Supporting information

Supporting information Assembly of lanthanide(iii) cubanes and dimers with single-molecule magnetism and photoluminescence Ho-Yin Wong, Wesley Ting Kwok Chan, Ga-Lai Law* Department of Applied Biology and Chemical Technology, The Hong Kong Polytechnic University, Hung Hom, Kowloon 99977, Hong Kong Email: ga-lai.law@polyu.edu.hk

Figure S1. FT-IR spectrum of 1-La 4. Figure S2. FT-IR spectrum of 2-Ce 4. 1

Figure S3. FT-IR spectrum of 3-Pr 4. Figure S4. FT-IR spectrum of 4-Nd 4. 2

Figure S5. FT-IR spectrum of 5-Sm 4. Figure S6. FT-IR spectrum of 6-Eu 4. 3

Figure S7. FT-IR spectrum of 7-Gd 4. Figure S8. FT-IR spectrum of 8-Tb 4. 4

Figure S9. FT-IR spectrum of 9-Dy 4. Figure S1. FT-IR spectrum of 1-Tb 2. 5

Figure S11. FT-IR spectrum of 11-Dy 2. Figure S12. FT-IR spectrum of 12-Ho 2. 6

Figure S13. FT-IR spectrum of 13-Er 2. Figure S14. FT-IR spectrum of 14-Tm 2. 7

Figure S15. FT-IR spectrum of 15-Yb 2. Figure S16. FT-IR spectrum of 16-Lu 2. 8

% weight % weight % weight % weight % weight % weight % weight % weight 1 1-La 4 1 2-Ce 4 8 8 6 6 4 4 4 6 8 4 6 8 Temperature ( o C) Temperature ( o C) 1 3-Pr 4 1 4-Nd 2 8 8 6 6 4 4 4 6 8 4 6 8 Temperature ( o C) Temperature ( o C) 1 5-Sm 4 1 6-Eu 4 8 8 6 4 6 4 4 6 8 4 6 8 Temperature ( o C) Temperature ( o C) 1 7-Gd 4 1 8-Tb 4 8 8 6 6 4 4 4 6 8 4 6 8 Temperature ( o C) Temperature ( o C) 9

% weight % weight % weight % weight % weight % weight % weight % weight 1 9-Dy 4 1 1-Tb 2 8 8 6 6 4 4 4 6 8 4 6 8 Temperature ( o C) Temperature ( o C) 1 11-Dy 2 1 12-Ho 2 8 8 6 6 4 4 4 6 8 4 6 8 Temperature ( o C) Temperature ( o C) 1 13-Er 2 1 14-Tm 2 8 8 6 6 4 4 4 6 8 4 6 8 Temperature ( o C) Temperature ( o C) 1 15-Yb 2 1 16-Lu 2 8 8 6 6 4 4 4 6 8 4 6 8 Temperature ( o C) Temperature ( o C) Figure S17. TGA curves of 1 16. 1

Figure S18. 1 H NMR (4 MHz, DMSO-d 6) spectrum of 1-La 4. Figure S19. 1 H NMR (4 MHz, DMSO-d 6) spectrum of 5-Sm 4. 11

Figure S. 1 H NMR (4 MHz, DMSO-d 6) spectrum of 6-Eu 4. 12

Tricapped trigonal prism (D 3h) Capped square antiprism (C 4v) Trigonal dodecahedron (D 2d) Bicapped trigonal prism (C 2v) Square antiprism (D 4d) Figure S21. Presentation of the coordination geometry of 1-La 4, 6-Eu 4, 1-Tb 2, 11-Dy 2 and 16-Lu 2 using polyhedron. 13

Figure S22. Packing diagram of 2-Ce 4 (top), 7-Gd 4 (middle) and 11-Dy 2 (bottom). For clarity, only the intermolecular π-interactions are marked. 14

