# Study of HgOH to Assess Its Suitability for Electron Electric Dipole Moment Searches

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## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Ground State Geometry Optimization

## 4. Method of Calculation

## 5. Results and Discussion

## 6. Other Prospective Polyatomic Molecules for EDM Measurements

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Pictorial representation of the bent geometry of the ground state of HgOH. Our finding shows ${\theta}_{Hg-O-H}=104.{83}^{\circ}$.

**Figure 2.**The extrapolation scheme from a linear to a bent geometry, for the permanent electric dipole moment (PDM) of HgOH. The data points in blue show the finite-field coupled-cluster (FFCC) values of the PDM in a non-relativistic framework, while for those in red, the point corresponding to the linear geometry is calculated, and the one pertaining to the bent geometry is the extrapolated value. We add that the distance that we have set between the points corresponding to the linear and bent geometries on the X-axis is arbitrary.

**Table 1.**List of the optimized geometry of the ground electronic state and three low-lying excited states of HgOH from various works. The unit of bond-lengths is angstrom (Å), while that of the bond angle is degrees.

State | ${\mathit{R}}_{\mathbf{Hg}-\mathit{O}}$ | ${\mathit{R}}_{\mathit{O}-\mathit{H}}$ | ${\mathit{\theta}}_{\mathbf{Hg}-\mathit{O}-\mathit{H}}$ | Reference |
---|---|---|---|---|

Ground | 2.091 | 0.966 | 104.1 | Ref. [50] |

2.25 | 0.99 | 106.8 | Ref. [32] | |

2.181 | - | - | Ref. [51] | |

2.2079 | 0.9691 | 103.6 | Ref. [33] | |

2.2294 | 0.9633 | 104.83 | This work | |

First-excited | 3.1458 | 0.9563 | 180 | This work |

Second-excited | 2.0766 | 0.9615 | 102.5 | This work |

Third-excited | 3.5482 | 1.0095 | 81.93 | This work |

**Table 2.**Table showing the calculated ${\mathcal{E}}_{\mathrm{eff}}$ (in GV/cm) and $\mu $ (in D) values in HgOH by assuming its hypothetical linear and the actual bent geometry ground state using the Dirac–Hartree–Fock (DHF) and RCCSD methods. We also give $\mu $ values from the previous calculations using density functional theory (DFT).

Geometry | ${\mathcal{E}}_{\mathbf{eff}}$ | $\mathit{\mu}$ | ||
---|---|---|---|---|

DHF | LECC | DHF | LECC | |

From this work | ||||

Linear | 107.24 | 109.02 | 1.57 | 1.04 |

Bent | 28.01 | 28.47${}^{\u2020}$ | 3.67 | 2.43 ${}^{\u2020}$ |

From other works | ||||

1.89 [32] | ||||

1.92 [51] | ||||

1.96 [33] |

^{†}Scaled results from the DHF and LECC values of the linear geometry calculations.

**Table 3.**Contributions from the individual relativistic coupled-cluster (RCC) terms to ${\mathcal{E}}_{\mathrm{eff}}$ (in GV/cm), $\mu $ (in D), and ${W}_{s}$ (in kHz) from both the linear and bent geometries of HgOH. O denotes the operator corresponding to the properties and h.c. means hermitian conjugate. Note that for the PDM, the term corresponding to the DHF contribution also accounts for the nuclear contribution in it.

