Ligand Effects on the Spin Relaxation Dynamics and Coherent Manipulation of Organometallic La(II) Potential Qudits

We present pulsed electron paramagnetic resonance (EPR) studies on three La(II) complexes, [K(2.2.2-cryptand)][La(Cp′)3] (1), [K(2.2.2-cryptand)][La(Cp″)3] (2), and [K(2.2.2-cryptand)][La(Cptt)3] (3), which feature cyclopentadienyl derivatives as ligands [Cp′ = C5H4SiMe3; Cp″ = C5H3(SiMe3)2; Cptt = C5H3(CMe3)2] and display a C3 symmetry. Long spin–lattice relaxation (T1) and phase memory (Tm) times are observed for all three compounds, but with significant variation in T1 among 1–3, with 3 being the slowest relaxing due to higher s-character of the SOMO. The dephasing times can be extended by more than an order of magnitude via dynamical decoupling experiments using a Carr–Purcell–Meiboom–Gill (CPMG) sequence, reaching 161 μs (5 K) for 3. Coherent spin manipulation is performed by the observation of Rabi quantum oscillations up to 80 K in this nuclear spin-rich environment (1H, 13C, and 29Si). The high nuclear spin of 139La (I = 7/2), and the ability to coherently manipulate all eight hyperfine transitions, makes these molecules promising candidates for application as qudits (multilevel quantum systems featuring d quantum states; d >2) for performing quantum operations within a single molecule. Application of HYSCORE techniques allows us to quantify the electron spin density at ligand nuclei and interrogate the role of functional groups to the electron spin relaxation properties.


■ INTRODUCTION
The chemistry of lanthanide (Ln) ions is largely dominated by the +3 oxidation state regardless of the number of 4f valence electrons because the +3 state constitutes a good balance between lattice and ionization enthalpies. 1 The 4f n Ln(III) to 4f n-1 Ln(II) reduction potentials of six lanthanides, Eu, Yb, Sm, Tm, Dy, and Nd, allow the isolation of compounds containing +2 ions, but Ln(II) complexes for the other lanthanides were not expected to be isolable. 2However, recent advances in organometallic chemistry have made stabilization of the +2 oxidation state possible for almost all elements of the 4f series, with tris(cyclopentadienyl) ligand environments being the first to stabilize Ln(II) complexes of all the other lanthanides except radioactive Pm. 3−5 Structural, spectroscopic and density functional theory (DFT) studies have indicated that the trigonal environment generated by three substituted cyclopentadienyl (Cp R ) ligands significantly stabilizes the (n+1) d z 2 orbital, such that a 4f n 5d 1 ground state configuration is typically found for the reduced Ln(II) ions, 3−5 and a 4d 1 configuration for Y(II). 6−10 One interest is in the electron spin relaxation properties, which has led to interest in a quantum information science (QIS) context.We recently reported the relaxation properties of a tris-cyclopentadienyl yttrium(II) complex. 9In common with several [Ln II X 3 ] − type complexes (X = Cp R , amide, aryloxide), the pseudo-C 3 symmetry enables direct mixing of the metal valence s-and d z 2 atomic orbitals, resulting in a large and near isotropic metal hyperfine interaction.This has the knock-on effect of retarding electron spin T 1 relaxation driven by spin−orbit coupling (which enables exchange of energy between the spin system and the lattice when the electronic energy levels are modulated by molecular motions) since the orbital angular momentum is largely quenched.The longer T 1 then does not limit T m , the phase memory time, and allows coherent manipulation of the spin to higher temperatures (up to room temperature for the Y(II) example). 9This near isotropic nature has some analogies with 2 S 1/2 state ions such as 171 Yb + used as hyperfine qubits in ion-trap QIS, 10 and has been exploited to observe clock transitions (transitions at avoided level crossings where relaxation is prolonged because the transition frequency becomes insensitive to magnetic field fluctuations) at X-band microwave frequencies. 11The combination of such strategies with other approaches that exploit, for example, atomic clock transitions, 12,13 removal of nuclear spins, 14,15 or reduction of vibrational relaxation, 16−18 could sufficiently enhance molecular qubit properties without the need to apply a too high degree of magnetic dilution.This is particularly important for quantum computation as its implementation demands bringing qubits together and to perform qubit gates (logic operations) 19−22 to carry out specific algorithms.