Normalized intensity Normalized intensity Normalized intensity Figure S23. Crystal structures of 2-Ce 4-15K (left) and 2-Ce 4-RT (right). 4-Nd 4 9-Dy 4 16-Lu 2 15-Yb 2 3-Pr 4 8-Tb 4 14-Tm 2 7-Gd 4 2-Ce 4 13-Er 2 6-Eu 4 12-Ho 2 1-La 4 5-Sm 4 11-Dy 2 1-La 4 simulated 6-Eu 4 simulated 1-Tb 2 1-Tb 2 Simulated 5 1 15 25 3 2θ ( o ) 5 1 15 25 3 2θ ( o ) 5 1 15 25 3 2θ ( o ) Figure S24. PXRD patterns for 1 4, 5 9 and 1 16. Further explanation for PXRD. The PXRD of compounds 1 16 were recorded to indicate the phase purity of the sample. For compounds 1 4 and 1 16, their PXRD patterns match the simulated patterns. However, for 5 9, the experimental PXRD patterns do not match the simulated pattern. This issue can be explained by the degradation of those single crystals. In the following figure, the newly grown single crystal of 6-Eu 4 and the degraded crystals after 1 day under an optical microscope are displayed: 15

Normalized intensity Many cracks appeared, and the transparency of the crystal was lost. Same behavior can be observed for compounds 5 9. The issue was examined by mounting the single crystal and the degraded crystal of 6- Eu 4 on a single crystal X-ray diffractometer (Bruker D8-Venture X-ray diffractometer) to record the powder XRD patterns, and the result is displayed below (red and pink lines): 6-Eu 4 Degraded SCXRD 6-Eu 4 Powder XRD 6-Eu 4 SCXRD 6-Eu 4 Simulated PXRD 5 1 15 25 3 2θ ( o ) Since the PXRD function of the Bruker D8-Venture X-ray diffractometer is still in its infancy, the patterns are not ideal, but the PXRD pattern of the 6-Eu 4 single crystal (red line) and the simulated pattern (black line) match. It indicates that the single crystal structure data is not problematic in terms of unit cell and space group determination. Also, there is no warning in Checkcif files. On the contrary, the PXRD pattern of the degraded crystal of 6-Eu 4 (pink line) matches that of the powder 6-Eu 4 (blue line, recorded by Bruker D8 Advance X-ray Diffractometer), and deviates from the simulated pattern. It shows that the crystal may undergo phase transition upon leaving the mother liquid, probably induced by the evaporation of the trapped acetone. Despite the deviation in PXRD for 5-9, these tetranuclear cubanes adequately characterized by CHN elemental analysis, 1 H NMR (5-Sm 4 and 6-Eu 4), FT-IR, TGA, UV-vis and DC magnetic susceptibility measurement. 16

4.5 6.5 4. 3-Pr 4 6. 4-Nd 4 M T/cm 3 mol -1 3.5 3. 2.5 2. 1.5 1..5. 5 1 15 25 3 Temperature/K M T/cm 3 mol -1 5.5 5. 4.5 4. 3.5 3. 2.5 2. 5 1 15 25 3 Temperature/K 27 26 12-Ho 2 M T/cm 3 mol -1 25 24 23 22 21 5 1 15 25 3 Temperature/K M T/cm 3 mol -1 26 24 22 18 16 14 12 1 8 6 13-Er 2 M T/cm 3 mol -1 4.8 4.6 4.4 4.2 4. 3.8 3.6 3.4 3.2 3. 15-Yb 2 5 1 15 25 3 Temperature/K 5 1 15 25 3 Temperature/K Figure S25. Temperature-dependence of magnetic susceptibilities, χ MT vs. T, for 3-Pr 4, 4-Nd 4, 12-Ho 2, 13-Er 2 and 15-Yb 2 at 1 Oe. Except 12-Ho 2 with ferromagnetic interactions, 3-Pr 4, 4-Nd 4, 13-Er 2 and 15-Yb 2 possess antiferromagnetic interaction between metal ions. 17

M "/cm 3 mol -1 M "/cm 3 mol -1 M '/cm 3 mol -1 M '/cm 3 mol -1 M "/cm 3 mol -1 M '/cm 3 mol -1 25 15 9 Hz 165 Hz 17 Hz 256 Hz 26 Hz 397 Hz 39 Hz 617 Hz 58 Hz 957 Hz 87 Hz 1488 Hz 1 5 1..8 8-Tb 4.6.4.2. 2 4 6 8 1 T/K 15 1 9 Hz 165 Hz 17 Hz 256 Hz 26 Hz 397 Hz 39 Hz 617 Hz 58 Hz 957 Hz 87 Hz 1488 Hz 14 12 1 8 6 9 Hz 165 Hz 17 Hz 256 Hz 26 Hz 397 Hz 38 Hz 617 Hz 58 Hz 957 Hz 87 Hz 1488 Hz 5 14 12 1 8 6 4 2 9-Dy 4 4 2 4 3 2 1 11-Dy 2 2 4 6 8 1 12 T/K 2 4 6 8 1 12 14 16 18 T/K Figure S26. Temperature dependence of the in-phase and out-of-phase ac susceptibility data for 8-Tb 4, 9-Dy 4 and 11-Dy 2 under zero dc field. The solid lines are the visual guide. 18

