The Impact of Ligand Field Symmetry on Molecular Qubit Coherence

Spin-Lattice Relaxation and Spin-Phonon Coupling of ns1 Metal Ions at the Surface

To use transition metal ions for spin-based applications, it is essential to understand fundamental contributions to electron spin relaxation in different ligand environments. For example, to serve as building blocks for a device, transition metal ion-based molecular qubits must be organized on surfaces and preserve long electron spin relaxation times, up to room temperature. Here we propose monovalent group 12 ions (Zn+ and Cd+) as potential electronic metal qubits with an ns1 ground state. The relaxation properties of Zn+ and Cd+, stabilized at the interface of porous aluminosilicates, are investigated and benchmarked against vanadium (3d1) and copper (3d9) ions. The spin-phonon coupling has been evaluated through DFT modeling and found to be negligible for the ns1 states, explaining the long coherence time, up to 2 μs, at room temperature. These so far unexplored metal qubits may represent viable candidates for room temperature quantum operations and sensing.

Evaluation of Point Group Symmetry in Lanthanide(III) Complexes: A New Implementation of a Continuous Symmetry Operation Measure with Autonomous Assignment of the Principal Axis

The structure of molecular systems dictates the physical properties, and symmetry is the determining factor for all electronic properties. This makes group theory a powerful tool in quantum mechanics to compute molecular properties. For inorganic compounds, the coordination geometry has been estimated as idealized polyhedra with high symmetry, which, through ligand field theory, provides predictive capabilities. However, real samples rarely have ideal symmetry, and although continuous shape measures (CShM) can be used to evaluate deviation from an ideal reference structure σideal, this often fails for lanthanide(III) complexes with high coordination numbers, no obvious choice of principal axes, and no obvious reference structure. In lanthanide complexes, the unique electronic structures and associated properties are intricately tied to the symmetry around the lanthanide center. Therefore, robust methodologies to evaluate and estimate point group symmetry are instrumental for building structure-property relationships. Here, we have demonstrated an algorithmic approach that orients a molecular structure Q in the best possible way to the symmetry axis of any given point group G and computes a deviation from the ideal symmetry σsym(G,Q). This approach does not compute the deviation from an ideal reference system, but the intrinsic deviation in the structure induced by symmetry operations. If the structure contains the symmetry operation, there is no deviation and σsym(G,Q) = 0. The σsym deviation is generated from all of the symmetry operation ÔS in a point group G using the most correct orientation of the sample structure in each group G. The best orientation is found by an algorithm that minimizes the orientation of the structure with respect to G. To demonstrate the methodology, we have investigated the structure and symmetry of 8- and 9-coordinated lanthanide(III) aqua complexes and correlated the luminescence from 3 europium(III) crystals to their actual symmetry. To document the methodology, the approach has been tested on 26 molecules with different symmetries. It was concluded that the method is robust and fully autonomous.

Alternant Hydrocarbon Diradicals as Optically Addressable Molecular Qubits

High-spin molecules allow for bottom-up qubit design and are promising platforms for magnetic sensing and quantum information science. Optical addressability of molecular electron spins has also been proposed in first-row transition-metal complexes via optically detected magnetic resonance (ODMR) mechanisms analogous to the diamond-nitrogen-vacancy color center. However, significantly less progress has been made on the front of metal-free molecules, which can deliver lower costs and milder environmental impacts. At present, most luminescent open-shell organic molecules are π-diradicals, but such systems often suffer from poor ground-state open-shell characters necessary to realize a stable ground-state molecular qubit. In this work, we use alternancy symmetry to selectively minimize radical-radical interactions in the ground state, generating π-systems with high diradical characters. We call them m-dimers, referencing the need to covalently link two benzylic radicals at their meta carbon atoms for the desired symmetry. Through a detailed electronic structure analysis, we find that the excited states of alternant hydrocarbon m-diradicals contain important symmetries that can be used to construct ODMR mechanisms leading to ground-state spin polarization. The molecular parameters are set in the context of a tris(2,4,6-trichlorophenyl)methyl (TTM) radical dimer covalently tethered at the meta position, demonstrating the feasibility of alternant m-diradicals as molecular color centers.

