# Extending NMR Quantum Computation Systems by Employing Compounds with Several Heavy Metals as Qubits

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{113}Cd,

^{199}Hg,

^{125}Te, and

^{77}Se assembled with the most common employed nuclei in NMR-QIP implementations (

^{1}H,

^{13}C,

^{19}F,

^{29}Si, and

^{31}P) could potentially be used in heteronuclear systems for NMR-QIP implementations. Hence, aiming to contribute to the development of future scalable heteronuclear spin systems, we specially designed four complexes, based on the auspicious qubit systems proposed in our previous work, which will be explored by quantum chemical calculations of their NMR parameters and proposed as suitable qubit molecules. Chemical shifts and spin–spin coupling constants in four complexes were examined using the spin–orbit zeroth-order regular approximation (ZORA) at the density functional theory (DFT) level, as well as the relaxation parameters (${T}_{1}$ and ${T}_{2}$). Examining the required spectral properties of NMR-QIP, all the designed complexes were found to be promising candidates for qubit molecules.

## 1. Introduction

^{1}H,

^{13}C,

^{19}F,

^{29}Si, and

^{31}P nuclei are the most employed ones [3,14,15]. Furthermore, other spin-1/2 nuclei, such as

^{113}Cd,

^{199}Hg,

^{125}Te, and

^{77}Se nuclei, have been investigated as qubits, and assembled with the most commonly employed nuclei, they could potentially be used in heteronuclear systems for NMR-QIP implementations.

_{2}derivative was found to be the best candidate for NMR quantum computations among the studied phosphorus heterocycles containing $\pi $-conjugated molecular skeletons.

^{31}P-

^{31}P couplings to provide tighter, more accurate control through large-scale NMR-QIP, proposing the application of TS J(

^{31}P,

^{31}P) coupling as a resource for universal logic in NMR quantum computers. From our results, PPN${}_{o}$–F, PPN${}_{o}$–ethyl, and PPN${}_{o}$–NH

_{2}were the best candidates for NMR-QIP, in which the large TS SSCCs could face the need for long-time quantum gates’ implementations. The following work [19] was devoted to evaluating and calculating directly the relaxation parameters of the suitable qubit molecules proposed in the last mentioned work. In NMR-QIP, decoherence is a key problem, and considering the unique qubits, the time of the coherence phase can be well measured by the time of transversal relaxation (${T}_{2}$), wherein the longer the relaxation time, the better the information processing is. The relaxation parameters (${T}_{1}$ and ${T}_{2}$) of the molecules PPN${}_{o}$–F, PPN${}_{o}$–ethyl, and PPN${}_{o}$–NH

_{2}were calculated, and the results supported and confirmed that they are suitable qubit molecules, which could improve the control accuracy through large-scale NMR-QIP, highlighted by the promising longer coherence time, avoiding a high decoherence rate.

^{113}Cd,

^{199}Hg,

^{77}Se, and

^{125}Te as qubits for NMR-QIP. We examined the NMR parameters of metal complexes with phosphine chalcogenide ligands (called MRE) using spin–orbit ZORA and four-component relativistic methods. We developed a computational design strategy for prescreening molecules that could enable many and heteronuclear qubits for NMR-QIP implementations. Particularly, the influence of different conformers, basis sets, functionals, and methods to treat the relativistic, as well as solvent effects was studied. The MRE complexes were found to be multiple-spin systems with Larmor frequencies appropriately dispersed, so well-defined qubits, allowing qubit addressability, together with an exceptionally large spin–spin coupling between the pair of spins, which enables the two-qubit operations.

^{113}Cd,

^{199}Hg,

^{77}Se, and

^{125}Te) as qubits. Supported by the findings of our last mentioned work [20], the use of heavy metals combined with the most frequently used qubits (

^{1}H,

^{13}C,

^{19}F,

^{29}Si, and

^{31}P) can boost the emergent scalable heteronuclear spin system in NMR-QIP. An NMR computer can be programmed electronically analogously to a quantum computer, but also, it can be implemented at room pressure and temperature using macroscopic liquid samples [21]. Thereby, we specially designed four complexes, still based on the auspicious qubit systems proposed in our work [20], which will be explored by quantum chemical calculations of their NMR parameters in order to investigate their suitability as qubit molecules.

