Relativistic Douglas-Kroll-Hess calculations of hyperfine interactions within first-principles multireference methods.

A relativistic magnetic hyperfine interaction Hamiltonian based on the Douglas-Kroll-Hess (DKH) theory up to the second order is implemented within the ab initio multireference methods, including spin-orbit coupling in the Molcas/OpenMolcas package. This implementation is applied to calculate relativistic hyperfine coupling (HFC) parameters for atomic systems and diatomic radicals with valence s or d orbitals by systematically varying active space size in the restricted active space self-consistent field formalism with restricted active space state interaction for spin-orbit coupling. The DKH relativistic treatment of the hyperfine interaction reduces the Fermi contact contribution to the HFC due to the presence of kinetic factors that regularize the singularity of the Dirac delta function in the nonrelativistic Fermi contact operator. This effect is more prominent for heavier nuclei. As the active space size increases, the relativistic correction of the Fermi contact contribution converges well to the experimental data for light and moderately heavy nuclei. The relativistic correction, however, does not significantly affect the spin-dipole contribution to the hyperfine interaction. In addition to the atomic and molecular systems, the implementation is applied to calculate the relativistic HFC parameters for large trivalent and divalent Tb-based single-molecule magnets (SMMs), such as Tb(III)Pc2 and Tb(II)(CpiPr5)2 without ligand truncation using well-converged basis sets. In particular, for the divalent SMM, which has an unpaired valence 6s/5d hybrid orbital, the relativistic treatment of HFC is crucial for a proper description of the Fermi contact contribution. Even with the relativistic hyperfine Hamiltonian, the divalent SMM is shown to exhibit strong tunability of HFC via an external electric field (i.e., strong hyperfine Stark effect).

Tb2O@C2(13333)-C74: A Non-Isolated Pentagon Endohedral Fullerene Containing a Nearly Linear Tb-O-Tb Unit.

Terbium has been added to the list of elements that form oxide clusters inside fullerene cages. Tb2O@C2(13333)-C74 has been isolated as a byproduct of the electric arc synthesis of the azafullerene Tb2@C79N. Cocrystallization of Tb2O@C2(13333)-C74 with Ni(OEP) (where OEP is the dianion of octaethylporphyrin) in toluene yielded black needles of Tb2O@C2(13333)-C74·NiII(OEP)·1.5C7H8 that have been examined by single-crystal X-ray diffraction. The resulting structure shows that a nearly linear Tb-O-Tb unit is contained in a C2(13333)-C74, which has two sites where pentagons share an edge to form pentalene units at opposite ends of the fullerene. Unlike the usual situations where metal atoms in fullerenes that do not obey the isolated pentagon rule are situated within the folds of the pentalene units, the Tb atoms in Tb2O@C2(13333)-C74 are positioned to the side of the pentalene units and near-neighboring hexagons. The magnetic properties of Tb2O@C2(13333)-C74 have been examined starting from the experimental geometry, using ab-initio multiconfigurational methods. The computations predict that Tb2O@C2(13333)-C74 will show strong axiality, which would make it a single-molecule magnet with a large magnetic anisotropy barrier.

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.

Electrically tuned hyperfine spectrum in neutral Tb(II)(CpiPr5)2 single-molecule magnet.

Molecular spin qubits with long spin coherence time as well as non-invasive operation methods on such qubits are in high demand. It was shown that both molecular electronic and nuclear spin levels can be used as qubits. In solid state systems with dopants, an electric field was shown to effectively change the spacing between the nuclear spin qubit levels when the electron spin density is high at the nucleus of the dopant. Inspired by such solid-state systems, we propose that divalent lanthanide (Ln) complexes with an unusual electronic configuration of Ln2+ have a strong interaction between the Ln nuclear spin and the electronic degrees of freedom, which renders electrical tuning of the interaction. As an example, we study electronic structure and hyperfine interaction of the 159Tb nucleus in a neutral Tb(ii)(CpiPr5)2 single-molecule magnet (SMM), which exhibits unusually long magnetization relaxation time, using the complete active space self-consistent field (CASSCF) method with spin-orbit interaction included within the restricted active space state interaction (RASSI). Our calculations show that the low-energy states arise from 4f8(6s,5dz2)1, 4f8(5dx2-y2)1, and 4f8(5dxy)1 configurations. We compute the hyperfine interaction parameters and the electronic-nuclear spectrum within our multiconfigurational approach. We find that the hyperfine interaction is about one order of magnitude greater than that for Tb(iii)Pc2 SMMs. This stems from the strong Fermi contact interaction between the Tb nuclear spin and the electron spin density at the nucleus that originates from the occupation of the (6s,5d) orbitals. We also uncover that the response of the Fermi contact term to electric field results in electrical tuning of the electronic-nuclear level separations. This hyperfine Stark effect may be useful for applications of molecular nuclear spins for quantum computing.

