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

Stable organic radical qubits and their applications in quantum information science

The past century has witnessed the flourishing of organic radical chemistry. Stable organic radicals are highly valuable for quantum technologies thanks to their inherent room temperature quantum coherence, atomic-level designability, and fine tunability. In this comprehensive review, we highlight the potential of stable organic radicals as high-temperature qubits and explore their applications in quantum information science, which remain largely underexplored. Firstly, we summarize known spin dynamic properties of stable organic radicals and examine factors that influence their electron spin relaxation and decoherence times. This examination reveals their design principles and optimal operating conditions. We further discuss their integration in solid-state materials and surface structures, and present their state-of-the-art applications in quantum computing, quantum memory, and quantum sensing. Finally, we analyze the primary challenges associated with stable organic radical qubits and provide tentative insights to future research directions.

Synergy of Experiment and Broadened Exploration of Ab Initio Calculations for Understanding of Lanthanide-Pentacyanidocobaltate Molecular Nanomagnets and Their Optical Properties

We present a synergistic experimental-theoretical methodology for the investigation of lanthanide-based single-molecule magnets (SMMs), demonstrated using the example of novel heterometallic molecules incorporating Nd3+/Ce3+ ions combined with three different, rarely explored, pentacyanidocobaltate(III) metalloligands, [CoIII(CN)5(azido/nitrito-N/iodido)]3-. The theoretical part of our approach broadens the exploration of ab initio calculations for lanthanide(III) complexes toward the convenient simulations of such physical characteristics as directional dependences of Helmholtz energy, magnetization, susceptibility, and their thermal and field evolution, as well as light absorption and emission bands. This work was conducted using newly designed SlothPy software (https://slothpy.org). It is introduced as an open-source Python library for simulating various physical properties from first-principles based on results of electronic structure calculations obtained within popular quantum chemistry packages. The computational results were confronted with spectroscopic and ac/dc-magnetic data, the latter analyzed using previously designed relACs software. The combination of experimental and computational methods gave insight into phonon-assisted magnetic relaxation mechanisms, disentangling them from the temperature-independent quantum tunneling of magnetization and emphasizing the role of local-mode processes. This study provides an understanding of the changes in lanthanide(III) magnetic anisotropy introduced with pentacyanidocobaltates(III) modifications, theoretically exploring also potential applications of reported compounds as anisotropy switches or optical thermometers.

Magnetic behaviour of a spin-canted asymmetric lanthanide quinolate trimer

An asymmetrical dysprosium trimer with a molecular formula of [Dy3(hq)7(hqH)(NO3)2(H2O)] was obtained through a reflux reaction employing as starting material Dy(NO3)3·nH2O and 8-quinolinoline as ligand. Magnetic susceptibility investigations show the system to be an SMM, which was corroborated by sub-Kelvin μSQUID studies. Upon cooling, the magnetic susceptibility also exhibits a decrease in the χMT product, which was confirmed to be due to intramolecular antiferromagnetic interactions. μSQUID measurements, moreover, reveal a marked magnetic behaviour in the angular dependence of the hysteresis loops. The latter is a direct consequence of the non-colinear spin arrangement of the anisotropy axes of each Dy(III) ion in [Dy3(hq)7(hqH)(NO3)2(H2O)] and the interaction between the ions, as also evidenced by CASSCF calculations. Our results evidence the effect of spin canting along with the intramolecular interactions, which can induce non-trivial magnetic behaviour in SMMs.

Spin-lattice relaxation with non-linear couplings: Comparison between Fermi's golden rule and extended dissipaton equation of motion

Fermi's golden rule (FGR) offers an empirical framework for understanding the dynamics of spin-lattice relaxation in magnetic molecules, encompassing mechanisms like direct (one-phonon) and Raman (two-phonon) processes. These principles effectively model experimental longitudinal relaxation rates, denoted as T1-1. However, under scenarios of increased coupling strength and nonlinear spin-lattice interactions, FGR's applicability may diminish. This paper numerically evaluates the exact spin-lattice relaxation rate kernels, employing the extended dissipaton equation of motion formalism. Our calculations reveal that when quadratic spin-lattice coupling is considered, the rate kernels exhibit a free induction decay-like feature, and the damping rates depend on the interaction strength. We observe that the temperature dependence predicted by FGR significantly deviates from the exact results since FGR ignores the higher order effects and the non-Markovian nature of spin-lattice relaxation. Our methods can be easily extended to study other systems with nonlinear spin-lattice interactions and provide valuable insights into the temperature dependence of T1 in molecular qubits when the coupling is strong.