Angular-resolved Rabi oscillations of orthorhombic spins in a Co(II) molecular qubit

Magnetic molecules are promising candidates for quantum information processing (QIP) due to their tunable electron structures and quantum properties. A high spin Co(II) complex, CoH2dota, is studied for its potential to be used as a quantum bit (qubit) utilizing continuous wave (CW) and pulsed electron paramagnetic resonance (EPR) spectroscopy at low temperature. On the X-band microwave energy scale, the system can be treated as an effective spin 1/2 with a strongly anisotropic g-tensor resulting from the significant spin-orbital coupling. An experimental and theoretical study is conducted to investigate the anisotropic Rabi oscillations of the two magnetically equivalent spin centres with different orientations in a single crystal sample, which aims to verify the relationship between the Rabi frequency and the orientation of the g-tensor. The findings of this study show that an effective quantum manipulation method is developed for orthorhombic spin systems.

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.

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.

Quantum spin coherence and electron spin distribution channels in vanadyl-containing lantern complexes

We herein investigate the heterobimetallic lantern complexes [PtVO(SOCR)4] as charge neutral electronic qubits based on vanadyl complexes (S = 1/2) with nuclear spin-free donor atoms. The derivatives with R = Me (1) and Ph (2) give highly resolved X-band EPR spectra in frozen CH2Cl2/toluene solution, which evidence the usual hyperfine coupling with the 51V nucleus (I = 7/2) and an additional superhyperfine interaction with the I = 1/2 nucleus of the 195Pt isotope (natural abundance ca. 34%). DFT calculations ascribe the spin density delocalization on the Pt2+ ion to a combination of π and δ pathways, with the former representing the predominant channel. Spin relaxation measurements in frozen CD2Cl2/toluene-d8 solution between 90 and 10 K yield Tm values (1-6 μs in 1 and 2-11 μs in 2) which compare favorably with those of known vanadyl-based qubits in similar matrices. Coherent spin manipulations indeed prove possible at 70 K, as shown by the observation of Rabi oscillations in nutation experiments. The results indicate that the heavy Group 10 metal ion is not detrimental to the coherence properties of the vanadyl moiety and that Pt-VO lanterns can be used as robust spin-coherent building blocks in materials science and quantum technologies.

Spin Qubit in a 2D GdIII NaI -Based Oxamato Supramolecular Coordination Framework

Qubits are the basic unit of quantum information and computation. To realize quantum computing and information processing, the decoherence times of qubits must be long enough. Among the studies of molecule-based electron spin qubits, most of the work focused on the ions with the spin S=1/2, where only single-bit gates can be constructed. However, quantum operations require the qubits to interact with each other, so people gradually carry out relevant research in ions or systems with S>1/2 and multilevel states. In this work, a two-dimensional (2D) oxygen-coordinated GdIII NaI -based oxamato supramolecular coordination framework, Na[Gd(4-HOpa)4 (H2 O)] ⋅ 2H2 O (1, 4-HOpa=N-4-hydroxyphenyloxamate), was selected as a possible carrier of qubit. The field-induced slow magnetic relaxation shows this system has phonon bottleneck (PB) effect at low temperatures with a very weak magnetic anisotropy. The pulse electron paramagnetic resonance studies show the spin-lattice and spin-spin relaxation times are T1 =1.66 ms at 4 K and Tm =4.25 μs at 8 K for its diamagnetically diluted sample (1Gd0.12 %). It suggested that the relatively long decoherence time is mainly ascribed to its near isotropic and the PB effect from resonance phonon trapped for pure sample, while the dilution further improves its qubit performance.

