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

Accepted Manuscript Title: Disentangling intricate web of interrelated notions at the interface between the physical (crystal field) Hamiltonians and the effective (spin) Hamiltonians Author: Czesław Rudowicz Mirosław Karbowiak PII: DOI: Reference:

S0010-8545(14)00339-7 http://dx.doi.org/doi:10.1016/j.ccr.2014.12.006 CCR 111972

To appear in:

Coordination Chemistry Reviews

Received date: Revised date: Accepted date:

28-8-2014 26-11-2014 8-12-2014

Please cite this article as: C. Rudowicz, M. Karbowiak, Disentangling intricate web of interrelated notions at the interface between the physical (crystal field) Hamiltonians and the effective (spin) Hamiltonians, Coordination Chemistry Reviews (2014), http://dx.doi.org/10.1016/j.ccr.2014.12.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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

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Czesław Rudowicz1* and Mirosław Karbowiak2 1 Institute of Physics, West Pomeranian University of Technology, Al. Piastów 17, 70–310 Szczecin, Poland 2 Faculty of Chemistry, University of Wrocław, ul. F. Joliot-Curie 14, 50-383 Wrocław, Poland * corresponding author: Tel.:+48 91 449-45-85, Fax:+48 91 449-41-81; E-mail address: [email protected]

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Contents 1. Introduction 2. Physical Hamiltonians for single transition ions 2.1. Free-ion Hamiltonians for transition ions 2.2. Hamiltonians for single transition ions in crystals 2.3. Crystal field (CF) and ligand field (LF) Hamiltonians 2.4. Quenching of the orbital angular momentum by the crystal field 2.5. Rare-earth ions with the Russell-Saunders ground multiplet 2.6. General comments concerning the CF (LF) effects and Hamiltonians 3. Effective Hamiltonians for single transition ions 3.1. Spin operators for single transition ions 3.1.1. True electronic spin 3.1.2. Effective spin 3.1.3. Fictitious 'spin' 3.2. Effective spin Hamiltonians for single transition ions 3.2.1. Generic „spin‟ Hamiltonians 3.2.2. Effective single-ion spin Hamiltonians 3.2.3.Microscopic SH (MSH) theory vs the generalized SH (GSH) theory 3.3. General comments concerning the effective single-ion spin Hamiltonians 4. Exchange coupled systems (ECS) of transition ions and single molecule magnets 4.1. Types of exchange interactions and Hamiltonians 4.2. Multispin (microscopic spin) Hamiltonians for ECS 4.3. Effective total (giant) spin Hamiltonians for ECS 4.4. Relationships between multispin Hamiltonians and giant spin Hamiltonians 4.5. General comments concerning Hamiltonians for ECS 5. Stevens, Wybourne, and other operators 5.1. Historical perspective and origin of the Stevens and Wybourne operators 5.2. Usual Stevens operators versus the extended Stevens operators (ESO) 5.3. Adoption of the Stevens operators and other notations in EMR studies 5.4. Hamiltonians versus operators 6. Forms of Hamiltonians and definitions of the associated parameters 6.1. Crystal field (ligand field) Hamiltonians 6.1.1. Forms of HCF (HLF) and the associated parameters 6.1.2. Relations between the CF (LF) parameters expressed in the Stevens and Wybourne notations 6.1.3. General comments concerning the forms of HCF (HLF) 6.2. Spin Hamiltonians and zero-field splitting (ZFS) Hamiltonians ~ ~ 6.2.1. Forms of H SH ( H ZFS ) and the associated parameters 6.2.2. Relations between the ZFS parameters expressed in the ESO and conventional notations 6.3. Higher-order terms in the generalized spin Hamiltonians (GSH) 7. Distinctions and interrelationships between the CF (LF) and SH (ZFS) quantities ~ ~ 7.1. Distinct physical nature of HCF (HLF) and H SH ( H ZFS ) 7.2. Interrelationships between the CF (LF) and SH (ZFS) parameters

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7.3. General comments concerning the Hamiltonians HCF (HLF), H SH ( H ZFS ), and the confusion of the type CF=ZFS and ZFS=CF 8. Current status of applications of the (extended) Stevens operators in recent literature 8.1. Importance of the ESOs 8.2. General comments concerning the usage of the ESOs 9. Generalized definitions of the full and restricted Hamiltonians versus the effective and fictitious ones 10. Conclusions and outlook Acknowledgments List of symbols and abbreviations Appendices References

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ABSTRACT

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This review provides a summary of distinct Hamiltonians used to describe magnetic and spectroscopic properties of paramagnetic and magnetic coordination compounds. Based on the origins and the underlying physical principles, clear recommendations are formulated on when these Hamiltonians are appropriate and how they relate to each other. The interface, denoted CF (LF)  SH (ZFS), encompasses the physical Hamiltonians, which include the crystal field (CF) [or equivalently ligand field (LF)] Hamiltonians, and the effective spin Hamiltonians (SH), which include the zero-field splitting (ZFS) Hamiltonians, as well as, to a certain extent, also the notion of magnetic anisotropy (MA). Survey of recent literature has revealed that the intricate web of interrelated notions has become dangerously entangled over the years. A given crucial notion is often referred to by one of the three names that are not synonymous: CF (LF), SH (ZFS), or MA, each having a well-defined and established meaning in optical spectroscopy, electron magnetic resonance (EMR), and magnetism, respectively. The terminological confusions occurring in literature call for in-depth clarifications, which are provided in this review. For this purpose, crucial notions are systematically defined and their logical interrelationships illustrated by concept maps. The operator types used to express the CF (LF) and SH (ZFS) Hamiltonians, i.e. the Stevens and Wybourne ones, are classified and their distinct properties discussed. Several key aspects are considered in the nutshell: basic forms of Hamiltonians and definitions of the associated parameters, distinct properties of the Stevens and Wybourne CF (LF) parameters and implications for conversion relations, distinctions and interrelationships between the CF (LF) and SH (ZFS) Hamiltonians or parameters, conversion relations between the Stevens ZFS parameters and conventional ones. The general focus is on the fundamental aspects underlying physics and chemistry of single transition (4fN and 3dN) ions in coordination compounds as well as the novel single-ion and polynuclear magnetic systems. This includes the single-ion magnets and the exchange coupled systems (ECS) based on transition ions, especially the single molecule magnets (SMM) or molecular nanomagnets (MNM). The general aim is to provide deeper understanding of the major intricacies involved in the CF (LF)  SH (ZFS) interface. The level of presentation is geared towards experimentalists with background in chemistry or physics. Keywords: Electron magnetic resonance (EMR); Optical spectroscopy; Magnetism; Crystal/Ligand field (CF/LF) Hamiltonian; Zero-field splitting (ZFS); Single molecule magnets (SMM). 1. Introduction This review provides background for the distinct types of Hamiltonians and related quantities that underlie interpretation of magnetic and spectroscopic properties of the paramagnetic and magnetic coordination compounds based on the transition metal (TM), 3dN, and rare-earth (RE), 4fN, ions, i.e. collectively, the transition ions. The single transition ions in various crystals or molecules, including the single-ion magnets, as well as the polynuclear magnetic systems, including the exchange coupled systems (ECS) based on transition ions, especially the single molecule magnets (SMM) or molecular nanomagnets (MNM), may serve as examples of such systems. The general focus is on the

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fundamental aspects underlying physics and chemistry of single transition ions in coordination compounds as well as the novel single-ion and polynuclear magnetic systems, which have been extensively studied in recent decades. Hence, we shall not discuss in details the topical content of pertinent publications or provide critical assessment of their content. Instead we refer the readers to specialized reviews as indicated whenever appropriate. The two major types of Hamiltonians are the physical Hamiltonians, which include the crystal field (CF) [or equivalently ligand field (LF)] Hamiltonians, denoted HCF (HLF), and the effective spin ~ Hamiltonians (SH), denoted H~ eff  H SH , which include as the major term the zero-field splitting ~

(ZFS) [or equivalently „fine structure‟] Hamiltonians, denoted H ZFS , as well as the Zeeman electronic ~

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(Ze) ones, H Ze , and some higher-order terms. The physical free ion Hamiltonians together with the CF (LF) Hamiltonians are fundamental in optical spectroscopy [1,2,3,4,5,6,7,8,9,10,11], whereas the effective SH (ZFS) Hamiltonians suitable for transition ions in crystals or their clusters are fundamental in electron magnetic resonance (EMR) spectroscopy [12,13,14,15,16,17,18]. Note that EMR is a more general name that encompasses nowadays electron paramagnetic resonance (EPR), electron spin resonance (ESR), and other related techniques, for which the underlying concept is the spin Hamiltonian [12-18]. Together with the notion of magnetic anisotropy (MA), various types of spin Hamiltonians and CF (LF) Hamiltonians are fundamental in magnetism of transition metal and rare earth ions [19,20,21,22,23,24, 25,26,27]. There are two major definitions of the word 'interface': (a) the notions, problems, and theories shared by two or more related areas of study form a specific interface, e.g. the interface between chemistry and physics, and (b) the 'interface' is also defined as a 'surface' regarded as the common boundary of two bodies or phases. In view of these commonly accepted definitions, the word 'interface' may encompass the notions, problems, and theories shared by the three areas, namely, optical spectroscopy, EMR, and magnetism. In this review, the 'interface' is specifically defined as a 'surface' regarded as the common boundary of the two distinct entities, i.e. the physical crystal (ligand) field Hamiltonians and the effective spin (ZFS) Hamiltonians. This interface is denoted for short as: CF (LF)  SH (ZFS), by referring to the major Hamiltonians involved. The CF (LF)  SH (ZFS) interface permeates vast areas of research, which include the emerging fields of, e.g. molecular magnetism and spintronics as well as the well established ones, e.g. optoelectronics, laser and magnetic materials, biological systems. Survey of recent literature has revealed that the CF (LF)  SH (ZFS) interface forms nowadays an intricate web of interrelated notions, which has become dangerously entangled over the years. A variety of conceptual problems has been identified at this interface. Our analysis indicates that these problems arise from considerable muddling of crucial notions, especially the CF parameters (CFPs) and the ZFS parameters (ZFSPs). Serious terminological confusions have lead to detrimental consequences, including pitfalls that bear on understanding of physical properties of magnetic systems. This situation calls for in-depth clarifications. Hence, the main rationale behind this review is the disentanglement of this web for the benefit of junior researchers, who on average, may be less familiar with the theoretical foundations of optical and EMR spectroscopy, and magnetism. For this purpose, the crucial notions and aspects concerning the CF (LF), SH (ZFS), and related quantities are presented in the nutshell to serve as a compendium for easy reference. This review concentrates on the two main groups of transition ions: (i) the RE ions with the ground electronic configuration [Xe]4fN (N = 0 - 14) and (ii) the TM ions with the configuration [Ar]3dN (N = 0 - 10). The adopted definitions are based on the prevailing presentations in the main textbooks [1-23] and the general reviews [28,29,30,31,32]. In view of the sheer amount of pertinent literature accumulated in the last two decades, it is hardly possible to provide an exhaustive coverage. Hence, we concentrate on the specialized reviews dealing with the RE-based compounds and the ECS of TM ions, which set the tone for nomenclature in the area of EMR and SMM (NMM). To facilitate subsequent considerations, the following general definition of confusion between two distinct notions: A and B (each being well-defined and predominantly established in a specific area), is adopted. The confusion of the type denoted A=B is defined as the cases of incorrect referral to the quantities associated with one notion B (e.g., ZFS) by the name A (e.g. CF). The quantities may be, e.g., effects, Hamiltonians, eigenfunctions, parameters, or energy level splitting. The CF=ZFS

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confusion, which pertains to the cases of labeling the true ZFS quantities as purportedly the CF (LF) quantities, has been discussed in the reviews [28-32] and most recently in [33]. The inverse ZFS=CF confusion, which pertains to the cases of labeling the true ZFS quantities as purportedly the CF (LF) quantities, has been discussed in [34]. The description of magnetic properties of the TM- or RE-based systems involves also the notion of magnetic anisotropy (MA), see, e.g. Refs [19-27]. The key observation is that a given crucial notion is often referred to by one of three names that are not synonymous: CF (LF), SH (ZFS), or MA. Since two of three names must be incorrectly used, such cases constitute confusion of the type: CF=ZFS, ZFS=CF, or MA=ZFS as well as in some cases a compounded confusion: MA=CF/LF=ZFS. Such confusions are unacceptable since each notion has a well-defined meaning in one of the major areas, i.e. optical spectroscopy, EMR, and magnetism, respectively. However, in view of the huge scope, terminological confusions related to the MA notion require separate consideration. In subsequent sections pertinent MA-related problems are pointed out only if they occur in the surveyed literature. The surveys [33,34] strongly indicate the need for systematization of nomenclature aimed at bringing order to the zoo of different Hamiltonians and the associated quantities. In this review we have embarked on this systematization with the aim to provide deeper understanding of the major intricacies identified at the CF (LF)  SH (ZFS) interface. In view of the conceptual problems and their consequences [33,34], we provide the basic definitions of the crucial notions underlying the three main research areas as well as clarify the logical interrelationships between these notions. A lucid and accessible overview of the nature, origin, and correct usage of the different relevant Hamiltonians and the associated wavefunctions would be of great benefit to researchers starting to work in these areas. It is not feasible to provide in a single review a comprehensive coverage of all pertinent topics. For a primer, it should be sufficient to concentrate on main aspects and deal with the most usual cases, i.e. ZFS of 3d ions with the quenched orbital angular momentum L, ZFS of exchange coupled 3d clusters in the giant spin and microscopic SH models, and CF splitting in 4f ions considering the Russell-Saunders ground multiplet. Specialist terms and symbols are defined and fundamental ideas are simply explained, while striving to keep the overall presentation jargon free. The level of presentation is geared towards experimentalists with background in chemistry or physics. Some general remarks on the presentation scheme adopted in this review are pertinent. The definitions of the crucial notions are introduced at a conceptual level in Section 2 to 4 to serve as signposts in subsequent sections. The 'equations' for the respective Hamiltonians provided in these Sections are not numbered. The symbol „≡‟ indicates the conceptual meaning of the Hamiltonians' definitions. Subsequently, these Hamiltonians are referred to by the defining symbols, whereas their listing is provided at the end. The most important aspects arising from the conceptual definitions are systematically presented in the consecutive subsections. To improve the readability of the text and avoid interruption of its continuity, all pertinent comments concerning the usage of particular terminology in the surveyed literature are collected in final subsections explicitly named in the Contents as 'general comments'. These subsections serve also as additional sources of relevant references for further reading and may be skipped on first reading. With the hindsight of the preceding Sections, the generalized definitions of the pertinent Hamiltonians are summarized in Section 9 from a global perspective. Then the distinctions between (i) the full and restricted forms of the physical CF (LF) Hamiltonians (Section 2), (ii) the effective Hamiltonians (Sections 3 and 4), and (iii) the fictitious ones (Section 3) are discussed in more details to make these notions clearer. 2. Physical Hamiltonians for single transition ions For better understanding of the crucial notions pertinent for the CF (LF)  SH (ZFS) interface for single transition ions, a concept map is provided in Fig. 1, followed by general definitions of the notions. Hamiltonians (H), associated energy levels and wavefunctions {}, which are involved in the free ion (FI), CF (LF), and SH (ZFS) theory for the transitions 3d N and 4fN ions, are indicated. The two routes for obtaining the (effective) SH introduced in Fig. 1, namely, the „derivational‟ one and the „constructional‟ one, are discussed in details in Section 3. Additional visualizations of the CF (LF)  SH (ZFS) interface as well as the interrelationships and distinctions between the pertinent notions are discussed in details in Sections 6 and 7.

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2.1. Free-ion Hamiltonians for transition ions

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The free-ion (FI) Hamiltonians for transition ions are conceptually defined as: HFI ≡ (HK + Hes) + (HSO + HSS) + HZe + {additional FI terms for 4fN and 3dN ions}. These Hamiltonians describe all physical interactions between the unpaired electrons of a free transition ion. The terms denote: the orbital part: Horb = HK + Hes, i.e. the kinetic energy of electrons, HK, and the electrostatic (es) Coulomb interactions (between electrons and nucleus and electronelectron interactions), Hes, the electronic spin dependent part: HSO + HSS, i.e. the electronic spin-orbit (SO) coupling (SOC), HSO, and the electronic spin-spin (SS) coupling, HSS, as well as the Zeeman electronic (Ze) interaction, HZe. The explicit forms of HFI vary for the 4fN and 3dN ions, and may include a plethora of other more sophisticated terms, e.g. relativistic, FI terms [21], discussion of which is beyond the level suitable for the intended readership. Note that these terms do not bear on the conceptual picture presented below. In the first step, Horb may be solved in the Hartree-Fock and central self-consistent field (HFCSCF) approximation, which yields the hydrogen-like one electron wavefunctions:  i (ri )  f nili (ri )Y i mi ( i ,i ) for the i-th electron, where f nili (ri ) is the modified radial part and Yli mi ( i , i ) is

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HF the orbital part, expressed in terms of the spherical harmonics. The solutions of each H CSCF (i) correspond to the degenerate energy levels described by the principal quantum number ni, like in the hydrogen atom, and the orbital quantum number  i . The set of quantum numbers {ni,  i } yields the

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energy: E   Eni i called the configuration nlN. In the second step, the difference between the full i 1

HF HF Horb and the approximated H CSCF , i.e. the residual electrostatic repulsion Hˆ es  ( Hˆ orb - H CSCF ) is

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taken into account using the Slater determinant wavefunctions. The matrix elements of Hˆ es within these wavefunctions can be factorized by some radial integrals, either the Slater parameters, or the Condon-Shortley ones, or the Racah parameters, e.g. for dN: (A, B, C). These integrals are rather not calculated but determined from optical spectroscopy of free ions. The second step yields the electronic energy levels as the 2S+1L terms, which are further split into 2S+1LJ multiplets by the spin-orbit interacion (S, L, and J are the quantum numbers of the electronic spin S, orbital L, and total J ≡ (L + S) angular momentum operators, respectively). Magnetic properties of transition ions are often determined, to a good approximation, by the lowest 2S+1LJ multiplet for 4fN ions and 2S+1L term for 3dN ions, whereas for description of spectroscopic properties in the visible range the whole configuration nlN must be considered. 2.2. Hamiltonians for single transition ions in crystals The total physical Hamiltonians for transition ions in crystals are conceptually defined as: Hphys ≡ HFI + HCF (HLF) + {other more sophisticated CF terms for 4fN and 3dN ions}. These Hamiltonians describe all interactions included in HFI and additionally the Hamiltonians that describe the physical interactions between the electrons of a paramagnetic transition ion and the surrounding diamagnetic ions, i.e. ligands, in crystals. As reviewed in [28,29], the name 'physical' has been often used in this context in literature. Moreover, this name is fully justified since it reflects the fact that such Hamiltonians describe the physical interactions and need to be distinguished from the effective (spin) Hamiltonians discussed in Section 3, which do not describe any physical interactions. Historically, the Hamiltonians HCF (HLF) were originally introduced by the name crystal field (CF) Hamiltonians and later become alternatively referred to by the equivalent name the ligand field (LF) ones [1-11]. A succinct overview of the development of the CF and LF theory is presented in [35]. The Hamiltonians, HCF (HLF), parameterize the effect of the electric field due to the surrounding n ligands (L) acting on a paramagnetic ion (M) in a given MLn complex in crystal or in a molecule (see, Fig. 2). For the 4fN and 3dN ions, HCF (HLF) may include other more sophisticated terms, e.g. relativistic, two-electron correlation CF, or spin-correlated CF terms [7,8,10,11], discussion of which

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is beyond the level suitable for the intended readership. Note that these terms do not bear on the conceptual picture presented below. The general forms of HCF (HLF) are discussed in Section 6.1. The relative strength of the SO and CF interactions, i.e. their effect on the energy levels splitting, differs for the 4fN and 3dN ions. This strength determines the most suitable scheme for construction of the basis of states, in which the total physical Hamiltonians may be conveniently solved. Diagonalization of Hphys yields the splitting of the electronic multiplets of the free 4f N and 3dN ions into levels, which states can be described, to a certain approximation, by the total J or orbital L angular momentum, respectively. Three schemes are available that enable taking into account all interactions included in Hphys together with HCF (HLF), namely, the strong, intermediate, and weak CF scheme [1-11]. For the 3dN ions, the intermediate (note that some authors use confusingly the name 'weak' instead) CF scheme applies since the effect of HCF is stronger than or comparable to that of HSO. Hence, first the energy levels described by the atomic orbitals are split due to the interelectron repulsion into the atomic terms 2S+1L (|L,S,ML,MS>) and then due to HCF into the CF terms 2S+1 (|,S,MS>). These CF energy levels may be classified according to the transformation properties of the respective states, which are determined by the irreducible representations (irreps)  of a given point-symmetry group (PSG) G (see, e.g. [36]) that describes the symmetry of the MLn complex and ' hence the symmetry of HCF. Next, taking into account HSO yields the CF multiplets,  (|(,S),‟>), classified by the double-valued irreps  of the double group G' of a given PSG G [36]. For the 4fN ions, the weak CF scheme applies since the effect of HCF is weaker than that of HSO. Thus HCF is considered as perturbation on the electronic levels of the free ion arising due to the interelectron repulsion and HSO. Hence, the first interaction yields the terms 2S+1L, which are then split due to HSO into the multiplets 2S+1LJ and subsequently due to HCF into the CF components (or Stark levels). For integer [half-integer] J one obtains (2J+1) [(J+1/2)] CF energy levels classified by the ' irreps  [  ] of G [G']. Note that if the effect of HCF is comparable to that of HSO, as for some 3dN ions, then alternatively HSO may be considered before HCF. To distinguish the resulting energy levels, Boča [21,23] proposed to employ the name „term‟ (e.g. CF terms, atomic terms) for levels obtained before inclusion of HSO, whereas the name 'multiplets' (e.g. CF multiplets, atomic multiplets) after inclusion of HCF. While the names 'atomic terms' (or 'free-ion terms') and 'multiplets' are generally accepted, in the case of the energy levels obtained due to HCF more commonly the names 'CF energy levels' ('CF components' or 'Stark levels') are adopted. This terminology is followed here, since it is most pertinent for the 4f N ions, for which HSO is always considered before HCF and hence such distinction is irrelevant.

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ip t cr us an M d Ac ce pt e Fig. 1. Concept map for the notions pertinent for single transition ions (M) in the MLn complex of n ligands (L); adapted from [32,37]. The abbreviations used stand for: HF-CSCF: Hartree-Fock and central self-consistent field approximation; LSS: local site symmetry, PSG: point-symmetry group, SO: spin-orbit coupling, SS: electronic spin-spin coupling; B: the magnetic induction; E: electric field intensity; I: nuclear spin.

2.3. Crystal field (CF) and ligand field (LF) Hamiltonians

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The nature of the CF (LF) as a phenomenon is visualized in Fig. 2 (upper part), whereas the schematic representation of energy levels and wavefunctions for a single 3d1 ion as well as historical origin of the cubic CF splitting are depicted in Fig. 2 (lower part). The distinction between the energy levels studied in the ultraviolet to visible range (UV-Vis) and those studied by EMR (EPR) is also illustrated to provide background for quantitative discussion of the CF (LF)  SH (ZFS) interface in subsequent sections.

Fig. 2. Visualization of the notion CF (LF) pertinent for single transition ions in crystals and molecules.

Since the free-ion 2S+1L(3dN) terms are split by HCF into the CF energy levels labeled by irreps  of a given PSG G [36], in general, the ground state of a transition-metal ion in crystal may be either an orbital singlet (denoted by irrep A or B), orbital doublet (E), or an orbital triplet (T). For example, in the case of Co2+ ion the ground state may be, depending on the site symmetry, either 4 A2(F) [4A2g(F)] or 4T1(F) [4T1g(F)] in the high-spin S = 3/2 configuration, whereas 1A1 in the low-spin S = ½ configuration. The energy levels of the orbital singlet ground states {|0>|S, MS>}, where |0> represents the orbital part and |S, MS> the electronic spin part, are labeled by irreps 2S+10. The higher lying states denoted {|>|S, MS>} are represented by suitable linear combinations of the orbital and 2S+1

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spin wavefunctions of the form { aMj L |L, ML>|S, MS>}, which transform according to a given irrep .

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Hence the higher lying energy levels are labeled by 2S+1. The ground level 2S+10 as well as the higher lying levels 2S+1 are split due to HSO into the SO energy levels, i.e. the CF multiplets, which ' are labeled by the double-valued irreps  . Examples of the ground and excited orbital states as well as the corresponding energy levels arising from Hphys are discussed in more details in Section 7. The left areas in Fig. 5 and 6 denoted “Total physical Hamiltonian” may be consulted for visualization of the action of HCF (HLF) on the schematically depicted energy levels for 3d2 (e.g., Ti2+, V3+) or 3d8 (e.g., Ni2+, Cu3+) ions and the S-state 4f7 (e.g., Gd3+, Eu2+) ions, respectively. The transitions between the spin levels within the orbital singlet ground state split due to the combined action of the CF (LF) and the SO coupling (and, to a lesser extent, the electronic SS coupling) are observed using EMR techniques, whereas those between the ground states and the states belonging to the higher lying multiplets are observed using optical spectroscopy techniques. In advance, we mention that to describe only the former transitions, without being bothered with the higher lying levels and the orbital parts of the wavefunctions, the concept of the effective single-ion ~ spin Hamiltonians H~ e f f  H SH has been introduced in literature. The concept of effective

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Hamiltonians is defined in details for the two major cases, i.e. (i) single transition ions and (ii) ECS of transition ions, in Section 3 and 4, respectively. To facilitate subsequent discussions, here we only state that the central issue in the two cases is the transition from a physical Hamiltonian, where parameters have a physical meaning, to an effective Hamiltonian, which reproduces the energies of the ground state and which allows for easy comparison of different systems, but where the physical origin of the splitting is apparently lost (see Fig. 5 and Fig. 6). The ground state considered in the case (i) and (ii) is the ground orbital singlet and the lowest total spin ST -multiplet, respectively. Importantly, Hphys acts within the basis of the physical states of the whole configuration nlN, including the orbital singlet {|0>|S, MS>}. However, the resulting energy levels (and the corresponding states) are often named as the CF levels (states), although in fact these quantities are due to the combined action of Hphys = HFI + HCF. For experimentalists the relationship between the observed spectra and theoretical energy levels is of prime importance. However, detailed discussion of these aspects is beyond the level suitable for the intended readership; the textbooks [1-11] may be consulted in this regard. For the 4fN ions the free-ion 2S+1LJ multiplets are split by HCF into the energy levels, which can be labeled by the quantum numbers MJ, although in fact the corresponding states are represented by suitable linear combinations of the |J, MJ> states. An axial CF splits each 2S+1LJ multiplet into (J + 1/2) Kramers‟ doublets for the 4fN ions with an odd number of electrons, whereas most of these states are doublets |J, MJ>, but at least one state is a singlet |J, MJ = 0>, for the 4fN ions with an even number of electrons. In a number of ion-host systems, the 3dN ions as well as the S-state 4f7 ions have an orbital singlet ground state {|0>|S, MS>}, hence its degeneracy is due only to the spin degeneracy (2S + 1). 2.4. Quenching of the orbital angular momentum by the crystal field The phenomenon of "quenching" of the orbital angular momentum L by the CF (LF) for TM ions in crystals refers to the experimentally observed fact that there is no orbital contribution, i.e. coming from L, to the magnetic moment  of TM ions in some crystals. This phenomenon occurs provided that the ground state of TM ion is a pure orbital singlet state, i.e. exhibits no orbital degeneracy. For these cases, the only contribution to the magnetic moment  comes from the spin angular momentum S. This quenching is a mathematical consequence of the fact that for TM ions in crystals the matrix elements of L: <0|L|0>, are exactly zero within any orbital singlet (ground or excited) state, which is given by the non-degenerate orbital |0> and the (2S+1)-degenerate spin wavefunctions {|S,MS>}. The general relation <0|L|0>  0 can be proved by group theory methods [36] and is responsible for the physical consequences of "quenching" of the orbital angular momentum that is observed for ions with an orbital singlet ground state {|0>|S, MS>}. The role of the quenching of the orbital angular momentum within the ground orbital singlet state in the emergence of the effective spin

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2.5. Rare-earth ions with the Russell-Saunders ground multiplet

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Hamiltonians for single transition ions is indicated in Fig. 1, whereas Fig. 5 and 6 (presented later) may be consulted for additional visualization of this emergence at the CF (LF)  SH (ZFS) interface. Note that the erroneous interpretations of the quenching of the orbital angular momentum L appear in some papers, e.g. quote:38 „Fe3+ belongs to d5 configuration with 6S (S = 5/2) as the ground state in the free ion, and no spin–orbit interaction (L = 0) exists„. In fact, the spin-orbit coupling does exist for all transition ions, including the S-state Fe3+(d5) ions. However, the orbital angular momentum L is quenched since the ground multiplet 2S+1SS is already given by the orbital singlet states {|0>|S, MS>} even in the absence of any CF. The spin-orbit coupling is, in fact, responsible for the (usually small) ZFS of the S-state d5 ions.

