# Finite-Field Calculations of Transition Properties by the Fock Space Relativistic Coupled Cluster Method: Transitions between Different Fock Space Sectors

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Pilot Applications to Transition Dipole Moment Calculations

#### 3.1. Transition Dipoles for Excitations of Closed-Shell Atoms

#### 3.2. Transition Dipole Moment Functions in ${\mathrm{I}}_{2}$ and $\mathrm{TlF}$

## 4. Concluding Remarks

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Perturbative Analysis of the Effective Property Operator and Effective Hamiltonian Derivative

## References

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**Figure 1.**Goldstone diagrams representing components of the wave operator contributing to $P\mathsf{\Omega}P$. Double arrows correspond to active holes/particles, while double horizontal lines denote cluster amplitudes.

**Figure 2.**$X{0}_{g}^{+}-B{0}_{u}^{+}$ transition dipole moment function of the ${\mathrm{I}}_{2}$ molecule: bold red line, empirical function from Tellinghuisen [55]; solid and dashed blue lines: finite-field (${D}_{X-B}^{\mathrm{FF}}$) and model-state (${D}_{X-B}^{\mathrm{MS}}$) FS RCC estimates, respectively; green dot-dashed line, quasirelativistic multireference perturbation theory data [56]; dotted violet line, the dependence of the largest amplitude in ${T}_{2}^{\left(0h0p\right)}$ ($\mathrm{max}|{t}_{2}^{\left(0h0p\right)}|$) on the internuclear separation.

**Figure 3.**Transition dipole moment function for the $X{0}^{+}-B1$ electronic transition in the $\mathrm{TlF}$ molecule: red circle, estimate based on the experimental lifetime measurements [2]; solid and dashed blue lines, ${D}_{X-B}^{\mathrm{FF}}\left(R\right)$ and ${D}_{X-B}^{\mathrm{MS}}\left(R\right)$ functions from the present FS RCC calculations; green dot-dashed line, MRCI + SO calculations [57].

**Table 1.**Transition dipole moments (a.u.) for excitations from the ground $5{p}^{6}{\phantom{\rule{0.277778em}{0ex}}}^{1}{S}_{0}$ state of the Xe atom according to the finite-field FS RCC calculations (${D}_{ij}^{\mathrm{FF}}$) and the corresponding dipole matrix elements between the model space vectors (${D}_{ij}^{\mathrm{MS}}$).

Excited State | Transition Wavenumber | ${\mathit{D}}_{\mathit{ij}}^{\mathbf{MS}}$ | ${\mathit{D}}_{\mathit{ij}}^{\mathbf{FF}}$ | ${\mathit{D}}_{\mathit{ij}}^{\mathbf{exptl}}$ | ||||
---|---|---|---|---|---|---|---|---|

FS RCC (Xe${}^{0}$) | FS RCC (Xe${}^{+}$) | Exptl. [49] | (Xe${}^{0}$) | (Xe${}^{+}$) | (Xe${}^{0}$) | (Xe${}^{+}$) | [49] | |

$5{p}^{5}{(}^{2}{P}_{3/2}^{\phantom{\rule{0.166667em}{0ex}}\mathrm{o}})6s\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.277778em}{0ex}}}^{2}{[3/2]}^{\mathrm{o}}$ | 68,147 | 67,230 | 68,045 | 0.637 | 0.494 | 0.634 | 0.630 | 0.654 |

$5{p}^{5}{(}^{2}{P}_{1/2}^{\phantom{\rule{0.166667em}{0ex}}\mathrm{o}})6s\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.277778em}{0ex}}}^{2}{[1/2]}^{\mathrm{o}}$ | 77,201 | 76,153 | 77,185 | 0.536 | 0.430 | 0.512 | 0.518 | 0.521 |

$5{p}^{5}{(}^{2}{P}_{3/2}^{\phantom{\rule{0.166667em}{0ex}}\mathrm{o}})5d\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.277778em}{0ex}}}^{2}{[1/2]}_{1}^{\mathrm{o}}$ | 80,259 | 79,306 | 79,987 | 0.074 | 0.050 | 0.101 | 0.090 | 0.120 |

$5{p}^{5}{(}^{2}{P}_{3/2}^{\phantom{\rule{0.166667em}{0ex}}\mathrm{o}})5d\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.277778em}{0ex}}}^{2}{[3/2]}_{1}^{\mathrm{o}}$ | 84,137 | 83,275 | 83,889 | 0.837 | 0.659 | 0.663 | 0.668 | 0.704 |

**Table 2.**Transition dipole moments (a.u.) for excitations from the ground $6{s}^{2}{\phantom{\rule{0.277778em}{0ex}}}^{1}{S}_{0}$ state of the Hg atom according to the finite-field FS RCC calculations (${D}_{ij}^{\mathrm{FF}}$) and the corresponding dipole matrix elements between the model space vectors (${D}_{ij}^{\mathrm{MS}}$).

Excited State | Transition Wavenumber | ${\mathit{D}}_{\mathit{ij}}^{\mathbf{MS}}$ | ${\mathit{D}}_{\mathit{ij}}^{\mathbf{FF}}$ | ${\mathit{D}}_{\mathit{ij}}^{\mathbf{exptl}}$ | ||||
---|---|---|---|---|---|---|---|---|

FS RCC (Hg${}^{0}$) | FS RCC (Hg${}^{+}$) | Exptl. [49] | (Hg${}^{0}$) | (Hg${}^{+}$) | (Hg${}^{0}$) | (Hg${}^{+}$) | [49] | |

$5{d}^{10}{(}^{1}S)6s6p\phantom{\rule{4pt}{0ex}}{}^{3}{P}_{1}^{\mathrm{o}}$ | 38,973 | 38,712 | 39,412 | 0.355 | 0.331 | 0.211 | 0.216 | 0.250 |

$5{d}^{10}{(}^{1}S)6s6p\phantom{\rule{4pt}{0ex}}{}^{1}{P}_{1}^{\mathrm{o}}$ | 54,284 | 53,813 | 54,069 | 2.009 | 1.655 | 1.383 | 1.375 | 1.527 |

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Zaitsevskii, A.; Oleynichenko, A.V.; Eliav, E.
Finite-Field Calculations of Transition Properties by the Fock Space Relativistic Coupled Cluster Method: Transitions between Different Fock Space Sectors. *Symmetry* **2020**, *12*, 1845.
https://doi.org/10.3390/sym12111845

**AMA Style**

Zaitsevskii A, Oleynichenko AV, Eliav E.
Finite-Field Calculations of Transition Properties by the Fock Space Relativistic Coupled Cluster Method: Transitions between Different Fock Space Sectors. *Symmetry*. 2020; 12(11):1845.
https://doi.org/10.3390/sym12111845

**Chicago/Turabian Style**

Zaitsevskii, Andréi, Alexander V. Oleynichenko, and Ephraim Eliav.
2020. "Finite-Field Calculations of Transition Properties by the Fock Space Relativistic Coupled Cluster Method: Transitions between Different Fock Space Sectors" *Symmetry* 12, no. 11: 1845.
https://doi.org/10.3390/sym12111845