# Significance of Non-Linear Terms in the Relativistic Coupled-Cluster Theory in the Determination of Molecular Properties

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## Abstract

**:**

## 1. Introduction

## 2. Theory and Implementation

## 3. Results and Discussions

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Depictions of Goldstone diagrams representing the linear terms of the expectation value evaluation expression using the LERCCSDmethod. The notations $i,j,k,\cdots $ denote the hole lines, while $a,b,c,\cdots $ denote the particle lines. Diagram (

**i**) corresponds to contribution from the DF method; (

**ii**) is from the $O{T}_{1}$ term; (

**iii**,

**iv**) are from ${T}_{1}^{\u2020}O{T}_{1}$; (

**v**–

**viii**) are diagrams for ${T}_{1}^{\u2020}O{T}_{2}$, with (

**v**,

**vii**) corresponding to direct terms and (

**vi**,

**viii**) corresponding to exchange terms. Sub-figures (

**ix**–

**xvi**) include direct and exchange diagrams from ${T}_{2}^{\u2020}O{T}_{2}$. We also note that the Hermitian conjugate diagrams of those given above are not explicitly sketched here.

**Figure 2.**The effective one-body terms representing particle-particle (p-p) diagrams considered in this work. $i,j,k,\cdots $ and $a,b,c,\cdots $ refer to holes and particles, respectively. The symbol of the operator, ${O}_{p-p}$, is not mentioned explicitly in the diagrams, and the property vertex is the dashed line ending with an “o” in each diagram.

**Figure 3.**The effective one-body terms representing the hole-hole (h-h) diagrams that are included in this work. The notations are the same as in the figure for the particle-particle diagrams. The property operator, ${O}_{h-h}$, is not explicitly mentioned in each of the diagrams, just as in Figure 2.

**Figure 4.**The list of the effective one-body terms representing the particle-hole (p-h) diagrams in this work. The notations are the same as in the particle-particle and the hole-hole diagrams.

**Figure 5.**Plot showing the scaling behavior of the program in the property evaluating expression for a representative system, SrF, with the number of processors of our computer. The X-axis gives the number of processors, while the Y-axis is the speedup, ${S}_{p}={t}_{1}/{t}_{p}$, where t is the time taken and the subscript denotes the number of processors. We used a double-zeta quality basis for this purpose and tested up to 192 processors, as the parallelism in our code was limited by the number of virtual orbitals, which was 208 in this case.

**Table 1.**Contributions from the Dirac–Fock (DF), LERCCSD, and nLERCCSDmethods to the ${\mathcal{E}}_{\mathrm{eff}}$s (in GV/cm) and permanent electric dipole moments (PDMs) (in Debye) of HgX, SrF, and BaF molecules from the present work (denoted as “This work” in the table). Comparison of the two properties from various works with our results are also presented. CASSCF, complete active space self-consistent field; RSPT2, second-order Rayleigh–Schrodinger perturbation theory; X2C-FSCC, exact two-component Hamiltonian–Fock space coupled-cluster.

