Disentangling intricate web of interrelated notions at the interface between the physical (crystal field) Hamiltonians and the effective (spin) Hamiltonians

Tailoring single-ion magnet properties of coordination polymer C11H18DyN3O9 (Dy-CP) using the radial effective charge model (RECM) and superposition model (SPM)

We investigate Dy-based coordination polymer C11H18DyN3O9 (Dy-CP) exhibiting single-ion magnet (SIM) properties, e.g., quantum tunnelling of magnetization (QTM), magnetic anisotropy, magnetic relaxation, and effective energy barrier (Ueff). To elucidate the underlying mechanisms, crystal field parameters (CFPs) for Dy3+ ions were modelled using the radial effective charge model (RECM) and superposition model (SPM), and the computational packages SIMPRE and SPECTRE. The modelled CFPs enable the prediction of the energy levels and associated wave functions, which successfully explain the field-induced Dy-CP SIM properties. The so-calculated magnetic susceptibility and isothermal magnetization match the experimental data reasonably well. The smaller energy separations of the first (Δ0-1 ∼ 31 cm-1) and the second (Δ0-2 = 74 cm-1) excited Kramers doublets suggest small Ueff = 65 cm-1 for Dy-CP. The magnetic moments of Dy3+ ions exhibit an easy-axis type magnetic anisotropy in the ground state, but change orientation in the excited states due to mixing of states from different Kramers doublets. Low-symmetry CF components play a crucial role in connecting different |±MJ〉 states within the ground multiplet, resulting in QTM and magnetic relaxation to the ground state occurring via the excited states. The RECM and SPM calculated CFP sets are standardized employing the 3DD package to enable meaningful comparison and assessing their mutual equivalence. The results demonstrate the correlation between structural and electronic features of the molecule and site symmetry and distortion of the local coordination polyhedra with SIM properties, offering insights for rational design of new SIMs. The importance of considering low-symmetry aspects in CFP modelling for accurate predictions of magnetic properties is highlighted. This study provides deeper understanding of field-induced behaviour in rare-earth-based SIMs and approaches for rationalization of experimentally measured SIMs' properties.

Eu3+ ions as a crystal-field probe for low-symmetry sites in doped phosphors - a case study: Eu3+ at triclinic sites in Li6RE(BO3)3 (RE = Y, Gd), YBO3 and ZnO and at trigonal sites in YAl3(BO3)4

We present crystal-field (CF) calculations of energy levels (Ei) of Eu3+ ions doped in various hosts aimed at exploring the low-symmetry properties of CF parameters (CFPs) and reliability of CFP modelling with decreasing site symmetry. The hosts studied are: Li6Y(BO3)3, Li6Gd(BO3)3, YBO3, and ZnO with Eu3+ at triclinic sites; YAl3(BO3)4 with Eu3+ ions at trigonal D3 symmetry. Two independent CFP modelling approaches utilizing the hosts' structural data are employed: the exchange charge model (ECM) and the superposition model (SPM). We adopt the Eu3+ actual site symmetry and not the approximated one. The Ei values calculated using CFPs modelled by the ECM and SPM mutually agree with the observed ones. For triclinic symmetry, the ECM/CFPs and SPM/CFPs were numerically distinct, yet turned out to be physically equivalent yielding identical rotational invariants, Sk (k = 2, 4, 6) and Ei. For trigonal symmetry, both CFP sets agree numerically, thus Sk and Ei are identical. This disparity poses a dilemma, since the modified crystallographic axis system was used in both approaches. The standardization of the triclinic CFPs using the 3DD package was performed to solve this dilemma. It has enabled discussing standardization aspects in experimental and computed CFP sets and elucidating intricate low-symmetry aspects inherent in CFP sets. Understanding of low-symmetry aspects in CF studies may bring about a better interpretation of the spectroscopic and magnetic properties of rare-earth ion doped host crystals. Thus, our study could provide more deep insights into the importance of clear definitions of axis systems and adequate treatment of actual site symmetry in the modelling of CFPs for low-symmetry cases which is essential for technological applications and engineering of rare-earth activated phosphor materials.

