Quantum phase transitions probed by EPR spectra in dimeric spin arrays with supramolecular couplings

Review Quantum phase transitions probed by EPR spectra in dimeric spin arrays with supramolecular couplings Rafael Calvo a,b, ⇑ , Rosana P. Sartoris b , Otaciro R. Nascimento c , Matúš Šedivy ´ d,e , Antonin Sojka d , Petr Neugebauer d , Vinicius T. Santana d, ⇑ a Instituto de Física del Litoral, Consejo Nacional de Investigaciones Científicas y Técnicas, Güemes 3450, 3000 Santa Fe, Argentina b Departamento de Física, Facultad de Bioquímica y Ciencias Biológicas, Universidad Nacional del Litoral, Ciudad Universitaria, 3000 Santa Fe, Argentina c Departamento de Física e Ciencias Interdisciplinares, Instituto de Física de São Carlos, Universidade de São Paulo, CP 369, 13560-970 São Carlos, SP, Brazil d CEITEC – Central European Institute of Technology, Brno University of Technology, Purkyn ˇova 123, 61200 Brno, Czech Republic e Department of Microelectronics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 10, 61600 Brno, Czech Republic article info Article history: Received 11 October 2022 Accepted 29 December 2022 Available online 23 January 2023 Keywords: Dimeric units Interdimeric exchange Phase transitions Triplet excitations EPR abstract Dimeric compounds with nearly isolated molecular units (d-units) having pairs of spins s 1 and s 2 coupled by antiferromagnetic (AFM) exchange H 0 ¼J 0 s 1 s 2 are non-trivial quantum spin systems having primary roles in magnetism. Weakly coupled infinite arrays of AFM d-units in crystal structures have an appealing spin dynamic arising from their energy-gapped level structure and display magnetic properties with important roles in materials science. They received great additional attention when it was discovered that the spin entanglement introduced by interdimeric couplings with magnitude J 1 (with J 1 jjjJ 0 j) gives rise to bosonic systems with novel properties and quantum phase transitions at high temperature (T). In this work, we collect recent advances in the interpretation of EPR spectral changes in terms of quantum phase transitions of arrays of d-units in the presence of weak interdimeric couplings. We review previous investigations of the problem and focused new experiments on the paradigmatic compound copper acetate monohydrate (CAH), collecting a detailed set of spectra in single-crystal and powder samples. The spectral features arising from this coupling are merging and narrowing of the peaks of the spectra of single crystals for specific magnetic field (B 0 ) orientations, and an extraordinary ‘‘Upeak” in the powder samples associated with the quantum phase transition of the dimeric spin array. Our historical overview collects studies of similar compounds with the same dimeric feature in which the U-peaks were generally misinterpreted as a double-quantum transition, or ignored. We describe procedures to identify and quantify the U-peak and the merging and narrowing phenomena, with a protocol to extract the interdimeric coupling magnitude. As a novel contribution, we explain the experimental results by proposing a spin model with a microscopic flip-flop mechanism involving the absorption and emission of two simultaneous spin-one excitations having energy jJ 0 j, connecting singlet and triplet levels of neighbor d-units and giving rise to a quantum phase displaying spin entanglement. This phase is tuned with the orientation of B 0 applied along directions within ‘‘magic rings”, the positions where the EPR peaks of the dimeric units intersect, that we propose as a phase diagram. Our model considers explicitly the role of energy conservation in the process and allows analyzing and simulating the features of the EPR spectra arising from the couplings, describing their roles in the spectral behavior and the magnetic phases. In conclusion, we review the history of dimeric compounds and the possibility of detecting interdimeric couplings in the EPR spectra experimentally, and we introduce a novel spin model for the dimeric array appropriate to analyze EPR data which allows us to understand the spin dynamics and the phase transitions arising from these couplings. Ó2023 The Authors. Published by Elsevier B.V. ThisisanopenaccessarticleundertheCC BYlicense(http:// creativecommons.org/licenses/by/4.0/). https://doi.org/10.1016/j.ccr.2022.215007 0010-8545/Ó2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). ⇑ Corresponding authors at: Instituto de Física del Litoral, Consejo Nacional de Investigaciones Científicas y Técnicas, Güemes 3450, 3000 Santa Fe, Argentina (R. Calvo). E-mail addresses: [email protected] (R. Calvo), [email protected] (V.T. Santana). Coordination Chemistry Reviews 480 (2023) 215007 Contents lists available at ScienceDirect Coordination Chemistry Reviews journal homepage: www.elsevier.com/locate/ccr Contents 1. Introduction . . . ........................................................................................................ 2 1.1. Magnetic properties and EPR spectra of dimeric compounds. . . . . . . ........................................................ 2 1.2. Changes in the EPR spectra arising from the interdimeric couplings ........................................................ 3 2. 3D-Structure of CAH: Interdimeric exchange paths. . . . . . . . . . .................................................................. 5 2.1. 3D structure of CAH . . . . ........................................................................................... 5 2.2. Interdimeric exchange paths . . . . . . . . . . . . . ........................................................................... 5 3. Theory: Coupled arrays of dimeric spin units . . . . . . . . . . . . . . .................................................................. 6 3.1. Intradimeric couplings and EPR spectra of the d-Units. . . . . . . . . . . . ........................................................ 6 3.2. Interdimeric couplings in an infinite 1D array of d-units. . . . . . . . . . ........................................................ 7 3.3. Spin algebra for single dimeric (d) and Double-Dimeric (dd) units . . ........................................................ 8 3.4. Perturbative calculation of transition probabilities. . . . . . . . . . . . . . . ....................................................... 10 3.5. Merging and narrowing of the EPR peaks and entangled phase at the magic rings . . . . . . . . .................................... 11 4. Experimental procedures. . . . . . . . . . . . .................................................................................... 12 4.1. Samples preparation . . . . .......................................................................................... 12 4.2. EPR measurements . . . . . .......................................................................................... 12 4.3. Spectral manipulation and computational methods . . . . . . . . . . . . . . ....................................................... 12 5. EPR results and discussion. . . . . . . . . . . .................................................................................... 13 5.1. Spectral features of AFM d-units arising from interdimeric couplings . . . . . . . . . . . . . . . . . . .................................... 13 5.2. Design of EPR measurements to characterize interdimeric couplings ....................................................... 13 5.3. Single-Crystal spectra at X, Qand W-bands. Data and analysis . . . . ....................................................... 14 5.4. Powder spectra and the U-Peak: Experimental results at low frequencies . . . . . . . . . . . . . . . .................................... 17 5.5. Powder spectra and the U-Peak at high frequencies. . . . . . . . . . . . . . ....................................................... 18 6. Conclusions. . . . ....................................................................................................... 19 Declaration of Competing Interest . . . . .................................................................................... 20 Acknowledgments . . . . . . . . . . . . . . . . . .................................................................................... 20 Appendix. . . . . . ....................................................................................................... 20 References . . . . ....................................................................................................... 22 1. Introduction 1.1. Magnetic properties and EPR spectra of dimeric compounds The magnetic properties of dimeric materials having nearly isolated molecular units with pairs of spins coupled by exchange have been studied from different points of view and using many experimental techniques. Copper acetate hydrate, here called CAH, is known for more than a century [1–6], and its magnetic properties appeared in the scene when Mookherjee (1945) [7] and Guha (1951) [8] reported paramagnetic anisotropy and magnetic susceptibility measurements. Their results disagreed with the predictions of the early crystal field theories of Bethe [9], Van Vleck [10], and Schlapp and Penney [11], a result that interested many contemporary researchers.Lancaster and Gordy [12] and Kumagai et al.