M "/cm 3 mol -1 M "/cm 3 mol -1 M "/cm 3 mol -1 M "/cm 3 mol -1 M '/cm 3 mol -1 M '/cm 3 mol -1 18 16 14 12 1 8 6 4 2 6 5 4 3 2 1 1 Hz 58 Hz 2 Hz 87 Hz 3 Hz 165 Hz 5 Hz 256 Hz 9 Hz 397 Hz 17 Hz 617 Hz 26 Hz 957 Hz 39 Hz 1488 Hz 2 4 6 8 1 12 T/K 7 6 5 4 3 2 1 3 9-Dy 4 2 1 1 Hz 58 Hz 2 Hz 87 Hz 3 Hz 165 Hz 5 Hz 256 Hz 9 Hz 397 Hz 17 Hz 617 Hz 26 Hz 957 Hz 39 Hz 1488 Hz 2 4 6 8 1 12 14 16 18 T/K 11-Dy 2 Figure S27. Temperature dependence of the in-phase and out-of-phase ac susceptibility data for 9-Dy 4 and 11-Dy 2 under Oe dc field. The solid lines are the visual guide. 6 5 4 3 2 1 1 1 1 1 Frequency/Hz 9-Dy 4 2K 2.5K 3K 3.5K 4K 4.5K 5K 5.5K 6K 6.5K 7K 7.5K 8K 8.5K 9K 9.5K 1K 1.5K 11K 11.5K 12K 2.5 2. 1.5 1..5. 1 1 1 1 Frequency/Hz 11-Dy 2 2K 2.5K 3K 3.5K 4K 4.5K 5K 5.5K 6K 6.5K 7K 7.5K 8K 8.5K 9K 9.5K 1K 1.5K 11K 11.5K 12K 12.5K 13K 13.5K 14K 14.5K 15K 15.5K 16K 16.5K 17K 17.5K 18K Figure S28. Frequency dependence of the out-of-phase ac susceptibility data for 9-Dy 4 and 11-Dy 2 under Oe dc field. The solid lines are the visual guide. 19

-2-4 9-Dy 4 zero-field 9-Dy 4 2k Oe -2 11-Dy 2 zero-field 11-Dy 2 2k Oe -4-6 ln ln -6-8 -8-1 -1-12.1.2.3.4.5..1.2.3.4.5 1/T (K -1 ) 1/T (K -1 ) Figure S29. Arrhenius plots of relaxation time data for 9-Dy 4 and 11-Dy 2 under Oe dc field. The solid lines correspond to fits to the equation τ -1 = τ -1 exp(-u eff/k BT) + CT n + τ QTM -1..6.4 8-Tb 4 M/M S.2. -.2 -.4 -.6-2 -1 1 2 μ /H (T) 2K 3K 4K 5K 6K 7K 8K 9K 1K M/M S.6.4.2. -.2 -.4 -.6 9-Dy 4 2K 3K 4K 5K 6K 7K 8K 9K 1K M/M S.4.2. -.2 -.4 11-Dy 2 2K 3K 4K 5K 6K 7K 8K 9K 1K -2-1 1 2 μ /H (T) -1 1 μ /H (T) Figure S3. Variable-field magnetization (M) data for 8-Tb 4, 9-Dy 4 and 11-Dy 2 collected from 2 to 1 K.