Limitations on the D-Parameter in Ni(II) Complexes

A number of hexacoordinate, pentacoordinate, and tetracoordinate Ni(II) complexes have been investigated by applying ab initio CASSCF + NEVPT2 + SOC calculations and Generalized Crystal Field Theory. The geometry of the coordination polyhedron covers D4h, D3h, D2h, D2d, C4v, C3v, and C2v symmetry. The calculated spin-Hamiltonian parameters D and E were compared to the available experimental data. The limiting values of the D-parameter in the class of Ni(II) complexes are identified. Magnetic anisotropy in Ni(II) complexes, expressed by the axial zero-field splitting parameter D, seriously depends upon the ground and first excited electronic states. In hexacoordinate complexes, the ground electronic term is nondegenerate 3B1g for the D4h symmetry; D is slightly positive or negative. In tetracoordinate systems, D is only positive when the electronic ground state is nondegenerate 3A or 3B; this diverges on the τ4 path when oblate bisphenoid approaches the prolate geometry and a level crossing with 3E occurs. In pentacoordinate systems, D could be extremely negative when approaching a trigonal bipyramid (Addison index τ5 ∼ 1, ground state 3E″). In pentacoordinate Ni(II) complexes with the D3h and C3v symmetry of the coordination polyhedron, the ground electronic term is orbitally doubly degenerate which causes the D-parameter stays undefined. It is emphasized that one has to inspect compositions of the spin-orbit multiplets from the spin states |MS⟩ and check whether the weights confirm the expected spin-Hamiltonian picture: with D > 0, the ground state contains a dominant part of |0⟩ (close to 100%) whereas with D < 0 the spin-orbit doublet is formed of |±1⟩ with high weights (approaching 50 + 50%). The calculations show that the situations are not black and white, and the mixing of the states might be more complex especially when the rhombic zero-field splitting parameter E is in the play. In the case of the 3E ground term, six spin-orbit multiplets are formed by mixing six |MS⟩ states from the ground and quasi-degenerate excited states.

Spectroscopic and Magnetic Studies of Co(II) Scorpionate Complexes: Is There a Halide Effect on Magnetic Anisotropy?

The observation of single-molecule magnetism in transition-metal complexes relies on the phenomenon of zero-field splitting (ZFS), which arises from the interplay of spin-orbit coupling (SOC) with ligand-field-induced symmetry lowering. Previous studies have demonstrated that the magnitude of ZFS in complexes with 3d metal ions is sometimes enhanced through coordination with heavy halide ligands (Br and I) that possess large free-atom SOC constants. In this study, we systematically probe this "heavy-atom effect" in high-spin cobalt(II)-halide complexes supported by substituted hydrotris(pyrazol-1-yl)borate ligands (Tp^tBu,Me and Tp^Ph,Me). Two series of complexes were prepared: [Co^IIX(Tp^tBu,Me)] (1-X; X = F, Cl, Br, and I) and [Co^IIX(Tp^Ph,Me)(Hpz^Ph,Me)] (2-X; X = Cl, Br, and I), where Hpz^Ph,Me is a monodentate pyrazole ligand. Examination with dc magnetometry, high-frequency and -field electron paramagnetic resonance, and far-infrared magnetic spectroscopy yielded axial (D) and rhombic (E) ZFS parameters for each complex. With the exception of 1-F, complexes in the four-coordinate 1-X series exhibit positive D-values between 10 and 13 cm^-1, with no dependence on halide size. The five-coordinate 2-X series exhibit large and negative D-values between -60 and -90 cm^-1. Interpretation of the magnetic parameters with the aid of ligand-field theory and ab initio calculations elucidated the roles of molecular geometry, ligand-field effects, and metal-ligand covalency in controlling the magnitude of ZFS in cobalt-halide complexes.

Possible Violation of the Spin-Hamiltonian Concept in Interpreting the Zero-Field Splitting

The majority of experimental data in electron spin resonance and molecular magnetism are interpreted in terms of the spin-Hamiltonian (SH) formalism. However, this is an approximate theory that requires a proper testing. In the older variant, the multielectron terms are used as a basis in which the D-tensor components are evaluated by employing the second-order perturbation theory (PT) for nondegenerate states; here, the spin-orbit interaction expressed via the spin-orbit splitting parameter λ serves for the perturbation. The model space is restricted only to the fictitious spin functions |S, M⟩. In the case of the orbital (quasi) degeneracy of the ground term, the PT tends to diverge and the subtracted D, E, and g parameters are false. In the second variant working in the "complete active space" (CAS), the spin-orbit coupling operator is involved by the variation method resulting in the spin-orbit multiplets (energies and eigenvectors) The multiplets can be evaluated either by applying ab initio CASSCF + NEVPT2 + SOC calculations or by using semiempirical generalized crystal-field theory (with the one-electron SOC operator depending upon ξ). The resulting states can be projected onto the subspace of the spin-only kets in the way that the eigenvalues stay invariant. Such an effective Hamiltonian matrix can be reconstructed using six independent components of the symmetric D-tensor from which the D and E values are obtained by solving linear equations. The eigenvectors of the spin-orbit multiplets in the CAS allow determining the dominating composition of the spin projection─cumulative weights of |±M⟩. These are conceptually different from those generated by the SH alone. It is shown that in some cases, the SH theory works satisfactorily for a series of transition-metal complexes; however, sometimes it fails. The ab initio calculations on the SH parameters are compared with the approximate generalized crystal-field theory conducted at the experimental geometry of the chromophore. In total, 12 metal complexes have been analyzed. One of the criteria that assesses the validity of SH is the projection norm N for spin multiplets (this has not to be far from 1). Another criterion is the gap in the spectrum of the spin-orbit multiplets that separates the hypothetical (fictitious) spin-only manifold from the rest of the states.