[13] reported anomalous EPR spectra of powder and single-crystal samples of CAH when compared with expectations for Cu II ions in typical molecular coordinations and mentioned the possibility of unusual couplings within dimeric molecular units (d-units, also called dinuclear units). These early magnetic and EPR measurements [7,8,12,13] were neither precise nor reproducible enough, and the molecular structure of CAH was not known at the time to allow a detailed explanation. In a big step forward, Bleaney and Bowers [14,15] (B&B) reported in 1952 EPR spectra of singlecrystal samples and demonstrated that the spectra and the susceptibility known at the time correspond to d-units containing pairs of Cu II ions with unpaired spins ½, s 1 and s 2 , coupled by isotropic AFM exchange [16,17], H ex ¼J 0 s 1 s 2 ð1Þ with J 0 =257 cm 1 according to B&B [14], and by much smaller and approximate axially symmetric anisotropic spin–spin interactions (jDj0.3 cm 1 ). Equation (1) describes dimeric molecules having an energy-gapped structure with a ground singlet state r , not contributing to the EPR spectra, and a triplet state s that for some purposes may be described as a spin-one state with activation energy jJ 0 jabove the singlet. In general in the literature, the EPR signals of these dimeric systems are considered as arising from S= 1 systems because the triplet is the only set of states with an EPR signal and describes the anisotropy of such systems (gand D tensors). However this differs from spin-one materials because the activation energy in the dimeric case changes the EPR response with Tand also allows singlet–triplet magnetic transitions to play relevant roles in the dynamical behavior. The ground singlet state (S= 0) has an effect only on the signal intensity that depends on the temperature according to the B&B model allowing to obtain the value of the coupling constant jJ 0 j. Meanwhile in a spin-one system, there is not and energy gap and the temperature behaviour follows straitghforward the Boltzmann statistics. Soon after B&B’s work, Van Niekerk and Schoening [18,19] reported the X-ray structure and found that CAH is made of nearly isolated d-units having pairs of copper ions at a distance of 2.64 Å in ‘‘paddlewheel” units (see Fig. 1). The studies of CAH reported by Bleaney and Bowers [14] and Van Niekerk and Schoening [19] were followed by publications dealing with magnetic, EPR [13,20–26], crystallographic [19,27,28], the nuclear magnetic resonance of protons [29,30] and 63 Cu [31], neutron inelastic scattering measurements [32] and more [33]. Examples of further experiments and theoretical ideas in CAH that contributed to significant progress are the studies by Valentine et al.[34] on the T-dependence of the EPR line broadening of chains of d-units, those of Rigamonti et al.[35], about the replacement of the apical water molecule of copper acetate by other molecules in the bridging of chains of tetracarboxylate dimeric molecules, the determination of the sign of the axial spin– spin D-coupling term by Ozarowski [36] using high-frequency EPR (the magnitude of Dwas originally reported by B&B [14,15]), its evaluation using a wave-function-based method by Maurice et al. [37], experimental and theoretical studies of charge density [28]. Many compounds with similar dimeric structures, particularly those with copper bridged by various geometries of carboxylate R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 2 ligands [38–44], were reported in the following years. The approximate axial molecular symmetry of CAH was absent in some of them. Besides coordination compounds, magnitudes of jJ 0 jsmaller than 0.1 cm 1 were observed in dimeric biradicals using EPR [45,46]. The studies of dimeric materials introduced a big impulse to the essence of magnetism and in the understanding of the superexchange interactions [16,17,47], giving rise to significant research in physics [17,48], chemistry [28,34,36,37,49,50], and biophysics [45,46,51,52]. They introduced the concept of ‘‘magnetic molecules” (added to ‘‘magnetic materials”), which contributed with ground bricks to the emerging field of Molecular Magnetism [16,17,47,53]. EPR has been a very appropriate and often selected technique that contributed to significant advances in the investigations of the magnetic properties of dimeric materials [54]. 1.2. Changes in the EPR spectra arising from the interdimeric couplings Magnetic studies of materials having weakly interacting arrays of AFM dimeric molecules received renewed great attention in the last 20 years since Bose-Einstein condensation (BEC) and other quantum phase transitions, plus outstanding contributions about magnetic excitations, were reported at high Tarising from weak interactions between neighboring d-units [55–64]. These phase transitions occur when an applied magnetic field B 0 (¼ l 0 H, where l 0 is the vacuum permeability) [65,66] crosses one energy level of the excited triplet state with the ground singlet in the presence of interdimeric couplings. According to the value J 0 =292 cm 1 obtained from magnetic susceptibility data [33] these energy crossings would require magnetic fields of 300 T, and are not expected in CAH as in many dimeric materials. However, investigating different but related characteristics of the problem, we show here that weak interdimeric couplings within dimeric arrays give rise to relevant changes in the EPR spectra of single-crystal and powder samples in much less extreme experimental conditions, with phase transitions consequence of the bosonic character of the dimeric structure, and driven by the orientation of B 0 . Interdimeric couplings have been reported to contribute to the EPR spectra of dimeric materials. However, these investigations were not yet conclusive, and it is desirable to review the existing EPR results and proceed with new experimental and theoretical steps forward, collecting new data and developing a consistent theoretical frame to analyze the available experimental information. We summarize here previous works on the subject, listing studies of compounds whose X-ray structure is known, and show spectra reflecting interdimeric couplings. In this direction, we found that reviewing EPR data related to the interdimeric coupling in dimeric arrays is a strenuous task, and there might be some missing contributions. Still, the collected information is extensive enough to allow describing the history, the situation, and the achievements in the area. We enrich that situation with a full set of EPR experiments for the paradigmatic dimeric CAH and use these and other results to introduce a spin model explaining the effects of interdimeric couplings in the EPR spectra. Quoting previous results published in the Russian language [67,68], Yablokov et al.[69,70] reported a peak at the center of the EPR spectra of powder samples of dimeric copper carboxylates which cannot be explained with the spin-Hamiltonian used for isolated dimeric Cu II molecules. This peak has a strong Tvariation, disappearing as Tdecreases, as occurs with the magnetization and EPR signal intensity in antiferromagnetic dimeric materials. They attributed their findings to interdimeric couplings without introducing specific microscopic mechanisms. In the following years, several other papers reported unexpected peaks in the spectra of other dimeric compounds, mostly with structures similar to copper acetate hydrate [70–97]. More than one source was proposed for these peaks. Using classical coupled Bloch equations [98] Galeev related the unexpected EPR peak observed in powder samples to spin–lattice relaxation [99] and to exchange narrowing phenomena [100]. The results in compounds with reported dimeric X-ray structure and EPR spectra displaying effects of interdimeric couplings are listed in Table 1, and they are discussed in this paper. EPR measurements in single crystals of dimeric materials reported by us since 2008 [78,79,83–85,90] provided new clues about the microscopic mechanisms of the interdimeric couplings responsible for the spectral changes, which are described in this work with a detailed spin model. Here we report EPR spectra d v 00 =dB 0 vs B 0 obtained in powder and single crystals samples of CAH at microwave frequencies between 9 and 375 GHz, in a wide range of T, and as a function of microwave power. We update the values of the spin Hamiltonian parameters for uncoupled d-units in CAH and go forward in the study of the spin dynamics arising from interdimeric couplings, extracting information from the merging and narrowing of the fine structure peaks in single crystals and their