Improving the molecular spin qubit performance in zirconium MOF composites by mechanochemical dilution and fullerene encapsulation

Enlarging the quantum coherence times and gaining control over quantum effects in real systems are fundamental for developing quantum technologies. Molecular electron spin qubits are particularly promising candidates for realizing quantum information processing due to their modularity and tunability. Still, there is a constant search for tools to increase their quantum coherence times. Here we present how the mechanochemical introduction of active spin qubits in the form of 10% diluted copper(ii)-porphyrins in the diamagnetic PCN-223 and MOF-525 zirconium-MOF polymorph pair can be achieved. Furthermore, the encapsulation of fullerene during the MOF synthesis directs the process exclusively toward the rare PCN-223 framework with a controllable amount of fullerene in the framework channels. In addition to the templating role, the incorporation of fullerene increases the electron spin-lattice and phase-memory relaxation times, T1 and Tm. Besides decreasing the amount of nuclear spin-bearing solvent guests in the non-activated qubit frameworks, the observed improved relaxation times can be rationalized by modulating the phonon density of states upon fullerene encapsulation.

T1 Anisotropy Elucidates Spin Relaxation Mechanisms in an S = 1 Cr(IV) Optically Addressable Molecular Qubit

Paramagnetic molecules offer unique advantages for quantum information science owing to their spatial compactness, synthetic tunability, room-temperature quantum coherence, and potential for optical state initialization and readout. However, current optically addressable molecular qubits are hampered by rapid spin-lattice relaxation (T1) even at sub-liquid nitrogen temperatures. Here, we use temperature- and orientation-dependent pulsed electron paramagnetic resonance (EPR) to elucidate the negative sign of the ground state zero-field splitting (ZFS) and assign T1 anisotropy to specific types of motion in an optically addressable S = 1 Cr(o-tolyl)4 molecular qubit. The anisotropy displays a distinct sin2(2θ) functional form that is not observed in S = 1/2 Cu(acac)2 or other Cu(II)/V(IV) microwave addressable molecular qubits. The Cr(o-tolyl)4 T1 anisotropy is ascribed to couplings between electron spins and rotational motion in low-energy acoustic or pseudoacoustic phonons. Our findings suggest that rotational degrees of freedom should be suppressed to maximize the coherence temperature of optically addressable qubits.

Template-directed 2D nanopatterning of S = 1/2 molecular spins

Molecular spins are emerging platforms for quantum information processing. By chemically tuning their molecular structure, it is possible to prepare a robust environment for electron spins and drive the assembly of a large number of qubits in atomically precise spin-architectures. The main challenges in the integration of molecular qubits into solid-state devices are (i) minimizing the interaction with the supporting substrate to suppress quantum decoherence and (ii) controlling the spatial distribution of the spins at the nanometer scale to tailor the coupling among qubits. Herein, we provide a nanofabrication method for the realization of a 2D patterned array of individually addressable Vanadyl Phthalocyanine (VOPc) spin qubits. The molecular nanoarchitecture is crafted on top of a diamagnetic monolayer of Titanyl Phthalocyanine (TiOPc) that electronically decouples the electronic spin of VOPc from the underlying Ag(100) substrate. The isostructural TiOPc interlayer also serves as a template to regulate the spacing between VOPc spin qubits on a scale of a few nanometers, as demonstrated using scanning tunneling microscopy, X-ray circular dichroism, and density functional theory. The long-range molecular ordering is due to a combination of charge transfer from the metallic substrate and strain in the TiOPc interlayer, which is attained without altering the pristine VOPc spin characteristics. Our results pave a viable route towards the future integration of molecular spin qubits into solid-state devices.

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.

Observation of slow magnetic relaxation phenomena in spatially isolated π-radical ions

The use of molecular spins as quantum bits is fascinating because it offers a wide range of strategies through chemical modifications. In this regard, it is very interesting to search for organic radical ions that have small spin-orbit coupling values. On the other hand, the feature of the magnetic relaxation of π-organic radical ions is rarely exploited due to the difficulty of spin dilution, and π-stacking interaction. In this study, we focus on N,N',N''-tris(2,6-dimethylphenyl)benzenetriimide (BTI-xy), where three xylene moieties connected to the imide groups cover the π-plane of the BTI core. As a result, BTI-xy radical anions without π-stacking interaction were obtained. This led to the slow magnetization relaxation, which is reported for the first time in organic radicals. Furthermore, the relaxation times in a solution state revealed the importance of spin interaction.

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.

