Symmetry of octa-coordination environment has a substantial influence on dinuclear Tb^III triple-decker single-molecule magnets

Abstract

Single-molecule magnet (SMM) properties of terbium(III)-phthalocyaninato and porphyrinato mixed ligand triple-decker complexes, [(TTP)Tb(Pc)Tb(TTP)] (1) and [(Pc)Tb(Pc)Tb(TTP)] type (2), were studied and were compared with those of the Tb^III homoleptic triple-decker complex [(obPc)Tb(obPc)Tb(obPc)] (3) in order to elucidate the relationship between octa-coordination environments and SMM properties (Tb^III = terbium(III), TTP²⁻ = tetraphenylporphyrinato, Pc²⁻ = phthalocyaninato, obPc²⁻ = 2,3,9,10,16,17,23,24-octabutoxyphthalocyaninato). By combining TTP²⁻ and Pc²⁻ with Tb^III ions, it is possible to make three octa-coordination environments: SP–SP, SAP–SP and SAP–SAP sites, where SAP is square-antiprismatic and SP is square-prismatic. The direction and magnitude of the ligand field (LF) strongly affect the magnetic properties. Complexes 2 and 3, which have SAP–SAP sites, undergo dual magnetic relaxation processes in the low temperature region in a direct current magnetic field. On the other hand, 1, which has an SP–SP environment, undergoes a single magnetic relaxation process, indicating that the octa-coordination environments strongly influence the SMM properties. The SMM behaviour of dinuclear Tb^III SMMs 1–3 were explained by using X-ray crystallography and static and dynamic susceptibility measurements. This work shows that the SMM properties can be fine-tuned by introducing different octa-coordination geometries with the same Tb^III–Tb^III distances.

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Introduction

The field involving single-molecule magnets (SMMs) began in 1993 with the synthesis of Mn₁₂ clusters.¹ In recent years, mononuclear metal complexes, not just polynuclear ones, have also been reported.² In other words, slow magnetic relaxation processes are observed for SMMs regardless of the type and number of metal ions in the complexes under certain conditions. In general, in crystals, d and/or f electrons forming the electronic states reflect the symmetry of the crystals. As a result, the crystal field (CF) or ligand field (LF) directly influences the magnetic properties. Recently, a few groups have published reviews on lanthanide(III) (Ln^III)-based SMMs,² which have several distinctive features, including slow magnetic relaxation and quantum tunnelling of the magnetization (QTM). Although the reviews show that molecular design is important to bring about SMM behaviour, controlling the magnetic relaxation pathway is still challenging. Since the tunneling gap (Δ_tunnel) is dependent on the establishment of QTM, most studies have focused on the influence of the spin–spin interactions and nuclear spins.² However, because molecular spintronics based on SMMs, which exhibit new quantum phenomena, will be the next generation devices,³ it is necessary to elucidate completely the magnetic relaxation processes of the SMMs used.

In 2003, the terbium(III)-phthalocyaninato stacked complex (TBA)⁺[TbPc₂]⁻ (Tb^III = terbium(III), Pc²⁻ = phthalocyaninato, TBA⁺ = tetrabutylammonium) was reported to exhibit SMM behaviour, which originates from the ligand field (LF).⁴ The magnetic properties of rare-earth metal ions are strongly related to the charge density distribution.² Tb^III and Dy^III ions exhibit uniaxial magnetic anisotropies of the easy axis, and splitting of the ground state multiplets is caused by the LF at the axial position. In this case, the energy gap between the ground and first excited states is associated with the energy barrier for the reversal of the magnetization (Δ), thereby causing slow relaxation of the magnetization. Consequently, the direction and magnitude of the LF strongly affect the magnetic properties.^1–4

Although lanthanide(III)-phthalocyaninato double-decker (LnPc₂) SMMs have been studied in detail, little is known about Ln₂Pc₃. Ishikawa and co-workers have reported a series of studies on the magnetic properties of the Ln^III-Pc triple-decker complex [(Pc)Ln(Pc)Ln(obPc)] (obPc²⁻ = 2,3,9,10,16,17,23,24-octabutoxyphthalocyaninato; Ln^III = Tb, Dy, and Y), which are the first reports on the dynamic magnetism of a coupled 4f system.⁵ More recently, we have reported that the Tb^III-Pc triple-decker SMM Tb₂(obPc)₃ (3) shows dual magnetic relaxation processes in the low temperature region in a dc magnetic field (H_dc).⁶ The dual magnetic relaxation behaviour has been observed not only for 3 but also for the spatially closed Tb^III-Pc quadruple- and quintuple-decker complexes.⁷ This is clear evidence that the magnetic relaxation mechanism depends heavily on the f–f interactions between the Tb^III ions in the dinuclear systems. Several SMMs with dual magnetic relaxation processes have been reported, and it has been shown that dual magnetic relaxation processes are observed for SMMs regardless of the type and number of metal ions in the complexes.⁸ It is well known that the magnetic relaxation rates reflect the local molecular symmetry and are extremely sensitive to tiny distortions in the coordination geometries. Our results suggest that the dual magnetic relaxation properties in the Tb^III-Pc multiple-decker systems have a magic number of “two”, i.e., the number of metal sites.⁷ However, further experiments are required to corroborate this hypothesis.

In this study, we focused on a porphyrin (H₂Por) ligand, which has a cyclic structure with four pyrroles in common with a phthalocyanine (H₂Pc). In the case of H₂Por, the four pyrroles in the macrocycle are linked by methine groups, whereas H₂Pc has a four isoindole molecules, which are pyrrole molecules fused with benzene rings, linked with nitrogen atoms. Both have planar structures with the π-conjugation spread over the entire molecule. However, the stacked coordination environments in complexes are significantly different depending on the presence or absence of π-electron repulsion. It is well-known that porphyrins form multiple-decker complexes when coordinated to lanthanide ions. A number of studies on Ln^III-Por²⁻ and Ln^III-Por²⁻/Pc²⁻ mixed multiple-decker type SMMs have been reported.⁹ From previous research, when using Por²⁻ ligands, the SMM properties are not as same as those of Pc²⁻ complexes due to the LF potential of coordination geometry. In addition, contractions of the square-antiprismatic (SAP) coordination environment affect the LF of the Ln^III ions, which in turn affects the SMM properties.^4,9a However, not only the effects of the ground state for different octa-coordination geometries but also dual magnetic relaxation processes have not been methodically discussed. In order to simplify the discussion, therefore, we designed dinuclear Tb^III triple-decker complexes with different octa-coordination environments but with similar intramolecular Tb^III–Tb^III distances (∼0.36 nm). The combination of Por²⁻ and Pc²⁻ with Ln^III ions can give rise to three different octa-coordination environments (Scheme 1).

Scheme 1 Schematic illustration of dinuclear Tb^III triple-decker complexes 1–3, which show the three octa-coordination environments: SP–SP, SAP–SAP* and SAP–SAP, respectively, where SAP = square-antiprismatic and SP = square-prismatic.

Herein we present the results of studies on the SMM properties of dinuclear Tb^III triple-decker mixed ligands complexes [(TTP)Tb(Pc)Tb(TTP)] (1) and [(Pc)Tb(Pc)Tb(TTP)] (2) (Scheme 1). The relationship between the octa-coordination environment and SMM properties of 1 and 2 are discussed in comparison to Tb₂(obPc)₃ (3).⁶ The SMM behaviours of dinuclear Tb^III complexes 1–3 were explained by using X-ray crystallography and static and dynamic susceptibility measurements. This work shows that the SMM properties can be fine-tuned by introducing different octa-coordination geometries while maintaining the same Tb^III–Tb^III distances.

