Ab Initio Prediction of High-Temperature Magnetic Relaxation Rates in Single-Molecule Magnets

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.

Halobenzene Adducts of a Dysprosocenium Single-Molecule Magnet

Dysprosium complexes with strong axial crystal fields are promising candidates for single-molecule magnets (SMMs), which could be used for high-density data storage. Isolated dysprosocenium cations, [Dy(CpR)2]+ (CpR = substituted cyclopentadienyl), have recently shown magnetic hysteresis (a memory effect) above the temperature of liquid nitrogen. Synthetic efforts have focused on reducing strong transverse ligand fields in these systems as they are known to enhance magnetic relaxation by spin-phonon mechanisms. Here we show that equatorial coordination of the halobenzenes PhX (X = F, Cl, Br) and o-C6H4F2 to the cation of a recently reported dysprosocenium complex [Dy(Cpttt)(Cp*)][Al{OC(CF3)3}4] (Cpttt = C5H2tBu3-1,2,4; Cp* = C5Me5) reduces magnetic hysteresis temperatures compared to that of the parent cation. We find that this is due to increased effectiveness of both one- (Orbach) and two-phonon (Raman) relaxation mechanisms, which correlate with the electronegativity and number of interactions with the halide despite κ1-coordination of a single halobenzene having a minimal effect on the metrical parameters of [Dy(Cpttt)(Cp*)(PhX-κ1-X)]+ cations vs the isolated [Dy(Cpttt)(Cp*)]+ cation. We observe unusual divergent behavior of relaxation rates at low temperatures in [Dy(Cpttt)(Cp*)(PhX)][Al{OC(CF3)3}4], which we attribute to a phonon bottleneck effect. We find that, despite the transverse fields introduced by the monohalobenzenes in these cations, the interactions are sufficiently weak that the effective barriers to magnetization reversal remain above 1000 cm-1, being only ca. 100 cm-1 lower than for the parent complex, [Dy(Cpttt)(Cp*)][Al{OC(CF3)3}4].

Synthesis and Magnetic Properties of Bis-Halobenzene Decamethyldysprosocenium Cations

The decamethyldysprosocenium cation, [Dy(Cp*)2]+ (Cp* = {C5Me5}), was a target single-molecule magnet (SMM) prior to the isolation of larger dysprosocenium cations, which have recently shown magnetic memory effects up to 80 K. However, the relatively short Dy···Cp*centroid distances of [Dy(Cp*)2]+, together with the reduced resonance of its vibrational modes with electronic states compared to larger dysprosocenium cations, could lead to more favorable SMM behavior. Here, we report the synthesis and magnetic properties of a series of solvated adducts containing bis-halobenzene decamethyldysprosocenium cations, namely [Dy(Cp*)2(PhX-κ-X)2][Al{OC(CF3)3}4] (X = F or Cl) and [Dy(Cp*)2(C6H4F2-κ2-F,F)(C6H4F2-κ-F)][Al{OC(CF3)3}4]. These complexes were prepared by the sequential reaction of [Dy(Cp*)2(μ-BH4)]∞ with allylmagnesium chloride and [NEt3H][Al{OC(CF3)3}4], followed by recrystallization from parent halobenzenes. The complexes were characterized by powder and single crystal X-ray diffraction, NMR and ATR-IR spectroscopy, elemental analysis, and SQUID magnetometry; experimental data were rationalized by a combination of density functional theory and ab initio calculations. We find that bis-halobenzene adducts of the [Dy(Cp*)2]+ cation exhibit highly bent Cp*···Dy···Cp* angles; these cations are also susceptible to decomposition by C-X (X = F, Cl, Br) activation and displacement of halobenzenes by O-donor ligands. The effective energy barrier to reversal of magnetization measured for [Dy(Cp*)2(PhF-κ-F)2][Al{OC(CF3)3}4] (930(6) cm-1) sets a new record for SMMs containing {Dy(Cp*)2} fragments, though all SMM parameters are lower than would be predicted for an isolated [Dy(Cp*)2]+ cation, as expected due to transverse ligand fields introduced by halobenzenes and the large deviation of the Cp*···Dy···Cp* angle from linearity promoting magnetic relaxation.

A highly anisotropic family of hexagonal bipyramidal Dy(III) unsaturated 18-crown-6 complexes exceeding the blockade barrier over 2700 K: a computational exploration

In the present work, we have explored a series of unsaturated hexa-18-crown-6 (U18C6) ligands towards designing highly anisotropic Dy(III) based single-ion magnets (SIMs) with the general formula [Dy(U18C6)X2]+ (where U18C6 = [C12H12O6] (1), [C12H12S6] (2), [C12H12Se6] (3), [C12H12O4S2] (4), [C12H12O4Se2] (5) and X = F, Cl, Br, I, OtBu and OSiPh3). By analysing the electronic structure, bonding and magnetic properties, we find that the U18C6 ligands prefer stabilising the highly symmetric eight-coordinated hexagonal bipyramidal geometry (HBPY-8), which is the source of the near-Ising type anisotropy in all the [Dy(U18C6)X2]+ complexes. Moreover, the ability of sulfur/selenium substituted U18C6 ligands to stabilize the highly anisotropic HBPY-8 geometry makes them more promising towards engineering the equatorial ligand field compared to substituted saturated 18C6 ligands where the exodentate arrangement of the S lone pairs results in low symmetry. Magnetic relaxation analysis predicts a record barrier height over 2700 K for [Dy(C12H12O6)F2]+ and [Dy(C12H12S6)X2]+ (where X = F, OtBu and OSiPh3) complexes, nearly 23% higher than those of the top performing Dy(III) based SIMs in the literature.

Optical Phenomena in Molecule-Based Magnetic Materials

Since the last century, we have witnessed the development of molecular magnetism which deals with magnetic materials based on molecular species, i.e., organic radicals and metal complexes. Among them, the broadest attention was devoted to molecule-based ferro-/ferrimagnets, spin transition materials, including those exploring electron transfer, molecular nanomagnets, such as single-molecule magnets (SMMs), molecular qubits, and stimuli-responsive magnetic materials. Their physical properties open the application horizons in sensors, data storage, spintronics, and quantum computation. It was found that various optical phenomena, such as thermochromism, photoswitching of magnetic and optical characteristics, luminescence, nonlinear optical and chiroptical effects, as well as optical responsivity to external stimuli, can be implemented into molecule-based magnetic materials. Moreover, the fruitful interactions of these optical effects with magnetism in molecule-based materials can provide new physical cross-effects and multifunctionality, enriching the applications in optical, electronic, and magnetic devices. This Review aims to show the scope of optical phenomena generated in molecule-based magnetic materials, including the recent advances in such areas as high-temperature photomagnetism, optical thermometry utilizing SMMs, optical addressability of molecular qubits, magneto-chiral dichroism, and opto-magneto-electric multifunctionality. These findings are discussed in the context of the types of optical phenomena accessible for various classes of molecule-based magnetic materials.

Unravelling the role of spin-vibrational coupling in designing high-performance pentagonal bipyramidal Dy(iii) single ion magnets