Another approach to the latter problem is to increase the dimension of the spin system (the Hilbert space) in individual molecules, rather than bringing multiple S = 1/2 qubits together.Such multistate systems are called qudits (d is the dimension of the Hilbert space). 19,23For an S = 1/2 molecule, the dimension of the spin system can be expanded by coupling to a nuclear spin via the hyperfine interaction.Hence, for the Y(II) example above the spin system has a dimension d = (2S +1) × (2I+1) = 4 due to interaction of the electron spin S = 1/ 2 with the 89 Y I = 1/2 nuclear spin, with the opportunity to encode information in the electronuclear states.Such an approach has been proven in molecular systems to implement, for example, Grover's algorithm in [Tb(phthalocyanine) 2 ] ( 159 Tb, I = 3/2) 24 and a quantum simulator using [Yb-(trensal)] ( 173 Yb, I = 5/2). 25−28 This can be regarded as an alternative route for the realization of multistate systems that open opportunity for larger and more complex calculation algorithms to be conducted within a single molecular unit. 29Such qudit systems have been proposed for diminished quantum error rates and simplified quantum logic, 30 and the potential advantages of molecules in this context have been reviewed recently elsewhere. 31Herein, we demonstrate that the nuclear Hilbert space of [Ln(Cp R ) 3 ] − can be expanded through the hyperfine interaction with the nuclear spin of the 139 La(II) isotope (I = 7/2; 99.95% natural abundance), thus leading to qudits with d = 16.We report coherent spin manipulations across the multiple electronuclear spin states in the family of complexes, 33 (3) (Figure 1; Cp′ = C 5 H 4 SiMe 3 ; Cp″ = C 5 H 3 (SiMe 3 ) 2 ; Cp tt = C 5 H 3 (CMe 3 ) 2 ) that contain one unpaired electron primarily residing in a low-energy s/d z2 orbital as described above.The three Cp R ligands feature one or two SiMe 3 or CMe 3 functional groups; these have been chosen to allow systematic study of the effect of differing sterics and electron-donating/withdrawing nature of the substituents within the same ligand framework, and hence of spin density distribution between ligand and metal, on the spin dynamics.

■ RESULTS AND DISCUSSION
Synthesis and structural characterization.The synthesis and structural characterization of 1, 32 2, 3 and 3 33 were performed according to the previously reported methods; their crystal structures are presented in the Supporting Information, Figures S1−S3.In these complexes, La(II) resides at the center of the plane formed by the centroids of the three cyclopentadienyl ligands leading to a nearly trigonal planar arrangement.Complexes 1−3 each displays a local pseudo-C 3 symmetry at the metal center, with the C 3 axis passing through the metal, and perpendicular to the plane made by the centroids of the three Cp R rings.The C 3 symmetry is important as it dictates the d z 2 orbital to be lowest in energy and thus able to accommodate the unpaired electron.The geometric parameters of the anions are very similar for the three different complexes; the mean La−C(Cp R ) distances are 2.853(9) Å, 2.88(2) Å and 2.90(8) Å for 1, 2 and 3, respectively, while the mean La-centroid lengths are 2.586 Å (1) 2.620 Å (2) and 2.638 Å (3), in line with the varying steric effects of ring substituents.
Electron paramagnetic resonance.Continuous-wave (CW) and echo-detected field-swept (EDFS) EPR spectra for 1-3 in MeTHF (Figure 2 and Figures S4−S8) display 8-line patterns due to the hyperfine interaction of the electron with the nuclear spin of 139 La.Spectra are nicely simulated with the parameters in Table 1, using EasySpin 34 and the spin Hamiltonian H ̂= gμ B BS + IAS, with g and A as the axial gtensor and the hyperfine coupling tensor, respectively.Simulations for 2 were further improved by adding a small amount of an unknown La(II) species (Figures S7 and S8).   2. Echo-detected field-swept data of 1 (blue), 2 (purple) and 3 (green) recorded at X-band (9.67 GHz) on frozen MeTHF solutions (see Table 1 for simulation parameters).−37 The pattern of gvalues is consistent with a 5d z 2 ground state with g z ≈ g e > g x,y , where g e is the free-electron g-value.The largest g-anisotropy is observed for 3, which is somewhat surprising given the heavier atoms in 1 and 2.