Experimental

All reagents were purchased from Wako chemicals, Tokyo Chemical Industry (TCI), and Sigma Aldrich and used without further purification. The compounds were prepared following reported procedures with slight modifications.¹⁰

Synthesis of [(TTP)Tb(Pc)Tb(TTP)] (1) and [(Pc)Tb(Pc)Tb(TTP)] (2)¹⁰

Tb(acac)₃·4H₂O (180 mg, 0.40 mmol) and H₂TTP (tetraphenylporphyrin) (150 mg, 0.25 mmol) were added to dry 1,2,4-trichlorobenzene (40 mL). The solution was refluxed under argon for 4 h. After cooling, Li₂Pc (158 mg, 0.60 mmol) was added to the mixture. Then the solution was refluxed for 12 h. After cooling, the reaction mixture was added to n-hexane (500 mL). The obtained solid was purified by using column chromatography on silica gel with chloroform as the eluent. [(TTP)Tb(Pc)Tb(TTP)] (1) was obtained from a deep brownish red fraction, which was the first fraction, by removing the solvent, and [(Pc)Tb(Pc)Tb(TTP)] (2) was obtained from the dark green second fraction. Column chromatography (C-200 silica gel, Wako and Sephadex G-10, Pharmacia Biotech) was used to remove the remaining impurities. Dark red fine block crystals of 1 were obtained from chloroform/n-hexane (50 mg). ESI-MS: m/z (%): 2055.46605 (100) [M⁺]; elemental analysis calcd (%) for C₁₂₀H₇₂N₁₆Tb₂·3.5CHCl₃: C 59.97, H 3.08, N 9.06; found: C 60.03, H 3.21, N 8.89. Black fine needle crystals of 2 were obtained from chloroform/n-hexane (40 mg). ESI-MS: m/z (%): 1855.30261 (100) [M⁺]; elemental analysis calcd (%) for C₁₀₈H₆₀N₂₀Tb₂·CHCl₃: C 63.09, H 2.96, N 13.50; found: C 62.91, H 3.28, N 13.36.

Physical property measurements

UV-Vis-NIR spectra for chloroform solutions of 1 and 2 were acquired on a SHIMADZU UV-3100PC in a quartz cell with a pathlength of 1 cm at 298 K (Fig. S1†). IR spectroscopy was performed on KBr pellets on a Jasco FT/IR-4200 spectrometer at 298 K (Fig. S2†). Electrospray ionization mass spectroscopy and elemental analyses were performed at the Research and Analytical Centre for Giant Molecules, Tohoku University. Magnetic susceptibility measurements were performed on Quantum Design SQUID magnetometers MPMS-3. Direct current (dc) measurements were performed in the temperature (T) range of 1.8–300 K and in dc magnetic fields (H_dc) of −70 000 to 70 000 Oe. Alternating current (ac) measurements were performed in the ac frequency (ν) range of 1–996 Hz with an ac field amplitude of 3 Oe in the presence of a dc field (zero–6000 Oe). Measurements were performed on randomly oriented powder samples of 1–3, which were placed in gel capsules and fixed with n-eicosane to prevent them from moving during measurements. All data were corrected for the sample holder, n-eicosane, and diamagnetic contributions from the molecules by using Pascal's constants.

X-Ray crystal structure analysis

A single crystal was mounted on a loop rod coated with Paratone-N (HAMPTON RESEARCH). Data collection was performed on a Rigaku Saturn 724+ CCD diffractometer with graphite-monochromated Mo-Kα radiation (λ = 0.71069 Å) at T = −180 ± 1 °C. An empirical absorption correction based on azimuthal scans of several reflections was applied. The data were corrected for Lorentz and polarization effects. All non-hydrogen atoms were refined anisotropically using a least-squares method, and hydrogen atoms were fixed at calculated positions and refined using a riding model. SHELXL-97 was used for structure refinement, and the structure was expanded using Fourier techniques.^12a The weighting scheme was based on counting statistics. Full-matrix least-squares refinements on F² based on unique reflections with unweighted and weighted agreement factors of R = ∑||F_o| − |F_c||/∑|F_o| (I > 2.00σ(I)), and wR = [∑w(F_o² − F_c²)²/∑w(F_o²)²]^1/2 were performed. Powder X-ray diffraction (PXRD) patterns of crushed polycrystalline samples of 1 and 2, loaded into capillaries (diameter: 0.8 mm, length: 80 mm, Hilgenderg) with their mother liquor, were collected at 298 K using a Rigaku X-ray diffractometer (AFC-7R/LW) operated at 50 kV and 300 mA in the diffraction angle (2θ) range of 3–60° in steps of 0.02° at 2 s per step (Fig. S3†). Visualization and analysis of the crystal structures and PXRD patterns were simulated from the single crystal data by using Mercury 3.0.^12b

Results and discussion

Synthesis and electronic structure

The structures of 1–3 are shown in Fig. 1 with several octa-coordination geometries for Tb^III dinuclear systems. In our studies, we used a tetraphenylporphyrinato (TTP²⁻) ligand with a phthalocyaninato (Pc²⁻) ligand because it has a high solubility, making crystallization easier.¹⁰ In order to control the spin orientation in the molecules, we designed 1–3 so that the two Tb^III ions were along the anisotropy axis. Mixed ligand triple-decker complexes 1 and 2 were synthesized in one-step using Tb(acac)₃·nH₂O, H₂TTP and Li₂Pc following a procedure reported by Weiss and co-workers.¹⁰ On the other hand, triple-decker 3 was synthesized in one step starting from Tb(acac)₃·4H₂O and H₂obPc following a procedure reported by Takahashi and co-workers.¹¹ Por²⁻ and Pc²⁻ can form neutral triple-decker complexes composed of two Tb^III ions and three obPc²⁻ ligands ([(obPc)Tb(obPc)Tb(obPc)] (3)) and in Por²⁻:Pc²⁻ ratios of 1:2 ([(Pc)Tb(Pc)Tb(Por)] (2)) and 2:1 ([(Por)Tb(Pc)Tb(Por)] (1)) with closed shell π electron systems.^5,6,9–11

Fig. 1 Crystal structure of (a) 1 (SP–SP), (b) 2 (SAP–SAP*) and (c) 3 (SAP–SAP): (left) side view and (right) top view. Proton and n-butoxy substituents of 3 omitted for clarity. Tb^III: pink, C: grey, and N: light blue.

Triple-decker complexes 1–3 are soluble in most organic solvents, except for alcohols and n-hexane. A chloroform solution of 1–3 showed 3–6 prominent absorption bands in the wavelength range of 250–1200 nm (Fig. S1†). The absorption spectrum of 1, which has an inner Pc²⁻ ligand strongly interacting with the two outer TTP²⁻ ligands, were similar to those of the [(TTP)Ln(Pc)Ln(TTP)] analogues and, therefore, were assigned on the basis of a previous report.¹⁰ The strong absorption bands at 419 nm were assigned to be Soret (S) bands, and the weak absorptions at 493 and 553 nm were assigned to be the Q bands of the TTP²⁻ ligands with a large contribution from the former tetrapyrroles, respectively. The bands at 353 (S band) and 606 nm (Q band) were attributed to the Pc²⁻ ligand. Furthermore, the weak broad bands in the near-IR region (850–1100 nm) were assigned to be electronic transitions between the dianionic ligands.