At the cutting edge of high-performance single-molecule magnets (SMMs) lie lanthanide-based complexes, renowned for their potent magnetic anisotropy. SMMs containing one metal centre are defined as single-ion magnets (SIMs). The performance of SMMs is measured generally via the barrier height for magnetisation reversal (Ueff) and blocking temperature (TB), below which the magnetisation is fully frozen. To enhance the Ueff and TB values in lanthanide-based SMMs, the static crystal field splitting of mJ levels has been effectively adjusted through ligand design, leveraging the oblate/prolate ground state 4f electron density shape. However, the maximum fine-tuning achievable through ligand design, known as the axial limit, has already been reached in this class of compounds. This necessitates new design principles to enhance SMM characteristics to better suit end-user applications. Among other avenues that can be explored to improve SMM characteristics, a deeper understanding of spin-phonon coupling is critical to advancing TB values. However, there are only a handful of examples where this has been deciphered. In this work, using a combination of DFT and ab initio CASSCF calculations, we have performed spin-phonon calculations on five classes of pentagonal bipyramidal Dy(iii) SIMs exhibiting TB values in the range of 4.5 K to 36 K ([Dy(bbpen)Br] (1, H2bbpen = N,N'-bis(2-hydroxybenzyl)-N,N'-bis(2-methylpyridyl)ethylenediamine), [Dy(OCMe3)Br(THF)5][BPh4] (2) [Dy(OSiMe3)Br(THF)5] [BPh4] (3), [Dy(LN5)(Ph3SiO)2](BPh4)·CH2Cl2 (4) and [L2Dy(H2O)5][I]3·L2·H2O (5, L = tBuPO(NHiPr)2)). Unlike the method employed elsewhere for the calculation of spin-phonon coupling, in this work, we have employed a set of criteria and intuitively selected vibrational modes to perform the spin-phonon coupling analysis. The approach provided here not only reduces the computational cost significantly but also suggests chemical intuition to improve the performance of this class of compounds. Our calculations reveal that low-energy vibrational modes govern the magnetisation relaxation in these SIMs. A flexible first coordination sphere found on some of the complexes was found to be responsible for low-energy vibrations that flip the magnetisation, reducing the TB values drastically (complexes 2 and 3). On the other hand, a rigid first coordination sphere and a stiff ligand framework move the spin-vibrational coupling that causes the relaxation to lie beyond the secondary coordination sphere, resulting in an increase in TB values. Our calculations also reveal that not only the atoms in the first coordination sphere but also those in the secondary coordination sphere affect the performance of the SMMs. Learning from this exercise, we have undertaken several in silico models based on these vibrations to improve the TB values. Some of these predictions were correlated with literature precedents, offering confidence in the methodology employed. To this end, our comprehensive investigation, involving twenty-three molecules/models and five sets of geometries for pentagonal bipyramidal Dy(iii) single-ion magnets (SIMs), unveils a treasure trove of chemically sound design clues, poised to enhance the TB values in this fascinating molecular realm.

A novel non-adiabatic spin relaxation mechanism in molecular qubits

The interaction of electronic spin and molecular vibrations mediated by spin-orbit coupling governs spin relaxation in molecular qubits. We derive an extended molecular spin Hamiltonian that includes both adiabatic and non-adiabatic spin-dependent interactions, and we implement the computation of its matrix elements using state-of-the-art density functional theory. The new molecular spin Hamiltonian contains a novel spin-vibrational orbit interaction with a non-adiabatic origin, together with the traditional molecular Zeeman and zero-field splitting interactions with an adiabatic origin. The spin-vibrational orbit interaction represents a non-Abelian Berry curvature on the ground-state electronic manifold and corresponds to an effective magnetic field in the electronic spin dynamics. We further develop a spin relaxation rate model that estimates the spin relaxation time via the two-phonon Raman process. An application of the extended molecular spin Hamiltonian together with the spin relaxation rate model to Cu(II) porphyrin, a prototypical S = 1/2 molecular qubit, demonstrates that the spin relaxation time at elevated temperatures is dominated by the non-adiabatic spin-vibrational orbit interaction. The computed spin relaxation rate and its magnetic field orientation dependence are in excellent agreement with experimental measurements.

Confining single Er3+ ions in sub-3 nm NaYF4 nanoparticles to induce slow relaxation of the magnetisation

Molecular systems known as single-molecule magnets (SMMs) exhibit magnet-like behaviour of slow relaxation of the magnetisation and magnetic hysteresis and have potential application in high-density memory storage or quantum computing. Often, their intrinsic magnetic properties are plagued by low-energy molecular vibrations that lead to phonon-induced relaxation processes, however, there is no straightforward synthetic approach for molecular systems that would lead to a small amount of low-energy vibrations and low phonon density of states at the spin-resonance energies. In this work, we apply knowledge accumulated over the last decade in molecular magnetism to nanoparticles, incorporating Er3+ ions in an ultrasmall sub-3 nm diamagnetic NaYF4 nanoparticle (NP) and probing the slow relaxation dynamics intrinsic to the Er3+ ion. Furthermore, by increasing the doping concentration, we also investigate the role of intraparticle interactions within the NP. The knowledge gained from this study is anticipated to enable better design of magnetically high-performance molecular and bulk magnets for a wide variety of applications, such as molecular electronics.

Bimetallic Synergy Enables Silole Insertion into THF and the Synthesis of Erbium Single-Molecule Magnets

The potassium silole K2 [SiC4 -2,5-(SiMe3 )2 -3,4-Ph2 ] reacts with [M(η8 -COT)(THF)4 ][BPh4 ] (M=Er, Y; COT=cyclo-octatetraenyl) in THF to give products that feature unprecedented insertion of the nucleophilic silicon centre into a carbon-oxygen bond of THF. The structure of the major product, [(μ-η8  : η8 -COT)M(μ-L1 )K]∞ (1M ), consists of polymeric chains of sandwich complexes, where the spiro-bicyclic silapyran ligand [C4 H8 OSiC4 (SiMe3 )2 Ph2 ]2- (L1 ) coordinates to potassium via the oxygen. The minor product [(μ-η8  : η8 -COT)M(μ-L1 )K(THF)]2 (2M ) features coordination of the silapyran to the rare-earth metal. In forming 1M and 2M , silole insertion into THF only occurs in the presence of potassium and the rare-earth metal, highlighting the importance of bimetallic synergy. The lower nucleophilicity of germanium(II) leads to contrasting reactivity of the potassium germole K2 [GeC4 -2,5-(SiMe3 )2 -3,4-Me2 ] towards [M(η8 -COT)(THF)4 ][BPh4 ], with intact transfer of the germole occurring to give the coordination polymers [{η5 -GeC4 (SiMe3 )2 Me2 }M(η8 -COT)K]∞ (3M ). Despite the differences in reactivity induced by the group 14 heteroatom, the single-molecule magnet properties of 1Er , 2Er and 3Er are similar, with thermally activated relaxation occurring via the first-excited Kramers doublet, subject to effective energy barriers of 122, 80 and 91 cm-1 , respectively. Compound 1Er is also analysed by high-frequency dynamic magnetic susceptibility measurements up to 106  Hz.

Endohedral metallofullerene molecular nanomagnets

Magnetic lanthanide (Ln) metal complexes exhibiting magnetic bistability can behave as molecular nanomagnets, also known as single-molecule magnets (SMMs), suitable for storing magnetic information at the molecular level, thus attracting extensive interest in the quest for high-density information storage and quantum information technologies. Upon encapsulating Ln ion(s) into fullerene cages, endohedral metallofullerenes (EMFs) have been proven as a promising and versatile platform to realize chemically robust SMMs, in which the magnetic properties are able to be readily tailored by altering the configurations of the encapsulated species and the host cages. In this review, we present critical discussions on the molecular structures and magnetic characterizations of EMF-SMMs, with the focus on their peculiar molecular and electronic structures and on the intriguing molecular magnetism arising from such structural uniqueness. In this context, different families of magnetic EMFs are summarized, including mononuclear EMF-SMMs wherein single-ion anisotropy is decisive, dinuclear clusterfullerenes whose magnetism is governed by intramolecular magnetic interaction, and radical-bridged dimetallic EMFs with high-spin ground states that arise from the strong ferromagnetic coupling. We then discuss how molecular assemblies of SMMs can be constructed, in a way that the original SMM behavior is either retained or altered in a controlled manner, thanks to the chemical robustness of EMFs. Finally, on the basis of understanding the structure-magnetic property correlation, we propose design strategies for high-performance EMF-SMMs by engineering ligand fields, electronic structures, magnetic interactions, and molecular vibrations that can couple to the spin states.