Effects of distortions in ion-host systems on optical spectra, crystal-field and spin-Hamiltonian parameters of Cr3+ ions doped pyrochlores Y2Ti2O7 and Y2Sn2O7: exchange charge model and superposition model calculations

Comparative modelling of the crystal-field parameters (CFPs), CF energy levels, and effective spin-Hamiltonian parameters (SHPs), i.e., the g-factors and zero-field splitting parameter (ZFSP), D, of the ground state ⁴A_2g of the Cr³⁺ dopant ions in Y₂Ti₂O₇ and Y₂Sn₂O₇ is carried out. The CFPs are calculated using XRD structural data by employing two semi-empirical models: the exchange charge model (ECM) and superposition model (SPM). This two-fold approach ensures increased reliability of CFP modelling and thus of the final results. The modelled CFPs serve as the input to the crystal-field analysis/microscopic spin-Hamiltonian (CFA/MSH) program to predict CF energy levels and wave functions, and to extract SHPs. Since the site symmetry of Cr³⁺ ions in these crystals is trigonal D_3d, a symmetry adapted axis system (SAAS) conforming to Watanabe convention is adopted for CFP modelling. The calculated CF energies and SHPs for Y₂Ti₂O₇:Cr³⁺ are in good agreement with the experimental results. Variations of the Racah parameter B, as well as ECM and SPM parameters for Y₂Sn_2-xCr_xO₇ are correlated with the chromium concentration (x), which affects distortions of CrO₆ structures. We find that the SHPs originate predominantly from the spin-orbit coupling, though contributions from spin-spin and spin-other-orbit couplings are also appreciable and thus important for analysis of lattice distortions and structural disorder. The uniqueness of the SAAS used for CFP modelling is also discussed. The present study enables exploring the influence of the radial and angular distortions of host clusters (Ti-O₆/Sn-O₆) introduced by Cr³⁺ ions on the structural and optical properties as well as the SHPs of Cr³⁺ ions doped in Y₂Ti₂O₇ and Y₂Sn₂O₇.

DFT computations combined with semiempirical modeling of variations with temperature of spectroscopic and magnetic properties of Gd3+-doped PbTiO3

The rare-earth or 3d transition metal dopants in perovskites have potential to induce interesting features, thus opening opportunities for investigations and applications. Hence, understanding some features, i.e., defect structure, site of incorporation, valence state, and mechanism of charge compensation, in a wide range of temperature is crucial for their technological applications. A comprehensive understanding of the mechanism of structural changes in PbTiO₃ doped with trivalent rare-earths is significant for their potential applications in photonics. To unravel the structural changes, we utilize the density functional theory (DFT) to optimize structural data, which then serve as input for the semiempirical superposition model (SPM) analysis of spectroscopic and magnetic properties of Gd³⁺-doped PbTiO₃. We compute the formation energies of the doped compounds with and without O-vacancy to determine the stable composition. Analysis of the Bader electron charges computed using DFT plus quantum theory of atoms in molecules enables elucidating the effects of the Gd dopant and O-vacancy on the ionic and covalent bonds and, thereby, chemical stability of the compositions. To explain and corroborate the zero-field splitting parameters (ZFSPs) measured by EMR and the lattice parameter changes obtained from XRD, we employ SPM. The optimized structures obtained from ab initio computations for various structural models of Gd³⁺ doped PbTiO₃ are utilized as input data for SPM calculations of ZFPs. This enables theoretical analysis of variations of ZFSPs from 5 to 780 K. The results were fine-tuned by matching with available experimental EMR data for Gd³⁺ probes in PbTiO₃ nanoparticles. Modeling has been carried out considering several possible structural models and the role of an O-vacancy around Gd³⁺ centers. The results show that the two-fold modeling approach, combining DFT and SPM, provides a reliable description of experimental data. Comparative analysis indicates that the Ti-site is less favorable for being replaced by Gd³⁺ with/without O-vacancy. This analysis confirms the plausibility of the Pb²⁺ site for Gd³⁺ dopants and sheds light on the changes of crystal structure during the phase transitions occurring in PbTiO₃ with decreasing temperature.