The absorption spectrum of 2 was similar to that of 1, but the absorption intensities and peak positions were different. From experimental and theoretical studies, this is clear evidence that excitonic interactions between the two Pc²⁻/Por²⁻ chromophores depend heavily on the molecular structures, in which the ligands are stacked in a staggered face-to-face fashion.¹⁰ The strong absorption bands at 341 nm with shoulders at 324 nm, and the weak absorptions at 617 and 669 nm were assigned to be S and Q bands of the Pc²⁻ ligands, respectively. The bands at 418 (S band) and those at 522 and 604 nm (Q bands) were attributed to the TTP²⁻ ligand. Furthermore, the broad band at 737 nm was assigned to be electronic transitions between the dianionic ligands. In IR spectra of 1 and 2, three strong peaks at 1330, 1062, 728 cm⁻¹, which were attributed to the Pc²⁻ ligand, were observed (Fig. S2†).

In the case of 3, the two absorption bands at 290 and 356 nm were assigned to be the S bands of the obPc²⁻ ligands,¹¹ and those at 577 and 655 nm were assigned to be the Q bands of the obPc²⁻ ligands.

Crystal structure and octa-coordination geometry

To evaluate the effects of the coordination geometry around the Tb^III ion in the triple-decker complexes 1–3, contractions of their octa-coordination environments were investigated by using single-crystal X-ray diffraction analysis (Fig. 1). The longitudinally contracted SAP coordination environment shares an edge with an undistorted complex, i.e., there is a twist angle (φ) between the upper and the lower Pc²⁻ ligands (Fig. S4†). Selected crystallographic data for 1–3 are compiled in Table S1.† The data clearly show that contractions of the octa-coordination environment have an influence on the LF of the Ln^III ions, which affects the SMM properties.^4,9,13

Complex 1, shown in Fig. 1a, crystallized with chloroform in the crystal lattice in the tetragonal space group I4/m, which has a SP (quasi-D_4h) coordination environment (Scheme 1 and Fig. S4d†). Complex 1 has two Tb^III ions, which a Pc²⁻ ligand coordinated between them. The inner Pc²⁻ ligand with four isoindole-nitrogen donor atoms (N_iso) and a center of symmetry strongly interacts with the two outer TTP²⁻ ligands. The center of the square formed by the four pyrrolic nitrogens of the Pc²⁻ ligand form a crystallographically imposed inversion center, making the two Tb^III ions and outer TTP²⁻ ligands equivalent. In addition, the inner Pc²⁻ ligand of 1 is located on a mirror plane. The intramolecular Tb^III–Tb^III distance was determined to be 0.372 nm (Tb^III–Tb^III distance was determined to be 0.352 nm in 3).⁶φ between the outer TTP²⁻ ligands and the inner Pc²⁻ ligand was determined to be 3.62°. Thus, the two Tb^III sites in 1 have a slightly distorted SP geometry (Fig. 1a). The Tb^III ions are unevenly spaced between the outer TTP²⁻ and inner Pc²⁻ ligands with distances of 0.239 nm (3: 0.234–0.237 nm)⁶ from the mean plane of the four N_iso of the outer TTP²⁻ ligands and 0.268 nm (3: 0.258–0.262 nm)⁶ from the mean plane of the four N_iso of the inner Pc²⁻ ligand.

Crystal-packing diagrams of 1 are shown in Fig. S5.† The intermolecular Tb^III⋯Tb^III distance along the a and c axes were determined to be 1.431 and 1.364 nm, respectively. Each molecule of 1 is rather well separated from neighboring molecules due to the tetraphenyl groups of TTP²⁻ ligands and chloroform molecules as crystal solvents. Furthermore, PXRD patterns for 1 at 293 K are similar to those simulated from X-ray single crystallographic data for 1 at 93 K (Fig. S3-1†).

Recently, Jiang and co-workers have reported the [(Por)Ln(Pc)Ln(Por)] type triple-decker SMM {(TCIPP)Dy[Pc(OBNP)₄]Dy(TCIPP)}, which has Dy^III ions in SAP (φ ≈ 7°) and SAP* (φ ≈ 42°) environments (TClPP = meso-tetrakis(4-chlorophenyl)porphyrinato; Pc(OBNP)₄ = tetrakis(dinaphtho[1,2-e:1′,2′-g]-1,4-dioxocine)[2,3-b;2′,3′-k;2′′,3′′-t;2′′′,3′′′-c′]phthalocyaninato).^9e The coordination environments are completely different from those in 1, which has two Tb^III ions in the slightly distorted SP environments (φ ≈ 4°).

On the other hand, complex 2, shown in Fig. 1b, crystallized with chloroform in the crystal lattice in the monoclinic space group P2₁/c. 2 has a SAP (quasi-D_4d) coordination environment (Scheme 1 and Fig. S4e†), and it has two Tb^III ions and an inner Pc²⁻ ligand, which strongly interact with each other. The Tb^III–Tb^III distance in 2 was determined to be 0.363 nm, and φ between the outer Pc²⁻ ligands and the center Pc²⁻ one was determined to be 38.40° (site B = SAP*). On the other hand, φ between the outer TTP²⁻ ligands and the inner Pc²⁻ one was determined to be 13.80° (site A = SAP). The Tb^III ions are unevenly spaced between the outer TTP²⁻ and inner Pc²⁻ ligands with distances of 0.237–0.240 nm from the mean plane of the four N_iso of the outer TTP²⁻ ligands and 0.270–0.274 nm from the mean plane of the four N_iso of the inner Pc²⁻ one. On the other hand, the Tb^III ions are unevenly spaced between the outer Pc²⁻ and inner Pc²⁻ ligands with distances of 0.235–0.237 nm from the mean plane of the four N_iso of the outer Pc²⁻ ligands and 0.257–0.260 nm from the mean plane of the four N_iso of the center Pc²⁻ one. The distance between the Tb^III ions and N_iso of 1–3 are essentially the same.

Crystal-packing diagrams of 2 are shown in Fig. S6.† The closest intermolecular Tb^III⋯Tb^III distance along the c axis was determined to be 1.162 nm. Each molecule of 2 is rather well separated from neighboring molecules due to the four phenyl groups of the TTP²⁻ ligands and chloroform as crystal solvents. PXRD patterns of 2 at 293 K were similar to the patterns simulated from X-ray single crystallographic data of 2 at 93 K (Fig. S3-2†).

A number of studies on [(Pc)Ln(Pc)Ln(Por)] mixed ligand triple-decker type SMMs have been reported.^9a,9d Ishikawa and co-workers reported the triple-decker complex {(Pc)Tb(Pc)Tb[T(p-OMe)PP]} [T(p-OMe)PP = 5,10,15,20-tetra(p-methoxyphenyl)porphyrinato], in which the Tb^III ions are in SP (φ ≈ 1°) and SAP (φ ≈ 45°) environments.^9a Jiang and co-workers reported the triple-decker complex {(TClPP)Dy[Pc(OPh)₈]Dy[Pc(OPh)₈]} [Pc(OPh)₈ = 2,3,9,10,16,17,23,24-octa(phenoxyl)phthalocyaninato] with Dy^III ions in SAP (φ ≈ 10°) and SAP* (φ ≈ 25°) environments.^9d From a comparison of the reported complexes, in 2, the Tb^III ions have SAP (φ ≈ 14°) and SAP* (φ ≈ 38°) environments, and the octa-coordination environments are completely different from those in 1 (SP–SP). In other words, both Tb^III sites in 2 exhibit slightly distorted SAP geometries with independent environments (Fig. 1b). Viewed from another direction, site A (φ ≈ 14°) appears to have a slightly distorted SP environment. Thus, the SMM properties cannot be predicted solely from the crystal structure, and it is necessary to study the SMM properties in detail (see Magnetic properties section).