Hyperfine interactions can involve two different mechanisms: the Fermi-contact (through bond) interaction and the anisotropic dipolar (through space) coupling. 38The metal hyperfine interaction in 1 -3 is dominated by the isotropic part in each case.This arises from electron spin density at the nucleus and can be related to the 6s-orbital character of the SOMO (which is predominantly 5d z 2 ).The comparable A iso values for complexes 1 and 2 (averaged anisotropic values of 426.7 and 391.7 MHz, respectively), that incorporate one and two SiMe 3 substituents respectively, suggest a similar degree of 6s-5d z 2 mixing in the SOMO, while that for 3 is substantially larger.The valence 6s-orbital spin population can be estimated by dividing A iso by the atomic isotopic hyperfine constant, here A = 6007 MHz for 139 La. 39 The results indicate that the SOMO has approximately 7.1% and 6.5% 6s character in 1 and 2, respectively, which is consistent with previous DFT calculations on [Ln(Cp′) 3 ] − (Ln = Y, La). 9 Complex 3 shows a larger A iso (636.7 MHz), from which we derive a 6sspin population of 10.6%, indicating a higher degree of 6s-5d z 2 mixing relative to 1 and 2. Hence, replacement of SiMe 3 in 2 with CMe 3 in 3 leads to a substantial increase in s-electron density at the La, i.e. the electronic properties of the ligands are a major factor in the orbital admixing at the metal ion.This finding is significant because the spin relaxation properties of the compounds relate to the population of the SOMO, with a more pronounced s-character of SOMO leading to prolonged electron spin relaxation. 8T calculations on the crystal structures of the anions in 1−3 provide full support for these findings, with A iso values of 390 MHz for 1, 360 MHz for 2, and 641 MHz for 3 (Table S13), also reproducing the experimental trend 3 ≫ 1 > 2 and the minimal hyperfine anisotropy.Loẅdin population analysis of the SOMO (Figure 3) in each case gives 7.7, 7.8 and 10.0% s-character for 1-3, respectively, in agreement with the experimentally derived values above, and reports 59, 59 and 61% d-character, respectively (Table S14).Loẅdin analysis of the spin density places 0.71 electron spin density on the La site in 1 and 2, but 0.76 spin density on the La site in 3, and hence ca.30% of the electron spin density is associated with the ligand scaffold.
Spin−lattice relaxation and spin coherence times.Observation of well resolved EDFS spectra up to ca. 100 K in frozen MeTHF solutions of 1−3 encouraged us to measure relaxation times, as rare earth complexes rarely show spectra at such a high temperature.The spin−lattice relaxation time constants (T 1 ) at different temperatures were derived by fitting inversion recovery data (π−t−π/2−τ−π−τ−echo, with variable t) 40 to a biexponential function (Figures 4a, Figures S9−S17, and Tables S1−S3).T 1 is strongly temperature dependent down to 5 K and reaches values of ca.17, 9, and 50 ms for 1, 2 and 3, respectively, measured on peaks corresponding to x,y orientations (B 0 ⊥ C 3 ).There is significant variation across measurement positions, with longer relaxation times found for the z-orientation (B 0 ∥ C 3 ), reaching as long as 98 ms for 3 (Tables S1−S3).Fitting of the data assuming a Raman-like dependence of T 1 with temperature (CT n ) 9,27 gave the Raman parameters in Table 1 (Figures S23, S27, and S30).The coefficient C is noticeably lower for 3.It is tempting to ascribe the faster T 1 relaxation for 1 and 2 than for 3 as being due to enhanced spin−orbit coupling (SOC) due to the heavier nuclei (Si) in the ligand set. 41However, as noted above, the ganisotropy, also dependent on SOC, is also greatest for complex 3. Hence, the slower relaxation of 3 must be due to another factor, and a likely candidate is the significant difference in electron spin distribution as evident from the Table 1.Extracted EPR parameters for 1, 2, and 3 (10 mM MeTHF) Journal of the American Chemical Society metal hyperfine interaction.The T 1 times correlate with the isotropic metal hyperfine, and hence with the metal valence orbital s-character of the SOMO, with 3 ≫ 1 > 2. Other possible factors affecting the relaxation times are the interactions of the electron with the environmental nuclear spins. 42,43Rotations of Me groups in particular cause spin relaxation through a mechanism known as spectral diffusion (SD). 42Fitting of the inversion recovery data of 1−3 included a temperature-dependent term associated with spectral diffusion (SD). 19The obtained T SD values are significantly smaller than T 1 and follows the variation in T 1 , i.e.T SD decreases as the temperature is raised, and is shortest for 2 (Tables S1−S3).This is consistent with spectral diffusion effects, and the steric demands of the Cp R ligands.SiMe 3 substituents in 2 are less hindered than CMe 3 in 3, because of longer C(Cp)-Si bonds, leading to larger separations between substituents and Cp rings in 2 than 3. Thus, Me groups in 2 are likely more efficient in triggering relaxation by spectral diffusion mechanism. 42hase memory time constants (T m ), measured with a Hahn echo decay sequence, 39 are temperature independent below ca. 10 K, and reach values of 2.0, 2.2, and 2.4 μs for 1, 2 and 3, respectively, at 5 K (Figures S18−S30 and Tables S4−S6).Similar T m values have been observed for related complexes 9 which, in common with 1-3, give electron spin echoes up to relatively high temperatures despite the 1 H and Me-group rich environment (and without optimization of concentration).In that work it was argued that T m was limited by nuclear spin diffusion, thought that spectral diffusion effects cannot be excluded.