Fine Structure and the Huge Zero-Field Splitting in Ni2+ Complexes

We perform a thorough study of the ground state magnetic properties of nickel-based 3d8 complexes. This includes an in-depth analysis of the contribution of the crystal field, spin exchange and spin-orbit interactions to the ground state magnetic properties. Of particular interest to the current investigation are the presence and occurrence of non-trivial zero-field splitting. The study focuses on the cases of Ni2+ ideal octahedral, trigonal bipyramidal, square planar and tetrahedral geometries. We provide results for the complete energy spectrum, the fine structure related to the ground state and the second set of excited states, low-field magnetic susceptibility and magnetization. In addition, we examine the zero-field fine structure in square pyramidal, trigonal pyramidal and trigonal planar complexes. The obtained results unequivocally show that a moderate or highly coordinated 3d8 complex can neither exhibit spin-orbit-driven large and giant magnetic anisotropy nor a huge zero-field splitting. Moreover, in the trigonal bipyramidal coordination, a fine structure associated to the ground state cannot result from the spin-orbit coupling alone.

Single-Ion Magnets with Giant Magnetic Anisotropy and Zero-Field Splitting

The design of mononuclear molecular nanomagnets exhibiting a huge energy barrier to the reversal of magnetization have seen a surge of interest during the last few decades due to their potential technological applications. More specifically, single-ion magnets are peculiarly attractive by virtue of their rich quantum behavior and distinct fine structure. These are viable candidates for implementation as single-molecule high-density information storage devices and other applications in future quantum technologies. The present review presents the comprehensive state of the art in the topic of single-ion magnets possessing an eminent magnetization-reversal barrier, very slow magnetic relaxation and high blocking temperature. We turn our attention to the achievements in the synthesis of 3d and 4f single-ion magnets during the last two decades and discuss the observed magnetostructural properties underlying the anisotropy behavior and the ensuing remanence. Furthermore, we highlight the fundamental theoretical aspects to shed light on the complex behavior of these nanosized magnetic entities. In particular, we focus on key notions, such as zero-field splitting, anisotropy energy and quantum tunneling of the magnetization and their interdependence.

Holmium(III) Single-Ion Magnet for Cryomagnetic Refrigeration Based on an MRI Contrast Agent Derivative

The coexistence of field-induced blockage of the magnetization and significant magnetocaloric effects in the low-temperature region occurs in a mononuclear holmium(III) diethylenetriamine-N,N,N',N″,N″-pentaacetate complex, whose gadolinium(III) analogue is a commercial MRI contrast agent. Both properties make it a suitable candidate for cryogenic magnetic refrigeration, thus enlarging the variety of applications of this simple class of multifunctional molecular nanomagnets.

Determination of the g-, hyperfine coupling- and zero-field splitting tensors in EPR and ENDOR using extended Matlab codes

The analysis of single crystal electron magnetic resonance (EMR) data has traditionally been performed using software in programming languages that are difficult to update, are not easily available, or are obsolete. By using a modern script-language with tools for the analysis and graphical display of the data, three MatLab® codes were prepared to compute the g, zero-field splitting (zfs) and hyperfine coupling (hfc) tensors from roadmaps obtained by EPR or ENDOR measurements in three crystal planes. Schonland's original method was used to compute the g- and hfc -tensors by a least-squares fit to the experimental data in each plane. The modifications required for the analysis of the zfs of radical pairs with S = 1 were accounted for. A non-linear fit was employed in a second code to obtain the hfc -tensor from EPR measurements, taking the nuclear Zeeman interaction of an I = ½ nucleus into account. A previously developed method to calculate the g- and hfc -tensors by a simultaneous linear fit to all data was used in the third code. The validity of the methods was examined by comparison with results obtained experimentally, and by roadmaps computed by exact diagonalization. The probable errors were estimated using functions for regression analysis available in MatLab. The software will be published at https://doi.org/10.17632/ps24sw95gz.1, Input and output examples presented in this work can also be downloaded from https://old.liu.se/simarc/downloads?l=en.

Molecular magnetism in the multi-configurational self-consistent field method

We develop a structured theoretical framework used in our recent articles (2019 Eur. Phys. J. B 92 93 and 2020 Phys. Rev. B 101 094427) to characterize the unusual behavior of the magnetic spectrum, magnetization and magnetic susceptibility of the molecular magnet Ni₄Mo₁₂. The theoretical background is based on the molecular orbital theory in conjunction with the multi-configurational self-consistent field method and results in a post-Hartree-Fock scheme for constructing the corresponding energy spectrum. Furthermore, we construct a bilinear spin-like Hamiltonian involving discrete coupling parameters accounting for the relevant spectroscopic magnetic excitations, magnetization and magnetic susceptibility. The explicit expressions of the eigenenergies of the ensuing Hamiltonian are determined and the physical origin of broadening and splitting of experimentally observed peaks in the magnetic spectra is discussed. To demonstrate the efficiency of our method we compute the spectral properties of a spin-one magnetic dimer. The present approach may be applied to a variety of magnetic units based on transition metals and rare Earth elements.