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For the 4fN ions the energy levels arising due to splitting of the 2S+1LJ free-ion multiplets by HCF may be calculated by full diagonalization of the total physical Hamiltonian Hphys. In the early days of development of CF theory, the interaction of RE ion with the electric field of ligands (see, Fig, 2), in fact only the non-spherical part of this interaction, was considered as a small perturbation of the whole configuration of the free 4fN ion. Calculations were carried out in two stages. First the matrix involving the free-ion interactions was constructed within the basis of |LS> states and diagonalized, yielding as the energy levels the free-ion multiplets. In the second stage, the CF interaction was considered thus yielding the energy levels corresponding to the |JMJ> wavefunctions obtained as linear combinations of the |LS> states obtained in the first stage. In view of the then available computers, this methodology was computationally more convenient than that employed at present, which is based on simultaneous diagonalization of the total physical Hamiltonian Hphys. Importantly, the former methodology could lead to incorrect results in the case of strong CF interaction. Moreover, the values of the free-ion parameters obtained in the first stage from analysis of the free-ion energy levels (without taking into account CF) may significantly differ from those obtained based on full diagonalization of Hphys, which pertain to the free-ion parameters suitable for an ion in crystal. In the latter case, all |LS> states are utilized to construct complete set of the |(LS)JMJ> states and the mixing of states with different J-values due to CF terms is taken into account (so-called J- mixing). Calculations within the basis of |(LS)JMJ> states require diagonalization of very large matrices, especially for the configurations f6 to f8 which nowadays are routinely achievable. Nevertheless, in practice, calculations are carried out in a judiciously selected sub-basis of states, since neglecting the states with very high energy has negligible effect on the states that determine the quantities experimentally measurable, e.g. using optical spectroscopy or magnetic measurements. Analysis of optical spectroscopy data aimed at determination of CFPs is usually carried out utilizing the basis of experimentally accessible |(LS)JMJ> states, i.e. in the case of classical UV-Vis spectroscopy - for energies below 50,000 cm-1. In practice, the states with energies higher by about 10,000 - 20,000 cm-1 than the highest lying experimentally determined state are taken into account in theoretical analysis. For the f-electron elements the splitting of the ground 2S+1LJ multiplet by CF reaches energies of about hundreds of cm-1, hence their magnetic properties are determined mainly by the corresponding energy levels and wavefunctions. Moreover, the excited states arising from higher lying 2S+1LJ multiplets are well separated in energy from the ground states. These facts justify the so-called Russell-Saunders approximation, i.e. replacing Hphys by HCF (HLF) and carrying out calculations only within the ground 2S+1LJ multiplet. This approach is equivalent to assumption of the pure LS coupling scheme, i.e. neglecting the admixture of higher lying |LS> states and the J- mixing. Since within the ground multiplet J is the good quantum number, the full form of HCF(4fN), which acts within the whole configuration 4fN, may be replaced by a simpler, albeit restricted, form HCF(J), which acts only within the states of the ground 2S+1LJ multiplet. The respective HCF forms are presented in Section 6.1. This procedure is based on the historical Stevens' operator-equivalent method and thus the form HCF(J) is customarily expressed in terms of the Stevens operators discussed in details in Section 5. In principle, the Hamiltonians HCF(J) could be expressed in terms of any arbitrarily defined operators (see, Section 5). The Russell-Saunders approximation enables simpler calculations based on the restricted HCF(J) form, however, it is mostly acceptable only for description of magnetic properties of 4fN ions, where consideration of the effect of CF interaction on the ground J-multiplet is sufficient. Note that, in

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general, for 4fN ions the CF terms with the operators of rank k equal 2, 4, and 6 (see, Section 6.1) are admissible. However, the ion Ce3+ presents a special case since the splitting of its ground multiplet 2 F5/2 depends only on the rank k = 2 and 4 CF terms and may be described by the restricted HCF(J) form that does not involve the sixth-rank CF terms. In general, the name 'restricted' Hamiltonian may be employed for a specific form of Hphys (or a given part of Hphys) that instead of its full basis of states acts only within a well-defined selected subbasis of states of Hphys. The case of HCF discussed in this section is supplemented in Section 9 by other pertinent examples, e.g. for the SOC. The concept of restricted Hamiltonians should be contrasted with that of effective Hamiltonians, which are introduced in Section 3 and 4 for single transition ions and ECS of transition ions, respectively. The generalized definitions of the full and restricted forms of the physical CF (LF) Hamiltonians, the effective Hamiltonians, and the fictitious ones (introduced in Section 3) as well as distinctions between these notions are discussed in details in Section 9. 2.6. General comments concerning the CF (LF) effects and Hamiltonians

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The effects of the electric field due to the surrounding ligands acting on the transition ions embedded in crystals were considered initially in the approximation of the point charge model, i.e. due only to the nearest neighbor ligands treated as point charges. This approximation soon proved to be insufficient, however, customarily the name „CF effects‟ has been employed to these days. Based on the point charge model, first conventional Hamiltonians HCF (HLF) were introduced to describe the effects of the electric field, i.e. the CF effects or alternatively the LF effects. The Hamiltonians HCF (HLF) are themselves 'physical' in nature, since they account for the physical, i.e. true, interactions between the electrons on a transition ion and those on the ligands. Hence, the CF (LF) effects and thus HCF (HLF) are electric in nature (see, Fig. 2). This feature is reflected openly in the equivalent names „crystal electric field‟ or „crystalline electric field interaction‟, which have also been used, mainly in inelastic neutron scattering [39] related literature, see, e.g., references concerning RE ions (RE = Er3+, Ho3+, Nd3+, Pr3+) in REBa2Cu3O7- high-Tc superconductors [40,41]. Incidentally, Zheng et al. [42] used the phrase: „the ligand (or electric) field‟ and also the name „crystal field‟ in the study of SMM planar {Dy4} clusters. The prevailing meaning attributed in literature to the name „LF‟ reflects the trend in evolution from the early point charge model approach to other approaches beyond this over simplistic model, which incorporate various other effects in the CF (LF) parameters, e.g. covalency, dipole charges on ligands, screening and polarization [7,8]. Hence some authors distinguish 'CF' and 'LF' along these lines, i.e. associating the name „CFPs‟ with the electrostatic origin only, whereas the name „LFPs‟ with the contributions from covalency and other effects (see, e.g. [17]). Generally, the LF theory may be considered as a more modern version of the CF theory. This 'evolutionary' view-point is evident in, e.g., the most recent review by Woodruff et al. [43], quote: 'Other groups have proposed a crystal field approach, or more precisely a ligand field approach, where the directionality and charge density of ligand donor atoms are accounted for, as well as the geometry produced by the traditional “point negative charges” of crystal field theory.'. The review [43] provides an extensive listing of lanthanide (Ln) SMM and comprehensive survey of their structural and magnetic properties. For illustration of the respective usage of the terms „CF‟ and „LF‟ and to establish to what extent the two notions may be considered as synonymous, we have surveyed a representative sample of the recent literature pertinent for SMM, concentrating especially on the RE-based compounds as well as the exchange coupled complexes of the TM ions, e.g. Fe4, Fe8, and Mn12 systems. The survey indicates a number of cases where the terms „CF‟ and/or „LF‟ were used in conjunction with either „effects‟, „theory‟, „analysis‟, „interaction‟, or „splitting‟. The meaning of the term either „CF‟ or „LF‟ was clearly defined in the papers in which the forms of Hamiltonians and/or symbols for parameters were explicitly provided. However, most often the terms „CF‟ and/or „LF‟ were mentioned without providing the form of Hamiltonian, i.e. in a descriptive way, nevertheless reflecting correctly their meaning. The term „CF‟ has been used most often, see, e.g. [44,45,46,47,48,49,50,51,52,53,54,55, 56,57,58,59,60,61,62,63], whereas the term „LF‟ less often, see, e.g. [64,65,66,67,68,69,70, 71 , 72 , 73 , 74 ]. It turns out that predominantly the two terms „CF‟ and „LF‟ have been used interchangeably in the same context and thus were treated as synonymous, see, e.g. the papers

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[42,75,76,77,78,79,80,81,82,83,84,85,86,87, 88,89,90,91,92,93,94,95, 96,97,98,99,100,101, 102,103,104,105]. Interestingly, synonymous usage of the terms „CF‟ and „LF‟ in text has also been encountered in the cases where the parameters were explicitly defined in one specific notation, e.g. the „LFPs‟ in the Stevens notation [106], whereas the „CFPs‟ in the Wybourne notation [107]. These two basic types of notations used for CFPs (LFPs) are defined in Sections 5 (operators) and 6.1 (Hamiltonians). Incidentally, in the majority of papers using the notions „CF‟ or „LF‟ and both „CF‟ and „LF‟, the term „ligand(s)‟ is also utilized and is referred to the ions surrounding the central metal ion, thus confirming that the true CF (LF) quantities were actually meant. Synonymous usage of the terms „CF‟ and „LF‟ has been adopted also in several reviews by, e.g., Anthon and Schäffer on nephelauxetism and computation of interelectronic repulsion in gaseous dq ions by Kohn–Sham version of the density functional theory (DFT), Porcher et al. [108] on various modeling techniques that enable calculations of the CFPs from the crystal structure data, Palii et al. [109] on exchange coupling in molecular magnets with unquenched orbital angular momenta, the tutorial review by Sorace et al. [110] on the magnetic properties of SMM based on the RE ions, and the most recent review by Zhang et al. [111] on recent advances in dysprosium-based SMM. In the reviews by Luzon and Sessoli [54] and Alonso et al. [112] the terms „CF‟ and „ligand(s)‟ are consistently utilized. The review [54] of lanthanides in molecular magnetism, including SMM systems, may be consulted for a succinct and instructive presentation of the true notion CF suitable for the 4fN ions in the approximation of the ground electronic 2S+1LJ multiplet. The table listing the ground state S, L, J values and the resulting 2S+1LJ multiplets of the Ln3+ ions given in the Supporting Information of Hamamatsu et al. [45] may be helpful. This survey indicates clearly that the two notions „CF‟ and 'LF' are used nowadays in the context which, to a great extent, implies synonymous meaning of both notions. It is worth to describe briefly the available computer programs, which take into account interactions included in Hphys, namely, programs developed for d- and f-shell by Schilder and Lueken: CONDON [65], for f-shell by Reid (see, Appendix 3 in Ref. [8] and the corrections to matrix elements of the electron spin–spin interaction (Hss) [113]), and for d-shell by Yeung and coworkers: the crystal field analysis (CFA) package [114,115]. These programs utilize Wybourne notation to express HCF (HLF) in terms of the spherical-tensor operators: the one electron operators Ckq(i) and the combined many-electron ones Ckq, as given in the general forms of HCF (HLF) discussed in Section 6.1. The distinctive feature of these programs is that HCF (HLF) acts in the whole configuration nlN, i.e. it is diagonalized in the complete basis of the respective states. Other earlier programs suitable for the configuration dN are listed by Boča according to the assumed levels of the theory, i.e. magnetotheoretical hierarchy (see, Table 2 in [23] and Table 6.2 in [116]). Reid's program is devoted to calculations of electronic structure of the energy levels of f-electron systems and intensity of the 4fN4fN and 4fN-4fN-15d1 transitions. This program is most widely used in the analysis of optical spectroscopy data of RE ions. Program CONDON has been developed more recently for studies of magnetic properties of mononuclear and exchange-coupled d- and f-systems. An additional advantage of this program is the capability to consider the effect of applied magnetic field on magnetic properties. Concluding Section 2: The notions defined herein may be now contrasted with the concept map in Fig. 1 and the visualization of the notion CF (LF) in Fig. 2. The origin of crystal field (CF), or ligand field (LF), together with the conceptual definitions of various pertinent physical Hamiltonians for single transition 3dN and 4fN ions, as well as associated wavefunctions and energy levels have been more deeply presented in text. This should enable a qualitative understanding of concepts indispensable for description of spectroscopic (optical) properties of single transition ions, especially those with an orbital singlet ground state in the MLn complex, for which the effective (spin) Hamiltonians are introduced in Section 3. 3. Effective Hamiltonians for single transition ions 3.1. Spin operators for single transition ions

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It is useful to explain first various existing types of „spin‟ and introduce different symbols for ~ each notion. In general, bold symbols in italics denote the respective „spin‟ operators (si, S, S , S′), ~ whereas normal fonts in italics (si, S, S , S′) – the respective „spin‟ quantum numbers. 3.1.1. True electronic spin The true electronic spin S of a single transition ion arises from the unpaired electrons with the N

spin si (si = ½) within the electronic configuration 3dN or 4fN: S   si . The distinction between the i

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Kramers versus non-Kramers ions is important for description of their spectroscopic properties studied by EMR. The Kramers ions with an uneven number of electrons have half-integer electronic spin. Hence their spin states are only doublets (so-called Kramers doublets), which can be split only by an external magnetic field. The non-Kramers ions with an even number of electrons have integer spin and their spin states are singlets or doublets. The non-Kramers doublets may be split even at zero B due to the zero-field splitting (ZFS) defined below.

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3.1.2. Effective spin

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The effective spin is defined as the spin angular momentum operator S that acts only within the basis of the (2 S~ + 1) effective spin states {| S~ , M~ S >} selected for a given orbital singlet ground state ~

M

of a single transition ion. It is utilized in the effective spin Hamiltonian H SH discussed below. To emphasize the different nature of the physical and effective quantities, the tilde (~) has initially been ~ ~ ~ introduced to distinguish the effective spin operator S and its states { S , M } from the true electronic S

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spin operator S and its states |S, MS>. Provided that the orbital angular momentum is fully quenched and the ground singlet subspace {|0>|S, MS>} is well separated in energy from the excited states, the part of the physical Hamiltonian (HSO + HSS + HZe) can be treated as a perturbation V. For these cases, ~ an effective Hamiltonian H eff can be derived, which acts in its own subspace {| S~ , M~ S >} instead of the subspace {|0>|S, MS>}. Basic conditions is that the parameters of H~ eff are adjusted in such a way as to faithfully and equivalently describe the eigenvalues of the physical perturbation V. For transition ions in crystals exhibiting an orbital singlet ground state with the true electronic spin S, the ~ ~ quantum number S is equal to the quantum number S associated with the effective spin operator S . ~ Importantly, the main difference between the operators S and the operators S is in their nature, which is determined by the respective distinct basis of states in which they act, i.e. |S, MS> and {| S~ , M~ S >}, respectively. This difference applies also to the respective Hamiltonians (see below) used to describe the system in the two spaces. In spite of this fact, often the two types of spins and the associated quantities are inappropriately mixed up in the EMR literature. 3.1.3. Fictitious 'spin'

The fictitious 'spin' S' may be ascribed to the specific subset of the ground states of a single transition ion or, in general, any selected subset of quantum states with a given degeneracy, i.e. multiplicity. The fictitious „spin‟ operator S’ is selected artificially to describe a particular subset of distinct N (N < Nt) lowest-lying energy levels of a paramagnetic ion, out of the total manifold of Nt energy levels. This subset of energy levels is regarded as equivalent to a 'spin' multiplet of a fictitious 'spin' operator S’, which is characterized by the „spin‟ quantum number S' and the magnetic quantum number MS' (-S'  MS'  +S'). Important point is that the fictitious „spin‟ S' is ascribed in such a way that the multiplicity (2S' + 1) is equal to the number N of the selected energy levels. Conversely, the fictitious 'spin' quantum number S' equals to (N-1)/2. This definition implies that, in general, the fictitious 'spin' quantum number S′ may have no direct relationship with that of the true electronic spin ~ S or the effective spin S of a paramagnetic ion (or spin system) in question. Hence, the natures of the ~ spin operators S, S , and S′ may be quite different from each other.

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The fictitious 'spin' S′ should be clearly distinguished from the momentum from which it originates, i.e. the momentum pertinent for the particular subset of states the fictitious 'spin' S′ is supposed to describe. Two basic cases may be realized: (1) the states described by the operator S’ ~ may originate due to the effective spin S or (2) the total angular momentum operator J = (L + S). Examples of the first case are provided by restricting the (2 S~ + 1) states {| S~ , M~ S >} to a smaller subsets, e.g. for S~ = 5/2 the lowest subset {| S~ =5/2, M~ S = 5/2>} or {| S~ =5/2, M~ S = 1/2>} may be described by S' = 1/2 and {| S' = 1/2, MS' = 1/2>}, whereas for S~ = 2 the lowest subset {| S~ =2, M~ S = 0, 1>} or {| S~ =2, M~ S = 2, +1>} by S' = 1 and {| S' = 1, MS' = 0, 1>}. Examples of the second case are provided by various TM ions with orbitally degenerate ground states in crystals. The simplest and most often encountered case of the fictitious 'spin' S′ is S‟ = ½, which may be employed for various systems exhibiting the two lowest energy levels, of any origin, which are well separated from the ~ higher levels. Importantly, the effective spin operators S act on the wavefunctions {| S~ , M~ S >}, whereas the fictitious 'spin' operators S′ on {|S', MS'>}). ~ It is prudent to distinguish the effective spin S from various specific types of the fictitious 'spin' S' for the following reasons. One has to keep in mind the origin and nature of the effective spin in the single-ion spin Hamiltonians discussed briefly above (and in more details below). The name fictitious may also be applied for operators and states associated purely with the orbital angular momentum L. The cases of the fictitious orbital angular momentum L' = 1 are discussed in Section 7.1 and 9. In these cases the name fictitious 'spin' is not appropriate. In general, the name fictitious may be used for any kind of 'fictitious' angular momentum X, based solely on the criterion of matching the number of energy levels: (2X+1) as introduced above for the fictitious 'spin' S'. Hence, apparently the 'effective spin' may be considered from a logical point of view as a particular case of a fictitious angular momentum. However, the opposite logical relationship occurs for the cases when ~ the fictitious spin S′ arises from a smaller subset of the effective spin S states as shown by the above examples. As an aside, we note that the fictitious „spin‟ is not a purely mathematical concept. The major reason for introduction of a fictitious „spin‟ is usually the experimental accessibility of EMR transitions. Usually only transitions between a limited number of low-lying states may be observed due to the spectrometer frequency range. A recent interesting case of usage of a fictitious spin, which was motivated by computational necessity, occurs in Schnack and Ummethum [117], quote: „The experimental magnetic studies of the {Gd12Mo4} were accompanied on the theoretical side by calculations that replaced the Gd spin of s = 7/2 by fictitious spins s = 5/2 since otherwise the calculation would not have been feasible in a reasonable time (of several weeks [sic!]).‟ Note that this example represents the case when the fictitious spin S′ = 5/2 is a particular case of the effective ~ spin S = 7/2. 3.2. Effective spin Hamiltonians for single transition ions 3.2.1. Generic „spin‟ Hamiltonians

The generic „spin‟ Hamiltonians (SH) HSH are conceptually defined as: HSH ≡ HZFS + HZe + {other more sophisticated terms for specific „spin‟ systems}. ~ By definition, any „spin‟ Hamiltonian HSH involves the operators of the generic „spin‟ only, e.g. S, S , or S′ (defined above) or the total spin ST ascribed to a given ECS (defined below), and, in general, the external fields, e.g. the magnetic field or the electric field. The nature of the „spin‟ operators, which depends on the way in which a given HSH is derived or postulated, determines the particular type of the „spin‟ Hamiltonian. The quotation marks used for the „spin‟ in the generic HSH signify the fact that the generic „spin‟ may mean different quantities. It may be related indirectly not only to the true ~ electronic spin S of a single transition ion - as it is the case of the effective spin S , but may involve contributions from the electronic orbital L and/or the total angular momentum operators J = L + S, or even may be not directly related to any of the operators J, L, or S - as it is the case of the fictitious 'spin' S'. Importantly, the nature of the effective SH defined in Section 3.2.2 and 4.3 indicates that

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such SH shall not be expressed explicitly in terms of the true electronic spin operators S but only S and ST, respectively. HSH includes two major terms: HZFS and HZe. The ZFS term HZFS describes the splitting within the basis of the „spin‟ states at zero B and hence is appropriately called the „zero-field splitting‟ (ZFS), or equivalently „fine structure splitting‟. Note that the latter name is strictly appropriate only for single transition ions, whereas rather inappropriate for ECS. The Zeeman electronic (Ze) term HZe describes the splitting of „spin‟ levels due to B. The specific name applicable to a given „spin‟ Hamiltonian: HSH = HZFS + HZe, depends on the type of „spin‟ involved. The second-rank ZFS term is most often used in literature in the bilinear form: SDS, where S is the 'spin' appropriate for a given „spin‟ system and D is a traceless 'tensor'. The general forms of the ZFS terms are discussed in Section 6.2. 3.2.2. Effective single-ion spin Hamiltonians

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~ ~ The effective single-ion spin Hamiltonians H~ eff  H SH associated with the effective spin S are conceptually defined as: ~ ~ ~ ~ N H eff  H SH ≡ H ZFS + H Ze + {Higher-order Zeeman or nuclear terms and specific terms for 4f and 3dN ions}. These Hamiltonians represents a specific class of the generic „spin‟ Hamiltonians that apply for the transition 3dN ions and the S-state 4fN ions exhibiting an orbital singlet ground state {|0>|S, MS>} ~ ~ with the electronic spin S in crystals. By definition H eff  H SH describes the splitting of the spin

M

energy levels within the ground orbital singlet {|0>|S, MS>}, which is due to the action of either the total physical Hamiltonian Hphys, if solved numerically, or the perturbation V = HSO + HSS + HZe, if solved in an approximated way. The Hamiltonian Hphys acts within the whole physical space {|>|S, MS>}, whereas V within the limited subspace of the selected lowest lying states of {|>|S, MS>}. ~ ~ However, the effective SH, H eff  H SH , acts only within its own subspace of states of the effective ~

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~ spin operator S , S , M , which is suitable for the given ground orbital singlet of a transition ion in S crystals.

3.2.3.Microscopic SH (MSH) theory vs the generalized SH (GSH) theory With the hindsight of the knowledge accumulated up-to-now, a general explanation of the origin and nature of the effective single-ion SH (ZFS) Hamiltonians may be stated as follows. Such Hamiltonians are introduced to reproduce a (relevant) subset of the total energy spectrum, which increases the ease of calculation, limits the number of parameters, and allows easy comparison of data at the expense of apparently losing the physical meaning of obtained parameters. The point at which this transition is made depends on the particular application of an effective Hamiltonian. In the case of the effective single-ion ZFS Hamiltonians the route from the physical picture to the effective one ~ ~ may be symbolically represented as the mapping: Hphys (or V)  H eff  H SH . The selected ndimensional subspace of states, here {|0>|S, MS>} with n = 2S + 1 and the energy levels Ei, arising from either the numerical solutions of the whole Hphys, or the approximated solutions of the perturbation V, is mapped onto the n-dimensional subspace of the adopted effective spin states {| S~ , ~ ~ ~ ~ ~ M S >} with n = 2 S + 1 and the energy levels i arising from H eff  H SH . Major advantage of H eff  ~ H SH is that it mimics the energy levels of the electronic spin states {|0>|S, MS>}, i.e. it yields in an

effective way the same energy levels as those of Hphys (or V). To understand better the relationships ~ ~ between Hphys (or V) and H eff  H SH , it is helpful to keep in mind the following points. Two routes from the physical picture to the effective one exist [12-18,21,23,28,29]: (i) the ‘derivational’ route, which belong to the realm of the microscopic spin Hamiltonian (MSH) theory, and (ii) the ‘constructional’ route, which belong to the realm of the generalized SH (GSH) theory (see, Fig. 1). Historically, the conventional SH was first obtained by Pryce [118] in 1950 for single

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TM ions with an orbital singlet ground state without invoking the effective spin concept [28,29]. Pryce [118] proposed an explicit „derivational‟ route based on the second-order perturbation theory (PT). This derivation has provided us with a Hamiltonian in the bilinear form: SDS, where D is a traceless 'tensor' (see Section 6.2). This Hamiltonian has later become known as the 'spin Hamiltonian' since it involves only the „spin‟ variables, whereas the spin S therein was explicitly ~ named as the effective spin S to distinguish it from the true electronic spin S [12-18,21,23,28,29]. Details of the conventional SH derivation and pertinent references, may be found in the textbooks [12-18,21,23] or the reviews [28,29] and thus are not reproduced here. Later the Pryce's method was extended by various authors to other cases. Various methods ~ employed in literature for „derivation‟ of H~ eff  H SH using MSH theory were classified [28] and an

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updated overview was provided [29]. Numerous explicit MSH expressions derived by the secondorder PT for various TM ions with an orbital singlet ground state may be found in [21,23]. Among others, the method based on tensor algebra [119,120] has enabled extension of the „derivational‟ route to the fourth-order PT [121,122]. This method has provided the ZFS terms, expressed for the first time directly in terms of the tensor operators, and the linear Ze terms. The explicit analytical expressions suitable for 3d4 and 3d6 ions with spin S = 2 within the 5D term split by axial and orthorhombic CF, were derived [119-122]. The fourth-order PT method combined with tensor algebra has enabled prediction of the 2nd- as well as 4th-rank ZFSPs and the gi components for, e.g. Cr2+, Mn3+, Fe4+ (3d4), and Fe2+ (3d6) ions in various crystals; see, [123,124,125,126,127,128] and references therein. A sample of these MSH expressions is presented in Section 7.2 to illustrate the interrelationships between the CF (LF) and SH (ZFS) parameters. A subtle point needs to be clarified concerning the meaning of „spin‟ in the SH obtained using the „derivational‟ route, i.e. one of the MSH PT-based methods like as those, e.g., in [118, 119-122]. In ~ brief, using these methods, H~ eff  H SH is derived from Hphys ≡ HFI + HCF for a given „spin‟ system for

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a selected subspace of the totality of states for a paramagnetic species. Importantly, the explicit SH forms are obtained using the MSH PT-based methods by integrating over the orbital variables [123,28,29]. The components of the g-tensor and the D-tensor emerging in such calculations are proportional to the familiar -tensor [12-18,21,23,28,29], which contains just the matrix elements of the type: <0|L|><|L|0> and suitable energy denominators. Hence, the „spin‟ operators in the so-obtained SH are indeed associated with the true electronic spin S, which remains without modification in perturbation calculations involving only the orbital angular momentum, and not the spin. However, the resulting ZFS term, SDS, no longer acts within the physical states {|0>|S, MS>} of the ground spin S-multiplet of single transition ions, but only within the spin states {|S, MS>}. To distinguish the former states and the latter, as discussed in Section 3.1.2, the tilde (~) is used: {|0>|S, MS>}  {| S~ , M~ S >}. This transition from the former basis of states to the latter is responsible for introduction of the concepts of the effective spin and the effective SH. These concepts acquire then the meaning defined in Section 3.1.2 and 3.2.2, respectively. Direct consequence of these concepts is the emergence of the interface defined in Introduction as the boundary of the two distinct entities, i.e. the physical (crystal field) Hamiltonians and the effective (spin) Hamiltonians. The CF (LF)  SH (ZFS) interface is graphically depicted later in Fig. 5 and 6. The MSH PT-based methods, like those in [118, 119-122], have serious shortcomings arising mainly from the tedious and approximate nature of PT calculations. In view of these cumbersome shortcomings, historical developments [28,29] have led to the ‘constructional’ route (see, Fig. 1), which was developed based on group theory [36]. In brief, in this route using various methods a ~ ~ generalized spin Hamiltonian (GSH), H eff  H SH , may be constructed for a given spin system, ~

~

characterized by an effective spin S , using symmetry arguments and based on the invariance of H SH under the elements of a given point-symmetry group. Here, one constructs an invariant combination ~ of the entities describing a given physical system, i.e. S and the magnetic induction (B) or the electric field intensity (E). Note that in principle, the ‘constructional’ route may be employed also for a fictitious 'spin' S′, however, it has not been used in practice, since only the ZFS and Zeeman terms are considered in these cases. Classification of various GSH methods (see, Fig. 1) to „construction‟ of

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~ ~ H eff  H SH existing in literature and pertinent references may be found in [28,29]. Importantly, the

GSH theory has enabled predictions, for the first time, of the forms of the higher-order terms in the GSH for transition ions, including the Zeeman terms that are discussed in Section 6.3, as well as the higher-order nuclear terms [28,29]. The former terms are field-dependent and their profound implications for high-magnetic field and high-frequency EMR (HMF-EMR; often named alternatively as HF EPR) measurements have recently been reviewed [37]. ~ A few important points shall be emphasized concerning the Hamiltonians H~ eff  H SH derived using the MSH theory or constructed using the GSH theory. Each resulting Hamiltonian term in H~ eff ~

~

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 H SH , i.e. H ZFS and H Ze , depends on the variables associated with the effective spin operators S . Hence, these Hamiltonians are magnetic in nature, unlike the physical Hamiltonians HCF (HLF), which are electric in nature. This profound distinction should be kept in mind when considering the CF (LF)  SH (ZFS) interface between the two types of Hamiltonians. It is a common practice in literature to represent for simplicity the spin operators in the effective single-ion ZFS Hamiltonians by the symbol S, which formally denotes the true electronic spin operators S. However, it should be kept ~ in mind that instead of the latter quantity, in fact, the effective spin operators S must be meant in this ~ case. It is only the quantum number S that is equal to the electronic spin quantum number S, ~ whereas the spin operators S and S have different properties.

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3.3. General comments concerning the effective single-ion spin Hamiltonians

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~ ~ Nowadays the notion of H eff  H SH profoundly underlies EMR and magnetic susceptibility, (T), studies. For a succinct and informative presentation of the CF (LF) Hamiltonians and the spin Hamiltonians as well as their interplay, especially the LF origin of SH parameters, the extensive review of magnetic circular dichroism spectroscopy as a probe of the geometric and electronic structure of non-heme ferrous enzymes by Solomon et al. [129] may be recommended. Using the present day computational techniques, the splitting of the orbital singlet ground state at zero B may be obtained by full diagonalization of the total physical Hamiltonian Hphys, which yields all energy levels and the corresponding wavefunctions, which are complicated combinations of the CF states of the type {|>|S, MS>}. Hence, the splitting in question has been appropriately named as the zero-field splitting (ZFS) or fine structure splitting. As discussed above, in the early days, instead of full diagonalization of Hphys, the ZFS was calculated using perturbation theory as the splitting of a given orbital singlet ground state due to the SOC (and eventually the electronic SS coupling) taken as a perturbation within the subset of higher lying CF states. Hence inclusion of parameters describing the SOC is indispensable for the emergence of the single-ion ZFS terms for the TM ions and the S-state RE ions. In the MSH methods, i.e. based either on perturbation theory or full diagonalization, the major ~ purpose of the ZFS Hamiltonian, H ZFS , is basically to reproduce the physical energy splitting caused by the SOC and SS couplings, by devising an effective Hamiltonian in terms of the effective spin ~ operators S . The parameters of the effective Hamiltonian are taken in such a way so as to faithfully represent the energy level splitting of the ground orbital singlet due to the physical perturbation: V = HSO + HSS. Hence, it should be kept in mind that for transition ions embedded in crystals the notion SH (ZFS) arises from consideration of the action of the physical Hamiltonian, which includes the free ion Hamiltonian and the true CF (LF) one, within the states of the configuration 4fN and 3dN. Moreover, the SH (ZFS) Hamiltonian acts only within the basis of its own effective spin states {| S~ , ~ M S >}, which correspond to the selected, usually lowest, states of the physical Hamiltonian. The projection or mapping of the energy levels of the physical Hamiltonian onto those of a suitably parameterized SH (ZFS) enables derivation of the microscopic relations for the ZFS parameters in terms of the parameters involved in the physical Hamiltonian (see Section 7). Hence, the notion CF (LF) logically precedes the notion ZFS and thus the ZFSPs depend implicitly on the CFPs.

