Spin Dynamics in Single-Molecule Magnets and Molecular Qubits

Over recent decades, much effort has been made to lengthen spin relaxation/decoherence times of single-molecule magnets and molecular qubits by following different chemical design rules as maximizing the total spin value, controlling symmetry, enhancing the ligand field or inhibiting key vibrational modes. Simultaneously, electronic structure calculations have been employed to provide an understanding of the processes involved in the spin dynamics of molecular systems and served to refine or introduce new design rules. This review focuses on contemporary theoretical approaches focused on the calculation of spin relaxation/decoherence times, highlighting their main features and scope. Fundamental aspects of experimental techniques for the determination of key Single Molecule Magnet/Spin Qubit properties are also reviewed.


Introduction
The chemical synthesis of paramagnetic molecules featuring increasingly longer magnetic relaxation and spin coherence times, is the main challenge for two intimately linked research areas: Single-Molecule Magnet (SMM)1-6 and Molecular Spin Qubit research.7,8 To achieve such objectives, spin dynamics of molecular systems must be tuned to suppress a series of relaxation mechanisms, depending on intra-and inter-molecular interactions. The main molecular factors influencing magnetic relaxation are the energy separation of electronic states, their magnetic anisotropy, molecular vibrations, and dipolar and hyperfine interactions. Environmental molecules can also affect spin relaxation though long-range spin-spin interactions (dipolar electron-electron and hyperfine electron-nuclear) and vibrational coupling. In the case of SMMs, uniaxial magnetic anisotropy must be enhanced, and electronic energy gaps must be as high as possible. Vibrations must be tuned to minimize vibronic coupling, either by symmetry control or by hampering molecular displacements in the magnetically relevant atoms. Detuning vibrational and electronic transition energy has been proposed as a convenient strategy for improving SMMs. On the other hand, molecular spin qubits can be derived from magnetically anisotropic9,10 or isotropic11-13 systems. For both SMMs and molecular spin qubits, hyperfine and dipolar interactions must be supressed to prolong demagnetization and coherence times. Providing a mechanistic description of spin dynamics is a necessarily complex task since a comprehensive picture must consider electronic and vibrational aspects of the molecular system and its interaction with surrounding nuclear and electronic spins. In this sense, the study of spin dynamics in molecular systems is a multidisciplinary field involving resonance spectroscopies, molecular magnetism and theoretical chemistry.

Spin Relaxation Times determined with Electronic Paramagnetic Resonance
Experimental practice for the determination of spin relaxation times is usually different for qubit molecules and SMMs. In the case of qubit molecules, electronic paramagnetic resonance (EPR) is usually employed.14-18 EPR measurements can be performed in single-crystal or powder samples also in frozen solution. The latter is often advantageous since solution samples present a lower concentration of magnetic molecules per unit volume, diminishing the interactions between magnetic centres and, increasing the spin relaxation time. The conventional approach is to use pulsed EPR, allowing the pulses to align the spin in a determined direction and to measure the time required to recover the initial orientation. There are two crucial parameters: the spin-lattice (or longitudinal) T1 and the spin-spin (or transversal) T2 relaxation times.17 The spin-lattice relaxation process involves the interaction of the spin with the vibrations of the surrounding lattice; consequently, T1 depends on the temperature. The energy exchange between the vibrations and the spin provides the mechanism that produces the spin relaxation. There are several methods that use the EPR technique, determining T1 time as the timescale for the evolution of magnetization along the z axis. 18 Taking the initial magnetization 0 , and subsequent spins in the +z direction aligned by an external magnetic field, the T1 time can be determined by applying a  microwave pulse inducing a 180º rotation of the magnetization. Analysis of the evolution time of the component until the system recovers the initial direction allows us to extract the T1 value using the Bloch equation: In many cases, a multi-pulse experiment is performed because the system does not follow the simple exponential Bloch equation due to the presence of other spin processes. These processes are usually called spectral diffusion, which encompasses several different contributions including motion of an anisotropic paramagnetic centre, electron-electron exchange, electron-nuclear cross relaxation and nuclear spin flip-flops.18 A general strategy for avoiding the presence of spectral diffusion effects is to increase the length of the saturating pulse until a constant relaxation time is reached, thus T1 is directly determined without any other superposed effects. In some cases, it is possible to improve the fitting with a simple modification of Equation 1 by introducing the effect of additional contributions through the  stretching parameter: If this does not improve the fitting, it is necessary to add in the mathematical expressions corresponding to the spectral diffusion contributions. In a multi-pulse experiment to determine T1, one initial  pulse is performed and after a delay of time T, a spin-echo sequence (−−− Hahn-echo19 see Figure 1) is used (echo-detected inversion recovery approach). A similar method can also be employed if the initial pulse is a long low-power pulse. This is called echo-detected saturation recovery method. Analysis of the recovering curve of the echo amplitude until equilibrium is reached, as a function of the time T, allows us to estimate the T1 value. The spin-spin relaxation involves the T2 time that follows the temporal evolution of magnetization in the xy plane until recovery of thermal equilibrium. Usually, T2 is considerably smaller than T1, thus the spin reorientation in the xy plane is much faster than in the z axis. However, as T1 decreases with increasing temperature, in some cases at higher temperatures both magnitudes could be similar. There are also several methods to determine T2. Again, the simplest method would be to apply a  pulse and follow the evolution of the component according to the following equation: Multi-pulse approaches are also employed for the xy spin relaxation, using the −−− Hahn echo, modifying the  time between the pulses to overcome the problems of fitting with a simple model (Eq. 3). 18 The spin echo dephasing time constant, usually known as phase memory decay time Tm, can be determined from the representation of the echo amplitude as a function of  time.20 The longer the delay of  time after the  pulse, the larger the number of dephased spins; a lesser number will be refocused with the  pulse, resulting in a decrease of the xy magnetization. An exponential decay of the echo signal with the  time is expected, and such decay is proportional to exp(-2/Tm). Thus, a fitting procedure allows us to determine the Tm magnitude, including all the processes that produce loss of electron spin phase coherence, T2 being one of them.