dependence on the orientation of B 0 , and from an additional U-peak observed in powder samples and its variation with Tand microwave power. We analyze the experimental information with a theoretical model of the weakly coupled dimeric array proposing a new mechanism for the magnetic excitations predicting spin dynamics thermally activated with the singlet–triplet exchange energy jJ 0 j, responsible for the strong changes of the spectra of single crystals observed for magnetic field orientations around the magic angles [101], where the anisotropic spin–spin interactions cancel out. We prove that the collapse and narrowing of the EPR peaks in single crystals of the dimeric array are produced by a flip-flop mechanism involving two triplet excitations (triplons, with spin one and energy jJ 0 j) involving simultaneous flipping of states r ? s and s ? r of neighbor d-units. This reminds the mechanism proposed by Anderson [102,103], in their pioneering work of exchange narrowing for energy-gapless monomeric materials, however replacing the Zeeman energies involved in the simultaneous flipping of spins ½ in monomeric compounds with the singlet–triplet flipping of the much higher energy jJ 0 jin the dimeric compound. The U-peak, a feature previously observed by other authors and by us (see Table 1), but unreported previously for CAH, is also explained here Fig. 1. Structure of the dimeric molecule of CAH made of two asymmetric unit cells with one of them emphasized with black bonds, plotted using the structural results of Bertolotti et al.[28]. R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 3 Table 1 Reported dimeric compounds with solved crystal structure, in which the EPR spectra displayed an ‘‘unexpected” U-peak. Microwave frequencies are indicated as X (9.4 GHz) or Q(35 GHz); EPR features related to interdimeric coupling observed in powder and/or single-crystal samples are indicated as pow and sc, and monomeric Cu II impurities observed in some cases are not included here. In cases where more than one compound is reported in the same paper, an id # is given. Compound intradimeric bridge data author’s suggestion note reference [Cu 2 (propionato) 4 ]∙2(p-Cl-aniline)∙H 2 O Tetrawheel X, pow, sc, Uinterdimeric exchange (Gavrilov et al., 1971) [70] Cu( l -adenine) 2 nH 2 O Tetrawheel X, pow, U not assigned (Goodgame et al., 1968) [71] [Cu(adipato) 4 H 2 O] 2 Tetrawheel X, pow, Udqt unclear spectrum (Sharrock et al.,1978) [82] [Cu(acetato) 2 (nicotinamide)∙H 2 O] 2 Tetrawheel Q, pow, Unot identified (Valko et al.,1993) [91] (Williams et al., 2003) [92] Cu 2 (formato) 4 (dimethylsulfoxide) 2t Tetrawheel X + Q, pow, Udqt (Sapiña et al., 1994) [93] [Cu(p-aminobenzoato)(4,7-1,10-phenanthroline)(H 2 O)] 2 (NO 3 ) 2 di-carboxylate X, pow, UCu(II) impurity (Zoroddu et al., 1996) [94] [Cu(suprofen) 2 H 2 O] 2 Tetrawheel Q, pow, Udqt (Kögerler et al., 1998) [95] Cu 2 (heptanoato) 4 Tetrawheel X, pow, Uinterdimeric exchange (Valko, et al., 2000) [96] Cu 2 (acetato) 4 (2-anilinopyridine) 2 Tetrawheel Q, pow, Udqt (Seco et al., 2002) [97] Cu 2 (acetato) 4 (2-methylaminopyridine) 2 Tetrawheel Q, pow, Udqt #1 (Barquín et al., 2004) [72] [Cu 2 (hexanoate) 4 ] n Tetrawheel X, pow, UInterdimeric exchange #1 Kozlevc ˇar et al. 2004) [73] [{Cu 2 (hexanoate) 4 (urea)} 2 ] Tetrawheel X, pow, UInterdimeric exchange #2 [Cu 2 (formato) 4 (pyrazine)] n Tetrawheel Q, pow, Udqt #2, EPR not shown, (Barquín et al., 2005) [74] Cu 2 (formato) 4 (2,6-dimethylpyrazine) 2 Tetrawheel Q, pow, Udqt #5, EPR not shown Cu 2 (formato) 4 (2-(phenylamino)pyridine) 2 Tetrawheel Q, pow, Udqt #1, EPR not shown (Barquín et al., 2006) [75] Cu 2 (formato) 4 (2-(methylamino)pyridine) 2 Tetrawheel Q, pow, Udqt #2 [Cu( l -adenine)(toluenesulfonate)-(phenantroline)] 2 2H 2 O Two N bonds of adenine X, Q, pow, UCu(II) impurity (García-Giménez et al.2007) [76] Cu 2 ( l -guanidinoacetato) 4 (NO 3 ) 2 Tetrawheel X, pow, Uinterdimeric exchange (Miranda et al., 2008) [77] Cu 2 (N-thiazol-2-yl-toluenesulfonamidato) 4 N bonds X + Q, pow, sc, U dqt, interdimeric exchange (Napolitano et al., 2008) [78] (Calvo et al., 2017) [79] {[Cu 2 ( l -acetato) 4 ] (methylpyrazine)} n Tetrawheel Q, pow, Udqt #3, EPR not shown (Barquín et al., 2010) [80] [Cu 2 ( l -acetato) 4 (2,3-dimethylpyrazine) 2 ] Tetrawheel Q, pow, Udqt #5 [Cu 2 ( l -acetato) 4 (2,6-dimethylpyrazine) 2 ] Tetrawheel Q, pow, Udqt #6, EPR not shown {[Cu 2 ( l -acetato) 4 ]( l -2,5dimethylpyrazine)} n Tetrawheel Q, pow, Udqt #7, EPR not shown [Cu 2 ( l -acetato) 4 (pyridazine) 2 ] Tetrawheel Q, pow, Udqt #1 (Barquín et al., 2010) [81]{[Cu 2 ( l -acetato) 4 ]( l -pyrimidine)} n Tetrawheel Q, pow, Udqt #3, EPR not shown [Cu 2 (trans-2-butenoico) 4 ] n tetrawheel X + Q, pow, Uinterdimeric exchange (Perec et al., 2010) [83] Cu 2 (flufenamato) 4 (dimethylformamide) 2 tetrawheel X + Q, sc, Udqt (Nascimento et al., 2011) [84] [Cu(acetato)(phenantroline)(H 2 O)] 2 (NO 3 ) 2 ∙4H 2 O di-carboxylate X + Q, pow, sc, U interdimeric exchange (Calvo et al., 2011) [85] {Cu 2 ( l -acetato) 4 }(acetamide) 2 tetrawheel X, Q, pow, sc, U interdimeric exchange (Paredes-García et al., 2013) [86] 3 1 [Cu I 2 Cu II 2 {H 2 O} 2 {3,30-(5,50-(thiophene-2,5-diyl)bis(3-methyl-4H1,2,4-triazole-5,4-diyl))dibenzoate} 2 ]Cl 2 tetrawheel X + Q, pow, Uinterdimeric exchange (Šiménas et al., 2015) [87] [Cu 2 (isophthalato) 4 (H 2 O)] n tetrawheel X + Q, pow, Uinterdimeric exchange (Šiménas et al., 2016) [88] Cu 2 ( l -acetato) 4 2H 2 O tetrawheel X, Q pow, UCu(II) impurity (Alter et al. 2019) [89] [{Cu(p-aminobenzoato)(1,10-phenanthroline)H 2 O} 2 NO3paminobenzoato2H 2 O] di-carboxylate X + Q, pow, sc no interdimeric exchange #1 (Sartoris et al., 2020) [90] [Cu 2 (p-aminobenzoato) 2 (1,10-phenanthroline) 2 pyrazine] n di-carboxylate X + Q, pow, sc, U interdimeric exchange #2 Cu 2 ( l -acetato) 4 2H 2 O tetrawheel X + Q + W, pow, sc, U interdimeric exchange This paper R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 4 using this model and developing simulation methods and graphical procedures that clarify the physical processes. The novel mechanism for the interdimeric coupling described above explains the observed changes in the EPR spectra in the presence of an anisotropic spin–spin coupling and of an energy gap arising from the intradimeric AFM exchange coupling J 0 , introducing a characteristic quantum phase transition in the dimeric array where the driving parameter is the orientation of the magnetic field. Thus, considering the results of our EPR measurements in single crystal and powder samples of CAH and other dimeric compounds, this behavior allows us to introduce here the ‘‘magic rings”, geometric place of the so-called magic angles for the magnetic field orientations as a phase diagram for the entangled quantum phase of the spins. Along this work, we prove that axial symmetry in the anisotropic intradimeric spin–spin and Zeeman couplings are important issues required to observe the contributions to the EPR spectra arising from interdimeric couplings. We analyze this condition theoretically, and since the relative weight of these non-axial contributions depends on the applied magnetic field, we collected data for CAH in a wide range of microwave frequencies. This work includes: i. A revision of previous experimental and theoretical results in compounds with known dimeric structures, which display EPR spectra with features we attribute to interdimeric couplings. ii. Introduces a theory for the behavior and EPR spectra of a coupled array of dimeric arrays with interdimeric interactions [78,79,83–85,90], highlighting the differences between these cases and the classical problem of exchange narrowing in coupled monomeric arrays of spin ½ [102,103]. iii. Explain why the magic angles play a relevant role in describing the phase transitions where entanglement occurs, introducing the magic rings, as a phase diagram where the orientation of the applied field B 0 is the driving parameter. The new EPR data allows for discussing the existence of double quantum spectral transitions suggested in previous works on the subject, and we discard this contribution in all cases where we have experimental results. Section 1 of the paper introduces the purpose and relevance of this and previous works on the subject. Section 2 discusses the dimeric structure and the main paths for interdimeric coupling in the supramolecular array of CAH. Section 3 describes the spin model for the interdimeric couplings and proposes a flip-flop mechanism that simultaneously creates and destroys triplet excitations with energy jJ 0 jand spins 1 connecting singlet and triplet levels of neighbor d-units in the dimeric array, which satisfactorily explains the data. The experimental procedures and the results for CAH are described in Sections 4 and 5, respectively, and we summarize the achievements of this investigation in Section 6. 2. 3D-Structure of CAH: Interdimeric exchange paths 2.1. 