Engineering electronic structure to prolong relaxation times in molecular qubits by minimising orbital angular momentum

The proposal that paramagnetic transition metal complexes could be used as qubits for quantum information processing (QIP) requires that the molecules retain the spin information for a sufficient length of time to allow computation and error correction. Therefore, understanding how the electron spin-lattice relaxation time (T1) and phase memory time (Tm) relate to structure is important. Previous studies have focused on the ligand shell surrounding the paramagnetic centre, seeking to increase rigidity or remove elements with nuclear spins or both. Here we have studied a family of early 3d or 4f metals in the +2 oxidation states where the ground state is effectively a 2S state. This leads to a highly isotropic spin and hence makes the putative qubit insensitive to its environment. We have studied how this influences T1 and Tm and show unusually long relaxation times given that the ligand shell is rich in nuclear spins and non-rigid.

in the main text).

Supplementary Figure 5.
Normalized Hahn echo signal intensities as a function of the interpulse delay 2t, for ~2% 1@5 (single crystal) at B0 = 349 mT (OP2 in Fig. 2b; B0^C3) and selected temperatures.  (Fig. 2b), and temperatures of 120 and 298 K. The solid line is a guide for the eye emphasizing the linear behaviour (B1 µ √P, where P is the microwave power).

Supplementary Figure 24.
Schematic representation for the binding of a Cp' ring in 1, and the direction of the molecular C3 axis.

Compound Metal atom
Non-metal   . 4 We note that molecules pack in the crystal in different ways, being classified in two groups: A and B, whose C3 axes (that is the axis perpendicular to the plane made by the centroids of the three Cp' ligands bound to Y 2+ ) are orthogonal. As such, we have chosen to rotate the crystal around the C3 axis of one of such molecules, in order to access such an orientation that molecules B would have their C3 axis parallel to the static B0 field (Supplementary Figure   3), while molecules A remained aligned with C3 perpendicular to B0. This allowed determination of both gz and gx,y components at the same time (  where k is the modulation depth, w is the Larmor angular frequency of a nucleus I coupled to the electron spin, f is the phase correction, X is the stretching parameter, Y(2t) is the echo integral for a pulse separation t, and Y(0) is the echo intensity extrapolated to t = 0. [9][10][11][12] The extracted Tm times for 1 and ~2% 1@5 are given in Supplementary Tables 1 and 2.
Owing to the 2p-ESE decays being dependent on experimental conditions, with longer (more selective) pulses resulting in longer relaxation decays (Supplementary Figure 6), comparison between the extracted Tm values at those temperatures must be regarded with caution. Figures 7 and 8) were acquired with a standard magnetisation inversion recovery sequence, where Y1 and YSD are the amplitudes, and TSD is the spectral diffusion time constant, 11 giving the results presented in Figs. 2e, 2f and S9, and Supplementary Tables 3 and 4. The presence of two decays is commonly attributed to the occurrence of both spectral diffusion and spin-lattice relaxation of which the latter is usually assigned as being the slower process. 12 We notice that the magnetization recovery curves do not reach full saturation below 15 K, indicating that the T1 spin-lattice relaxation time is very long. Fitting such curves to an exponential model is likely to introduce some inaccuracy in the determination of the T1 values at these temperatures.

Transient Nutation Experiments (Rabi oscillations). The transient nutation data (Figs. 2c
and 2d, and Supplementary Figures 11 to 22) were acquired with a three-pulse nutation sequence, tp-tw-p/2-t-p-t-echo. 8,13 The length of the tipping pulse, tp, pulse was varied in 2 ns increments, whilst those of the pulses p/2 and p were kept fixed at the optimal values needed to generate a maximum echo intensity, for tp = 0. The tw and t delays were kept constant at 6 µs (chosen to be much longer than Tm) and 200 ns, respectively. The Rabi frequency, ΩR, was determined by zero-filling the Rabi oscillation curves, followed by Fast Fourier Transform (FFT). ΩR is expected to vary linearly with B1, as opposed to nuclear modulations that are insensitive to the strength of B1 ( Supplementary Figures 16 and 23) (11).