Molecule | Method | PDM | ${\mathcal{E}}_{\mathbf{eff}}$ |
---|---|---|---|

SrF | CASSCF-MRCI [46] | 3.36 | |

CASSCF-RSPT2 [46] | 3.61 | ||

Z-vector [47] | 3.45 | ||

LERCCSD [9,33] | 3.6 | 2.17 | |

FFCCSD [9] | 3.62 | 2.16 | |

X2C-MRCI [48] | 3.20 | ||

X2C-FSCC [48] | 3.46 | ||

DF (This work) | 2.99 | 1.54 | |

LERCCSD (This work ) | 3.57 | 2.15 | |

nLERCCSD (This work) | 3.60 | 2.16 | |

Experiment [26] | 3.4676(1) | ||

BaF | MRCI [49] | 2.96 | |

LERCCSD [33] | 3.4 | 6.50 | |

FFCCSD [9] | 3.41 | 6.46 | |

X2C-MRCI [48] | 2.90 | ||

X2C-FSCC [48] | 3.23 | ||

Z-vector [50] | 3.08 | ||

ECP-RASSCF [51] | 7.5 | ||

RASCI [52] | 7.28 | ||

MRCI [53] | 5.1 | ||

MRCI [54] | 6.1 | ||

DF (This work) | 2.61 | 4.81 | |

LERCCSD (This work) | 3.32 | 6.45 | |

nLERCCSD (This work) | 3.37 | 6.39 | |

Experiment (PDM) [55] | 3.17(3) | ||

HgF | CI [56] | 4.15 | 99.26 |

LERCCSD [57] | 2.61 | ||

MRCI [53] | 68 | ||

MRCI [54] | 95 | ||

DF (This work) | 4.11 | 105.69 | |

LERCCSD [8] | 115.42 | ||

FFCCSD [9] | 2.92 | 116.37 | |

LERCCSD (This work) | 3.25 | 114.93 | |

nLERCCSD (This work) | 3.45 | 113.77 | |

HgCl | CI [58] | 3.28 | |

LERCCSD [57] | 2.72 | ||

LERCCSD [8] | 113.56 | ||

FFCCSD [9] | 2.96 | 114.31 | |

DF (This work) | 4.30 | 104.33 | |

LERCCSD (This work) | 3.26 | 112.51 | |

nLERCCSD (This work) | 3.45 | 110.94 | |

HgBr | CI [58] | 2.62 | |

LERCCSD [57] | 2.36 | ||

LERCCSD [8] | 109.29 | ||

FFCCSD [9] | 2.71 | 109.56 | |

DF (This work) | 4.14 | 99.72 | |

LERCCSD (This work) | 2.62 | 109.38 | |

nLERCCSD (This work) | 2.94 | 107.42 | |

HgI | LERCCSD [57] | 1.64 | |

LERCCSD [8] | 109.3 | ||

FFCCSD [9] | 2.06 | 109.56 | |

DF (This work) | 3.61 | 99.27 | |

LERCCSD (This work) | 1.50 | 110.00 | |

nLERCCSD (This work) | 2.01 | 107.38 |

**Table 2.**Individual correlation contributions to the effective electric fields (in GV/cm) of mercury monohalides (HgX; X = F, Cl, Br, and I), SrF, and BaF, from the LERCCSD (abbreviated as “L”) and nLERCCSD (denoted by “nL”) methods. In the first column, A could be O (which corresponds to LERCCSD diagrams) or ${O}_{x-y}$ (which is associated with nLERCCSD diagrams), where “x” and “y” could stand for the corresponding particle or hole line for a given term. The values are all rounded-off to two decimal places for HgX, while numbers that are extremely small in the case of SrF and BaF are denoted in the scientific notation instead.

Molecule | HgF | HgCl | HgBr | HgI | SrF | BaF | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Term | Diagram | L | nL | L | nL | L | nL | L | nL | L | nL | L | nL |

DF | Figure 1i | 105.69 | 104.33 | 99.72 | 99.27 | 1.54 | 4.81 | ||||||

$A{T}_{1}$ | Figure 1ii | 17.09 | 13.11 | 17.05 | 12.21 | 19.83 | 14.76 | 23.85 | 15.71 | 0.63 | 0.61 | 1.79 | 1.60 |

${T}_{1}^{\u2020}A{T}_{1}$ | Figure 1iii | −1.85 | −0.28 | −2.01 | −0.25 | −2.65 | −0.62 | −3.66 | −0.41 | −1.86$\times {10}^{-2}$ | −1.00$\times {10}^{-3}$ | −7.65$\times {10}^{-2}$ | −2.00$\times {10}^{-4}$ |

Figure 1iv | −1.41 | 0.16 | −1.40 | 0.28 | −1.21 | 0.47 | −1.56 | 1.16 | −9.01$\times {10}^{-3}$ | 4.80$\times {10}^{-4}$ | −6.47$\times {10}^{-2}$ | 7.60$\times {10}^{-3}$ | |

${T}_{1}^{\u2020}A{T}_{2}$ | Figure 1v | 1.19 | 0.93 | 0.65 | 0.29 | 0.38 | −0.11 | 0.38 | −0.27 | 2.73$\times {10}^{-3}$ | 1.02$\times {10}^{-3}$ | 9.46$\times {10}^{-3}$ | 2.51$\times {10}^{-3}$ |

Figure 1vi | 0.05 | 0.08 | 0.06 | 0.05 | −0.01 | −0.07 | −0.03 | −0.09 | −4.91$\times {10}^{-4}$ | −7.93$\times {10}^{-4}$ | −1.49$\times {10}^{-3}$ | −2.39$\times {10}^{-3}$ | |

Figure 1vii | 0.61 | 0.58 | 0.92 | 0.85 | 0.66 | 0.32 | 0.57 | 0.19 | 1.43$\times {10}^{-2}$ | 1.48$\times {10}^{-2}$ | 7.04$\times {10}^{-2}$ | 7.13$\times {10}^{-2}$ | |