Investigation of the Electronic Structure and Optical, EPR, and ODMR Spectroscopic Properties for 171Yb3+-Doped Y2SiO5 Crystal: A Combined Theoretical Approach

Various spectroscopic properties of Yb³⁺-doped Y₂SiO₅ crystal have been extensively investigated due to its promising application in quantum information processing. However, the local structure, electronic structure of Yb³⁺:Y₂SiO₅ crystal, and its optical and magnetic properties have not been comprehensively studied from a theoretical viewpoint. In this work, the geometric and electronic structures of Yb³⁺ that replaces two crystallographic Y³⁺ sites in the Y₂SiO₅ crystal are first obtained by the method of density functional theory (DFT). Then, the optical, electron paramagnetic resonance (EPR), and optically detected magnetic resonance (ODMR) spectra for ¹⁷¹Yb³⁺ (nuclear spin I = 1/2) at such two sites are simultaneously calculated in the framework of the complete diagonalization (of energy) matrix (CDM) based on the optimized local structure around ¹⁷¹Yb³⁺ ion by DFT. The various calculated spectroscopic properties by such combined theoretical approach are consistent with the experimental ones, which demonstrates that CDM is effective and particularly suitable for calculating hyperfine A-tensors under zero, low, and intermediate magnetic field. More importantly, based on the obtained accurate hyperfine structure of ¹⁷¹Yb³⁺ in Y₂SiO₅ crystal, the possible "clock transitions", which can enhance the optical coherence time, can be assigned or predicted by the present approach. This study successfully explains the spectroscopic properties of ¹⁷¹Yb³⁺-doped Y₂SiO₅ and provides a feasible method to design and search for practical rare-earth-doped quantum information materials for the community.

Importance of Axial Symmetry in Elucidating Lanthanide-Transition Metal Interactions

In this paper, we experimentally study and model the electron donating character of an axial diamagnetic Pd²⁺ ion in four metalloligated lanthanide complexes of formula [PPh₄][Ln{Pd(SAc)₄}₂] (SAc^- = thioacetate, Ln = Tb, Dy, Ho, and Er). A global model encompassing inelastic neutron scattering, torque magnetometry, and dc magnetometry allows to precisely determine the energy level structure of the complexes. Solid state nuclear magnetic resonance reveals a less donating character of Pd²⁺ compared to the previously reported isostructural Pt²⁺-based complexes. Consequently, all complexes invariably show a lower crystal field strength compared to their Pt²⁺-analogues. The dynamic properties show an enhanced single molecule magnet behavior due to the suppression of quantum tunneling, in agreement with our model.

Importance of the fourth-rank zero field splitting parameters for Fe2+ (S = 2) adatoms on the CuN/Cu(100) surface evidenced by their determination based on DFT and experimental data

Equations allow to determine 2nd- and 4th-rank ZFSPs (B_k^q) based on spin energy levels (λi) at B = 0. This method is applied to Fe²⁺ (S = 2) adatoms on CuN/Cu(100) surface using DFT and experimental data. Relative importance of ZFSPs is analyzed.

Role of Ab Initio Calculations in the Design and Development of Organometallic Lanthanide-Based Single-Molecule Magnets

Single-molecule magnets based on lanthanides are very attractive due to their potential applications proposed in the area of microelectronic devices. Very recent advances in this area are due to the blend of conventional lanthanide chemistry with organometallic ligands, and several breakthrough achievements are attained with this combination. Ab initio methods based on multi-reference CASSCF calculations are playing a vital role in the design and development of such molecules. In this minireview, we aim to appraise various contributions in the area of organometallic lanthanide complexes (those containing lanthanide-carbon bonds) and describe how these robust wavefunction-based methods have played a constructive role not only in rationalizing the observed magnetic properties but also proven to be a potential predictive tool with some selected examples.

The Multiple Faces, and Phases, of Magnetic Anisotropy

The notion of magnetic anisotropy is very central to the field of molecule-based magnetism, where it is considered to be a key quantity that must be rationalized and controlled in order to improve the performances of, e.g., single-molecule magnets. A rough classification of the magnetic properties is widely done in terms of the qualitative descriptors of magnetic anisotropy: "easy-axis" and "easy-plane". They can be based on different physical properties, in casu: free energy, magnetization, or magnetic susceptibility. However, this degree of freedom leads in some cases, including very simple ones like [V(H₂O)₆]³⁺, to incommensurate descriptions of a system being simultaneously easy-axis and easy-plane, depending only on the choice of the physical quantity on which the descriptor is based. Moreover, it has recently been pointed out that the magnetic anisotropy of a chemical system can be addressed and switched using external stimuli like temperature and magnetic field. These external parameters are, though, not the only ones capable of triggering anisotropy switching for actual chemical systems under experimentally relevant conditions. Indeed, this applies also to pressure, as discussed here. In this paper, we try to illustrate the multifaceted nature of magnetic anisotropy and assist the overview using anisotropy phase diagrams.