We have already reported a crystal structure of 3.⁶ Complex 3 crystallized with ethanol in the crystal lattice in the triclinic space group P1̄, as shown in Fig. 1c. Complex 3 has two Tb^III ions sandwiched between three obPc²⁻ ligands with four N_iso donor atoms and a center of symmetry. The intramolecular Tb^III–Tb^III distance was determined to be 0.352 nm. φ between the outer obPc²⁻ ligands and the inner one was determined to be 32°. This means that both Tb^III sites in 3 have the same slightly distorted SAP geometry (quasi-D_4d). The intermolecular Tb^III⋯Tb^III distance along the a axis was determined to be 1.098 nm. Each molecule of 3 is rather well separated from neighbouring molecules due to the n-butoxy chains. This is supported by the dc and ac magnetic measurements on an isomorphous sample of 3 diluted with diamagnetic Y₂(obPc)₃, which show that the intermolecular magnetic interactions are negligible. In other words, the magnetic properties of 1–3 are due to intramolecular Tb^III–Tb^III interactions and the symmetry of the octa-coordination geometry (vide infra). Selected molecular geometry information for dinuclear Ln^III triple-decker mixed ligand complexes are listed in Table 1.

Table 1 Selected crystallographic data for dinuclear Ln^III triple-decker mixed ligand complexes

Complexes	Ln–Ln distance/nm	φ A/°	φ B/°	Site geometry
a This work; see main text in this manuscript.
[(TTP)Tb(Pc)Tb(TTP)] 1^a	0.372	4	4	SP–SP
[(Pc)Tb(Pc)Tb(TTP)] 2^a	0.363	14	38	SAP–SAP*
[(obPc)Tb(obPc)Tb(obPc)] 3^6	0.352	32	32	SAP–SAP
{(Pc)Tb(Pc)Tb[T(p-OMe)PP]}^9a	0.360	1	45	SP–SAP
{(TCIPP)Dy[Pc(OBNP)4]Dy(TCIPP)}^9e	0.371	7	42	SAP–SAP*
{(TCIPP)Dy[Pc(OPh)8]Dy[Pc(OPh)8]}^9d	0.361	10	25	SAP–SAP*

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Static magnetic properties

In order to understand the SMM properties of 1 (SP–SP) and 2 (SAP–SAP*), we compared them with those of SMM 3 (SAP–SAP).⁶ The dc magnetic susceptibilities (χ_M) of powder samples of 1 and 2 were measured in the temperature (T) range of 1.8–300 K in a dc magnetic field (H_dc) of 500 Oe by using a superconducting quantum interference device (SQUID) magnetometer.

The χ_MT value at 300 K (23.6 cm³ K mol⁻¹) corresponds to two free Tb^III ions (⁷F₆, S = 3, L = 3, g = 4/3) (Fig. 2a). The χ_M of 1 obeyed the Curie–Weiss law, giving a Curie constant (C) of 23.3 cm³ K mol⁻¹ with a positive Weiss constant (θ) of 0.15 K over the T range of 180–300 K (Fig. S7a†). With a decrease in T, the χ_MT value of 1 gradually increased and reached ∼25 cm³ K mol⁻¹ at 1.8 K. In the case of 2, the χ_MT value at 300 K (23.6 cm³ K mol⁻¹) corresponds to two free Tb^III ions (Fig. 2a). The χ_M obeyed the Curie–Weiss law, giving a C value of 23.6 cm³ K mol⁻¹ with a positive θ of 0.02 K over the entire T range (Fig. S7b†). With a decrease in T, the χ_MT value of 2 slightly increased and reached 34 cm³ K mol⁻¹ at 1.8 K. The behaviour of χ_MT clearly shows that the magnetic behaviour depends heavily on the magnetic dipole–dipole interactions between the Tb^III ions in the molecule under 10 K. Behaviour similar to those of triple-decker 3 was observed, as shown in Fig. 2a.⁶χ_MT versus T plots for 3 increased with a decrease in T and reached a maximum of 36 cm³ K mol⁻¹ at 1.8 K, which indicates the existence of ferromagnetic interactions between the Tb^III ions.^5–71 has a lower χ_MT value at 1.8 K than 2 and 3 do. The deviation in the χ_MT values occur due to a combination of a few factors, such as the intermetallic Tb^III–Tb^III distance, LF effects with significant magnetic anisotropy, and/or depopulation of the excited states.^{2b–e,g,h,5–7,9a}

Fig. 2 (a) Temperature (T) dependence of χ_MT measured on powder samples of 1 (SP–SP), 2 (SAP–SAP*) and 3 (SAP–SAP) at 500 Oe. (b) Normalised magnetization versus field (M/M_sversus H) performed on powder samples of 1–3. The solid lines are guides for eyes.

To understand the deviation in the χ_MT value of 1, we first studied the influence of the Tb^III–Tb^III distance. The intramolecular Tb^III–Tb^III distance of complexes 1–3 were determined to be in the range of 0.352–0.372 nm, indicating that there were strong magnetic interactions between Tb^III ions in the molecules. This is clear evidence that the χ_MT value of Tb^III dinuclear systems depends heavily on the magnetic dipole–dipole (f–f) interactions between the Tb^III ions.^5–7 Therefore, the ferromagnetic dipole–dipole interactions (D_ij ∝ 1/r_ij³, where r_ij is the distance between spin i and j) between the Tb^III ions in 1 are similar to those in triple-decker complex 3 because r_ij between the Tb^III ions of 1 (0.372 nm) is similar to that of 3 (0.352 nm) (ca. 1/1.1).⁶ In other words, the χ_MT value of 1 cannot be explained solely on the basis of the Tb^III–Tb^III distance, and it is possible that the magnetic anisotropies of 1–3 are due to the octa-coordination geometries of the Tb^III ions.

Next, we focused on the different octa-coordination environments of 1–3. To study the effects on the magnetic properties of the triple-decker complexes, contractions of the coordination environments in 1 and 2 were investigated. From XRD analysis, it was determined that the octa-coordination geometries of 1 (SP–SP), 2 (SAP–SAP*), and 3 (SAP–SAP) had pseudo-4-fold axes (direction of the uniaxial magnetic anisotropy) perpendicular to the tetrapyrrole rings. In addition, contractions of the SAP coordination environments affect the LF of the Ln^III ions, which affects the SMM properties.^4,9a,13 It has been reported that the LF parameters (B^q_k) of an ideal SAP coordination geometry with D_4d symmetry is different from B^q_k of an SP coordination geometry with D_4h symmetry.^4,9a,13 Differences in the SMM properties are thought to be due to the presence or absence of particular off-diagonal LF terms. Thus, SAP geometry only occurs in the terms B⁰₂ (α₂A⁰₂r²), B⁰₄ (α₄A⁰₄r⁴) and B⁰₆ (α₆A⁰₆r⁶), which are LF parameters for the axial anisotropy, where the α_kA^q_kr^k are LF parameters.¹³ The LF Hamiltonian can be written as Ĥ_LF = ∑∑^k_q=−k B^q_kO^q_k, where O^q_k are spin operators.¹³ On the other hand, SP geometry not only contributes to the LF of axial parameters but also to the terms B⁴₄ (α₄A⁴₄r⁴) and B⁴₆ (α₆A⁴₆r⁶), which are LF parameters for the transverse anisotropy. The fourth-range extradiagonal parameters (B⁴₄ and B⁴₆) enable mixing between the ground state of the |+6〉 and |−6〉 doublet in a zero magnetic field.^4,9a,13 It exerts a greater effect on both the hyperfine structure of the ground state and the ground state multiplets structure.

