# The Role of Magnetic Dipole—Dipole Coupling in Quantum Single-Molecule Toroics

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## Abstract

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## 1. Introduction

## 2. Quantum Heisenberg Triangle with Strong On-Site Magnetic Anisotropy

#### 2.1. The Semi-Classical Ising Picture

#### 2.2. The Role of Magnetic Dipole–Dipole Coupling in Quantum SMTs

#### 2.3. Tunnelling of the Ground-State Toroidal Moment in a Non-Kramers System

## 3. Spin Frustration in Molecular Triangles with Vanishing Magnetic Anisotropy

## 4. Magnetic Dipole–Dipole Interactions in Extended Heisenberg Rings

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SMM | Single-molecule magnet |

SMT | Single-molecule toroic |

## References

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**Figure 1.**(

**a**) Schematic depiction of the molecular triangle with on-site magnetic anisotropy axes (blue arrows) arranged tangentially about the vertices of the triangle and canted by an angle $\theta $ from $\widehat{\mathbf{z}}$. (

**b**) Ground and first excited exchange multiplets of the semi-classical Ising states of the triangle with idealised in-plane magnetic anisotropy $\theta ={90}^{\circ}$. (

**c**,

**d**) Energy as a function of canting angle $\theta $ of the lowest lying Ising exchange manifolds in the semi-classical on-site spin $S=3/2$ molecular triangle for ratios of $|{J}_{\mathrm{ex}}|/{R}^{3}=0$ cm${}^{-1}$ Å${}^{-1}$ (solid), $0.25$ cm${}^{-1}$ Å${}^{-1}$ (dashed) and 1 cm${}^{-1}$ Å${}^{-1}$ (fine dashed) for (

**c**) ferromagnetic and (

**d**) antiferromagnetic exchange coupling ${J}_{\mathrm{ex}}$.

**Figure 2.**(

**a**,

**b**) Expectation value of the toroidal moment as a function of canting angle $\theta $ for the on-site $S=3/2$ molecular triangle with a fixed ratio of $|D/{J}_{\mathrm{ex}}|=10$ and (

**a**) ferromagnetic and (

**b**) antiferromagnetic exchange coupling. (

**c**,

**d**) Base 10 logarithm of the tunnelling rate between toroidal states in the molecular triangle induced by a stray magnetic dipole. (

**c**) Tunnelling rates in the ferromagnetically coupled triangle with magnetic axis canting angle $\theta ={20}^{\circ}$. (

**d**) Tunnelling rates in the antiferromagnetically coupled triangle with magnetic axis canting angle $\theta ={80}^{\circ}$. The effect of intramolecular magnetic dipole–dipole coupling between paramagnetic ions in the SMT on the stray-dipole-assisted tunnelling rate is highlighted by vanishing (solid line), intermediate (dashed line) and strong (fine dashed line) intramolecular dipole–dipole coupling.

**Figure 3.**(

**a**,

**b**) Toroidal moment expectation value computed on the quasi-degenerate ground manifold of the triangle with ferromagnetic (red) and antiferromagnetic (blue) Heisenberg exchange coupling for (

**a**) $|D/{J}_{\mathrm{ex}}|=5$ and (

**b**) $|D/{J}_{\mathrm{ex}}|=10$. (

**c**,

**d**) Tunnelling rate of the ground-state toroidal moment as a function of canting angle $\theta $ in the molecular triangle with on-site spin $S=1$ for a range of magnetic anisotropy strengths $|D/{J}_{\mathrm{ex}}|$ and (

**c**) ferromagnetic and (

**d**) antiferromagnetic exchange coupling.

**Figure 4.**(

**a**,

**b**) Toroidal moment expectation value computed on the quasi-degenerate ground manifold of the triangle with (

**a**) ferromagnetic and (

**b**) antiferromagnetic exchange coupling for varying strengths of magnetic dipole–dipole interaction. (

**c**,

**d**) Tunnelling rate of the ground-state toroidal moment as a function of canting angle $\theta $ for the on-site $S=1$ molecular triangle with a fixed ratio of $|D/{J}_{\mathrm{ex}}|=10$ and with magnetic dipole–dipole coupling included between paramagnetic sites in the ring. The tunnel splitting is shown for several dipole–dipole coupling strengths and for (

**c**) ferromagnetic and (

**d**) antiferromagnetic Heisenberg exchange coupling.

**Figure 5.**Energy levels for $N=3$-, 5-, 7- and 9-membered antiferromagnetically coupled Heisenberg rings with ${J}_{\mathrm{ex}}=10{D}_{\mathrm{DM}}$. Degenerate levels that support toroidal moments are labelled in red, whilst those that do not are labelled in black. Note that only for $N=3$ are toroidal moments present in the ground state.

**Figure 6.**Ground state expectation value of the toroidal moment operator in a five-membered (

**a**) and a seven-membered (

**b**) Heisenberg ring inclusive of intramolecular magnetic dipole–dipole interactions and ferromagnetic (red) or antiferromagnetic (blue) isotropic exchange coupling ${J}_{\mathrm{ex}}$. Insets depict the expectation values of the on-site spin for one of the ring ground states below the critical circumscribed radius ${R}_{*}$, where a non-zero value of the toroidal moment is observed.

**Figure 7.**Critical circumscribed radius ${R}_{*}$ at which toroidal ground states manifest in five-membered on-site $S=1/2$ molecular wheels as function of Dzyaloshinskii–Moriya coupling strength ${D}_{\mathrm{DM}}$ in Heisenberg rings with ferromagnetic (red) and antiferromagnetic (blue) isotropic exchange coupling. Inset shows the dependence of the ground-state toroidal moment on the orientation of the ${\mathbf{D}}_{\mathrm{DM}}$ vector for a 5-membered $S=1/2$ ring with $|{J}_{\mathrm{ex}}|/R=1$ cm${}^{-1}$ Å${}^{-1}$ and ${D}_{\mathrm{DM}}/\left|{J}_{\mathrm{ex}}\right|=-0.1$ with ferromagnetic (red) and antiferromagnetic (blue) isotropic exchange coupling.

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**MDPI and ACS Style**

Hymas, K.; Soncini, A.
The Role of Magnetic Dipole—Dipole Coupling in Quantum Single-Molecule Toroics. *Magnetochemistry* **2022**, *8*, 58.
https://doi.org/10.3390/magnetochemistry8050058

**AMA Style**

Hymas K, Soncini A.
The Role of Magnetic Dipole—Dipole Coupling in Quantum Single-Molecule Toroics. *Magnetochemistry*. 2022; 8(5):58.
https://doi.org/10.3390/magnetochemistry8050058

**Chicago/Turabian Style**

Hymas, Kieran, and Alessandro Soncini.
2022. "The Role of Magnetic Dipole—Dipole Coupling in Quantum Single-Molecule Toroics" *Magnetochemistry* 8, no. 5: 58.
https://doi.org/10.3390/magnetochemistry8050058