Spin Relaxation Times determined with AC Susceptibility
Analysis of the spin relaxation time for single-molecule magnets is usually performed by measuring alternating current (ac) magnetic susceptibility with, for instance, a SQUID device. 1 + 2(wt ) 1-a sin(pa / 2) + (wt ) 2-2a (5) From this kind of measurement it is possible to fit the τ values, temperature and external magnetic field21 that can be modified in the experiment. It is worth noting the papers of Ho and Chibotaru which provide an analysis of the results for systems showing two maxima in the ac susceptibility that can appear even in mononuclear systems.22,23

Comparison Spin Relaxation Times using different Techniques
In principle,  can be identified with the spin-lattice T1 relaxation time determined by EPR. The comparison between these magnitudes is not straightforward because in most cases the EPR samples are diluted to avoid decoherence. Such a process is induced by spin-spin interactions, either electron-electron or electron-nuclear. Hence, dilution in diamagnetic matrices or solution using solvents with elements with or without reduced nuclear spin, for instance CS2 or deuterated solvents, help to increase T2 values determined by EPR. 24 The use of solutions complicates the comparison with ac-susceptibility relaxation times because such measurements are usually taken in powder, and the molecular environments are completely different. Dilution in diamagnetic matrices is also a problem for susceptibility measurements due to the low percentage of magnetic contributions modified in the diluted samples, but also some spin relaxation mechanisms, such as quantum tunnelling.

Complexes with Long Coherence Times
For the practical application of the qubits, it is vital to reach a reasonable coherence time, usually

Spin Relaxation Mechanism through Phonons
Due to the multitude of interactions associated with spin relaxation, several magnetic relaxation The hyperfine coupling producing an admixture of states can be effective in breaking such a principle. 63 For the direct spin relaxation mechanism for a Kramers system, the rate can be expressed by the following equation: where A is an adjustable parameter, B is the external magnetic field and T is the temperature, while an equivalent term with the relaxation rate proportional to B2 is used for non-Kramers species.64 Raman Process. This is a mechanism involving a two-phonon process with a virtual excited state with an energy smaller than the Debye temperature. 65 The relaxation rate can be expressed as, where C is a constant parameter, T the temperature and n is 7 and 9 for non-Kramers and Kramers systems, respectively. In practice, n is treated as an adjustable parameter and can strongly deviate from its theoretical temperature dependence. The energy in the transfer from the spin system to the lattice is the difference between absorption and emission to the virtual state (see Fig. 2 Orbach Process. Like the Raman mechanism, this is also a two-phonon process, but the intermediate state involves an electronic excited state of the system:17 The temperature dependence of the Orbach mechanism is exponential.
Where A is a constant parameter, E the energy of the excited state and T the temperature. The approximation is valid for E >> . Thus, from the fitting of 1 −1 at different temperatures, it is possible to extract the energy barrier involved in this process, corresponding to the excited state allowing for spin relaxation. The spin inversion process can be due to a complete jump of the energy barrier through the highest low-spin excited state of the multiplet or helped by an efficient quantum tunnelling relaxation in one of the intermediate excited states. It is worth mentioning that in S=1/2 systems with only two degenerate states, the Orbach process is ill-defined.
Local-mode Process. This is a similar mechanism to the two-phonon processes previously described, but here, the involved state is a vibrational excited state of the electronic ground state.69 The mathematical expression is similar to that for the Orbach process, but the energy involved corresponds to a vibrational energy ℏ of the system: ℎ being a constant. The main difference between this relaxation contribution and Raman, localmode and Orbach is the dependence of the Larmor frequency. See Fig. 2 for a pictorial description of the spin relaxation mechanisms through spin-lattice processes.

