# Coulomb (Velocity) Gauge Recommended in Multiconfiguration Calculations of Transition Data Involving Rydberg Series

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. MultiConfiguration Calculations

#### 2.2. Transition Parameters

## 3. Computational Methodology—Optimization of the Orbital Basis

#### 3.1. C IV

#### 3.2. C III

## 4. Results

## 5. Discussion

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The uncertainty $dT$ of the computed transition rates for transitions between the $2p{\phantom{\rule{3.33333pt}{0ex}}}^{2}{P}_{1/2}^{\circ}$ state and Rydberg $ns{\phantom{\rule{3.33333pt}{0ex}}}^{2}{S}_{1/2}$ states of increasing principal quantum number n in C IV. The black squares and magenta diamonds, respectively, correspond to the results from the conventional and the alternative strategies for optimizing the radial orbitals. (

**b**) Same as the first panel, but for transitions between the $8s{\phantom{\rule{3.33333pt}{0ex}}}^{2}{S}_{1/2}$ state and successive Rydberg $np{\phantom{\rule{3.33333pt}{0ex}}}^{2}{P}_{1/2}^{\circ}$ states in C IV.

**Figure 2.**(

**a**) The uncertainty $dT$ of the computed transition rates for transitions between the $2{s}^{2}{\phantom{\rule{3.33333pt}{0ex}}}^{1}{S}_{0}$ state and Rydberg $2snp{\phantom{\rule{3.33333pt}{0ex}}}^{1}{P}_{1}^{\circ}$ states of increasing principal quantum number n in C III. The black squares and magenta diamonds, respectively, correspond to the results from the conventional and alternative strategies for optimizing the radial orbitals. (

**b**) Same as the first panel, but for transitions between the $2s7p{\phantom{\rule{3.33333pt}{0ex}}}^{1}{P}_{1}^{\circ}$ state and successive Rydberg $2sns{\phantom{\rule{3.33333pt}{0ex}}}^{1}{S}_{0}$ states in C III.

**Figure 3.**(

**a**) The transition rates A in the Babushkin (circles) and the Coulomb (triangles) gauges for the $2p{\phantom{\rule{3.33333pt}{0ex}}}^{2}{P}_{1/2}^{\circ}-8s{\phantom{\rule{3.33333pt}{0ex}}}^{2}{S}_{1/2}$ transition in C IV, as a function of the increasing number of correlation layers. The transition rates computed in the conventional and the alternative manner are respectively shown in black and magenta. (

**b**) Same as the first panel, but for the $2{s}^{2}{\phantom{\rule{3.33333pt}{0ex}}}^{1}{S}_{0}-2s7p{\phantom{\rule{3.33333pt}{0ex}}}^{1}{P}_{1}^{\circ}$ transition in C III.

**Figure 4.**(

**a**) The transition rates A in the Babushkin and the Coulomb gauges for the $2p{\phantom{\rule{3.33333pt}{0ex}}}^{2}{P}_{1/2}^{\circ}-8s{\phantom{\rule{3.33333pt}{0ex}}}^{2}{S}_{1/2}$ transition in C IV, as a function of the square root of the upper integration bound R in the radial transition integrals (11) and (12) involving correlation orbitals. The radial transition integrals involving spectroscopic orbitals extend to their full values, so that $R=0$ corresponds to transition rates computed from wave functions exclusively built from spectroscopic orbitals. The wave functions are produced by the alternative computational strategy. (

**b**) The spectroscopic $2p$ and $8s$ radial orbitals in C IV as a function of $\sqrt{r}$. The two orbitals occupy different regions in space and their overlap is minor. The $8s$ orbital extends far out from the atomic core.

**Figure 5.**(

**a**) Same as Figure 4a, but for the $7p{\phantom{\rule{3.33333pt}{0ex}}}^{2}{P}_{1/2}^{\circ}-8s{\phantom{\rule{3.33333pt}{0ex}}}^{2}{S}_{1/2}$ transition in C IV. (

**b**) The spectroscopic $7p$ and $8s$ radial orbitals in C IV as functions of $\sqrt{r}$. Both orbitals occupy nearly the same regions in space, overlapping to a great extent.