Isolation of a Bent Dysprosium Bis(amide) Single-Molecule Magnet

The isolation of formally two-coordinate lanthanide (Ln) complexes is synthetically challenging, due to predominantly ionic Ln bonding regimes favoring high coordination numbers. In 2015, it was predicted that a near-linear dysprosium bis(amide) cation [Dy{N(SiiPr3)2}2]+ could provide a single-molecule magnet (SMM) with an energy barrier to magnetic reversal (Ueff) of up to 2600 K, a 3-fold increase of the record Ueff for a Dy SMM at the time; this work showed a potential route to SMMs that can provide high-density data storage at higher temperatures. However, synthetic routes to a Dy complex containing only two monodentate ligands have not previously been realized. Here, we report the synthesis of the target bent dysprosium bis(amide) complex, [Dy{N(SiiPr3)2}2][Al{OC(CF3)3}4] (1-Dy), together with the diamagnetic yttrium analogue. We find Ueff = 950 ± 30 K for 1-Dy, which is much lower than the predicted values for idealized linear two-coordinate Dy(III) cations. Ab initio calculations of the static electronic structure disagree with the experimentally determined height of the Ueff barrier, thus magnetic relaxation is faster than expected based on magnetic anisotropy alone. We propose that this is due to enhanced spin-phonon coupling arising from the flexibility of the Dy coordination sphere, in accord with ligand vibrations being of equal importance to magnetic anisotropy in the design of high-temperature SMMs.

Molecular nanomagnets: a viable path toward quantum information processing?

Molecular nanomagnets (MNMs), molecules containing interacting spins, have been a playground for quantum mechanics. They are characterized by many accessible low-energy levels that can be exploited to store and process quantum information. This naturally opens the possibility of using them as qudits, thus enlarging the tools of quantum logic with respect to qubit-based architectures. These additional degrees of freedom recently prompted the proposal for encoding qubits with embedded quantum error correction (QEC) in single molecules. QEC is the holy grail of quantum computing and this qudit approach could circumvent the large overhead of physical qubits typical of standard multi-qubit codes. Another important strength of the molecular approach is the extremely high degree of control achieved in preparing complex supramolecular structures where individual qudits are linked preserving their individual properties and coherence. This is particularly relevant for building quantum simulators, controllable systems able to mimic the dynamics of other quantum objects. The use of MNMs for quantum information processing is a rapidly evolving field which still requires to be fully experimentally explored. The key issues to be settled are related to scaling up the number of qudits/qubits and their individual addressing. Several promising possibilities are being intensively explored, ranging from the use of single-molecule transistors or superconducting devices to optical readout techniques. Moreover, new tools from chemistry could be also at hand, like the chiral-induced spin selectivity. In this paper, we will review the present status of this interdisciplinary research field, discuss the open challenges and envisioned solution paths which could finally unleash the very large potential of molecular spins for quantum technologies.

Understanding Single-Molecule Magnet properties of lanthanide complexes from 4f orbital splitting

We present an approach for connecting the magnetic anisotropy of lanthanide mononuclear complexes with their f-orbital splitting for both idealized and real coordination environments. Our proposal is straightforward to apply and provides sensible estimations of the energy spacing of the ground multiplet for axial magnetic systems. This energy splitting controls Single-Molecule Magnet properties of lanthanide complexes, determining key parameters such as the demagnetization energy barrier (Ueff). Importantly, this approach is consistent with the current paradigm of oblate and prolate preferences for the distribution of the f-electron density, but delivers a finer description for ions belonging to the same group (e.g. the oblates TbIII and DyIII). The model provides simple explanations for some general trends observed experimentally (e.g. the low barriers for ErIII complexes in comparison to DyIII or the large barriers observed for cyclopentadienyl DyIII complexes in comparison with other ligands based on organometallic rings), contributing as a valuable tool to expand our description of ligand field effects in lanthanide-based SMMs.

Vibronic effects on the quantum tunnelling of magnetisation in Kramers single-molecule magnets

Single-molecule magnets are among the most promising platforms for achieving molecular-scale data storage and processing. Their magnetisation dynamics are determined by the interplay between electronic and vibrational degrees of freedom, which can couple coherently, leading to complex vibronic dynamics. Building on an ab initio description of the electronic and vibrational Hamiltonians, we formulate a non-perturbative vibronic model of the low-energy magnetic degrees of freedom in monometallic single-molecule magnets. Describing their low-temperature magnetism in terms of magnetic polarons, we are able to quantify the vibronic contribution to the quantum tunnelling of the magnetisation, a process that is commonly assumed to be independent of spin-phonon coupling. We find that the formation of magnetic polarons lowers the tunnelling probability in both amorphous and crystalline systems by stabilising the low-lying spin states. This work, thus, shows that spin-phonon coupling subtly influences magnetic relaxation in single-molecule magnets even at extremely low temperatures where no vibrational excitations are present.

Spin-Vibronic Dynamics in Open-Shell Systems beyond the Spin Hamiltonian Formalism

Vibronic coupling has a dramatic influence over a large number of molecular processes, ranging from photochemistry to spin relaxation and electronic transport. The simulation of vibronic coupling with multireference wave function methods has been largely applied to organic compounds, and only early efforts are available for open-shell systems such as transition metal and lanthanide complexes. In this work, we derive a numerical strategy to differentiate the molecular electronic Hamiltonian in the context of multireference ab initio methods and inclusive of spin-orbit coupling effects. We then provide a formulation of open quantum system dynamics able to predict the time evolution of the electron density matrix under the influence of a Markovian phonon bath up to fourth-order perturbation theory. We apply our method to Co(II) and Dy(III) molecular complexes exhibiting long spin relaxation times and successfully validate our strategy against the use of an effective spin Hamiltonian. Our study sheds light on the nature of vibronic coupling, the importance of electronic excited states in spin relaxation, and the need for high-level computational chemistry to quantify it.

Thermally Stable Terbium(II) and Dysprosium(II) Bis-amidinate Complexes

The thermostable four-coordinate divalent lanthanide (Ln) bis-amidinate complexes [Ln(Piso)2] (Ln = Tb, Dy; Piso = {(NDipp)2CtBu}, Dipp = C6H3iPr2-2,6) were prepared by the reduction of parent five-coordinate Ln(III) precursors [Ln(Piso)2I] (Ln = Tb, Dy) with KC8; halide abstraction of [Ln(Piso)2I] with [H(SiEt3)2][B(C6F5)] gave the respective Ln(III) complexes [Ln(Piso)2][B(C6F5)]. All complexes were characterized by X-ray diffraction, ICP-MS, elemental analysis, SQUID magnetometry, UV-vis-NIR, ATR-IR, NMR, and EPR spectroscopy and ab initio CASSCF-SO calculations. These data consistently show that [Ln(Piso)2] formally exhibit Ln(II) centers with 4fn5dz21 (Ln = Tb, n = 8; Dy, n = 9) valence electron configurations. We show that simple assignments of the f-d coupling to either L-S or J-s schemes are an oversimplification, especially in the presence of significant crystal field splitting. The coordination geometry of [Ln(Piso)2] is intermediate between square planar and tetrahedral. Projecting from the quaternary carbon atoms of the CN2 ligand backbones shows near-linear C···Ln···C arrangements. This results in strong axial ligand fields to give effective energy barriers to magnetic reversal of 1920(91) K for the Tb(II) analogue and 1964(48) K for Dy(II), the highest values observed for mononuclear Ln(II) single-molecule magnets, eclipsing 1738 K for [Tb(C5iPr5)2]. We tentatively attribute the fast zero-field magnetic relaxation for these complexes at low temperatures to transverse fields, resulting in considerable mixing of mJ states.

Analytical nonadiabatic coupling and state-specific energy gradient for the crystal field Hamiltonian describing lanthanide single-ion magnets

Paramagnetic molecules with a metal ion as an electron spin center are promising building blocks for molecular qubits and high-density memory arrays. However, fast spin relaxation and decoherence in these molecules lead to a rapid loss of magnetization and quantum information. Nonadiabatic coupling (NAC), closely related to spin-vibrational coupling, is the main source of spin relaxation and decoherence in paramagnetic molecules at higher temperatures. Predicting these couplings using numerical differentiation requires a large number of computationally intensive ab initio or crystal field electronic structure calculations. To reduce computational cost and improve accuracy, we derive and implement analytical NAC and state-specific energy gradient for the ab initio parametrized crystal field Hamiltonian describing single-ion molecular magnets. Our implementation requires only a single crystal field calculation. In addition, the accurate NACs and state-specific energy gradients can be used to model spin relaxation using sophisticated nonadiabatic molecular dynamics, which avoids the harmonic approximation for molecular vibrations. To test our implementation, we calculate the NAC values for three lanthanide complexes. The predicted values support the relaxation mechanisms reported in previous studies.