There have been many attempts to elucidate the relationship between the coordination geometry and LF parameters both experimentally and theoretically. The LF parameters have been shown to have a significant influence on the ground state multiplets structure and magnetic anisotropy. The effects of the LF parameters on the uniaxial anisotropy are described below.

Gatteschi and co-workers have reported that the twist angle (φ) has a big effect on the ground state and the first excited state of the octa-coordination geometries with C₄ symmetry axes for Ln^III complexes, since the structures deviate from D_4d symmetry. This deviation appears in the fourth-range extradiagonal parameters, and the distortion has an effect on both axial and transverse parameters, which has been shown by using theoretical calculations.^13e In addition, the angles (α) between the C₄ axis and the direction of Ln^III–N_iso coordination bond have a strong influence on the LF parameters.^13f

Ishikawa and co-workers have reported that the values of the energy barrier for the reversal of the magnetization (Δ) of the ground state multiplets of [TbPc₂]^+/− complexes are changed depending on the difference in the closest Pc–Pc distance.⁴ Longitudinal contraction of the SAP structure causes the changes in Δ and increases the LF splitting of the ground multiplets of Ln^III ions in the cation structure.⁴ⁱ Furthermore, Ishikawa and co-workers have reported the LF parameters of the Y^III–Tb^III triple-decker complex [(Pc)Y(Pc)Tb(obPc)] which has quasi-D_4d geometry like 3.^5c They do not report φ for the Tb^III site, but it should be ca. 32° on the basis of the structure of a similar complex.^5d The value of A⁰₂r² decreases with a decrease in φ from 414 cm⁻¹ for [TbPc₂]⁻ with φ ≈ 45° to 293 cm⁻¹ for [(Pc)Y(Pc)Tb(obPc)] with φ ≈ 32°. It should be noted that the A⁴₄r⁴ dramatically increases from 10 to 863 cm⁻¹, and A⁴₆r⁶ increases from a negligible value to 357 cm⁻¹.^4,5c,13j The Δ value for [(Pc)Y(Pc)Tb(obPc)] with φ ≈ 32° has been determined to be 300 cm⁻¹ from the LF parameters.^5c On the other hand, Δ for [TbPc₂]⁻ with φ ≈ 45° has been determined to be ca. 420 cm⁻¹ from the LF parameters.⁴ Theoretical calculations support the relationship between the LF parameters and Δ.^13g,h Furthermore, they demonstrate that they can reversibly switch between the non-SMM protonated form [TbH(TTP)₂] with a hepta-coordinate environment to the SMM deprotonated form [Tb(TTP)₂]⁻ (H-DBU)⁺ with D_4d symmetry.^9b They argue that the change in SMM properties is due to the off-diagonal LF term and/or that the lowest energy level is no longer J_z = ±6 in the hepta-coordinate environment. Therefore, there is a loss of uniaxial anisotropy, and [TbH(TTP)₂] does not exhibit SMM behaviour.

Recently, Chibotaru, Tong and co-workers have reported strong evidence for the relationship between the symmetry and Δ of SMMs.^13g They demonstrated that when Δ of an Dy^III SMM was decreased by changing symmetry from quasi-D_5h to quasi-O_h. They report that Δ changes by two orders of magnitude between the quasi-D_5h (305 cm⁻¹) and quasi-O_h structures (negligible value). The axial parameters B⁰₂ and B⁰₄ of quasi-D_5h structure are large, whereas the values of the non-axial parameters B²₂, B⁴₄ and B³₄ of quasi-O_h structure are the same or slightly larger than those of the axial ones. In addition, the Dy^III site in the quasi-D_5h structure exhibits very strong axial magnetic anisotropy. On the other hand, the Dy^III site in the quasi-O_h structure shows weak axial magnetic anisotropy due to the non-axial parameters. In other words, the uniaxial anisotropy of the quasi-D_5h structure is stronger than that of quasi-O_h structure on the basis of the g tensor.

Considering the above research results, it is possible that the transverse anisotropy parameters (B^q_k; q ≠ 0) have an effect on the uniaxial magnetic anisotropies and the ground state multiplets structure. The coefficient values show that there are interactions between the metal orbitals and the ligands, and they can be determined experimentally.^2,4,5,9,13 However, we discuss this point in the section “Dynamic magnetic properties”.

In order to qualitatively estimate the magnetic anisotropy, reduced magnetization measurements were performed on 1 (SP–SP), 2 (SAP–SAP*), and 3 (SAP–SAP) (Fig. S8-1†). If molecules have large magnetic anisotropies, M versus HT⁻¹ plots do not overlap on a single curve, and they did not show saturation at high fields (up to 70 kOe).^1,2M of 3 was not synchronised, whereas those of 2 and 1 slightly unaligned, indicating the presence of significant magnetic anisotropy.^13h To elucidate the ground state of 1–3, as shown in Fig. 2b, we focused on the initial increase in M in a weak dc magnetic field (M versus H). M of 1 gradually increased, whereas those of 2 and 3 sharply increased. The M–H curve reflects the overall effect of the LF and magnetic interactions. As the ground states are rather differently mixed, it is very difficult to infer the strength of the magnetic interactions in various LFs from the M data which have strong magnetic anisotropies. It is indicated that the ground state of 1 is different from that of 2 and 3. This result agrees with other information concerning the ground state (see Dynamic magnetic properties and Magnetic relaxation mechanism). For diluted samples of 1–3, behaviour similar to that of 1–3 (powder sample) were observed, thus confirming that the magnetic properties were not due to the intermolecular Tb^III⋯Tb^III interactions, as shown in Fig. S9.†^6,7 Therefore, the different magnetic behaviours observed for 1–3 is thought to be due to the symmetry of the octa-coordination environment.^4–6,9,13 Below, the relationship between the symmetry of coordination environments and the ν dependence of the ac susceptibilities are discussed.

Dynamic magnetic properties

To elucidate the details of the relaxation dynamics, ac magnetic susceptibility measurements were performed on powder samples of 1 (SP–SP) and 2 (SAP–SAP*) in a 3 Oe oscillating ac field. The in-phase (χ_M′) and out-of-phase (χ_M′′) signals changed in different T ranges depending on the ν in the range of 1–996 Hz, indicating that 1 and 2 are SMMs (Fig. 3 and 4, S10–S28†). The values of the energy barrier for the reversal of the magnetization (Δ) and pre-exponential factor (τ₀: the average relaxation time in response to thermal fluctuation) can be estimated from Arrhenius equation: τ = τ₀ exp(Δ/k_BT),^1,2 where τ is the magnetic relaxation time, k_B is Boltzmann constant. This linear relation between ln(τ) and T⁻¹ indicates that the Orbach process (spin–phonon interaction which approximately corresponds to the Δ for thermal relaxation for the reversal of the magnetic moment) is dominant in high-T ranges. This method is highly credible for the high-T region.

Fig. 3 Frequency (ν) and temperature (T) dependence of the ac magnetic susceptibilities (χ_M′′: out-of-phase) of 1 (SP–SP). The measurements were performed in a 3 Oe ac magnetic field at the indicated frequencies in an H_dc of (a) zero. The solid lines are guides for eyes. (b) χ_M′′/χ_M′ versus T⁻¹ (13–18 K) plot at the given ν (198–996 Hz) of the ac susceptibility data by using the Kramers–Kronig equation (eqn (S1)–(S3)†). The solid lines were fitted as described in Table S2.† (c) ν dependence of χ_M′′ versus T in an H_dc of 1500 Oe. The solid lines are guides for eyes.