## 2. Computational Details

_{3})

_{4}, TMS) for

^{1}H,

^{13}C, and

^{29}Si; CFCl

_{3}for

^{19}F; PH

_{3}for

^{31}P; SeMe

_{2}for

^{77}Se; TeMe

_{2}for

^{125}Te; CdMe

_{2}for

^{113}Cd; HgMe

_{2}for

^{199}Hg. All these references compounds were optimized and calculated at the same level as mentioned before. Table 1 shows the individual absolute shieldings. The chemical shifts ($\delta $ in ppm) of the nucleus (N) were then calculated as:

^{31}P and

^{113}Cd chemical shifts were subsequently corrected by the gas phase experimental value of CdMe

_{2}, −706.15 ppm [36], and PH

_{3}, −266.1 ppm [37], as they are not the reference compounds of the standard chemical shift scale for Cd and P. Comparing our calculated absolute shielding constant for the carbon in TMS, 191.3 ppm, with a value of 188.1 ppm from a semi-experimentally derived absolute shielding scale [38] showed good agreement, taking into account that our value is for the equilibrium geometry, while the experimentally derived value is still for 300 K. In addition, one should recall that calculated chemical shifts or differences in Larmor frequencies, as we study here, are normally more accurate due to error cancellations.

## 3. Results and Discussion

_{2}[((CH

_{3})

_{3}SiCH

_{2})(CH

_{3})Te][(C

_{4}H

_{8}N)(CH

_{3})HPTe],

_{2}[((CH

_{3})

_{3}SiCH

_{2})(CH

_{3})Se][(C

_{4}H

_{8}N)(CH

_{3})HPSe],

_{2}[((CH

_{3})

_{3}SiCH

_{2})(CH

_{3})Te][(C

_{4}H

_{8}N)(CH

_{3})HPTe], and

_{2}[((CH

_{3})

_{3}SiCH

_{2})(CH

_{3})Se][(C

_{4}H

_{8}N)(CH

_{3})HPSe],

#### 3.1. Conformational Flexibility

#### 3.2. Spectroscopic Parameters: Chemical Shift Values

#### 3.3. Spectroscopic Parameters: Spin–Spin Coupling Constant Values

#### 3.4. Correlation Time and Spectral Density

#### Validation of the Theoretical Methodology

#### 3.5. Spectroscopic Parameters: Relaxation Times

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Studied complexes HgF2[((CH3)3SiCH2)(CH3)Te][(C4H8N)(CH3)HPTe] (complex 1), HgF2[((CH3)3SiCH2)(CH3)Se][(C4H8N)(CH3)HPSe] (complex 2), CdF2[((CH3)3SiCH2) (CH3)Te][(C4H8N)(CH3)HPTe] (complex 3), and CdF2[((CH3)3SiCH2)(CH3)Se][(C4H8N)(CH3)HPSe] (complex 4).

**Table 1.**

^{1}H,

^{13}C,

^{29}Si,

^{19}F,

^{31}P,

^{77}Se,

^{125}Te,

^{113}Cd, and

^{199}Hg NMR absolute shieldings, $\sigma $ (ppm) of the ((Si(CH

_{3})

_{4}, TMS), CFCl

_{3}, PH

_{3}, SeMe

_{2}, TeMe

_{2}, CdMe

_{2}, and HgMe

_{2}reference molecules. Calculated using the two-component ZORA approach with the PBE0 exchange-correlation functional and the TZ2P basis sets.

Nucleus | Shielding | Nucleus | Shielding |
---|---|---|---|

$\sigma $ Hg (HgMe_{2}) | 9197.834 | $\sigma $ Si (TMS) | 368.108 |

$\sigma $ Cd (CdMe_{2}) | 3561.291 | $\sigma $ F (CFCl_{3}) | 184.172 |

$\sigma $ Te (TeMe_{2}) | 3557.070 | $\sigma $ C (TMS) | 191.333 |

$\sigma $ Se (SeMe_{2}) | 1930.599 | $\sigma $ H (TMS) | 31.553 |

$\sigma $ P (PH_{3}) | 607.566 |

**Table 2.**Chemical shifts (M =

^{113}Cd or

^{199}Hg and E =

^{77}Se or

^{125}Te), $\delta $ (ppm) calculated using the two-component ZORA approach with the PBE0 exchange-correlation functional and the TZ2P basis sets.