Intensity (a.u.) Normalized absorbance Normalized absorbance 9-Dy 4 16-Lu 2 8-Tb 4 15-Yb 2 7-Gd 4 14-Tm 2 6-Eu 4 5-Sm 4 13-Er 2 4-Nd 4 12-Ho 2 3-Pr 4 11-Dy 2 2-Ce 4 1-Tb 2 1-La 4 25 3 35 Wavelength (nm) 25 3 35 Wavelength (nm) Figure S31. UV-vis absorption spectra of 1-16 in acetonitrile, 1 um. 7 F 5 L 6 8-Eu 4 em = 614 nm 7 F 5 D 2 Intensity (a.u.) 8-Tb 4 em = 541 nm 5 D4 4 F 6 25 3 35 4 45 5 Wavelength (nm) 25 3 35 4 45 5 Wavelength (nm) Figure S32. Excitation spectra of 6-Eu 4 and 8-Tb 4 at room temperature. 21

Intensity (a.u.) Intensity (V) Intensity (V) Intensity (V) 7 6 5 5 D 7 F 2 ex = 337 nm = 674 s 8 6 5 D 4 4 F 5 ex = 337 nm = 9.1 s 4 3 4 2 1 2-1 1 3 4 Time ( s) 4 6 8 1 Time ( s) Figure S33. Luminescence decay curves of 6-Eu 4 (left) and 8-Tb 4 (right). The solid lines are the monoexponential fit curve. 1-La 4 ex = 35 nm 4 3 ex = 337 nm em = 449 nm = 88.9 ms 2 1 4 45 5 55 6 65 Wavelength (nm) 4 6 8 Time ( s) Figure S34. Emission spectrum and luminescence decay curve of 1-La 4 at K. 22

J = 2 Intensity (a.u.) Intensity (V) J = 1 J = 6 J = 3 Intensity (a.u.) J = 4 J = 5 5 D 4 4 F J 1-Tb 2 45 5 55 6 65 7 Wavelength (nm) 6 1-Tb 2 em = 541 nm 5 4 3 5 D 4 4 F 5 ex = 337 nm = 854 s 5 D4 4 F 6 2 1 25 3 35 4 45 5 Wavelength (nm) 1 3 4 5 6 7 Time ( s) Figure S35. Emission and excitation spectra and luminescence decay curve of 1-Tb 2 at K. Emission measurement condition: λ ex = 35 nm, slit = 4..5 nm, longpass filter = 4 nm. 23

J = Intensity (a.u.) Intensity (V) J = 3 J = 1 Intensity (a.u.) J = 4 J = 2 5 D 7 F J 6-Eu 4 ex = 35 nm 56 6 64 68 7 Wavelength (nm) 6-Eu 4 em = 613 nm 8 5 D 7 F 2 ex = 337 nm 5 L6 7 F 5 D2 7 F 6 4 2 = 763 s 25 3 35 4 45 5 Wavelength (nm) 1 3 4 Time ( s) Figure S36. Emission and excitation spectra and luminescence decay curve of 6-Eu 4 at K. Emission measurement condition: λ ex = 35 nm, slit = 2..3 nm, longpass filter = 4 nm. 24

J = 3 J = 2 Intensity (a.u.) Intensity (V) J = 1 Intensity (a.u.) J = 6 J = 4 J = 5 5 D 4 4 F J 8-Tb 4 ex = 35 nm 45 5 55 6 65 7 Wavelength (nm) 8-Tb 4 em = 541 nm 5 4 3 5 D 4 4 F 5 ex = 337 nm = 155 s 5 D4 4 F 6 2 1 25 3 35 4 45 5 Wavelength (nm) 4 6 8 Time ( s) Figure S37. Emission and excitation spectra and luminescence decay curve of 8-Tb 4 at K. Emission measurement condition: λ ex = 35 nm, slit = 2..3 nm, longpass filter = 4 nm. 25

Figure S38. HRMS of fragment ion of 2-Ce 4 [Ce 4(OH) 4+3tfa+4hfa+4phen] +. Figure S39. HRMS of fragment ion of 3-Pr 4 [Pr 4(OH) 4+3tfa+4hfa+4phen] +. Figure S4. HRMS of fragment ion of 4-Nd 4 [Nd 4(OH) 4+3tfa+4hfa+4phen] +. 26

Figure S41. HRMS of fragment ion of 5-Sm 4 [Sm 4(OH) 4+3tfa+4hfa+4phen] +. Figure S42. HRMS of fragment ion of 7-Gd 4 [Gd 4(OH) 4+3tfa+4hfa+4phen] +. 27