Fig. 4 Frequency (ν) and temperature (T) dependence of the ac magnetic susceptibility (χ_M′′: out-of-phase) of 2 (SAP–SAP*). The measurements were performed in a 3 Oe ac magnetic field at the indicated ν. (a) ν dependence of χ_M′′ measured between 1.8 and 20 K in an H_dc of zero. The solid lines are guides for eyes. (b) Arrhenius plots for 2 in an H_dc of zero. Solid lines were fitted by using the Arrhenius equation (see main text). (c) ν dependence of χ_M′′T versus T at H_dc of 1000 Oe. The solid lines are guides for eyes.

χ _M′′ versus T plots for both 1 and 2 in the ν range of 1–996 Hz barely showed any tailing, and the expected maximum value due to blocking could not be observed down to 1.8 K in an applied H_dc of zero (Fig. 3a, S10 and S11†). The χ_M′′ did not disappear in the T region below 2 K, indicating that the magnetic moment was not frozen and that a different relaxation process, which is likely a ground state QTM process, became dominant in the low-T region. Since we have no basis for comparing Δ and τ₀ in a zero H_dc, we determined Δ for 1 and 2 from χ_M′′/χ_M′ versus T⁻¹ (13–18 K) plots of the ac susceptibility data at various ν (198–996 Hz) by using the Kramers–Kronig equation (eqn (S1)–(S3)† and Fig. 4b).¹⁴ By fitting the data, the Δ was determined to be 18–24 cm⁻¹ with τ₀ = 10⁻⁶ to 10⁻⁷ s for 1 and 36–38 cm⁻¹ with τ₀ ≈ 10⁻⁶ s for 2 in a zero H_dc. The estimated parameters for 1 and 2 are summarized in Tables S2 and S3,† respectively.

In the case of 1 (SP–SP), in detailed H_dc dependent ac magnetic susceptibility (χ_M′ and χ_M′′) measurements at ν of 996 Hz in the T range of 2–30 K, χ_M′′ showed a single maximum at 11 K in an H_dc of 1500 Oe (Fig. 3c and S12†). The shapes of the peaks of 1 drastically changed, and the peaks shifted to the low-T side when an H_dc large enough to suppress QTM was applied. The χ_M′ and χ_M′′ peaks were clearly shifted in different T ranges dependent on the ν (1–996 Hz) in H_dc of 1500 and 3000 Oe, indicating that 1 is a field-induced SMM (Fig. S13†).

In contrast to 1, 2 (SAP–SAP*) showed clear H_dc field dependent character and two χ_M′′ peaks appeared at 11 K and 27 K at 996 Hz and did not show H_dc dependence in the range of 250–5000 Oe, indicating that 2 is a field-induced SMM (Fig. S14 and S15†). However, a χ_M′′ peak top for 2 was not clearly observed in the measured T range (Fig. S15b†). In order to clarify the T dependence, we prepared χ_M′T and χ_M′′T graphs (Fig. 4c and S16†). The shape of the two peaks of 2 were drastically different from those of 1 and clearly shifted in different T ranges depending on ν in an H_dc of 1000 Oe.

This behaviour is similar to those of other dinuclear Tb^III complexes with slightly different structures relative to 2, such as {(Pc)Tb(Pc)Tb[T(p-OMe)PP]} and [(Pc)Tb(Pc)Tb(obPc)], with different coordination environments around the asymmetric Tb^III sites reported by Ishikawa and co-workers.^5,9a Our results clearly show that the two Tb^III ion sites (SAP and SAP* sites) are inequivalent and are in agreement with the structure of 2 (SAP–SAP*) determined using XRD and the ac susceptibility measurements. From a χ_M′′T versus T plot, the intensity of the χ_M′′ peak top at 27 K (the SAP* site of 2) was lower in an H_dc of 3000 Oe. On the other hand, the intensity of the χ_M′′ peak top at 11 K (the SAP site of 2) showed ν dependence, and this behaviour is similar to that observed in a χ_M′′T versus T plot for 2 in a zero H_dc (Fig. S17 and S18†). It was attributed to an Orbach process, which was observed when an H_dc large enough to suppress QTM (τ ≈ 10⁻⁴ s) was applied. The Orbach process for the SAP* environment is inhibited by fast QTM and/or direct magnetic relaxation process (τ ≈ 10⁻³ s).^13g,15 See the details in “the Magnetic relaxation mechanism” section.

In the case of 3 (SAP–SAP), in a χ_M′′ versus T plot at ν of 996 Hz, only a single peak around 24 K was observed in an H_dc of zero.⁶ These results indicate that the two χ_M′′T peak top T values (11 and 27 K) at 996 Hz for 2 with an SAP–SAP* site correspond to a state that is a combination of 1 with an SP–SP site (11 K) and 3 with an SAP–SAP site (24 K), respectively (Fig. 5). Thus, it is important to study the properties of dinuclear Tb^III triple-decker type SMMs on the bases of crystal structure features, such as molecular contractions, i.e., LF and CF of the central metal ions.

Fig. 5 Temperature (T) dependence of the ac magnetic susceptibilities (χ_M′′: out-of-phase) of 1–3 (χ_M′′T versus T plot). The measurements were performed in a 3 Oe ac magnetic field at the indicated frequencies (996 Hz) in an H_dc of 1000 Oe for 1 (SP–SP), 1500 Oe for 2 (SAP–SAP*), and zero for 3 (SAP–SAP), respectively. In order to suppress QTM, we applied an H_dc for 1 and 2 (see main text). The solid lines are guides for eyes.

To elucidate the details of the relaxation dynamics of 1 and 2, the ν dependence of the χ_M′ and χ_M′′ signals in the range of 10–996 Hz were studied at each T. Argand plots (χ_M′′ versus χ_M′) for 1 were partially semicircular in shape in an H_dc of zero in the T range of 4–16 K (Fig. S19†). In the case of the χ_M′′ signals for 1 in the range of 10–996 Hz, the expected maximum value due to blocking could not be observed down to 1.8 K in an applied H_dc of zero (Fig. S19b†). On the contrary, below 10 K, a small shoulder peak in the range of 10–100 Hz was observed. A χ_M′′ peak top T of 5 K was observed when H_dc > 200 Oe (Fig. S20b†). The shape of the χ_M′′ peaks for 1 drastically changed and shifted to the low-ν side when an H_dc large enough to suppress QTM was applied, which will be described in detail below (Fig. 8).

The obtained τ values of 1 are plotted as a function of T⁻¹ in Fig. S21.† We observed QTM (T-independent regime for τ) with a fast τ below 10 K. These results confirm that QTM via the presence of off-diagonal LF terms in SP–SP environments is not suppressed in sharp contrast to that of 3 (SAP–SAP).⁶ Ishikawa and co-workers have reported that the ground state of an SP site has a high probability of QTM in comparison to an SAP one due to the presence of off-diagonal LF terms.^4,9a,13 In the above case, the τ values at 5 K for 1–3 in an H_dc of zero follow the order: 1 with SP–SP site (4.1 × 10⁻⁵ s) < 2 with SAP–SAP* sites (5.6 × 10⁻⁴ s) < 3 with SAP–SAP sites (1.8 × 10⁻² s) (Table S4†). This coordination relationship is also reflected in the τ of the QTM process at low-T region.

