#
Electronic Structure of Lr^{+} (Z = 103) from Ab Initio Calculations

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Method

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Specification of the generalized active space (GAS) scheme used for the calculations for the Lu${}^{+}$ and Lr${}^{+}$ ions (see the text for details).

GAS | Accumulated Electrons | Number of Spinors | Characters ${}^{\mathit{a}}$ | |
---|---|---|---|---|

Min ${}^{\mathit{b}}$ | Max | |||

1 | 10 − x | 10 | 10 | (n − 2)d |

2 | 18 − y | 18 | 8 | (n − 1)s, (n − 1)p |

3 | 32 − z | 32 | 14 | (n − 2)f |

4 | 32 | 34 | 18 | ns, (n − 1)d, np |

5 | 34 | 34 | (<30 au) ${}^{c}$ | Virtual |

^{a}For Lu

^{+}and Lr

^{+}, n = 6 and 7, respectively.

^{b}x, y and z are variables that control the electron excitation process attributed to the selective GAS. In the calculations, we defined the following: x = 0, y = 2 and z = 1, in line with the previous presentation [31].

^{c}This includes all the virtual spinors up to an energy of 30 atomic units.

**Table 2.**Calculated energies (in cm${}^{-1}$) of the ground and the low-lying excited states of the Lu${}^{+}$ ions obtained from the MRCI model using the double- (2), triple- (3) and quadruple-zeta (4) basis sets and the energy values derived at the complete basis set limit (∞), together with the final energy values (Final) that take into consideration the energy corrections obtained for the Breit (${\Delta}_{B}$) and QED (${\Delta}_{B+QED}$) contributions, compared with the experimental data (Exp.) and the FSCC results.

Levels | MRCI | Corrections | Final | Reference | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Config. | State | J | (2) | (3) | (4) | (∞) | ${\Delta}_{\mathit{B}}$ | ${\Delta}_{\mathit{B}+\mathit{Q}\mathit{E}\mathit{D}}$ | Exp. ${}^{\mathit{a}}$ | FSCC ${}^{\mathit{b}}$ | |

6s${}^{2}$ | ${}^{1}$S | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

5d${}^{1}$6s${}^{1}$ | ${}^{3}$D | 1 | 12,227 | 12,213 | 12,172 | 12,145 | 92 | −104 | 12,041 (41) | 11,796 | 12,354 |

2 | 12,698 | 12,669 | 12,626 | 12,598 | 91 | −88 | 12,510 (43) | 12,435 | 12,985 | ||

3 | 13,946 | 13,907 | 13,866 | 13,838 | 88 | −24 | 13,814 (41) | 14,199 | 14,702 | ||

${}^{1}$D | 2 | 16,817 | 16,656 | 16,583 | 16,535 | 98 | −44 | 16,491 (73) | 17,333 | 17,892 | |

6s${}^{1}$6p${}^{1}$ | ${}^{3}$P | 0 | 27,712 | 28,004 | 28,462 | 28,752 | 63 | −88 | 28,664 (456) | 27,264 | 27,091 |

1 | 28,886 | 29,208 | 29,646 | 29,923 | 64 | −77 | 29,846 (438) | 28,503 | 28,440 | ||

2 | 32,650 | 33,127 | 33,599 | 33,899 | 60 | −36 | 33,863 (472) | 32,453 | 32,294 | ||

${}^{1}$P | 1 | 38,071 | 38,402 | 38,453 | 38,484 | 101 | −51 | 38,433 (51) | 38,223 | 38,464 |

**Table 3.**Calculated energies (in cm${}^{-1}$) of the ground and the low-lying excited states of the Lr${}^{+}$ ion obtained from the MRCI model using the double- (2), triple (3) and quadruple-zeta (4) basis sets, and the energy values derived at the complete basis set limit (∞), together with the final energy values (Final) that take into consideration the energy corrections obtained for the Breit (${\Delta}_{B}$) and QED (${\Delta}_{B+QED}$) contributions, compared with the FSCC results.

Levels | MRCI | Corrections | Final | Reference | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Config. | State | J | (2) | (3) | (4) | (∞) | ${\Delta}_{\mathit{B}}$ | ${\Delta}_{\mathit{B}+\mathit{Q}\mathit{E}\mathit{D}}$ | FSCC ${}^{\mathit{a}}$ | |

7s${}^{2}$ | ${}^{1}$S | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

6d${}^{1}$7s${}^{1}$ | ${}^{3}$D | 1 | 21,796 | 21,768 | 21,696 | 21,649 | 219 | −86 | 21,563 (72) | 20,265 |

