# Relativistic Fock Space Coupled Cluster Method for Many-Electron Systems: Non-Perturbative Account for Connected Triple Excitations

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Pilot Applications

#### 3.1. Atomic Energy Levels of Thallium and Lead

#### 3.2. Electronic States of TlH

#### 3.3. Static Dipole Polarizability of Lead

## 4. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Diagrammatic representation of cluster operators in the (0h,0p), (0h,1p) and (0h,2p) Fock space sectors at the CCSDT level of theory. Double arrows refer to active (valence) lines. Cluster amplitudes with all valence indices are excluded for the ${T}_{1}^{(0h,1p)}$ and ${T}_{2}^{(0h,2p)}$ diagrams to ensure these operators to be open regarding model space.

**Figure 2.**Diagrams of the most time-consuming terms in the CCSDT amplitudes equations for the (0h,0p), (0h,1p) and (0h,2p) Fock space sectors. Time complexities of the corresponding tensor contractions are given, N is the total number of spinors, A stands for the number of active (valence) spinors.

**Figure 3.**Leading terms in the right-hand side of amplitude Equation (4) for the $\left(0h1p\right)$ and $\left(0h2p\right)$ sectors used to estimate the triple amplitudes within the FS-CCSDT-1 approximation.

**Figure 4.**Folded diagrams contributing to the equations for triple amplitudes within the FS-CCSDT approximation in the $(0h,1p)$ (

**a**) and $(0h,2p)$ (

**b**–

**e**) Fock space sectors. Empty rectangular blocks correspond to the effective interactions, $\tilde{H}-{H}_{0}$.

**Table 1.**Deviations of the calculated ionization potentials (IP) and excitation energies (EE) of neutral thallium and lead and lead cation (cm${}^{-1}$) from the experimental values. FS-RCCSD/LB+T/SB stands for the combined scheme (8).

State | Exptl | IH-FS- | FS- | FS-RCCSD/LB + T/SB | ||||||
---|---|---|---|---|---|---|---|---|---|---|

[84] | RCCSD [47] | RCCSD/LB | SDT-1 | SDT-1’ | SDT-2 | SDT-3 | SDT | |||

Tl, ground state$6{s}^{2}6p$ ${}^{2}{P}_{1/2}$ | ||||||||||

IP | 49,266 | −56 | −38 | −38 | −204 | −151 | −32 | |||

EE | $6{s}^{2}6p$ | ${}^{2}{P}_{3/2}$ | 7793 | −112 | 23 | 23 | 1 | 9 | −31 | |

Pb${}^{+}$, ground state $6{s}^{2}6p$ ${}^{2}{P}_{1/2}$ | ||||||||||

IP | 121,245 | −168 | −143 | −28 | −28 | −190 | −158 | −59 | ||

EE | $6{s}^{2}6p$ | ${}^{2}{P}_{3/2}$ | 14,081 | −196 | −136 | 25 | 25 | 12 | 14 | −42 |

Pb, ground state $6{s}^{2}6{p}^{2}$ ${}^{3}{P}_{0}$ | ||||||||||

IP | 59,819 | −543 | 364 | −44 | −285 | −347 | −336 | 7 | ||

EE | $6{s}^{2}6{p}^{2}$ | ${}^{3}{P}_{1}$ | 7819 | −288 | −302 | 76 | 5 | −4 | −3 | −28 |

${}^{3}{P}_{2}$ | 10,650 | −343 | −235 | 130 | 129 | 97 | 102 | 13 | ||

${}^{1}{D}_{2}$ | 21,458 | −605 | −394 | 215 | 203 | 158 | 167 | 5 | ||

${}^{1}{S}_{0}$ | 29,467 | −208 | 414 | 170 | 248 | 293 | 302 | 173 |

**Table 2.**Equilibrium internuclear distances ${r}_{e}$, vibrational constants ${\omega}_{e}$ and term energies ${T}_{e}$ for the $X{0}^{+}$ and $A{0}^{+}$ states of TlH.

$\mathit{X}{0}^{+}$ | $\mathit{A}{0}^{+}$ | |||||
---|---|---|---|---|---|---|

${\mathit{r}}_{\mathit{e}}$, Å | ${\mathit{\omega}}_{\mathit{e}}$, cm${}^{-\mathbf{1}}$ | ${\mathit{T}}_{\mathit{e}}$, cm${}^{-\mathbf{1}}$ | ${\mathit{r}}_{\mathit{e}}$, Å | ${\mathit{\omega}}_{\mathit{e}}$, cm${}^{-\mathbf{1}}$ | ${\mathit{T}}_{\mathit{e}}$, cm${}^{-\mathbf{1}}$ | |

FS-RCCSD/LB | 1.775 | 1800 | 0 | 1.749 | 1572 | 15,914 |

FS-RCCSD/LB + SDT-1/SB | 1.862 | 1378 | 0 | 1.812 | 1222 | 17,899 |

FS-RCCSD/LB + SDT/SB | 1.840 | 1500 | 0 | 1.801 | 1302 | 17,501 |

RKR($X{0}^{+}$) + FS-RCCSD/LB | 1.909 | 1018 | 15,948 | |||

RKR($X{0}^{+}$) + FS-RCCSD/LB + SDT-1/SB | 1.829 | 1218 | 17,931 | |||

RKR($X{0}^{+}$) + FS-RCCSD/LB + SDT/SB | 1.846 | 1138 | 17,567 | |||

Exptl. [86,87,90] | 1.873 | 1391 | 0 | 1.840 | 1043 | 17,723 |

State | FS- | FS-RCCSD/LB + T/SB | ||||||
---|---|---|---|---|---|---|---|---|

RCCSD/LB | SDT-1 | SDT-1’ | SDT-2 | SDT-3 | SDT | Experiment | ||

Pb${}^{2+}$ | $6{s}^{2}$${}^{1}{S}_{0}$ | 13.8 | 13.4 | 13.4 | 13.4 | 13.8 | 13.7 | 13.62(8) [94] 13.38(2) [95] |

Pb${}^{+}$ | $6{s}^{2}6{p}^{1}$${}^{2}{P}_{1/2}$ | 22.7 | 22.5 | 22.5 | 22.6 | 22.8 | 22.7 | |

Pb | $6{s}^{2}6{p}^{2}$${}^{3}{P}_{0}$ | 43.6 | 47.7 | 47.1 | 46.8 | 47.0 | 47.0 | 47.1(7.1) [96,97] |

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## Share and Cite

**MDPI and ACS Style**

Oleynichenko, A.V.; Zaitsevskii, A.; Skripnikov, L.V.; Eliav, E.
Relativistic Fock Space Coupled Cluster Method for Many-Electron Systems: Non-Perturbative Account for Connected Triple Excitations. *Symmetry* **2020**, *12*, 1101.
https://doi.org/10.3390/sym12071101

**AMA Style**

Oleynichenko AV, Zaitsevskii A, Skripnikov LV, Eliav E.
Relativistic Fock Space Coupled Cluster Method for Many-Electron Systems: Non-Perturbative Account for Connected Triple Excitations. *Symmetry*. 2020; 12(7):1101.
https://doi.org/10.3390/sym12071101

**Chicago/Turabian Style**

Oleynichenko, Alexander V., Andréi Zaitsevskii, Leonid V. Skripnikov, and Ephraim Eliav.
2020. "Relativistic Fock Space Coupled Cluster Method for Many-Electron Systems: Non-Perturbative Account for Connected Triple Excitations" *Symmetry* 12, no. 7: 1101.
https://doi.org/10.3390/sym12071101