Crystal environment of impurity Nd3+ ion in yttrium and scandium orthosilicate crystals

The ESEEM spectroscopy was used to determine positions of the ¹⁴³Nd³⁺ impurity ions in Y₂SiO₅ single crystal. It is established that neodymium ions substitute yttrium ions in the Y2 positions with seven nearest oxygen ions. Crystal field parameters of ¹⁴³Nd³⁺ impurity centers in isotopically pure Y₂²⁸SiO₅ and Sc₂²⁸SiO₅ single crystals were determined using data of CW EPR spectroscopy and the known energy level schemes.

Exact Mapping from Many-Spin Hamiltonians to Giant-Spin Hamiltonians

Thermodynamic and spectroscopic data of exchange-coupled molecular spin clusters (e.g. single-molecule magnets) are routinely interpreted in terms of two different models: the many-spin Hamiltonian (MSH) explicitly considers couplings between individual spin centers, while the giant-spin Hamiltonian (GSH) treats the system as a single collective spin. When isotropic exchange coupling is weak, the physical compatibility between both spin Hamiltonian models becomes a serious concern, due to mixing of spin multiplets by local zero-field splitting (ZFS) interactions ('S-mixing'). Until now, this effect, which makes the mapping MSH→GSH ('spin projection') non-trivial, had only been treated perturbationally (up to third order), with obvious limitations. Here, based on exact diagonalization of the MSH, canonical effective Hamiltonian theory is applied to construct a GSH that exactly matches the energies of the relevant (2S+1) states comprising an effective spin multiplet. For comparison, a recently developed strategy for the unique derivation of effective ('pseudospin') Hamiltonians, now routinely employed in ab initio calculations of mononuclear systems, is adapted to the problem of spin projection. Expansion of the zero-field Hamiltonian and the magnetic moment in terms of irreducible tensor operators (or Stevens operators) yields terms of all ranks k (up to k=2S) in the effective spin. Calculations employing published MSH parameters illustrate exact spin projection for the well-investigated [Ni(hmp)(dmb)Cl]₄ ('Ni₄ ') single-molecule magnet, which displays weak isotropic exchange (dmb=3,3-dimethyl-1-butanol, hmp^- is the anion of 2-hydroxymethylpyridine). The performance of the resulting GSH in finite field is assessed in terms of EPR resonances and diabolical points. The large tunnel splitting in the M=± 4 ground doublet of the S=4 multiplet, responsible for fast tunneling in Ni₄ , is attributed to a Stevens operator with eightfold rotational symmetry, marking the first quantification of a k=8 term in a spin cluster. The unique and exact mapping MSH→GSH should be of general importance for weakly-coupled systems; it represents a mandatory ultimate step for comparing theoretical predictions (e.g. from quantum-chemical calculations) to ZFS, hyperfine or g-tensors from spectral fittings.

Analyses of sizable ZFS and magnetic tensors of high spin metallocomplexes

The fictitious spin-1/2 Hamiltonian approach is the putative method to analyze the fine-structure/hyperfine ESR spectra of high spin metallocomplexes having sizable zerofield splitting (ZFS), thus giving salient principal g-values far from around g = 2 without explicitly providing their ZFS parameters in most cases. Indeed, the significant departure of the g-values from g = 2 is indicative of the occurrence of their high spin states, but naturally they never agree with true g-values acquired by quantum chemical calculations such as sophisticated DFT or ab initio MO calculations. In this work, we propose facile approaches to determine the magnetic tensors of high spin metallocomplexes having sizable ZFS, instead of performing advanced high-field/high-frequency ESR spectroscopy. We have revisited analytical expressions for the relationship between effective g-values and true principal g-values for high spins. The useful analytical formulas for the g^eff-g^true relationships are given for S's up to 7/2. The genuine Zeeman perturbation formalism gives the exact solutions for S = 3/2, and for higher S's it is much more accurate than the pseudo-Zeeman perturbation approach documented so far (A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Metal Ions, 1970; J. R. Pilbrow, J. Magn. Reson., 1978, 31, 479; F. Trandafir et al., Appl. Magn. Reson., 2007, 31, 553; M. Fittipaldi et al., J. Phys. Chem. B, 2008, 112, 3859), in which the E(S_x² - S_y²) term is putatively treated to the second order. To show the usefulness of the present approach, we exploit Fe^III(Cl)OEP (S = 5/2) (OEP: 2,3,7,8,12,13,17,18-octaethylporphyrin) and Co^IIOEP (S = 3/2) well magnetically diluted in the diamagnetic host crystal lattice of Ni^IIOEP. The advantage of single-crystal ESR spectroscopy lies in the fact that the molecular information on the principal axes of the magnetic tensors is crucial in comparing with reliable theoretical results. In high spin states of metallocomplexes with sizable ZFS in pseudo-octahedral symmetry, their fine-structure ESR transitions for the principal z-axis orientation appear in the lower field far from g = 2 at the X-band, disagreeing with the putative intuitive picture obtained using relevant ESR spectroscopy. A Re^III,IV dinuclear complex in a mixed valence state exemplifies the cases, whose fine-structure/hyperfine ESR spectra of the neat crystals have been analyzed in their principal-axis system. The DFT-based/ab initio MO calculations of the magnetic tensors for all the high spin entities in this work were carried out.

