Magnetic Anisotropy Trends along a Full 4f-Series: The fn+7 Effect

The combined experimental and computational study of the 13 magnetic complexes belonging to the Na[LnDOTA(H2O)] (H4DOTA = tetraazacyclododecane-N,N′,N″,N‴-tetraacetic acid and Ln = Ce–Yb) family allowed us to identify a new trend: the orientation of the magnetic anisotropy tensors of derivatives differing by seven f electrons practically coincide. We name this trend the fn+7 effect. Experiments and theory fully agree on the match between the magnetic reference frames (e.g., the easy, intermediate, and hard direction). The shape of the magnetic anisotropy of some couples of ions differing by seven f electrons might seem instead different at first look, but our analysis explains a hidden similarity. We thus pave the way toward a reliable predictivity of the magnetic anisotropy of lanthanide complexes with a consequent reduced need of computational and synthetical efforts. We also offer a way to gain information on ions with a relatively small total angular momentum (i.e., Sm3+ and Eu3+) and on the radioactive Pm3+, which are difficult to investigate experimentally.

S4 Figure S2. Description of the first coordination sphere around the Ln ion. The reported angles are averaged on the four oxygen (or nitrogen) atoms and along the entire series. Figure S3. View of the packing. The labels of the oxygen atoms correspond to the labels reported in Figure S4. O5 and O6 are the closest oxygen atoms belonging to neighboring molecules (orange and yellow). Color code: Ln: green, O: red, N: blue, C: gray, Na: violet. Figure S4. Distances between the oxygen of the apical water molecule (Ow) and a) the two closest carboxylic oxygens of the neighboring molecules and b) the four coordinated carboxylic oxygens. The lines are a guide to the eye.
S5 Figure S5. Shortest Ln-Ln distance in the crystal lattice. The line is a guide to the eye.

Experimental determination of the easiest direction
We performed the measurements using a home made two-legged CuBe cantilever separated by 0.1 mm from a gold plate. The cantilever was inserted into an Oxford Instruments MAGLAB2000 platform with automated rotation in a vertical magnet. The capacitance was detected with an Andeen-Hagerling 2500A Ultra Precision Capacitance Bridge.  Figure S6. Fits (with fictitious b2 0 and b2 2 parameters) of the experimental torque for all the derivatives with previously unknown direction of the easy axis. The fit for Eu was performed using a J = 1 ground state. Rot2 for Nd did not give a detectable signal. The magnetic field (B) at the beginning of Rot1 and Rot2 was along (-0.989,0.150,0) and (0.150,0.989,0), respectively. The rotation axis (Y) for Rot1 and Rot2 was (-0.150,-0.989,0) and (-0.989,0.150,0), respectively. The only exceptions are: Rot2 for Ce (B = (0,-1,0) and Y = (-0.643, 0, -0.766)), and Rot2 for Gd (B = (0.150,0.989,0) and Y = (0,0,-1)).
The Hamiltonian used for the fits in Figure S6 was composed by a crystal field and a Zeeman term: Where b2 0 (O2 0 ) and b2 2 (O2 2 ) are the 2 nd order axial and rhombic parameter (operator), gJ is the appropriate Landè g-factor, B is the Bohr magneton, Ĵ is the total angular momentum operator and B is the magnetic field. S9 EPR X-band EPR spectra (ν  9.5 GHz, exact frequency reported in the caption of each spectrum) were recorded on a Bruker Elexsys E500 spectrometer equipped with an ESR900 (Oxford Instruments) continuous-flow 4 He cryostat to work at low temperature and a SHQ resonator. The crystalline powder of each sample was ground and pressed in pellet to avoid preferential orientation and then placed in 4 mm diameter quartz tubes. EPR spectra (ν  94 GHz) of Gd were recorded on a Bruker E600 continuous-wave spectrometer with cylindrical cavity equipped with a split-coil superconducting magnet that generates a horizontal magnetic field (Oxford Instruments). Temperature was controlled with a continuous-flow cryostat (Oxford CF935), operating from room temperature down to 4.2 K. Ground powder was dispersed in wax to avoid preferential orientation of the micro-crystallites due to magnetic torque, and inserted in 0.8 mm diameter quartz tube to perform the measurement. Notwithstanding this procedure, spectra clearly shows the additional contribution of a single individual crystal which is not powder averaged.
Simulations of the derivatives containing anisotropic Kramers' ions providing an EPR signal (Ce, Nd, Er, Yb) were performed using EasySpin 2 on the basis of an effective doublet spin Hamiltonian, including hyperfine coupling where needed (i.e. except for Ce): For Gd, a Spin Hamiltonian acting on the S = 7/2 ground multiplet was considered, with isotropic g value (g = 1.995), rhombic second order contributions and only axial fourth order terms (Stevens' notation) 3 : Best simulation parameters are reported in Table S4, and comparison between experimental and simulated spectra are reported in Figure S7-S11. For the sake of clarity, in Figure S12 we report simulation of Gd obtained by assuming a reverse sign for the Spin Hamiltonian parameters with respect to the best fit ones reported on main text. While this set still provides the correct resonance fields, it clearly fails to correctly reproduce the temperature dependence of the spectra. In all cases, pseudo-Voigtian lines were assumed, with isotropic H-Strain superimposed on them to account for unresolved hyperfine coupling. S10 Figure S7. EPR X-band (ν = 9.4308 GHz) experimental spectrum (red trace) of Ce measured at 5 K and best simulation (black trace) obtained with parameters reported in Table S4. Figure S8. EPR X-band (ν = 9.4270 GHz) spectrum of Nd measured at 5 K and best simulation obtained with parameters reported in Table S4. S11 Figure S9. EPR X-band (ν = 9.6398 GHz) spectrum of YEr (doping level 5%) measured at 5 K and best simulation obtained with parameters reported in Table S4. Figure S10. EPR X-band (ν = 9.4046 GHz) spectrum of Yb measured at 5 K and best simulation obtained with parameters reported in Table S4.
S12 Figure S11. EPR X-(freq = 9.4200 GHz) and W-band (freq = 94.320 GHz) experimental spectra (red traces) of GdY (doping level 6 %) at different temperatures and best simulation (black traces). Figure S12. Comparison between experimental (red traces) and simulated (black traces) EPR spectra assuming opposite signs of the Spin Hamiltonian parameters with respect to those reported in Table S4 and Figure 3.

