Intramolecular bridging strategies to suppress two-phonon Raman spin relaxation in dysprosocenium single-molecule magnets

Dy(iii) bis-cyclopentadienyl (Cp) sandwich compounds exhibit extremely strong single-ion magnetic anisotropy which imbues them with magnetic memory effects such as magnetic hysteresis, and has put them at the forefront of high-performance single-molecule magnets (SMMs). Owing to the great success of design principles focused on maximising the anisotropy barrier, ever higher Ueff values have been reported leading to significant slow down of single-phonon Orbach spin relaxation. However, anisotropy-based SMM design has largely ignored two-phonon Raman spin relaxation, which is still limiting the temperatures at which a memory effect can be observed. In this work, we study the suppression of Raman relaxation through covalent bridging of the Cp ligands by alkyl chains, testing the hypothesis that increasing the rigidity of the ligand framework results in a blue shift of low frequency vibrations in the first coordination sphere of the Dy(iii) ion. This reshaping of the vibrational low-energy density of states (DOS) results in lower occupation of pseudo-acoustic phonons available to drive Raman relaxation at low temperatures. We simulate Orbach and Raman spin relaxation in a series of zero-, mono-, di- and tri-bridged [Dy(Cpttt)2]+ analogues fully ab initio, using a quantum mechanics (QM)/molecular mechanics (MM) condensed phase embedding protocol in a periodic solvent matrix as a generic and experimentally testable environment model that can include (pseudo-)acoustic phononic degrees of freedom. We show that this approach can simulate magnetic relaxation dynamics in the condensed phase for the existing non-bridged [Dy(Cpttt)2]+ compound with quantitative experimental accuracy. Subsequently, we find a significant slowing down of Raman relaxation can be achieved for the singly-bridged SMM, while the introduction of further bridges leads to faster relaxation. A key result being that we find the two-phonon Raman rates correlate with the purity of the first-excited Kramers doublet in terms of its mJ = ±13/2 content. Even though the bridging design principle is successful at progressively reshaping the low-energy DOS, the introduction of linker atoms in the equatorial plane successively degrades magnetic anisotropy, suggesting the importance of refined design of the linker chemistry. The accuracy of our results emphasises the value of a generic periodic solvent embedding model, such that it permits the modelling of molecular spin dynamics in the condensed phase without knowledge of a crystal structure. This allows the study of hypothetical molecules or aggregates under real-world conditions, which we expect to have utility beyond the field of molecular magnetism.


S1 Classical force field interactions
The OPLS-AA forcefield is employed to describe all bonded and nonbonded interactions.
Since quantum mechanics (QM) and molecular mechanics (MM) region solely interact through nonbonded atomic van der Waals (vdW) and coulomb interactions, only the inter-molecular solvent interactions include harmonic covalent bond and angle stretching contributions involving force constants k r,θ and equilibrium bond lengths and angles r 0 and θ 0 , see Table S1.
All nonbonded interactions expressed in Equation 2 involve atomic charges {q i } to describe Coulomb interactions, and {ϵ ij } and {σ ij } parameters to describe vdW interactions using the geometric mean combination rule for hetero-atomic interactions between atom types i and j (ϵ ij = √ ϵ ii ϵ jj and σ ij = √ σ ii σ jj ).All nonbonded parameters are compiled in Table S2.
where r ij are the interatomic distances and the notation ⟨i, j⟩ indicates pairs of atoms being restricted to different molecules.

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Figure S2: Barrier diagrams depicting Orbach transition rates between the magnetic sub-levels at 100 K.

S4 Characterisation of the crystal field states
Table S3: Kramers doublets of [Dy(Cp ttt ) 2 ] + (0) under application of a magnetic field B z = 0.2 mT as obtained from the crystal field parameterisation of a (9,7)-CASSCF calculation (see main text Section 2 for details).transformation of the spin-phonon coupling matrix elements from atomic to mass-frequency scaled normal mode coordinates defined in ref [1] ⟨m| V (1e)

E/cm
this discrepancy can be linked to the √ ω j −1 and √ M i −1 factors appearing in Equation 3, where ω j and M i are radial frequency of mode j and atomic mass of atom i, respectively, and is a result of transformation 3 being non-orthogonal and hence lacking the norm-preserving property; while the matrix DL (the orthogonal transformation L which diagonalises the mass-weighted Hessian within the non-translating coordinate frame D) exhibits this property, the mode and atom specific factors named above render the overall transformation nonorthogonal.The observation that transforming from atomic to normal mode coordinates, the basis relevant for spin-phonon coupling applications, skews the maximum of the total coupling strength magnitude as a function of the number of bridges n towards higher n (Figure S5), further implies that with increasing n, magnitude of the coupling strength is increasingly shifted onto (i) lower frequency vibrations which (ii) naturally involve the motion of heavier atoms, yielding overall lower total spin-phonon coupling strengths at high n.

S5Figure S3 :Figure S4 :
FigureS3: Decomposition of the condensed phase vibrational density of states (DOS) computed with a bandwidth of 10 cm −1 into rigid body, inter-and intra-fragment contributions of the singlemolecule magnet (SMM) (coloured solid lines) as well as rigid body and intra-molecular vibrations of the solvent molecules (coloured dashed lines) enveloped by the total DOS (black solid line).The total DOS is slightly offset for visualisation purposes.The inset shows the contribution of SMM motion to the low-energy DOS below 300 cm−1

Figure S5 :
Figure S5: Total coupling strength metric in the normal mode (a) and atomic (b) basis.(a) corresponds to the integral of the spectral density shown in Figure 6 in the main text.

Table S1 :
DCM bonded force field parameters.

Table S4 :
Kramers doublets of [Dy(Cp ttb ) 2 ] + (1) under application of a magnetic field B z = 0.2 mT as obtained from the crystal field parameterisation of a (9,7)-CASSCF calculation (see main text Section 2 for details).

Table S5 :
Kramers doublets of [Dy(Cp tbb ) 2 ] + (2) under application of a magnetic field B z = 0.2 mT as obtained from the crystal field parameterisation of a (9,7)-CASSCF calculation (see main text Section 2 for details).

Table S6 :
Kramers doublets of [Dy(Cp bbb ) 2 ] + (3) under application of a magnetic field B z = 0.2 mT as obtained from the crystal field parameterisation of a (9,7)-CASSCF calculation (see main text Section 2 for details).