Table S1. Summary of crystal data and structure refinement. 1-La 4 2-Ce 4 3-Pr 4 4-Nd 4 Formula La 4C 76H 4N 8O F 36 Ce 4C 76H 4N 8O F 36 Pr 4C 76H 4N 8O F 36 Nd 4C 76H 4N 8O F 36 Formula weight 2624.8 2629.64 2632.8 2646.12 Temperature (K) 293(2) 293(2) 297(2) 293(2) Wavelength (Å).7173.7173.7173.7173 Crystal system Tetragonal Tetragonal Tetragonal Tetragonal Space group I4 1/a I4 1/a I4 1/a I4 1/a a (Å) 14.2215(3) 14.21(3) 14.224(5) 14.2443(5) b (Å) 14.2215(3) 14.21(3) 14.224(5) 14.2443(5) c (Å) 47.65(2) 47.6166() 47.478(3) 47.38(3) α (deg) 9 9 9 9 β (deg) 9 9 9 9 γ (deg) 9 9 9 9 V (Å 3 ) 9628.1(5) 9614.9(5) 965.8(7) 9613.5(7) Z 4 4 4 4 D c (Mg/m 3 ) 1.811 1.817 1.821 1.828 μ (mm -1 ) 1.876 1.995 2.13 2.262 F() 556 572 588 514 a R 1 [I > 2σ(I)].369.365.439.368 b wr 2 [I > 2σ(I)].85.88.131.863 a R 1 (all data).522.473.758.571 b wr 2 (all data).964.897.1223.993 GOF c 1.114 1.82 1.99 1.86 5-Sm 4 6-Eu 4 7-Gd 4 8-Tb 4 Formula Sm 4C 82H 52N 8O 22F 36 Eu 2C 24H 12N 2O 6F 18 Gd 2C 24H 12N 2O 6F 18 Tb 2C 24H 12N 2O 6F 18 Formula weight 2786.72 1396.58 1349.8 1352.42 Temperature (K) 15(2) 567(2) 297(2) 297(2) Wavelength (Å).7173.7173.7173.7173 Crystal system Tetragonal Orthorhombic Orthorhombic Orthorhombic Space group I4 1/a Pbcn Pbcn Pbcn a (Å) 14.1851(4) 16.3568(9) 16.8678(12) 16.8319(15) b (Å) 14.1851(4) 22.5395(11) 22.5425(15) 22.5311(19) c (Å) 47.548(26) 25.9278(14) 25.915(19) 25.818(2) α (deg) 9 9 9 9 β (deg) 9 9 9 9 γ (deg) 9 9 9 9 V (Å 3 ) 9567.5(6) 9558.9(9) 9854.(12) 9791.3(15) Z 4 8 8 8 D c (Mg/m 3 ) 1.935 1.941 1.819 1.835 μ (mm -1 ) 2.563 2.733 2.791 2.989 F() 5392 548 5168 5184 a R 1 [I > 2σ(I)].319.612.1255.995 b wr 2 [I > 2σ(I)].76.1279.2282.2163 a R 1 (all data).372.197.2767.1986 b wr 2 (all data).795.1521.3344.2714 GOF c 1.57 1.3 1.4 1.3 28