Dc measurements
The samples were measured in form of ground and pressed crystalline powders wrapped in Teflon. The mass of the samples was of the order of tens of mg. For the dc measurements and the ac measurements at low frequency, a Quantum Design MPMS SQUID magnetometer (0.1 Hz -1 kHz) was used. For the ac measurements at high frequency, Quantum Design PPMS equipped with AC susceptibility probe (10 Hz-10 kHz) was used.

Ab initio model
All the calculations in this study were performed on experimentally determined structures (the only exception being the radioactive Pm, for which the structure of Sm was used). The position of the water's hydrogen atoms was kept fixed on the positions optimized for Dy considering the interactions with neighboring chemical groups, as previously discussed. 6 Indeed, analyzing the crystal packing along the whole structural series, the oxygens atoms framework around the apical water molecule is maintained for all the derivatives (see Crystallographic section). Second-order Douglas-Kroll-Hess Hamiltonian has been employed in all the calculations to consider scalar relativistic corrections.    z  2  21  4  7  2  1  y  2  21  5  12  69  45  x  1  6  2  11  69  45  Table S12. Angle between the ab initio reference frame of the 6 couples differing by 7 f-electrons.
All the values are expressed in degrees. Figure S41. 4f CASSCF/RASSI-SO orbitals for all the derivatives except Gd. 4f orbitals are associated with different ml values, assuming the pseudo-C4 axis of the complex as the quantization axis. The association between colours of the squares and ml values is the following: red/ml=1, blue/ml=2, green/ml=3, yellow/ml=0. Squares are around the occupied orbitals in the first half of the series, and around the doubly occupied ones in the second half. Figure S42. Magnetic susceptibility tensor of Ce, Pr, Nd, Tb, Dy and Ho calculated at T = 100 K and B = 0.1 T.  Table S15. Principal values of the susceptibility tensors (cm 3 mol -1 ) calculated at T = 100 K. The z axis corresponds to the highest susceptibility axis for an easy comparison.
S48 Figure S43. Ab initio computed electronic structure of all the derivatives. Figure S44. Electron density of the electrons in the 4f orbitals (contour value 5·10 -4 e -/a0 3 ). The shown density for Tb-Yb was obtained subtracting the Gd electron density, computed within each corresponding molecular geometry, to highlight the similarity between the 4f n and the 4f n+7 ions. The negative electron density (blue areas) is an artefact that arises from such subtraction.