The effect of such processes on relaxation can be suppressed by dynamical decoupling experiments using the Carr-Purcell-Meiboom-Gull (CPMG) pulse sequence, 44 which involves an initial π/2 followed by a series of π pulses.Indeed, such experiments on 1-3 gave much longer time constants than twopulse Hahn echo experiments (Figure 4, Figures S31−S36, and Tables S7−S9).For 1, 2 and 3, respectively, we find CPMG time constants as high as 36, 49, and 161 μs at 5 K, with the higher values observed for z orientations.The reason for the much greater enhancement for 3 than for 1 and 2 is not immediately clear.
Coherent manipulation of spins by microwave pulses.Coherent manipulation of the electron spin in 1−3 was studied by transient nutation experiments, using a 3-pulse sequence that involves an incremented tipping pulse, t p , followed by Hahn echo detection. 27,45This corresponds to creation and measurement of arbitrary superposition states.The oscillation in the echo intensity as a function of the duration of the initial pulse is known as a Rabi oscillation. 46abi oscillations were detected for all eight hyperfine transitions of the three complexes (Figure 5a     ).This confirms their origin as Rabi oscillations with Rabi frequency Ω R .There is also a sharp B 1 -independent peak observed, which corresponds to the 1 H Larmor frequency, 27 due to interactions with 1 H nuclei on ligands and/or solvent.
The nutation data provide information on the properties of 1-3 as potential qubits.The time period between a maximum and adjacent minimum of the oscillation corresponds to the flipping time of the spin, meaning the time required for executing a logical operation. 47Crucially, the operation time parameter needs to be notably shorter than the lifetime of the qubit in order the qubit to be functional.This operation time is 78, 115, and 105 ns for 1, 2 and 3, respectively, from the nutation data recorded under 16 dB applied mw power (Figure 5a).This time reduces significantly for higher power pulses.The qubit figure of merit, Q M , is defined as 2Ω R T 2 and represents the number of coherent single−qubit NOT computational operations. 48For complexes 1, 2 and 3 the Q M value is 156, 128 and 130, respectively.These values are significant for rare earth qubits, 26 and comparable to the Q M values of other molecular systems. 49With T 2 CPMG = 161 μs (3), we predict unprecedented Q M = 8720 for lanthanide complexes.The fact that such manipulations can be performed at 80 K on all eight hyperfine transitions indicates that this system has robust quantum properties and great ability to form a qudit with d = 16.