Hyperfine and quadrupole interactions for Dy isotopes in DyPc2 molecules.

Nuclear spin levels play an important role in understanding magnetization dynamics and implementation and control of quantum bits in lanthanide-based single-molecule magnets. We investigate the hyperfine and nuclear quadrupole interactions for 161Dy and 163Dy nuclei in anionic DyPc2 (Pc = phthalocyanine) single-molecule magnets, using multiconfigurational ab initio methods (beyond density-functional theory) including spin-orbit interaction. The two isotopes of Dy are chosen because the others have zero nuclear spin. Both isotopes have the nuclear spin I = 5/2, although the magnitude and sign of the nuclear magnetic moment differ from each other. The large energy gap between the electronic ground and first-excited Kramers doublets, allows us to map the microscopic hyperfine and quadrupole interaction Hamiltonian onto an effective Hamiltonian with an electronic pseudo-spin [Formula: see text] that corresponds to the ground Kramers doublet. Our ab initio calculations show that the coupling between the nuclear spin and electronic orbital angular momentum contributes the most to the hyperfine interaction and that both the hyperfine and nuclear quadrupole interactions for 161Dy and 163Dy nuclei are much smaller than those for the 159Tb nucleus in TbPc2 single-molecule magnets. The calculated separations of the electronic-nuclear levels are comparable to experimental data reported for 163DyPc2. We demonstrate that hyperfine interaction for the Dy Kramers ion leads to tunnel splitting (or quantum tunneling of magnetization) at zero field. This effect does not occur for TbPc2 single-molecule magnets. The magnetic field values of the avoided level crossings for 161DyPc2 and 163DyPc2 are found to be noticeably different, which can be observed from the experiment.

Nature of Hyperfine Interactions in TbPc2 Single-Molecule Magnets: Multiconfigurational Ab Initio Study.

Lanthanide-based single-ion magnetic molecules can have large magnetic hyperfine interactions as well as large magnetic anisotropy. Recent experimental studies reported tunability of these properties by changes of chemical environments or by application of external stimuli for device applications. In order to provide insight onto the origin and mechanism of such tunability, here we investigate the magnetic hyperfine and nuclear quadrupole interactions for a 159Tb nucleus in TbPc2 (Pc = phthalocyanine) single-molecule magnets using multiconfigurational ab initio methods including spin-orbit interaction. Since the electronic ground and first-excited (quasi)doublets are well separated in energy, the microscopic Hamiltonian can be mapped onto an effective Hamiltonian with an electronic pseudospin S = 1/2. From the ab initio calculated parameters, we find that the magnetic hyperfine coupling is dominated by the interaction of the Tb nuclear spin with electronic orbital angular momentum. The asymmetric 4f-like electronic charge distribution leads to a strong nuclear quadrupole interaction with significant transverse terms for the molecule with low symmetry. The ab initio calculated electronic-nuclear spectrum including the magnetic hyperfine and quadrupole interactions is in excellent agreement with the experiment. We further find that the transverse quadrupole interactions significantly influence the avoided level crossings in magnetization dynamics and that the molecular distortions affect mostly the Fermi contact terms as well as the transverse quadrupole interactions.

Toward Long-Range Entanglement between Electrically Driven Single-Molecule Magnets.

Over the past two decades, several molecules have been explored as possible building blocks of a quantum computer, a device that would provide exponential speedups for a number of problems, including the simulation of large, strongly correlated chemical systems. Achieving strong interactions and entanglement between molecular qubits remains an outstanding challenge. Here, we show that the TbPc2 single-molecule magnet has the potential to overcome this obstacle because of its sensitivity to electric fields stemming from the hyperfine Stark effect. We show how this feature can be leveraged to achieve long-range entanglement between pairs of molecules using a superconducting resonator as a mediator. Our results suggest that the molecule-resonator interaction is near the edge of the strong-coupling regime and could potentially pass into it given a more detailed, quantitative understanding of the TbPc2 molecule.

Multireference Ab Initio Studies of Magnetic Properties of Terbium-Based Single-Molecule Magnets.