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~ The notion of the effective Hamiltonian H~ eff  H SH has basically been introduced in order to

simplify the description of the observed ZFS within the physical states {|0>|S, MS>} of the lowest ~ spin S multiplet of single transition ions. In many EMR studies various terms in H SH are just adopted in a phenomenological way (see, Fig. 1) as the starting point to account for the experimental ~ observations. A common misconception in some studies is that H ZFS 'splits' the (effective) spin states {| S~ , M~ S >}, most often denoted as {|S, MS>}. This may appear true only from purely mathematical point of view, whereas from physical point of view one should keep in mind that the ZFS has not been created out of nothing, but it is due to the action of specific physical interactions and hence H~ eff 

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~ ~ ~ H SH = H ZFS + H Ze merely describes the experimentally observed ZFS in an effective way. The ~ detachment of H SH from its physical origin is clearly noticeable in some magnetism papers, where ~ either the true H ZFS is named as the 'CF' Hamiltonian or the true ZFSPs as „CFPs‟ [30,31]. This is a

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common case in many theoretical studies of various 'spin models', where no practical applications to specific ion-host systems are considered [32]. Most recent cases of the CF=ZFS confusion have been discussed in the review [33]. The notion of spin Hamiltonian is also used in a wider sense [55,90, 130 ]. For specific applications, it includes not only the ZFS Hamiltonian and the Zeeman electronic (Ze) one but also various types of the exchange interactions (EI), which are discussed in Section 4, as well as the hyperfine coupling, the nuclear quadrupole interaction, and the nuclear Zeeman effect. A note of caution is pertinent concerning the relationships between the EI terms and ZFS ones. As reviewed [33], a misconception exists in literature, whereby the ZFS term SDS is referred to as an 'interaction' or 'coupling' term. However, these names are not applicable for any ZFS terms, since the same spin, ~ irrespective of its nature (S [in fact S ], S′, or ST) cannot obviously „interact‟ or 'couple' with itself. In general, the notion 'interaction' or 'coupling' implies the existence of two distinct entities, say A and B, which may be either singular or collective. The ZFS form SDS only superficially resembles an interaction term, like that for the true EI between two spins localized on different ions (i and j): Hex = Si  Dij  S j (see Section 4). However, this resemblance does not allow for considering the ZFS term

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as an 'interaction' or 'coupling', unlike, e.g. the hyperfine interaction (or coupling) term (Hamiltonian), which represents the true physical interaction between the electronic and nuclear spins. Such terminological misconceptions represent specific cases of the confusion between the EI and the ZFS quantities, denoted EI=ZFS [30,32,33]. The notion SH is so ubiquitous in the EMR and magnetism areas that it is often invoked without explanations of its meaning, origin, or the range of its applicability, as in the case of several reviews, e.g. [54,112,131,132]. An informative introduction to the basic SH theory and applications geared for chemists may be found in the most recent reviews [133,134], which utilize the proper terminology for the CF terms as well as the major SH terms, i.e. the Ze terms and the ZFS or equivalently fine structure ones. Symmetry aspects concerning SH are also briefly covered [134], whereas 'the spin angular momentum, S, a quantum vector property' is explicitly named as the 'electron spin'. However, the symbol S is rather used implicitly in text in the sense of a generic „spin‟, i.e. encompassing other ~ types of the spin operators ( S , S′; ST). In view of the nature of the pertinent quantities, the effective spin Hamiltonians of the two types discussed herewith shall not be expressed explicitly in terms of the true electronic spin operators S of a single transition ion. As exemplified and discussed in more details in the reviews [28,29], the notions: physical Hamiltonian versus effective Hamiltonian as well as true electronic spin versus effective spin versus fictitious 'spin', are not well defined and are often confused with each other in the EMR-related literature. The notion of the effective spin is used often in literature also in the cases where the notion of the fictitious 'spin' would be more appropriate. Opposite replacement of the two names also occur. Selected examples are discussed below. In the review of prediction of molecular properties and molecular spectroscopy with the DFT, Neese [131] states, quote: „EPR and NMR experiments are parameterized by an effective spinHamiltonian (SH) that only contains spin-degrees of freedom.‟ and provides „for an isolated spinsystem with total spin S‟ the SH with the ZFS term written as S  D  S , Eq. (121) in [131], where S

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„refers to the spin-operators for the fictitious total spin of the system.‟ In view of the above definitions, the name effective spin would be more appropriate than the fictitious 'spin' used in a general context in [131]. Lukens and Walter [55] correctly describe the total Hamiltonian for magnetically isolated Kramers‟ and non-Kramers‟ f-ions as well as the resulting energy levels and the corresponding states, which may be doublets and singlets. However, the authors [55] state, quote: „These doublets and singlets can be described using a fictitious effective spin, S, equal to 0.5 for the doublets and 0 for the singlets.‟ and then „The spins of the f-electrons will be referred to as “true spins” to distinguish them from the effective spins.‟. Lack of distinction between the two notions: the fictitious 'spin' and the effective spin, may create an additional confusion. Most often utilized cases of the fictitious 'spin' are for the lowest Kramers doublets of various transition ions. For example, Co2+(3d7) ion exhibits the lowest Kramers doublet with multiplicity equal to 2, which may be described by the fictitious 'spin' S‟ = ½. Alonso et al. [112] describe, with a reference to the book [12], the behavior of the ground state Kramers doublet of FeIII in the low-spin configuration using the hole formalism based on „the fictitious 'spin' formalism that allows us to handle the ground state Kramers doublet as S‟ = ½ system...in a more compact and simple way‟. Here, the name fictitious 'spin' is also most appropriate. Several examples of the fictitious 'spin' S‟ pertinent for 3dN and 4fN ions in various crystals will be reviewed [135]. The categorization [135] of the fictitious 'spin' cases occurring in literature may help understanding the subtle differences between the fictitious „spin‟ and the effective spin. Alternatively, the fictitious 'spin' S‟ = ½ arising from the lowest Kramers doublet of 4fN ions or CoII and low-spin FeIII clusters in various crystals was named as the „pseudospin‟ („pseudo-spin‟) [70,96,97,109,136,137], which also adequately reflects the nature of this quantity. Recently, Chibotaru and Ungur [102] have utilized the name „pseudospin‟ and „pseudospin Hamiltonians‟ in the sense encompassing both the fictitious 'spin' and the effective spin [28,29]. Chilton et al. [61], 'based on the pseudospin S~ = 1/2 formalism', have performed the CASSCF and RASSI procedures for DyIII ions in Dy2 dimmer to calculate 'the anisotropic g-tensors for the 50 lowest spin-orbit and crystal-field Kramers doublets'. As an aside, we note that Guo et al. [58] in the study of 'an asymmetric Dy2 SMM' stated that 'each III Dy ion is described as S = 1/2' without an explanation of the nature of the spin 'S'. Likewise, Boulon et al. [100] used for DyIII ion the name 'an effective Seff = 1/2'. Both cases [58,100] present an implicit usage of the fictitious 'spin' S‟ = ½ for the lowest Kramers doublets arising for DyIII ion from the ground multiplet 6H15/2 with J = 15/2. Concluding Section 3: The notions defined herein may be now contrasted with the concept map in Fig. 1. The conceptual definitions of various meanings of 'spin', associated wavefunctions, and energy levels as well as the ~ origin of the effective spin Hamiltonians H SH have been elucidated. This has enabled to establish the qualitative connection between the physical Hamiltonians, involving the operators acting on the orbital and spin part of the wavefunctions {|>|S, MS>}, and the effective SH (ZFS) Hamiltonians ~ involving the effective spin operators S acting on the wavefunctions {| S~ , M~ S >}. This connection forms the interface between the physical and effective Hamiltonians. The interrelationships and distinctions between the quantities related to the CF and ZFS have also been succinctly overviewed in Fig. 1. A deeper global discussion of these aspects is postponed till Section 7, i.e. after details on ECS, the Stevens and Wybourne operators, and forms of Hamiltonians and definitions of the associated parameters are presented in Sections 4, 5, and 6, respectively. These Sections provide comprehensively the background knowledge indispensable for description of magnetic and spectroscopic properties for single transition 3dN and 4fN ions with an orbital singlet ground state in the MLn complex. 4. Exchange coupled systems (ECS) of transition ions and single molecule magnets For better understanding of the crucial notions pertinent for the CF (LF)  SH (ZFS) interface for the ECS of transition ions, a concept map is provided in Fig. 3, followed by general definitions of the notions. The two types of SH sketched in Fig. 3, namely, the multispin Hamiltonian (MH) and the total spin or giant spin (GS) Hamiltonian are discussed in details in text.

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4.1. Types of exchange interactions and Hamiltonians

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It is useful to categorize the various existing types of the exchange interactions and the respective Hamiltonians pertinent for ECS. The interrelationships and distinctions between the notions are also discussed below. The exchange interactions (EI) of various types may exist within a magnetic molecule exhibiting the individual spins Si (i = 1, 2,…, N) localized on separate N sites, where N is the total number of sites. Since, the spin operators involved in the exchange interaction terms are magnetic in nature, such Hamiltonian terms have also magnetic nature. Such systems or clusters are collectively called the exchange coupled systems (ECS). Most ubiquitous examples of the ECS are formed by clusters of two (dimmers), three (trimmers), four (tetramers), or more transition ions, each with the spin Si (i = 1, 2,…, N). The molecular magnets like SMM (MNM) may be considered as a special class of ECS, which exhibit macroscopic quantum tunneling, or equivalently quantum tunneling of magnetization, see, e.g. [24]. As revealed by literature survey, the ECS that have recently been most extensively studied are the heterometallic 4fN-3dN ion complexes as well as molecules forming SMM, like, e.g. Fe4, Fe8, Mn12 and Mn4 systems. Numerous references to the original literature dealing with these systems may be found in the reviews [33,34].

Fig. 3. Concept map for the notions pertinent for the exchange coupled systems formed by paramagnetic species (i, j) localized on separate N sites; for specific explanations, see text.

The most basic type of the exchange interactions are the Heisenberg isotropic exchange interactions: Hiso-ex ≡ JijSiSj, where Jij is the exchange constant. Other conventions to represent this Hamiltonian exist: Hiso-ex ≡ -2JS1S2 or Hiso-ex ≡ -JS1S2 which require rescaling the values or redefining the sign of the exchange constants, respectively. Most often the same symbol J is used in each of these conventions. Hence, the definitions actually used for Hiso-ex need to be checked when comparing the J values taken from various sources. The Hamiltonians Hiso-ex are just as much effective Hamiltonians operating on the effective spins as the single-ion ZFS Hamiltonians. For example, the energy splitting between say the singlet and triplet spins states for a spin 1/2 dimmer is

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due to the physical interactions, which can be formulated in terms of the electron coordinates (the orbital part of the wavefunction), but which can be described in an effective spin space by a Hamiltonian formulated in terms of spin operators. Hence, formally the tilde mark (~) should be also used for Hiso-ex and Si, but most commonly it is omitted for convenience. Apart from the isotropic Heisenberg terms, Hiso-ex, other anisotropic exchange interaction terms are considered, namely: the bilinear terms: Haniso-ex ≡ SiDijSj, and the antisymmetric (or Dzialoszynski-Moriya) terms: Hantisym-ex ≡ dijSiSj, as well as other more sophisticated types of the EI, like the biquadratic ones: Jij(SiSj)2, have been invoked in the studies of ECS [15,19,24,55,90,138,139,140,141,142,143]. Note that in general the tensors Dij and dij involved in the anisotropic exchange interaction terms may be decomposed into specific components [21]. Detailed discussion of these aspects is beyond the level suitable for the intended readership. A useful introduction to ECS and their magnetic properties has been presented by Kahn [144], who used as an example dinuclear complexes with the electronic spin S = ½. Note that in Eq. (2) in [144] the third term dSASB, which represents Hantisym-ex, should be replaced by dSASB. A generalized exchange SH has been developed for high-nuclearity magnetic clusters [145]. Along with the isotropic exchange term (called 'the Heisenberg-Dirac-Van Vleck (HDVV) term'), the authors [145] have also considered, quote: 'the higher-order isotropic exchange terms (biquadratic exchange), as well as the anisotropic terms (anisotropic and antisymmetric exchange interactions and local single-ion anisotropies)'. Note that Hamiltonian in Eq. (13) in [145] represents the summation of the true single-ion ZFS terms and hence the name 'the single-ion anisotropy Hamiltonian' constitutes the MA=ZFS confusion. Recently, an informative introduction to the exchange interactions for dinuclear TM (d block) coordination complexes in the context of high-frequency and -field EPR (HF EPR) studies has been provided by Telser et al. [146]. The most general exchange interactions Hamiltonian Hex is conceptually defined as: Hex ≡ Hiso-ex + Haniso-ex + Hantisym-ex + {other specific terms for 4fN and 3dN ion systems}. This Hamiltonian describes collectively all exchange interactions existing in a given ECS. Note that the dipole-dipole interactions (see, e.g. [55,140]), which have different nature than the exchange interactions, are sometimes also included as a part of Hex. It is important to note that the exchange interactions of various types are sometimes named as the spin-spin interactions and/or spin-spin couplings. However, in this context, the two terms do not encompass the electronic spin-spin interactions between the unpaired electrons on a given transition ion, which have already been included in the free-ion Hamiltonian HFI. 4.2. Multispin (microscopic spin) Hamiltonians for ECS The multispin Hamiltonians HMH (or, equivalently, the microscopic spin Hamiltonians Hmicro) are conceptually defined as: HMH ( Hmicro) 

N ~ + ( H ( i , j )  H ZFS ( i ) +  ex N

i j

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~ H Ze ( i ) ).

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This Hamiltonian is used to represent, at the level of the constituent parts of the system, all Hamiltonians involved in description of ECS. Hence, the name multispin Hamiltonian suitably reflects the nature of such combined Hamiltonians that include all exchange interactions, Hex, and all ~ ~ ~ effective single-ion spin Hamiltonians H SH (≡ H ZFS + H Ze ) for the constituent transition ions. The Hamiltonians HMH play an equivalent role for ECS as the total physical Hamiltonians Hphys ≡ HFI + HCF (HLF) for single transition ions in crystals, so strictly speaking HMH is a 'quasi-physical' ~T Hamiltonian, which serves as a basis for a more 'effective' H SH defined below. Since the Hamiltonians HMH for magnetic molecules involve the operators of the individual spins Si (i = 1, 2,…, N), in the 'uncoupled' (see below) basis of states the respective wavefunctions may be symbolically represented as {| S1, MS1;.....; SN, MSN>}. A brief overview of the alternative names used for the Hamiltonians of the type HMH is pertinent. The name 'microscopic' spin Hamiltonian, Hmicro, is alternatively used for HMH as illustrated by a few examples. Waldman and Güdel [139] used 'the microscopic spin Hamiltonian describing a magnetic spin cluster of N spin centers', whereas Carretta et al. [142] 'microscopic spin Hamiltonian

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that contains all the main interactions present in the cluster'. Chaboussant et al. [ 147 ] used 'microscopic Hamiltonian for exchange interactions between individual Mn ions in the Mn12-acetate cluster' and then included in their Hmicro also 'the single-ion anisotropy terms', i.e., in fact the singleion ZFS terms defined in Section 3. Feng et al. [148] used „microscopic SH‟ to describe HMF-EMR spectra and magnetic data for Mn3 complexes. It should be emphasized that the so defined name 'microscopic SH' for ECS has a different meaning than that used in the realm of the microscopic spin ~ Hamiltonian (MSH) theory, which was originally employed for derivation of the effective SH H SH for single transition ions (see Section 3 and 7). Hence, the name 'multispin' Hamiltonian seems more adequate and preferable than 'microscopic SH', since it avoids potential confusion with the historically proceeding and well-established term 'MSH' defined in Section 3. As illustrated by a few examples, instead of using an explicit name for the Hamiltonians of the type HMH, they are sometimes described as Hamiltonians that apply for a cluster of individual transition ions each with a given spin Si. Kulik et al. [149] for the Mn4OxCa cluster used, quote: 'SH of a system with n coupled Mn ions' including the term SiDiSi and properly named Di as 'the zerofield splitting (ZFS) tensor for the electron spin of the i-th Mn ion'. Nakano and Oshio [150] used for their HMH-like Hamiltonian the name: 'cluster SH', which is also suitable, whereas for the ZFSP of the cluster - the name: „the molecular zfs parameter Dmol‟. Accorsi et al. [151] used 'the SH of the Fe4 cluster' including the term that 'accounts for single-ion (magnetocrystalline) anisotropies', i.e., in fact the true single-ion ZFS term of the form HZFS = SiDiSi and the term that 'describes anisotropic spinspin interactions' of the form Hex = SiDijSj. Note that other non-specific names are also often used for Hamiltonians representing, in fact, HMH, e.g. 'model Hamiltonian' [138] or 'essential SH describing a magnetic ring' [141].

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4.3. Effective total (giant) spin Hamiltonians for ECS

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This approach is based on the concept of the total spin (or giant spin) of ECS. The total spin ST may be ascribed to a given ECS formed by transition ions coupled by various types of the EI according to the adopted models of ferromagnetic and/or antiferromagnetic interactions between the spin Si localized on the constituent transition ions (i = 1, 2,…, N). The spin ST and the spins (S1, S2, ..., N) have, in general, different values. More importantly, the two types of spins do have distinct physical origin and thus properties, and hence, must be clearly distinguished. Since the total spin ST for most SMM discovered in the last two decades, e.g. ST = 10 Fe8 and Mn12 systems, is larger than the highest single transition ion spin, i.e. S = 7/2 for Gd3+, the name giant spin [143,148,152,153,154,155,156,157, 158,159,160,161] has been coined for ST in SMM literature.

~T

The effective total (giant) spin Hamiltonians, HGS  H SH , for ECS describe the ground total spin ST-multiplet of a given system in terms of the total spin operators ST and are conceptually defined as:

~T ~T ~ HGS  H SH  H ZFS + HTZe + {Higher-order ZFS and Ze terms} ~

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T T The higher-order ZFS and Ze terms in H SH are discussed in Section 6.3. The Hamiltonians H SH ( ~ ~ HGS) play an equivalent role for ECS as the H eff  H SH ones for single transition ions and hence they

~

T share some common characteristics. H SH acts only within the basis of states of the total spin STmultiplet {|ST, MST>}, which is the ground (g) state of the magnetic molecule. The basis of states {|ST, MST>} represents effectively the lowest lying states that originate from coupling of the spins operators represented in the 'uncoupled' basis of states {| S1, MS1;.....; SN, MSN>}. This coupling yields also the states of all excited spin multiplets, up to the highest (h) multiplet, which may be symbolically represented as |(S1,....SN); STa, MSTa>. The index a runs over all possible combinations (a = g, ..., h) of the total spin STa obtainable from addition, i.e. vector coupling, of the individual spin operators Si (i = 1, 2,…, N) arising from the assumed model of ferromagnetic and/or antiferromagnetic interactions between the spin Si. This includes the ground total spin ST states (a = g) for which the index a is customarily omitted and thus implicitly ST = STg. ~ In practice it is very difficult to fully solve HMH ( Hmicro), hence, in analogy, with H SH for the

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types of the Hamiltonians for ECS is in their origin that determines the spin operators in which they ~ ~T ~ are expressed: H SH  ( S i), whereas H SH  (ST). One of the basic conditions for a valid application of ~T is the existence of strong EI between the constituent ions forming the molecule of a given ECS or H SH

SMM, since then the total spin ST may be considered as a good quantum number. 4.4. Relationships between multispin Hamiltonians and giant spin Hamiltonians ~

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T To understand better the relationships between HMH and HGS  H SH and the nature of the two types of the Hamiltonians, it is helpful to keep in mind the following points. The route from the quasi-physical picture to the effective total one may be symbolically represented as the mapping: HMH ~T ( Hmicro)  HGS  H SH . This mapping involves projection of the energy levels and the associated states of the lowest spin ST multiplet arising from HMH onto the corresponding energy levels and states ~T of H SH for ECS. This approach, if achievable, would enable derivation of explicit relations for the ~T total ZFS parameters appearing in H SH  (ST) in terms of the parameters involved in HMH  (Si, Sj), i.e. the single-ion ZFSPs and the exchange interaction constants between the individual ions. ~T However, the mapping HMH ( Hmicro)  HGS  H SH for ECS, especially for the complex SMM, is ~ ~ much more difficult than the mapping of Hphys (or V)  H eff  H SH considered for single transition

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T ions. In general, derivation of the relationships between the total ZFSPs of H SH  (ST) and the parameters of HMH  (Si, Sj) is very difficult and thus not straightforward. On the other hand, full diagonalization of HMH ( Hmicro) is rather prohibitive, nevertheless, if achievable, it should yield all energy levels and the corresponding wavefunctions of the lowest spin ST multiplet as well as of all n higher lying spin STn-multiplets, e.g. for ST = 10, the multiplets with STn = 9, 8,...,0. In the absence of B, the splitting of the lowest spin ST multiplet of HMH has the meaning of T the total (spin ST) zero-field splitting. This ZFS can be effectively described by the term H~ ZFS in the T T (effective) total spin Hamiltonian H SH . The parameters of H~ ZFS must be taken in such a way so as to faithfully represent the energy level splitting of the lowest spin ST -multiplet due to the quasi-physical HMH. T The notion of the effective total spin Hamiltonian H~ ZFS has basically been introduced in order to simplify the description of the observed ZFS within the lowest spin ST multiplet of ECS. In many ~T SMM studies various terms in H ZFS are just adopted in a phenomenological way as the starting point to account for the experimental observations. A common misconception in some studies is that this Hamiltonian 'splits' the (effective) total spin states {|ST, MST>}; see, e.g. 'Most MNMs have well defined spin ground states. The splitting of the ground state by ZFS...' [159]. This may appear true only from purely mathematical point of view. However, from physical point of view one should keep ~T T in mind that the total ZFS as a phenomenon is merely described by the ZFS terms H~ ZFS in H SH .

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the total ZFS, as a phenomenon, is due to the action of specific terms included in HMH ( Hmicro). T Hence H~ ZFS merely describes the experimentally observed ZFS in an effective way but is not the cause of the total ZFS. 4.5. General comments concerning Hamiltonians for ECS ~

T A brief overview of the alternative names used for the Hamiltonians of the type H SH ( HGS) is pertinent. In analogy to the alternative name „giant spin‟ for the total spin ST of ECS or SMM, the ~T approach utilizing the effective total spin Hamiltonian H SH is also referred to as the giant spin Hamiltonian, or giant spin model, or giant spin approximation (GSA). Examples may be found in the reviews, e.g., dealing with: (i) magnetic anisotropies in paramagnetic polynuclear metal complexes by Nakano and Oshio [150] (ii) the origin of quantum tunneling of magnetization (QTM)

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in SMM studied by HMF-EMR by Feng et al. [160], and (iii) the magnetization reversal in nanoclusters by Wernsdorfer [ 162 ]. Importantly, some combined names, which may lead to additional confusion, are also employed. For example, Lampropoulos et al. [163] to analyze highfrequency EPR spectra of SMM in truly axial symmetry [Mn12O12(O2CCH2But)16 (MeOH)4]·MeOH used, quote: 'the effective giant-spin Hamiltonian' with S (i.e. ST) = 10 and parameters: 'D parametrizes the dominant axial anisotropy, while B40 and B44 parametrize the fourth-order anisotropy terms'. The description of the relationships [163] between the true total ZFS terms, i.e. here D, B40 and

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B44 , and the „anisotropy terms‟ implies identification of the two distinct notions. It is inappropriate in view of the definition of the notion of magnetic anisotropy (MA), see, e.g. Refs [19-27]. In general, MA may be due to the anisotropic exchange interactions as well as the single-ion contributions, which ~ originate for the TM ions and the S-state RE ions from the single-ion ZFS terms, H ZFS , whereas for other RE ions from the CF effects within a given ground J(fN)-multiplet. The MA of the latter origin is known as the single-ion anisotropy (SIA), or equivalently the magnetocrystalline anisotropy (MCA). ~T For ECS, instead of the single-ion ZFS terms, the effective total ZFS terms in H SH acting within the ground total spin ST-multiplet are to be considered. The MA phenomenon is quantifiable in terms of the macroscopic quantity called the magnetic anisotropy energy (MAE), which is expressed by the magnetic anisotropy constants Ki. Regardless of the MA origin, MAE is defined as the part of the free energy FE of the magnetic system that depends on the direction of the magnetization M in crystal. ~ ~ The key point is that neither the ZFS terms associated with the single-ion effective spin S , H~ ZFS  ( S ), T nor the ZFS associated with the effective total spin ST of ECS (SMM), H~ ZFS  (ST), defined in Section 3 and 4, respectively, should be referred to as the MA terms. Such identification is, however, common in magnetism literature and has serious implications for interpretation of the ZFSPs. Detailed considerations of the magnetic anisotropy (and anisotropic magnetic properties of magnetic systems) and their relationships with the true single-ion and total ZFS terms are beyond the scope and the size limit of this review and will be provided elsewhere [164]. ~T Incidentally, the SH equivalent to H SH was also referred to as 'single-spin Hamiltonian' [159], which may be misleading when comparing with 'single-ion spin' Hamiltonians, whereas the SH equivalent to HMH was named [160] as 'coupled single-ion Hamiltonian', which is acceptable. The name 'single-ion Hamiltonian' used [161] for the Hamiltonian equivalent to HMH is rather inappropriate since HMH involves summations over single ions. For the tetranuclear Mn cluster in photosystem II centers Peloquin et al. [165] used, quote: 'the general spin Hamiltonian for a system containing n electron and n nuclear magnetic moments, Huncoupled', which was referred to as 'the uncoupled spin Hamiltonian because the individual spin operators are present' - such Huncoupled is equivalent to HMH. The authors [165] have also used 'Hcoupled' referred to as 'the coupled spin Hamiltonian because the individual spin operators are coupled into the total spin operator' - such ~T Hcoupled is equivalent to H SH . The true ZFS terms appearing in the two types of Hamiltonians were properly named [165] and the single-ion spins Si were clearly distinguished from the total spin ST of the system. However, the name 'uncoupled' for the HMH-like Hamiltonians seems not strictly precise, since such Hamiltonians contain also the EI terms, which, due to their nature, do represent „exchange coupling‟ between the individual spins. ~T Kulik et al. [149] for the Mn4OxCa cluster used the SH equivalent to H SH ( HGS), which arisen from their SH equivalent to HMH and was referred to as 'SH of a system with n coupled Mn ions' rewritten, quote: 'in the coupled representation for each exchange multiplet'. That SH included the term HZFS = STDTST , which was properly named as 'effective ZFS term'. Hence the logical link ~T between HMH and H SH ( HGS) is well presented [149]. Often non-specific general names, e.g. „SH‟ or ~T „effective SH‟, were used for Hamiltonians representing, in fact, either HMH or H SH . For example, the name „effective SH‟ was used in, e.g., the reviews by Gatteschi et al. [166] on SR investigations of ~T molecular magnets and by McInnes [167] on spectroscopic studies of SMM for H SH , whereas by Amoretti et al. [168] on INS investigations of MNM for HMH.

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T The relationships between the total ZFSPs of H SH  (ST) and the parameters of HMH  (Si, Sj) may be obtained for particular less complicated cases of ECS, see, e.g. [15,143,144,169]. To illustrate the origin of the ZFSP DT for a simple ECS, e.g. the triplet ST = 1 state arising in a dinuclear complex of two ions with the electronic spin SA = SB = ½, one may consider Eq. (22) provided by Kahn [144]. In this case DT (originally denoted as D) is expressed as: D  [(gz )2 Jx y , x2 - y 2] , where „gz is the deviation of the component gz of the Zeeman factor for the monomeric fragment referred to ge= 2.0024. This deviation comes from the second-order effect of the spin-orbit coupling. Jx y , x2 - y 2 is the magnitude of the interaction between the xy-type ground state of an ion and the x2-y2 type excited state of the other ion.‟ It is clear that the SOC does not directly affect the ZFSP D (= DT) but only ~T indirectly via gz. Note that the relationships between the SOC and H SH  (ST) appear to be misinterpreted in some SMM studies. These aspects will be discussed elsewhere. For description of the properties of ECS, the interplay between the total physical Hamiltonian, (HFI + HCF)i, and that describing the EI, Hex, between the constituent single ions in a given system, is important. Generally, it is difficult to solve the Hamiltonian ((HFI + HCF )i + Hex) and various techniques may be utilized, usually limited to the ground state typically determined by EPR spectroscopy [15,19,24]. Lukens and Walter [55] utilized the method of quantifying exchange coupling in f-ion pairs using the diamagnetic substitution method, which allows the information about the CF contained in the susceptibility of the magnetically isolated analog to be used to analyze the coupling between f-ions without having to determine the CFPs. To describe 'the electronic states of the FeIII-MII exchange pairs' in cyanide-bridged FeIII-CN-MII (M = Cu, Ni) SMM, Atanasov et al. [170] in DFT and LF study utilized the Hamiltonian, their Eq. (1), which contains the physical Hamiltonian terms: Hphys  (HSO + HLF + HZe) for the FeIII ions as well as 'the operator which takes account of the possible single-center anisotropy in the s2 = 1 ground state of NiII', originally denoted ~ as quote [170]: ' HSFS2  DNi ( sz22  32 ) '. Such 'HSFF2', in fact, represents an effective single-ion H ZFS for the NiII ions. Interestingly, the former and the latter Hamiltonian terms belong to different categories, i.e. the physical Hamiltonians and effective ones, respectively. ~T Usage of the Hamiltonians of the type HMH ( Hmicro) and H SH ( HGS) is overviewed below to illustrate their relationships. To explain the magnetic behavior of Ni(II) dimmer [Ni2(en)4Cl2]Cl2 Herchel et al. [130] postulated the „SH‟ that represents the multispin Hamiltonian HMH. For a cluster containing N interacting paramagnetic centers, Liviotti et al. [140] first defined 'the spin Hamiltonian ': H ≡ H0 + Hcf + Hdip, 'where H0 is 'the isotropic Heisenberg–Dirac interaction', Hcf describes the local crystal field and Hdip includes both the intracluster dipole-dipole interaction and the anisotropic exchange'. Such SH represents a multispin Hamiltonian HMS, whereas the ill-named „CF‟ Hamiltonian, Hcf, was explicitly given as the sum SiDiSi, hence, in fact, it represents the true single~ ion ZFS Hamiltonian, H ZFS . In spite of the incorrect name „CF‟, the tensor Di was properly named [140] as 'the zero field splitting tensor'. This HMS was subsequently replaced 'by an effective Hamiltonian HS written in terms of the total spin operator S and whose parameters can be expressed ~T as a function of the single-ion spin Hamiltonian parameters', which in fact represents H SH . The SH

terms were given up to the second order as: H S = S  D  S = 31

q 2

B O

q 2

q 2

q 2

, where Okq are 'the Stevens

q k

operator equivalents defined in the total spin space and B are the corresponding parameters'. These operators are discussed in Section 5 and 8. Maurice et al. [143] for a binuclear system with two Si = 1/2 magnetic centers in the copper

ˆ MS  JSaSb + Sa D Sb + dSaSb, and the acetate complex used both the 'multispin Hamiltonian': H ab

ˆ GS  Sˆ D Sˆ (original notation for the the tensors D is used here). The latter effective 'giant SH': Η ZFS Hamiltonian, i.e. HGS, for this ECS is associated with the total spin ST = 1 and acts only within the space of a total spin multiplet. Similarly, Maurice et al. [171] for a binuclear centrosymmetric Ni(II) complex with the individual spins Si = 1 used both Hamiltonians, i.e. 'multispin Hamiltonian' and the 'giant SH' with the total spin ST = 2, i.e. 'the spin of the ground state of the whole molecule, i.e., the giant spin', expressed in terms of 'the standard Stevens equivalent operators' Okm . Note the

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true ZFS 'tensors' appearing in both types of Hamiltonians were properly named [171] and the singleion spins Si were distinguished from the total spin of the complex denoted as S. For molecular magnetic clusters, such as SMM, characterized by spin ground states with welldefined total spin S and exhibiting ZFS, Waldman and Güdel [139] have derived the MSH relations ~T that represent the total ZFSPs in H SH  (ST) in terms of the ZFSPs for the individual ions as well as the exchange interaction constants appearing in HMH  (Si, Sj). The analytical results [139] illustrate the degree of complexity involved in such relations. Three examples of applications of the MSH relations [139], i.e. for an antiferromagnetic heteronuclear dimmer, the Mn-[3x3] grid molecule, and the SMM Mn12, were considered. The CF=ZFS confusion [28-33] is evident in the papers [138,139,140,141,142], since the SH terms, which in fact represent the true single-ion ZFS terms, were named as the 'crystal field' or 'ligand field' ones. ~T Special cases of the Hamiltonians of the type HMH and H SH , which represent a mixture of both types, have also been utilized. In the HMF-EMR studies of heterometallic [Ln2Ni] complexes, N N Okazawa et al. [172,173] have adopted HMH ≡  H ex (i, j ) +  H~ SH (i) . However, since the HMF-EPR i