Modelling spin Hamiltonian parameters of molecular nanomagnets

Molecular nanomagnets encompass a wide range of coordination complexes possessing several potential applications. A formidable challenge in realizing these potential applications lies in controlling the magnetic properties of these clusters. Microscopic spin Hamiltonian (SH) parameters describe the magnetic properties of these clusters, and viable ways to control these SH parameters are highly desirable. Computational tools play a proactive role in this area, where SH parameters such as isotropic exchange interaction (J), anisotropic exchange interaction (Jx, Jy, Jz), double exchange interaction (B), zero-field splitting parameters (D, E) and g-tensors can be computed reliably using X-ray structures. In this feature article, we have attempted to provide a holistic view of the modelling of these SH parameters of molecular magnets. The determination of J includes various class of molecules, from di- and polynuclear Mn complexes to the {3d-Gd}, {Gd-Gd} and {Gd-2p} class of complexes. The estimation of anisotropic exchange coupling includes the exchange between an isotropic metal ion and an orbitally degenerate 3d/4d/5d metal ion. The double-exchange section contains some illustrative examples of mixed valance systems, and the section on the estimation of zfs parameters covers some mononuclear transition metal complexes possessing very large axial zfs parameters. The section on the computation of g-anisotropy exclusively covers studies on mononuclear Dy(III) and Er(III) single-ion magnets. The examples depicted in this article clearly illustrate that computational tools not only aid in interpreting and rationalizing the observed magnetic properties but possess the potential to predict new generation MNMs.

Analysis on the local structures for 3d1 impurities (Ti3+ and V4+ ) in KTiPO4

Making use of the perturbation formulae for 3d¹ ions (Ti³⁺ and V⁴⁺ ) under orthorhombically compressed octahedra, the spin Hamiltonian parameters (g factors: g_x , g_y , g_z and hyperfine structure constants: A_x , A_y , A_z ) and local structures of the 3d¹ impurity centres C₁ , C₂ , and C₃ in KTiOPO₄ crystals are theoretically analyzed in a consistent way. The remarkable local distortions (i.e., the relative axial compression ratios 11.2%, 7.0%, and 5.5% along Z axis and the relative planar bond length variation ratios 15.9%, 7.0%, and 6.0%) are obtained for the [Ti2O₆ ]^9- cluster on Ti2 site and [VO₆ ]^8- clusters on Ti1 and Ti2 sites, respectively, in view of the Jahn-Teller effect. The above local orthorhombic distortion parameters in the impurity centres are found to be more significant than the host Ti1 and Ti2 sites in pure KTiOPO₄ . The sequences (C₁  > C₂  > C₃ ) of the local orthorhombic distortion parameters ρ and τ are in accordance with those of the axial and perpendicular anisotropies Δg and δg of g factors, respectively.

The geometric phase of Zn- and T-symmetric nanomagnets as a classification toolkit

We derive the general form of the non-trivial geometric phase resulting from the unique combination of point group and time reversal symmetries. This phase arises e.g. when a magnetic adatom is adsorbed on a non-magnetic C_n crystal surface, where n denotes the fold of the principal axis. The energetic ordering and the relevant quantum numbers of the eigenstates are entirely determined by this quantity. Moreover, this phase allows to conveniently predict the protection mechanism of any prepared state, shedding light onto a large number of experiments and allowing a classification scheme. Owing to its robustness this geometric phase also has great relevance for a large number of applications in quantum computing, where topologically protected states bearing long relaxation times are highly desired.

Behaviour of DFT-based approaches to the spin–orbit term of zero-field splitting tensors: a case study of metallocomplexes, MIII(acac)3 (M = V, Cr, Mn, Fe and Mo)

Zero-field splitting tensors of M^III(acac)₃ complexes are calculated using ab initio and DFT methods.

SIMPRE1.2: Considering the hyperfine and quadrupolar couplings and the nuclear spin bath decoherence

SIMPRE is a fortran77 code which uses an effective electrostatic model of point charges to predict the magnetic behavior of rare-earth-based mononuclear complexes. In this article, we present SIMPRE1.2, which now takes into account two further phenomena. First, SIMPRE now considers the hyperfine and quadrupolar interactions within the rare-earth ion, resulting in a more complete and realistic set of energy levels and wave functions. Second, and to widen SIMPRE's predictive capabilities regarding potential molecular spin qubits, it now includes a routine that calculates an upper-bound estimate of the decoherence time considering only the dipolar coupling between the electron spin and the surrounding nuclear spin bath. Additionally, SIMPRE now allows the user to introduce the crystal field parameters manually. Thus, we are able to demonstrate the new features using as examples (i) a Gd-based mononuclear complex known for its properties both as a single ion magnet and as a coherent qubit and (ii) an Er-based mononuclear complex. © 2016 Wiley Periodicals, Inc.