HYSCORE spectroscopy.As noted above, stabilization of the +2 oxidation state of the lanthanum ion implies an electronic configuration change from 4f 0 5d 0 to 4f 0 5d 1 upon reduction of La(III) to La(II). 3,32,33However, the assumption that the unpaired electron of La(II) purely resides in the lowest energy 5d z 2 orbital is inconsistent with the small anisotropy of the g-and hyperfine A-tensors.This implies significant electron spin density on the ligands.In the context of QIP, weak electron−nuclear interactions can contribute to decoherence, shortening qubit lifetime. 28Given the CPMG results, and observation of 1 H interaction in the nutation experiments, we probed the weak ligand hyperfine interactions in 1, 2 and 3 by using the two-dimensional hyperfine sublevel correlation (HYSCORE) technique. 50This hyperfine method allows quantifying small (down to sub-MHz) hyperfine couplings to nuclear spins such as 1 H, 13 C or 29 Si. 51,52YSCORE uses a four-pulse electron spin−echo sequence (π/2 -τπ/2 -t 1 -π -t 2 -π/2 -τ − echo, with fixed τ and variable t 1 and t 2 ), which creates correlations between the nuclear frequencies in the α and β electron spin manifolds in a 2D experiment.For each of 1−3 we observe signals centered on the 1 H and 13 C Larmor frequencies of the HYSCORE spectra (Figure 6

and Figures S64−S74).
To model the spectra we have taken an approach similar to that described elsewhere on a related molecule. 53We initially focused on the 13 C carbon region, because the main metal− ligand bonding interaction involves the 2p π orbitals of the cyclopentadienyl ligands.The observed hyperfine coupling (A) for a given 13 C nucleus of the Cp rings is taken as the sum of contributions from spin density at that carbon n (A Cn ) and the point dipole (through space) interaction with spin density at other atoms (A dip ).The dipolar contributions, based on a point dipolar model, are given by where gand g n 1 are the electronic and nuclear g matrices (g n is the nuclear g-value; 1 is the unit matrix), β e and β n are the electron and nuclear magnetons, ρ k is the spin population at the atom k, n k a is the n•••k unit vector (expressed in the molecular frame), r k is n...k distance, h is Planck's constant and μ 0 the vacuum permittivity.
Assuming dominant spin density on the La(II) ions (ρ La = 1), the A dip interactions of each carbon position in the cyclopentadienyl ring were calculated based on the crystallographic coordinates.The molecular axis system was defined in reference to the g-tensor, with the g z component lying along the pseudo-C 3 axis (Figure 7).
Calculated spectra based purely on these calculated A dip matrices do not match the experimental data (Figures S64− S73).Hence, an additional contribution to the 13 C hyperfines was introduced via A Cn .We assumed this additional contribution to be axial with the unique axis aligned with the 2p π direction (i.e., in the molecular xy plane), giving A ∥ and A ⊥ as parameters for each C atom.

=
, where P p is the electron nuclear dipolar coupling parameter for unit population (ρ p = 1) of a 13 C 2p orbital.With P p = 268 MHz, 39 and the hyperfine values deducted by 13 C HYSCORE, we obtained the population at C atoms on cyclopentadienyl rings in 1−3 (Table 2).
DFT calculations using the crystal structures of 1-3 provide estimates of 13 C hyperfine coupling and C spin density on the Cp R rings (Tables S15−S20), which are on the order of the values determined from the model of the HYSCORE data here.Notably, a large A ∥ is obtained for C 5 in 1 and for C 2 in 2 and 3, in agreement with the HYSCORE model.However, these calculations account for the asymmetry of the solid-state structures, which have significant influences on the calculated spin densities and ligand hyperfine couplings and we also observe some rhombic 13 C hyperfine couplings.Hence, we have optimized the geometries of 1-3 in the gas-phase and recalculated the hyperfine coupling and atomic spin densities (Tables S21−S28).There are very little changes when it comes to the 139 La hyperfine, and some 13 C couplings on the Cp R rings are still very rhombic.But overall, for complexes 2 and 3 which have the most sterically demanding ligands, we observe that the ligand hyperfine is not much different than that obtained with the crystal structures, and that the three ligands now show more symmetric values.For 1 however, there is a larger deviation between the two calculations, presumably owing to greater molecular flexibility, and we also find that the calculated values for one ligand in the optimized geometry are quite different than for the other two ligands, likely a result of the particular conformational minimum obtained in the optimization.Given the significant dependence of the calculated hyperfine coupling and spin densities on the structural details and that the experiments are performed on amorphous frozen solution samples with inherent molecular distributions, it is not assured that a simple model such as the one employed here should work.That it manages to capture similar results compared to the DFT calculations is quite remarkable.