Term | ${\mathcal{E}}_{\mathbf{eff}}$ (GV/cm) | $\mathit{\mu}$ (D) |
---|---|---|

O (DHF) | 107.24 | 1.57 |

$O{T}_{1}+$h.c. | 9.50 | −0.42 |

${T}_{1}^{\u2020}O{T}_{1}$ | −2.76 | −0.15 |

${T}_{1}^{\u2020}O{T}_{2}+$h.c. | −0.38 | 0.12 |

${T}_{2}^{\u2020}O{T}_{2}$ | −4.58 | −0.11 |

**Table 4.**Contributions from different atomic orbital (AO) mixing to the DHF value of ${\mathcal{E}}_{\mathrm{eff}}$ (in GV/cm), where the AO in the left hand side is a small component AO and that in the right hand side is a large component AO. Non-zero contributions may come only from odd-parity AO mixings $\langle {\left(AO\right)}_{1}^{S}\left|\widehat{O}\right|{\left(AO\right)}_{2}^{L}\rangle ,$ where superscript S, and L stand for small component and large component AOs respectively. Results are given for both the linear and the actual bent geometry HgOH molecule.

Atom | AOs | Linear | Bent |
---|---|---|---|

Hg | ${s}_{1/2}^{S}-{p}_{1/2}^{L}$ | 378.40 | 100.11 |

${p}_{1/2}^{S}-{s}_{1/2}^{L}$ | −270.26 | −71.81 | |

${p}_{1/2}^{S}-{d}_{3/2}^{L}$ | −31.40 | −8.07 | |

${d}_{3/2}^{S}-{p}_{3/2}^{L}$ | 30.19 | 7.77 | |

${d}_{5/2}^{S}-{f}_{5/2}^{L}$ | 0.79 | 0.19 | |

${f}_{5/2}^{S}-{d}_{5/2}^{L}$ | −0.78 | −0.18 | |

O | ${s}_{1/2}^{S}-{p}_{1/2}^{L}$ | 2.78 | 1.44 |

${p}_{1/2}^{S}-{s}_{1/2}^{L}$ | −2.77 | −1.44 |

**Table 5.**Comparison of measured (ThO, HfF${}^{+}$, and YbF) and projected sensitivities (from appropriate references as given in the table) offered by different molecules for EDM experiments. For molecules where measurements are not available, the sensitivity is estimated with appropriate N, T, $\tau $, and $\eta $ values. The unit chosen for $\delta {d}_{e}$ is e-cm, while the effective electric field is given in GV/cm.

Molecule | ${\mathcal{E}}_{\mathbf{eff}}$ | $\mathit{\delta}{\mathit{d}}_{\mathit{e}}$ | Reference(s) |
---|---|---|---|

ThO | 79.9 [12] | $1.1\times {10}^{-29}$ | Ref. [14] |

HfF${}^{+}$ | 22.5 [13] | $1.3\times {10}^{-28}$ | Ref. [15] |

YbF | 23.1 [10] | $1.06\times {10}^{-27}$ | Ref. [16] |

HgOH | 28.47 | $5\times {10}^{-30}$ | This work |

HgCH${}_{3}$ | 75.07 | $2\times {10}^{-30}$ | This work |

HgCF${}_{3}$ | 60.95 | $2\times {10}^{-30}$ | This work |

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**MDPI and ACS Style**

Mitra, R.; Prasannaa, V.S.; Sahoo, B.K.; Hutzler, N.R.; Abe, M.; Das, B.P. Study of HgOH to Assess Its Suitability for Electron Electric Dipole Moment Searches. *Atoms* **2021**, *9*, 7.
https://doi.org/10.3390/atoms9010007

**AMA Style**

Mitra R, Prasannaa VS, Sahoo BK, Hutzler NR, Abe M, Das BP. Study of HgOH to Assess Its Suitability for Electron Electric Dipole Moment Searches. *Atoms*. 2021; 9(1):7.
https://doi.org/10.3390/atoms9010007

**Chicago/Turabian Style**

Mitra, Ramanuj, V. Srinivasa Prasannaa, Bijaya K. Sahoo, Nicholas R. Hutzler, Minori Abe, and Bhanu Pratap Das. 2021. "Study of HgOH to Assess Its Suitability for Electron Electric Dipole Moment Searches" *Atoms* 9, no. 1: 7.
https://doi.org/10.3390/atoms9010007