Nucleus | Complex 1 | Complex 2 | Complex 3 | Complex 4 |
---|---|---|---|---|

$\delta M$ | −1513.02 | −1387.93 | 326.71 | 303.00 |

$\delta E2$ | −46.23 | 58.59 | −188.55 | −10.28 |

$\delta E3$ | −565.29 | −195.80 | −700.37 | −250.82 |

$\delta P4$ | −18.67 | 25.03 | −22.51 | 24.25 |

$\delta Si5$ | 4.41 | 3.00 | 4.50 | 2.32 |

$\delta F6$ | −202.22 | −220.33 | −231.50 | −244.14 |

$\delta F7$ | −148.50 | −145.68 | −175.97 | −185.88 |

$\delta C9$ | 1.59 | 20.69 | −2.00 | 19.18 |

$\delta C10$ | −1.01 | −0.34 | −0.92 | 1.26 |

$\delta C11$ | 0.06 | −1.01 | −0.23 | −1.77 |

$\delta C12$ | −0.65 | −1.88 | −0.74 | −1.03 |

$\delta C13$ | −6.62 | 15.57 | −8.88 | 14.59 |

$\delta C14$ | 20.48 | 19.14 | 19.63 | 18.47 |

$\delta C15$ | 53.53 | 50.95 | 53.88 | 50.77 |

$\delta C16$ | 28.63 | 29.00 | 28.71 | 29.31 |

$\delta C17$ | 30.96 | 30.70 | 30.23 | 30.09 |

$\delta C18$ | 51.55 | 51.47 | 50.72 | 50.66 |

$\delta C19$ | 1.38 | 1.47 | 1.39 | 1.73 |

$\delta C20$ | 4.22 | 3.71 | 3.43 | 3.66 |

$\delta C21$ | −0.03 | 0.27 | 0.35 | −0.03 |

$\delta C22$ | 0.95 | −0.05 | −0.03 | 0.16 |

$\delta C23$ | −0.13 | −0.02 | −0.03 | 0.04 |

$\delta C24$ | 1.71 | −0.08 | 0.02 | 0.02 |

$\delta C25$ | −0.14 | −0.14 | −0.03 | −0.22 |

$\delta C26$ | −0.19 | 1.16 | 0.51 | 2.50 |

$\delta C27$ | −0.03 | −0.15 | −0.14 | 0.07 |

$\delta C28$ | 0.26 | 1.56 | 1.98 | 0.24 |

$\delta C29$ | −0.02 | −0.01 | −0.22 | 0.02 |

$\delta C30$ | 1.38 | 1.59 | 1.47 | 1.85 |

$\delta C31$ | 1.44 | 1.72 | 1.42 | 1.84 |

$\delta C32$ | 6.07 | 5.64 | 5.27 | 3.82 |

$\delta C33$ | 7.62 | 7.09 | 6.34 | 5.75 |

$\delta C34$ | 1.66 | 1.54 | 1.67 | 1.53 |

$\delta C35$ | 2.63 | 2.01 | 2.66 | 2.04 |

$\delta C36$ | 7.76 | 8.30 | 7.57 | 8.15 |

$\delta C37$ | 2.45 | 2.62 | 2.43 | 2.57 |

$\delta C38$ | 3.36 | 3.62 | 3.54 | 3.85 |

$\delta C39$ | 1.45 | 1.46 | 1.40 | 1.47 |

$\delta C40$ | 3.76 | 3.98 | 4.03 | 3.90 |

$\delta C41$ | 2.11 | 2.09 | 2.14 | 2.07 |

$\delta C42$ | 1.62 | 1.65 | 1.66 | 1.70 |

$\delta C43$ | 4.50 | 4.60 | 4.55 | 4.82 |

$\delta C44$ | 2.74 | 2.85 | 2.76 | 2.85 |

**Table 3.**Spin–spin coupling constants for directed coupled nuclei (M =

^{113}Cd or

^{199}Hg and E =

^{77}Se or

^{125}Te), J (Hz), and differences in the Larmor frequencies (for a 16.5 T magnet) between the coupled nuclei, $\Delta {\omega}_{0}$ (Hz).