3D structure of CAH The earliest crystallographic study of CAH seems to be the optical measurements reported by Brooke in 1823 [2]. Hull [6] reported in 1938 the unit cell parameters obtained from an X-ray study. The dimeric molecular structure was first reported by Van Niekerk and Schoening [18,19] in 1953, with the limitations of the state of the art of X-ray crystallography at that time (R= 21 %). Musumeci and Frost reported the spectroscopic and thermoanalytical properties of CAH in its natural form (baptized hoganite) [104]. Many updated structural reports of CAH with a greater resolution are found nowadays in the Cambridge Structural Database [105], and here we use the results of Bertolotti et al.[28], having an R-factor of 1.5 %. CAH crystallizes in the monoclinic C2/c group with a= 13.0834(1) Å, b= 8.5028(1) Å, c= 13.7315(1) Å and b= 116.865(1)°. The structural figures that follow are constructed using these results. The eight symmetry-related copper ions in the unit cell conform to four dimeric molecules [Cu(CH 3 COO) 2 H 2 O] 2 having a centrosymmetric paddlewheel configuration, in which four acetate bridges connect pairs of coppers at d 0 = 2.611 Å with approximate C 4v symmetry, and water molecules at the apical positions (Fig. 1). The four d-units in the crystal cell are symmetry-related, with three of them obtained from the first by aC 2 rotation around the b-axis (C 2b ), an inversion I,oraC 2b rotation followed by an inversion. Since d-units that differ by an inversion are magnetically equal, we call Aand Bthe magnetically different pairs of sites related by a rotation C 2b . Thus, the variation with the orientation of B 0 of the EPR spectra of d-units in sites Aand Bdiffer by the same operation. The angle between the normal to the squares of ligands to coppers in d-units of types Aand Bis 63.5°, as previously obtained by B&B [14] from the EPR spectra. 2.2. Interdimeric exchange paths Ad-unit in the lattice of CAH has six nearest neighbor d-units. Two of the same type (AA or BB pairs) are at distances d 0 AA = d 0 BB = 5.558 Å between their centers of symmetry, along n//(a+b) for the AA coupling, or along g //(a–b) for the BB coupling. The AA and BB paths (see Fig. 2a) have a total length d 1 AA =d 1 BB = 7.027 Å and involve two H-bonds related by a C 2b rotation. This unit is also connected to two neighboring d-units of the different types (AB path, see Fig. 2b) on each side along the fc-axis at a distance d 0 AB = 5.173 Å between their centers of symmetry, through two inversion-related H-bonds Cu-O ap -H...O eq -Cu paths, similar to the AA paths, have three non-magnetic atoms and a total Cu-Cu path length d 1 AB = 6.867 Å. The magnitude of the interdimeric exchange between pairs type AA (or BB)ofd-unit neighbors along the b-direction with longer distances d 0 b = 8.508 Å between centers of symmetry, and with a total length d 1 b = 10.203 Å, through a more complex chemical path containing five atoms provides weaker couplings and is discarded in our analysis. Fig. 2 may be compared with Scheme 1 of the paper by Rigamonti et al.[35]. The interconnected 3D array of d-units shown in Figs. 3 and 4 displays the Fig. 2. Chemical paths connecting copper ions in neighboring d-units along chains in CAH. (a) Transversal coupling between neighboring A-units (or B-units) in chains along n//(a+b) (or g //(a–b)). (b) Longitudinal coupling between Aand Bneighboring units along f//c. In both cases the paths involve two symmetry-related H-bonds; O5D—H7 ... O1 in (a) and O5—H8 ... O2D in (b), where O5 ... H belongs to an apical water molecule of a d-unit, and O1 or O2D are equatorial carboxylate oxygens ligands of coppers in neighbor d-units in the chain. R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 5 chains of d-units coupled by the transversal (T) and longitudinal (L) paths for interdimeric couplings described in Fig. 2. Doubledimeric (dd) units containing pairs AA (red), BB (blue), and AB (red and blue) of neighboring d-units and four copper spins (½s A;1; c ;k ;s A;2; c ;k ;s B;1; c ;k ;s B;2; c ;k for AB pairs) are emphasized by ellipses in Figs. 3 and 4a,b. The d-units are identified in the 3D structure by their longitudinal and transversal coordinates ( c ,k), where c is measured along the z-direction of the chains and the vector kgives the position in the transversal plane. Unit vectors (n, g ,f) along (a+b), (a-b), and cdirections, respectively, define a nonorthogonal coordinate system appropriate to describe the positions of the d-units in the dimeric array. Thus, according to Figs. 3 and 4 the 3D molecular array in CAH is interconnected by ‘‘transversal couplings” (T, in planes perpendicular to z) between magnetically equal molecules AA or BB, and by ‘‘longitudinal couplings” (L, along z) between rotated neighbor d-units Aand B, defining ‘‘spincommunication chains”, and giving rise to different physical processes and behavior. 3. Theory: Coupled arrays of dimeric spin units In this section, we model the behavior of the spins in an array of d-units that resemble the structure of CAH (Fig. 3). Minor changes would allow to apply similar ideas to other dimeric arrays. Subsection 3.1 reviews the spin Hamiltonian H 0 containing the intradimeric contributions of single d-units that allow simulating the EPR spectra in the absence of interdimeric coupling [14]. The differences between the experimental results and these spectral simulations which are outside of the experimental uncertainty are attributed to the interdimeric couplings. Subsection 3.2 describes formally the interdimeric couplings within the infinite dimeric chains described in Section 2 for CAH, which also occur in other dimeric copper tetracarboxylates. This treatment is simplified in 3.3 to finite dd-units made of pairs of neighbor units d 1 and d 2 , the smallest spin array that allows considering interdimeric couplings. The spin algebra used in this work has common issues with the exact calculations reported by Grimaudo et al. [106,107] of the dynamics of interacting two-level systems. In 3.4 we use the Fermi golden rule [108] to treat perturbatively the transition probabilities or flipping frequencies between states of the dd-units arising from the interdimeric matrix elements, and in 3.5 we describe the resulting dynamics of the spin system and the related modifications of the EPR spectra, that are used later to analyze the experimental results. Part of this theory strictly applies to axially symmetric d-units, a condition approximately valid for CAH that simplifies the treatment, without changing physical concepts. 3.1. Intradimeric couplings and EPR spectra of the d-Units In the spin s= ½ representation, the spin Hamiltonian H 0 describing the EPR spectra of CAH of a magnetic unit cell contains ad-unit type AA with identical spins s A;1 and s A;2 , and one type BB with rotated spins s B;1 , and s B;2 . The AFM isotropic exchange H ex (eq (1)) with J 0 =292.2 cm –1 [33] and the anisotropic spin–spin coupling H D (including dipole–dipole and anisotropic exchange couplings) occur between spins of the same type (AA or BB dunits), and the Zeeman couplings H Z occur between each spin and the field B 0 [101]: H 1=2ðÞ 0 ¼ðH ex þH Z þH D Þ A þðH ex þH Z þH D Þ B ¼J 0 s A;1 s A;2 þs B;1 s B;2 ½ þ l B B 0 g A s A;1 þs A;2 ðÞþg B s B;1 þs B;2 ðÞ½ þs A;1 2D A ðÞs A;2 þs B;1 2D B ðÞs B;2 ð2Þ H ex splits the four states of each d-unit in a ground singlet r and an excited triplet s [ s 1 , s 0 , s -1 ] with energy jJ 0 jabove r . The components of the triplet s are further split by the much smaller contributions of H D and H Z . The Zeeman and the anisotropic spin–spin terms H Z and H D in eq (2) are much smaller than the intradimeric exchange H ex , and the inversion symmetry of the dunits prevents contributions of antisymmetric Dzyaloshinskii– Moriya exchange couplings [101] within the d-units. Contributions arising from hyperfine couplings with the nuclear spins of the copFig. 3. Three-dimensional magnetic array of d-units types Aand Bin CAH, displayed as red and blue arrow pairs, respectively. Magnetic dd-units AA (or BB), and AB are emphasized with red and green elliptic shapes, respectively. The couplings connecting spins in neighbor d-units are described as double-arrows, orange for the transversal couplings (J T )AA along the unit vectors n//(a+b), or BB along g //(a– b), described in Fig. 2a, or light green for the longitudinal coupling (J L )AB along the direction f//cdisplayed, described in Fig. 2b. These couplings give rise to uniform dimeric chains AA and BB, or alternate dimeric chains AB. Fig. 4. Cartoon displaying magnetic chains supporting interdimeric coupling of dunits in CAH. (a) Transversal (T) chains connecting AA (or BB) neighboring d-units at 5.558 Å. (b) Longitudinal (L) chains, with AB neighboring d-units at 5.173 Å. The ovals indicate double-dimeric ‘‘dd-units” in AA (a) and AB (b) chains, the smallest spin units where interdimeric couplings occur. R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 6 per ions, observed in CAH and many dimeric systems when the peaks narrow at low T, are not included in H 0 of eq (2). In the original work of B&B they showed that the hyperfine structure of the EPR spectra provides a detailed fingerprint of dimeric behavior [14,101,109]. Interdimeric couplings change these hyperfine interactions by merging and broadening this structure, as was reported by Farach et al.