Figure 1viii | −1.31 | −1.27 | −1.24 | −1.18 | −0.91 | −0.63 | −1.26 | −0.98 | 9.63$\times {10}^{-3}$ | 9.91$\times {10}^{-3}$ | −2.49$\times {10}^{-2}$ | −2.32$\times {10}^{-2}$ | |

${T}_{2}^{\u2020}A{T}_{2}$ | Figure 1ix | −2.50 | −2.46 | −2.54 | −2.49 | −2.68 | −2.65 | −2.93 | −2.89 | 8.58$\times {10}^{-3}$ | 6.15$\times {10}^{-3}$ | 3.22$\times {10}^{-2}$ | 2.17$\times {10}^{-2}$ |

Figure 1x | −0.17 | −0.17 | −0.15 | −0.14 | −0.14 | −0.13 | −0.13 | −0.11 | −2.17$\times {10}^{-3}$ | −1.93$\times {10}^{-3}$ | −6.87$\times {10}^{-3}$ | −6.83$\times {10}^{-3}$ | |

Figure 1xi | −1.22 | −1.40 | −1.50 | −1.47 | −1.65 | −1.85 | −1.96 | −1.99 | −1.96$\times {10}^{-2}$ | −2.17$\times {10}^{-2}$ | −7.54$\times {10}^{-2}$ | −7.71$\times {10}^{-2}$ | |

Figure 1xii | −0.17 | −0.17 | −0.15 | −0.14 | −0.14 | −0.13 | −0.13 | −0.11 | −2.17$\times {10}^{-3}$ | −1.93$\times {10}^{-3}$ | −6.87$\times {10}^{-3}$ | −6.83$\times {10}^{-3}$ | |

Figure 1xiii | −1.64 | −1.57 | −1.67 | −1.57 | −1.70 | −1.58 | −1.84 | −1.69 | −1.38$\times {10}^{-3}$ | −1.96$\times {10}^{-3}$ | −3.20$\times {10}^{-4}$ | −9.61$\times {10}^{-4}$ | |

Figure 1xiv | −0.10 | −0.10 | −0.10 | −0.10 | −0.10 | −0.10 | −0.10 | −0.10 | −5.39$\times {10}^{-4}$ | −5.53$\times {10}^{-4}$ | −2.28$\times {10}^{-3}$ | −2.33$\times {10}^{-3}$ | |

Figure 1xv | 0.77 | 0.74 | 0.36 | 0.37 | 0.08 | 0.12 | −0.37 | −0.21 | 4.42$\times {10}^{-3}$ | 4.51$\times {10}^{-3}$ | 2.82$\times {10}^{-3}$ | 3.18$\times {10}^{-3}$ | |

Figure 1xvi | −0.10 | −0.10 | −0.10 | −0.10 | −0.10 | −0.10 | −0.10 | −0.10 | −5.39$\times {10}^{-4}$ | −5.53$\times {10}^{-4}$ | −2.28$\times {10}^{-3}$ | −2.33$\times {10}^{-3}$ | |

Total | 114.93 | 113.77 | 112.51 | 110.94 | 109.38 | 107.42 | 110.00 | 107.38 | 2.15 | 2.16 | 6.45 | 6.39 |

**Table 3.**Correlation contributions to the PDMs (in Debye) of mercury monohalides (HgX; X = F, Cl, Br, and I), SrF, and BaF. The notation is the same as in Table 2. The entry “NC” stands for nuclear contribution to the PDM.

Molecule | HgF | HgCl | HgBr | HgI | SrF | BaF | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Term | Diagram | L | nL | L | nL | L | nL | L | nL | L | nL | L | nL |

DF | Figure 1i | −767.04 | −925.61 | −1002.31 | −1075.83 | −375.75 | −578.39 | ||||||

$A{T}_{1}$ | Figure 1ii | −0.60 | −0.78 | −0.83 | −1.01 | −1.26 | −1.54 | −1.92 | −2.33 | 0.63 | 0.65 | 0.80 | 0.82 |

${T}_{1}^{\u2020}A{T}_{1}$ | Figure 1iii | 0.21 | 0.04 | 0.26 | 0.06 | 0.34 | 0.11 | 0.48 | 0.23 | 0.14 | −0.01 | 0.19 | −0.02 |

Figure 1iv | −0.45 | 0.05 | −0.48 | 0.07 | −0.62 | 0.13 | −0.79 | 0.26 | −0.18 | −0.01 | −0.23 | −0.02 | |