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experiments were performed at sufficiently low temperatures, Ln ions were regarded as Ising spins and then the „full Hamiltonian of the system‟ has been reduced to „an effective Hamiltonian‟, in which z the J Ln value was fixed to that of the ground state. The existence of two (or more) different transition ions in the [Ln2Ni] complexes naturally necessitates distinguishing the symbols for the spins as well as the ZFSPs used in the respective Hamiltonians. For this purpose, Okazawa et al. [172,173] z introduced the symbols: J Dy , z , S Niz , SNi, DNi, ENi. Liu et al. [174] have carried out HMF-EMR 1 J Dy 2

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and QTM measurements on single-crystal samples of a mixed-valent SMM, for short Mn4, based on ~T two MnII and two MnIII ions. Both H SH („GSA‟ with „an S = 9 spin ground state‟) and HMH („multispin‟) have been invoked. However, to simplify the calculations the Mn4 system was modeled as a trimmer consisting of a dimmer with „a central spin sA = 4 formed by the two s = 2 MnIII ions‟ and the two MnII (sB1 = sB2 = 5/2) ions. Hence, their „multi-spin‟ Hamiltonian [174] represents, in fact, a mixed Hamiltonian, where only the two MnII ions are described each by an axial ZFSP, i.e. a true HMH, ~T whereas the two MnIII (each with spin s = 2) are described by H SH corresponding to the dimmer with the spin sA = 4. The advantage of this approximation is that: „By doing so, „the dimension of the multi-spin Hamiltonian is reduced from 900900 to 324324, which allows much faster computations.‟ [174]. ~T In several studies both types of Hamiltonian HMH ( Hmicro) and H SH ( HGS) have been invoked as illustrated by selected examples. For simulations of the HMF-EPR spectra of SMM Mn12tBuAc Barra et al. [157] used „the giant-spin Hamiltonian (GSH) compatible with the tetragonal symmetry of the crystal and including up to the sixth-order terms‟ as well as „multispin Hamiltonian‟ assuming a „simplified five-spin model‟. Both Hamiltonians were denoted by the generic symbol H, however, clear distinction between the spins involved was maintained: HMH  {si, i = 1 - 5} and HGS  {S  ST }. For SMM Mn3 complex containing three MnIII ions each with spin s = 2 yielding the giant spin S ( ST) = 6 Feng et al. [148] used HMH named as „microscopic SH‟ for MnIII ions and „giant SH (S = 6)‟ for the whole Mn3 complex. The respective ZFSPs were well distinguished and properly named as „ZFS‟ parameters. Liu et al. [175] for Mn3 complex first used the giant SH introduced as: ' The GSA treats the total spin S associated with the ground state of a molecule to be exact. For Mn 3, this results in 2S + 1 (=13) multiplet states that can be described by the following effective spin Hamiltonian', whereas the multispin Hamiltonian was described subsequently as 'a more physical model, which takes into account the zfs tensors of individual ions and the coupling between them'. The true ZFS terms in HGS were named as: 'the so-called zero-field splitting (zfs) anisotropy', which is inappropriate [164], whereas those in HMH as 'the second-order zfs in the local coordinate frame of each MnIII ion', which is appropriate. Liu et al. [175] considered 'mapping the spectrum of a Mn3III SMM obtained via diagonalization of a multispin (three s = 2 spins) Hamiltonian onto that of a giant-spin model with spin S = 6'. In principle, such procedure may enable derivation of the expressions relating the global

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parameters in HGS for the whole molecule with the parameters in HMH for the constituent ions. However, no explicit expressions were provided since only numerical solutions were feasible. For clarity and to avoid misconceptions, it is very important to distinguish the single-ion spins Si, i = 1, 2,…,N, localized on the ions that constitute a given ECS and the total spin ST of the system, as well as the respective associated ZFSPs. The importance of this distinction, evident in the papers [148,157,172,173,174,175], may be further reinforced by additional examples of good practices. To distinguish the respective associated ZFSPs, Nakano and Oshio [150] denoted them appropriately as, respectively, „local zfs parameter Dion‟ and „the molecular zfs parameter Dmol‟. Alternatively, sometimes the small letter si or the capital letter Si (and the associated ZFSPs, e.g. Di, Ei), whereas the capital letter S (and the associated ZFSPs, e.g. D, E) are used for this purpose [138,176,177]. The spins Si and ST (and the associated ZFSPs) are clearly distinguished also in the reviews devoted to SMM [152,160,168,169,178 , 179 ]. Other examples may be found in the studies of single-chain magnets based on, e.g., MnIII-FeIII-MnIII trinuclear SMM with an ST = 9/2 spin ground state [180], or heterometallic MnIII-NiII chain with ST = 3 magnetic units [181], and heterometallic SMM quasi-linear {Mn2Ni3} clusters [182], or magnetostructural correlations in tetrairon(III) SMM with ST = 5 [177], and SMM Mn3 ST = 6 complexes [148] or Ni2Ln complexes [ 183 ] as well as the ECS, e.g. [LFeGd(NO3)3]2O complexes [184], the Mn4OxCa cluster [149], binuclear Ni(II) complex [171], and the tetranuclear Mn cluster [165]. The total spin of SMM (MNM) is most commonly denoted as S, especially if the constituent single-ion spins are not discussed, as, e.g., in the review [159]. Lack of distinction between the single-ion spins and the total spin, as well as between the respective ZFSPs associated with each type of spins, may hamper analysis of experimental data. Inconsistent usage of the small and capital letters for the ZFSPs, as e.g. in [168,185], may lead to ambiguities. The notation [168,185] denoting the q ZFSPs associated with the single-ion spins si as bk (i ) , whereas the ZFSPs associated with the total q

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spin S as Bkq , creates an ambiguity since the symbols bk together with the scaling factors are welldefined in EMR area (see Section 6.2) and have different meaning. Concerning the literature dealing with the Haldane gap systems, especially those based on the Ni2+ ions, it is also important to distinguish the total spin ST (ST = 1) arising from the exchange interactions within the whole system and the single transition ion spin Si. Since the value of the quantum number ST equals that for the Ni2+ (Si = 1), sometimes ambiguities occur concerning the actual meaning of the spin considered in the studies of the Haldane gap systems as reviewed [186]. Hence, comparing the values of the respective ZFSPs from various sources one must verify to which spin these ZFSPs refer to. Finally, we note that several other misconceptions have been identified in the SMM-related literature, which arise due to two major factors. First factor is related to mixing up the properties of the ZFS associated with the total spin ST of SMM with those of the ZFS associated with the singleion spins Si. Such misconceptions bear on the interpretation of the energy barrier for the ~T magnetization reversal in SMM. Second factor is due to a detachment of the Hamiltonian H SH from its physical origin. This factor is clearly noticeable in the SMM-related papers, in which the Hamiltonians of this type are named purportedly as 'CF' Hamiltonians. Most recent cases of the CF=ZFS confusion of this category are discussed in the review [33]. Importantly, such detachment ~T affects also the considerations of the global symmetry of H SH , which is determined, in principle, by the symmetry of a given single magnet molecule as well as by the local site symmetry of the single ions forming the SMM. The methodology for its determination seems to be not fully understood as yet. Doubts arise concerning the considerations of the symmetry of the Hamiltonians of the type HMH ~T ( Hmicro) and H SH ( HGS) in the most recent reviews by Hill [187] on the QTM in Mn12 SMM and Liu et al. [188] on a microscopic and spectroscopic view of QTM in SMM and in the paper by Liu and Hill [189]. These misconceptions as well as the doubts concerning the symmetry aspects in [187,188,189] require detailed considerations, which may be carried out elsewhere. As an aside, we ~T note that comparison of the terminology used for the Hamiltonians of the type HMH and H SH ( HGS) in the reviews devoted to QTM in SMM reveals how this terminology has evolved and become more meaningful and widely accepted. The examples are provided, e.g. by the recent reviews [187,188]

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and an earlier one by del Barco et al. [190], who used non-specific names for HGS, e.g. 'the simplest zero-field Hamiltonian'. Concluding Section 4: The notions defined herein may be now contrasted with the concept map in Fig. 3. The conceptual definitions of various Hamiltonians and their origin, i.e. the exchange interactions Hamiltonian Hex, ~T the multispin Hamiltonian HMH, and the effective total SH or giant SH: H SH  HGS, succinctly overviewed in Fig. 3, as well as the interrelationships and distinctions between these notions have been elucidated. This has enabled to establish the qualitative connection between the Hamiltonians: ~T HMH and H SH  HGS as well as their respective wavefunctions. This connection forms an analogous interface as that between the physical and effective Hamiltonians discussed in Section 3. This knowledge should provide a comprehensive background for description of magnetic and spectroscopic properties for the ECS of transition ions and SMM (MNM).

cr

5. Stevens, Wybourne, and other operators

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To overview this section, in the diagram shown in Fig. 4 we illustrate the two major types of ~ operators, i.e. the Stevens and Wybourne ones used to express the Hamiltonians: HCF (HLF) or H SH ~

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T ( H ZFS ) and HMH or H SH ( HGS), as well as the distinct properties and applications of these operators.

Fig. 4. Summary diagram for the Stevens and Wybourne operators.

5.1. Historical perspective and origin of the Stevens and Wybourne operators A brief account of the historical development of the two major operator notations, i.e. the Stevens notation and Wybourne one, and the related notations used in literature is pertinent. Both notations have originated in CF (LF) studies, whereas the Stevens notation precedes the Wybourne one. Notably only the Stevens notation has later been adopted in EMR studies. The reviews [28,29] may

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be consulted for details and references. Some remarks on the origin of these operators are pertinent to expose their general characteristics, which bears on (i) the conversion relations between the ZFS parameters expressed in the Stevens notation and the conventional ones discussed in Section 6.2 and (ii) the distinct properties of the CF (LF) parameters and the ZFS ones expressed in the respective notations discussed in Section 7. The notation introduced in the CF theory, as historically first, had represented the CF Hamiltonian explicitly in terms of the Cartesian coordinates: HCF  (x, y, z). This notation was arising from the initial application of the (crude) point charge model to the nearest neighbor ligands, initially in octahedral environments exhibiting cubic CF and later in axial symmetry complexes (see Fig, 2). To simplify calculations of the matrix elements of HCF in the basis of states within a given ground J(fN)multiplet of the RE ions ({M  |J, MJ>}, where J is a good quantum number), Stevens [191] proposed in 1952 a replacement of the Cartesian coordinates (x, y, z) in HCF by the Cartesian components (Jx, Jy, Jz) of the total angular momentum operator J = (L + S). This replacement requires taking into account the commutation relations between the operators, which do not exist between the Cartesian components. By analogy, this replacement has later been adopted also to express HCF within a given ground L(dN)-term for the TM ions in terms of the components (Lx, Ly, Lz) of L. This method became known as the operator equivalents method, whereas the particular combinations of the operators (Jx, Jy, Jz) or (Lx, Ly, Lz) appearing in HCF were later named as the Stevens operators and most commonly denoted as Okq with the coefficients Bkq [28,29]. Importantly, due to their nature the Stevens operators apply only within a given ground J(fN)-multiplet of the RE ions or L(dN)-term of the TM ions and do not act within the states of the whole 4fN and 3dN configuration (see Section 7). Importantly, before the advent of the CF theory, based on the general angular momentum theory, Racah [ 192 ] in 1942 has introduced in the theory of atomic spectra the „irreducible-tensor operators‟, denoted originally as Tkq. The Racah operators Tkq(J) are functions of the generic angular momentum operator J. It turns out that these operators are actually first example of the operator equivalents to the spherical-harmonics, hence, they belong to the class of the spherical-tensor operators (STO) [28,29]. Independently, in the 1965 book Wybourne [193] has introduced into the CF theory the one electron operators Ckq(i) and the combined many-electron ones Ckq, which were later named as the Wybourne operators. Both the Racah operators and the Wybourne ones act within the states of the whole 4fN and 3dN configuration and also belong to the STO class. As reviewed [28,29], several other types of operators of the STO class have independently been developed and used in EMR and optical spectroscopy area. By the virtue of the Wigner-Eckhart theorem [36], all operators of the STO class, are simply related to the Racah operators [192] by the respective reduced matrix elements. 5.2. Usual Stevens operators versus the extended Stevens operators (ESO) The operators Okq originally introduced by Stevens [191] comprised the rank k = 2, 4, and 6 and the positive components 0  q  +k only [12-18]. As reviewed [28,29], this set of the Stevens operators was sufficient for the then considered site symmetry cases with orthorhombic being the lowest symmetry. It turns out that the operators [191] Okq with 0  q are the operator equivalents to c the real tesseral-harmonics, Z kq , and hence belong to the class of the tesseral-tensor operators (TTO)

[28,29]. Various sets of the TTO operators of the Stevens type Okq (X), X = S, J, or L, which included also some negative q components, have been introduced in literature as reviewed [194,28,29]. Hence, to distinguish these operator sets from the original Stevens set [191] the latter set is denoted as the usual Stevens operators. In 1985 the usual Stevens operators Okq (X), 0  q  +k, and the existing partial listings with negative q were systematized and extended to all q components: -k  q  +k for k = 1 to 6 [194]. The extended Stevens operators (ESOs) Okq (X) form a complete set the TTO of this type [194]. Importantly, the transformation properties of the ESO Okq (X), which were indispensable for practical applications in EMR and CF theory, were also derived by computer algebra [194].

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To facilitate the experimentalists‟ appreciation, the explicit definitions of the ESOs Okq (X) for k = 2, 4, and 6, adapted from Newman and Urban [195] (note that the abbreviation “XY – YX“in [195] should read “XY + YX“; see also the listing in book by Misra [18]) are provided in Table A1 in Appendix 1. Importantly, the ES operators [194] Okq (X) have been generalized [196] to any rank k and any quantum number X of the angular momentum operator X = S, J, or L. For completeness and verification, other recent listings of the ESOs Okq (X) are surveyed below. Ryabov [ 197 ] has provided further extensions and clarifications on the Stevens operator equivalents used in EMR. Corrections to the listings of the matrix elements of the ESO available in literature have been provided [196,197]. Independently, albeit using different symbols, Rotter [198] and Chilton et al. [199] have provided a complete list of the ESO up to k = 6, which have been incorporated into the computer program McPhase and Phi, respectively. Analysis of the respective listings confirms that the operators provided by Rotter [198] and Chilton [199] coincide with those listed in Table S1. Importantly, Chibotaru and Ungur [103] have recently developed ab initio method for calculation of anisotropic magnetic properties of complexes and provided unique definition of the so-called 'pseudospin' Hamiltonians and their derivation. In view of the existing approaches, several aspects concerning the terminology and interpretation of methodology in [103] require separate comments, which are, however, beyond the scope of this review. Here, we mention only aspects related to the operator notations. In Eq. (42) of [103] 'the other Stevens operators' have been defined as quote: 'the Stevens operators On m to be proportional to Yn m with the same proportionality coefficient kn for all m'. The authors [103] have commented that 'other versions of these operators have also been proposed in the last years.74, 75'; their Ref. 74 and 75 corresponds to [197] and [196] here, respectively. Also the proportionality coefficients between the irreducible tensor operators Onm and  mn introduced in [103] and 'the conventional Stevens operators Onm (St) and  mn (St)' have been provided. Explicit

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listings of the second-rank operators, O2 m , which resemble closely the ESOs, were provided in Eq. (50) of [103]. However, these operators represent another version of the Stevens operator equivalents that belong to the TTO class with the same O20 , while other components differ from the ESO by normalization coefficients as indicated in [103]. Hence, the existing conventions reviewed in [28,29] and the distinction between the usual Stevens operators and the ESO [194,196] seem to be not applied in [103,102]. 5.3. Adoption of the Stevens operators and other notations in EMR studies The notation introduced in the EMR and magnetism literature, as historically first, had ~ ~ represented the SH (ZFS) Hamiltonian, H SH ( H ZFS ), in terms of the Cartesian components of the „spin‟ ~ operator S (Sx, Sy, Sz), i.e. in fact, the effective spin S . This notation was later labeled the conventional SH notation [28,29] to distinguish it from the tensor operator notations introduced ~ subsequently. The explicit form of H ZFS  (Sx, Sy, Sz) was arising from its original derivation based on the microscopic spin Hamiltonian (MSH) theory outlined in Section 3. The nature of the „spin‟ operator S in SH has later been generalized and, in fact [28,29], „spin‟ may mean either the effective ~ spin S or the fictitious 'spin' S' for single transition ions as well as the total spin ST ascribed to a given ECS. By analogy with the developments in the CF (LF) theory, in the EMR area the ~ components (Sx, Sy, Sz) in the conventional H ZFS were later replaced by particular combinations of the usual Stevens operators being functions of (Sx, Sy, Sz). The introduction of this operator notation in the EMR area has led to specific conversion relations between the conventional ZFSPs and those expressed in the Stevens notation (see Section 6.2). In the initial two decades, the usual Stevens operators [191] Okq (k = 2, 4, 6; 0  q) have gained dominant popularity in the EMR and magnetism literature [12-18]. However, these Stevens operators, unlike the STOs, suffer from serious drawbacks. First general drawback is the lack of

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normalization, which is not critical since it affects only the conversion relations. This drawback has prompted various attempts to introduce the normalized sets of the TTOs, which correspond more directly to the STOs [28,29]. However, applications of various types of the normalized TTOs in the EMR area have meet with limited success so far [28,29]. Second drawback was then the lack of a full set of the Stevens operators, which would have included also the operator equivalents of the imaginary tesseral harmonics, Z kqs , i.e. the negative q components. These components are indispensable for site symmetry lower than orthorhombic, i.e. monoclinic and triclinic [28,29]. The introduction [194] of the ESO, Okq (X), has removed the second drawback. As reviewed [28,29], in order to overcome these drawbacks, an abundance of other operators of the STO and TTO type have independently been introduced in the EMR area since the early 1960's. One type of such operators was introduced independently by Buckmaster and by Smith and Thornley (BST) - for references, see the reviews [28,29]. However, the BST operators, most commonly denoted as Oq( k ) with the coefficients Bqk , were used only occasionally in the early EMR studies [28,29]. It turns out that the BST operators also belong to the STO class and are mathematically equivalent to the Wybourne operators Ckq, which have been used exclusively in the CF studies [3-8] and not in the EMR studies [12-18]. In the hindsight, the present situation regarding the operators existing in the EMR area may be described as a maze difficult to follow, especially by experimentalists. To simplify the navigation in this maze, classification of the various operators used in the pertinent literature and their specific properties have been summarized in the reviews [28,29]. 5.4. Hamiltonians versus operators

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Correlation between the general properties of the operators discussed in Sections 5.1 and 5.2 with ~ those of the Hamiltonians HCF (HLF) and H ZFS outlined in Sections 2 and 3, respectively, may be helpful for deeper understanding of the major points. One important point to keep in mind is that mathematically any type of operators forms an independent set, which exists in its own rights and has distinct properties as discussed below. The major difference between the Stevens operators and the Wybourne operators resides in the class of operators, i.e. the TTO and STO, respectively, which determines their distinct properties [28,29]. The usual [191] Stevens operators Okq originated as the

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operator equivalents due to replacement in HCF  {(x, y, z)  (Jx, Jy, Jz)}. The usual [191] and extended [28,29,194] Stevens operators Okq used in the CF (LF) area are functions of either J: Okq (J) - within a given J(fN)-multiplet of RE ions or L: Okq (L) within a given L(dN)-term of the TM ions. The Wybourne operators originated via a different route and have become dominant in the optical spectroscopy area since they are applicable within the whole 4fN and 3dN configuration. The Stevens operators play a smaller role in CF (LF) area since they are less versatile to express ~ ~ ~T HCF, whereas have become most widely used to express H SH ( H ZFS ) as well as H SH ( HGS), which underlie the EMR [12-18] and magnetism areas [20,24]. In the CF (LF) Hamiltonians Okq (X), X = J or L, may appear only, whereas in the SH (ZFS) Hamiltonians X must be replaced by the respective ~ „spin‟ angular momentum operator: X = S [in fact: S ], S′, or ST. Unfortunately in many EMR papers, the nature of the operator X is not explicitly indicated. This leads to an apparent identification in the EMR literature of the CF (LF) Hamiltonians with the SH (ZFS) ones and thus contributes to the CF=ZFS confusion discussed in [28-32] and most recently in [33]. To avoid such confusion it is important to keep in mind that as argued in Sections 2, HCF (HLF) are physical Hamiltonians being ~ ~ ~T electric in nature, whereas the ZFS terms in H SH ( H ZFS ) and H SH ( HGS) are effective Hamiltonians being magnetic in nature, irrespective of the type of operators in which these Hamiltonians are expressed. Concluding Section 5: The major points discussed herein may be now contrasted with the concept maps in Fig. 1 and 3 and the summary diagram in Fig. 4. This should provide a comprehensive conceptual background for the crucial notions and the operators used to express them and thus may facilitate considerations in the

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subsequent sections. An overview of the current status of applications of the (extended) Stevens operators as well as discussion of various other problems concerning the usage of the ESO in recent literature are provided in Section 8. 6. Forms of Hamiltonians and definitions of the associated parameters 6.1. Crystal field (ligand field) Hamiltonians 6.1.1. Forms of HCF (HLF) and the associated parameters

cr

ip t

For reference, we provide below the most general forms of HCF (HLF), i.e. suitable for the lowest triclinic (C1 and Ci) symmetry cases, expressed in terms of the Stevens and Wybourne notations. In terms of the ESOs Okq (J or L), defined in Section 5, within a given J(fN)-multiplet or L(dN)-term of a transition ion, HCF (HLF) may be expressed, respectively, as [2-13]:

H CF (ESO)   BkqOkq ( J or L )  Akq r k θk Okq   Ckqθk Okq , k,q

k,q

k,q

(1)

q

q

d

M

an

q

us

where the CFPs Bk ( Ak , C k ) are all real, whereas the so-called multiplicative Stevens factors k = , , and  for the rank k = 2, 4, and 6, respectively, are tabulated [2-13]. The summation in Eq. (1) includes all q components: -k  q  +k, whereas specific limits on the non-zero components q are governed by the local site symmetry and group theory [2-13]. The limit to the ranks k (k  2l) for the operators acting within the ground multiplet and the associated CFPs arises from the orbital quantum number (l) of a given configuration, namely, J(fN): l = 3 yields k = 2, 4, and 6 or L(dN): l = 2 yields k = 2 and 4. Usually the even k terms are only considered, whereas the odd k terms are discussed in Section 7.1. The implicit dependence of the ESOs Okq in HCF (HLF) on J or L should be always kept in mind. Note that Eq. (1) given in the review [110] is basically equivalent to Eq. (1) above, however, its interpretation requires additional explanations provided in the review [33]. In view of different forms of HCF and variety of CFP symbols used in the nominally Stevens q

q

q

Ac ce pt e

notation, the reference CFP symbols, conforming to the prevailing conventions [3-8]: Bk , Ak , C k , are defined in Eq. (1). It is advisable to check the original notations, since some authors use the CFP symbols in a different context. Note that adoption of the convention [8] k  1 requires recalculations of the numerical CFP values, as discussed for numerous cases found in literature [200,201,202,203,204,205,40]. Chilton et al. [61,199] have recently utilized the ESOs Okq (L) to express HCF(ESO) for d- and f-block complexes. It turns out that their CFPs denoted as Bkqi are q

equivalent to the CFPs C k defined in Eq. (1). Hence, proper rescaling must be performed when comparing the CFPs obtained from the program Phi [61,199] with literature data. In terms of the Wybourne (Wyb) operators [193] C q(k ) (Ckq), HCF (HLF) within the whole dN or fN configuration may be represented in two general forms, i.e. compact and expanded. In the compact form [28,29], i.e. with -k  q  +k, the triclinic HCF (HLF) is given in several equivalent representation, e.g. most commonly as [3-8,206]: H CF (Wyb )  BkqCq( k )  BkqCkq  Bqk Cq( k ) (2)

 kq

 kq

 kq

where the CF (LF) parameters Bkq ( B ) are in general complex, except for q = 0, and Cq( k ) ' s (Ckq‟s) are to k q

be summed over all unpaired electrons of the unfilled shell of the RE (TM) ion, i.e.: Cq( k ) 

C  ,  k q

i

i

i

The same limits on the rank k and the non-zero components q apply as for Eq. (1). The third alternative representation [206] in Eq. (2) is also widely used, however, since the CFP symbols Bqk resemble closely those in the Stevens notation in Eq. (1) this representation is not recommended. In the expanded form [28,29] i.e. with 0  q  +k, the triclinic HCF is equivalently given as [7,8]:

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H CF (Wyb ) 

 k

 (k )  Bk 0C0  

 Re B C k

kq

(k ) q

q 1

 q q   1 Cq( k )  i Im Bkq C( kq )   1 Cq( k )  . 







(3)

Often in Eq. (3) the real parts ReBkq and the imaginary ImBkq parts of the complex CFPs Bkq in Eq. (2) are replaced by the symbols Bkq and Bk-q, respectively, yielding a simplified form (with 0  q  +k):

H CF (Wyb ) 

 k

 (k )  Bk 0C0  

 B C k

kq

(k ) q

q 1

 q q   1 Cq( k )  iBk q C( kq )   1 Cq( k )  







(4)



cr



ip t

The representation in Eq. (4) is not recommended, since it introduces unintended ambiguities. Various existing crystal field parameterizations [3-8] have been reviewed by Görller-Walrand and Binnemans [207]. Caution is needed since as reviewed in [208] incorrect or inconsistent expressions for CF Hamiltonians as well as various conventions for the phase factors in Eq. (3) and (4) are also employed in literature. It should be noticed that, in general, the signs of the low symmetry CFPs with q odd depend on the sequence of operators in the two terms in the round brackets used in Eqs. (3) and (4). As reviewed [208], an alternative phase convention: Cq( k )   1q C( qk ) is used in some low symmetry CF studies, whereas other mixed conventions are also occasionally adopted, e.g., like









q

q

us

Bkq Cq( k )   1 C( kq)  iBk q C( kq)   1 Cq( k ) [8]. The relations between the Wybourne CFPs and N

the conventional ones for 3d ions in axially symmetric crystal fields were discussed in [209].

an

6.1.2. Relations between the CF (LF) parameters expressed in the Stevens and Wybourne notations

Ac ce pt e

d

M

As outlined above, different forms of CF (LF) Hamiltonians [28,29,207,208] and variety of CFP symbols [1-13] as well as confusing relationships (occurring, e.g. in the review [110] as discussed in [33]) have been used in literature. This may lead to confusion and hampers direct comparison of CFPs from various sources. These problems are compounded by the ambiguities and usage in literature of the CFP symbols with incorrect meaning. For completeness, the conversion relations between the CF (LF) parameters expressed in the ESO notation in Eq. (1) and those in the Wybourne notation in Eq. (3) and (4) are provided in Table A2 in Appendix 2 together with the full set of the conversion factors for k = 2, 4, 6 and all pertinent values of |q|. It should be emphasized that these relations shall be employed exclusively for conversions between the CFPs expressed in the Stevens and Wybourne notations. Crucial point is that for the reasons discussed in the reviews [33,34] these relations shall never be employed for crossconversions between the CFPs (defined in Section 6.1) and the ZFSPs (defined in Section 6.2 below) or for inter-conversions between any types of the ZFSPs. 6.1.3. General comments concerning the forms of HCF (HLF) Apart from the Stevens and Wybourne CFP notations, various types of notations, nowadays called the conventional CFP notations [1,2,5,9] have also been developed. However, the origin of these notations differs from that of the conventional SH notation described in Section 5.3. The conventional CFP notations originated by ascribing the sets of some unique conventional CFPs directly to the specific energy level splitting for particular 3dN ions at trigonal, tetragonal, and orthorhombic symmetry sites. If a given ground J(fN)-multiplet of the RE ions or L(dN)-term of the TM ions is only considered, the matrix elements of HCF(Jx, Jy, Jz) or HCF(Lx, Ly, Lz), respectively, may be projected from those of HCF expressed in the Wybourne (or Racah) tensor operators within the whole basis of states of the 4fN and 4dN configuration to those in the restricted basis of a given ground multiplet. This yields the specific Stevens factors [12,13] k in Eq. (1). Hence, specific conversion relations arise between the CFPs expressed in the respective notations: the (usual) Stevens, Racah, and Wybourne operators, whereas even more complicated relations arise between those CFPs and the conventional CFPs. Comprehensive derivation of the pertinent relations would be worthwhile, however, such relations must be established on the case by case basis. Incidentally, the survey of the usage of the terms „CF‟ and „LF‟ discussed in Section 2 has also indicated the perception concerning the Stevens and Wybourne notations, quote [64]: „In addition to

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33

ip t

the most commonly used Stevens notation50 with an equivalent operator method, the ligand-field term in CONDON can be described by Wybourne notation51 as well.‟ A pertinent note of caution concerning the Stevens and Wybourne notations by Troć et al. [50], who employed the Wybourne „CF‟ parameters denoted Bkq, may be quoted: „The Bkq parameters should not be mixed up with those used in the more common equivalent Stevens operators method according to which the CF Hamiltonian acts on the two-electron functions restricted to the ground [3H4] multiplet in the pure RussellSaunders coupling scheme.‟. In both papers [50,64], however, the references of historical value were only provided, i.e. for the Stevens notation [191] and Wybourne one [193]. Note that a variety of symbols, other than the prevailing ones Bkq defined in Eq. (2) to (4), is being used for the Wybourne operators and the associated CFPs. For example, Novák et al. [52,210] denoted CFPs as Bq(k ) , whereas Kebaili and Dammak [53] and Apostolidis et al. [50] as Bqk , which

cr

resembles the Stevens CFPs. Ryabov [211] has introduced a new notation for the CFPs Bkqc and Bkqs in terms of the cosine and sine real components of the Racah‟s spherical harmonics. Although, these CFPs are simply related to the Wybourne CFPs, denoted in [211] as Bkq , the need for yet another CFP

us

notation is questionable. 6.2. Spin Hamiltonians and zero-field splitting (ZFS) Hamiltonians ~

~

an

6.2.1. Forms of H SH ( H ZFS ) and the associated parameters

~

~

M

For reference, we provide below the most general forms of H SH ( H ZFS ), i.e. suitable for the lowest triclinic (C1 and Ci) symmetry cases, expressed in the ESO notation [194,196] discussed in Section 5 and the conventional one. In terms of the ESOs, Okq , the general SH form including the Zeeman electronic (Ze) term and the ZFS term is given as [12-21,28,29,212]: ~ ~ ~ H  H Ze  H ZFS   B B  g  S   Bkq Okq S x , S y , S z   B B  g  S   f k bkq Okq S x , S y , S z , (5)









d

where g is the spectroscopic splitting factor,  B - Bohr magneton, B - the magnetic induction, and

Bkq and bkq are the ZFSPs associated with the ESOs Okq . The summation in Eq. (5) includes all q

Ac ce pt e

components: -k  q  +k, whereas specific limits on the non-zero components q are governed by the local site symmetry and group theory [12-21,28,29,212]. The limits for the spin operators and ZFSPs to the ranks up to k = 2S and the even k are discussed in Section 6.3 and 7.1, respectively. The „spin‟ ~ S may have the meaning [28,29] of either the effective spin S , the fictitious 'spin' S', or the total spin ST (see Section 3 and 4). As pointed out [212], in some papers the scaling factors defined as [28,29]: f k = 1/3, 1/60, and 1/1260 for k = 2, 4, and 6, respectively, are missing in SH. This creates an ambiguity concerning the meaning of the reported values of the ZFSPs, which could represent Bkq q if mistakenly small bkq were given in SH instead of Bkq or the true bk were meant - if the symbols q f k were mistakenly omitted from SH. As mentioned in Section 3 and 6.2, the notation bk (i ) without the factors f k used by Amoretti et al. [168] and Carretta et al. [185] arises from different

considerations and appears to represent, in fact, Bkq (i) . ~

~

The tilde (~), which distinguishes the effective nature of H SH ( H ZFS ) from the physical nature of HCF (HLF), is most often omitted for convenience in literature [28,29]. It should be kept in mind that the symbols Bkq in Eq. (5) represent Bkq (ZFS/ESO), whereas those in Eq. (1) represent Bkq (CF/ESO). However, this explicit way of indicating the distinction between the CF parameters and the ZFS ones expressed in the ESO notation [194,196] is impractical. Instead, below the meaning of the symbols will be sufficiently indicated in text or it may be unambiguously inferred from the context. The general second-rank ZFS Hamiltonian [28,29] in the conventional SH notation is expressed as [12-21]: ~ (6) H ZFS = S  D  S .