Ab initio calculations as a quantitative tool in the inelastic neutron scattering study of a single-molecule magnet analogue

Ab initio calculations of the electronic structure of Na₉[Tb(W₅O₁₈)₂] have allowed interpretation of the inelastic neutron scattering spectra, revealing the subtle relationship between molecular geometry and magnetic properties of two isostructural species.

Measuring giant anisotropy in paramagnetic transition metal complexes with relevance to single-ion magnetism

“Giant magnetic anisotropy” is a phenomenon identified in certain coordination complexes of nd- and nf-block ions. The strengths and weaknesses of multiple methods used to measure it are evaluated.

CW-EPR Spectral Simulations: Solid State

This chapter summarizes the core concepts underlying the simulation of EPR spectra from biological samples in the solid state, from a user perspective. The key choices and decisions that have to be made by a user when simulating an experimental EPR spectrum are outlined. These include: the choice of the simulation model (the network of spins and the associated spin Hamiltonian), the dynamic regime (solid, liquid, slow motion), the level of theory used in the simulation (matrix diagonalization, perturbation theory, etc.), the treatment of orientational order and disorder (powder, crystal, partial ordering), the inclusion of the effects of structural disorder (strains), the effects of other line broadening mechanisms (unresolved hyperfine couplings, relaxation), and the inclusion of experimental distortions (field modulation, power saturation, filtering). Additionally, the salient aspects of utilizing least-squares fitting algorithms to aid the analysis of experimental spectra with the help of simulations are outlined. Although drawing from the experience gained from implementing EasySpin and from interacting with EasySpin's user base, this chapter applies to any EPR simulation software.

Magnetic properties of a Kramers doublet. An univocal bridge between experimental results and theoretical predictions

The magnetic response of a Kramers doublet is analyzed in a general case taking into account only the formal properties derived from time reversal operation. It leads to a definition of a matrix G (gyromagnetic matrix) whose expression depends on the chosen reference frame and on the Kramers conjugate basis used to describe the physical system. It is shown that there exists a reference frame and a suitable Kramers conjugate basis that gives a diagonal form for the G-matrix with all non-null elements having the same sign. A detailed procedure for obtaining this canonical expression of G is presented when the electronic structure of the KD is known regardless the level of the used theory. This procedure provides a univocal way to compare the theoretical predictions with the experimental results obtained from a complete set of magnetic experiments. In this way the problems arising from ambiguities in the g-tensor definition are overcome. This procedure is extended to find a spin-Hamiltonian suitable for describing the magnetic behavior of a pair of weakly coupled Kramers systems in the multispin scheme when the interaction between the two moieties as well as the individual Zeeman interaction are small enough as compared with ligand field splitting. Explicit relations between the physical interaction and the parameters of such a spin-Hamiltonian are also obtained.

EMR studies of the internal motion of Mn(4+) ions in the Sr overdoped (La(1-x)Sr(x))(Ga(1-y)Mn(y))O3 (x/y up to 8) supplemented by magnetic and optical spectroscopy measurements

The effect of the Sr doping on electronic structure in single crystals of (La(1-x)Sr(x))(Ga(1-y)Mn(y))O3 solid solutions (LSGM) is investigated by means of electron magnetic resonance (EMR). The EMR results are supplemented by magnetic susceptibility and optical spectroscopy measurements. The compositions with small concentration of Mn doping (y<1%) and overdoped content of Sr (the ratio x(Sr)/y(Mn) up to 8) are used to maximally enhance the role of divalent doping. The experimental results provide evidence of the holes delocalization in the overdoped compound (x(Sr)/y(Mn)>1). This delocalization is accompanied by appearance of the new charge transfer transitions in the optical spectrum and dynamical valence change of manganese atoms. Additionally we observe the thermally activated narrowing of resonance EMR lines due to the internal motion, which is characterized by the energy barrier depending strongly on the ratio x(Sr)/y(Mn). The energy barrier is found to be associated with the charge carrier (hole) self-trapped energy. Fitting the EMR spectra in three orthogonal planes to an orthorhombic spin Hamiltonian enables extracting the zero-field splitting (ZFS) parameters and the Zeeman g-factors for Mn(4+) (S=3/2) ions in LSGM. The experimental ZFS parameters are modeled using superposition model analysis based on an orthorhombic symmetry approximation.

Improving f-element single molecule magnets

Historical developments, trends, pitfalls and strategies in improving f-element single molecule magnets are described.