For the 1 H HYSCORE spectra, the point dipole-only model again failed to reproduce the experimental data, although dipolar interactions are larger because of the larger magnetic moment of 1 H cf. 13 C. Thus, in order to reproduce the spectra we added a hyperfine contribution due to the spin density on the carbon to which it is bound via spin polarization. 55The hyperfine matrix to an α proton in organic π-radicals typically takes the form [α iso /2, α iso , 3α iso /2], where the smallest component is oriented along the C−H bond, the middle one along the 2p π direction, and the largest along the cross-product of the 2p π and C−H orientation.With this model, with a single variable (α iso ) for each 1 H, excellent simulations of the 1 H spectra were obtained (Figure 6e−h).For 1, we found α iso = − 1.227 MHz for H 2,5 and α iso = −0.927MHz for H 3,4 (Figure 6d).For complexes 2 (Figure 6g,h) and 3 (Figure S72 data were satisfactory simulated with α iso = −2.037and −1.41 MHz for H 2 , respectively (contributions from the protons attached to C 4,5 are negligible due to the negligible densities at these  Journal of the American Chemical Society carbons, and C 1,3 have no attached protons).The isotropic hyperfine constant α iso at the α-proton is linked to the spin density at the associated C 2p π orbital by the simple McConnell relationship, α iso = Q CH •ρ P , 56 where Q CH is the 1 H hyperfine coupling that would be observed for ρ P = 1.With Q CH = −84 MHz from studies of Cp radicals, 57 we obtained spin densities at C which agree with those derived from the 13 C data (Table 2), providing a consistent analysis.Summing up the HYSCORE results, there are considerable differences in the electron spin density measured at ligand atoms.The total measured carbon-2p π spin population on the three ligands is estimated to be 15.6%, 11.4% and 7.4% for 1, 2 and 3, respectively, which is in agreement with the trend predicted by DFT calculations, of ca.13.5% (1) > 12.0% (2) ≫ 5.97% (3) (Tables S15−S23).These results are at first unexpected as 1 and 2 contain one and two SiMe 3 groups, respectively, while 3 contains two CMe 3 groups.As the silyl substituents are electron withdrawing in cyclopentadienyl rings the order 2 > 1 ≫ 3 might have been predicted.On the other hand, comparing 2 and 3 which have the same substitution pattern, we find that there is substantially greater ligand spin population in 2 than in 3. Recalling that the metal hyperfine for 2 is much smaller than that for 3, the conclusion is that the electron-withdrawing nature of the SiMe 3 groups in 2 (cf. to electron-donating CMe 3 groups in 3) leads to a greater ligand spin density and a smaller metal spin density.This is not what one would predict based on a simplistic MO argument for [M(Cp) 3 ] complexes, 58 where the decrease in energy of the Cp R π-orbitals would lead to a poorer energy match with the metal 5d z 2 that dominated the SOMO and hence less ligand character in the a 1 SOMO.It is more difficult to make arguments for complex 1 because the different substitution pattern affects the pattern of spin distribution.It is interesting that despite the greater ligand character in 2 (and 1) than in 3, the electron spin relaxation times for 3 are longer.This would be consistent with the metal s-character being more important.

■ CONCLUSIONS
Pulsed EPR studies on three organometallic La II complexes based on Cp′ (1), Cp″ (2), and Cp tt (3) are reported.By modifying the chemical structure of the molecules, we investigated the effect of various substituents on the spin dynamics.We found the spin−lattice relaxation time T 1 and the electronic coherence T CPMG times to vary in line with the 6s-orbital character of SOMO, that is 6.5% (2) < 7.1% (1) ≪ 10.6% (3).At 5 K, T 1 is incredibly long, varying from ca. 7 ms (2) to 19 ms (1) and, more importantly, to 50 ms for 3. T CPMG reaches 36 and 49 μs at 5 K for 1 and 2 respectively, increasing to 161 μs for 3. Notably, the same trend is observed at 80 K, with the highest T CPMG value (4 μs) being measured for 3, which has the smallest spin delocalization onto ligands and the largest 6s character of La(II) single occupied molecular orbital (SOMO) most likely due to the strong electron-donating nature of CMe 3 substitutes.T 1 is also largest for 3, whose SOMO has a significant s-orbital character (10.6%), which reduces the effect of SOC via metal.Complex 1 also has a longer T 1 than complex 2, and has greater La 6s-orbital and ligand character than 2. Phase memory time values display only small differences between the three complexes, and the largest value (2.4 μs) is measured for 3. We find T m to be limited by interactions with the environmental nuclear spins, and used advanced HYSCORE techniques to quantify such interactions.We measure significant spin density at the 1 H protons of Cp rings, and 29 Si of SiMe 3 groups (1), indicating that these nuclei participate to decoherence.Coherent spin manipulations were probed for up to eight hyperfine transitions for 1−3, with Rabi oscillations observed up to 80 K. Coherence times can be extended by CMPG methods.Additionally, similar NOT operation times and Q M values were determined for all complexes, with 1 implementing a faster inversion of the qubit and thus, allowing a higher number of single−qubit NOT computational operations in a given time.