Accurate and Efficient Spin-Phonon Coupling and Spin Dynamics Calculations for Molecular Solids

Molecular materials are poised to play a significant role in the development of future optoelectronic and quantum technologies. A crucial aspect of these areas is the role of spin-phonon coupling and how it facilitates energy transfer processes such as intersystem crossing, quantum decoherence, and magnetic relaxation. Thus, it is of significant interest to be able to accurately calculate the molecular spin-phonon coupling and spin dynamics in the condensed phase. Here, we demonstrate the maturity of ab initio methods for calculating spin-phonon coupling by performing a case study on a single-molecule magnet and showing quantitative agreement with the experiment, allowing us to explore the underlying origins of its spin dynamics. This feat is achieved by leveraging our recent developments in analytic spin-phonon coupling calculations in conjunction with a new method for including the infinite electrostatic potential in the calculations. Furthermore, we make the first ab initio determination of phonon lifetimes and line widths for a molecular magnet to prove that the commonplace Born-Markov assumption for the spin dynamics is valid, but such "exact" phonon line widths are not essential to obtain accurate magnetic relaxation rates. Calculations using this approach are facilitated by the open-source packages we have developed, enabling cost-effective and accurate spin-phonon coupling calculations on molecular solids.

Peroxido-bridged chiral double-decker dysprosium macrocycles

Lanthanide peroxides show high reactivity in oxidative coupling of methane (OCM). However, the number of isolated and structurally characterized molecular species remains relatively small. To the best of our knowledge, homochiral molecule-based lanthanide peroxides have not been reported. Herein, two pairs of side-on peroxido-bridged dinuclear hexaazamacrocyclic dysprosium enantiomers with formulas [Dy2(LES/R)2L2O2](BPh4)2·CH3OH·CH3CN (where LE is derived from the condensation reaction between (1S,2S)/(1R,2R)-1,2-diphenylethylenediamine and 2,6-diformylpyridine; HL = 2,6-diphenylphenol) (1/2) and [Dy2(LES/R)2Cl2O2](BPh4)2·2CH3CN (3/4) are specially designed and created with the help of hydrogen peroxide. The out-of-phase alternating-current magnetic susceptibility of 1/2 gives rise to frequency-dependent peaks between 6 and 32 K under a zero applied direct current (dc) field, while no peak at any temperature and frequency was observed for 3/4 implying the presence of a weak axial crystal field (CF).

Understanding electrostatics and covalency effects in highly anisotropic organometallic sandwich dysprosium complexes [Dy(CmRm)2] (where R = H, SiH3, CH3 and m = 4 to 9): a computational perspective

In this article, we have thoroughly studied the electronic structure and 4f-ligand covalency of six mononuclear dysprosium organometallic sandwich complexes [Dy(CmRm)2]n+/- (where R = H, SiH3, CH3; m = 4 to 9; n = 1, 3) using both the scalar relativistic density functional and complete active space self-consistent field (CASSCF) and N-electron valence perturbation theory (NEVPT2) method to shed light on the ligand field effects in fine-tuning the magnetic anisotropy of these complexes. Energy decomposition analysis (EDA) and ab initio-based ligand field theory AILFT calculations predict the sizable 4f-ligand covalency in all these complexes. The analysis of CASSCF/NEVPT2 computed spin-Hamiltonian (SH) parameters indicates the stabilization of mJ |±15/2〉 for [Dy(C4(SiH3)4)2]- (1), [Dy(C5(CH3)5)2]+ (2) and [Dy(C6H6)2]3+ (3) complexes with the Ucal value of 1867.5, 1621.5 and 1070.8 cm-1, respectively. On the other hand, we observed mJ |±9/2〉 as the ground state for [Dy(C7H7)2]3- (4) and [Dy(C8H8)2]- (5) complexes with significantly smaller Ucal values of 237.1 and 38.6 cm-1 respectively. For the nine-membered ring [Dy(C9H9)2]+ (6) complex, we observed the stabilization of the mJ |±1/2〉 ground state, with the first excited state being located ∼29 cm-1 higher in energy. AILFT-NEVPT2 ligand field splitting analysis indicates that the presence of π-type 4f-ligand interactions in complexes 1-3 help generate the axial-ligand field, while the δ-type interactions in complexes 4-5 generate the equatorial ligand field despite the ligands approaching from the axial direction. As the ring size increases, φ-type interactions dominate, generating a pure equatorial ligand field stabilising mJ |±1/2〉 as the ground state for 6. Calculations suggest that the nature of the ligand field mainly governs the Ucal values in the following order: 4f-Lσ > 4f-Lπ > 4f-Lδ > 4f-Lφ. Calculations were performed by replacing ligands with CHELPG charges to access the crystal field (CF) effects which suggests the stabilization of pure mJ |±15/2〉 in all the charge-embedded models (1Q-6Q). Our findings point out that the crystal field and ligand field effects complement each other and generate a giant barrier for magnetic relaxation in the small ring complexes 1-3, while a relatively weak crystal field and adverse 4f-Lδ/4f-Lφ interactions diminish the SMM behaviour in the large ring complexes 4-6.

Theoretical Magnetic Relaxation and Spin-Phonon Coupling Study in a Series of Molecular Engineering Designed Bridged Dysprosocenium Analogues

A detailed computational study of hypothetical sandwich dysprosium double-decker complexes, bridged by various numbers of aliphatic linkers, was performed to evaluate the effect of the structural modifications on their ground-state magnetic sublevels and assess their potential as candidates for single-molecule magnets (SMMs). The molecular structures of seven complexes were optimized using the TPSSh functional, and the electronic structure and magnetic properties were investigated using the complete active space self-consistent field method (CASSCF). Estimates of the magnetic moment blocking barrier (Ueff) and blocking temperatures (TB) are reported. In addition, a new method based on computed derivatives of effective demagnetization barriers Ueff with respect to vibrational normal modes was introduced and applied to evaluate the impact of spin-phonon coupling on the SMM properties. On the basis of the computed parameters, we have identified promising candidates with properties superior to those of the existing single-molecule magnets.

AtomAccess: A Predictive Tool for Molecular Design and Its Application to the Targeted Synthesis of Dysprosium Single-Molecule Magnets

Isolated dysprosocenium cations, [Dy(CpR)2]+ (CpR = substituted cyclopentadienyl), have recently been shown to exhibit superior single-molecule magnet (SMM) properties over closely related complexes with equatorially bound ligands. However, gauging the crossover point at which the CpR substituents are large enough to prevent equatorial ligand binding, but small enough to approach the metal closely and generate strong crystal field splitting has required laborious synthetic optimization. We therefore created the computer program AtomAccess to predict the accessibility of a metal binding site and its ability to accommodate additional ligands. Here, we apply AtomAccess to identify the crossover point for equatorial coordination in [Dy(CpR)2]+ cations in silico and hence predict a cation that is at the cusp of stability without equatorial interactions, viz., [Dy(Cpttt)(Cp*)]+ (Cpttt = C5H2tBu3-1,2,4, Cp* = C5Me5). Upon synthesizing this cation, we found that it crystallizes as either a contact ion-pair, [Dy(Cpttt)(Cp*){Al[OC(CF3)3]4-κ-F}], or separated ion-pair polymorph, [Dy(Cpttt)(Cp*)][Al{OC(CF3)3}4]·C6H6. Upon characterizing these complexes, together with their precursors, yttrium and yttrium-doped analogues, we find that the contact ion-pair shows inferior SMM properties to the separated ion-pair, as expected, due to faster Raman and quantum tunneling of magnetization relaxation processes, while the Orbach region is relatively unaffected. The experimental verification of the predicted crossover point for equatorial coordination in this work tests the limitations of the use of AtomAccess as a predictive tool and also indicates that the application of this type of program shows considerable potential to boost efficiency in exploratory synthetic chemistry.

