SDT substitutions. The three thick lines correspond to the spectroscopic $1s$ (red, no node), $2s$ (blue, one node), and $2p$ (green, no node) orbitals.

**Figure 4.**Convergence of hyperfine constants (

**a**) $A(1{s}^{2}2s{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{S}_{1/2})$, (

**b**) $A(1{s}^{2}2p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2}^{\phantom{\rule{4pt}{0ex}}o})$, (

**c**) $A(1{s}^{2}2p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}^{\phantom{\rule{4pt}{0ex}}o})$ and (

**d**) $B(1{s}^{2}2p{\phantom{\rule{3.33333pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{3/2}^{\phantom{\rule{4pt}{0ex}}o})$ of ${}^{7}$Li (all in MHz) from the energy-driven layer-by-layer calculations (red squares), from the energy-driven fully variational calculations using the MCDFGME code (green circles), and from calculations with merged polarization orbitals (blue crosses). The difference in the convergence patterns between the energy-driven layer-by-layer GRASP calculations and the energy-driven fully variational calculations with MCDFGME is probably due to the fact that the former were performed by simultaneously optimizing the three states in the extended optimal level (EOL) approach, whereas in the latter case, each state was separately optimized (OL).

**Table 1.**Hyperfine interaction constants (in MHz) for the $1{s}^{2}2s{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{S}_{1/2}$ and $1{s}^{2}2p{\phantom{\rule{4pt}{0ex}}}^{2}\phantom{\rule{-0.166667em}{0ex}}{P}_{1/2,\phantom{\rule{0.166667em}{0ex}}3/2}^{\phantom{\rule{0.166667em}{0ex}}o}$ states in ${}^{7}$Li from weighted average DF calculations and weighted average MCDHF calculations including spin and orbital polarization effects with four layers of $s,d\text{-},d$ orbitals, i.e., orbitals set up to $\{6s,6d\text{-},6d\}$. The nuclear parameters for ${}^{7}$Li are $I=3/2$, ${\mu}_{I}=3.256424{\mu}_{N}$, and $Q=-0.0406$ barn.

$\mathit{A}({}^{2}\phantom{\rule{-0.166667em}{0ex}}{\mathit{S}}_{1/2})$ | $\mathit{A}({}^{2}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{1/2}^{\phantom{\rule{0.166667em}{0ex}}\mathit{o}})$ | $\mathit{A}({}^{2}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{3/2}^{\phantom{\rule{0.166667em}{0ex}}\mathit{o}})$ | $\mathit{B}({}^{2}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{3/2}^{\phantom{\rule{0.166667em}{0ex}}\mathit{o}})$ | |
---|---|---|---|---|

DF | 290.249 | 32.356 | −6.469 | $-0.2235$ |

$\{3s,3d\text{-},3d\}$ | 374.047 | 44.741 | $-4.910$ | $-0.2238$ |

$\{4s,4d\text{-},4d\}$ | 380.692 | 42.461 | $-2.619$ | $-0.1968$ |

$\{5s,5d\text{-},5d\}$ | 380.341 | 42.610 | $-2.766$ | $-0.1968$ |

$\{6s,6d\text{-},6d\}$ | 380.342 | 42.611 | $-2.766$ | $-0.1968$ |

Exp. | 401.752043 ${}^{a}$ | 45.914(25) ${}^{b}$ | −3.055(14) ${}^{b}$ |

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## Share and Cite

**MDPI and ACS Style**

Li, Y.; 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**, *11*, 4.
https://doi.org/10.3390/atoms11010004

**AMA Style**

Li Y, Jönsson P, Godefroid M, Gaigalas G, Bieroń J, Marques JP, Indelicato P, Chen C.
Independently Optimized Orbital Sets in GRASP—The Case of Hyperfine Structure in Li I. *Atoms*. 2023; 11(1):4.
https://doi.org/10.3390/atoms11010004

**Chicago/Turabian Style**

Li, Yanting, Per Jönsson, Michel Godefroid, Gediminas Gaigalas, Jacek Bieroń, José Pires Marques, Paul Indelicato, and Chongyang Chen.
2023. "Independently Optimized Orbital Sets in GRASP—The Case of Hyperfine Structure in Li I" *Atoms* 11, no. 1: 4.
https://doi.org/10.3390/atoms11010004