For future studies on 139 La, there are significant advantages to growing magnetically diluted crystals of 1−3 as measurement of relaxation times and manipulation of the electron and nuclear spins should be achievable up to room temperature for these prototype qudits.Using Davis ENDOR one can explore the coherent spin properties of 139 La nuclear spins, including nuclear spin Rabi oscillations, which will provide significant insight into the dynamics of the nuclear spins, allowing us to also measure the nuclear spin relaxation and the nuclear spin coherence times.These experiments are feasible and were used to study the coherent dynamics of 171 Yb ions in yttrium orthosilicate. 64EXPERIMENTAL SECTION Experimental materials and methods.All manipulations and syntheses were conducted with rigorous exclusion of air and water using standard Schlenk line and glovebox techniques under an argon or dinitrogen atmosphere.The preparation and characterization of complexes 1−3 3,32,33 and KC 8 59 followed previously reported methods.All glassware was flame-dried under vacuum or stored overnight in a hot oven prior to use.Argon and dinitrogen were passed from cylinders through columns of activated 3 Å molecular sieves and Cu catalyst prior to use.MeTHF was refluxed over molten K for 3 days, distilled, and stored over activated 4 Å molecular sieves.As expected for early metal organometallic complexes, 1−3 are highly air-and moisture-sensitive. 2 In addition, all three complexes are temperature-sensitive and ethereal solutions have been reported to decompose rapidly above −20 °C, 3,32,33 in common with other similar trigonal planar La(II) complexes. 5,6All samples for frozen solution EPR spectroscopy were prepared under strict anaerobic conditions and measured in flame-sealed quartz EPR tubes to avoid oxidation of La 2+ to La 3+ , and to enable safe investigation.Two different sample preparation procedures were followed.For samples of 1 and 2, aliquots of 10 mM MeTHF solutions of the respective complex at −30 °C were transferred to precooled quartz EPR tubes (to a sample height of ca. 5 cm) within an Ar glovebox.These tubes were quickly attached to an appropriate set of apparatus to make an airtight seal, removed from the glovebox, and the solutions flash-frozen in a Dewar containing liquid nitrogen.The frozen solutions were placed under vacuum on a Schlenk line to the point at which the apparatus could maintain a constant static pressure of less than 5 × 10 −3 mbar, and then flame-sealed under a dynamic vacuum.For the sample of 3, a solution of [La(Cp tt ) 3 ] in MeTHF in a sample vial in a glovebox at −30 °C was treated with KC 8 and stirred for 10 min at this temperature to give a ca. 10 mM solution of 3.An aliquot of the suspension was transferred to a precooled quartz EPR tube (to a sample height of ca. 5 cm), and flame-sealed under dynamic vacuum following the same method as above.Frozen solution samples were stored in liquid nitrogen until measurement.