Complex 1 | Complex 2 | Complex 3 | Complex 4 | |||||
---|---|---|---|---|---|---|---|---|

${}^{\mathbf{1}}J$ | $\mathbf{\Delta}{\mathbf{\omega}}_{\mathbf{0}}$ | ${}^{\mathbf{1}}J$ | $\mathbf{\Delta}{\mathbf{\omega}}_{\mathbf{0}}$ | ${}^{\mathbf{1}}J$ | $\mathbf{\Delta}{\mathbf{\omega}}_{\mathbf{0}}$ | ${}^{\mathbf{1}}J$ | $\mathbf{\Delta}{\mathbf{\omega}}_{\mathbf{0}}$ | |

M1–E2 | 3302.70 | 96,802,165 | −835.01 | 8,435,780 | −481.12 | 66,742,136 | 105.44 | 21,582,334 |

M1–E3 | 5102.88 | 96,686,569 | −1485.00 | 8,401,607 | −878.27 | 66,628,154 | 233.79 | 21,614,648 |

M1–F6 | −2621.28 | 535,132,459 | −2483.84 | 535,104,716 | 783.72 | 505,084,765 | 677.95 | 505,080,108 |

M1–F7 | −2405.23 | 535,167,977 | −2330.77 | 535,154,067 | 784.66 | 505,121,481 | 691.68 | 505,118,628 |

E2–C9 | −15.82 | 46,039,162 | −83.82 | 42,314,827 | 195.52 | 46,008,099 | −81.18 | 42,323,811 |

E2–C13 | −24.25 | 46,040,612 | −69.38 | 42,313,921 | 163.74 | 46,009,316 | −69.04 | 42,322,999 |

E3–P4 | 1411.66 | 61,967,771 | −607.71 | 150,249,393 | 1504.62 | 61,996,763 | −660.24 | 150,256,563 |

P4–C14 | 20.86 | 107,887,999 | 28.34 | 107,900,668 | 22.47 | 107,887,060 | 29.75 | 107,900,565 |

P4–H36 | 422.22 | 417,965,157 | 448.08 | 417,953,108 | 428.45 | 417,966,118 | 451.89 | 417,953,223 |

Si5–C9 | −34.38 | 37,025,106 | −34.41 | 37,028,676 | −33.74 | 37,024,459 | −32.89 | 37,028,505 |

Si5–C10 | −41.49 | 37,024,645 | −37.33 | 37,024,961 | −38.53 | 37,024,649 | −39.00 | 37,025,340 |

Si5–C11 | −41.75 | 37,024,835 | −42.01 | 37,024,843 | −40.35 | 37,024,771 | −42.09 | 37,024,804 |

Si5–C12 | −37.53 | 37,024,710 | −41.88 | 37,024,688 | −41.38 | 37,024,681 | −39.32 | 37,024,935 |