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34

Historically, this form has appeared in literature earlier than that in Eq. (5) [28,29]. Two conventional ~ orthorhombic H ZFS forms exist [12-21, 28,29]:

~ H ZFS  D[S z2  S3 (S  1)]  E (S x2  S y2 )  H ZFS  Dx S x2  Dy S y2  Dz S z2 .

(7)

ip t

The forms in Eq. (7) are physically equivalent, since they yield the same splitting of energy levels. However, one of the three ZFSPs appearing in the second form is interrelated with the remaining two ZFSPs. Hence the number of independent ZFSPs is equal to two, i.e. the same in both forms. ~ Importantly, the truncated forms of the conventional orthorhombic H ZFS , i.e. with one of the three Diterms in Eq. (7) arbitrarily omitted, were shown invalid [213]. Various conventional expressions for the 4th- and 6th-rank ZFS terms for orthorhombic or higher symmetry cases as well as for the secondrank ZFS terms for lower symmetry cases existing in literature have been reviewed [28,29].

Dyy   B22  B20 , Dxz  Dzx  B21 2,

Dzz  2 B20 , Dyz  Dzy  B21 2 .

(9)

M

Dxx  B22  B20 , Dxy  Dyx  B22 ,

an

us

cr

6.2.2. Relations between the ZFS parameters expressed in the ESO and conventional notations The orthorhombic ZFSPs (D, E) in Eq. (7) are related to those in the ESO notation in Eq. (5) as: 2 0 0 2 D = 3 B2 = b2 , E = B2 = 1/3 b2 . (8) The incorrect relationships between the conventional ZFSPs Dij in Eq. (6) and those in the ESO notation ( Bkq and bkq ) for orthorhombic and lower symmetry appearing in pre-2000 literature were reviewed [212]. Note that the conversion relations for the conventional triclinic ZFSPs Dij given in Eq. (5) in the review [110] turn out also to be incorrect as pointed out in [33]. Hence, for completeness the correct ones [214,215,216] are reproduced here:

6.3. Higher-order terms in the generalized spin Hamiltonians (GSH)

Ac ce pt e

d

Two types of the higher-order terms may be included in the GSH [28,29]: (i) the higher-order ZFS (HOZFS) terms and (ii) the higher-order field-dependent (HOFD). The true HOZFS terms arise for the ECS with the total spin ST greater than S = 7/2 and must be distinguished from the 'usual' well-known ZFS terms with the rank k = 2, 4, and 6 that exist for the highest single-ion spin S = 7/2 for Gd3+ or Eu3+ [12-21,28,29]. Note that for the reasons discussed in Section 7.1, the odd-rank ZFS terms with k = 1, 3, 5 are not admissible. The HOFD terms in GSH apply both for single transition ions and ECS. The studies of the HOFD terms for single transition ions and their implications for HMF-EMR measurements have recently been reviewed [37]. Concerning the higher-order terms of the type (i), we recall that from the point of view of ~ group theory the systems with spin S require the ZFS terms in the effective SH H ZFS with the spin operators of the even ranks up to k = 2S. This yields the limit k = 2, 4, and 6 for the S-state ions with S = 7/2. Hence, in principle, the studies of high-spin complexes, like Fe8 and Mn12 with ST = 10 or Fe19 with ST = 33/2, may require the respective HOZFS terms in the giant SH. For example, for ST = 10 the spin operators of the even ranks up to k = 20, including the 'usual' ZFS terms and additionally the HOZFS terms beyond k  8 may be invoked. However, the definitions of the higher ranks operators, greater than S = 8, and thus listings of their matrix elements have not been available as yet. Only recently the general theoretical framework for the higher-rank ESO has been developed [196,197]. A survey of recent SMM literature indicates what follows. In the majority of SMM studies the giant ZFS terms of the second-rank and, to a lesser extent, of the fourth-rank have been included. A note of caution is pertinent concerning the truncated fourth-rank ZFS terms being occasionally used in SMM studies. A few pertinent examples, like the conventional expressions: AS z4 or C (S4  S4 ) , or equivalent ones in the ESO notation, are discussed below. In general, such truncations lead to specific relationships between various second- and fourth-rank ZFSPs, which are often not realized when comparing the ZFSP values. Thus usage of truncated fourth-rank ZFS terms may have detrimental implications for interpretation of the ZFSP values, as pointed out, e.g., for Fe3+ ions in BaTiO3 [217] and Fe2+ (S = 2) ions in K2FeF4 [218]. However, a full scale survey of various truncated ZFS

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SH, as:

Ac ce pt e

d

M

an

us

cr

ip t

Hamiltonians of the second- and fourth-rank used in EMR and magnetism studies as well as their consequences is beyond the scope of this review. ~T The HOZFS terms in the GSH for ECS beyond the rank k = 4, either at the level of HMH or H SH , have rarely been even mentioned in SMM literature, and never included for practical purposes. This is mainly due to the lack of suitable theoretical framework as well as the fact that such terms would present considerable computational challenges on top of the already very cumbersome calculations incorporating the second- and some fourth-rank giant ZFS terms. Gatteschi [219] mentioned the HOZFS terms in the GSH for SMM, quote: 'In fact for a spin with S = 10 the zero field splitting Hamiltonian should include terms up to the order 20 and the high-order terms determine sizable deviations from the regular pattern.' and surmised that such terms could better reproduce the single crystal EMR spectra for SMM with ST = 10. Waldman and Güdel [139] have discussed the 'higher-order spin terms in the effective SH' for SMM Mn12 but only of the type S z4 . The authors [139] remarked that 'higher-order spin terms, most of which are experimentally inaccessible and thus disregarded, have a crucial influence' and 'the higher-order spin terms originate from either microscopic anisotropy terms such as single-ion ligand-field terms, or from a mixing between the different spin multiplets...'. It is important to note that invoking in [139] 'the ligand-field interactions / terms' in the context of the SH for MNM is a serious case of the CF=ZFS confusion as discussed in the review [33]. Survey of the pertinent references [140,185,220,221], cited in [139], reveals what follows. Liviotti et al. [140] have presented 'a quite general and straightforward procedure to study the contributions to the high-order anisotropy terms due to the Smixing in magnetic clusters', but only the the fourth-rank terms have been considered. Wernsdorfer and Sessoli [220] invoked only the fourth-rank terms C (S4  S4 ) . Prokof‟ev and Stamp [221] noted only that 'even very small higher-order transverse couplings (up to the 20th order in S+ and S-) can make important contributions to ∆10', i.e. 'tunneling matrix element ∆10'. Carretta et al. [185] mentioned that 'in order to evaluate S-mixing effects,... the system can be still described as an Q effective spin S = 10, provided the spin-Hamiltonian (3)[Hsub  ( BK , K = 2, Q = 0, 2; K = 4, Q = 0, 2, 4) + HZe ] is properly modified: the parameters of the Stevens operators are renormalized, and new higher rank (K > 4) terms are added.' This procedure has led to terms 'K  8 and even, and K  Q  K' [185]. ~T In the review [159] the HOZFS terms [185] have been included symbolically in H SH , i.e. the giant



k  4 ,q

Bkq Okq and described as, quote [159]: 'The last term is the sum of higher (than second)

order ZFS terms, which were shown to parametrize the effects of the ZFS induced mixing between spin states on the level splitting within the ground multiplet (S-mixing) [31]', i.e. Ref. [185] here. Note that, in general, the mixing of spin states is rather the result and not the cause of the HOZFS terms [159,185]. The mechanisms of such „induced mixing‟ and thus the origin of the HOZFS terms ~T may be better understood by taking into account the nature of H SH and group theory requirements. Such considerations are, however, beyond the scope of this review. ~T Liu et al. [174] have included in 'the zero-field Hamiltonian', i.e. the giant SH HGS ( H SH ), the term that 'represents higher order zero-field splitting (ZFS) anisotropies (n-order) where q is related to the symmetry of anisotropies'. This HOZFS term was symbolically written as:

n

 n

k 0

Bnq Onq , n

= 4, 6, 8,.... ; note that the index k should be replaced by q. Similarly, in the most recent reviews by Hill [187] and Liu et al. [188] and the paper by Liu and Hill [189], the HOZFS terms were invoked in 2S

HGS as:

p

 p

q 0

Bpq Opq expressed 'in terms of so-called extended Stevens operators'. Hill [187]

described this term as: 'the quantum tunneling perturbation, Hˆ QT ', whereas Liu and Hill [189] as 'the off-diagonal perturbations ( Hˆ QT ) that mix mS states'. However, Liu et al. [188] described the same HOZFS term as: „the effective zfs Hamiltonian H zfs ‟. The name 'quantum tunneling (QT)' does not

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36

reflect the physical origin of the term Hˆ QT and pertains rather to its presumed effect. The HOZFS parameters Bpq were described [187,188,189] as: „the associated phenomenological (or effective) ZFS parameters‟. For the reasons discussed in [164] the terminology 'ZFS anisotropy' [174] and „the spinorbit zero-field splitting (ZFS) anisotropy‟[187] is inappropriate. ~T The odd-rank HOZFS terms invoked by some authors in H SH are somewhat controversial. Hill [187] and Liu and Hill [189] explicitly specified for Hˆ QT , quote: 'The subscript p denotes the order of

Ac ce pt e

d

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an

us

cr

ip t

the operator, and must be even due to the time-reversal invariance of the SO interaction; the order is also limited by the total spin of the molecule (p  2S). The superscript q ( p) denotes the rotational ~T symmetry of the operator about the z-axis.' Exclusion of the odd-rank HOZFS terms from H SH on the ground of 'the time-reversal invariance of the SO interaction' [187,189] runs contrary to the widely accepted reasons [28,29] applicable for exclusion of the odd-rank ZFS terms with k = 1, 3, 5 for transitions ions discussed in Section 7.1. The misconceptions concerning the role of the SOC in SMM and QTM that occur in the reviews [187,188] and the paper [189] require detailed considerations, which may be carried out elsewhere. Concerning the higher-order terms of the type (ii), we mean by the HOFD terms in GSH ~ [28,29] any higher-rank terms in the 'spin' operators, S [in fact: S ], S′, or ST, and/or the nuclear spin, I, which are non-linear in the magnetic induction, B. Group theory predicts that, in general, the terms of the type SaIbBc are admissible, where (a + b+ c) must be an even integer due to time reversal symmetry [222]. Hence, the terms of the type: B2S2, B3S, B5S or B2I2, B3I, B5I as well as the mixed terms like, e.g. SIB2, may be invoked for the highest single-ion spin S = 7/2 [37]. For the high-spin complexes with ST greater than 7/2 additional terms are also admissible, thus increasing tremendously the complexity of GSH. Since the theoretical framework for the HOFD terms has not been developed as yet, these terms have been barely explored in studies of transition ions in crystals so far. The availability for HMF-EMR measurements of the pulsed magnetic fields with the induction strength up to 55 T with the frequencies up to about 2 THz [223,224] and fields of up to even 100 T with a shorter pulse duration time [225] offers increased incentives for studies of the HOFD terms. Major advantage of applications of such high magnetic fields is that, in comparison with the usual linear Zeeman term BgS studied at moderate B values, the HOFD terms with higher powers in B become significant and presumably likely detectable, even if the associated parameters may be small. The review [37] provides also a blueprint for future theoretical and experimental studies of the HOFD terms in GSH for single transition ions, taking into account their implications for HMF-EMR measurements. The above sampling of SMM related literature indicates that the studies of the HOFD terms in the effective GSH for single transition ions as well as the higher-order ZFS and HOFD terms in the

~T

effective total SH H SH for SMM with the giant spin ST appear to be an important and largely unexplored area. Comprehensive consideration of the HOZFS terms and the HOFD terms is, however, beyond the scope of this review. Deeper studies in this area promise interesting and new results, however, they require a lot of experimental and theoretical efforts. The key aspects are theoretical predictions of the experimental signatures of the HOZFS and HOFD terms and development of the computational software packages for analysis of experimental high-magnetic field and high-frequency EMR data. Moreover, in view of the recent advances in HMF-EMR (HF EPR) techniques presented in the reviews [159,146,226,227,228,229,230,231,232], studies of the higher-order terms of the type (i) and (ii) become timely and of prime scientific significance. 7. Distinctions and interrelationships between the CF (LF) and SH (ZFS) quantities ~

~

7.1. Distinct physical nature of HCF (HLF) and H SH ( H ZFS ) The general mathematical forms of the two types of Hamiltonians, i.e. HCF (HLF) in Eq. (1) and

~ H ZFS in Eq. (5), and thus their specific forms for a given point-symmetry group G, are apparently

identical. This is due to the fact that both Hamiltonians must be invariant under the group G that represents the local site symmetry of the paramagnetic ion in the MLn complex under consideration

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[12-18,28,29]. q k

Importantly, the second and third form of HCF (HLF) in Eq. (1), which involves the ~

q

CFPs A and C k , respectively, are not applicable to H ZFS under any circumstances. This is why the relation for the CFPs: Bkq   k Akq  r k  invoked in the review [110] does not apply to the ZFSPs as ~

M

an

us

cr

ip t

discussed in details in [33]. Due to the major distinctions between HCF (HLF) in Eq. (1) and H ZFS in Eq. (5) discussed below, their mathematical similarity does not entail their identification, which is the major cause of the CF=ZFS confusion discussed in the reviews [28-32] and most recently in [33]. ~ One seemingly 'minor' distinction between HCF (HLF) and H ZFS , which has profound implications, concerns the odd-order terms. As a consequence of the electrical nature of CF discussed in Section 2, the odd-order CF terms, i.e. those characterized by k = 1, 3 or 5 in Eqs. (2) and (3), are admissible, but only for the point-symmetry groups G that are characterized by a non-centrosymmetric pointgroup symmetry, wherein spatial inversion is not allowed. Then, the non-zero matrix elements of the odd-order CF terms may exist but only between states of different parity, i.e. belonging to different configurations as, e.g., 3dN and 3dN-14s. Hence, the odd-order CF terms are not applicable to HCF (HLF) in Eq. (1). In practice, the odd-order CF terms have been used in all but a few cases [28,29]. On the other hand, the odd-order ZFS terms in SH, i.e. those characterized by k = 1, 3 or 5 in Eq. (5), are not admissible. The experimental and theoretical grounds for introduction of such terms given by Buckmaster and coworkers [ 233 , 234 , 235 ] were critically discussed and refuted independently by Rudowicz and Bramley [236] and Grachev [237]. However, renewed attempts to justify the presence of odd-order ZFS terms in the SH for the S-state ions have later appeared ~ [238,239]. As discussed in the reviews [28,29], the erroneous identification of H ZFS and HCF, i.e. the CF=ZFS confusion discussed in [33], may have contributed to the misconceptions [233,234,235] on the odd-order terms in a magnetic resonance GSH for the S-state ions. The possible presence of such odd-order terms [233,234,235] has been dismissed [236,28,29] based on the properties of the axial ~ ~ and time-odd spin operators S (S′) involved in the GSH H ZFS , which behave as the magnetic ~

d

induction vector B. The exclusion of the odd-order terms from H ZFS for transition ions is commonly taken for granted without providing any explanation [12-18,28,29]. Although no rigorous proof is available in literature, it may be expected that the same arguments shall also apply to the spin ~T operators ST involved in H SH ( HGS) for SMM and ECS, thus limitting the HOZFS terms to even k-

Ac ce pt e

~

T ranks only. In view of the discussion of the odd-rank HOZFS terms in H SH in Section 6.3, the arguments put forward in [187,188,189] are controversial and require detailed considerations, which may be carried out elsewhere. ~ Other major distinctions between HCF (HLF) and H ZFS are as follows. (i) The operators, in which Hamiltonians are expressed, are functions of the components, ( = x, y, z), of different angular momentum - for HCF of the orbital L and total angular momentum J for 3dN and 4fN ions, ~ ~ respectively, whereas for H SH ( H ZFS ) - the „spin‟ angular momentum S (effective or fictitious) for ~ given „spin‟ systems [28,29]. (ii) The nature of HCF (HLF) and that of H ZFS is different since it is determined by the respective distinct basis of states. (iii) HCF (HLF) acts within the basis of states of ~ ~ the type {|>|S, MS>}, whereas H SH ( H ZFS ) act exclusively within the basis of its own effective spin states {| S~ , M~ S >}, which correspond to the selected, usually lowest, states of the physical

Hamiltonian {|0>|S, MS>}. (iv) The measurable parameters involved in HCF (HLF) are only indirectly ~ related with those in H ZFS as discussed in Section 7.2. ~

~

In order to visualize the distinct physical nature of HCF (HLF) and H SH ( H ZFS ), the corresponding energy level schemes and the CF (LF)  SH (ZFS) interface are depicted in Fig. 5 and 6 for representative 3dN and 4fN ions, respectively. The physical nature of the two types of Hamiltonians bears on the actual interrelationships between the respective parameters discussed in Sections 7.2.

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ip t cr us an M

Ac ce pt e

d

Fig. 5. Hamiltonians (H), wavefunctions (), and energy levels (EL) pertinent for the CF (LF)  SH (ZFS) interface for 3d2 (e.g., Ti2+, V3+) and 3d8 (e.g., Ni2+, Cu3+) ions. The solutions obtained at various stages at the level of the total physical Hamiltonian are sequentially indicated in columns (a) to (d), whereas the solutions obtained at the level of the effective spin Hamiltonian with D > 0 and with D < 0 in column (e1) and (e2), respectively. ' denotes the irreps of the double group, e.g. D '4 .

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ip t cr us an M

Ac ce pt e

d

Fig. 6. Hamiltonians (H), wavefunctions (), and energy levels (EL) pertinent for the CF (LF)  SH (ZFS) interface for the S-state 4f7 (e.g., Gd3+, Eu2+) ions. The solutions obtained at various stages at the level of the total physical Hamiltonian are sequentially indicated in columns (a) to (c), whereas the solution obtained at the level of the effective spin Hamiltonian (for brevity, with D > 0 only) in column (d). ' denotes the irreps of the double group. Only the leading components are indicated for the free-ion multiplets i(MJ).

~

2S+1

LJ and the CF levels

~

Fig. 5 and 6 visualize important distinctions between HCF (HLF) and H SH ( H ZFS ) and illustrate how ~ H SH arises at the interface between the physical energy levels and the effective ones. The

Hamiltonians HCF (HLF) describe spectra measured by the optical spectroscopy, i.e. in the visible ~ (optical) range, which are due to transitions between the CF (LF) states. The Hamiltonians H SH ~

( H ZFS ) describe spectra measured by the EMR (EPR/ESR) spectroscopy techniques, i.e. in the microwave and millimeter/sub-millimeter range, which are due to transitions between the „spin‟ states ~ accounted for by the given „spin‟ operator: S [in fact: S ], S′, or ST. The transitions between the states actually measured by a given technique should be the decisive factor for assigning the name to the Hamiltonians, which are supposed to describe the particular spectra. For comparison, the NMR spectra are measured in the radiowave range and due to transitions between the nuclear spin I states. ~ ~ The fact that the Hamiltonians HCF (HLF) and H SH ( H ZFS ) pertain to transitions between different states is often forgotten, resulting in the CF=ZFS confusion between the CF (LF) and SH (ZFS) quantities discussed in the reviews [28-32] and most recently in [33]. Concerning the Hamiltonians invoked in Fig. 5 (and 6) at the left and right hand side of the CF ~ ~ ~ (LF)  SH (ZFS) interface: Hphys ≡ HFI + HCF (HLF) and H SH ≡ H Ze + H ZFS , respectively, the following general distinctions may also be noted. (a) The former Hamiltonians describe the true physical interactions, whereas the later one does not represent any physical interactions, it merely accounts for the splitting within the lowest subset of levels of a given „spin‟ system, i.e. the ZFS. (b)

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ip t

The values of the true CFPs and the true ZFSPs as well as the respective transitions differ significantly. The transitions between the CF energy levels for transition ions may be measured by optical spectroscopy, inelastic neutron scattering (INS), or Raman spectroscopy and are of the order of magnitude of several hundreds to several thousands of cm-1. However, the transitions between the spin levels may be measured by the EMR techniques and are in the range of fractions of cm-1 to a few cm-1 in the case of X- or Q-band EMR, whereas up to several tens of cm-1 to about hundred cm-1 in the case of HMF-EMR. Importantly, some experimental techniques used extensively in magnetism, e.g. magnetic susceptibility, magnetic specific heat (or heat capacity), INS, Mössbauer spectroscopy, can measure either the true CFPs or the true ZFSPs. However, each type of parameters may be measured in distinct physical systems by different techniques and not in the same physical system by the same technique. It is worth to mention the cases of proper description of the nature of HCF (HLF) expressed in terms ~ of the ESOs Okq (J or L) within a given J- (or L-) multiplet and those of H ZFS expressed in terms of

M

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cr

the ESOs Okq (S) within a given S-multiplet. Such cases have been provided, e.g, by Ryabov [197], who also emphasized that the CFPs should not be confused with the ZFSPs. Distinction between HCF ~ ~ (HLF) and H SH ( H ZFS ) has been also well presented by Ryabov [211] in the EPR study of Cr3+ centers with triclinic local symmetry in forsterite crystals. A brief explanation concerning the conventional-like forms of HCF (HLF) is also pertinent. Such ~ ~ forms are akin to those commonly used in the conventional notation for H SH ( H ZFS ) and hence may be confused by inexperienced researchers. In the conventional notations, as discussed in Section 5 and 6, Hamiltonians are expressed explicitly in terms of the respective angular momentum operators (Xx, Xy, Xz), which differ for the two Hamiltonians. The conventional-like forms of HCF (HLF) that resemble ~ the H ZFS in Eq. (7) have rarely been employed. An exception is the case of a subset of CF states with the fictitious orbital angular momentum L' = 1 arising, e.g. for Fe2+ ion with an orbitally degenerate ground state in crystals. For description of the Fe2+(5Tg) multiplet, the form: ~ H LF   ax[ L2z  13 L( L  1)]   rh ( L2x  L2y ) has been used [12]. Such HLF resembles H ZFS in Eq. (7), but

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d

it represents a form of HCF (HLF) in Eq. (1) in a restricted (see Section 9) basis of states. The distinct ~ physical nature of this HLF and that of H ZFS in Eq. (7) should be kept in mind. Recent examples of similar HLF (L' = 1) may be found in the study of the Cr(III) ion in the octahedral ligand field by Goswami and Misra [88] and of vanadium(III) trisoxalate in hydrated compounds by Kittilstved et al. ~ [240]. The interplay between the two types of Hamiltonians: HLF (L' = 1) and H ZFS has been well presented in [88,240]. Colacio et al. [241] have used similar HLF (L' = 1) for Co2+ ions in Co-Y SMM. Lloret et al. [74] have used an axial HLF (L' = 1) in the study of the magnetic susceptibility of sixcoordinated high-spin Co(II) complexes, both mononuclear and polynuclear. However, as discussed in [34], the notion ZFS has been confusingly used in [74] in double meaning. 7.2. Interrelationships between the CF (LF) and SH (ZFS) parameters As discussed in Section 2, for transition ions with an orbital singlet ground state embedded in ~ ~ crystals the effective spin Hamiltonian H SH ( H ZFS ) arises from consideration of the action of the physical Hamiltonian Hphys = HFI + HCF within the 4fN and 3dN states. Moreover, keeping in mind ~ the distinct nature of Hphys and that of H SH outlined in Section 7.1, it is evident that the notion SH (ZFS) logically arises from the CF (LF) notion and not other way round. Hence, there is no direct simple interrelationship between the CFPs and the ZFSPs. This important aspect has been overlooked in the review [110] thus leading, as discussed in [34], to an implied usage of the invalid conversion relations for the ZFS parameters expressed in the Stevens and conventional notations. The CFPs and the ZFSPs may be indirectly related via the MSH theory outlined in Section 3. Apart from the PT-derivation of MSH employed in [118,119-122], the major method used nowadays in the MSH theory is the projection of the energy levels, Ea, of the physical Hamiltonian, obtainable ~ ~ by full diagonalization of Hphys, onto the energy levels, b, of a suitably parameterized H SH ( H ZFS )

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calculated in a parametric way within its own basis of effective spin states {| S~ , M~ S >}. Succinctly, the MSH theory provides methods for derivation of analytical expressions or numerical relationships amenable for computer programming, which enable to express the experimental parameters {EPs}, measured by EMR and related techniques, in terms of more fundamental microscopic parameters {MPs}, obtainable from other independent spectroscopic techniques. For transition-metal ions the set of EPs includes the ZFSPs { Bkq (ZFS)} and the Zeeman electronic factors {gij(Ze)}, whereas the set of MPs - the parameters involved in the total physical Hamiltonian Hphys, i.e. the free ion (FI) parameters, e.g. Racah ones {B, C}, describing the electrostatic Coulomb interactions, the SOC parameter: {} or {} (    / 2S ), and either the CF parameters expressed in the ESO notation { Bkq (CF)} [or the Wybourne {Bkq(CF)} notation] or more often the CF energy levels Ei, which are obtained after diagonalization of Hphys and hence are directly related to the CFPs. Hence, in the case of the effective SH parameters for transition-metal ions such analytical relations may be conceptually represented as: {EPs}  { Bkq (ZFS), gij(Ze)} 

us

{FIPs: B, C; SOCP:  (or ); CFPs: ( Bkq (CF) or CF energy levels Ei}  {MPs}.