[110] using a formalism introduced by Sack [111]. However, detailed experiments dealing with this problem are complex to deal with interdimeric couplings because relaxation and other broadening and narrowing sources contribute, and Tplays a primary role. Neuman et al.[112] studied the hyperfine coupling in a sample where dimeric copper molecules are diluted in an isomorphous Zn lattice. In that case the magnitude of the interdimeric coupling is a stochastic distribution depending on the Cu/Zn ratio in the lattice. We did not find detailed sets of hyperfine data in a pure dimeric compound where the effect of interdimeric coupling is described and preferred to leave this treatment for another work. In particular CAH, our target compound would allow such a study only at low T<J 0 jj =k B , for field orientation ranges where the hyperfine coupling is largest, but the EPR signals are very weak. The sums S A and S B of the spins in d-units Aand B, and the differences between them, u A and u B , are defined as: S A ¼s A;1 þs A;2 ;S B ¼s B;1 þs B;2 ;u A ¼s A;1 s A;2 ;u B ¼s B;1 s B;2 ð3Þ Using eqs (3),H 0 in the spin 1 representation is: H ð1Þ 0 ¼1 2J 0 S A S A þ1ðÞ 1 2J 0 S B S B þ1ðÞ þ l B B 0 g A S A þg B S B ½þS A D A S A þS B D B S B ð4Þ independent of u A and u B .S A and S B in eq (4) may have values 0 and 1 and only the components of the excited triplet states with S= 1 and energy jJ 0 jabove the singlet state contribute to the EPR transitions of sites Aand B. Equations (2) and (4) predict spectra that sum those for sites Aand B, each having two allowed EPR peaks, s 1 M s 0 and s 0 M s -1 , and a weaker forbidden peak s 1 M s -1 at half magnetic field [14,101,109]. Besides these allowed and forbidden transitions, it has been shown with a second-order perturbative calculation that transitions s 1 M s -1 occur in an S= 1 system with the simultaneous absorption or emission of two photons [101,113–115]. This second-order contribution to the spectra called ‘‘double quantum transitions” (dqt), may be identified experimentally because its peak intensity increases quadratically with microwave power. It is neither related to the interdimeric couplings, nor perturbed by them, but, as discussed later, it may be wrongly attributed to these couplings due to common features of their contributions. The crystal symmetry of CAH relates the principal axes of the gmatrices g A and g B , and the anisotropic spin–spin coupling matrices D A and D B of eqs (2) and (4) by operation C 2b . Thus, the spectra of sites Aand Bare coincident for B 0 in the ca* plane. Also, since H D cancels out and the energy distance between the states s 1 and s 0 , and s 0 and s -1 are equal, the two allowed peaks of each d-unit intersect at the ‘‘magic angle” orientation a = 54.74°given by the condition ð3cos 2 a 1Þ¼0, where a is the angle between B 0 and the axis of symmetry of the sites [101]. The parameters of H 0 may be calculated by fitting eqs (2) or (4) to the data on the position of the spectral peaks of Aand Bsites as a function of the orientation of B 0 . They are calculated later for uncoupled d-units in CAH from our experimental results as described in the experimental section and are given in Table 2. These values may be considered as an update of many results reported since B&B [14,15], which are needed in this work to distinguish the features of the spectra arising from the interdimeric couplings. The differences between observed spectra and simulations obtained with these parameters allow the detection of the spectral features arising from the interdimeric interactions. The contributions to the EPR data of the intradimeric spin–spin (D-terms) in eqs (2) or (4) do not change with the magnetic field B 0 (or microwave frequency m ), but the Zeeman coupling changes with B 0 and m . Thus, to study the role of non-axial contributions of these terms in the possibility of detecting effects of the interdimeric couplings in the EPR spectra our experiments in CAH were performed in the widest relevant range of frequencies (9–375 GHz), and the conclusions are discussed later in the paper. 3.2. Interdimeric couplings in an infinite 1D array of d-units Neglecting interdimeric coupling, the infinite 1D array of dunits in CAH shown in Fig. 3 is described in the spin ½ representation by the Hamiltonian: H ð1=2Þ 0 ¼ R c ;k l B B 0 g A s A;1; c ;k þs A;2; c ;k  þg B s B;1; c ;k þs B;2; c ;k  J 0 s A;1; c ;k s A;2; c ;k þs B;1; c ;k s B;2; c ;k  þs A;1; c ;k ð2D A Þs A;2; c ;k þs B;1; c ;k ð2D B Þs B;2; c ;k ð5Þ where ( c ,k) are the coordinates of the dimeric units along c( c ), and in the perpendicular plane (kis a 2-dimensional vector). As described in 2.2 and displayed by Figs. 3 and 4a,b, there are interdimeric couplings J T between neighbor pairs AA or BB of d-units along uniform chains in transversal planes, and J L between pairs AB along the alternate chains in the longitudinal z-direction. These couplings are described by: H ð1=2Þ T ¼ R c ;k J T s A;2; c ;k s A;1; c ;kþn þs B;2; c ;k s B;1; c ;kþ g  ð6Þ H ð1=2Þ L ¼ R c ;k J L s A;2; c ;k s B;1; c ;k þs B;2; c ;k s A;1; c þf;k  ð7Þ Equations (6) and (7) have matrix elements between triplet states and between singlet and triplet states of neighbor d-units. For a generic pair of spins ( l , v ) in neighbor d-units, eqs (6) and (7) can be written as: H 1=2ðÞ 1 l ; m ; a ;bðÞ¼J 1 ½s z l ;2; a s z m ;1;b þ1 2ðs þ l ;2; a s  m ;1;b þs  l ;2; a s þ m ;1;b Þ ð8Þ Table 2 (a) Spin Hamiltonian parameters corresponding to uncoupled units obtained fitting the Hamiltonian of eq (4) to the data on the angular variation of the position of the EPR peaks at Q-band shown in Fig. 10a,b,c. The principal g-values (g1<g2<g3) of the matrices gAand gB, the principal components (Dand E)[101] of the D-matrices DA and DB, and the eigenvectors Vof the matrices gand D, are considered approximately to be the same. Releasing this condition allows calculating eigenvectors differing within the tabulated experimental uncertainties. Upper and lower signs in the eigenvectors corresponding to sites Aand B. The Euler angles corresponding to the eigenvectors are also given. The sign of Dwas taken from Ozarowski [36]. The eigenvectors Vof gand Dare given in the a*bc orthogonal laboratory coordinate system. Eigenvalues eigenvectors g 1 2.0563 ± 0.0023 V 1g,D =[0.302, ±0.839, 0.451] g 2 2.0775 ± 0.0027 V 2g,D =[0.8287, 0.002, 0.559] g 3 2.3671 ± 0.0027 V 3g,D =[0.470, 0.543, 0.696] D(10160 ± 40) MHz = (0.339 ± 0.002) cm 1 E(251 ± 20) MHz = (0.0084 ± 0.0010) cm 1 a [°] ±(49.07 ± 0.17) b[°](45.95 ± 0.10) c [°] ±(51.12 ± 2.3) r fit [mT] 4.4 z normal (sites A/B) from structure [28]. [0.4300, 0.5270, 0.7335] R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 7 where J 1 stays for J T or J L . Since J 1 jjjJ 0 j, and J 1 jj<jDj, the first (z) term of eq (8) transforms the energy states in energy bands with width jJ 1 j; the second term induces simultaneous flip-flop transitions [116] of copper spins ½ in neighbor d-units ( l , v ) [116], introducing a spin dynamics in the chains with an ‘‘exchange frequency” x ex that depends on J 1 (J L or J T ) and other variables of the energy-gapped dimeric system. This x ex has a role similar to that introduced by Anderson and Weiss [102,117] for monomeric spin arrays assuming Gaussian behavior for the spin correlation functions. As they explained, the exchange frequency x ex is the result of the superposition of the effects of rather small exchange coupling distributed over a large number of neighbors. In dimeric arrays it involves statistical averages of J 1 over the magnetic moments of the d-units in the array, varying with Tas the average magnetic moment [79]. Since the couplings of eq (8) do not produce other simultaneous changes, energy conservation requires differences between the energies of the flipping pairs of transitions smaller than the energy bandwidth jJ 1 jarising from the first term of eq (8), a condition that holds only at magic angles. When coupled neighbor d-units in the array are identical, as for the T-type coupling of Fig. 4a(AA or BB pairs) in d-units axially symmetric around z, mainly the case treated in this section, the magic angle occurs at the same B 0 in any plane containing the z-axis. Meanwhile, in AB pairs of d-units, as for the L -type coupling in Fig. 4b, the magic angle occurs at different fields when varying this plane, with important consequences on the effect of the interdimeric coupling in the EPR spectra as discussed later when presenting the data. Thus, even if similar physical processes occur in uniform (T) or alternate (L) dimeric chains of CAH, the possibilities of observing the effects of the interdimeric coupling in the EPR spectra are reduced in L chains, as it is observed experimentally (see later). With appropriate analysis, we will introduce a formalism in the next subsection that replaces eq (8) with a convenient model that considers the detailed characteristics of the dimeric array. 