C15–C16 | 26.44 | 4398 | 26.40 | 3878 | 26.27 | 4447 | 26.22 | 3791 |

C16–C17 | 23.53 | 410 | 23.53 | 300 | 23.64 | 268 | 23.76 | 138 |

C17–C18 | 26.76 | 3637 | 26.82 | 3670 | 26.85 | 3620 | 26.73 | 3634 |

C9–H19 | 123.68 | 525,852,016 | 122.91 | 525,848,702 | 123.92 | 525,852,654 | 124.00 | 525,849,153 |

C9–H20 | 135.65 | 525,854,011 | 136.23 | 525,850,273 | 133.03 | 525,854,085 | 132.86 | 525,850,508 |

C10–H21 | 112.77 | 525,851,482 | 111.88 | 525,851,574 | 112.46 | 525,851,732 | 112.68 | 525,851,083 |

C10–H22 | 117.10 | 525,852,170 | 112.07 | 525,851,354 | 112.23 | 525,851,464 | 112.86 | 525,851,213 |

C10–H23 | 112.22 | 525,851,414 | 111.99 | 525,851,373 | 111.84 | 525,851,467 | 112.27 | 525,851,130 |

C11–H24 | 119.40 | 525,852,519 | 112.53 | 525,851,447 | 112.84 | 525,851,377 | 112.07 | 525,851,650 |

C11–H25 | 111.88 | 525,851,220 | 112.21 | 525,851,409 | 112.37 | 525,851,343 | 110.23 | 525,851,481 |

C11–H26 | 110.67 | 525,851,181 | 118.19 | 525,852,321 | 115.41 | 525,851,725 | 120.02 | 525,853,393 |

C12–H27 | 112.02 | 525,851,417 | 111.54 | 525,851,552 | 111.82 | 525,851,358 | 112.22 | 525,851,556 |

C12–H28 | 111.68 | 525,851,624 | 119.15 | 525,852,752 | 119.44 | 525,852,844 | 113.36 | 525,851,678 |

C12–H29 | 112.20 | 525,851,429 | 111.67 | 525,851,649 | 110.88 | 525,851,304 | 113.30 | 525,851,523 |

C13–H30 | 132.24 | 525,853,467 | 131.56 | 525,849,693 | 132.76 | 525,853,929 | 133.17 | 525,850,045 |

C13–H31 | 129.49 | 525,853,505 | 127.68 | 525,849,787 | 128.90 | 525,853,895 | 128.17 | 525,850,038 |

C13–H32 | 151.11 | 525,856,759 | 150.06 | 525,852,540 | 150.00 | 525,856,598 | 144.53 | 525,851,434 |

C14–H33 | 134.29 | 525,853,058 | 133.34 | 525,852,924 | 133.78 | 525,852,312 | 133.42 | 525,852,103 |

C14–H34 | 123.44 | 525,848,870 | 123.40 | 525,849,024 | 123.77 | 525,849,030 | 123.26 | 525,849,137 |

C14–H35 | 121.84 | 52,5849,553 | 121.41 | 525,849,354 | 121.87 | 525,849,729 | 120.86 | 525,849,499 |

C15–H37 | 129.93 | 525,843,591 | 129.48 | 525,844,164 | 130.37 | 525,843,517 | 129.55 | 525,844,160 |

C15–H38 | 128.27 | 525,844,233 | 129.07 | 525,844,865 | 128.31 | 525,844,291 | 129.42 | 525,845,060 |

C16–H39 | 121.40 | 525,847,283 | 121.31 | 525,847,231 | 121.09 | 525,847,240 | 121.22 | 525,847,184 |

C16–H40 | 129.67 | 525,848,911 | 130.16 | 525,848,998 | 130.18 | 525,849,087 | 129.51 | 525,848,891 |

C17–H41 | 126.03 | 525,847,341 | 125.78 | 525,847,371 | 125.49 | 525,847,492 | 124.91 | 525,847,466 |

C17–H42 | 116.70 | 525,846,995 | 116.72 | 525,847,064 | 116.85 | 525,847,149 | 117.18 | 525,847,205 |

C18–H43 | 141.85 | 525,845,384 | 141.07 | 525,845,467 | 138.72 | 525,845,563 | 137.77 | 525,845,763 |

C18–H44 | 129.44 | 525,844,147 | 129.09 | 525,844,233 | 129.06 | 525,844,302 | 129.05 | 525,844,379 |

Structures | Correlation Times (fs) | |
---|---|---|

TFE | 19 | 47.71 |

Complex 1 | 28 | 36.13 |

Complex 2 | 24 | 41.37 |

Complex 3 | 8 | 137.52 |

Complex 4 | 8 | 137.15 |

${\mathit{T}}_{1}$ (s) | ${\mathit{T}}_{2}$ (s) | ${\mathit{R}}_{1}$ (s${}^{-1}$) | ${\mathit{R}}_{2}$ (s${}^{-1}$) | ||
---|---|---|---|---|---|

TFE (F1–F3) | Theoretical | 5.29 | 0.19 | 0.18 | 5.26 |

Experimental ${}^{1}$ | 5.37 | 0.14 | 0.19 | 7.14 | |

TFE (F1–F2) | Theoretical | 5.25 | 0.15 | 0.19 | 6.66 |

Experimental ${}^{1}$ | 5.56 | 0.12 | 0.18 | 8.33 |

^{1}From [53].