HYSCORE (Hyperfine sub-level correlation) Measurements. The HYSCORE spectra
were recorded with a four-pulse sequence, p/2-t-p/2-t1-p-t2-p/2-echo, 8  where g and gn1 are the electron and nuclear g (3x3) matrices (gn is a scalar; 1 is the unit matrix), βe and βn are the electron and nuclear magnetons, ρk is the electron spin population at atom k (0 ≤ ρk ≤ 1), rk is n…k distance, nk and ñk are the n…k unit vector expressed in the molecular frame (a column vector) and its transpose, h is the Plank's constant, and μ0 is the vacuum permittivity. It is also assumed that gz lies along the C3 unique axis spin density transferred from the metal ion will be in these C 2pp -orbitals. 10,15 Thus, for each matrix A Cn (assumed to be axial) we fix the unique axis to be oriented along the 2pp direction (i.e. in the molecular xy plane), which allows to determine Az and Ax,y per each C site ( Table 2, main text). The 2pπ spin population (ρp) at the individual carbon positions can be derived from Equation (5) reproduce the experimental data. We then added a contribution from the C 2pπ spin density on the Cp' ligands, which can occur via spin polarisation of the C-H bond. 10,16 Generally, the hyperfine coupling of an α-proton in a π radical has its principal values oriented with the smallest component along the C-H vector, one along the 2pπ direction, and the largest component orthogonal to the 2pπ and C-H directions. 17 17 and aiso = -0.7 MHz, we get ρp = 0.00833 (0.83 %) for C 2,5 in excellent agreement with ρp = 0.008 from analysis of the 13 C data. This gives a total of ~ 6 % spin population on the three Cp' rings.

ENDOR (Electron nuclear double resonance) Measurements. Davies-ENDOR data
(Supplementary Figure 28) were acquired with the standard pulse sequence, p-pRF-p/2t-p-t-inverted echo, [19] with microwave pulses p/2 and p of 128 and 256 ns, respectively. Figure 29) were recorded by using a stimulated-echo sequence, p/2-t-p/2-pRF-p/2-t-stimulated echo, [19] based on three non-selective π/2 pulses (16 ns). In both cases, a radiofrequency pulse pRF of 12 µs was used. In order to correct the effect of potential blind spots, Mims-ENDOR spectra were collected at different cases. We also found no significant changes using the ZORA Hamiltonian (Supplementary   Table 12), nor when accounting for the electrostatic potential of the crystalline environment by using a sphere of 30 Å radius of point charges located at the K and Y lattice sites, with charges of +1 and -1, respectively (Supplementary Table 13).

Mims-ENDOR data (Supplementary
As there was no significant effect on the obtained hyperfine parameters for the anion in 1 upon change of metal basis set, functional, relativistic Hamiltonian or inclusion of the crystalline electrostatic potential, we subsequently calculated the hyperfine parameters for both crystalline and optimised structures for the anions of 1-4 in the gas phase using the PBE functional, the appropriate def2-TZVP basis set, and the DKH Hamiltonian (Supplementary Table 14). Overall we find excellent agreement with the experimental data ( Table 1 in main text), where all metal hyperfine coupling parameters and the g-values are nearly isotropic.
The orbital breakdown of the spin densities for 2-4 are remarkably similar to that described in the main text for 1; only 50 -80% of the spin density is located on the metal atom, in predominantly s and d functions (Supplementary Table 14). Despite the significant dcomponent and "dz 2 " appearance of the spin density ( Figure S30), the anisotropies of the calculated metal hyperfines (|Az Y -Ax,y Y |/|Aiso Y |) are only on the order of 1 -7% (Supplementary Table 14).
CASSCF calculations. State-averaged CASSCF calculations for the anion in 1 were performed with MOLCAS 8.0 (21). We used basis sets from the ANO-RCC library (22,23) with VTZP quality for the Y ion, VDZP quality for the Cp ring carbon atoms, and VDZ quality for all other atoms. The two electron integrals were Cholesky decomposed with a threshold of 10 -8 . We employed an active space of 7 electrons in 16 orbitals (Supplementary Figures   31 to 33 and Supplementary Table 15), which was optimised for the 10 lowest-lying states (Supplementary Table 16 Figure 51 and Supplementary   Table 33), which is quite different to the gas-phase calculation, and thus the electrostatic crystalline potential has lowered the first 4d function by ca. 10,000 cm -1 . States 7 and 8 are diffuse ligand functions like for the gas-phase results ( Supplementary Figures 52 and 53 and Supplementary Tables 34 and 35), however now states 9 and 10 are π* orbitals on the Cp' ring that is proximate to the K + counter ion ( Supplementary Figures 54 and 55 and Supplementary Tables 36 and 37); these charge transfer states are now much lower in energy due to the stabilisation from the positive charge of the K + cation. Despite these differences in the higher energy states, the ground state SOMO and the diffuse character of the low-lying ligand-based excited states remains, and appears to be an intrinsic feature of this molecule.