In order to study the field-induced SMM properties, the ν dependence of the χ_M′ and χ_M′′ signals for 1 in the range of 10–996 Hz were measured at each T (4–16 K) in H_dc of 1500 and 3000 Oe (Fig. S22 and S23†). From Argand plots in H_dc of 1500 and 3000 Oe, which were made using a generalized Debye model (eqn (S4)–(S6)†), the magnetic relaxations of 1 with SP–SP geometry showed a single component process.¹⁶ The obtained τ values are plotted as a function of T⁻¹ in Fig. S24.† The T-independent regime for QTM was restrained, and there were clear signs of the T-dependent regime for the Orbach process. The Δ was estimated from a χ_M′′ versus ν plot to be 32 cm⁻¹, and a τ₀ value of 3.97 × 10⁻⁶ s was obtained from an Arrhenius plot using τ = τ₀ exp(Δ/k_BT) for 1 in an H_dc of 1500 and 3000 Oe in the T range of 11–16 K (Fig. S24†). These results confirm that the QTM is suppressed in an H_dc and that the direct process is enhanced. This is clear evidence that τ of 1 depends heavily on T and H_dc.

On the other hand, as shown in Fig. S25,† Argand plots for 2 were semicircular in an H_dc of zero in the T range of 1.8–20 K and could be analyzed using a generalized Debye model (eqn (S4)–(S6)†).¹⁶ The obtained τ values of 2 are plotted as a function of T⁻¹ in Fig. 4b. We did not observe pure QTM (T-independent regime for τ), and the T-dependent regime for τ occurred under 10 K. These results confirm that pure QTM via the SAP–SAP* environments of 2 is suppressed as it is in 3 with SAP–SAP sites.⁶ The solid line represents a least-squares fit by using an Arrhenius equation with the following kinetic parameters: Δ = 15 cm⁻¹ and τ₀ = 4.8 × 10⁻⁵ s (Fig. 4b). These values have similar orders of magnitude as those obtained by using the Kramers–Kronig equation (Table S3†).

On the other hand, the T-dependent regime for τ in the low-T region appears to involve thermally assisted QTM via a doublet ground state, a direct process and/or a Raman process, which are difficult to separate from each other.^13d,15 Together with the Arrhenius plot of the data from χ_M′′T versus T (Fig. S16b†) and χ_M′′ versus ν plots for 2 in an H_dc of 1000 Oe (Fig. 4c and S26†), Δ was estimated to be 60 cm⁻¹ with τ₀ of 3.48 × 10⁻⁶ s (SAP site: blue circle in Fig. 6) and 162 cm⁻¹ with 3.16 × 10⁻⁸ s (SAP* site: red open circle in Fig. 6) in the T range of 18–29 K. Furthermore, Argand plots for 2 were partially semicircular in shape for the magnetic relaxation process in an H_dc of 3000 Oe in the T range of 4–17 K (Fig. S27†). Similar irregularity was observed in the Argand plots when T < 6 K but in an applied H_dc > 500 Oe (Fig. S28†).

Fig. 6 An Arrhenius plot of 2 (SAP–SAP*), where the magnetic relaxation time τ (blue and red circles) were obtained from χ_M′′ versus ν plots (Fig. S26b†) at T between 1.8 and 20 K in an H_dc of 1000 Oe and the red open circles were obtained from peak top T of χ_M′′T versus T plots (Fig. 4c) at T between 22 and 27 K in an H_dc of 1000 Oe. τ was obtained from the least-squares fitting using a generalized Debye model (eqn (S4)–(S6)†) and an extended Debye model (eqn (S8)–(S10)†).

In order to elucidate the relationship between the geometry and LF, we examined the correlation between the twist angle (φ) and Δ (Fig. 7). Δ versus φ plots for 1–3 and [TbPc₂]⁻ complex showed linear relationships. Above 10 K, the lowest excitation in the energy gap between the ground and first excited state levels of each SMM are approximately the same magnitude as the obtained Δ values because the intramolecular Tb^III–Tb^III interactions have little effect on the LF (Fig. S29†). As previously stated, the molecular geometry is related to the mixing of off-diagonal LF terms.^4,5,9,13 In other words, the degree of mixing of the off-diagonal LF terms increases from the D_4d ([TbPc₂]⁻ complex with φ ≈ 45°) to D_4h (1 with φ ≈ 4°). Thus, Δ for the D_4h geometry becomes narrower than that for the D_4d one due to the influence of LF parameters. This agrees with earlier studies.^{2,4,5,9,13i,j} In addition, there is a slight deviation from linearity in the plot of Δ due to distortions in the molecular geometry, such as longitudinal contraction of the coordination environment.^4e,i Selected geometry and SMM information for 1–3 and related SMM are listed in Table S5.†

Fig. 7 The relationship between the twist angle (φ) and the energy barrier for the reversal of the magnetization (Δ). The experimental data for [TbPc₂]⁻ complex with D_4d geometry from ref. 4c and 3 from ref. 7e. In order to suppress QTM, we applied an H_dc for 1 and 2 (see main text). The solid line is a guide for eyes.

Magnetic relaxation mechanism

The magnetic relaxation processes in dc magnetic fields can be explained by applying the Landau–Zener–Stükelberg (LZS) model of the ground state ±J_z levels.^1,2,17 Probability P of the adiabatic change can be obtained from the relationship between Δ_tunnel and the magnetic field sweep rate. The fast spin reversal between the ±J_z levels due to the adiabatic process is QTM. Relaxation via a direct process causes the release of a phonon (h_ω) when the spin flips in a nonadiabatic process. The probability of nonadiabatic process (1 − P) shows that the spin is not maintained with changes in the magnetic field sweep rate. This is thought to be the basis for the magnetic relaxation mechanism of 1–3 in the low-T region.

In a previous study on SMM 3 (SAP–SAP), we determined the ground state from the magnetic heat capacity (C_m) in an H_dc (Fig. S30†).⁶ When two Tb^III ions are in close proximity to each other, ferromagnetic (|±6, ±6〉) and antiferromagnetic states (|±6, ∓6〉) form, and these states are energetically separated by the f–f interactions between the Tb^III ions. Broadening of the C_m peak with an increase in H_dc shows that the J_z = |−6, −6〉, |+6, +6〉 and J_z = |−6, +6〉, |+6, −6〉 energy levels of the dinuclear Tb^III system are split by Zeeman energy. The magnetic axes of the ground state at the Tb^III sites are parallel to each other and have ferromagnetic dipole–dipole interactions among the magnetic moments (J_z = |±6, ±6〉). The dependence of the C_m of 3 on H_dc aligned along the magnetic easy axis could be reproduced theoretically (Fig. S30d†). The degenerate J_z = |±6, ±6〉 is split into J_z = |−6, −6〉 and J_z = |+6, +6〉 in an H_dc (Zeeman splitting). Thus, the crossing point at around ±4000 Oe is due to level crossing of the J_z = |±6, ∓6〉 state and J_z = |+6, +6〉 state. Furthermore, in the micro-SQUID experiments, 3 clearly exhibited a butterfly-shaped hysteresis loop with the field applied along the easy magnetization axis (Fig. S30b†).⁶ In the up sweep, the magnetization jump around ±3500 Oe is due to level crossing of the doublet ground state between the J_z = |±6, ∓6〉 state and J_z = |+6, +6〉 state.