(10)

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M

an

The physical meanings of the respective parameters used in Eq. (10) correspond to the definitions of ~ ~ HCF (HLF) and H SH ( H ZFS ) provided in Section 2 and 3, respectively. Various modeling techniques for analysis and interpretation of EMR data for transition ions at low symmetry sites in crystals have been reviewed [242]. Examples of the MSH relations as in Eq. (10) for the ZFSPs and the Zeeman factors for various 3d ions, usually derived only up to second-order of PT, may be found in several EMR textbooks [12-19], whereas extensive listings of the MSH relations are provided by Boča [21,23]. Most recently a handbook of magnetochemical formulae has appeared [116], where numerous explicit MSH relations for TM ions with an orbital singlet ground state are systematically listed. The MSH theory within the approximation of the states of the ground term 5D has been worked out up to the fourth-order PT for each of the four possible energy levels schemes with a distinct orbital singlet ground state arising for the non-Kramers 3d4 and 3d6 (S = 2) ions at orthorhombic symmetry sites (point groups: C2v, D2, D2h) [128]. To illustrate Eq. (10), below we provide sample MSH expressions [128] suitable for the Fe2+ ions in reduced rubredoxin and desulforedoxin; see Ref. [128] for the full set of the MSH expressions. For the tetragonal case (denoted in [128] as αTE) obtained by an ascent in symmetry from the orthorhombic case (denoted in [128] as αOI1) the two selected contributions due to the SOC parameter  are:

2 2 4 B20 (λ ) =  ( 1  4 ) , B20 (λ ) = 3  yz  x 2  y 2

4 1  43 21 27 496 . For the orthorhombic case αOII1 the contribution B 0 (λ2) is: [ 2 (   ) 3 ] 2 21  yz  yz  x 2  y 2  z 2 x 2 y2

1  V 2 V 2 8q 2  , where  are the 5D energy levels [128], whereas U = 3 cosθ  sinθ ij  B02 2  2      6   yz  xz  xy 

 

and V = cosθ  3 sinθ are the combinations of the mixing coefficients defined as q  cosθ and s  sinθ Note that, instead of the four individual terms for the tetragonal αTE case, the contribution B20 (λ4) for the orthorhombic αOI1 case comprises sixteen terms, which are more involved and hence are not reproduced here. The major contribution to the fourth-order ZFSP B40 (λ4) is similarly quite complicated:  V 2  3V 2 V 2  4 16q 2  1 3U 2  V 2  V 2  4 3V 2 16q 2  1 3U 2        2         2        yz  xz  xy  m   xz   yz  xz  xy  m   0 4 4 1  yz  B4      280  16q 2  V 2  1 V 2  1 8q 2 8s 2   34q 2  6s 2 2U U VV  32qs  U V U V  1               2          xz  xy  m   yz  xz   xy m  xz  xy  yz xy m  yz  

 

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The involved nature and complexity of the interrelationship between the CFPs and the ZFSPs is well indicated by the fact that q and s, and hence the angle θ, are related [128] to the CFPs Bkq (CF) in Eq. (1) as: q  1 / 1  2 , 

3 ( B22  3B44 )

. The sign of s may

1 3{( B  5B  4 B }  [( B  5B  B )  ( B22  3B42 ) 2 ]1 / 2 } 3 0 2

0 4

4 4

0 2

0 4

4 2 4

an

us

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be both positive and negative, whereas that of q can be always kept positive (q2 + s2 = 1) as implemented in the package MSH/VBA [128] to facilitate computations. In the review of high-frequency EPR that revisited LF theory, Gatteschi et al. [243] mentioned MSH relations for the S =2 Mn3+ and Cr2+ ions, quote: „The theory that relates the ZFS parameters to those of the ligand field is well established [9]' (i.e. Krzystek et al. [244]). This quote reflects correctly the interrelationship between the CFPs and the ZFSPs, unlike in the cases of the CF=ZFS confusion occurring in [24, 110], which were discussed in [33]. The non-Kramers ions with the integer spin have generated considerable interest due to possibility of detection of EMR signals in the parallel mode not available for the Kramers ions. Examples of such ions are provided by the 3d4 and 3d6 ions with the spin S = 2, especially Fe2+, Mn3+, and Fe4+ ions, in various systems [243,244,245], as well as the 3d2 (V3+) and 3d8 (Ni2+) ions with the spin S = 1. These ions seem potentially most suitable for HMF-EMR studies. The availability of new experimental EMR data [244,246,247,248,249] makes the MSH calculations using the computer package MSH/VBA [128] more attractive since it enables easy predictions of the ZFSPs and the g-factors. ~

~

M

7.3. General comments concerning the Hamiltonians HCF (HLF), H SH ( H ZFS ), and the confusion of the type CF=ZFS and ZFS=CF

Ac ce pt e

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Lack of proper distinction between the CF (physical) related quantities and the ZFS (effective) ones, either at the level of the respective Hamiltonians, parameters, energy level splitting, or transitions, has lead to the confusion the type CF=ZFS as well as ZFS=CF. As exemplified thoroughly in the reviews [28,29,30,33,34], the major general causes of both types of confusion may be due to the similar mathematical forms of the two types of Hamiltonians as well as the fact that HCF ~ (HLF) in Eq. (1) and H ZFS in Eq. (5) may be expressed in terms of the same (only mathematically!) tensor operators. The erroneous identification of the CF (LF) and SH (ZFS) related quantities leads to some nomenclature pitfalls. Such identification is unacceptable in view of the distinct physical nature ~ ~ of HCF (HLF) in Eq. (1) and H SH ( H ZFS ) in Eqs. (5-7) exposed in Section 7.1 as well as the interrelationships between the CF (LF) and SH (ZFS) parameters discussed in Section 7.2. Misinterpretation of the CF and ZFS quantities could have been avoided if the meanings of the quantities involved were clearly defined, while keeping in mind their physical distinctions. However, it has not been the case in a number of papers, which imply identification of the CF and ZFS quantities. Over the years, the CF=ZFS confusion has become widely spread in EMR [28-30] as well as magnetism studies [31,32,213] of specific compounds [31] and theoretical models of spin systems [32]. In order to visualize the distinction between the true ZFS quantities and the true CF ones, the cases of correct usage of the true CF quantities in magnetism studies have also been reviewed in [30]. The CF=ZFS confusion may be easily identified and thus avoided using a rule of thumb: "In most practical situations, to recognize if the name 'CF' is correctly used in a given context, one needs to verify (i) the type of operators in which a given 'CF-like' Hamiltonian is expressed and (ii) the corresponding basis of states. If these operators depend on the spin angular momentum and act only within the respective spin states, then such Hamiltonian and the associated quantities must be considered as the ZFS-related quantities, regardless of being named incorrectly as the „CF‟-related quantities." The various degrees of the CF=ZFS confusion occurring in EMR and magnetism literature [2832] may be categorized (in an increasing order of severity) as follows [28]: (1) calling the ZFS parameters the 'CF' parameters; (2) calling the ZFS Hamiltonian the 'CF' Hamiltonian; (3) using the nomenclature of the strong, intermediate and weak CF schemes [1-6], which are applicable to HCF

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(HLF) and HSO, when comparing the relative strength of H ZFS versus HZe; (4) using the notation commonly accepted for ZFS parameters with respect to the CF parameters; (5) adopting a point charge model (or any other equivalent one) of the true CF parameters for the ZFS parameters. The categories (4) and (5) are quite serious since they affect numerical results. Pertinent examples in earlier literature have been discussed in the reviews [28-32], whereas the examples occurring in the review [110] and other recent literature are dealt with in details in [33,34]. At present another category (6), identified in most recent literature, may be added, which follows up from the category (4) and represents mostly the confusion of the type ZFS=CF. The category (6) comprises the cases of the factual or implied usage of the invalid conversion relations, which are applicable only to the true CFPs, for the inter-conversions between the true ZFS parameters and the CF ones. This category has most detrimental consequences, including errors of substance, and thus calls for in-depth clarifications. Examples of cases of the category (6), which pertain to the implied usage of the invalid conversion relations, have been identified in [105,110,250,251,252,253,254,199, 255] and discussed in details in [34]. Specific problems concerning the factual usage of the invalid conversion relations occurring in [105,250-253] and [256] have been discussed in details in [257] and [258], respectively. 8. Current status of applications of the (extended) Stevens operators in recent literature

an

8.1. Importance of the ESOs

M

The importance of the ESOs in the eyes of the great pioneer has been emphasized by Prof. K.W.H. Stevens in the 1997 book [20]. The ESOs are indispensable for low site symmetry, e.g. monoclinic and triclinic, and thus have been well recognized in magnetism, including the SMM area, and EMR literature, see, e.g. the reviews by Sorace et al. [110], Nakano and Oshio [150], Fittipaldi et al. [152], and Santini et al. [259]. Most recently the ESOs were invoked in the reviews by Hill [187] ( Opq ) and

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Liu et al. 188 ( Onq ) to express the HOZFS terms in HGS (discussed in Section 6.3). The reviews [110, 150,152, 187,188,259] set a good example in this regard by providing for the ESOs with -k  q  +k references to the original paper [191] the most recent ones [196,197] and the textbook [12]. The selected examples of recent applications discussed below illustrate the versatility and usefulness of the ~ ~T ESOs to express HCF (HLF) as well as H ZFS and H SH . Importantly, the ESOs Okq (S) have been incorporated into several the computational packages. Stoll and Schweiger [260] have developed the package EasySpin for spectral simulation and analysis in EPR, which is gaining an increasing popularity. Rotter and coworkers [41,198,261,262] have utilized the ESOs (listed in [198] up to k = 6) and their transformation properties [194] in the computer program McPhase for the calculation of magnetic and orbital excitations in rare-earth based systems. Similarly, Chilton et al. [61] have utilized the ESOs (listed in [199] also up to k = 6) in the computer program Phi for analysis of anisotropic monomeric and exchange-coupled polynuclear dand f-block complexes. The ESOs Okq (J) have been used to express HCF(J) in the software package SIMPRE recently developed by Baldoví et al. [105] for calculations of CFPs, energy levels, and magnetic properties of mononuclear Ln-complexes. ~ A suitable H ZFS has been used [263] to describe the low-symmetry effects in EPR of Gd3+ in EuAlO3 with CS point symmetry - here the ESOs, Okq (S), are functions of the electronic spin operator S for Gd3+(S = 7/2) ions. Cardona-Serra et al. [264] used an appropriate terminology for q H CF expressed in terms of the „operator equivalents Ok expressed as polynomials of the total angular momentum operators21‟. It is commendable to indicate the nature of the ESOs, i.e. Okq (J), and provide two pertinent sources for the ESOs [264], i.e. an early paper [265] and the basic source [194]. Van Wüllen [266] utilized the transformation properties of the Stevens operators with respect to a rotation of the coordinate frame [194], whereas cited Orbach [265] for the operators' definition and additionally listed explicitly four operators Okq (S). Pointillart et al. [59] have developed own

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program for CF calculations within the ground multiplet states of Ln-complexes with C2v symmetry, which incorporates the ESOs Okq (J) referred to the papers [265,194]. Liu et al. [60] have expressed k k H CF in terms of the ESO denoted Oq (referred to the papers [194,102]) and the CFPs Bq to study

the DyIII ions in the single-ion magnets [Zn–Dy–Zn] complexes. The authors [60] have used the SINGLE_ANISO software incorporated into the MOLCAS267 quantum chemistry package, which allows for ab initio calculations of the CFPs in Ln complexes. The application of this package and the implementation of the ESO in [60] require separate consideration in view of the definitions of the irreducible tensor operators provided by Chibotaru and Ungur [102] (see Section 5).

ip t

8.2. General comments concerning the usage of the ESOs

cr

For the cases of higher site symmetry, i.e. not involving the negative q components, only the usual Stevens operators [28,29] need to be included, then reference to the textbook [12] is sufficient. Ishikawa et al. [75,76,87] have used HCF, which is equivalent to HCF in Eq. (1) and adequately ~ referred Okq to the textbook [12]. The higher-order terms in H ZFS (defined in Section 6.3) were [28]

us

expressed in the review [155], quote: „using the so-called Stevens operator equivalents



n ,k

Bnk Onk

where the Bnk are parameters …and Onk are operators of power n in the spin angular momentum.

an

The value of k is restricted to 0  k  n.„ In this case only the usual Stevens operators [28,29] are explicitly included and the source [12] (Ref. 28 in [155]) is adequate. In the review of multifrequency and high-field EPR in high-spin transition metal coordination complexes, Krzystek et al. ~ m [132] represented tetragonal H ZFS using the unnamed operators denoted ' O4 ', which were referred to q

M

the source [12]. This implicitly indicates the usual Stevens operators. The ZFSPs Bkq were used,

Ac ce pt e

d

while the notation bk , like that defined in Eq. (5), was only commented on as rarely used in the literature. The usage of the ESOs as well as the symbols for operators and associated parameters in some the papers require comments and explanations. Survey of a representative sample of the most recent literature dealing with the 4fN and 3dN ions based compounds has also indicated the following general pitfalls: (i) often the symbols Okq (or equivalent) are used for the Stevens operators without providing adequate references for the low symmetry components [194,196] with q: -k  q  0; (ii) the nature of Okq (X): X meaning either S, J, or L, is not specified, thus contributing to the wide spread CF=ZFS confusion discussed in Section 7, and (iii) adequate explanations on the ranks k and the components q are not provided, thus contributing to an ambiguity concerning the type of operators and the symmetry cases actually used. Representative examples of these pitfalls are discussed below. Doubts may arise concerning the operators used by Yang et al. [268], who for Gd3+ ion in the q tetragonal symmetry used the 'effective spin Hamiltonian' with the small bk ZFSPs like in Eq. (5). However, the 'spin operators Omn ' were referred [268] to Buckmaster and Shing [269], who utilized the operators belonging to the STO category [28,29]. Nevertheless, judging by the ZFSP symbols q [268] bk , it may be assumed that these ZFSPs are given in the Stevens notation. The tetragonal SH for Gd3+ ion, identical with that in Yang et al. [268] is widely used, see, e.g. Gorlov [270], who named this SH notation as 'standard' and referred it to the textbook [12]. Note that instead of Ref. [12] some Russian authors more often refer to the textbook [13] and denote the usual Stevens parameters and operators (0  q  +k) as bkq and Okq, respectively, whereas the extended Stevens parameters (-k  q  +1) as ckq without any specific symbol for the ESOs, see, e.g. Astryan et al. [271] and Vazhenin et al. [272,273]. It should be kept in mind that Stevens [191] and Abragam and Bleaney [12] have provided only the usual Stevens operators [194] Okq (0  q  +k), which are sufficient only for orthorhombic and some higher symmetry cases [28,29]. Importantly, for triclinic site symmetry all components [194]

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45

~

(-k  q  +k) must be included, as indicated in HCF in Eq. (1) and H ZFS in Eq. (5), whereas for monoclinic and axial type II symmetry cases [30,274] some negative components (-k  q  +1) are indispensable. Hence, the most often cited sources [12,191] are not sufficient for low symmetry cases as well as for any general forms of the ZFS or CF Hamiltonians expressed in terms of Okq with arbitrary rank k and components q (often denoted by a variety of symbols). Such forms are often invoked [50,64,178] without stating explicitly the limits of the summation and citing only the sources [12,191] for the operators Okq . Then a problem arises since such forms shall be considered as

cr

ip t

implicitly including all q components, i.e. -k  q  +k [194]. Representative examples of other questionable usage of the ESO, identified also in the survey, are discussed below. Luzon and Sessoli [54] utilized HCF 'acting on the basis of the 2J + 1 functions of the 2S+1LJ ground multiplet' in the form equivalent to Eq. (11) and referred 'the Stevens operators' Oˆ kl to the sources [12,191]. In the study of butterfly {Fe3LnO2} (Ln = Dy and Gd) SMM, Badia-Romano q q ˆ et al. [67] expressed the LF „single-ion Hamiltonian‟ as: Η LF   Βp Οp in terms of the „Steven‟s q

p

‟ without providing any references for their definition. Magnani et al. [68] have

us

operators O

M J

expressed HLF acting on the ith ion in terms of the „Stevens operator equivalents Ο2k (i) ‟ referred to Newman and Ng [8]. Mart nez-P rez et al. [ 275 ] for Gd-based SMM with tunable magnetic ˆ m referred only to Stevens [191]. Waldman and anisotropy used „Stevens equivalent spin operator‟ O

an

q

n

Güdel [139] used the fourth-rank terms: i,m B (i ) Oˆ 4m (i ) , expressed in terms of the 'Stevens m 4

operators acting on the ith spin Oˆ 4m (i ) ' and referred them to the textbook [12]. However, the

M

calculations [139] were carried out using the irreducible tensor operators Tˆq( k ) and the conversion factors were provided for m = 0, 2, and 4 only. Liviotti et al. [140] expressed the effective total SH q 2

B O

q 2 q k

q 2

q 2

, where Okq are 'the Stevens operator equivalents31

d

~T H SH for ECS as: H S = S  D  S =

Ac ce pt e

defined in the total spin space and B are the corresponding parameters'. However, Hutchings [276] (Ref. 31 in [140]) does not cover all q components, but q = 0 and 2 for k = 2 only. Usage of the operators and CFPs by Palii et al. [89] shows that even experienced researchers are struggling with selection of a suitable notation. The Hamiltonian HLF(5D) for the trigonal complex within the given 5D- term was represented in terms of the operators Ypm(5D) representing 'the

~

spherical harmonics equivalents', which were equal to 'the Racah operators' denoted O pm multiplied by a numerical factor. The CFPs B pm used in [89] closely resemble Bkq (CF/ESO) defined in Section

~

6.1. The operators O pm were described in [89] as 'closely related to the operator equivalents defined by Stevens30 (see ref 31 for the details)', where Ref. 30 and 31 in [89] corresponds to Stevens [191] and Lindgård and Danielsen [277], respectively. Their selection was motivated by the fact that, quote: 'the Racah operators represent the irreducible tensors. This fact makes them preferable for computer calculations.'. This implicitly reflects the drawback of the Stevens operators, i.e. the lack of normalizations (see Section 5). A preliminary analysis of the notations used by Palii et al. [89] and comparison with the general definitions reveal that the CFPs [89] B pm and Bkq (CF/ESO) are not related in any simple way. Hence, derivation of the conversion relations between the two types of CFPs seems rather cumbersome. Some additional problems with the definitions of the operators and parameters have been identified in the studies of the high-spin SMM. Several authors [157,160,163,166,167,278,279,280,281,282,283] have explicitly listed the operators, which were ~T needed to express the respective Hamiltonians representing H SH ( HGS), in terms of the components (S+, S-, Sz). However, neither the operators' name nor source references were provided, while in some

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cases the book [12] was cited in passing but not directly for the operators' definitions. Resorting to such explicit definitions may help avoiding ambiguities, however, this practice makes confusingly an appearance of some newly defined operators and thus should rather be avoided. Nevertheless, judging by symbols used, e.g. Oˆ 40 , Oˆ 42 , and Oˆ 44 , which resemble Okq , presumably the usual Stevens

1 3 B4 [ S z ( S 3  S 3 )  ( S 3  S 3 ) S z ] . An inspection of Table A1 in Appendix 1 reveals that this term 4

an

H trans 

us

cr

ip t

operators with 0  q  k have been used in all cases encountered so far. This may be confirmed by direct comparison of the ad hoc defined operators with those listed in Table S1. Note that in some cases the constant term '3S2(S + 1)2' in Oˆ 40 has been omitted. A more serious problem occurs in the papers, where the notion of the Stevens operators is taken for granted and no references are provided for the existing operators' definitions. This problem is evident, e.g., in the review by van Slageren [159]. In view of the existing abundance of the operator and parameter notations used in EMR, optical spectroscopy and magnetism literature [28,29] as well as the wide spread confusion with regard to the pertinent notations, such practices are basically inappropriate and should rather be avoided. Other problems are posed by mixed notations involving both conventional terms and the tensor operator ones. For example, for Fe4 SMM clusters Mannini et al. [94] used the conventional CFPs: the cubic one denoted 'Dq' (i.e. properly 'Dq' [1-6]) and low symmetry ones Dt and Ds, whereas a conventional form of 'the second-order terms in the spin Hamiltonian, which describe magnetic anisotropy', with the usual D and E. However, additionally a quasi-conventional form of 'the three-fold fourth-order term' was also used, i.e. the ZFS term:

Ac ce pt e

d

M

is equivalent to the term B43O43 in the Stevens notation. Such mixed notations should be avoided in view of the well-defined Stevens operators and the respective parameters. Various problems concerning the meaning of symbols used for the operators and CFPs appearing in HCF expressed nominally in the Stevens (and/or Wybourne) notation have been identified in the studies of high temperature superconductors. These problems have prompted comparative analysis and standardization of CFPs obtained from: (i) INS and related studies of RE ions (RE = Er3+, Ho3+, Nd3+, Pr3+) in REBa2Cu3O7- [40], (ii) Mössbauer spectroscopy for Tm3+ ions in Tm2BaXO5 (X = Co, Cu, Ni) [205], (iii) spectroscopic data for Eu3+ and Er3+ ions in RE2BaXO5 (X = Co, Cu, Ni, Zn) [204], and (iv) Mössbauer spectroscopy studies of Tm3+ ions in TmBa2Cu4O8 and TmBa2Cu3O7- [203]. Some ambiguities concern the type of operators and the associated CFPs that have actually been utilized in the calculations based on the exchange charge model proposed by Malkin [284]. The operators defined in [284] were originally related both to the Wybourne operators and the Stevens ones. Usually, as in the original notation [284], the operators are denoted as O pk , whereas the total k k k CFPs as Bp  Bpq  BpS . These CFPs include two contributions arising from: (i) the point charges k k B pq and (ii) the effects due to covalent bonding and exchange interactions B pS . However, various

authors have provided different descriptions of the operators used in the exchange charge model calculations, which bears on the actual meaning of the CFPs as discussed below. Brik et al. [285] named O pk as 'the linear combinations of irreducible tensor operators acting k upon the angular parts of an impurity ion wave functions', whereas B p as 'CFPs' without mentioning

either the Wybourne or Stevens notation. Similar description of O pk was used by Brik and Avram k [286], whereas B p were named as 'CFP containing all information about geometrical arrangement

of the ligands around the central ion' without specifying the notation. In view of the variety of symbols being used in literature for the CFPs in both notations, an ambiguity arises since the description 'the irreducible tensor operators' [285,286] suggests that the Wybourne CFPs were k k ˆ considered. On the other hand, Srivastava and Brik [287] used HCF defined as Η    Β p Ο p p

p  2 ,4 k  -p

and stated that O „are the suitably chosen linear combinations of the irreducible tensor operators ...‟, k p

whereas in Table 2 in [287] the CFPs B pk were described as „in Stevens normalization‟.

Similar

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47

description of O pk and B pk was used by Brik et al. [288]. Explicit indication of the type CFP notation ( or 'normalization') [287,288] is commendable, since it removes ambiguity, however, the usage [285-288] of the name 'irreducible tensor operators' for the ESO [191,194,195,196,197] remains questionable. In view of the above ambiguities, we have carried out analysis of the notations and the expressions for the exchange charge model contributions to the CFPs defined by Malkin [284] as well as those used in several related papers. Subsequent comparison of the definitions of the tesseral harmonics used by Malkin [284] with the general ones given by Prather [289]: Z pk , k = c and s - to

ip t

which the ESO [191,194,195,196,197] are the operator equivalents, reveals additional problems. It turns out that the tesseral harmonics used by Malkin [284] differ from those defined by Prather [289]. k The parameters of the electrostatic field of the point lattice [284], B pq , are defined in Eq. (2.7) of

cr

Malkin [284] in such a way that they represent the Stevens CFPs Akq r k in Eq. (1) above, assuming

us

the correspondence of indices: lower p = k and upper k = q, respectively. Hence, the CFPs calculated [285-288] based on the exchange charge model [284] do not correspond directly to the Stevens CFPs Bkq in Eq. (1) but rather to Akq r k . This observation necessitates a comprehensive survey of the

an

papers utilizing the exchange charge model to establish unambiguously the actual meaning of the reported CFPs. However, this is beyond the scope of this review. Ghosh et al. [69] have carried out the multi-frequency EPR study of a mononuclear holmium in SMM based on the polyoxometalate [HoIII(W5O18)2]9− with D4d point symmetry. The 'LF Hamiltonian' was described in [69] as expressed in terms of the 'extended Stevens Operators' Okq and

M

'associated coefficients Bkq '. The ESOs Okq were referred to Refs [196] and [260] and appropriately described as functions of the total angular momentum operators J for Ho3+(J = 8) ions. The experimental values of the pure axial CFPs Bkq with q = 0 and k = 2, 4, and 6 were determined from magnetic measurements for the HoIII compound [69]. However, another Hamiltonian has been k



Bkq Okq  J .A.I   B B0 .g .J , and used in the program

d

explicitly given in their Eq. (1): H 

Ac ce pt e

k 2 ,4 ,6 q 0

EasySpin [260] to simulate EPR spectra [69]. Doubts arise since neither the type of the operators: Okq (J) or Okq (S), nor the value of J or S used in this Hamiltonian [69] have been provided. It seems that the authors [69] might have used HCF (HLF) to simulate EPR spectra and confused the properties ~ ~ of HCF (HLF) and H SH ( H ZFS ). If it was the case, the procedure used by Ghosh et al. [69] would be inappropriate and it would represent the inverse ZFS=CF confusion discussed in Section 7.3. To verify this assertion sample test calculations for Ho3+ ions using the program EasySpin [260] must be carried out. Hence, a closer analysis of the methodology [69] is beyond the scope of this review. To summarize this section we note that the above survey indicates that (i) a variety of disparate ~ symbols is being used in literature for the operators and/or parameters to express HCF (HLF) and H ZFS in terms of „Stevens operators‟ and (ii) proper source references for the given notations are often not provided. As indicated by the examples discussed above even experienced researchers have problems with the definitions of the operator and parameter notations. Hence, such situation may be difficult to comprehend even more so by young experimentalists entering the respective fields of study. The possible remedy to the problems outlined in this section may be an international unification of the operator and parameter notations utilized in the EMR and optical spectroscopy areas. 9. Generalized definitions of the full and restricted Hamiltonians versus the effective and fictitious ones There exists a controversy in literature concerning the meaning of the Hamiltonians in question, since various names are often used in different context by various authors. Hence, based on the previous Sections, we provide some generalized definitions of the pertinent Hamiltonians to further

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48

elucidate the two subtle points. First point concerns distinction between the effective Hamiltonians and the restricted forms of Hamiltonians, which are sometimes confused in literature. Second point concerns the degree of 'effectiveness' of particular Hamiltonians. Two types of effective Hamiltonians considered here are: (1) the effective single-ion spin ~ ~ Hamiltonians, H~ eff  H SH , that include H ZFS for a given orbital singlet ground state, and (2) the T T effective total (giant) spin Hamiltonians for an ECS of transition ions, HGS  H SH , that include H~ ZFS for a given ground ST-multiplet. These Hamiltonians were defined in Section 3 and 4, respectively. The restricted forms of Hamiltonians were introduced in Section 2.5 in the context of CF (LF) Hamiltonians HCF (HLF) for RE ions within the approximation of the Russell-Saunders ground multiplet. The concept of the fictitious 'spin' and the corresponding fictitious 'spin' Hamiltonians were defined in Section 3.1 in the context of the effective single-ion SH. As we show below this concept is applicable also to HCF (HLF) for transition ions. Generalizing the earlier considerations, the central issue for the concept of effective Hamiltonians of the type (1) or (2) may be viewed as transition from a physical or 'parent' Hamiltonian, where parameters have a physical or higher-level meaning, respectively, to an effective Hamiltonian, which reproduces the energies of the specific ground state. Such transition allows for easy comparison of different systems, but where the physical origin of the splitting is apparently lost. The ground state considered in the two cases is the ground orbital singlet and the lowest total spin ST-multiplet, respectively. It is worth to emphasize that the loss of the physical origin of the given splitting is only apparent not actual. In most cases the physical origin is either not presented explicitly or the given splitting is considered confusingly as caused by the given effective Hamiltonian. In fact, the splitting of the given ground state is caused by specific terms in the respective physical Hamiltonian and is merely described by the given effective Hamiltonian. ~ In the case of the effective single-ion ZFS Hamiltonians, H ZFS , the ZFS of a given orbital singlet ground state, as a phenomenon, is due to the combined action of the CF (LF) and the SO coupling (and, to a lesser extent, the electronic SS coupling). The link between Hphys ≡ HFI + HCF (HLF) + HSO ~ ~ ~ + HSS and H eff  H SH  H ZFS is provided either by the perturbation theory or full diagonalization of

M

an

us

cr

ip t

~

Ac ce pt e

d

Hphys. The perturbation calculations are carried out within the subset of selected higher-lying CF states and may yield analytical relations for the ZFSPs of an effective Hamiltonian that involves only ~ the effective spin operators S . Likewise, the full diagonalization of Hphys combined with the projection method, which reproduces the physical energy splitting caused by the SO and SS couplings within the effective spin states, may yield numerical relations for the ZFSPs. Importantly, no simple ~ relations exist between the parameters in the effective H ZFS and those involved in the physical Hphys (see, Section 7). T In the case of the effective (giant) total ZFS Hamiltonians for ECS, H~ ZFS , the ZFS of a given lowest total spin ST -multiplet, as a phenomenon, is due to the combined action of the (physical) exchange interactions and the (effective) single-ion ZFS terms:

N

 H ex (i, j ) + i j

N



~ H ZFS ( i ) (see, Section

i

4.2). These two major terms are included in the multispin Hamiltonian HMH, which here plays the role ~T T of the quasi-physical, i.e. 'parent', Hamiltonian. The link between HMH and HGS  H SH  H~ ZFS is provided in this case by advanced computational techniques, since due to the complexity of the problem and the large basis of states, the total ZFSPs can hardly be obtained analytically. Various computational techniques are employed that are based on the projection method, which reproduces the ~T physical energy splitting in terms of the parameters of an effective Hamiltonian H ZFS involving only ~

T the effective total spin operators ST. The relationships between HMH and H SH were briefly discussed in Section 4.4. Derivation of numerical relations between the parameters in the effective total ZFS ~T one, H ZFS , and those involved in HMH is extremely difficult. Hence, instead of derivation of any

~

~

~

T T T relations between parameters of HMH and those of HGS  H SH  H ZFS , most often H ZFS is just postulated and considered as a tool to parameterize the experimental data.

Page 49 of 66

49

Ac ce pt e

d

M

an

us

cr

ip t

Based on the features of the respective Hamiltonians recapped above, the global criteria for a Hamiltonian to be considered as an effective one may be generalized as follows. (1) The effective Hamiltonian acts only within an own specific basis of states, which is not directly related to the basis of states of the physical or 'parent' Hamiltonian. (2) The effective spin Hamiltonian is expressed in ~ ~ terms of the effective spin operators X , either S (most often denoted as S) or ST (often also denoted ~ ~ as S), whereas the quantum number X associated with the effective spin operator X shall reflect the ~ number of states of the physical or 'parent' Hamiltonian (2 X + 1), for description of which the effective SH is employed. In the case of the effective single-ion ZFS Hamiltonians, the quantum ~ number S is equal to the electronic spin quantum number S. In the case of the effective (giant) total ZFS Hamiltonians for ECS, the number of states in the lowest ST-multiplet of ECS is equal to that corresponding to the quantum number of the total spin ST, i.e. (2ST + 1). (3) The effective Hamiltonian itself does not represents any physical interaction or effect, regardless of the nature of the physical or 'parent' Hamiltonian. (4) No simple relations exist between the parameters in the effective Hamiltonian and those involved in the physical or 'parent' Hamiltonian. Such relations have a character of complicated functions and cannot be obtained by just only considering the matrix elements of the operators involved in each of the two types of Hamiltonians, as it is the case for the restricted and full Hamiltonians discussed below. Consequently, there exists an interface between the physical Hamiltonians, or 'parent' ones, and the effective Hamiltonians. In this case we adopt the definition of the interface given in Introduction, i.e. as the boundary of the respective two distinct ~ entities. The above criteria fit the effective single-ion ZFS Hamiltonians, H ZFS , origin and nature of which have been discussed in Section 3, as well as the effective total (giant) spin Hamiltonians for ECS discussed in Section 4. Likewise, the global criteria for a Hamiltonian to be considered as restricted with respect to the 'parent' full Hamiltonian may be generalized as follows. (1) The restricted Hamiltonian acts only within a selected restricted basis of states of the full Hamiltonian, in which certain quantum mechanical quantities associated with specific operators may be considered as good quantum numbers, e.g. the orbital angular momentum L and/or the spin angular momentum S. (2) The simplified equivalent form of the full Hamiltonian within the restricted basis of states may be found if expressed in terms of the specific total operators, e.g. L, S or J. (3) The restricted Hamiltonian represents still the same physical interaction or effect as the full one, albeit in the selected restricted basis of states. (4) The relations between the parameters appearing in the full Hamiltonian and those in the restricted Hamiltonian may be obtained by considering the matrix elements of the operators involved in each of the two types of Hamiltonians. Such relations have a character of conversion relations, since such relations are usually represented by specific numerical coefficients. Consequently, no interface (in the meaning defined in Introduction) between the 'parent' full Hamiltonians and the restricted ones exists, since these two entities are not distinct but the latter is part of the former. Two examples of often used Hamiltonians may be provided for illustration of the global criteria formulated above: the SOC Hamiltonian, HSO, and HCF (HLF). For these Hamiltonians both the 'parent' full form and the corresponding restricted form are widely employed. Two forms of the HSO are used for transition ions: (i) within the whole 3dN configuration: HSO =   l i si and (ii) within the i

lowest 2S+1L term: HSO = L  S . Each form involves a distinct SOC parameter (for references, see, e.g., [290, 291]): (i)  (alternatively denoted as ), which is always positive for all N and (ii) , which is positive for electron configurations less than half-full, as for, e.g. Cr3+ and Fe4+, whereas negative for electron configurations more than half-full, as for, e.g. Co2+ and Ni2+. Note that the second form cannot be used for Fe3+ or Mn2+ with the half-filled d-shells, since in these cases the ground multiplet is 5S with the quantum number L = 0. The two SOC parameters are simply interrelated as:    / 2S . Hence, according to the above criteria, the (i)-form represents the full HSO, whereas the (ii)-form - the restricted HSO. Both forms of the SOC Hamiltonians describe the same physical interaction. The second example concerns the CF (LF) Hamiltonians. One case of the full and restricted HCF (HLF) forms has already been discussed in Section 2.5. Due to the lack of well established conventions,

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Ac ce pt e

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an

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some authors consider the restricted HCF (HLF) acting within a 2S+1LJ ground multiplet of RE3+ ions as an effective Hamiltonian with respect to the full HCF (HLF) acting within the whole 4fN configuration. To evaluate appropriateness of such terminology, a few points should be kept in mind. Note that the two types of CFPs, namely, Bkq(Wyb) appearing in the true full HCF (HLF) acting within the whole 4fN q configuration and Bk (ESO) appearing in the true restricted HCF (HLF) acting only within a given ground 2S+1L (TM ions) or 2S+1LJ (RE ions) multiplet, are related by specific well-defined (so-called Stevens) coefficients tabulated for each transition ion [1-11]. Moreover, the true restricted HCF (HLF) still describes the same physical interactions between the electrons of a paramagnetic transition ion and the surrounding diamagnetic ligands, which give rise to the electric field, i.e. the CF (LF) field. To make clearer the distinction between the full and restricted Hamiltonians versus the effective and fictitious ones, the interrelationships between the four notions are graphically presented in Fig. 7 using the criteria formulated above. Concerning the meaning of the fictitious 'spin' S' (as defined in Section 3.1.3), this concept is applicable both to the effective Hamiltonians as well as the restricted ones. As depicted in Fig. 7, the simplest case of the fictitious 'spin' S′, i.e. the case of S‟ = ½, may be ascribed to the lowest two states within a 2S+1LJ ground multiplet of RE3+ ions as well as to the lowest ~ two states of an effective SH with the effective spin operator being either S or ST.