Multireference Study of Optically Addressable Vanadium-Based Molecular Qubit Candidates

Molecular electron spin qubits with optical manipulation schemes are some of the most promising candidates for modern quantum technologies. Key values that determine a compound's viability for optical-spin initialization and readout include its singlet-triplet gap and zero-field splitting (ZFS) parameters. Generally, these values are very small in magnitude and are thus difficult to reproduce with theoretical methods. Here, we study a previously identified optically addressable molecular qubit, (C₆F₅)₃trenVCN^tBu (tren = tris(2-aminoethyl)amine), using the complete active space self-consistent field (CASSCF) and post-CASSCF methods: complete active space second-order perturbation theory (CASPT2), multiconfiguration pair-density functional theory (MC-PDFT), and hybrid MC-PDFT (HMC-PDFT). Of those methods, we successfully reproduce the singlet-triplet gap and ZFS parameters with reasonable accuracy using 0.5 HMC-PDFT and CASPT2. Four additional V³⁺ complexes with differing ligands were also investigated. We found that the ligands have minimal effect on the spin properties of the molecule and propose them to be optically addressable qubit candidates. These potential qubits are further analyzed in terms of ab initio ligand field theory (AILFT) to understand the influence of the ligands on the singlet-triplet gap and ZFS parameters.

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.

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.

Analysis of vibronic coupling in a 4f molecular magnet with FIRMS

Vibronic coupling, the interaction between molecular vibrations and electronic states, is a fundamental effect that profoundly affects chemical processes. In the case of molecular magnetic materials, vibronic, or spin-phonon, coupling leads to magnetic relaxation, which equates to loss of magnetic memory and loss of phase coherence in molecular magnets and qubits, respectively. The study of vibronic coupling is challenging, and most experimental evidence is indirect. Here we employ far-infrared magnetospectroscopy to directly probe vibronic transitions in [Yb(trensal)] (where H₃trensal = 2,2,2-tris(salicylideneimino)trimethylamine). We find intense signals near electronic states, which we show arise due to an "envelope effect" in the vibronic coupling Hamiltonian, which we calculate fully ab initio to simulate the spectra. We subsequently show that vibronic coupling is strongest for vibrational modes that simultaneously distort the first coordination sphere and break the C₃ symmetry of the molecule. With this knowledge, vibrational modes could be identified and engineered to shift their energy towards or away from particular electronic states to alter their impact. Hence, these findings provide new insights towards developing general guidelines for the control of vibronic coupling in molecules.

Isotopomeric Elucidation of the Mechanism of Temperature Sensitivity in 59Co NMR Molecular Thermometers

Understanding the mechanisms governing temperature-dependent magnetic resonance properties is essential for enabling thermometry via magnetic resonance imaging. Herein we harness a new molecular design strategy for thermometry─that of effective mass engineering via deuteration in the first coordination shell─to reveal the mechanistic origin of ⁵⁹Co chemical shift thermometry. Exposure of [Co(en)₃]³⁺ (1; en = ethylenediamine) and [Co(diNOsar)]³⁺ (2; diNOsar = dinitrosarcophagine) to mixtures of H₂O and D₂O produces distributions of [Co(en)₃]³⁺-d_n (n = 0-12) and [Co(diNOsar)]³⁺-d_n (n = 0-6) isotopomers all resolvable by ⁵⁹Co NMR. Variable-temperature ⁵⁹Co NMR analyses reveal a temperature dependence of the ⁵⁹Co chemical shift, Δδ/ΔT, on deuteration of the N-donor atoms. For 1, deuteration amplifies Δδ/ΔT by 0.07 ppm/°C. Increasing degrees of deuteration yield an opposing influence on 2, diminishing Δδ/ΔT by -0.07 ppm/°C. Solution-phase Raman spectroscopy in the low-frequency 200-600 cm^-1 regime reveals a red shift of Raman-active Co-N₆ vibrational modes by deuteration. Analysis of the normal vibrational modes shows that Raman modes produce the largest variation in ⁵⁹Co δ. Finally, partition function analysis of the Raman-active modes shows that increased populations of Raman modes predict greater Δδ/ΔT, representing new experimental insight into the thermometry mechanism.