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

Application of the Adiabatic Connection Random Phase Approximation to Electron-Nucleus Hyperfine Coupling Constants

The electron-nucleus hyperfine coupling constant is a challenging property for density functional methods. For accurate results, hybrid functionals with a large amount of exact exchange are often needed and there is no clear "one-for-all" functional which describes the hyperfine coupling interaction for a large set of nuclei. To alleviate this unfavorable situation, we apply the adiabatic connection random phase approximation (RPA) in its post-Kohn-Sham fashion to this property as a first test. For simplicity, only the Fermi-contact and spin-dipole terms are calculated within the nonrelativistic and the scalar-relativistic exact two-component framework. This requires to solve a single coupled-perturbed Kohn-Sham equation to evaluate the relaxed density matrix, which comes with a modest increase in computational demands. RPA performs remarkably well and substantially improves upon its Kohn-Sham (KS) starting point while also reducing the dependence on the KS reference. For main-group systems, RPA outperforms global, range-separated, and local hybrid functionals─at similar computational costs. For transition-metal compounds and lanthanide complexes, a similar performance as for hybrid functionals is observed. In contrast, related post-Hartree-Fock methods such as Møller-Plesset perturbation theory or CC2 perform worse than semilocal density functionals.

Electronic Structure of a Neodymium(III) Tris(oxidiacetate) Complex from Luminescence Data and Ab Initio Calculations

Neodymium(III) is a near-infrared emissive and magnetic ion, which has found use in various high-technology applications. Yet, accurate predictions of the luminescent and magnetic properties of neodymium(III) based on the coordination environment remain to be done. Guidelines exist, but to build structure-property relationships for this element, more data are needed. Herein, we present a high-symmetry starting point. The tris(oxidiacetate) complex of neodymium(III) was prepared and crystallized, and access to the experimentally determined structure allowed us to quantify the symmetry of the compound and to perform calculations directly on the same structure that is investigated experimentally. The luminescent properties were determined and the electronic structure was computed using state-of-the-art ab initio methods. All electronic transitions in the range from 490 to 1400 nm were mapped experimentally. Using a Boltzmann population analysis, the combination of the excitation and emission spectra revealed the crystal field splitting of the 18 lowest-energy Kramers levels that experimentally could be unambiguously resolved. This assignment was supported by ab initio calculations, and the crystal field splitting was well reproduced. The electronic structure reported for the tris(oxidiacetate) complex was used to deduce the coordination structure in aqueous solution. Finally, the results are compared to empirical trends in the literature for the electronic structure of neodymium(III).

Hyperfine interactions for small systems including transition-metal elements using self-interaction corrected density-functional theory

The interactions between the electronic magnetic moment and the nuclear spin moment, i.e., magnetic hyperfine (HF) interactions, play an important role in understanding electronic properties of magnetic systems and in realizing platforms for quantum information science applications. We investigate the HF interactions for atomic systems and small molecules, including Ti or Mn, by using Fermi-Löwdin orbital (FLO) based self-interaction corrected (SIC) density-functional theory. We calculate the Fermi contact (FC) and spin-dipole terms for the systems within the local density approximation (LDA) in the FLO-SIC method and compare them with the corresponding values without SIC within the LDA and generalized-gradient approximation (GGA), as well as experimental data. For the moderately heavy atomic systems (atomic number Z ≤ 25), we find that the mean absolute error of the FLO-SIC FC term is about 27 MHz (percentage error is 6.4%), while that of the LDA and GGA results is almost double that. Therefore, in this case, the FLO-SIC results are in better agreement with the experimental data. For the non-transition-metal molecules, the FLO-SIC FC term has the mean absolute error of 68 MHz, which is comparable to both the LDA and GGA results without SIC. For the seven transition-metal-based molecules, the FLO-SIC mean absolute error is 59 MHz, whereas the corresponding LDA and GGA errors are 101 and 82 MHz, respectively. Therefore, for the transition-metal-based molecules, the FLO-SIC FC term agrees better with experiment than the LDA and GGA results. We observe that the FC term from the FLO-SIC calculation is not necessarily larger than that from the LDA or GGA for all the considered systems due to the core spin polarization, in contrast to the expectation that SIC would increase the spin density near atomic nuclei, leading to larger FC terms.

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).

Hilbert Space in Isotopologue Dy(III) SMM Dimers: Dipole Interaction Limit in [163/164Dy2(tmhd)6(tape)]0 Complexes

Single-molecule magnets are molecular complexes proposed to be useful for information storage and quantum information processing applications. In the quest for multilevel systems that can act as Qudits, two dysprosium-based isotopologues were synthesized and characterized. The isotopologues are [164Dy2(tmhd)6(tape)] (1(I=0)) and [163Dy2(tmhd)6(tape)] (2(I=5/2)), where tmhd = 2,2,6,6-tetramethylheptandionate and tape = 1,6,7,12-tetraazaperylene. Both complexes showed slow relaxation at a zero applied magnetic field with dominant Orbach and Raman relaxation mechanisms. μSQUID studies at milli-Kelvin temperatures reveal quasi-single ion loops, in contrast with the expected S-shape (near zero field) butterfly loops, characteristic of antiferromagnetically coupled dimeric complexes. Through analysis of the low-temperature data, we find that the interaction operating between Dy(III) is small, leading to a small exchange biasing from the zero-field transition. The resulting indirectly coupled nuclear states are degenerate or possess a small energy difference between them. We, therefore, conclude that for the creation of Qudits with enlarged Hilbert spaces, shorter Dy(III)···Dy(III) distances are deemed essential.