We investigate how different chemical environments influence magnetic properties of terbium(III) (Tb)-based single-molecule magnets (SMMs), using first-principles relativistic multireference methods. Recent experiments showed that Tb-based SMMs can have exceptionally large magnetic anisotropy and that they can be used for experimental realization of quantum information applications, with a judicious choice of chemical environment. Here, we perform complete active space self-consistent field calculations including relativistic spin-orbit interaction for representative Tb-based SMMs such as TbPc2 and TbPcNc in three charge states. We calculate the low-energy electronic structure from which we compute the Tb crystal-field (CF) parameters and construct an effective pseudospin Hamiltonian. Our calculations show that the ligand type and fine points of molecular geometry do not affect the gap between the ground-state and first-excited doublets, whereas the latter varies weakly with oxidation number. On the other hand, higher-energy levels have a strong dependence on all these characteristics. For neutral TbPc2 and TbPcNc molecules, the Tb magnetic moment and ligand spin are parallel to each other and the coupling strength between them does not depend much on the ligand type and details of the atomic structure. However, ligand distortion and molecular symmetry play a crucial role in transverse CF parameters which lead to tunnel splitting. The tunnel splitting induces quantum tunneling of magnetization by itself or by combining with other processes. Our results provide insights into the mechanisms of magnetization relaxation in the representative Tb-based SMMs.

Author Correction: Two-magnon scattering in the 5d all-in-all-out pyrochlore magnet Cd2Os2O7.

A correction to this article has been published and is linked from the HTML version of this article.

Two-magnon scattering in the 5d all-in-all-out pyrochlore magnet Cd2Os2O7.

5d pyrochlore oxides with all-in-all-out magnetic order are prime candidates for realizing strongly correlated, topological phases of matter. Despite significant effort, a full understanding of all-in-all-out magnetism remains elusive as the associated magnetic excitations have proven difficult to access with conventional techniques. Here we report a Raman spectroscopy study of spin dynamics in the all-in-all-out magnetic state of the 5d pyrochlore Cd2Os2O7. Through a comparison between the two-magnon scattering and spin-wave theory, we confirm the large single ion anisotropy in this material and show that the Dzyaloshinskii-Moriya and exchange interactions play a significant role in the spin-wave dispersions. The Raman data also reveal complex spin-charge-lattice coupling and indicate that the metal-insulator transition in Cd2Os2O7 is Lifshitz-type. Our work establishes Raman scattering as a simple and powerful method for exploring the spin dynamics in 5d pyrochlore magnets.Pyrochlore 5d transition metal oxides are expected to have interesting forms of magnetic order but are hard to study with conventional probes. Here the authors show that Raman scattering can be used to measure magnetic excitations in Cd2Os2O7 and that it exhibits complex spin-charge-lattice coupling.

Bulk magnetoelectricity in the hexagonal manganites and ferrites.

Improper ferroelectricity (trimerization) in the hexagonal manganites RMnO3 leads to a network of coupled structural and magnetic vortices that induce domain wall magnetoelectricity and magnetization (M), neither of which, however, occurs in the bulk. Here we combine first-principles calculations, group-theoretic techniques and microscopic spin models to show how the trimerization not only induces a polarization (P) but also a bulk M and bulk magnetoelectric (ME) effect. This results in the existence of a bulk linear ME vortex structure or a bulk ME coupling such that if P reverses so does M. To measure the predicted ME vortex, we suggest RMnO3 under large magnetic field. We suggest a family of materials, the hexagonal RFeO3 ferrites, also display the predicted phenomena in their ground state.

Direct visualization of magnetoelectric domains.

The coupling between the magnetic and electric dipoles in multiferroic and magnetoelectric materials holds promise for conceptually novel electronic devices. This calls for the development of local probes of the magnetoelectric response, which is strongly affected by defects in magnetic and ferroelectric ground states. For example, multiferroic hexagonal rare earth manganites exhibit a dense network of boundaries between six degenerate states of their crystal lattice, which are locked to both ferroelectric and magnetic domain walls. Here we present the application of a magnetoelectric force microscopy technique that combines magnetic force microscopy with in situ modulating high electric fields. This method allows us to image the magnetoelectric response of the domain patterns in hexagonal manganites directly. We find that this response changes sign at each structural domain wall, a result that is corroborated by symmetry analysis and phenomenological modelling, and provides compelling evidence for a lattice-mediated magnetoelectric coupling. The direct visualization of magnetoelectric domains at mesoscopic scales opens up explorations of emergent phenomena in multifunctional materials with multiple coupled orders.

Imaging and control of surface magnetization domains in a magnetoelectric antiferromagnet.

We report the direct observation of surface magnetization domains of the magnetoelectric Cr(2)O(3) using photoemission electron microscopy with magnetic circular dichroism contrast and magnetic force microscopy. The domain pattern is strongly affected by the applied electric field conditions. Zero-field cooling results in an equal representation of the two domain types, while electric-field cooling selects one dominant domain type. These observations confirm the existence of surface magnetization, required by symmetry in magnetoelectric antiferromagnets.