Trityl-Based Lanthanide-Supramolecular Assemblies Exhibiting Slow Magnetic Relaxation

The triphenylmethane (trityl) group has been recognized as a supramolecular synthon in crystal engineering, molecular machine rotors and stereochemical chirality inductors in materials science. Herein we demonstrate for the first time how it can be utilized in the domain of molecular magnetic materials through shaping of single molecule magnet (SMM) properties within the lanthanide complexes in tandem with other non-covalent interactions. Trityl-appended mono- (HL1 ) and bis-compartmental (HL2 ) hydrazone ligands were synthesized and complexated with Dy(III) and Er(III) triflate and nitrate salts to generate four monometallic (1-4) and two bimetallic (5, 6) complexes. The static and dynamic magnetic properties of 1-6 were investigated, revealing that only ligand HL1 induces assemblies (1-4) capable of showing SMM behaviour, with Dy(III) congeners (1, 2) able to exhibit the phenomenon also under zero field conditions. Theoretical ab initio studies helped in determination of Dy(III) energetic levels, magnetic anisotropic axes and corroborated magnetic relaxation mechanisms to be a combination of Raman and quantum tunnelling in zero dc field, the latter being cancelled in the optimum non-zero dc field. Our work represents the first study of magneto-structural correlations within the trityl Ln-SMMs, leading to generation of slowly relaxing zero-field dysprosium complexes within the hydrogen-bonded assemblies.

Spin-phonon coupling and magnetic relaxation in single-molecule magnets

Electron-phonon coupling is important in many physical phenomena, e.g. photosynthesis, catalysis and quantum information processing, but its impacts are difficult to grasp on the microscopic level. One area attracting wide interest is that of single-molecule magnets, which is motivated by searching for the ultimate limit in the miniaturisation of binary data storage media. The utility of a molecule to store magnetic information is quantified by the timescale of its magnetic reversal processes, also known as magnetic relaxation, which is limited by spin-phonon coupling. Several recent accomplishments of synthetic organometallic chemistry have led to the observation of molecular magnetic memory effects at temperatures above that of liquid nitrogen. These discoveries have highlighted how far chemical design strategies for maximising magnetic anisotropy have come, but have also highlighted the need to characterise the complex interplay between phonons and molecular spin states. The crucial step is to make a link between magnetic relaxation and chemical motifs, and so be able to produce design criteria to extend molecular magnetic memory. The basic physics associated with spin-phonon coupling and magnetic relaxation was outlined in the early 20th century using perturbation theory, and has more recently been recast in the form of a general open quantum systems formalism and tackled with different levels of approximations. It is the purpose of this Tutorial Review to introduce the topics of phonons, molecular spin-phonon coupling, and magnetic relaxation, and to outline the relevant theories in connection with both the traditional perturbative texts and the more modern open quantum systems methods.

Ab initio prediction of key parameters and magneto-structural correlation of tetracoordinated lanthanide single-ion magnets

Single-molecule magnets (SMMs) have great potential in becoming revolutionary materials for micro-electronic devices. As one type of SMM and holding the performance record, lanthanide single-ion magnets (Ln-SIMs) stand at the forefront of the family. Lowering the coordination number (CN) is an important strategy to improve the performance of Ln-SIMs. Here, we report a theoretical study on a typical group of low-CN Ln-SIMs, i.e., tetracoordinated structures. Our results are consistent with those of experiments and they identify the same three best Ln-SIMs via a concise criterion, i.e., the co-existence of long τ_QTM and high U_eff. Compared to the record-holding dysprosocenium systems, the best SIMs here possess τ_QTM values that are shorter by several orders of magnitude and U_eff values that are lower by ∼1000 Kelvin (K). These are important reasons for the fact that the tetracoordinated Ln-SIMs are clearly inferior to dysprosocenium. A simple but intuitive crystal-field analysis leads to several routes to improve the performance of a given Ln-SIM, including compression of the axial bond length, widening the axial bond angle, elongation of the equatorial bond length and usage of weaker equatorial donor ligands. Although these routes are not brand-new, the most efficient option and the degree of improvement resulting from it are not known in advance. Consequently, a theoretical magneto-structural study, covering various routes, is carried out for the best Ln-SIM here and the most efficient route is shown to be widening the axial ∠O-Dy-O angle. The most optimistic case, having a ∠O-Dy-O of 180°, could have a τ_QTM (up to 10³ s) and U_eff (∼2400 K) close to those of the record-holders. Subsequently, a blocking temperature (T_B) of 64 K is predicted to be possible for it. A more practical case, with ∠O-Dy-O being 160°, could have a τ_QTM of up to 400 s, U_eff of around 2200 K and the possibility of a T_B of 57 K. Although having an inherent precision limit, these predictions provide a guide to performance improvement, starting from an existing system.

Tuning symmetry and magnetic blocking of an exchange-coupled lanthanide ion in isomeric, tetrametallic complexes: [LnCl6(TiCp2)3]

The synthesis and magnetic properties of two pairs of isomeric, exchange-coupled complexes, [LnCl₆(TiCp₂)₃] (Ln = Gd, Tb), are reported. In each isomeric pair, the central lanthanide ion adopts either a pseudo-octahedral (O-Ln) or trigonal prismatic geometry (TP-Ln) yielding complexes with C ₁ or C _3h molecular symmetry, respectively. Ferromagnetic exchange coupling is observed in TP-Ln as indicated by the increases in χ _m T below 30 K. For TP-Gd, a fit to the susceptibility reveals ferromagnetic coupling between the Gd³⁺ ion and the Ti³⁺ ions (J = 2.90(1) cm^-1). In contrast to O-Tb, which shows no single-molecule magnetic behavior, the TP-Tb complex presents slow magnetic relaxation with a 100s-blocking temperature of 2.3 K and remanent magnetization at zero field up to 3 K. The calculated electronic structures of both compounds imply that trigonal prismatic geometry of TP-Tb is critical to the observed magnetic behavior.

Modular Approach to Creating Functionalized Surface Arrays of Molecular Qubits

The quest for developing quantum technologies is driven by the promise of exponentially faster computations, ultrahigh performance sensing, and achieving thorough understanding of many-particle quantum systems. Molecular spins are excellent qubit candidates because they feature long coherence times, are widely tunable through chemical synthesis, and can be interfaced with other quantum platforms such as superconducting qubits. A present challenge for molecular spin qubits is their integration in quantum devices, which requires arranging them in thin films or monolayers on surfaces. However, clear proof of the survival of quantum properties of molecular qubits on surfaces has not been reported so far. Furthermore, little is known about the change in spin dynamics of molecular qubits going from the bulk to monolayers. Here, a versatile bottom-up method is reported to arrange molecular qubits as functional groups of self-assembled monolayers (SAMs) on surfaces, combining molecular self-organization and click chemistry. Coherence times of up to 13 µs demonstrate that qubit properties are maintained or even enhanced in the monolayer.

Influence of pressure on a dysprosocenium single-molecule magnet

The effects of external pressure on a high-performing dysprosocenium single-molecule magnet are investigated using a combination of X-ray diffraction, magnetometry and theoretical calculations. The effective energy barrier (U_eff) decreases from ca. 1300 cm^-1 at ambient pressure to ca. 1125 cm^-1 at 3 GPa. Our results indicate that compression < 1.2 GPa has a negligible effect on the Orbach process, but magnetic relaxation > 1 GPa increases via Raman relaxation and/or quantum tunnelling of magnetisation.

Computational Biomaterials: Computational Simulations for Biomedicine

With the flourishing development of material simulation methods (quantum chemistry methods, molecular dynamics, Monte Carlo, phase field, etc.), extensive adoption of computing technologies (high-throughput, artificial intelligence, machine learning, etc.), and the invention of high-performance computing equipment, computational simulation tools have sparked the fundamental mechanism-level explorations to predict the diverse physicochemical properties and biological effects of biomaterials and investigate their enormous application potential for disease prevention, diagnostics, and therapeutics. Herein, the term "computational biomaterials" is proposed and the computational methods currently used to explore the inherent properties of biomaterials, such as optical, magnetic, electronic, and acoustic properties, and the elucidation of corresponding biological behaviors/effects in the biomedical field are summarized/discussed. The theoretical calculation of the physiochemical properties/biological performance of biomaterials applied in disease diagnosis, drug delivery, disease therapeutics, and specific paradigms such as biomimetic biomaterials is discussed. Additionally, the biosafety evaluation applications of theoretical simulations of biomaterials are presented. Finally, the challenges and future prospects of such computational simulations for biomaterials development are clarified. It is anticipated that these simulations would offer various methodologies for facilitating the development and future clinical translations/utilization of versatile biomaterials.