Fig. 7. Visualization of distinction between full, restricted, effective, and fictitious Hamiltonians presented by the respective energy levels.

The above comparison of the origin and nature of the Hamiltonians in question indicates clearly that the degree of 'effectiveness' existing between the full form of HCF (HLF) and the restricted form is ~ completely different than that existing between Hphys (see, Section 2) and H ZFS (see, Section 3.2) as ~T well as that between HMH (see, Section 4.2) and H ZFS (see, Section 4.3). Importantly, in the case of the CF (LF) Hamiltonians the form of HCF (HLF) acting within a 2S+1LJ ground multiplet of RE3+ ions ~ ~T explicitly satisfies the criteria for the restricted Hamiltonian. In the case of H ZFS or H ZFS the criteria

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for a Hamiltonian that is effective with respect to the physical or 'parent' Hamiltonian, i.e. Hphys or HMH, respectively, are satisfied. 10. Conclusions and outlook

Ac ce pt e

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This review focuses on the fundamental aspects underlying physics and chemistry of single transition (4fN and 3dN) ions in coordination compounds as well as the novel single-ion and polynuclear magnetic systems. Examples of such systems are provided by the single-ion magnets as well as the exchange coupled systems (ECS) based on transition ions, especially the single molecule magnets (SMM) or molecular nanomagnets (MNM). Due to unique and intriguing properties, these systems have been extensively studied in recent decades. Nowadays their experimental and theoretical investigations form a rapidly developing field. The pace of experimental developments in this field is staggering as indicated by the shear amount of relevant papers. However, the urgent needs for theoretical interpretations of the results, which become accumulated in the studies of magnetic and spectroscopic properties of transition ions and other paramagnetic or spin species in various systems, have created some undesirable problems. The emphasis is put on elucidation of the problems encountered at the interface between the crystal field (CF) Hamiltonian, i.e. equivalently the ligand field (LF) one, and the spin Hamiltonian (SH), which incorporates the zero-field splitting (ZFS) Hamiltonian. A variety of conceptual problems has been revealed in an extensive survey of recent EMR, ECS and SMM (MNM) related literature. The misinterpretations of the crucial notions have created serious terminological confusions, which have led to pitfalls and errors of substance that bear on understanding of physical properties of magnetic systems. This review attempts to provide adequate in-depth clarifications in order to enable a deeper understanding of the intricacies involved in the CF (LF)  SH (ZFS) interface. The presentation has been kept at the level comprehensible to experimentalists with background in chemistry or physics. Hence, specialist terms and symbols were defined and fundamental ideas simply explained, while striving to keep the overall presentation jargon free. The aim is to disentangle the web of interdependencies arising at the CF (LF)  SH (ZFS) interface and, in turn, help experimentalists to better negotiate the murky presentations occurring in several textbooks and review articles and thus to avoid the existing pitfalls. The background theory provided here should be helpful to researchers in several ways. It may enable better interpretation of experimental results and extraction of useful structural information inherent in the fitted and theoretical CF parameters or ZFS ones, especially for orthorhombic, monoclinic, and triclinic site symmetry cases. It may also help to distinguish the cases of sloppy usage of terminology, which has no direct consequence for the published results, and the cases involving obvious lack of understanding of the physical principles, which are obviously more serious. To summarize our efforts in disentangling intricate web of interrelated notions at the interface between the physical (crystal field) Hamiltonians and the effective (spin) Hamiltonians, a table that compares different terminologies is compiled (Table 1). The terminologies discussed above, based on the prevailing definitions in the main textbooks [1-23] and the general reviews [28-34], which are deemed by the authors as most appropriate and correct, are listed in Table 1 on the left-hand side, whereas various incorrect terminologies appearing in the literature - on the right-hand side. Such table would be beneficial to the readers, since together with the precise definitions provided in text, such compilation facilitates grasp of the physical principles involved. It may be hoped that this review makes a significant impact on the scientific community, hopefully, leading to greater acceptance of correct terminologies in the long terms. Table 1. Summary of terminologies for key physical entities listed by categories in consecutive blocks (#): (1) spins; (2) physical Hamiltonians; (3) effective single-ion Hamiltonians; (4) Hamiltonians for ECS. Equivalent names (EN) are also listed. # Symbol

Correct name as defined in text

Incorrect names for associated quantities1) used in various contexts in literature

Confusion type (remarks)

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~ H eff ~  H SH ~ H ZFS

~ H Ze

effective single-ion zero-field splitting Hamiltonian (EN: fine structure Hamiltonian)

spin Hamiltonian; crystal or ligand field Hamiltonian; single-ion anisotropy; crystal field Hamiltonian; ligand field Hamiltonian; single-ion anisotropy; spin-spin interaction; electronelectron coupling Zeeman electronic Hamiltonian spin Hamiltonian; uncoupled spin Hamiltonian

d

effective single-ion Zeeman electronic Hamiltonian 4 HMH multispin Hamiltonian (EN: microscopic spin Hamiltonian [MSH]3)) HGS ( effective giant (EN: total) spin ~T Hamiltonian for ECS ) H SH (EN: giant-spin Hamiltonian; total-spin Hamiltonian) ~T effective giant (EN: total) zeroH ZFS field splitting Hamiltonian (EN: giant-spin ZFS Hamiltonian; total-spin ZFS Hamiltonian)

zero-field splitting Hamiltonian

spin Hamiltonian; coupled spin Hamiltonian crystal or ligand field Hamiltonian; single-ion anisotropy crystal or ligand field Hamiltonian; single-ion anisotropy

Ac ce pt e

1)

ip t

total spin for ECS (EN: effective spin for ECS) free ion Hamiltonian crystal field Hamiltonian (EN: ligand field Hamiltonian) total physical Hamiltonian for single-ion ( HFI + HCF) effective single-ion spin Hamiltonian

(S may be confused with ~ S or ST ) (inappropriate name) (inappropriate names) (generic name only) (S is commonly used for ~ S) (S may be confused with ~ S or S' ) ZFS=CF

cr

ST

electron spin; fictitious spin; fictitious effective spin; spin angular momentum; spin S (no specific name) spin S (no specific name)

us

effective spin for single-ion

spin S (no specific name); effective spin S

an

~ S

2 HFI HCF (HLF) Hphys 3

electronic spin for single-ion fictitious 'spin' for a given 'spin' system (EN: pseudospin)

M

1 S S′

(generic name only) CF=ZFS; MA=ZFS2) CF=ZFS; MA=ZFS2) EI=ZFS

(generic name only) (inappropriate name) (generic name only) (inappropriate name) CF=ZFS; MA=ZFS2) CF=ZFS; MA=ZFS2)

The quantities associated with a given notion referred to by incorrect names may be, e.g., effects, Hamiltonians, eigenfunctions, parameters, or energy level splitting. 2) In some cases a compounded confusion: MA=CF/LF=ZFS occurs. 3) In view of the different primary meaning of the notion 'MSH' existing in literature (see text), the name MSH in this context is not recommended to avoid confusion.

While working on this review, it has also transpired that it would be worthwhile to survey the computer programs for simulation and fitting of (i) the CFPs obtained from optical spectroscopy or INS data and (ii) the SH (ZFS) parameters obtained from EMR spectroscopy or magnetic susceptibility data, ~T as well as (iii) the computer programs that enable relating the total ZFSPs appearing in H SH  (ST) with the parameters appearing in HMH  (Si, Sj), i.e. the ZFSPs and the exchange interaction constants for the individual ions. In view of the deficiencies and ambiguities in presentation of various notations for the operators and parameters that occur in several papers, as discussed in earlier sections, such survey could provide answer to an important question: which notation has actually been used in a given computer program? This question boils down to the central problem, i.e. how the matrix elements of the given operators have been calculated and to what extent these quantities were reliably incorporated into the subroutines for diagonalization of the respective matrices. However, answer to this question requires checking the source codes, which usually are not available to other users.

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The tremendous progress has been made in recent decades in development of several advanced quantum mechanical ab initio methods, e.g. DFT. Subsequently, the computer codes based on these methods have nowadays become more widely available. This has enabled prediction of spectroscopic properties of transition ions, including in some cases determination of the CF (LF) parameters as well as the SH (ZFS) ones. In the course of working on this review, some conceptual problems have also been identified in this area. Another aspect worth considering is bridging the gap between DFT and related methods and the conventional approaches discussed here and in many other reviews cited above, including the review by Sorace et al. [110]. Importantly, understanding of the fundamental aspects underlying physics and chemistry of single transition (4fN and 3dN) ions in coordination compounds in magnetic systems discussed in this review is crucial for meaningful application of ab initio methods and proper interpretation of results. However, detailed discussion of ab initio methods is left out from this review for two reasons. First, this vast area of studies requires a separate extensive review and second, this topic is beyond the level suitable for the intended readership. Other relevant, so highly specialized, topics that are worth reviewing include, e.g. modeling of the CF (LF) parameters with disregard of local site symmetry and modeling of the ZFS parameters with „hidden‟ CF parameters. On the closing note, we cannot agree more with the Sorace et al. [110] concluding sentences on the importance of the experimental techniques discussed therein and especially, quote: „Finally much more must be done in optical methods which can provide insights into new fundamental physics, especially in the field of magneto optics.‟ Recent research on the determination of crystal field energy levels and temperature dependence of magnetic susceptibility for Dy3+ in [Dy2Pd] heterometallic complex [292] is following this recommendation. Concerning outlook, a philosophical remark is pertinent. The situation at the CF (LF)  SH (ZFS) interface, as revealed by surveys of recent literature [33,34], has started to resemble the ancient Babel tower case, since researchers increasingly more often appear to be talking in multiple different languages, i.e. using confusing and incoherent terminology. Such situation violates the basic tenet of natural sciences that stipulates development of precise terminology. As a consequence, the crucial notions at each side of the CF (LF)  SH (ZFS) interface, which are interrelated, have become disconnected, thus leading to serious terminological confusions [33,34]. By systematization of nomenclature and bringing order to the zoo of different Hamiltonians and the associated quantities, proliferation of terminological confusions may be prevented in future literature. This would significantly reduce the instances of pitfalls and errors of substance that have detrimental consequences for understanding of physical properties of the single transition ions in various crystals or molecules as well as the exchange coupled systems of transition ions, especially the single molecule magnets or molecular nanomagnets. Acknowledgments

The authors are extremely grateful to the colleagues, especially Prof. Z. Sojka, who have kindly provided their valuable feedback, which helped improving the presentation and thus understanding of the theoretical aspects. Thanks are also due to anonymous referees for constructive criticism and helpful comments as well as for bringing to our attention several pertinent papers. We also acknowledge with thanks helpful correspondence with Prof. S.K. Misra as well as providing recent preprints in advance by Prof. S. Hill, Prof. Z. Sojka, Prof. R. Boča, and Dr Telser. We are also grateful to Mrs H. Dopierała for technical help with references. List of symbols and abbreviations (T) magnetic susceptibility (as a function of temperature T) B magnetic induction CF crystal field CFPs crystal field (or equivalently ligand field) parameters DFT density functional theory E electric field intensity ECS exchange coupled systems

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EI exchange interactions EMR electron magnetic resonance EPs experimental parameters ESO extended Stevens operator(s) FI free ion GS giant spin GSA giant spin approximation GSH generalized spin Hamiltonian ~ ~ H eff  H SH effective single-ion spin Hamiltonian

ip t

~ H ZFS effective single-ion zero-field splitting Hamiltonian ~ H Ze effective single-ion Zeeman electronic Hamiltonian ~T HGS ( H SH ) giant spin Hamiltonian

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HMH multispin Hamiltonian ~T (HGS) total spin Hamiltonian H SH ~T H ZFS total zero-field splitting Hamiltonian HF EPR high-frequency and -field EPR (alternatively: HMF-EMR) HMF-EMR high-magnetic field and high-frequency EMR (alternatively: HF EPR) HOFD higher-order field-dependent (terms) HOZFS higher-order ZFS (terms) INS inelastic neutron scattering irreps irreducible representations J ≡ (L + S) total angular momentum operator L orbital angular momentum operator LF ligand field LFPs ligand field (or equivalently crystal field) parameters Ln lanthanide (ions) LSS local site symmetry M magnetization MA magnetic anisotropy MCA magnetocrystalline anisotropy MNM molecular nanomagnets MPs microscopic parameters MS multispin/multi-spin (Hamiltonian or model) MSH microscopic spin Hamiltonian (theory) PSG point symmetry group PT perturbation theory RE rare-earth (ions) S true electronic spin S′ fictitious 'spin' ~ S effective spin SIA single-ion anisotropy SH spin Hamiltonian SMM single molecule magnets SO spin-orbit SOC spin-orbit coupling SS (electronic) spin-spin (coupling) TM transition metal (ions) TTO tesseral-tensor operators STO spherical-tensor operators Ze Zeeman electronic ZFS zero-field splitting ZFSPs zero-field splitting parameters

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Appendix 1. The extended Stevens operators Table A1. Explicit listing of the extended Stevens operators Okq (S). The following abbreviations are used: S n = Sn + Sn , S2n = Sn(S + 1)n and {X, Y} = XY + YX. The operators Ok q are

ip t

obtained simply by changing the definition of S n to [  i(Sn  Sn ) ]. For specific applications the ~ generic symbol S should be replaced by one of the actual operators defined in text, e.g. S, L, J, or S , S′, ST , as appropriate.

O20 = 3 S 2z - S2 1 {Sz, S } 4 1 O22 = S 2 2

us

cr

O21 =

O40 = 35 S4z - (30S2 - 25) S2z - 6S2 + 3S4 1 {7 S3z - 3S2Sz - Sz, S } 4 1 O42 = {7 S2z - S2 - 5, S 2 } 4 1 3 O4 = {Sz, S 3 } 4 1 4 4 O4 = S 2

d

M

an

O41 =

O61 = O62 =

O63 = O64 = O65 =

O66 =

Ac ce pt e

O60 = 231 S6z - 105(3S2 - 7) S4z + (105S4S2 + 294) S2z - 5S6 + 40S4 - 60S2 1 {33 S5z - (30S2 -15) S3z 4 1 {33 S4z - (18S2 + 123) 4 1 {11 S3z - (3S2 + 59)Sz, 4 1 {11 S2z - S2 - 38, S 4 } 4 1 {Sz, S 5 } 4 1 6 S 2

+ (5S4 - 10S2 +12)Sz, S }

S2z + S4 + 10S2 + 102, S 2 } S3 }

Appendix 2. Conversions relations The conversions relations between the crystal-field (or ligand field) parameters expressed in the extended Stevens operator notation defined in Eq. (1) in text and those in the Wybourne notation defined in Eq. (3): kq Akq r k  Re Bkq for q  0 and k q Akq r k  Im Bk q for q < 0. (A1)

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The conversion factors kq for k = 2, 4, 6 and all pertinent values of |q| are provided in Table A2. Note, that all factors kq are positive only if the phase convention (see text): Ck q   1q Ckq is





used as in Eq. (3), otherwise the signs of some parameters should be changed appropriately. Since the form in Eq. (4) has often been used, we also provide the conversions relations applicable in this case: kq Akq r k  Bkq for q  0 and k q Akq r k  Bkq for q < 0. (A2) Table A2. The conversion factors kq for the relations in Eq. (A1) and (A2).

2

3

4

5

k 2

1/ 6

2/ 6

4

8

2/ 5

4 / 10

2 / 35

8 / 70

6

16

8 / 42

16 / 105

8 / 105

16 / 3 14

8 / 3 77

16 / 231

an

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2

6

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1

cr

q 0

References

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M

[1] R.B. Burns, Mineralogical Applications of Crystal Field Theory, Cambridge University Press, Cambridge, 1993. [2] A.B.P. Lever, E.I. Solomon, Ligand Field Theory and the Properties of Transition Metal Complexes, Inorganic Electronic Structure and Spectroscopy, Eds. E.I. Solomon, A.B.P. Lever, Wiley, New York, 1999. [3] C.A. Morrison, Crystal Fields for Transition-Metal Ions in Laser Host Materials, Springer, Berlin, 1992. [4] R.C. Powell, Physics of Solid-State Laser Materials, Springer, New York, 1998. [5] B.N. Figgis, M.A. Hitchman, Ligand Field Theory and its Applications, Wiley-VCH, New York, 2000. [6] B. Henderson, R.H. Bartram, Crystal-Field Engineering of Solid-State Laser Materials, Cambridge Univ. Press, Cambridge, 2000. [7] J. Mulak, Z. Gajek, The Effective Crystal Field Potential, Elsevier, Amsterdam, 2000. [8] D.J. Newman, B. Ng, (eds.) Crystal Field Handbook, Cambridge Univ. Press, Cambridge, 2000. [9] M. Wildner, M. Andrut, C. Rudowicz, in: A. Beran, E. Libowitzky, (eds.) Spectroscopic Methods in Mineralogy - EMU Notes Mineralogy. Vol. 6, Ch. 3, p 93-143, Eötvös Univ. Press, Budapest, 2004. [10] G. Liu, B. Jacquier (eds.) Spectroscopic Properties of Rare Earths in Optical Materials, Tsinghua University Press and Springer, Berlin, 2005. [11] B.G. Wybourne, L. Smentek, Optical Spectroscopy of Lanthanides: Magnetic and Hyperfine Interactions, CRC Press, Taylor & Francis Group, Boca Raton, 2007. [12] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Clarendon Press, Oxford, 1970; Dover, New York, 1986. [13] S. Altshuler, B.M. Kozyrev, Electron Paramagnetic Resonance in Compounds of Transition Elements, Wiley, New York, 1974. [14] J.E. Wertz, J.R. Bolton, Electron Spin Resonance Elementary Theory and Practical Applications, McGraw-Hill, New York, 1972; J.A. Weil, J.R. Bolton, J.E. Wertz, Electron Paramagnetic Resonance, Elemental Theory and Practical Applications, Wiley, New York, 1994; J.A. Weil, J. R.Bolton, Electron Paramagnetic Resonance, Elemental Theory and Practical Applications, Wiley, New York, 2007. [15] A. Bencini, D. Gatteschi, EPR of Exchange Coupled Systems, Springer, Berlin, 1990. [16] J.R. Pilbrow, Transition-Ion Electron Paramagnetic Resonance, Clarendon Press, Oxford, 1990.

Page 57 of 66

57

Ac ce pt e

d

M

an

us

cr

ip t

[17] F.E. Mabbs, D. Collison, Electron Paramagnetic Resonance of d Transition-Metal Compounds, Elsevier, Amsterdam, 1992. [18] S.K. Misra, Ed., Multifrequency Electron Paramagnetic Resonance, Wiley-VCH, Weinheim, 2011. [19] O. Kahn, Molecular Magnetism,Wiley-VCH, New York, 1993. [20] K.W.H. Stevens, Magnetic Ions in Crystals, Princeton Univ. Press, Princeton, 1997. [21] R. Boča, Theoretical Foundations of Molecular Magnetism. Elsevier, Amsterdam, 1999. [22] K.H.J. Buschow, F.R. de Boer, Physics of Magnetism and Magnetic Materials, Kluwer Academic, New York, 2003. [23] R. Boča, Magnetic parameters and magnetic functions in mononuclear complexes beyond the spin-Hamiltonian formalism, Struct. Bond. 117 (2006) 1. [24] D. Gatteschi, R. Sessoli, J. Villain, Molecular Nanomagnets, Oxford Univ Press, Oxford, 2006. [25] R. Skomski, Simple Models of Magnetism, Oxford University Press, Oxford, 2008. [26] J. Jensen, A.R. Mackintosh, Rare Earth Magnetism, Clarendon Press, Oxford, 1991. [27] R.M. White, Quantum Theory of Magnetism, Springer, Berlin, 2007. [28] C. Rudowicz, Magn. Reson. Rev. 13 (1987) 1; Erratum, ibidem 13 (1988) 335. [29] C. Rudowicz, S.K. Misra, Appl. Spectrosc. Rev. 36 (2001) 11. [30] C. Rudowicz, H.W.F. Sung, Physica B 300 (2001) 1. [31] C. Rudowicz, Physica B 403 (2008) 1882. [32] C. Rudowicz, Physica B 403 (2008) 2312. [33] C. Rudowicz, M. Karbowiak, Physica B 451 (2014) 134. [34] C. Rudowicz and M. Karbowiak, Physica B (2014) - in press: DOI: 10.1016/j.physb.2014.09.011. [35] C. Anthon, C. E. Schäffer, Coord. Chem. Rev. 226 (2002) 17. [36] B.S. Tsukerblat, Group Theory in Chemistry and Spectroscopy, Academic Press, London, 1994. [37] C. Rudowicz, Nukleonika 58 (2013) 341. [38] M. Acikgoz, S. Kazan, F.A. Mikailov, E. Kerimova, B. Aktas, Cryst. Res. Technol. 43 (2008) 863. [39] G. Schaack, in: M. Cardona, G. Günthcrodt, (eds.) Light Scattering in Solids VII Crystal-Field and Magnetic Excitations, Springer, Berlin, 2000. [40] C. Rudowicz, M. Lewandowska, J. Alloys Compd. 540 (2012) 279. [41] C. Mazumdar, M. Rotter, M. Frontzke, H. Michor, M. Doerr, A. Kreyssig, M. Koza, A. Hiess, J. Voigt, G. Behr, L.C. Gupta, M. Prager, M. Loewenhaupt, Phys. Rev. B 78 (2008) 144422. [42] J.Z. Zheng, Y. Lan, C.E. Anson, A.K. Powell, Inorg. Chem. 47 (2008) 10813. [43] D.N. Woodruff, R.E.P. Winpenny, R.A. Layfield, Chem. Rev. 113 (2013) 5110. [44] M. Andruh, J.P. Costes, C. Diaz, S. Gao, Inorg. Chem. 48 (2009) 3342. [45] T. Hamamatsu, K. Yabe, M. Towatari, S. Osa, N. Matsumoto, N. Re, A.Pochaba, J. Mrozinski, J. L. Gallani, A.Barla, P. Imperia, C. Paulsen, J.P. Kappler, Inorg. Chem. 46 (2007) 4458. [46] F. Tuna, C.A. Smith, M. Bodensteiner, L. Ungur, L.F. Chibotaru, E.J.L. McInnes, R.E.P. Winpenny, D. Collison, R.A. Layfield, Angew. Chem. Int. Ed. 51 (2012) 6976. [47] K. Bernot, J. Luzon, L. Bogani, M. Etienne, C. Sangregorio, M. Shanmugam, A. Caneschi, R. Sessoli, D. Gatteschi, J. Am. Chem. Soc. 131 (2009) 5573. [48] P.H. Lin, T.J. Burchell, L. Ungur, L.F. Chibotaru, W. Wernsdorfer, M. Murugesu, Angew. Chem. Int. Ed. 48 (2009) 9489. [49] R.A. Layfield, J.J.W. McDouall, S.A. Sulway, F. Tuna, D. Collison, R.E.P. Winpenny, Chem. Eur. J. 16 (2010) 4442. [50] C. Apostolidis, A. Morgenstern, J. Rebizant, B. Kanellakopulos, O. Walter, B. Powietzka, M. Karbowiak, H. Reddmann, H.D. Amberger, Z. Anorg. Allg. Chem. 636 (2010) 201. [51] R. Troć, Z. Gajek, A. Pikul, H. Misiorek, Phys. Rev. B 88 (2013) 24416. [52] P. Novák, K. Knizek, J. Kuneš, Phys. Rev. B 87 (2013) 205139. [53] I. Kebaili, M. Dammak, J. Lumin. 132 (2012) 2092. [54] J. Luzon, R. Sessoli, Dalton Trans. 41 (2012) 13556. [55] W.W. Lukens, M.D. Walter, Inorg. Chem. 49 (2010) 4458. [56] F. Borsa, Y. Furukawa, A. Lascialfari, Inorg. Chim. Acta 361 (2008) 3777. [57] L. Zhang, P. Zhang, L. Zhao, S.Y. Lin, S. Xue, J. Tang, Z. Liu, Eur. J. Inorg. Chem. (2013) 1351.

Page 58 of 66

58

Ac ce pt e

d

M

an

us

cr

ip t

[58] Y.N. Guo, G.F. Xu, W. Wernsdorfer, L. Ungur, Y. Guo, J. Tang, H.J. Zhang, L.F. Chibotaru, A.K. Powell, J. Am. Chem. Soc. 133 (2011) 11948. [59] F. Pointillart, B. Le Guennic, S. Golhen, O. Cador, O. Maury, L. Ouahab, Inorg. Chem. 52 (2013)1610. [60] J.L. Liu, Y.C. Chen, Y.Z. Zheng, W.Q. Lin, L. Ungur, W. Wernsdorfer, L.F. Chibotaru, M.L. Tong, Chem. Sci. 4 (2013) 3310. [61] N. F. Chilton, R. P. Anderson, L. D. Turner, A. Soncini, K. S. Murray, J. Comp. Chem. 34 (2013) 1164. [62] N.F. Chilton, D. Collison, E.J.L. McInnes, R.E.P. Winpenny, A. Soncini, Nat. Comm. 4 (2013) 2551. [63] L. Ungur, L.F. Chibotaru, Phys. Chem. Chem. Phys. 13 (2011) 20086. [64] S.D. Jiang, S.S. Liu, L.N. Zhou, B.W. Wang, Z.M. Wang, S. Gao, Inorg. Chem. 51 (2012) 3079. [65] H. Schilder, H. Lueken, J. Magn. Magn. Mater. 281 (2004) 17. [66] S.D. Jiang, S.S. Liu, L.N. Zhou, B.W. Wang, Z.M. Wang, S. Gao, Inorg. Chem. 51 (2012) 3079. [67] L. Badia-Romano, F. Bartolomé, J. Bartolomé, J. Luzon, D. Prodius, C. Turta, V. Mereacre, F. Wilhelm, A. Rogalev, Phys. Rev. B 87 (2013)184403. [68] N. Magnani, E. Colineau, R. Eloirdi, J.C. Griveau, R. Caciuffo, S.M. Cornet, I. May, C.A. Sharrad, D. Collison, R.E.P. Winpenny, Phys. Rev. Lett. 104 (2010) 197202. [69] S. Ghosh, S. Datta, L. Friend, S. Cardona-Sierra, A. Gaita-Arino, E. Coronado, S. Hill, Dalton Trans. 41 (2012) 13697. [70] P.L.W. Tregenna-Piggott, D. Sheptyakov, L. Keller, S.I. Klokishner, S.M. Ostrovsky, A.V. Palii, O.S. Reu, J. Bendix, T. Brock-Nannestad, K. Pedersen, H. Weihe, H. Mutka. Inorg. Chem. 48 (2009) 128. [71] Q. Zhou, F. Yang, B. Xin, G. Zeng, X. Zhou, K. Liu, D. Ma, G. Li, Z. Shi, S. Feng, Chem. Commun. 49 (2013) 8244.. [72] M. Ren, S.S. Bao, N. Hoshino, T. Akutagawa, B. Wang, Y.C. Ding, S. Wie,L. M. Zheng, Chem. Eur. J. 19 (2013) 9619. [73] S.K. Langley, N.F. Chilton, B. Moubaraki, K.S. Murray, Dalton Trans. 41 (2012) 1033. [74] F. Lloret, M. Julve, J. Cano, R. Ruiz-Garcia, E. Pardo, Inorg. Chim. Acta 361 (2008) 3432. [75] N. Ishikawa, M. Sugita, T. Okubo, N. Tanaka, T. Lino, Y. Kaizu, Inorg. Chem. 42 (2003) 2440. [76] N. Ishikawa, M. Sugita, W. Wernsdorfer, J. Am. Chem. Soc. 127 (2005) 3650. [77] J.D. Rinehart, J.R. Long, Dalton Trans. 41 (2012)13572. [78] N.F. Chilton, S.K. Langley, B. Moubaraki, A. Soncini, S.R. Batten, K.S. Murray, Chem. Sci. 4 (2013) 1719. [79] C. R. Ganivet, B. Ballesteros, G. de la Torre, J.M. Clemente-Juan, E. Coronado, T. Torres, Chem. Eur. J. 19 (2013) 1457. [80] Y. Bi, Y.N. Guo, L. Zhao, Y. Guo, S.Y. Lin, S.D. Jiang, J. Tang, B.W. Wang, S. Gao, Chem. Eur. J. 17 (2011) 12476. [81] J.D. Rinehart, K.R. Meihaus, J.R. Long, J. Am. Chem. Soc. 132 (2010) 7572. [82] M.A. AlDamen, J.M. Clemente-Juan, E. Coronado, C. Martí-Gastaldo, A. Gaita-Arino, J. Am. Chem. Soc. 130 (2008) 8874. [83] S.D. Jiang, B.W. Wang, G. Su, Z.M. Wang, S. Gao, Angew. Chem. Int. Ed. 49 (2010) 7448. [84] J.D. Rinehart, J.R. Long, J. Am. Chem. Soc.131 (2009) 12558. [85] D.P. Mills, F. Moro, J. McMaster, J. van Slageren, W. Lewis, A.J. Blake, S.T. Liddle, Nat. Chem. 3 (2011) 454. [86] Y. Gao, G.F. Xu, L. Zhao, J. Tang, Z. Liu, Inorg. Chem. 48 (2009) 11495. [87] N. Ishikawa, Polyhedron 26 (2007) 2147. [88] T. Goswami, A. Misra, J. Phys. Chem. A 116 (2012) 5207. [89] A.V. Palii, J.M. Clemente-Juan, E. Coronado, S.I. Klokishner, S.M. Ostrovsky, O.S. Reu, Inorg. Chem. 49 (2010) 8073. [90] L. Sorace, C. Sangregorio, A. Figuerola, C. Benelli, D. Gatteschi, Chem. Eur. J. 15 (2009) 1377. [91] A. Cornia, M. Mannini, P. Sainctavit, R. Sessoli, Chem. Soc. Rev. 40 (2011) 3076. [92] J.D. Rinehart, J.R. Long, Chem. Sci. 2 (2011) 2078.