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.

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.

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.

Spectroscopic techniques to probe magnetic anisotropy and spin-phonon coupling in metal complexes

Magnetism of molecular quantum materials such as single-molecule magnets (SMMs) has been actively studied for potential applications in the new generation of high-density data storage using SMMs and quantum information science. Magnetic anisotropy and spin-phonon coupling are two key properties of d- and f-metal complexes. Here, phonons refer to both intermolecular and intramolecular vibrations. Direct determination of magnetic anisotropy and experimental studies of spin-phonon coupling are critical to the understanding of molecular magnetism. This article discusses our recent approach in using three complementary techniques, far-IR and Raman magneto-spectroscopies (FIRMS and RaMS, respectively) and inelastic neutron scatterings (INS), to determine magnetic excited states. Spin-phonon couplings are observed in FIRMS and RaMS. DFT phonon calculations give energies and symmetries of phonons as well as calculated INS spectra which help identify magnetic peaks in experimental INS spectra.

Determining the key vibrations for spin relaxation in ruffled Cu(ii) porphyrins via resonance Raman spectroscopy

Pinpointing vibrational mode contributions to electron spin relaxation (T1) constitutes a key goal for developing molecular quantum bits (qubits) with long room-temperature coherence times. However, there remains no consensus to date as to the energy and symmetry of the relevant modes that drive relaxation. Here, we analyze a series of three geometrically-tunable S = ½ Cu(ii) porphyrins with varying degrees of ruffling distortion in the ground state. Theoretical calculations predict that increased distortion should activate low-energy ruffling modes (∼50 cm-1) for spin-phonon coupling, thereby causing faster spin relaxation in distorted porphyrins. However, experimental T1 times do not follow the degree of ruffling, with the highly distorted copper tetraisopropylporphyrin (CuTiPP) even displaying room-temperature coherence. Local mode fitting indicates that the true vibrations dominating T1 lie in the energy regime of bond stretches (∼200-300 cm-1), which are comparatively insensitive to the degree of ruffling. We employ resonance Raman (rR) spectroscopy to determine vibrational modes possessing both the correct energy and symmetry to drive spin-phonon coupling. The rR spectra uncover a set of mixed symmetric stretch vibrations from 200-250 cm-1 that explain the trends in temperature-dependent T1. These results indicate that molecular spin-phonon coupling models systematically overestimate the contribution of ultra-low-energy distortion modes to T1, pointing out a key deficiency of existing theory. Furthermore, this work highlights the untapped power of rR spectroscopy as a tool for building spin dynamics structure-property relationships in molecular quantum information science.

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.

Increasing the Magnetic Blocking Temperature of Single-Molecule Magnets

The synthesis of single-molecule magnets (SMMs), magnetic complexes capable of retaining magnetization blocking for a long time at elevated temperatures, has been a major concern for magnetochemists over the last three decades. In this review, we describe basic SMMs and the different approaches that allow high magnetization-blocking temperatures to be reached. We focus on the basic factors affecting magnetization blocking, magnetic axiality and the height of the blocking barrier, which can be used to group different families of complexes in terms of their SMM efficiency. Finally, we discuss several practical routes for the design of mono- and polynuclear complexes that could be applied in memory devices.