Journal of the American Chemical Society
The c.w. EPR spectra were recorded at X-band (ca.9.4 GHz) mw frequency using a Bruker EMX 300 EPR spectrometer equipped with a 1.8 T electromagnet and a Stinger closed-cycle helium gas cryostat.Field corrections were applied using Bruker strong pitch (g = 2.0028) as a reference, and spectra were baselined before simulation.EPR spectra were simulated using the EasySpin toolbox implemented within Matlab. 34The simulation model has assumed S = 1/2 for 1−3 (5d 1 ), with axial (g x = g y ≠ g z ) g-tensors.Where hyperfine coupling was resolved, an anisotropic hyperfine coupling, A = [A x , A x , A 3 ], was included in the model, with A and g tensors assumed to be collinear.Line broadening is the result of a distribution of g-values, unresolved hyperfine coupling/a distribution of A-values, and dipolar coupling, among other effects.Anisotropic line broadenings have been modeled with phenomenological g-strains (distribution of gvalues) to account for all broadening effects.Pulsed EPR Xband studies were performed on a Bruker ElexSys E580 spectrometer.The primary Hahn echo sequence (π/2-τ−π−τecho) 40 was used for the two-pulse electron spin echo measurements, with initial π/2 and π pulse of 16 and 32 ns, respectively.For the relaxation time measurements, T m studies were made by incrementing the τ time in the Hahn echo sequence (longer pulses were used to suppress the 1 H modulation).T 1 was measured by the inversion recovery sequence (π-t-π/2-τ−π−τ-echo) 40 with π/2 and π pulses of 16 and 32 ns, respectively, with τ = 300 ns and varying t.The dynamic decoupling measurements were carried out by a CPMG sequence (π/2-τ-(π-2τ) n -π−τ-echo) with n = 300, π/2 and π pulse of 16 and 32 ns, respectively, and using 16-step phase cycling.HYSCORE measurements were performed using the four-pulse sequence (π/2-τ-π/2-t 1 -π-t 2 -π/2-echo), 40 π/2 and π pulse of 16 and 32 ns, respectively, initial times t 1,2 = 0.1 μs and τ values of 136 and 200 ns.
Computational methods.Density functional theory (DFT) calculations were performed with the TPSSH functional and the ZORA relativistic Hamiltonian 60 with Orca 5.0.4. 61The ZORA-def2-TZVP basis set 62 was used for all nonmetal atoms, and the SARC-ZORA-TZVP basis set 63 was used for La.Calculations were performed either on nonoptimized crystal structures, or optimized geometries using the above method in conjunction with the COSMO solvent model for THF.Hyperfine couplings were calculated using picture change corrections.We find Loẅdin population analysis for the La components is the closest to the traditional EPR analysis methods and is consistent with our usual MO analysis techniques, however we find that the Mulliken spin density values of the ligand atoms correlate with their DFT-calculated hyperfine couplings (where the equivalent Loẅdin values do not) and so we use the latter there.
Molecular structures of complexes 1, 2 and 3, continuous-wave electron paramagnetic resonance spectra, additional echo-detected field-swept spectra, spin− lattice relaxation measurements, phase memory time measurements, additional transient nutation data, HYSCORE spectra and simulations, and DFT data are available in the Supporting Information (PDF) ■ AUTHOR INFORMATION Corresponding Authors

Figure 3 .
Figure 3. Renderings of the SOMO for (a) 1; (b) 2; and (c) 3, from DFT calculations on the crystal structures of the anions.

Figure 4 .
Figure 4. (a) Temperature dependence of T 1 , T 2 and T 2 CPMG for 1−3.(b) Decay of the echo intensity for 3 at temperatures between 5 and 60 K, measured at 366.2 mT with a CPMG sequence.Solid lines are exponential fits (parameters in TableS9).

Figure 5 .
Figure 5. (a) Nutation (Rabi oscillations) data for 1, 2 and 3 measured at 16 dB at 20 K; (b) Fourier transforms of the nutation data at different microwave powers; and (c) B 1 dependence of the Rabi frequency (Ω R ); the solid line is a guide of the eye emphasizing the linear dependence.

Figure 6 .
Figure 6.X-band HYSORE spectra: (a) 13 C and 29 Si region for 1 at a static field of B 0 = 360.8mT (at g xy ); (b) 13 C region for 1 at B 0 = 353.8mT (at g z ); (c) 13 C region for 2 at B 0 = 333.3mT (at g xy ); (d) 13 C region for 3 at B 0 = 366.2mT (at g xy ); (e) 1 H region for 1 at a static field of B 0 = 345.3mT (at g xy ); (f) 1 H region for 1 at a static field of B 0 = 353.8mT (at g z ); (g) 1 H region for 2 at a static field of B 0 = 361.7 mT (at g xy ); (h) 1 H region for 2 at a static field of B 0 = 352.8mT (at g z ), with the simulation in red based on the model described in the text.

Figure 7 .
Figure 7. Schematic representation of the (a) La-Cp′ (in 1), (b) La-Cp″ (in 2) and (c) La-Cp tt (in 3) along with the labels used for the HYSCORE simulations and the orientation of the C 3 axis.