2 | 22,494 | 22,459 | 22,375 | 22,320 | 218 | −61 | 22,259 (84) | 21,623 | ||

3 | 24,761 | 24,723 | 24,633 | 24,574 | 211 | 56 | 24,630 (90) | 26,210 | ||

${}^{1}$D | 2 | 28,883 | 28,721 | 28,570 | 28,472 | 230 | 32 | 28,504 (151) | 31,200 | |

7s${}^{1}$7p${}^{1}$ | ${}^{3}$P | 0 | 29,825 | 30,072 | 31,006 | 31,600 | 144 | −81 | 31,519 (934) | 29,487 |

1 | 32,114 | 32,360 | 33,222 | 33,770 | 150 | −60 | 33,710 (862) | 31,610 | ||

2 | 43,428 | 43,809 | 44,783 | 45,402 | 152 | 49 | 45,451 (974) | 43,513 | ||

${}^{1}$P | 1 | 47,908 | 48,135 | 48,794 | 49,212 | 205 | 33 | 49,245 (659) | 47,819 |

^{a}Taken from [23].

**Table 4.**Calculated Einstein coefficients ${A}_{E1}$ (in 1/s) and branching ratios $\beta $ for the electric dipole’s allowed transitions in Lu${}^{+}$, obtained from the MRCI transition dipole moment matrix and the $\Delta E$ (in cm${}^{-1}$) from the complete basis set limit, compared with the reference experimental values.

Levels | MRCI | Reference ${}^{\mathit{a}}$ | |||
---|---|---|---|---|---|

Upper | Lower | $\Delta \mathit{E}$ | ${\mathit{A}}_{\mathit{E}\mathbf{1}}$ | $\mathbf{\beta}$ | ${\mathit{A}}_{\mathit{E}\mathbf{1}}$ (NIST) |

${}^{3}$P${}_{1}$ (6s${}^{1}$6p${}^{1}$) | ${}^{1}$S${}_{0}$ (6s${}^{2}$) | 29,924 | 6.10 × 10${}^{6}$ | 0.08 | 1.25 × 10${}^{7}$ |

${}^{1}$P${}_{1}$ (6s${}^{1}$6p${}^{1}$) | 38,474 | 3.74 × 10${}^{8}$ | 0.90 | 4.53 × 10${}^{8}$ | |

${}^{3}$P${}_{0}$ (6s${}^{1}$6p${}^{1}$) | ${}^{3}$D${}_{1}$ (5d${}^{1}$6s${}^{1}$) | 16,609 | 4.38 × 10${}^{7}$ | 1.00 | |

${}^{3}$P${}_{1}$ (6s${}^{1}$6p${}^{1}$) | 17,779 | 1.39 × 10${}^{7}$ | 0.19 | ||

${}^{3}$P${}_{2}$ (6s${}^{1}$6p${}^{1}$) | 21,750 | 1.14 × 10${}^{6}$ | <0.01 | ||

${}^{1}$P${}_{1}$ (6s${}^{1}$6p${}^{1}$) | 26,329 | 4.00 × 10${}^{5}$ | <0.01 | ||

${}^{3}$P${}_{1}$ (6s${}^{1}$6p${}^{1}$) | ${}^{3}$D${}_{2}$ (5d${}^{1}$6s${}^{1}$) | 17,326 | 5.36 × 10${}^{7}$ | 0.72 | 9.90 × 10${}^{6}$ |

${}^{3}$P${}_{2}$ (6s${}^{1}$6p${}^{1}$) | 21,297 | 2.06 × 10${}^{7}$ | 0.16 | ||

${}^{1}$P${}_{1}$ (6s${}^{1}$6p${}^{1}$) | 25,876 | 3.48 × 10${}^{7}$ | 0.08 | ||

${}^{3}$P${}_{2}$ (6s${}^{1}$6p${}^{1}$) | ${}^{3}$D${}_{3}$ (5d${}^{1}$6s${}^{1}$) | 20,058 | 1.09 × 10${}^{8}$ | 0.82 | |

${}^{3}$P${}_{1}$ (6s${}^{1}$6p${}^{1}$) | ${}^{1}$D${}_{2}$ (5d${}^{1}$6s${}^{1}$) | 13,385 | 1.04 × 10${}^{6}$ | 0.01 | |

${}^{3}$P${}_{2}$ (6s${}^{1}$6p${}^{1}$) | 17,356 | 1.82 × 10${}^{6}$ | 0.01 | ||

${}^{1}$P${}_{1}$ (6s${}^{1}$6p${}^{1}$) | 21,935 | 7.95 × 10${}^{6}$ | 0.01 |

**Table 5.**Calculated Einstein coefficients ${A}_{E1}$ (in 1/s) and branching ratios $\beta $ for the electric dipole’s allowed transitions in Lr${}^{+}$, obtained from the MRCI transition dipole moment matrix and the $\Delta E$ (in cm${}^{-1}$) from the complete basis set limit, compared with the reference theoretical data.