Influence of ligand substitution and the solvent effect on the structures and magnetic properties of dinuclear Dy2 supramolecular architectures constructed with the bis-β-diketonate-Dy2 building block as a metalloligand

Based on the bis-β-diketonate-Dy₂ metalloligand [Dy₂(pbth)₄]·2Et₃N (1, pbth = (3z,3'z)-4,4'-(1,3-phenylene)bis(1,1,1-trifluoro-4-hydroxybut-3-en-2-one)), six dinuclear complexes with eight-coordinated geometries were synthesized solvothermally through different capping N-donor coligands or solvent systems. These complexes are namely [Dy₂(pbth)₃(Phen)₂]·2C₂H₅OH (2), [Dy₂(pbth)₃(BPhen)₂]·2C₂H₅OH (3), [Dy₂(pbth)₃(Dppz)₂]·2C₂H₅OH (4), [Dy₂(pbth)₃(Dppz)₂]·2CH₃OH (4a), [Dy₂(pbth)₃(4-Dmbp)₂]·CH₃OH·C₂H₅OH (5) and [Dy₂(pbth)₃(5-Dmbp)₂]·CH₃OH (6) (Phen = 1,10-phenanthroline, BPhen = 4,7-diphenyl-1,10-phenanthroline, dppz = dipyrido [3,2-a:2',3'-c] phenazine, 4-Dmbp = 4,4'-dimethyl-2,2'-bipyridyl, 5-Dmbp = 5,5'-dimethyl-2,2'-bipyridyl), respectively. In the synthetic processes of 2-6, one of four bis-β-diketonate ligands in the metalloligand is replaced by two capping N-donor coligands. The coordination geometries, metal distances and M-L-M torsion angles of the synthesized complexes are perceptibly fine-tuned by the modification of the capping N-donor coligands or the latticed solvent molecules. Systematic magnetic investigations indicate the different magnetic relaxation dynamics of 1-6. Complex 1 displays no characteristics of single-molecule magnets (SMMs), while complexes 2-6 exhibit SMM behaviours in the absence of a static magnetic field. Complexes 2 and 3 possess effective energy barriers (U_eff) of 110.18 (2) K and 133.21 (4) K, respectively. Theoretical analysis based on ab initio calculation provides some interpretations of experimental observation.

Chiral All-Nitrogen-Coordinated Dysprosium Single-Molecule Magnets

Two pairs of chiral end-on azido-bridged dinuclear hexaazamacrocycles, [Dy₂ (L^N6 _R/S )₂ (N₃ )₂ Cl₂ ](BPh₄ )₂ (1R/1S) and [Dy₂ (L^N6 _R/S )₂ (N₃ )₄ ]Cl₂ (2R/2S) (L^N6 _R/S is hexaazamacrocyclic neutral Schiff base ligand derived from 2,6-diformylpyridine and (1R, 2R)/(1S, 2S)-diaminocyclohexane), were constructed by adjusting the molar ratio of sodium azide to Dy(III) macrocycle precursor. Structural analyses reveal that all Dy(III) centers in complexes 1R/1S and 2R/2S are nine-coordinate with hula-loop coordination geometry, and the differences between 1R/1S and 2R/2S are the terminal coordination anion and counter anion. Magnetic studies indicate that complex 2S displays typical SMM behaviors under a zero dc field, whereas 1S just shows slow relaxation of magnetization resulting from a relatively weak axial crystal field. Significantly, complex 2R/2S represents the first homochiral all-nitrogen-coordinated lanthanide single-molecule magnet.

Back to the future of organolanthanide chemistry

At the dawn of the development of structural organometallic chemistry, soon after the discovery of ferrocene, the description of the LnCp₃ complexes, featuring large and mostly trivalent lanthanide ions, was rather original and sparked curiosity. Yet, the interest in these new architectures rapidly dwindled due to the electrostatic nature of the bonding between π-aromatic ligands and 4f-elements. Almost 70 years later, it is interesting to focus on how the discipline has evolved in various directions with the reports of multiple catalytic reactivities, remarkable potential in small molecule activation, and the development of rich redox chemistry. Aside from chemical reactivity, a better understanding of their singular electronic nature - not precisely as simplistic as anticipated - has been crucial for developing tailored compounds with adapted magnetic anisotropy or high fluorescence properties that have witnessed significant popularity in recent years. Future developments shall greatly benefit from the detailed reactivity, structural and physical chemistry studies, particularly in photochemistry, electro- or photoelectrocatalysis of inert small molecules, and manipulating the spins' coherence in quantum technology.

Theoretical Investigation of Single-Molecule-Magnet Behavior in Mononuclear Dysprosium and Californium Complexes

Early-actinide-based (U, Np, and Pu) single-molecule magnets (SMMs) have yet to show magnetic properties similar to those of highly anisotropic lanthanide-based ones. However, there are not many studies exploring the late-actinides (more than half-filled f shells) as potential candidates for SMM applications. We computationally explored the electronic structure and magnetic properties of a hypothetical Cf(III) complex isostructural to the experimentally synthesized Dy(dbm)₃(bpy) complex (bpy = 2,2'-bipyridine; dbm = dibenzoylmethanoate) via multireference methods and compared them to those of the Dy(III) analogue. This study shows that the Cf(III) complex can behave as a SMM and has a greater magnetic susceptibility compared to other experimentally and computationally studied early-actinide-based (U, Np, and Pu) magnetic complexes. However, Cf spontaneously undergoes α-decay and converts to Cm. Thus, we also explored the isostructural Cm(III)-based complex. The computed magnetic susceptibility and g-tensor values show that the Cm(III) complex has poor SMM behavior in comparison to both the Dy(III) and Cf(III) complexes, suggesting that the performance of Cf(III)-based magnets may be affected by α-decay and can explain the poor performance of experimentally studied Cf(III)-based molecular magnets in the literature. Further, this study suggests that the ligand field is dominant in Cf(III), which helps to increase the magnetization blocking barrier by nearly 3 times that of its 4f congener.

Strong Axiality in a Dysprosium(III) Bis(borolide) Complex Leads to Magnetic Blocking at 65 K

Substituted dysprosocenium complexes of the type [Dy(Cp^R)₂]⁺ exhibit slow magnetic relaxation at cryogenic temperatures and have emerged as top-performing single-molecule magnets. The remarkable properties of these compounds derive in part from the strong axial ligand field afforded by the cyclopentadiene anions, and the design of analogous compounds with even stronger ligand fields is one promising route toward identifying new single-molecule magnets that retain a magnetic memory at even higher temperatures. Here, we report the synthesis and characterization of a dysprosium bis(borolide) compound, [K(18-crown-6)][Dy(BC₄Ph₅)₂] (1), featuring the dysprosocenate anion [Dy(BC₄Ph₅)₂]^- with a pseudoaxial coordination environment afforded by two dianionic pentaphenyl borolide ligands. Variable-field magnetization data reveal open magnetic hysteresis up to 66 K, establishing 1 as a top-performing single-molecule magnet among its dysprosocenium analogues. Ac magnetic susceptibility data indicate that 1 relaxes via an Orbach mechanism above ∼80 K with U_eff = 1500(100) cm^-1 and τ₀ = 10^-12.0(9) s, whereas Raman relaxation and quantum tunneling of the magnetization dominate at lower temperatures. Compound 1 exhibits a 100 s blocking temperature of 65 K, among the highest reported for dysprosium-based single-molecule magnets. Ab initio spin dynamics calculations support the experimental U_eff and τ₀ values and enable a quantitative comparison of the relaxation dynamics of 1 and two representative dysprosocenium cations, yielding additional insights into the impact of the crystal field splitting and vibronic coupling on the observed relaxation behavior. Importantly, compound 1 represents a step toward the development of alternatives to substituted dysprosocenium single-molecule magnets with increased axiality.