Hg | Cd | Te | Se | P | Si | F | C | H | |
---|---|---|---|---|---|---|---|---|---|

Complex 1 | |||||||||

${T}_{1}$ (s) | 21.27 | - | 7.17 | - | 1.20 | 44.80 | 1.22 | 1.95 | 1.05 |

${T}_{2}$ (s) | 2.04 | - | 1.02 | - | 0.63 | 3.52 | 1.12 | 1.49 | 0.46 |

${R}_{1}$ (s${}^{-1}$) | 0.04 | - | 0.14 | - | 0.83 | 0.02 | 0.82 | 0.51 | 0.95 |

${R}_{2}$ (s${}^{-1}$) | 0.49 | - | 0.98 | - | 1.58 | 0.28 | 0.89 | 0.67 | 2.17 |

Complex 2 | |||||||||

${T}_{1}$ (s) | 32.77 | - | - | 30.44 | 2.77 | 11.87 | 2.12 | 2.58 | 1.30 |

${T}_{2}$ (s) | 1.51 | - | - | 2.59 | 1.28 | 9.34 | 1.60 | 1.27 | 0.45 |

${R}_{1}$ (s${}^{-1}$) | 0.03 | - | - | 0.03 | 0.36 | 0.08 | 0.47 | 0.38 | 0.76 |

${R}_{2}$ (s${}^{-1}$) | 0.66 | - | - | 0.39 | 0.78 | 0.11 | 0.63 | 0.78 | 2.22 |

Complex 3 | |||||||||

${T}_{1}$ (s) | - | 7.82 | 9.22 | - | 2.01 | 13.99 | 2.72 | 2.56 | 0.95 |

${T}_{2}$ (s) | - | 1.05 | 2.05 | - | 0.37 | 1.10 | 1.97 | 1.25 | 0.35 |

${R}_{1}$ (s${}^{-1}$) | - | 0.13 | 0.11 | - | 0.50 | 0.07 | 0.37 | 0.39 | 1.05 |

${R}_{2}$ (s${}^{-1}$) | - | 0.95 | 0.49 | - | 2.70 | 0.91 | 0.81 | 0.80 | 2.85 |

Complex 4 | |||||||||

${T}_{1}$ (s) | - | 6.06 | - | 21.56 | 2.70 | 15.89 | 1.67 | 2.53 | 1.20 |

${T}_{2}$ (s) | - | 2.09 | - | 1.83 | 0.58 | 9.97 | 1.08 | 1.24 | 0.38 |

${R}_{1}$ (s${}^{-1}$) | - | 0.17 | - | 0.05 | 0.37 | 0.06 | 0.59 | 0.36 | 0.83 |

${R}_{2}$ (s${}^{-1}$) | - | 0.48 | - | 0.55 | 1.72 | 0.10 | 0.93 | 0.81 | 2.63 |

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**MDPI and ACS Style**

Lino, J.B.d.R.; Gonçalves, M.A.; Sauer, S.P.A.; Ramalho, T.C.
Extending NMR Quantum Computation Systems by Employing Compounds with Several Heavy Metals as Qubits. *Magnetochemistry* **2022**, *8*, 47.
https://doi.org/10.3390/magnetochemistry8050047

**AMA Style**

Lino JBdR, Gonçalves MA, Sauer SPA, Ramalho TC.
Extending NMR Quantum Computation Systems by Employing Compounds with Several Heavy Metals as Qubits. *Magnetochemistry*. 2022; 8(5):47.
https://doi.org/10.3390/magnetochemistry8050047

**Chicago/Turabian Style**

Lino, Jéssica Boreli dos Reis, Mateus Aquino Gonçalves, Stephan P. A. Sauer, and Teodorico Castro Ramalho.
2022. "Extending NMR Quantum Computation Systems by Employing Compounds with Several Heavy Metals as Qubits" *Magnetochemistry* 8, no. 5: 47.
https://doi.org/10.3390/magnetochemistry8050047