3.3. Spin algebra for single dimeric (d) and Double-Dimeric (dd) units Now we describe the methods used to deal with the intraand interdimeric coupling within finite arrays of d-units that are inspired by the spin Hamiltonian approach described before and used since the original papers of B&B on CAH in the case of single d-units [14,15]. We do that by introducing a procedure that may help to deal with systems of higher complexity, as in molecular magnetism. The simplest AFM d-unit is described as s 1 J 0 s 2 ð9Þ The 2 2 matrices for the components s q ( q =x,y,z) of the single ½ spins, the shift operators s þ ¼s x þis y , and s  ¼s x is y , and the 2 2 unit matrix Iare given in units of g by: s x ¼01=2 1=20  ,s y ¼0i=2 i=20  ,s z ¼1=20 01=2  s þ ¼01 00  ,s  ¼00 10  ,I¼10 01  The base of states products of monomeric-spin ½ states jþi and ji for s 1 and s 2 is: u d 1 ¼jþþi; u d 2 ¼jþi; u d 3 ¼jþi; u d 4 ¼ji ð10Þ Next, we employ the Kronecker product algebra of spin operators, a procedure briefly described by Poole and Farach in their EPR textbook [118], and with higher detail in a paper by Fernández [119], which is implemented in Matlab [120]. Indicating the Kronecker matrix product by , the components s d i; q (i= 1, 2) of the spins s 1 and s 2 within ad-unit described by eq (9) are the operators: s d 1; q ¼s q I;s d 2; q ¼Is q ð11Þ that may be expressed as 4 4 matrices in the uncoupled-spin dimeric base of eq (10). Similar definitions hold for the shift operators and the components S d q of the total spin S d of a d-unit, its modulus, S d  2 , and the exchange coupling H d ex;0 within the dunits are, S d q ¼s d 1; q þs d 2; q S d  2 ¼S d x S d x þS d y S d y þS d z S d z ð12Þ H d ex;0 ¼J 0 ðs d 1;x s d 2;x þs d 1;y s d 2;y þs d 1;z s d 2;z Þð13Þ where þand indicate standard sums and products of matrices. The operators S d  2 and H d ex;0 (eqs (12) and (13)) commute and have common eigenvectors. The matrices of S d  2 and H d ex;0 in the states of eq (10), their eigenvalues e S d ðÞ 2 and e H d ex;0 , and the common eigenvectors given by the columns of the matrices V S d ðÞ 2 ¼V H d ex;0 obtained using eqs (11),(12), and (13) are: S d  2 ¼ 2000 0110 0110 0002 0 B B B @ 1 C C C A H d ex;0 ¼J 0 1=40 0 0 01=41=20 01=21=40 0001=4 0 B B B @ 1 C C C A ðe S d ðÞ 2 Þ0 ¼ 0;2;2;2ðÞ; e H d ex;0  0 ¼J 0 3=4;1=4;1=4;1=4ðÞ V S d ðÞ 2 ¼V H d ex;0 ¼ 0100 ffiffiffiffiffiffiffiffi 1=2 p0ffiffiffiffiffiffiffiffi 1=2 p0 ffiffiffiffiffiffiffiffi 1=2 p0ffiffiffiffiffiffiffiffi 1=2 p0 0001 0 B B B @ 1 C C C A So, for J 0 <0, the exchange coupling breaks the degeneracy of the spin states of eq (10) into a ground singlet r , and an excited triplet s , with a splitting jJ 0 j, and the eigenvectors, exchange energy, and spins are: r¼ffiffiffiffiffiffiffiffi 1=2 p u d 2 þ u d 3  ;3J 0 =4;S¼0;S z ¼0ð14aÞ s 1 ¼ u d 1 ;J 0 =4;S¼1;S z ¼1ð14bÞ s 0 ¼ffiffiffiffiffiffiffiffi 1=2 p u d 2 þ u d 3  ;J 0 =4;S¼1;S z ¼0ð14cÞ s 1 ¼ u d 4 ;J 0 =4;S¼1;S z ¼1ð14dÞ a well-known result for d-units. The smallest finite spin arrays allowing to consider interdimeric couplings are dd-units made of two neighbor AA,BB,orAB pairs of d-units d 1 and d 2 , and four spins ½, s 1 ,s 2 ,s 3 , and s 4 , as those in the red and green ovals in Fig. 3, indicated as: s 1 J 0 s 2 J 1 s 3 J 0 s 4 ð15Þ where J 1 is the interdimeric exchange coupling, that has a base of 16 monomeric-product states u dd i , R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 8 u dd 1 ¼jþþþþi; u dd 2 ¼jþþþi; u dd 3 ¼jþþþi; u dd 4 ¼jþþi; u dd 5 ¼jþþþi; u dd 6 ¼jþþi; u dd 7 ¼jþþi; u dd 8 ¼jþi; u dd 9 ¼jþþþi; u dd 10 ¼jþþi; u dd 11 ¼jþþi; u dd 12 ¼jþi; u dd 13 ¼jþþi; u dd 14 ¼jþi; u dd 15 ¼jþi; u dd 16 ¼ji ð16Þ As the s d i; q defined for the d-units in eq (11), we introduce the operators s dd i; q for the components q =x,y, and zof the individual spins s dd i (i = 1, ..., 4) in the dd-unit that in the base of eq (16) are: s dd 1; q ¼s q III;s dd 2; q ¼Is q II; s dd 3; q ¼IIs q I;s dd 4; q ¼IIIs q ð17Þ and the components S dd q of the total spin modulus S dd  2 , the intradimeric H dd ex;0 , and the interdimeric H dd ex;1 exchange couplings of the dd-unit are in terms of the s dd i of eq (17): S dd q ¼s dd 1; q þs dd 2; q þs dd 3; q þs dd 4; q ð18Þ ðS dd Þ 2 ¼S dd x S dd x þS dd y S dd y þS dd z S dd z ð19Þ H dd ex;0 ¼J 0 ½s dd 1;x s dd 2;x þs dd 1;y s dd 2;y þs dd 1;z s dd 2;z  þs dd 3;x s dd 4;x þs dd 3;y s dd 4;y þs dd 3;z s dd 4;z  ð20Þ H dd ex;1 ¼J 1 ðs dd 2;x s dd 3;x þs dd 2;y s dd 3;y þs dd 2;z s dd 3;z Þð21Þ where again the symbols þand in eqs (18-21) indicate standard sums and products of the operators. The 16 16 matrices corresponding to S dd  2 ,H dd ex;0 , and H dd ex;1 of eqs (19), (20) and (21),in the states of eq (16) obtained with Matlab [120] are given in the Appendix. Since S dd  2 ;H dd ex;0  ¼0 and S dd  2 ;H dd ex;1  ¼0, but ½H dd ex;0 ;H dd ex;1 –0, we choose the common base of eigenvectors for S dd  2 and H dd ex;0 (not including H dd ex;1 ) given in Table 3. Angular momentum shift operators for d-units as well as dd-units were used to obtain the eigenstates of the exchange couplings H d ex;0 and H dd ex;0 . For AFM dd-units (J 0 <0), they are a ground singlet R 0 with energy 3J 0 =2 and S= 0, degenerate triplet states, t 1,i and t 2,i (S= 1, and i = 1, 0, 1) with energy J 0 =2, and nine states with energy J 0 =2, including one quintet q j (S= 2, and j = 2, 1, 0, 1,- 2), one triplet t 3,i (S= 1 and i = 1, 0, 1), and one singlet R 1 (S= 0), that in the base of states of eq (16) are: q 2 ¼1000000000000000½ q 1 ¼01=21=201=20001=20000000½ q 0 ¼0001=ffiffiffi 6 p01=ffiffiffi 6 p1=ffiffiffi 6 p001=ffiffiffi 6 p1=ffiffiffi 6 p01=ffiffiffi 6 p000 hi q 1 ¼00000001=20001=201=21=20½ q 2 ¼0000000000000001½ t 3;1 ¼01=21=20 1=2000 1=20000000½ t 3;0 ¼0001=ffiffiffi 2 p00000000 1=ffiffiffi 2 p000 hi t 3;1 ¼00000001=20001=20 1=21=20½ P 1 ¼0001=ffiffiffi 3 p01=ffiffiffiffiffiffi 12 p1=ffiffiffiffiffiffi 12 p00 1=ffiffiffiffiffiffi 12 p h1=ffiffiffiffiffiffi 12 p01=ffiffiffi 3 p000i t 2;1 ¼01=ffiffiffi 2 p1=ffiffiffiffiffiffi 12 p0000000000000 hi t 2;0 ¼00000 1=21=200 1=21=200000½ t 2;1 ¼0000000000000 1=ffiffiffi 2 p1=ffiffiffi 2 p0 hi t 1;1 ¼0000 1=ffiffiffi 2 p0001=ffiffiffi 2 p0000000 hi t 1;0 ¼00000 1=21=2001=21=200000½ t 1;1 ¼0000000 1=ffiffiffi 2 p0001=ffiffiffi 2 p0000 hi P 0 ¼000001=21=200 1=21=200000½ ð22Þ A transformation matrix Tbetween the common eigenvectors of S dd  2 and H dd ex;0 and the states of eq (16), with the eigenvectors given in columns, is given in the Appendix. These eigenstates are also given in Table 3 in terms of products of the eigenvectors of the individual d 1 and d 2 units in the dd-unit. The Zeeman and Table 3 Exchange eigenstates of the dd-units (a) in terms of the monomeric states of eq (16), and (b) their relationship with the d-states of d 1 and d 2 ; (c) spin Sand the z-component of the eigenstate; (d) exchange contribution to the state energy E ex , according to subsection 3.3. (a) Eigenstates (b) Relation with states of d 1 and d 2 (c) S,S z (d) E ex q 2 ¼u dd 1 ( s 1 ) 1 ( s 1 ) 2 2, 2 J 0 jj=2 q 1 ¼ðu dd 2 þu dd 3 þu dd 5 þu dd 9 Þ=2[( s 1 ) 1 ( s 0 ) 2 +( s 0 ) 1 ( s 1 ) 2 ]/p22,1J 0 jj =2 q 0 ¼u dd 4 þu dd 6 þu dd 7 þu dd 10 þu dd 11 þu dd 13  =ffiffiffi 6 p[( s 1 ) 1 ( s -1 ) 2 +( s 0 ) 1 ( s 0 ) 2 +( s -1 ) 1 ( s 1 ) 2 ]/p32,0 J 0 jj =2 q 1 ¼ðu dd 8 þu dd 12 þu dd 14 þu dd 15 Þ=2[(( s -1 ) 1 ( s 0 ) 2 +( s 0 ) 1 ( s -1 ) 2 ]/p2 2,-1 J 0 jj =2 q 2 ¼u dd 16 ( s -1 ) 1 ( s -1 ) 2 2,-2 J 0 jj =2 t 3;1 ¼ðu dd 2 þu dd 3 u dd 5 u dd 9 Þ=2[( s 1 ) 1 ( s 0 ) 2 -( s 0 ) 1 ( s 1 ) 2 ]/ p2 1,1 J 0 jj=2 t 3;0 ¼ðu dd 4 u dd 13 Þ=ffiffiffi 2 p[-( s 1 ) 1 ( s 2 ) 2 +( s 2 ) 1 ( s 1 ) 2 ]/ p2 1,0 J 0 jj=2 t 3;1 ¼ðu dd 8 þu dd 12 u dd 14 u dd 15 Þ=2[( s 0 ) 1 ( s -1 ) 2 -( s -1 ) 1 ( s 0 ) 2 ]]/p2 1,-1 J 0 jj =2 R2 ¼u dd 4 u dd 6 =2u dd 7 =2u dd 10 =2u dd 11 =2þu dd 13  =ffiffiffi 3 p[( s 1 ) 1 ( s -1 ) 2 +( s -1 ) 1 ( s 1 ) 2 -( s 0 ) 1 ( s 0 ) 2 ]/p30,0 J 0 jj =2 t 2;1 ¼ðu dd 2 þu dd 3 Þ=ffiffiffi 2 p( r ) 1 ( s 1 ) 2 1,1 J 0 jj =2 t 2;0 ¼ðu dd 6 þu dd 7 u dd 10 þu dd 11 Þ=2( r ) 1 ( s 0 ) 2 1,0 J 0 jj =2 t 2;1 ¼ðu dd 14 þu dd 15 Þ=ffiffiffi 2 p( r ) 1 ( s -1 ) 2 1,-1 J 0 jj =2 t 1;1 ¼ðu dd 5 þu dd 9 Þ=ffiffiffi 2 p( s 1 ) 1 ( r ) 2 1,1 J 0 jj=2 t 1;0 ¼ðu dd 6 u dd 7 þu dd 10 þu dd 11 Þ=2( s 0 ) 1 ( r ) 2 1,0 J 0 jj =2 t 1;1 ¼ðu dd 5 þu dd 9 Þ=ffiffiffi 2 p( s -1 ) 1 ( r ) 2 1,-1 J 0 jj =2 R0 ¼ðu dd 6 u dd 7 u dd 10 þu dd 11 Þ=2( r ) 1 ( r ) 2 0, 0 3J 0 jj=2 R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 