# Calculation of Polarizabilities for Atoms with Open Shells

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Polarizabilities

#### 2.2. The RPA Method

#### 2.3. The CIPT Method

## 3. Results and Discussions

#### 3.1. Ytterbium

#### 3.2. Erbium and Thulium

#### 3.3. Further Developments and Applications

## Funding

## Conflicts of Interest

## References

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${}^{1}$S${}_{0}$ | ${}^{3}$P${}_{0}^{\mathbf{o}}$ | Source |
---|---|---|

143 | 340 | this work |

141(6) | 302(14) | Ref. [21] |

139(6) | Ref. [1], Recommended value | |

From 134.4(1.0) to 144.2(1.0) | From 280.1(1.0) to 289.9(1.0) | Ref. [22], Experimental values |

**Table 2.**Energies of the lowest states of Er and Tm (cm${}^{-1}$) which contribute the most to the polarizabilities. Er states belong to the $4{f}^{12}6s6p$ configuration while Tm states belong to the $4{f}^{13}6s6p$ configuration. ${J}_{n}={J}_{a},{J}_{a}\pm 1$, where ${J}_{a}$ the the total angular momentum of the clock states; ${J}_{a}=6,4$ for Er and ${J}_{a}=7/2,5/2$ for Tm.

Er | Tm | ||||
---|---|---|---|---|---|

${\mathit{J}}_{\mathit{n}}$ | NIST | CIPT | ${\mathit{J}}_{\mathit{n}}$ | NIST | CIPT |

6 | 0 | 0 | 7/2 | 0 | |

4 | 6958 | 6370 | 5/2 | 8771 | 8350 |

3 | 22,269 | 23,280 | 3/2 | 19,132 | 19,574 |

4 | 18,816 | 18,595 | 5/2 | 17,752 | 18,336 |

5 | 17,347 | 16,581 | 7/2 | 16,742 | 15,799 |

6 | 17,073 | 15,500 | 9/2 | 17,613 | 18,195 |

7 | 17,157 | 16,315 |

**Table 3.**Static scalar and tensor polarizabilities of the ground and clock states of Er and Tm (in a.u.).

${}^{3}$H${}_{6}$ | ${}^{3}$F${}_{4}$ | ${}^{2}$F${}_{7/2}^{\mathit{o}}$ | ${}^{2}$F${}_{5/2}^{\mathit{o}}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{J}}_{\mathit{n}}$ | ${\mathit{\alpha}}_{\mathbf{0}}$ | ${\mathit{\alpha}}_{\mathbf{2}}$ | ${\mathit{J}}_{\mathit{n}}$ | ${\mathit{\alpha}}_{\mathbf{0}}$ | ${\mathit{\alpha}}_{\mathbf{2}}$ | ${\mathit{J}}_{\mathit{n}}$ | ${\mathit{\alpha}}_{\mathbf{0}}$ | ${\mathit{\alpha}}_{\mathbf{2}}$ | ${\mathit{J}}_{\mathit{n}}$ | ${\mathit{\alpha}}_{\mathbf{0}}$ | ${\mathit{\alpha}}_{\mathbf{2}}$ |

5 | 47.10 | − 47.10 | 3 | 42.73 | −42.73 | 5/2 | 38.37 | −38.37 | 3/2 | 34.20 | −34.20 |

6 | 55.28 | 86.88 | 4 | 55.02 | 77.03 | 7/2 | 50.77 | 67.70 | 5/2 | 50.86 | 58.12 |

7 | 64.29 | −40.41 | 5 | 67.23 | −34.23 | 9/2 | 63.87 | −29.81 | 7/2 | 68.22 | −24.36 |

Total | 166.67 | −0.63 | 164.98 | 0.07 | 153.02 | −0.48 | 153.28 | −0.45 | |||

Ref. [9] | 150.2 | 150.2 | 144.3 | 144.3 | |||||||

Ref. [1] | 150(10) | 144(15) |

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Dzuba, V.
Calculation of Polarizabilities for Atoms with Open Shells. *Symmetry* **2020**, *12*, 1950.
https://doi.org/10.3390/sym12121950

**AMA Style**

Dzuba V.
Calculation of Polarizabilities for Atoms with Open Shells. *Symmetry*. 2020; 12(12):1950.
https://doi.org/10.3390/sym12121950

**Chicago/Turabian Style**

Dzuba, Vladimir.
2020. "Calculation of Polarizabilities for Atoms with Open Shells" *Symmetry* 12, no. 12: 1950.
https://doi.org/10.3390/sym12121950