Spin-Spin Relaxation Mechanisms and Quantum Tunnelling
The second main group of spin relaxation mechanisms are those involving the spin-spin interactions, which consider both hyperfine (electron-nuclear spin) and dipolar (electron-electron) terms.70,71 Some of these interactions will be intramolecular (i.e. by the presence of I0 nuclei or more than one paramagnetic centre in the same molecule) or intermolecular (by magnetic centres belonging to neighbour molecules). Strategies to supress these interactions include magnetic dilution (replacement of a fraction of crystallized magnetic molecules by diamagnetic analogs)72, replacement of atoms with I=0 isotopes73,74 or the preparation of frozen solution samples with nuclear spin-free solvents.12 The latter approach will supress both hyperfine and spin dipolar relaxation. The spin-spin relaxation time (T2) is employed to quantify these relaxation processes.
This parameter is of prime importance for the coherence time, which is the key parameter in the design of molecular spin qubits.26 For an isolated Kramers doublet system, quantum tunnelling is forbidden by the time-reversal symmetry principle.62 However, the presence of spin-spin interactions causes a dipolar broadening of the energy levels, allowing spin relaxation though quantum tunnelling. 75  A external magnetic field induces an energy gap in the tunnelling doublet, supressing relaxation.
Field dependence of tunnelling relaxation time is usually adjusted to the following expression: where 1 and 2 are constant parameters. Spin relaxation by tunnelling is not only associated with the ground state, but excited states can also exhibit relaxation associated to this mechanism, in a process called thermally assisted quantum tunnelling. 78 where −1 corresponds to the characteristic phonon collision rate of the lattice that shows an exponential temperature dependence, is a perturbation that quantifies the mixing between ±mJ ground-state wave functions, ℏ is the tunnelling gap and i is the field exponent. A small tunnelling gap (around 10-3-10-5 cm-1) indicates an efficient tunnelling relaxation, thus, the experimental fitting of such parameters allows to determine its magnitude. Furthermore, large tunnelling gap values (1 cm-1 ≈30 GHz) might be suitable to be determined by pulsed EPR and it is a key ingredient for long coherence times and insensitivity to magnetic noise in molecular qubits.51

Determination of Parameters of the Spin Relaxation Mechanisms
The spin relaxation mechanisms provide the mathematical framework to fit the experimental spin relaxation times obtained using EPR or ac-susceptibility measurements. Usually, susceptibility measurements of powder samples can be used, according to Eq. 5, to determine the relaxation while for S = ½, the Orbach term is not considered, As mentioned above (see section 3.1) Brons-van Vleck field dependence of the Raman contribution is mostly employed in S=1/2 systems despite that it was originally included to study S=1/2 and S=3/2 compounds.62 Thus, the second term of Eq. 14 could be replaced by the equivalent field-dependent contribution of Eq. 15. In most reported studies of SMM systems, only the Orbach term is considered to extract the energy barrier that corresponds to the energy of the excited state involved in the spin relaxation process.1-3,5 Thus, with just the slope of the representation of log() vs 1/T in the linear dependence region it is possible to determine the energy barrier.
Nevertheless, despite over parametrization problems, the determination of the parameters has often been performed. However, as mentioned above, the local-mode term is not usually included in the fitting of the data. It is worth noting the analysis of susceptibility data performed by Zheng and others.79 They studied a family of mononuclear DyIII complexes for the dependence of the relaxation rate on temperature and the external field, using ac and dc measurements. The fielddependent data was extracted from the time evolution of dc magnetization using a stretched exponential decay model. The analysis of a large amount of relaxation rate data allowed for an accurate determination of the main relaxation mechanisms for the different temperatures and external fields.