Phonon-assisted nearly pure spin current in DNA molecular chains: a multifractal analysis

Motivated by the development of molecular spintronics, we studied the phonon-assisted spin transport along a DNA chain in the presence of environmental-induced dephasing using multifractal analysis. The results demonstrate that a nearly pure spin current is generated in the presence of the voltage gate. The pure spin current is enhanced by increasing thermal effects. The vibration modes due to the thermal phonon bath assist in generating the spin current, so the spin state is more delocalized in strong electron-phonon coupling. The phonon chirality can translate to the electron spin to create a nontrivial spin texture, including spin currents. The spin states become more extended by increasing the phonon temperature. On the other hand, the spin states are less localized in longer chains as the spin selectivity is higher in longer chains than in short ones. Therefore, we can engineer a molecular spintronic device by controlling phonon effects on the storage and transport of binary digits.

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.

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.

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.

Temperature-Dependent Spin-Lattice Relaxation of the Nitrogen-Vacancy Spin Triplet in Diamond

Spin-lattice relaxation within the nitrogen-vacancy (NV) center's electronic ground-state spin triplet limits its coherence times, and thereby impacts its performance in quantum applications. We report measurements of the relaxation rates on the NV center's |m_{s}=0⟩↔|m_{s}=±1⟩ and |m_{s}=-1⟩↔|m_{s}=+1⟩ transitions as a function of temperature from 9 to 474 K in high-purity samples. We show that the temperature dependencies of the rates are reproduced by an ab initio theory of Raman scattering due to second-order spin-phonon interactions, and we discuss the applicability of the theory to other spin systems. Using a novel analytical model based on these results, we suggest that the high-temperature behavior of NV spin-lattice relaxation is dominated by interactions with two groups of quasilocalized phonons centered at 68.2(17) and 167(12) meV.

Direct determination of high-order transverse ligand field parameters via µSQUID-EPR in a Et4N[160GdPc2] SMM

The development of quantum technologies requires a thorough understanding of systems possessing quantum effects that can ultimately be manipulated. In the field of molecular magnetism, one of the main challenges is to measure high-order ligand field parameters, which play an essential role in the relaxation properties of SMMs. The development of highly advanced theoretical calculations has allowed the ab-initio determination of such parameters; however, currently, there is a lack of quantitative assessment of how good the ab-initio parameters are. In our quest for technologies that can allow the extraction of such elusive parameters, we develop an experimental technique that combines the EPR spectroscopy and µSQUID magnetometry. We demonstrate the power of the technique by performing EPR-µSQUID measurement of a magnetically diluted single crystal of Et₄N[GdPc₂], by sweeping the magnetic field and applying a range of multifrequency microwave pulses. As a result, we were able to directly determine the high-order ligand field parameters of the system, enabling us to test theoretical predictions made by state-of-the-art ab-initio methods.

Thermalization of Nuclear Spins in Lanthanide Molecular Magnets

Single-molecule magnets (SMMs) distinguish themselves in the field of quantum magnetism through the ability to combine fundamental research with promising applications. The evolution of quantum spintronics in the last decade exemplifies the potential held by molecular-based quantum devices. Notably, the readout and manipulation of the nuclear spin states embedded in a lanthanide-based SMM hybrid device were employed in proof of principle studies of quantum computation at the single-molecule level. In the quest for further understanding of the relaxation behavior in SMMs for their integration in novel applications, herein, we study the relaxation dynamics of the ¹⁵⁹Tb nuclear spins in a diluted molecular crystal employing the recently acquired understanding of the nonadiabatic dynamics of TbPc₂ molecules. Through numerical simulation, we find that phonon-modulated hyperfine interaction opens a direct relaxation channel between the nuclear spins and the phonon bath. The mechanism is of potential importance for the theory of spin bath and the relaxation dynamics of the molecular spins.

Modeling spin relaxation in complex radical systems using MolSpin

Spin relaxation is an important aspect of the spin dynamics of free radicals and can have a significant impact on the outcome of their spin-selective reactions. Examples range from the use of radicals as spin qubits in quantum information processing to the radical pair reactions in proteins that may allow migratory birds to sense the direction of the Earth's magnetic field. Accurate modeling of spin relaxation, however, is non-trivial. Bloch-Redfield-Wangsness theory derives a quantum mechanical master equation from system-bath interactions in the Markovian limit that provides a comprehensive framework for describing spin relaxation. Unfortunately, the construction of the master equation is system-specific and often resource-heavy. To address this challenge, we introduce a generalized and efficient implementation of BRW theory as a new feature of the spin dynamics toolkit MolSpin which offers an easy-to-use approach for studying systems of reacting radicals of varying complexity.