9-Dy 4 1-Tb 2 11-Dy 2 12-Ho 2 Formula Dy 2C 24H 12N 2O 6F 18 Tb 2C 44H 22N 4O 1F 24 Dy 2C 44H 22N 4O 1F 24 Ho 2C 44H 22N 4O 1F 24 Formula weight 1359.58 154.5 1547.66 1552.52 Temperature (K) 297(2) K 297(2) 294(2) 297(2) Wavelength (Å).7173.7173.7173.7173 Crystal system Orthorhombic Triclinic Triclinic Triclinic Space group Pbcn P-1 P-1 P-1 a (Å) 16.8732(17) 1.3394(1) 1.1922(6) 1.23(6) b (Å) 22.518(2) 11.1742(11) 11.1245(7) 11.1418(7) c (Å) 25.766(3) 13.487(13) 13.5282(11) 13.4877(8) α (deg) 9 98.836(3) 67.679(2) 99.3141(18) β (deg) 9 112.35(3) 67.883(2) 112.67(17) γ (deg) 9 17.855(3) 73.194(2) 16.9388(18) V (Å 3 ) 9789.6(17) 139.5(2) 1295.28(15) 1297.6(14) Z 8 1 1 1 D c (Mg/m 3 ) 1.845 1.953 1.984 1.987 μ (mm -1 ) 3.153 2.822 3.8 3.172 F() 5 74 742 744 a R 1 [I > 2σ(I)].1134.435.299.367 b wr 2 [I > 2σ(I)].2282.928.687.785 a R 1 (all data).2179.585.413.474 b wr 2 (all data).2892.145.762.875 GOF c 1.4 1.143 1.89 1.166 13-Er 2 14-Tm 2 15-Yb 2 16-Lu 2 Formula Er 2C 44H 22N 4O 1F 24 Tm 2C 44H 22N 4O 1F 24 Yb 2C 44H 22N 4O 1F 24 Lu 2C 44H 22N 4O 1F 24 Formula weight 1557.18 156.52 1568.74 1572.6 Temperature (K) 15(2) 296(2) 293(2) 296(2) Wavelength (Å) 1.54178 1.54178.7173.7173 Crystal system Triclinic Triclinic Triclinic Triclinic Space group P-1 P-1 P-1 P-1 a (Å) 1.478(5) 1.1954(5) 1.1832(5) 1.172(19) b (Å) 1.8332(5) 11.1354(5) 11.1168(6) 11.1176(22) c (Å) 13.4724(6) 13.4749(6) 13.4137(7) 13.4267(27) α (deg) 1.3258(13) 99.6783(16) 99.8561(15) 99.94(52) β (deg) 111.7733(11) 112.6(14) 111.8722(13) 111.934(45) γ (deg) 15.319(12) 16.5382(16) 16.513(14) 16.2756(48) V (Å 3 ) 1249.3(1) 1293.13(1) 1283.89(12) 1284.9(4) Z 1 1 1 1 D c (Mg/m 3 ) 2.7 2.4 2.29 2.32 μ (mm -1 ) 7.45 7.598 3.767 3.966 F() 746 748 75 752 a R 1 [I > 2σ(I)].283.458.224.199 b wr 2 [I > 2σ(I)].756.116.556.512 a R 1 (all data).294.514.277.226 b wr 2 (all data).763.1166.582.531 GOF c 1.27 1.62 1.81 1.83 29

2-Ce 4-15K 2-Ce 4-RT Formula La 4C 76H 4N 8O F 36 Ce 4C 76H 4N 8O F 36 Formula weight 2745.8 2629.64 Temperature (K) 15(2) 294(2) Wavelength (Å).7173.7173 Crystal system Tetragonal Tetragonal Space group I4 1/a I4 1/a a (Å) 14.49(14) 14.68(3) b (Å) 14.49(14) 14.68(3) c (Å) 47.7658(99) 47.6434(24) α (deg) 9 9 β (deg) 9 9 γ (deg) 9 9 V (Å 3 ) 9417(2) 9616.(6) Z 4 4 D c (Mg/m 3 ) 1.937 1.816 μ (mm -1 ) 2.43 1.995 F() 5328 572 a R 1 [I > 2σ(I)].258.39 b wr 2 [I > 2σ(I)].662.61 a R 1 (all data).316.46 b wr 2 (all data).691.669 GOF c 1.64 1.155 a R1 = Fo Fc / Fo ; b wr = { [w(fo 2 Fc 2 )] 2 }/ [w(fo 2 )2]} 1/2. c Goodness-of-fit = { [w(fo 2 Fc 2 )] 2 }/(Nobs Nparam)} 1/2, based on all data. 3