Strategies to quench quantum tunneling of magnetization in lanthanide single molecule magnets

Enhancing blocking temperature (T_B) is one of the holy grails in Single Molecule Magnets(SMMs), as any future potential application in this class of molecules is directly correlated to this parameter. Among many factors contributing to a reduction of T_B value, Quantum Tunnelling of Magnetisation (QTM), a phenomenon that is a curse or a blessing based on the application sought after, tops the list. Theoretical tools based on density functional and ab initio CASSCF/RASSI-SO methods have played a prominent role in estimating various spin Hamiltonian parameters and establishing the mechanism of magnetization relaxation in this class of molecules. Particularly, various strategies to quench QTM effects go hand-in-hand with experiments, and different methods proposed to quell QTM effects are scattered in the literature. In this perspective, we have explored various approaches that are proposed in the literature to quench QTM effects, and these include the role of (i) local symmetry of lanthanides, (ii) super-exchange interaction in {3d-4f} complexes, (iii) direct-exchange interaction in {radical-4f} and metal-metal bonded complexes to suppress the QTM, (iv) utilizing external stimuli such as an electric field or pressure to modulate the QTM and (v) avoiding QTM effects by stabilising toroidal states in 4f and {3d-4f} clusters. We believe the strategies summarized here will help to design new-generation SMMs.

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.

Rare-earth based tetrapyrrolic sandwiches: chemistry, materials and applications

This review summarises advances in chemistry of tetrapyrrole sandwiches with rare earth elements and highlights the current state of their use in single-molecule magnetism, organic field-effect transistors, conducting materials and nonlinear optics.

Hyperfine Coupling Constants in Local Exact Two-Component Theory

We present a highly efficient implementation of the electron-nucleus hyperfine coupling matrix within the one-electron exact two-component (X2C) theory. The complete derivative of the X2C Hamiltonian is formed, that is, the derivatives of the unitary decoupling transformation are considered. This requires the solution of the response and Sylvester equations, consequently increasing the computational costs. Therefore, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The finite nucleus model is employed for both the scalar potential and the vector potential. Two-electron picture-change effects are modeled with the (modified) screened nuclear spin-orbit approach. Our implementation is fully integral direct and OpenMP-parallelized. An extensive benchmark study regarding the Hamiltonian, the basis set, and the density functional approximation is carried out for a set of 12-17 transition-metal compounds. The error introduced by DLU is negligible, and the DLU-X2C Hamiltonian accurately reproduces its four-component "fully" relativistic parent results. Functionals with a large amount of Hartree-Fock exchange such as CAM-QTP-02 and ωB97X-D are generally favorable. The pure density functional r²SCAN performs remarkably and even outperforms the common hybrid functionals TPSSh and CAM-B3LYP. Fully uncontracted basis sets or contracted quadruple-ζ bases are required for accurate results. The capability of our implementation is demonstrated for [Pt(C₆Cl₅)₄]^- with more than 4700 primitive basis functions and four rare-earth single-molecule magnets: [La(OAr*)₃]^-, [Lu(NR₂)₃]^-, [Lu(OAr*)₃]^-, and [TbPc₂]^-. Here, the results with the spin-orbit DLU-X2C Hamiltonian are in an excellent agreement with the experimental findings of all Pt, La, Lu, and Tb molecules.

Electron-Nucleus Hyperfine Coupling Calculated from Restricted Active Space Wavefunctions and an Exact Two-Component Hamiltonian

Exact two-component (X2C) relativistic nuclear hyperfine magnetic field operators were incorporated in X2C ab initio wavefunction calculations at the multireference restricted active space (RAS) level for calculations of nuclear hyperfine magnetic properties. Spin-orbit coupling was treated via RAS state interaction (SO-RASSI). The method was tested by calculations of electron-nucleus hyperfine coupling constants. The approach, implemented in the OpenMolcas program, overcomes a major limitation of a previous SO-RASSI implementation for hyperfine coupling that relied on nonrelativistic hyperfine operators [J. Chem. Theor. Comput. 2015, 11, 538-549] and therefore had limited applicability. Results from calculations on systems with light and heavy main group elements, transition metals, lanthanides, and one actinide complex demonstrate reasonably good agreement with experimental data, where available, as long as the active space can generate sufficient spin polarization.

A quasilinear hydrazone-based mononuclear dysprosium compound with C4v symmetry exhibiting field-induced complex magnetic relaxation

A C4v symmetrical mononuclear dysprosium(iii) compound has been successfully isolated using a new quasilinear single pyrazinyl hydrazone ligand. Single-ion behavior and the short-range intermolecular magnetic dipolar interaction are essential to the complex magnetic relaxation.

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.