Levels | MRCI | Reference ${}^{\mathit{a}}$ | |||
---|---|---|---|---|---|

Upper | Lower | $\Delta \mathit{E}$ | ${\mathit{A}}_{\mathit{E}\mathbf{1}}$ | $\mathbf{\beta}$ | ${\mathit{A}}_{\mathit{E}\mathbf{1}}$(CI+MBPT) |

${}^{3}$P${}_{1}$ (7s${}^{1}$7p${}^{1}$) | ${}^{1}$S${}_{0}$ (7s${}^{2}$) | 33,783 | 2.97 × 10${}^{7}$ | 0.49 | 6.36 × 10${}^{7}$ |

${}^{1}$P${}_{1}$ (7s${}^{1}$7p${}^{1}$) | 49,221 | 7.93 × 10${}^{8}$ | 0.87 | 8.34 × 10${}^{8}$ | |

${}^{3}$P${}_{0}$ (7s${}^{1}$7p${}^{1}$) | ${}^{3}$D${}_{1}$ (6d${}^{1}$7s${}^{1}$) | 9966 | 1.54 × 10${}^{7}$ | 1.00 | 5.44 × 10${}^{6}$ |

${}^{3}$P${}_{1}$ (7s${}^{1}$7p${}^{1}$) | 12,134 | 6.91 × 10${}^{6}$ | 0.11 | 2.42 × 10${}^{6}$ | |

${}^{3}$P${}_{2}$ (7s${}^{1}$7p${}^{1}$) | 23,764 | 2.44 × 10${}^{6}$ | <0.01 | 9.41 × 10${}^{5}$ | |

${}^{1}$P${}_{1}$ (7s${}^{1}$7p${}^{1}$) | 27,572 | 1.07 × 10${}^{6}$ | <0.01 | 1.36 × 10${}^{6}$ | |

${}^{3}$P${}_{1}$ (7s${}^{1}$7p${}^{1}$) | ${}^{3}$D${}_{2}$ (6d${}^{1}$7s${}^{1}$) | 11,463 | 2.38 × 10${}^{7}$ | 0.39 | 4.66 × 10${}^{6}$ |

${}^{3}$P${}_{2}$ (7s${}^{1}$7p${}^{1}$) | 23,093 | 4.03 × 10${}^{7}$ | 0.17 | 9.70 × 10${}^{6}$ | |

${}^{1}$P${}_{1}$ (7s${}^{1}$7p${}^{1}$) | 26,901 | 4.98 × 10${}^{7}$ | 0.06 | 1.63 × 10${}^{7}$ | |

${}^{3}$P${}_{2}$ (7s${}^{1}$7p${}^{1}$) | ${}^{3}$D${}_{3}$ (6d${}^{1}$7s${}^{1}$) | 20,839 | 1.93 × 10${}^{8}$ | 0.81 | 3.43 × 10${}^{7}$ |

${}^{3}$P${}_{1}$ (7s${}^{1}$7p${}^{1}$) | ${}^{1}$D${}_{2}$ (6d${}^{1}$7s${}^{1}$) | 5307 | 2.51 × 10${}^{4}$ | <0.01 | |

${}^{3}$P${}_{2}$ (7s${}^{1}$7p${}^{1}$) | 16,937 | 2.68 × 10${}^{6}$ | 0.01 | 3.19 × 10${}^{5}$ | |

${}^{1}$P${}_{1}$ (7s${}^{1}$7p${}^{1}$) | 20,745 | 6.60 × 10${}^{7}$ | 0.07 | 1.68 × 10${}^{7}$ |

^{a}Theoretical values obtained by using CI plus many-body perturbation theory (MBPT) in [23].

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**MDPI and ACS Style**

Ramanantoanina, H.; Borschevsky, A.; Block, M.; Laatiaoui, M.
Electronic Structure of Lr^{+} (*Z* = 103) from Ab Initio Calculations. *Atoms* **2022**, *10*, 48.
https://doi.org/10.3390/atoms10020048

**AMA Style**

Ramanantoanina H, Borschevsky A, Block M, Laatiaoui M.
Electronic Structure of Lr^{+} (*Z* = 103) from Ab Initio Calculations. *Atoms*. 2022; 10(2):48.
https://doi.org/10.3390/atoms10020048

**Chicago/Turabian Style**

Ramanantoanina, Harry, Anastasia Borschevsky, Michael Block, and Mustapha Laatiaoui.
2022. "Electronic Structure of Lr^{+} (*Z* = 103) from Ab Initio Calculations" *Atoms* 10, no. 2: 48.
https://doi.org/10.3390/atoms10020048