Theoretical Approaches for Spin Relaxation
Spin dynamics have been extensively studied by means of classical approaches such as the

Landau-Lifshitz-Bloch equation of motion.80 This methodology is phenomenological and allows
to estimate the time evolution of magnetization in terms of the gyromagnetic ratio and a damping parameter. However, quantum effects are not considered in this method and it cannot be directly applied to the spin relaxation of single-molecule magnets and molecular qubits. Some attempts to include such effects have been carried out for single-molecule magnets.81

Direct Calculation of the Spin Relaxation Times (T2 and Tm)
The direct calculation of the spin relaxation times T1 and T2 from electronic structure methods is where Δ is the energy gap between the two given magnetic states and E 2 (E 2 ) is the nuclear proposed as a convenient molecular descriptor for T2. This parameter is molecular since it depends on the interaction of the electronic spin with nuclear spins from the same molecule. The effect of counterion nuclear spins was also considered, showing an important role of spin dephasing.

Ab Initio Calculations for Magnetic Anisotropy and Spin-Lattice Relaxation
As mentioned above, strong uniaxial magnetic anisotropy is absolutely necessary to obtain high performance SMMs, while this requirement is not so strict for molecular spin qubits. Accordingly, theoretical methods describing SMMs must be particularly accurate when describing magnetic  Orbach regime with a fast-thermal excitation and slower excited-state tunnelling, the demagnetization rate for each Kramers doublet is expressed as, where Ei is the excitation energy of the i doublet determined at CASSCF level, and Z is the partition function. We can define the effective demagnetization barrier as, where M is the number of Kramers doublets involved in the spin relaxation and Nk is a normalization factor for the tunnelling rate. In Fig. 4 Where Ueff as calculated from Eq. 18, was equal to 100 s, following the usual convention107 and 0 is the Orbach prefactor. For the studied systems, 0 as always in a narrow range between 10-11-10-12 s, so the denominator in Eq. 19 was always close to 28. This value can be considered as a convenient approximation for the relation between the effective demagnetization barrier and the Orbach limiting temperature.

Spin-Phonon Coupling Constants.
To progress from the calculation of demagnetization barriers to directly estimating the relaxation time at a given temperature, the effect of vibrations must be considered. In recent years, different groups have developed proposals in this direction.79,108-112 The common strategy is to determine the vibrational modes and corresponding energies with DFT calculations, using periodic or discrete models for the molecules. Once the vibrational modes are determined, the second step is to calculate the spin-phonon coupling for each vibrational mode using a multiconfiguration method such as CASSCF, CASPT2 or NEVPT2. Thus, it is possible to determine which are the most relevant vibrational modes affecting spin relaxation. Normally, this information is fed into a master matrix which eigenvalues are the characteristic rates of the system.  These methods tend to be computationally demanding since they require several CASSCF calculations along every normal mode. Lower cost approaches to determining spin-phonon coupling constants have been recently proposed.122,123 In these approaches, only one CASSCF calculation is required to determine energies and wave functions of the central point, whose crystal-field parameters are extracted using the SIMPRE code,124 based on effective charge electrostatic calculations. Spin-phonon coupling parameters are determined using the evolution of the Stevens parameters along the distortion induced by each vibrational mode, using the effective charge electrostatic calculations. The comparison shows a reasonable agreement of the spinphonon constants between this simplified approach122 and those obtained through CASSCF calculations of the distorted geometries for each vibrational mode.103 Basically, the new approach allows to identify the key vibrational modes involved in spin relaxation, even though the spinphonon constants are at energies that differ slightly from the full method.

Concluding Remarks
To conclude, it is clear that fundamental theoretical challenges in the study of spin relaxation in molecular systems still remain unsolved. Current efforts point to several directions as: (i) direct and accurate calculation of the relaxation times T1 and T2 for molecular qubits; (ii) calculations of the blocking temperature for single-molecule magnets, and development of models that would allow the rationalization of magnitudes that are not correlated with the spin-inversion energy barrier; (iii) a better description of the Raman process with, a clear description of the states involved on such a relaxation process. Furthermore, our understanding of spin-photon coupling will be fundamental to the rationalization of processes in qubits, based on microwave resonators using molecular qubits.