The critical role of ultra-low-energy vibrations in the relaxation dynamics of molecular qubits

Improving the performance of molecular qubits is a fundamental milestone towards unleashing the power of molecular magnetism in the second quantum revolution. Taming spin relaxation and decoherence due to vibrations is crucial to reach this milestone, but this is hindered by our lack of understanding on the nature of vibrations and their coupling to spins. Here we propose a synergistic approach to study a prototypical molecular qubit. It combines inelastic X-ray scattering to measure phonon dispersions along the main symmetry directions of the crystal and spin dynamics simulations based on DFT. We show that the canonical Debye picture of lattice dynamics breaks down and that intra-molecular vibrations with very-low energies of 1-2 meV are largely responsible for spin relaxation up to ambient temperature. We identify the origin of these modes, thus providing a rationale for improving spin coherence. The power and flexibility of our approach open new avenues for the investigation of magnetic molecules with the potential of removing roadblocks toward their use in quantum devices.

Magnetic Anisotropy and Relaxation of Pseudotetrahedral [N2 O2 ] Bis-Chelate Cobalt(II) Single-Ion Magnets Controlled by Dihedral Twist Through Solvomorphism

The methanol solvomorph 1 ⋅ 2MeOH of the cobalt(II) complex [Co(L^Sal,2-Ph )₂ ] (1) with the sterically demanding Schiff-base ligand 2-(([1,1'-biphenyl]-2-ylimino)methyl)phenol (HL^Sal,2-Ph ) shows the thus far largest dihedral twist distortion between the two chelate planes compared to an ideal pseudotetrahedral arrangement. The cobalt(II) ion in 1 ⋅ 2MeOH exhibits an easy-axis anisotropy leading to a spin-reversal barrier of 55.3 cm^-1 , which corresponds to an increase of about 17 % induced by the larger dihedral twist compared to the solvent-free complex 1. The magnetic relaxation for 1 ⋅ 2MeOH is significantly slower compared to 1. An in-depth frequency-domain Fourier-transform (FD-FT) THz-EPR study not only allowed the direct measurement of the magnetic transition between the two lowest Kramers doublets for the cobalt(II) complexes, but also revealed the presence of spin-phonon coupling. Interestingly, a similar dihedral twist correlation is also observed for a second pair of cobalt(II)-based solvomorphs, which could be benchmarked by FD-FT THz-EPR.

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.

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.

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 Insights into Electronic Excitations, Spin-Orbit Coupling Effects, and Spin Decoherence in Cr(IV)-Based Molecular Qubits

The great success of point defects and dopants in semiconductors for quantum information processing has invigorated a search for molecules with analogous properties. Flexibility and tunability of desired properties in a large chemical space have great advantages over solid-state systems. The properties analogous to point defects were demonstrated in the Cr(IV)-based molecular family, Cr(IV)(aryl)₄, where the electronic spin states were optically initialized, read out, and controlled. Despite this kick-start, there is still a large room for enhancing properties crucial for molecular qubits. Here, we provide computational insights into key properties of the Cr(IV)-based molecules aimed at assisting the chemical design of efficient molecular qubits. Using the multireference ab initio methods, we investigate the electronic states of Cr(IV)(aryl)₄ molecules with slightly different ligands, showing that the zero-phonon line energies agree with the experiment and that the excited spin-triplet and spin-singlet states are highly sensitive to small chemical perturbations. By adding spin-orbit interaction, we find that the sign of the uniaxial zero-field splitting (ZFS) parameter is negative for all considered molecules and discuss optically induced spin initialization via non-radiative intersystem crossing. We quantify (super)hyperfine coupling to the ⁵³Cr nuclear spin and to the ¹³C and ¹H nuclear spins, and we discuss electron spin decoherence. We show that the splitting or broadening of the electronic spin sub-levels due to superhyperfine interaction with ¹H nuclear spins decreases by an order of magnitude when the molecules have a substantial transverse ZFS parameter.

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.