Table S2. Results from the shape analysis for compounds 1 16. Compounds Shape measure ( o ) Tricapped trigonal prism (D 3h) Capped square antiprism (C 4v) 1-La 4 8.14 9.42 2-Ce 4 8.13 9.21 3-Pr 4 8.11 9.7 4-Nd 4 7.99 8.9 5-Sm 4 8. 8.47 6-Eu 4 7.87 6.13 1.73 5.5 7-Gd 4 8.4 6.15 1.99 5.68 8-Tb 4 8.36 6.33 1.66 5.96 9-Dy 4 8.58 6.5 1.79 6. Shape measure ( o ) Compounds Bicapped trigonal Trigonal dodecahedron Square antiprism (D 4d) prism (C 2v) (D 2d) 1-Tb 2 8.58 8.13 7.8 11-Dy 2 8.22 8.9 8.32 12-Ho 2 8.13 8.14 8.29 13-Er 2 7.3 8.17 9.67 14-Tm 2 7.84 8. 8.67 15-Yb 2 7.89 8.3 8.77 16-Lu 2 7.66 8.17 8.95 Table S3. Temperature-dependence of magnetic susceptibilities of Ln III clusters at 3 K and 1 Oe. Clusters Experimental χ MT (cm 3 mol -1 K) Expected χ MT (cm 3 mol -1 K) 3-Pr 4 4. 6.4 4-Nd 4 6. 6.5 12-Ho 2 26.2 28.1 13-Er 2 23.7 23. 15-Yb 2 4.7 5.1 Table S4. Comparison of magnetic properties of some Ln III hydroxo dimeric clusters. Compounds Dy OH (Å) a Dy O Dy ( o ) a O Dy O ( o ) a Dy Dy distance (Å) a SMM b Ueff (K) c Ref 11-Dy2 [Ln2(μ-OH)2(hfa)4(phen)2] 2.239 ±.14 11.1 69.9 3.668 Yes 82.5 This work [Dy2(12-C-4)2(ClO4)4(OH)2(H2O)2 2.229 ±.2 19.29 7.71 3.635 Yes 35 (1k Oe) 1 [Dy2(μ- OH)2(H2bpte)2Cl2(MeOH)2]Cl2 2.28 ±.3 19.58 7.42 3.726 Yes 59 (2.5k Oe) 2 [Dy2(tepa)2 (μ2-oh)2cl2]3[as3se6]2 2.261 ±.3 111.38 68.6 3.735 Yes 9.7 (1k Oe) 3 [Dy(μ-OH)(DBP)2(THF)]2 2.259 ±.2 11.49 69.51 3.712 Yes 754 4 [Dy2(apca)4(µ2-OH)2(H2O)2]n 2.264 ±.4 19.52 7.48 3.698 Yes 61.2 5 31

Table S5. Fitting parameters of 9-Dy 4 and 11-Dy 2 for the equation of Raman and Orbach processes and QTM model. Parameter 9-Dy 4 9-Dy 4 (2K Oe) 11-Dy 2 11-Dy 2 (2K Oe) τ 1.22 x 1-8 1.31 x 1-8 1.73 x 1-7 3.15 x 1-7 U eff 67. 78.7 82.6 86.1 C 9.25 1.9 2.57.16 n 2.94 3.82 3.5 4.16 τ QTM.18 -.1 - Fitting function τ -1 = τ -1 exp(-ueff/kbt) + CT n + τqtm -1 Table S6. Parameters of 9-Dy 4 obtained from Cole-Cole plots using the generalized Debye model. Temperature (K) χ S (cm 3 mol -1 K) χ T (cm 3 mol -1 K) α 2 1.578 36.763.168 2.5 1.928 28.615.177 3 2.98 23.1.177 3.5 2.163 19.18.168 4 2.163 16.28.155 4.5 2.129 14.159.137 5 2.78 12.514.119 5.5 2.22 11.189.1 6 1.938 1.1.91 6.5 1.945 9.233.82 7 1.941 8.481.87 Fitting function y =.5*(χS-χT)/tan((1-α)*1.577)+sqrt((x-χS)*(χT-x)+.25*(χT-χS)^2/(tan((1-α)*1.577))^2) Table S7. Parameters of 11-Dy 2 obtained from Cole-Cole plots using the generalized Debye model. Temperature (K) χ S (cm 3 mol -1 K) χ T (cm 3 mol -1 K) α 2 2.143 14.461.266 2.5 1.742 11.468.27 3 1.484 9.452.269 3.5 1.315 8.28.263 4 1.193 6.976.256 4.5 1.1 6.176.248 5 1.45 5.541.233 5.5 1.85 4.77.168 6.983 4.569.185 6.5.949 4.4.162 7.926 3.891.135 Fitting function y =.5*(χS-χT)/tan((1-α)*1.577)+sqrt((x-χS)*(χT-x)+.25*(χT-χS)^2/(tan((1-α)*1.577))^2) 32

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