**Table 1.**The mean radii $\langle r\rangle $ (a.u.) of the spectroscopic and correlation orbitals that belong to the s and p symmetries in C IV. The correlation orbitals result from two different optimization schemes, the conventional and the alternative, and they occupy different regions in space.

Spectroscopic | Correlation | Spectroscopic | Correlation | ||||||
---|---|---|---|---|---|---|---|---|---|

Conventional | Alternative | Conventional | Alternative | ||||||

$1s$ | 0.27 | $9s$ | 0.51 | 1.12 | $9p$ | 0.44 | 0.87 | ||

$2s$ | 1.31 | $10s$ | 0.43 | 0.92 | $2p$ | 1.28 | $10p$ | 0.41 | 1.03 |

$3s$ | 3.00 | $11s$ | 0.42 | 0.84 | $3p$ | 2.95 | $11p$ | 0.40 | 1.00 |

$4s$ | 5.55 | $12s$ | 0.46 | 0.87 | $4p$ | 5.64 | $12p$ | 0.45 | 1.18 |

$5s$ | 8.81 | $13s$ | 0.56 | 0.87 | $5p$ | 8.99 | $13p$ | 0.48 | 2.59 |

$6s$ | 12.82 | $14s$ | 0.40 | 1.26 | $6p$ | 13.10 | $14p$ | 0.77 | 5.94 |

$7s$ | 17.58 | $7p$ | 17.94 | ||||||

$8s$ | 23.09 | $8p$ | 23.54 |

**Table 2.**Same as Table 1, but for radial orbitals in C III. The correlation orbitals $8s$ and $8p$, which are introduced to account for the $LS$-term dependencies, are the same in both optimization schemes and fairly diffuse in comparison with the rest of the correlation orbitals.

Spectroscopic | Correlation | Spectroscopic | Correlation | ||||||
---|---|---|---|---|---|---|---|---|---|

Conventional | Alternative | Conventional | Alternative | ||||||

$1s$ | 0.26 | $9s$ | 1.05 | 4.86 | $9p$ | 1.00 | 3.56 | ||

$2s$ | 1.28 | $10s$ | 1.48 | 3.67 | $2p$ | 1.23 | $10p$ | 1.24 | 3.37 |

$3s$ | 3.57 | $11s$ | 1.88 | 3.25 | $3p$ | 3.74 | $11p$ | 1.56 | 3.37 |

$4s$ | 6.63 | $12s$ | 1.87 | 8.40 | $4p$ | 7.04 | $12p$ | 1.52 | 9.04 |

$5s$ | 10.80 | $5p$ | 11.37 | ||||||

$6s$ | 15.95 | $6p$ | 16.71 | ||||||

$7s$ | 22.10 | $7p$ | 23.04 | ||||||

term corr. | term corr. | ||||||||

$8s$ | 8.22 | $8p$ | 5.55 |

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**MDPI and ACS Style**

Papoulia, A.; Ekman, J.; Gaigalas, G.; Godefroid, M.; Gustafsson, S.; Hartman, H.; Li, W.; Radžiūtė, L.; Rynkun, P.; Schiffmann, S.; Wang, K.; Jönsson, P. Coulomb (Velocity) Gauge Recommended in Multiconfiguration Calculations of Transition Data Involving Rydberg Series. *Atoms* **2019**, *7*, 106.
https://doi.org/10.3390/atoms7040106

**AMA Style**

Papoulia A, Ekman J, Gaigalas G, Godefroid M, Gustafsson S, Hartman H, Li W, Radžiūtė L, Rynkun P, Schiffmann S, Wang K, Jönsson P. Coulomb (Velocity) Gauge Recommended in Multiconfiguration Calculations of Transition Data Involving Rydberg Series. *Atoms*. 2019; 7(4):106.
https://doi.org/10.3390/atoms7040106

**Chicago/Turabian Style**

Papoulia, Asimina, Jörgen Ekman, Gediminas Gaigalas, Michel Godefroid, Stefan Gustafsson, Henrik Hartman, Wenxian Li, Laima Radžiūtė, Pavel Rynkun, Sacha Schiffmann, Kai Wang, and Per Jönsson. 2019. "Coulomb (Velocity) Gauge Recommended in Multiconfiguration Calculations of Transition Data Involving Rydberg Series" *Atoms* 7, no. 4: 106.
https://doi.org/10.3390/atoms7040106