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cr

ip t

[93] J. Bartolomé, G. Filoti, V. Kuncser, G. Schinteie, V. Mereacre, C.E. Anson, A.K. Powell, D. Prodius, C. Turta, Phys. Rev. B 80 (2009) 014430. [94] M. Mannini, F. Pineider, C. Danieli, F. Totti, L. Sorace, P. Sainctavit, M.A. Arrio, E. Otero, L. Joly, J.C. Cezar, A. Cornia, R. Sessoli, Nature 468 (2010) 417. [95] V. Mereacre, A. Baniodeh, C.E. Anson, A. K. Powell, J. Am. Chem. Soc. 133 (2011) 15335. [96] K.S. Pedersen, M. Schau-Magnussen, J. Bendix, H. Weihe, A.V. Palii, S.I. Klokishner, S. Ostrovsky, O.S. Reu, H. Mutka, P.L.W. Tregenna-Piggott, Chem. Eur. J. 16 (2010) 13458. [97] A.V. Palii, O.S. Reu, S.M. Ostrovsky, S.I. Klokishner, B.S. Tsukerblat, M. Hilfiger, M. Shatruk, A. Prosvirin, K.R. Dunbar, J. Phys. Chem. A 113 (2009) 6886. [98] D. Visinescu, L.M. Toma, O. Fabelo, C. Ruiz-Perez, F. Lloret, M. Julve, Inorg. Chem. 52 (2013) 1525. [99] F. Volcker, Y. Lan, A.K. Powell P.W. Roesky, Dalton Trans. 42 (2013) 11471. [100] E.M. Boulon, G. Cucinotta, J. Luzon, C. Degl‟Innocenti, M. Perfetti, K. Bernot, G. Calvez, A. Caneschi, R. Sessoli, Angew. Chem. 125 ( 2013) 368; Angew. Chem. Int. Ed. 52 (2013) 350. [101] D. Aravena, E. Ruiz, Inorg. Chem. 52 (2013) 13770. [102] L. Ungur, W. Van den Heuvel, L.F. Chibotaru, New J. Chem. 33 (2009)1224. [103] L.F. Chibotaru, L. Ungur, J. Chem. Phys. 137 ( 2012) 64112. [104] J.M. Clemente-Juan, E. Coronado, A. Gaita-Arino, Chem. Soc. Rev. 41 (2012) 7464. [105] J.J. Baldoví, S. Cardona-Serra, J.M. Clemente-Juan, E. Coronado, A. Gaita-Arino, A. Palii, J. Comput. Chem. 34 (2013) 1961. [106] N. Ishikawa, M. Sugita, W. Wernsdorfer, Angew. Chem. Int. Ed. 44 (2005) 2931. [107] H. Souissi, S. Kammoun, J. Lumin. 131 (2011) 2515. [108] P. Porcher, M. Couto Dos Santos, O. Malta, Phys. Chem. Chem. Phys. 1 (1999) 397. [109] A. Palii, B. Tsukerblat, S. Klikishner, K.R. Dunbar, J.M. Clemente-Juan, E. Coronado, Chem. Soc. Rev. 40 (2011) 3130. [110] L. Sorace, C. Benelli, D. Gatteschi, Chem. Soc. Rev. 40 (2011) 3092. [111] P. Zhang, Y.N. Guo, J. Tang, Coord. Chem. Rev. 257 (2013) 1728. [112] P.J. Alonso, J.I. Martinez, I. Garcia-Rubio, Coord. Chem Rev. 251 (2007) 12. [113] Y. Y. Yeung, P. A. Tanner, J. Alloys Compd. 575 (2013) 54. [114] Y. Y. Yeung, C. Rudowicz, Comput. Chem. 16 (1992) 207. [115] Z. Y. Yang, Y. Hao, C. Rudowicz, Y. Y. Yeung, J. Phys.: Cond. Mat. 16 (2004) 3481. [116] R. Boča, A Handbook of Magnetochemical Formulae. Elsevier, Amsterdam, 2012. [117] J. Schnack, J. Ummethum, Polyhedron 66 (2013) 28. [118] M.H.L. Pryce, Proc. Phys. Soc. A63 (1950) 25. [119] C. Rudowicz, Acta Phys. Polon. A43 (1973) 551. [120] C. Rudowicz, Acta Phys. Polon. A44 (1973) 103. [121] C. Rudowicz, B. Konkiewicz, Acta Phys. Polon. A47 (1975) 291. [122] C. Rudowicz, Acta Phys. Polon. A47 (1975) 305. [123] C. Rudowicz, Acta Phys. Polon. A51 (1977) 515. [124] C. Rudowicz, Biochimica et Biophysica Acta 490 (1977) 301. [125] C. Rudowicz, J. Phys. Chem. Solids 38 (1977) 1243. [126] C. Rudowicz, Y.Y. Zhou, J. Mag. Mag. Mat. 111 (1992) 153. [127] C. Rudowicz, Y. Zhou, Computers & Chemistry 21 (1997) 45. [128] C. Rudowicz, H.W.F. Sung, Physica B 337 (2003) 204. [129] E.I. Solomon, E.G. Pavel, K.E. Loeb, C. Campochiaro, Coord. Chem. Rev. 144 (1995) 369. [130] R. Herchel, R. Boča, J. Krzystek, A. Ozarowski, M. Duran, J. van Slageren, J. Am. Chem. Soc. 129 (2007) 10306. [131] F. Neese, Coord. Chem. Rev. 253 (2009) 526. [132] J. Krzystek, A. Ozarowski, J. Telser, Coord. Chem. Rev. 250 (2006) 2308. [133] P. Pietrzyk, K. Podolska, Z. Sojka, Molecular interpretation of EPR parameters – computational spectroscopy approaches, in Electron Paramag. Reson. ed. by B.C. Gilbert, D.M. Murphy, V. Chechik, 23 (2013) 264.

Page 60 of 66

60

Ac ce pt e

d

M

an

us

cr

ip t

[134] P. Pietrzyk, T. Mazur, Z. Sojka, Electron paramagnetic resonance spectroscopy of inorganic materials, in Local Structural Characterisation, ed. by D.W. Bruce, D. O‟Hare, R. I. Walton, Wiley & Sons, (2014) 225. [135] C. Rudowicz, D. Piwowarska, P. Gnutek, Electron magnetic resonance (EMR) and related investigations of the Co 2+(3d7) ions in crystals with the effective spin S = 3/2 and S = ½ – the origin of the ground Kramers doublets with the fictitious „spin‟ S‟ = ½ revisited, in preparation (2014). [136] A. Venugopal, F. Tuna, T.P. Spaniol, L. Ungur, L.F. Chibotaru, J. Okuda, R.A. Layfield, Chem. Comm. 49 (2013) 901. [137] M. Lahoubi, J. Phys. Conf. Series 340 (2012) 012068. [138] T. Guidi, S. Carretta, P. Santini, E. Liviotti, N. Magnani, C. Mondelli, O. Waldman, L.K. Thompson, L. Zhao, C.D. Frost, G. Amoretti, R. Caciuffo, Phys. Rev. B 69 (2004) 104432. [139] O. Waldman, H.U. Güdel, Phys. Rev. B 72 (2005) 094422. [140] E. Liviotti, S. Carretta, G. Amoretti, J. Chem. Phys. 117 (2002) 3361. [141] M. Affronte, T. Guidi, R. Caciuffo, S. Carretta, G. Amoretti, J. Hinderer, I. Sheikin, A.G.M. Jansen, A. A. Smith, R.E.P. Winpenny, J. van Slageren, D. Gatteschi, Phys. Rev. B 68 (2003) 104403. [142] S. Carretta, J. van Slageren, T. Guidi, E. Liviotti, C. Mondelli, D. Rovai, A. Cornia, A.L. Dearden, F. Carsughi, M. Affronte, C.D. Frost, R.E.P. Winpenny, D. Gatteschi, G. Amoretti, R. Caciuffo, Phys. Rev. B 67 (2003) 094405. [143] R. Maurice, K. Sivalingam, D. Ganyushin, N. Guihery, C. de Graff, F. Neese, Inorg. Chem. 50 (2011) 6229. [144] O. Khan, Angew. Chem. Int. Ed. 24 (1985) 834. [145] J.J. Borrás-Almenar, J.M. Clemente-Juan, E. Coronado, B.S. Tsukerblat, Inorg. Chem. 38 (1999) 6081. [146] J. Telser, A. Ozarowski, J. Krzystek, High-frequency and -field electron paramagnetic resonance of transition metal ion (d block) coordination complexes, in Electron Paramag. Reson., ed. by B.C. Gilbert, D.M. Murphy, V. Chechik, 23 (2013) 209. [147] G. Chaboussant, A. Sieber, S. Ochsenbein, H.U. Güdel, M. Murrie, Phys. Rev. B 70 (2004) 104422. [148] P.L. Feng, C. Koo, J.J. Henderson, P. Manning, M. Nakano, E. del Barco, S. Hill, D.N. Hendrickson, Inorg. Chem. 48 (2009) 3480. [149] L.V. Kulik, B. Epel, W. Lubitz, J. Messinger, J. Am. Chem. Soc. 129 (2007) 13435. [150] M. Nakano, H. Oshio, Chem. Soc. Rev. 40 (2011) 3239. [151] S. Accorsi, A.L. Barra, A. Caneschi, G. Chastanet, A. Cornia, A.C. Fabretti, D. Gatteschi, C. Mortalo, E. Olivieri, F. Parenti, P. Rosa, R. Sessoli, L. Sorace, W. Wernsdorfer, L. Zobbi, J. Am. Chem. Soc.128 (2006) 4742. [152] M. Fittipaldi, L. Sorace, A.L. Barra, C. Sangregorio, R. Sessoli, D. Gatteschi, Phys. Chem. Chem. Phys. 11 (2009) 6555. [153] S. Carretta, T. Guidi, P. Santini, G. Amoretti, O. Pieper, B. Lake, J. van Slageren, F. El Hallak, W. Wernsdorfer, H. Mutka, M. Russina, C.J. Milios, E. K. Brechin, Phys. Rev. Lett. 100 (2008) 157203. [154] A. Wilson, J. Lawrence, E.C. Yang, M. Nakano, D.N. Hendrickson, S. Hill, Phys. Rev. B 74 (2006) 140403. [155] D. Gatteschi, R. Sessoli, Angew. Chem., Int. Ed. 42 (2003) 268. [156] W. Wernsdorfer, Molecular Nanomagnets, in Advanced Magnetic Nanostructures, D. Sellmyer, R. Skomski (eds), Springer, Berlin, 2006. [157] A. L. Barra, A. Caneschi, A. Cornia, D. Gatteschi, L. Gorini, L.P. Heiniger, R. Sessoli, L. Sorace, J. Am. Chem. Soc. 129 (2007) 10754. [158] W.Wernsdorfer, in Handbook of Magnetism, Advanced Magnetic Materials, ed. by H. Kronmuller, S. Parkin, Wiley, New York, 2007. [159] J. van Slageren, in M. Drescher, G. Jeschke (eds), EPR Spectroscopy. Applications in Chemistry, Biology, Springer, Berlin Heidelberg; Topics in Current Chemistry, 321 (2012) 199. [160] P.L. Feng, C.C. Beedle, C. Koo, J. Lawrence, S. Hill, Inorg. Chim. Acta 361 (2008) 3465.

Page 61 of 66

61

Ac ce pt e

d

M

an

us

cr

ip t

[161] J.J. Henderson, C. Koo, P.L. Feng, E. del Barco, S. Hill, I.S. Tupitsyn, P.C.E. Stamp, D.N. Hendrickson, Phys. Rev. Lett. 103 (2009) 017202. [162] W. Wernsdorfer, Advances in Chemical Physics, ed. by I. Prigogine, S.A. Rice, Wiley & Sons, Chichester, 2001. [163] C. Lampropoulos, M. Murugesu, A.G. Harter, W. Wernsdorfer, S. Hill, N.S. Dalal, A.P. Reyes, P.L. Kuhns, K.A. Abboud, G. Christou, Inorg. Chem. 52 (2013) 258. [164] C. Rudowicz, M. Karbowiak, Interface between of the magnetic anisotropy related quantities and the effective zero-field splitting quantities, in preparation (2014). [165] J. M. Peloquin, K.A. Campbell, D.W. Randall, M.A. Evanchik, V.L. Pecoraro, W.A. Armstrong, R.D. Britt, J. Am. Chem. Soc. 122 (2000) 10926. [166] D. Gatteschi, P. Carretta, A. Lascialfari, Physica B 289-290 (2000) 94. [167] E.J.L. McInnes, Struct. Bond. 122 (2006) 69. [168] G. Amoretti, R. Caciuffo, S. Carretta, T. Guidi, N. Magnani, P. Santini, Inorg. Chim. Acta 361 (2008) 3771. [169] R. Basler, C. Boskovic, G. Chaboussant, H.U. Güdel, M. Murrie, S.T. Ochsenbein, A. Sieber, ChemPhysChem. 4 (2003) 910. [170] M. Atanasov, P. Comba, C.A. Daul, Inorg. Chem. 47 (2008) 2449. [171] R. Maurice, C. de Graaf, N. Guihery, Phys. Rev. B 81 (2010) 214427. [172] A. Okazawa, T. Nogami, H. Nojiri, T. Ishida, Inorg.Chem. 47 (2008) 9763. [173] A. Okazawa, T. Shimada, N. Kojima, S. Yoshii, H. Nojiri, T. Ishida, Inorg. Chem. 52 (2013) 13351. [174] J. Liu, C.C. Beedle, H.M. Quddusi, E. del Barco, D.N. Hendrickson, S. Hill, Polyhedron 30 (2011) 2965. [175] J. Liu, E. del Barco, S. Hill, Phys. Rev. B 85 (2012) 12406. [176] S. Carretta, P. Santini, E. Liviotti, N. Magnani, T. Guidi, R. Caciuffo, G. Amoretti, Eur. Phys. J. B 36 (2003) 169. [177] L. Gregoli, C. Danieli, A.L. Barra, P. Neugebauer, G. Pellegrini, G. Poneti, R. Sessoli, A. Cornia, Chem. Eur. J. 15 (2009) 6456. [178] R. Sessoli, Inorg. Chim. Acta 361 (2008) 3356. [179] D. Gatteschi, A.L. Barra, A. Caneschi, A. Cornia, R. Sessoli, L. Sorace, Coord. Chem. Rev. 250 (2006) 1514. [180] M. Farbinteanu, H. Miyasaka, W. Wernsdorfer, K. Nakata, K. Sugiura, M. Yamashita, C. Coulon, R. Clerac, J. Am. Chem. Soc. 127 (2005) 3090. [181] R. Clerac, H. Miyasaka, M. Yamashita, C. Coulon, J. Am. Chem. Soc. 124 (2002) 12844. [182] A. Das, K. Gieb, Y. Krupskaya, S. Demeshko, S. Deckert, R. Klingeler, V. Kataev, B. Buchner, P. Muller, F. Meyer, J. Am. Chem. Soc. 133 (2011) 3433. [183] V. Chandrasekhar, B. Murugesa Pandian, R. Boomishankar, A. Steiner, J.J. Vittal, A. Houri, R. Clerac, Inorg. Chem. 47 (2008) 4918. [184] J.P. Costes, F. Dahan, F. Dumestre, J.M. Clemente-Juan, J. Garcia-Tojal, J.P. Tuchagues, Dalton Trans. (2003) 464. [185] S. Carretta, E. Livotti, N. Magnani, P. Santini, G. Amoretti, Phys. Rev. Lett. 92 (2004) 207205. [186] C. Rudowicz, P. Gnutek, S. Kimura, M. Açıkgöz, Y.Y. Yeung, Appl. Mag. Res. 44 (2013) 899. [187] S. Hill, Polyhedron 64 (2013) 128. [188] J. Liu, E. del Barco, S. Hill, in: J. Bartolome, J. Fernandez, F. Luis (Eds.), Molecular Nanomagnets: Physics and Applications, Springer, in press, http://dx.doi.org/10.1016/j.poly.2013.03.018. [189] J. Liu, S. Hill, Polyhedron 66 (2013) 147. [190] E. del Barco, A.D. Kent, S. Hill, J.M. North, N.S. Dalal, E.M. Rumberger, D.N. Hendrickson, N. Chakov, G. Christou, J. Low Temp. Phys. 140 (2005) 119. [191] K.W.H. Stevens, Proc. Phys. Soc. 65 (1952) 209. [192] G. Racah, Phys. Rev. 62 (1942) 438. [193] B.G. Wybourne, Spectroscopic Properties of Rare Earth, Wiley, New York, 1965. [194] C. Rudowicz, J. Phys. C: Solid State Phys. 18 (1985) 1415; Erratum: ibidem C 18 (1985) 3837. [195] D.J. Newman, W. Urban, Adv. Phys. 24 (1975) 793.

Page 62 of 66

62

Ac ce pt e

d

M

an

us

cr

ip t

[196] C. Rudowicz, C.Y. Chung, J. Phys. Cond. Matter. 16 (2004) 5825. [197] I.D. Ryabov, Appl. Magn. Reson. 35 (2009) 481. [198] M. Rotter, Comp. Mater. Sci. 38 (2006) 400. [199] N.F. Chilton, PHI, User Manual v1.7 (2013). [200] C. Rudowicz, J. Qin, J. Alloys Compd. 389 (2005) 256. [201] C. Rudowicz, J. Qin, Physica B 364 (2005) 239. [202] C. Rudowicz, P. Gnutek, M. Karbowiak, Phys. Rev. B 76 (2007) 125116. [203] C. Rudowicz, P. Gnutek, M. Lewandowska, M. Orłowski, J. Alloys Compd. 467 (2009) 98. [204] C. Rudowicz, P. Gnutek, M. Lewandowska, J. Alloys Compd. 467 (2009) 106. [205] C. Rudowicz, M. Lewandowska, P. Gnutek, J. Alloys Compd. 497 (2010) 32. [206] G.W. Budrick, M. F. Reid, Mol. Phys. 102 (2004) 1141. [207] C. Görller-Walrand, K. Binnemans, Rationalization of crystal-field parametrization, in: K.A. Gschneidner, Jr., L. Eyring (eds.), Handbook on the Physics and Chemistry of Rare Earths, v. 23, ch. 155, p. 121, Elsevier, Amsterdam, 1996. [208] C. Rudowicz, P. Gnutek, M. Karbowiak, Opt. Mat. 33 (2011) 1557. [209] Z.Y. Yang, Q. Wei, Physica B 370 (2005) 137. [210] P. Novák, K. Knizek, M. Marysko, Z. Jirak, J. Kuneš, ar xiv (1306.5948) 1 (2013) 1. [211] I.D. Ryabov, Phys. Chem. Minerals 38 (2011) 177. [212] C. Rudowicz, J. Phys.: Condens. Matter. 12 (2000) L417. [213] C. Rudowicz, J. Mag. Magn. Mater. 321 (2009) 2946. [214] T.H. Yeom, C. Rudowicz, S.H. Choh, D.G. McGavin, Physica Status Solidi (b) 198 (1996) 839. [215] C. Rudowicz, P. Gnutek, Physica B 403 (2008) 2349. [216] P. Gnutek, C. Rudowicz, Opt. Mater. 31 (2008) 391. [217] C. Rudowicz, P. Budzyński, Phys. Rev. B 74 (2006) 054415. [218] C. Rudowicz, D. Piwowarska, Solid State Commun. 151 (2011) 855. [219] D. Gatteschi, J. Phys. Chem. B 104 (2000) 9780. [220] W. Wernsdorfer, R. Sessoli, Science 284 (1999) 133. [221] N.V. Prokof‟ev, P.C.E. Stamp, Phys. Rev. Lett. 80 (1998) 5794. [222] S. Kuroda, M. Motokawa, M. Date, J. Phys. Soc. Japan 44 (1978) 1797. [223] M. Hagiwara, T, Kashiwagi, S. Kimura, Z. Honda, K. Kindo, J. Mag. Magn. Materials 310 (2007) 1272. [224] S. Kimura, M. Hagiwara, T.Takeuchi, H. Yamaguchi, H. Ueda, Y. Ueda, K. Kindo, Phys. Rev. B 83 (2011) 214401. [225] Y. Imanaka, N. Miura, H. Nojiri, S. Luther, M.V. Ortenberg, T. Yokoo, Physica B 246-247 (1998) 561. [226] M. Motokawa, Rep. Prog. Phys. 67 (2004) 1995. [227] A.L. Barra, D. Gatteschi, R. Sessoli, L. Sorace, Magn. Reson. Chem. 43 (2005) S183. [228] M. Bennati, T.F. Prisner, Rep. Prog. Phys. 68 (2005) 411. [229] K.K. Andersson, P.P. Schmidt, B. Katterle, K.R. Strand, A.E. Palmer, S.K. Lee, E.I. Solomon, A. Graslund, A. L. Barra, J. Biol. Inorg. Chem. 8 (2003) 235. [230] J. van Slageren, S. Vongtragool, B. Gorshunov, A.A. Mukhin, N. Karl, J. Krzystek, J. Telser, A. Muller, C. Sangregorio, D. Gatteschi, M. Dressel, Phys. Chem. Chem. Phys. 5 (2003) 3837. [231] T. Sakurai, K. Fujimoto, R. Goto, S. Okubo, H. Ohta, Y. Uwatoko, J. Magn. Reson. 223 (2012) 41. [232] L.C. Brunel, J. van Tol, A. Angerhofer, S. Hill, J. Krzystek, A.L. Maniero, Biol. Magn. Reson. 22 (2004) 465. [233] M.R. Smith, R. Chatterjee, H.A. Buckmaster, J. Magn. Reson. 25 (1977) 499. [234] H.A. Buckmaster, R. Chatterjee, J.A. Tuszynski, J. Chem. Phys. 83 (1985) 4001. [235] J.A. Tuszynski, H.A. Buckmaster, R. Chatterjee, J.M. Boteler, J. Magn. Reson. 63 (1985) 241. [236] C. Rudowicz, R. Bramley, J. Phys. C20 (1987) L77. [237] V.G. Grachev, Sov. Phys., JETP 65 (1987) 1029. [238] R. Chatterjee, H.A. Buckmaster, J. Phys. C3 (1991) 7079. [239] H.A. Buckmaster, R. Chatterjee, Phys. Stat. Sol. (b) 209 (1998) 433.

Page 63 of 66

63

Ac ce pt e

d

M

an

us

cr

ip t

[240] K.R. Kittilstved, L.A. Sorgho, N. Amstutz, P.L. W. Tregenna-Piggott, A. Hauser, Inorg. Chem. 48 (2009) 7750. [241] E. Colacio, J. Ruiz, E. Ruiz, E. Cremades, J. Krzystek, S. Carretta, J. Cano. T. Guidi, W. Wernsdorfer, E.K. Brechin, Angew. Chem. Int. Ed. 52 (2013) 9130. [242] C. Rudowicz, P. Gnutek, Physica B 404 ( 2009) 3582. [243] D. Gatteschi, L. Sorace, R. Sessoli, A.L. Barra, Appl. Magn. Reson. 21 (2001) 299. [244] J. Krzystek, J. Telser, L.A. Pardi, D.P. Goldberg, B.M. Hoffman, L.C. Brunel, Inorg. Chem. 26 (1999) 6121. [245] C. Rudowicz, H.W.F. Sung, J. Phys. Soc. Japan 72 Supplement B (2003) 61. [246] J. Krzystek, J.H. Park, M.W. Meisel, M.A. Hitchman, H. Stratemeier, L.C. Brunel, Inorg. Chem. 41 (2002) 4478. [247] J. Krzystek, G.J. Yeagle, J.H. Park, R.D. Britt, M.W. Meisel, L.C. Brunel, J. Telser, Inorg. Chem. 42 (2003) 4610. [248] J. Krzystek, A.T. Fiedler, J.J. Sokol, A. Ozarowski, S.A. Zvyagin, T.C. Brunold, J.R. Long, L.C. Brunel, J. Telser, Inorg. Chem. 43 (2004) 5645. [249] J. Krzystek, J. England, K. Ray, A. Ozarowski, D. Smirnov, L. Que, Jr., J. Telser, Inorg. Chem. 47 (2008) 3483. [250] J.J. Baldov , S. Cardona-Serra, J.M. Clemente-Juan, E. Coronado, A. Gaita-Ari o, A. Palii, Inorg. Chem. 51 (2012) 12565. [251] J.J. Baldoví, J.J. Borrás-Almenar, J.M. Clemente-Juan, E. Coronado, A. Gaita-Ariño, Dalton Trans. 41 (2012) 13705. [252] J.J. Baldoví, S. Cardona-Serra, J.M. Clemente-Juan, E. Coronado, A. Gaita-Ariño, Chem. Sci. 4 (2013) 938. [253] J.J. Baldoví, J.M. Clemente-Juan, E. Coronado, A. Gaita-Ari o, Polyhedron 66 (2013) 39. [254] A.Yamashita, A. Watanabe, S. Akine, T. Nabeshima, M. Nakano, T, Yamamura, T. Kajiwara, Angew. Chem. Int. Ed. 50 (2011) 4016. [255] J. M. Clemente-Juan, E. Coronado, A. Gaita-Arino, Chem. Soc. Rev. 41 (2012) 7464. [256] S. Pandey, R. Kripal, Acta Phys. Polonica A123 (2013) 101. [257] M. Karbowiak, C. Rudowicz, J. Comput. Chem. (2014), DOI: 10.1002/jcc.23700 [258] C. Rudowicz, M. Karbowiak, Acta Phys. Polonica 125 (2014) 1215. [259] P. Santini, S. Carretta, G. Amoretti, R. Caciuffo, N. Magnani, G.H. Luer, Rev. Modern Phys. 81 (2009) 807. [260] S. Stoll, A. Schweiger, J. Magn. Reson. 178 (2006) 42. [261] M. Rotter, J. Magn. Magn. Mater. 272-276 (2004) e481. [262] M. Rotter, D.M. Le, L.G. Pascut, T. Hoffmann, M. Doerr, R. Schedler, P.F. Hoffmann, S. Rotter, M. Banks, McPhase Users Manual version 4.7, June 28, 2012; http://www.mcphase.de. [263] S.L. Andronenko, R.R. Andronenko, S.K. Misra, Physica B 407 (2012) 1203. [ 264 ] S. Cardona-Serra, J. M. Clemente-Juan, E. Coronado, A. Gaita-Ari o, A. Cam n, M. Evangelisti, F. Luis, M. J. Mart nez-P rez and J. Ses , J. Am. Chem. Soc. 134 (2012) 14982. [265] R. Orbach, Proc. Phys. Soc. A 264 (1961) 458. [266] C. van Wüllen, J. Chem. Phys. 130 (2009) 194109. [267] F. Aquilante, L. De Vico, N. Ferre, G. Ghigo, P.A. Malmqvist, P. Neogrady, T.B. Pedersen, M. Pitonak, M. Reiher, B.O. Roos, L. Serrano-Andres, M. Urban, V. Veryazo, R. Lindh, J. Comput. Chem. 31 (2010) 224. [268] W.Q. Yang, Y. Zhang, Y. Lin, W.C. Zheng, J. Magn. Reson. 227 (2013) 62. [269] H.A. Buckmaster, Y.H. Shing, Phys. Status Solidi A 12 (1972) 325. [270] A.D. Gorlov, Phys. Solid State 55 (2013) 960. [271] G.R. Astryan, M. Nikl, V.A. Vazhenin, A.P. Potapov, Phys. Solid State 55 (2013) 116. [272] V.A. Vazhenin, A.P. Potapov, A.V. Fokin, M.Y. Artyomov, Phys. Solid State 54 (2012) 2450. [273] V.A. Vazhenin, A.P. Potapov, A.V. Fokin, M.Y. Artyomov, Phys. Solid State 55 (20130 1214. [274] C. Rudowicz and J. Qin, Phys. Rev. B 67 (2003) 174420. [275] M. J. Mart nez-P rez, S. Cardona-Serra, C. Schlegel, F. Moro, P. J. Alonso, H. Prima-García, J. M. Clemente-Juan, M. Evangelisti, A. Gaita-Ari o, J. Ses , J. van Slageren, E. Coronado and F. Luis, Phys. Rev. Lett. 108 (2012) 247213.

Page 64 of 66

64

Ac ce pt e

d

M

an

us

cr

ip t

[276] M. T. Hutchings, Solid State Phys., ed. by F. Seitz and D. Turnbull, Academic Press, New York, 1964. [277] P.A. Lindgård, O. Danielsen, J. Phys.C: Solid State Phys. 7 (1974) 1523. [278] A.L. Barra, Appl. Magn. Reson. 21 (2001) 619. [279] A. Forment-Aliaga, E. Coronado, M. Feliz, A. Gaita-Ariño, R. Llusar, F.M. Romero, Inorg. Chem. 42 (2003) 8019. [280] I. Mirebeau, M. Hennion, H. Casalta, H. Andres, H. U. Güdel, A. V. Irodova, A. Caneschi, Phys. Rev. Letters 83 (1999) 628. [281] R. Bircher, G. Chaboussant, A. Sieber, H. U. Güdel, H. Mutka, Phys. Rev. B 70 (2004) 212413. [282] A.L. Barra, F. Bencini, A. Caneschi, D. Gatteschi, C. Paulsen, C. Sangregorio, R. Sessoli, L. Sorace, ChemPhysChem. 2 (2001) 523. [283] A.L. Barra, D. Gatteschi, R. Sessoli, Chem. Eur. J. 6 (2000) 1608. [284] B.Z. Malkin, in Spectroscopy of Solids Containing Rare-Earth, ed. by A.A. Kaplyanskii, B.M. Macfarlane, North-Holland, Amsterdam, 1987. [285] M.G. Brik, A. El-Korashy, M. Almokhtar, J. Alloys Compd. 459 (2008) 71. [286] M.G. Brik, C.N. Avram, J. Lumin. 131 (2011) 2642. [287] A.M. Srivastava, M.G. Brik, J. Lumin. 132 (2012) 579. [288] M.G. Brik, H. Teng, H. Lin, S. Zhou, N.M. Avram, J. Alloys Compd. 506 (2010) 4. [289] J.L. Prather, Atomic Energy Levels in Crystals, U.S. Government Printing Office, Washington, 1961. [290] O. Atakol, R. Boča, I. Ercan, F. Ercan, H. Fuess, W. Haase, R. Herchel, Chem. Phys. Letter 423 (2006) 192. [291] J. Titiš, R. Boča, L‟. Dlháń, T. Ďurčeková, H. Fuess, R. Ivaniková, V. Mrázová, B. Papánková, I. Svoboda, Polyhedron 26 (2007) 1523. [292] M. Karbowiak C. Rudowicz, T. Ishida, Inorg. Chem. 52 ( 2013) 13199.

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*Highlights (for review)

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Intricacies at interface: crystal field Hamiltonians  spin Hamiltonians elucidated Crucial notions systematically defined & their logical interrelationships illustrated Nature, origin, and usage of physical, effective and fictitious Hamiltonians outlined Basic theory for interpretation of optical spectroscopy, EMR & magnetic data provided Focus on transition ions in coordination compounds and single ion/molecule magnets

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