Magnetodielectric Effect in a Triangular Dysprosium Single‐Molecule Toroics

Abstract Single‐molecule toroics are molecular magnets with vortex distribution of magnetic moments. The coupling between magnetic and electric properties such as the magnetodielectric effect will provide potential applications for them. Herein, the observation of significant magnetodielectric effect in a triangular Dy3 crystal with toroidal magnetic moment and multiple magnetic relaxations is reported. The analysis of magnetic and electric properties implies that the magnetodielectric effect is closely related to the strong spin‐lattice coupling, magnetic interactions of Dy3+ ions, as well as molecular packing models.


Table of Contents
1. Computational details page 3

Computational details
Trinuclear complex Dy3 has three types of magnetic center Dy 3+ ions indicated as Dy1, Dy2 and Dy3.Complete-active-space self-consistent field (CASSCF) calculations on individual Dy 3+ fragments for complex Dy3 (see Figure S5) on the basis of single-crystal X-ray determined geometry have been carried out with OpenMolcas [2] program package.Each of individual Dy 3+ fragments in Dy3 was calculated keeping the experimentally determined structures of the corresponding compound while replacing the other Dy 3+ ions with diamagnetic Lu 3+ .
The basis sets for all atoms are atomic natural orbitals from the ANO-RCC library: ANO-RCC-VTZP for Dy 3+ ; VTZ for close O and N; VDZ for distant atoms.The calculations employ the second order Douglas-Kroll-Hess Hamiltonian, where scalar relativistic contractions are taken into account in the basis set and the spin-orbit couplings are handled separately in the restricted active space state interaction (RASSI-SO) procedure. [3,4]Active electrons in 7 active orbitals include all f electrons (CAS (9 in 7for Dy 3+ )) in the CASSCF calculation.To exclude all the doubts, we calculate all the roots in the active space.We have mixed the maximum number of spin-free state which is possible with our hardware (all from 21 sextets, 128 from 224 quadruplets, 130 from 490 doublets) for each fragment.SINGLE_ANISO [5][6][7] program is used to obtain the energy levels, g tensors, magnetic axes, et.al. based on the above CASSCF/RASSI-SO calculations.
To fit the exchange interactions in complex Dy3, we took two steps to obtain them.Firstly, we calculated individual Dy 3+ fragments using CASSCF/RASSI-SO to obtain the corresponding magnetic properties.Then, the exchange interaction between the magnetic centers was considered within the Lines model, [8] while the account of the dipole-dipole magnetic coupling was treated exactly.The lines model is effective and has been successfully used widely in the research field of d and f-elements single-molecule magnets. [9,10]r Dy3, there are three types of  ̃ and the Ising exchange Hamiltonian is: The  ̃1 = 25 cos   1 , where  is the angle between the anisotropy axes on sites Dy1 and Dy2, and  1 is the Lines exchange coupling parameter.The other two coupling constants of  ̃2 and  ̃3 have the similar expressions.The y D S = 1/2 is the ground pseudospin on the Dy 3+ site.
̃ is the parameter of the total magnetic coupling constant ( ̃ =  ̃ +  ̃ℎ ) between magnetic center ions.]

Results and Discussion
Figure S1 The thermogravimetric curve of Dy3.Table S1 The crystallographic parameters of Dy3.

Formula C20H43Dy3N8O20 C20H43Dy3N8O20
Formula weight 1203.12 1203.12Temperature (K) 30( 2   Table S3 Fitted exchange couplings exch J , the calculated dipole-dipole interactions dip J and the total constants total J between magnetic center ions in Dy3 (cm −1 ).The intermolecular interaction zJ´ of Dy3 was fitted to −0.02 cm −1 .
Table S4 Exchange energies E (cm −1 ), the transversal magnetic moments Δt (µB) and the main values of the gz for the lowest four exchange doublets of complex Dy3.

Figure S1 Thermogravimetric curve of Dy3 page 4 Figure S2 Powder XRD patterns of Dy3 page 4 Figures 5 Figure S5 5 Figure S6 6 Figure S7 7 Figure S8 8 Figure S9 9 Figure S10
Figure S1 Thermogravimetric curve of Dy3 page 4 Figure S2 Powder XRD patterns of Dy3 page 4 Figures S3 and S4 Additional molecular structures page 5 Figure S5 Calculated model structures of individual Dy 3+ fragments in Dy3 page 5 Figure S6 The best fit results with the modified general Debye function for Dy3 page 6 Figure S7 The best fit to piecewise power law equations page 7 Figure S8 The crystal orientation data for Dy3 page 8 Figure S9 Magnetic anisotropy of the single crystal sample for Dy3 page 9 Figure S10 Dielectricity of the single crystal sample for Dy3 page 10

Figure S6
Figure S6The best fit results with the modified general Debye function for Dy3 at the temperatures between 3.6 K and 15 K, respectively.

Figure S7
Figure S7The solid red lines represent the best fit to piecewise power law equations.

Figure S8
Figure S8The crystal orientation of the single crystal of Dy3.The larger crystal face of (001 ̅ ) is named ab plane for clarity.

Figure S9
Figure S9 Magnetic anisotropy of the single crystal sample for Dy3.Zero-field cooling and field cooling curves and magnetic hysteric loops with H // ab plane and H ⊥ ab plane.

Figure S10
Figure S10 Dielectricity and the tangent loss of the single crystal sample for Dy3 in the temperature range of 2-300 K.No dielectric anormalies indicates that there is no structure phase transation.

Table S1
Crystallographic parameters of Dy3

Table S2
CSM calculations for three Dy sites in Dy3

Table S2
CSM calculations for three Dy sites in Dy3.

Table S7
Angles between the main magnetic axes on Dy 3+ ions in their ground KDs and the angles between the magnetic axes and the Dy3 plane for Dy3.