Experimental assignment of long-range magnetic communication through Pd & Pt metallophilic contacts

Record-breaking magnetic exchange interactions have previously been reported for 3d-metal dimers of the form [M(Pt(SAc)₄)(pyNO₂)]₂ (M = Ni or Co) that are linked in the solid state via metallophilic Pt⋯Pt bridges. This contrasts the terminally capped monomers [M(Pt(SAc)₄)(py)₂], for which neither metallophilic bridges nor magnetic exchange interactions are found. Computational modeling has shown that the magnetic exchange interaction is facilitated by the pseudo-closed shell d⁸⋯d⁸ metallophilic interaction between the filled Pt²⁺ 5d _{z ²} orbitals. We present here inelastic neutron scattering experiments on these complexes, wherein the dimers present an oscillatory momentum-transfer-dependence of the magnetic transitions. This allows for the unequivocal experimental assignment of the distance between the coupled ions, which matches exactly the coupling pathway via the metallophilic bridges. Furthermore, we have synthesized and magnetically characterized the isostructural palladium-analogues. The magnetic coupling across the Pd⋯Pd bridge is found through SQUID-magnetometry and FD-FT THz-EPR spectroscopy to be much weaker than via the Pt⋯Pt bridge. The weaker coupling is traced to the larger radial extent of the 5d _{z ²} orbitals compared to that of the 4d _{z ²} orbitals. The existence of a palladium metallophilic interaction is evaluated computationally from potential surface cuts along the metal stretching direction. Similar behavior is found for the Pd⋯Pd and Pt⋯Pt-systems with clear minima along this coordinate and provide estimates for the force constant for this distortion. The estimated M⋯M stretching frequencies are found to match experimental observed, polarized bands in single-crystal Raman spectra close to 45 cm^-1. This substantiates the existence of energetically relevant Pd⋯Pd metallophilic interactions. The unique properties of both Pt²⁺ and Pd²⁺ constitutes an orthogonal reactivity, which can be utilized for steering both the direction and strength of magnetic interactions.

Unraveling the Contributions to Spin-Lattice Relaxation in Kramers Single-Molecule Magnets

The study of how spin interacts with lattice vibrations and relaxes to equilibrium provides unique insights into its chemical environment and the relation between electronic structure and molecular composition. Despite its importance for several disciplines, ranging from magnetic resonance to quantum technologies, a convincing interpretation of spin dynamics in crystals of magnetic molecules is still lacking due to the challenging experimental determination of the correct spin relaxation mechanism. We apply ab initio spin dynamics to a series of 12 coordination complexes of Co²⁺ and Dy³⁺ ions selected among ∼240 compounds that largely cover the literature on single-molecule magnets and well represent different regimes of spin relaxation. Simulations reveal that the Orbach spin relaxation rate of known compounds mostly depends on the ions' zero-field splitting and little on the details of molecular vibrations. Raman relaxation is instead found to be also significantly affected by the features of low-energy phonons. These results provide a complete understanding of the factors limiting spin lifetime in single-molecule magnets and revisit years of experimental investigations by making it possible to transparently distinguish Orbach and Raman relaxation mechanisms.

Axial Ligand as a Critical Factor for High-Performance Pentagonal Bipyramidal Dy(III) Single-Ion Magnets

The choice of axial ligands is of great importance for the construction of high-performance Dy-based single-molecule magnets (SMMs). Here, combining axial ligands Ph₃SiO^- (anion of triphenylsilanol) and 2,6-dichloro-4-nitro-PhO^- (the anion of 2,6-dichloro-4-nitrophenol) with a neutral macrocyclic ligand 2,14-dimethyl-3,6,10,13,19-pentaazabicyclo[13.3.1]nonadeca-1(19),2,13,15,17-pentaene (L₂^N5) generates two new pentagonal bipyramidal Dy(III) complexes [Dy^III(L₂^N5) (X)₂](BPh₄) (X = Ph₃SiO^-, 1; 2,6-dichloro-4-nitro-PhO^-, 2) with strong axial ligand fields. Magnetic characterizations show that 1 possesses a large energy barrier above 1000 K and a magnetic hysteresis up to 9 K, whereas 2 only displays field-induced peaks of alternating-current susceptibilities without the hysteresis loop, even though 2 has a similar coordination geometry with 1. Detailed Ab initio calculations indicate an apparent difference in the axial negative charge between both complexes, which is caused by the diverse electron-donating properties of the axial ligands. The present work provides an efficient strategy to enhance the SMMs' properties, which highlights that the electron-donating property of the axial ligands is especially important for constructing the high-performance Dy-based SMMs.

Analytic Linear Vibronic Coupling Method for First-Principles Spin-Dynamics Calculations in Single-Molecule Magnets

Accurate modeling of vibronically driven magnetic relaxation from ab initio calculations is of paramount importance to the design of next-generation single-molecule magnets (SMMs). Previous theoretical studies have been relying on numerical differentiation to obtain spin-phonon couplings in the form of derivatives of spin Hamiltonian parameters. In this work, we introduce a novel approach to obtain these derivatives fully analytically by combining the linear vibronic coupling (LVC) approach with analytic complete active space self-consistent field derivatives and nonadiabatic couplings computed at the equilibrium geometry with a single electronic structure calculation. We apply our analytic approach to the computation of Orbach and Raman relaxation rates for a bis-cyclobutadienyl Dy(III) sandwich complex in the frozen-solution phase, where the solution environment is modeled by electrostatic multipole expansions, and benchmark our findings against results obtained using conventional numerical derivatives and a fully electronic description of the whole system. We demonstrate that our LVC approach exhibits high accuracy over a wide range of coupling strengths and enables significant computational savings due to its analytic, "single-shot" nature. Evidently, this offers great potential for advancing the simulation of a wide range of vibronic coupling phenomena in magnetism and spectroscopy, ultimately aiding the design of high-performance SMMs. Considering different environmental representations, we find that a point charge model shows the best agreement with the reference calculation, including the full electronic environment, but note that the main source of discrepancies observed in the magnetic relaxation rates originates from the approximate equilibrium electronic structure computed using the electrostatic environment models rather than from the couplings.

Computational design of magnetic molecules and their environment using quantum chemistry, machine learning and multiscale simulations

Having served as a playground for fundamental studies on the physics of d and f electrons for almost a century, magnetic molecules are now becoming increasingly important for technological applications, such as magnetic resonance, data storage, spintronics and quantum information. All of these applications require the preservation and control of spins in time, an ability hampered by the interaction with the environment, namely with other spins, conduction electrons, molecular vibrations and electromagnetic fields. Thus, the design of a novel magnetic molecule with tailored properties is a formidable task, which does not only concern its electronic structures but also calls for a deep understanding of the interaction among all the degrees of freedom at play. This Review describes how state-of-the-art ab initio computational methods, combined with data-driven approaches to materials modelling, can be integrated into a fully multiscale strategy capable of defining design rules for magnetic molecules.

Single-Molecule Magnetism in Linear Fe(I) Complexes with Aufbau and Non-Aufbau Ground States

With the ongoing efforts on synthesizing mononuclear single-ion magnets (SIMs) with promising applications in high-density data storage and spintronics devices, the linear or quasi-linear Fe(I) complexes emerge as the enticing candidates possessing large unquenched angular momentum. Herein, we have studied five experimentally synthesized linear Fe(I) complexes to uncover the origin of single-molecule magnetic behavior of these complexes. To begin with, we benchmarked the methodology on the experimentally and theoretically well-studied complex [Fe(C(SiMe₃)₃)₂]^-1 (1) (SiMe₃ = trimethylsilyl), which is characterized with a large spin-reversal barrier of 226 cm^-1. Subsequently, the spin-phonon coupling coefficients are calculated for the low-frequency vibrational modes to understand the relaxation mechanism of the complex. Furthermore, the two Fe(I) complexes, that is, [Fe(cyIDep)₂]⁺¹ (2) (cyIDep = 1,3-bis(2',6'-diethylphenyl)-4,5-(CH₂)₄-imidazole-2-ylidene) and [Fe(sIDep)₂]⁺¹ (3) (sIDep = 1,3-bis(2',6'-diethylphenyl)-imidazolin-2-ylidene), are studied that are experimentally reported with no SIM behavior under ac or dc magnetic fields; however, they exhibit large opposite axial zero field splitting (-62.4 and +34.0 cm^-1, respectively) from ab initio calculations. We have unwrapped the origin of this contrasting observation between experiment and theory by probing their magnetic relaxation pathways and the pattern of d orbital splitting. Additionally, the two experimentally synthesized Fe(I) complexes, that is, [(η⁶-C₆H₆)FeAr*-3,5-Pr₂ⁱ] (4) (Ar*-3,5-Pr₂ⁱ = C₆H-2,6-(C₆H₂-2,4,6-Pr₃ⁱ)₂-3,5-Pr₂ⁱ) and [(CAAC)₂Fe]⁺¹ (5) (CAAC = cyclic (alkyl) (amino)carbene), are investigated for SIM behavior, since there is no report on their magnetic anisotropy. To this end, complex 4 presents itself as the possible candidate for SIM.