${T}_{1}^{\u2020}A{T}_{2}$ | Figure 1v | 0.10 | 0.11 | 0.13 | 0.13 | 0.20 | 0.19 | 0.30 | 0.29 | 2.44$\times {10}^{-2}$ | 2.53$\times {10}^{-2}$ | 3.04$\times {10}^{-2}$ | 3.13$\times {10}^{-2}$ |

Figure 1vi | 0.01 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | −1.99$\times {10}^{-3}$ | 2.13$\times {10}^{-3}$ | 2.40$\times {10}^{-3}$ | 2.55$\times {10}^{-3}$ | |

Figure 1vii | 0.01 | 0.01 | 0.01 | −0.01 | −0.01 | −0.03 | 0.01 | −0.01 | 9.48$\times {10}^{-3}$ | 9.15$\times {10}^{-3}$ | 9.46$\times {10}^{-3}$ | 9.42$\times {10}^{-3}$ | |

Figure 1viii | 0.02 | 0.01 | 0.01 | 0.01 | 0.02 | 0.03 | 0.04 | 0.04 | 1.47$\times {10}^{-3}$ | 1.45$\times {10}^{-3}$ | −4.02$\times {10}^{-3}$ | 4.10$\times {10}^{-3}$ | |

${T}_{2}^{\u2020}A{T}_{2}$ | Figure 1ix | 1.19 | 1.19 | 1.48 | 1.47 | 1.66 | 1.66 | 1.84 | 1.85 | 0.82 | 0.82 | 0.98 | 0.97 |

Figure 1x | −0.01 | −0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 9.92$\times {10}^{-4}$ | 1.57$\times {10}^{-3}$ | −2.43$\times {10}^{-3}$ | −2.99$\times {10}^{-3}$ | |

Figure 1xi | 1.14 | 1.16 | 1.40 | 1.41 | 1.57 | 1.62 | 1.73 | 1.81 | 0.79 | 0.78 | 0.95 | 0.94 | |

Figure 1xii | −0.01 | −0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 9.22$\times {10}^{-4}$ | 1.57$\times {10}^{-3}$ | −2.43$\times {10}^{-3}$ | −2.99$\times {10}^{-3}$ | |

Figure 1xiii | −1.26 | −1.25 | −1.53 | −1.52 | −1.72 | −1.70 | −1.91 | −1.89 | −0.84 | −0.84 | −1.01 | −1.01 | |

Figure 1xiv | 0.01 | 0.01 | 0.01 | 0.01 | 0.00 | 0.00 | 0.01 | 0.00 | 7.35$\times {10}^{-3}$ | 7.37$\times {10}^{-3}$ | −8.38$\times {10}^{-3}$ | 0.01 | |

Figure 1xv | −1.23 | −1.21 | −1.51 | −1.48 | −1.70 | −1.67 | −1.91 | −1.85 | −0.83 | −0.83 | −0.99 | −0.98 | |

Figure 1xvi | 0.01 | 0.01 | 0.01 | 0.01 | 0.00 | 0.00 | 0.01 | 0.00 | 7.35$\times {10}^{-3}$ | 7.37$\times {10}^{-3}$ | −8.38$\times {10}^{-3}$ | 0.01 | |

NC | 771.15 | 929.91 | 1006.45 | 1079.44 | 378.74 | 581.00 | |||||||

Total | 3.25 | 3.45 | 3.26 | 3.45 | 2.62 | 2.94 | 1.50 | 2.01 | 3.57 | 3.60 | 3.32 | 3.37 |

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**MDPI and ACS Style**

Prasannaa, V.S.; Sahoo, B.K.; Abe, M.; Das, B.P.
Significance of Non-Linear Terms in the Relativistic Coupled-Cluster Theory in the Determination of Molecular Properties. *Symmetry* **2020**, *12*, 811.
https://doi.org/10.3390/sym12050811

**AMA Style**

Prasannaa VS, Sahoo BK, Abe M, Das BP.
Significance of Non-Linear Terms in the Relativistic Coupled-Cluster Theory in the Determination of Molecular Properties. *Symmetry*. 2020; 12(5):811.
https://doi.org/10.3390/sym12050811

**Chicago/Turabian Style**

Prasannaa, V. Srinivasa, Bijaya K. Sahoo, Minori Abe, and Bhanu P. Das.
2020. "Significance of Non-Linear Terms in the Relativistic Coupled-Cluster Theory in the Determination of Molecular Properties" *Symmetry* 12, no. 5: 811.
https://doi.org/10.3390/sym12050811