Ishikawa and co-workers have obtained the Zeeman plots, where in the presence of the nuclear spin of I_z = ±3/2, ±1/2 with J_z = ±6 of the dinuclear Tb^III system, pure QTM in an H_dc of zero is forbidden because of the absence of mixing between the |±6, ∓6〉 and |±6, ±6〉 states.^9a Therefore, there is no crossover between the QTM and direct processes. At around ±3500 Oe, although QTM is possible, the direct process is a possible relaxation pathway in such a strong H_dc at low-T (Fig. 8c and S30b†). Therefore, we have reported that 3 shows pure QTM and a direct process in H_dc of around 3500 Oe, which is consistent with the Zeeman diagrams. Furthermore, we can explain the dual magnetic relaxation mechanism of 2 (SAP–SAP*) using Zeeman diagrams.

Fig. 8 τ versus H for (a) 1 (SP–SP), (b) 2 (SAP–SAP*), and (c) 3 (SAP–SAP)⁶ at 5 K. τ was obtained from the least-squares fitting using an extended Debye model (eqn (S8)–(S10) and Fig. S20 and S28†). The order of the maximum values of τ (red circle) are different from each other. See main text. The solid lines are guides for eyes.

On the other hand, below 18 K, the Argand plots for 2 showed two sets of irregular semicircular shapes for the dual magnetic relaxation processes in an H_dc of 1000 Oe (Fig. S26†). In other words, the magnetic relaxation splits from a one-component system into a two-component system (τ₁: high-frequency part and τ₂: low-frequency part) in an H_dc similar to the case of 3. In order to understand the different relaxation mechanisms corresponding to the two observed peaks, an extended Debye model (eqn (S8)–(S10)†) was used to fit the values of τ₁ and τ₂.^6,8aτ₁ and τ₂ are plotted as a function of T⁻¹ in Fig. 6. There are two different T regions. Above 18 K, the relaxation follows a thermally-activated mechanism, such as the Orbach process, whereas at lower temperatures (<10 K), a gradual crossover to a T-independent regime for τ₁ and T-dependent regime for τ₂ occurs. Similar behaviour was observed for 3 (SAP–SAP), and this is clear evidence that the magnetic relaxation mechanism depends heavily on the ground state properties of the Tb^III ions in the dinuclear systems below 10 K (Fig. 2).^6,7

To corroborate the relationship between the dual magnetic relaxation process in the dinuclear Tb^III complexes 1–3 in the low-T region and the octa-coordination environments, the ν dependence of χ_M′ and χ_M′′ in the range of 10–996 Hz were conducted at 5 K in several H_dc.^6,7 In the case of the Argand plots for 2 in an H_dc up to 6000 Oe at 5 K (Fig. S28†), the magnetic relaxation splits from a one-component system into a two-component system (τ₁ and τ₂). τ₁ (∼10⁻⁴ s) did not change in the range of 0–6000 Oe. On the other hand, τ₂ fluctuated with an “M” shaped curve (Fig. 8b) in the same ν range. τ₂ increased from 250 Oe and became a maximum of ∼10⁻² s in an H_dc of 1000 Oe (H_τ2,max). These results confirm that the QTM is suppressed in an H_dc of 1000 Oe, and the direct process is enhanced. When H_dc > 1000 Oe, the magnitude of τ₂ decreased to a minimum value (∼10⁻³ s) in H_dc of ∼3000 Oe (H_τ2,min). Then τ₂ began to increase again to a second maximum in an H_dc of 5000 Oe, followed by another decrease. These results confirm that the QTM is induced in the H_dc range of 1000–5000 Oe, and the direct process and QTM are enhanced due to Zeeman splitting of J_z = |±6, ±6〉 and level crossing of the doublet ground state between the J_z = |±6, ∓6〉 state and J_z = |+6, +6〉 state around 3000 Oe, respectively. The two levels of the lines in the energy diagrams are broadened because the coupled states are finely split by hyperfine coupling (I–J) and the interactions between the spins (f–f), which are due to absorption and/or emission of phonons (spin–phonon interactions).^4,6 Thus, the width of level crossing of the doublet ground state between the J_z = |±6, ∓6〉 and J_z = |+6, +6〉 states expanded in the H_dc range of 1000–5000 Oe.^7e Therefore, we believe that the magnetic relaxation process in an H_dc can be explained using Zeeman diagrams. By comparing the H_dc dependence of τ for the dinuclear Tb^III system, we found that the octa-coordination geometries with SAP–SAP* and SAP–SAP sites affected the H_dc regions where QTM was active.

In contrast to 2 and 3 (Fig. 8b and c), 1 with SP–SP sites exhibited only a single magnetic relaxation process (Fig. 8a), which could be described by using a generalized Debye model (eqn (S4)–(S6)†). The order of τ (10⁻⁴ s) increased from 250 Oe, reaching a maximum value of ∼10⁻³ s in an H_dc of 1500 Oe (H_τ,max). These results confirm that the QTM is suppressed in an H_dc = 1500 Oe and that the direct process is enhanced. When H_dc > 1500 Oe, the order of magnitude of τ decreased to a minimum value of ∼10⁻⁴ s in an H_dc of 6000 Oe. These results confirm that the QTM is suppressed up to an H_dc of 6000 Oe and that the direct process is enhanced. The ground state properties of 1 are completely different from those of 2 and 3. The ground state of 2 and 3 undergo a level crossing to give rise to a dual magnetic relaxation phenomenon, whereas 1 does not (Fig. 8a and S30†). This is consistent the differences in the values of Δ_tunnel in the ground states of the SP and SAP environments.^9a The differences are due to the differences in the ground state brought about by the symmetry of octa-coordination geometries and can be explained on the bases of the Zeeman diagrams.

Conclusions

The relationship between the octa-coordination environment and the SMM properties of 1 (SP–SP) and 2 (SAP–SAP*) with different octa-coordination environments are discussed in comparison to 3 (SAP–SAP). The SMM behaviours of the three complexes were explained by using X-ray crystallography and static and dynamic susceptibility measurements.

The relaxation dynamics of 1 and 2 indicate that they are field-induced SMMs because the QTM is enhanced by the presence of off-diagonal LF terms (B^q_k; q ≠ 0). Both 2 and 3 undergo dual magnetic relaxation processes in an H_dc, whereas 1 undergoes only a single magnetic relaxation process. The differences are due to the differences in the ground state of the respective octa-coordination geometries and can be explained on the bases of the Zeeman diagrams. Thus, the dual magnetic relaxation properties of dinuclear Tb^III complexes can be tuned by changing the strength of the mixing of fourth-range extradiagonal parameters using the octa-coordination environments, and thus, we must carefully design the dinuclear Tb^III triple-decker complexes to control the dual magnetic relaxation mechanisms.

In addition, we demonstrated that there was a linear relationship between the twist angle (φ) and the energy barrier for the reversal of the magnetization (Δ). Qualitatively, this result is consistent with the degree of the mixing of off-diagonal LF terms due to the symmetry of the octa-coordination environment. Detailed theoretical calculations will make it possible to fully elucidate the relationship between LF and SMM properties in detail.

Finally, the dual magnetic relaxation mechanisms are related the two energy gap in the doublet ground state in the dinuclear Tb^III ions with ferromagnetic dipolar interactions and the octa-coordination geometry because the SAP–SAP sites have axial components in the LF terms and because the SP–SP sites have not only the axial LF terms but also the off-diagonal LF terms.

Acknowledgements

This work was financially supported by a Grant-in-Aid for Scientific Research (S) (Grant No. 20225003, MY) and Grant-in-Aid for Young Scientists (B) (grant No. 24750119, KK) and Scientific Research (C) (grant No. 15K05467, KK) from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) and CREST, JST, Japan.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available. CCDC 989974 for 1 and 1041202 for 2. For ESI and crystallographic data in CIF or other electronic format see DOI: 10.1039/c5sc04669f

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