Lanthanide single-molecule magnets with high anisotropy barrier: where to from here?

High-temperature lanthanide single-molecule magnets with large coercive fields are being pursued.

Synthesis, crystal structure and magnetic properties of mer-tricyanidoiron(III) precursor-based 1D heterobimetallic complexes

Three new cyanide-bridged heterometallic complexes {{[Cu(S,S-Chxn)2][Fe(bbp)(CN)3]}2·2 H2O} n (1), {{[Cu(R,R-Chxn)2][Fe(bbp)(CN)3]}2·2 H2O} n (2) and {{[Cu(Cycam)][Fe(bbp)(CN)3]}·CH3OH·2 H2O} n (3) (bbp = bis(2-benzimidazolyl)pyridine dianion, Chxn = 1,2-diaminocyclo hexane, cyclam = 1,4,8,11-tetraazacyclodecane) have been assembled from the rarely used mer-tricyanidoiron(III) building block [PPh4]2[Fe(bbp)(CN)3] and three copper(II) compounds. The complexes have been characterized by elemental analysis, IR spectroscopy and single crystal X-ray diffraction. For the chiral enantiomers 1 and 2, the circular dichroism (CD) spectrum was also investigated. X-ray structural analyses revealed that the structures of the cyanide-bridged Fe-Cu complexes 1 and 2 are characterized by two crystallographically independent but structurally very similar homochiral neutral chains, each consisting of the repeating units {[Cu(S,S-Chxn)2][Fe(bbp)(CN)3]} (1) or {[Cu(R,R-Chxn)2][Fe(bbp)(CN)3]} (2). The crystal structure of 3 likewise is build up of chains consisting of {[Cu(Cyclam)][Fe(bbp)(CN)3]} building blocks. The temperature-dependent magnetic susceptibility and field dependent magnetization of the complexes showed antiferromagnetic interactions in complex 1 between the Fe(III) and Cu(II) ions, while complex 3 is ferromagnetic, indicating that the magnetic coupling through cyanide linkage is very sensitive to the structure parameters around the paramagnetic metal ions. These results have been further confirmed by fitting of the experimental data using a uniform chain model, leading to the coupling constants J = −6.35 cm−1, g = 2.08, R = 4.42 × 10−4 and J = 1.24 cm−1, g = 2.09, R = ∑(χ obsd T − χ cald T)2/∑(χ obsd T)2 = 4.67 × 10−4 for complexes 1 and 3, respectively.

Toward exact predictions of spin-phonon relaxation times: An ab initio implementation of open quantum systems theory

Spin-phonon coupling is the main driver of spin relaxation and decoherence in solid-state semiconductors at finite temperature. Controlling this interaction is a central problem for many disciplines, ranging from magnetic resonance to quantum technologies. Spin relaxation theories have been developed for almost a century but often use a phenomenological description of phonons and their coupling to spin, resulting in a nonpredictive tool and hindering our detailed understanding of spin dynamics. Here, we combine time-local master equations up to the fourth order with advanced electronic structure methods and perform predictions of spin-phonon relaxation time for a series of solid-state coordination compounds based on both transition metals and lanthanide Kramers ions. The agreement between experiments and simulations demonstrates that an accurate, universal, and fully ab initio implementation of spin relaxation theory is possible, thus paving the way to a systematic study of spin-phonon relaxation in solid-state materials.

Strong Relativistic Effects in Lanthanide-Based Single-Molecule Magnets

Lanthanide-based single-molecule magnets (SMMs) are promising building blocks for quantum memory and spintronic devices. Designing lanthanide-based SMMs with long spin relaxation time requires a detailed understanding of their electronic structure, including the crucial role of the spin-orbit coupling (SOC). While traditional calculations of SOC using the perturbation theory applied to a solution of the nonrelativistic Schrödinger equation are valid for light atoms, this approach is questionable for systems containing heavy elements such as lanthanides. We investigate the accuracy of the perturbation estimates of SOC by variationally solving the Dirac equation for the [DyO]+ molecule, a prototype of a lanthanide-based SMM. We show that the energy splittings between the M J states involved in spin relaxation depend on the interplay between strong SOC and dynamic electron correlation. We demonstrate that this interplay affects the resonances between the spin and vibrational transitions and, therefore, the spin relaxation time.

Ligand Fluorination to Mitigate the Raman Relaxation of DyIII Single-Molecule Magnets: A Combined Terahertz, Far-IR and Vibronic Barrier Model Study

The fast Raman relaxation process via a virtual energy level has become a puzzle for how to chemically engineer single-molecule magnets (SMMs) with better performance. Here, we use the trifluoromethyl group to systematically substitute the methyl groups in the axial position of the parent bis-butoxide pentapyridyl dysprosium(III) SMM. The resulting complexes-[Dy(OL^A )₂ py₅ ][BPh₄ ] (L^A =CH(CF₃ )₂ ^- 1, CH₂ CF₃ ^- 2, CMe₂ CF₃ ^- 3)-show progressively enhanced T_B ^hys (@100 Oe s^-1 ) from 17 K (for 3), 20 K (for 2) to 23 K (for 1). By experimentally identifying the varied under barrier relaxation energy in the 5-500 cm^-1 regime, we are able to identify that the C-F bond related vibration energy of the axial ligand ranging from 200 to 350 cm^-1 is the key variant for this improvement. Thus, this finding not only reveals a correlation between the structure and the Raman process but also provides a paradigm for how to apply the vibronic barrier model to analyze multi-phonon relaxation processes in lanthanide SMMs.

Chemical control of spin-lattice relaxation to discover a room temperature molecular qubit

The second quantum revolution harnesses exquisite quantum control for a slate of diverse applications including sensing, communication, and computation. Of the many candidates for building quantum systems, molecules offer both tunability and specificity, but the principles to enable high temperature operation are not well established. Spin-lattice relaxation, represented by the time constant T ₁, is the primary factor dictating the high temperature performance of quantum bits (qubits), and serves as the upper limit on qubit coherence times (T ₂). For molecular qubits at elevated temperatures (>100 K), molecular vibrations facilitate rapid spin-lattice relaxation which limits T ₂ to well below operational minimums for certain quantum technologies. Here we identify the effects of controlling orbital angular momentum through metal coordination geometry and ligand rigidity via π-conjugation on T ₁ relaxation in three four-coordinate Cu²⁺ S = ½ qubit candidates: bis(N,N'-dimethyl-4-amino-3-penten-2-imine) copper(ii) (Me₂Nac)₂ (1), bis(acetylacetone)ethylenediamine copper(ii) Cu(acacen) (2), and tetramethyltetraazaannulene copper(ii) Cu(tmtaa) (3). We obtain significant T ₁ improvement upon changing from tetrahedral to square planar geometries through changes in orbital angular momentum. T ₁ is further improved with greater π-conjugation in the ligand framework. Our electronic structure calculations reveal that the reduced motion of low energy vibrations in the primary coordination sphere slows relaxation and increases T ₁. These principles enable us to report a new molecular qubit candidate with room temperature T ₂ = 0.43 μs, and establishes guidelines for designing novel qubit candidates operating above 100 K.

In silico design criteria for high blocking barrier uranium (III) SIMs

A combination of DFT and ab initio CASSCF/PT2 calculations on U(III) fictitious models and numerous reported X-ray structures unveils several geometries from coordination number 1 to 12 that can be targeted to design potential U(III) SIMs with attractive barrier heights. Among the geometries studied, the T-shaped and capped pentagonal antiprism geometries yield values exceeding 1500 cm^-1 - a value that is elusive for any uranium SIMs.