9 angular ranges, and show the strong narrowing of the signals in the same orientation ranges of all the merging of Fig. 12a-d. From the line width data in Fig. 12e-h and using the equation C ¼ C 0 þ1 2 dB calc 0  2  h x ex =g l B ¼ C 0 þ1 2 dB calc 0  2 B ex we determine B ex ¼J 1 jj =g l B ¼ h x ex =g l B in magnetic field units, and the parameters of the fits are collected in Table 4. Line crossings of peaks of d-units pairs AB were also studied in the plane cb at Q-band (Fig. 10c), but they did not show merging or narrowing of the peaks due to the non-axial characteristics of these crossings. The integrated intensity I(T) of the EPR spectrum of a dimeric material is proportional to the population of the triplet level. As Table 4 Values of the exchange field Bex in (mT) calculated from the Anderson plots of Fig. 12, obtained at room temperature, at the line crossings Qca 1 ,Qca 2 ,Qcb, and Wcb indicated with colored circles and ellipses in Figs. 10 and 11. Crossing B ex ¼—hx dd ex =ghi l B [mT] Anderson Lw parabola Qca 1 10 ± 2 9 ± 2 Qca 2 11 ± 2 11 ± 2 Qcb 8±2 8±2 Wcb 10 ± 2 7 ± 2 Fig. 13. (a) Positions B 0 of the peaks as a function of angle uin the ca* plane at 298 K, allowing to find the orientation where B 0 is largest (u=49°, indicated with an arrow), where the T-dependence of the intensity of this peak was measured. (b) Some of the spectra observed as a function of Tbetween 20 and 298 K. (c) Plot of the observed ITðÞTvsT(the signal intensity IðTÞis proportional to its area), and best fit of the Bleaney and Bowers equation [14,17] shown in the figure, to the experimental results (blue symbols and line), allowing to obtain J 0 . Fig. 12. Merging and narrowing of the EPR peaks in the Qca 1 , (a) and (e), Qca 2 , (b) and (f), and Qcb, (c) and (g) angular ranges at Q-band, and in the Wca angular range at W-band (d) and (h). The results for B 0 and C in Figs. 10 and 11 are plotted as indicated in the text. The calculated dashed lines allow us to evaluate the indicated interdimeric exchange frequencies given in Table 4. Fig. 14. (a) EPR spectra of powder samples of CAH at Q-band observed at selected values of T. The experimental results are shown in blue, and the simulations are in red. At 305 K the simulation is decomposed as the sum of the contribution of all orientations outside the magic rings (green, labeled as bulk), and those within them (U-peak, magenta, labeled as U-peak). The magnetic field range where the U-peak should be observed is shown with a yellow shade. This magnetic field span stays within the U-peak width reflecting the negligible non-axial anisotropy in the gtensor at this frequency. (b) Observed T-dependence of the ratio between the integrated intensities of the U-peak and the total observed signal. R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 16 reported by B&B, ITðÞTcan be written for dimeric compounds as [14,17], ITðÞT/3þexp J 0 k B T  1 ð29Þ The value of J 0 is well known for CAH but to have a reference value for the intensity of the U-peak as a function of T, we also measured the intensity of the single-crystal high-field peak d v 00 =dB 0 vs T at X-band with B 0 in the ca* plane (see Fig. 13). This was done for the field orientation where this peak is at the highest field in this plane, indicated with an arrow in Fig. 13a that displays a set of spectra at X-band as that shown at Q-band in Fig. 9.By selecting a single crystal peak we avoid errors caused by the presence of an eventual baseline in broad swept spectra. The solid line for the angular variation shown in Fig. 13a is calculated with the parameters in Table 2 obtained at the Q-band, indicating again the quality of our data, compared across different frequencies in this work. For this orientation of B 0 , we collected spectra at various Tbetween 20 and 298 K (see Fig. 13b), calculated their intensities IðTÞby double integration, and corrected these intensities considering the signal of the simultaneously measured Cr III : MgO marker introduced together with the sample, which was assumed to follow a paramagnetic 1/TCurie law. The corrected T-dependence of ITðÞTof CAH is plotted in Fig. 13c. Fitting eq (29) to these results we obtain J 0 =(306 ± 10) cm 1 , in reasonable agreement with more accurate results (e.g., J 0 =292.2 cm 1 ) obtained from susceptibility measurements [33]. In general, EPR is not the most accurate technique to evaluate J 0 from absolute intensity measurements. The integrated intensity of the allowed EPR peak as a function of T(Fig. 13c) is, however, useful for this work as a reference for the intensity of peaks arising from interdimeric couplings. 5.4. Powder spectra and the U-Peak: Experimental results at low frequencies Fig. 14a displays spectra d v 00 =dB 0 vs B 0 of CAH at Q-band collected in powder samples at various Tbetween 107 and 305 K, and Fig. 15a,b,c display these spectra at X-, Kand W-bands at 298 K. All peaks of the powder spectra are labeled according to the principal directions of the gand D-matrices, B q ( q =x,yand z). These powder spectra show the ‘‘U” peak emphasized with a yellow band, which is not expected from uncoupled d-units [83] and arises from the interdimeric interaction. This U-peak is a consequence of the merging and narrowing of the peaks at the magic angle in single crystals described in 5.3 for CAH (Fig. 12) and previously for other similar compounds [79,83,85,90], that collects the signal arising from microcrystals with B 0 oriented along the magic angles where the allowed transitions s 1 M s 0 and s 0 M s -1 intersect. As observed here for CAH (see Fig. 12a,b,c,d), for either Aor Bsites these peaks merge and, when summed over all Fig. 15. Measured and simulated powder spectra of CAH in room Tat different microwave frequencies. (a) W-band, (b) K-band, and (c) X-band. In each case, the experimental result is blue, the simulated spectrum is red, the simulated contribution of the U-peak is magenta, and that of the rest of the sample, excluding the Upeak is green. Fig. 16. Microwave power attenuation of the intensity of the peaks of the powder spectra of CAH. (a) EPR spectra for various power attenuations. (b) The integrated intensity of the peaks vs microwave power attenuation. It is observed that the intensity of the allowed, forbidden ( D m¼2), and U-peaks change equally with applied microwave power. The Cr 3+ :MgO was used as a field marker. R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 17 microcrystals in a powder sample, accumulate in a narrow field range, and grow further in intensity because of the observed narrowing. This behavior is particularly important in the case of axially-symmetric d-units (E0) [101] because the signal accumulates in a narrower range, and its peak height is consequently larger. Fig. 14b displays as a function of Tat Q-band the ratio between the integrated intensities of the U-peak and that of the full powder sample, both obtained by double integration. This ratio decreases to zero below 180 K and stays constant above this T. So, the U-peak behaves as arising from an activated process that differs at low Tfrom that predicted for the population of the triplet state (and the magnetic susceptibility given by the B&B result of eq (29)). When the peaks crossing at the magic angles belong to equal sites (AA or BB crossings in CAH) the theoretical situation is simple because energy conservation is easily obtained. In the case of the crossing of peaks of Aand Bsites, the axial symmetry of the crossing is broken, and the broader U-peak tends to be weaker or disappear, as is observed in our single-crystal and powder measurements. Fig. 16a displays the spectra d v 00 =dB 0 vs B 0 of a powder sample, for various values of the microwave power attenuation. The calculated integrated intensities of the individual spectral peaks D m jj ¼1 and D m jj ¼2, and the U-peak of these spectra, are plotted in Fig. 16b as a function of microwave power, to show the power saturation curves of these peaks. The result indicates that, within the experimental uncertainty, the saturation of the U-peak with microwave power is identical to that of the rest of the spectra, allowing to discard the possibility of a contribution of a double quantum transition to this peak. 5.5. Powder spectra and the U-Peak at high frequencies Fig. 17a displays a two-dimensional map B 0 ; m ½of the EPR data obtained in a powder sample of CAH at microwave frequencies, v between 180 and 375 GHz. The map is divided into two sections covering the lower (between 180 GHz and 240 GHz with a resolution of 15000 3612 [frequency, field] points) and higher ranges (from 260 GHz to 375 GHz with a resolution of 11500 7457 [frequency, field] points), displayed as colored dots (blue for positive peaks and red for negative peaks) the positions of the signal peaks observed as a function of B 0 for different m . The black dash-dotted lines indicate fixed frequencies (horizontal lines) or fixed fields (vertical lines) where individual spectra are shown (up (b) and down (c) for fixed microwave frequencies m , 182 and 375 GHz, and on the right side (for fixed magnetic fields (d) 6.8 T and (e) 9.9 T). The grey dotted lines in (a) are the positions of the EPR peaks as pairs ½B 0 ; m of the dimeric spectra (eq (4)) simulated using the parameters from Table 2 and show an excellent agreement all over the measured frequency range, including X-, Q-, and K-band. The evolution of the peaks with frequency in (a) evidences the gmatrix’s anisotropy. In (b) and (c) we indicate the corresponding orientations of the resonance positions and the anisotropy jg x g y jbecomes clearer, especially in comparison with Fig. 15. These results illustrate experimentally the importance of highfrequency measurements for resolving the small anisotropies that translate also in a spread of the magnetic fields at the magic angles where the U-peak is collected (yellow band in Fig. 17a). From the map, we extracted the spectrum at 375 GHz and 240 K shown in Fig. 18 to simulate and demonstrate the effect of the anisotropy Fig. 17. Map of high-frequency spectra. (a) Two-dimensional frequency-field ½B 0 ; m maps of spectra of CAH in the frequency range from 180 GHz to 240 GHz (resolution of 15000 3612 [frequency, field] points) and from 260 GHz to 375 GHz resolution of 11500 7457 [frequency, field] points). The light red shades are background noise, the red color corresponds to maximum positive values of d v 00 =dB 0 and the blue color to maximum negative values. The extracted field and frequency domain spectra (b-e) are indicated by black dash-dotted lines in the maps. The dotted grey lines in (a) are the positions of the EPR peaks as pairs ½B 0 ; m of the dimeric spectra (eq (4)) simulated using the parameters from Table 2 using Easyspin. The yellow shades indicate the magnetic field range corresponding to the width of the U-peak based on the maximum and minimum values of the magnetic field at the magic angles simulated for a single crystal sample in the three orthogonal planes a*b,ca*, and cb. R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 18 in the U-peak, which is broader and split in two as a consequence of the difference in the magnetic field position at the magic angle in the single crystal caused by jg x g y j. 6. Conclusions The interdimeric coupling in dimeric materials has been studied with EPR, but until recently with incomplete experimental information and without detailed explanations and modeling of the observations. Meanwhile, the last two decades fascinating results dealing with the collective behavior of dimeric compounds arising from these couplings, as quantum phase transitions [48], including Bose-Einstein condensation at high T[64], elementary excitations, and more, have been obtained using other magnetic techniques or methodologies [48,61–63,126,143]. We reviewed the existing EPR information about interdimeric coupling and advance the subject by adding new data, explanations, and theoretical ideas. Along with this work, we collected earlier achievements in this problem which are listed in Table 1. To elucidate them we introduce here a spin model that points out the roles of the interdimeric interactions in the EPR spectra and allows suggesting appropriate characterization measurements. We claim with our interpretation that the interdimeric couplings produce for certain orientations of the applied magnetic field an entanglement of the dimeric arrays that changes the spectra of single crystal and powder samples. We describe the results with a spin model considering magnetic excitations with spin 1 (triplons) that diffuse in the crystal lattice, and a quantum phase transition that is driven by the orientation of the applied magnetic field. We propose a phase diagram, that we call ‘‘the magic rings”, that divides the sphere of field orientations into two phases. One phase contains quantum-independent d-units, where the role of the interdimeric exchange is negligible compared to the intradimeric anisotropic spin–spin interaction (D-term of the spin-Hamiltonian of eqs (2) and (4)). The other phase includes dimeric units that are quantum entangled by weak interdimeric exchange couplings J 1 (with jJ 1 =J 0 j10 5 , when compared with an intradimeric exchange coupling J 0 ) between neighbor units, by a flip-flop process where two simultaneous triplet excitations connecting the singlet state of a d-unit with the triplet state of a neighbor unit in the dimeric array are absorbed and emitted. The magic rings where the entangled phase occurs collect an angular region around the magic angles [101] where the D-term within the dunits is comparable to J 1 . The transitions between the phases is produced changing the orientation of the applied magnetic field B 0 . This condition should be compared with the quantum phase transitions observed in dimeric arrays [48,55,64] when a change of the magnitude of the applied magnetic field produces an energy crossing between one state of the excited triplet state and the ground singlet state of an antiferromagnetic dimeric array. The EPR spectra of the d-units in the entangled phase display merging and narrowing effects of the spectral peaks that are observed in oriented single crystal samples, a process that in turn gives rise to the extraordinary U-peak in the spectra of a powder sample that has been observed since 1962 [67–70] and explained in 2010 [83]. Our theoretical results indicate that for CAH and other previously studied dimeric compounds [79,83,85,90], joint EPR studies of merging and narrowing of the peaks in single crystal samples and U-peak in powder samples are required to obtain full information about the effects of the interdimeric couplings on the collective magnetic behavior of the dimeric arrays. The observed merging and narrowing of the EPR peaks in single crystals parallel the phenomenon of exchange narrowing introduced early by Anderson [102,103] and observed experimentally [130,144,145] for interacting monomeric spins. However, the presence of a large singlet–triplet energy gap jJ 0 jin the individual d-units of dimeric arrays produces big differences with the behavior of monomeric arrays. Our results for the merging and narrowing of the EPR peaks in single crystal samples of AFM dimeric materials allow the evaluation of an appropriate interdimeric exchange frequency x dd ex that explains the spin dynamics produced by the interdimeric coupling that involves singlet–triplet transitions, and explain qualitatively and quantitatively the U-peak in powder samples [83], that is secondary to the observations in single crystals and provides a simple view of the quantum phases described above. The intrinsic properties of the dimeric units given by the spin-Hamiltonian parameters in Table 2 for CAH, influence these effects and allow us designing the appropriate experiments to characterize the interdimeric coupling J 1 and to understand the related phase transitions. In the EPR observations of interdimeric coupling, one needs to optimize the experimental procedures and maximize the EPR signal-to-noise ratio to observe the merging and narrowing of the peaks in single crystals (as in Fig. 12), and the U-peak in powder samples (as in Figs. 14-16). The microwave frequency has to be chosen to avoid the superposition of the U-peak’s magnetic field with that of the standard peaks of the dimeric EPR spectra. The intrinsic linewidth is an important factor contributing to the resolution of the different peaks necessary for our investigation. In this direction, we show in Fig. 8 simulations of powder EPR spectra of CAH in the wide range of accessible microwave frequencies, plotted vs B 0 = m . In this figure, we indicate with a yellow band the field where the U-peak should appear. The most appropriate microwave frequencies are between 35 and 100 GHz, where the U-peak and the rest of the powder spectra are well separated, and above 350 GHz where they separate again. We showed the importance of the ratio Rbetween the integrated intensity of the U-peak and that of the dimeric signal to quantify the interdimeric coupling because it reflects the fraction of d-units in the entangled phase and is the consequence of accumulated EPR signal in the angular range around the magic angles where the distance between the two merging allowed resonance peaks is smaller than the interdimeric coupling J 1 and may be quantified by the fraction of solid angle within a sphere occupied by the magic rings shown in Fig. 6. Non-axially symmetric contributions to the Hamiltonian of eqs (2) or (4) of the dimeric units spread the signals attributed to interdimeric coupling so decreasing the signal amplitude. These Fig. 18. HFEPR powder spectrum of CAH at 375 GHz and 240 K extracted from the map in Fig. 17a. The simulated (red) spectrum was obtained with the parameters from Table 2 and summed with the theoretically predicted U-peak spectrum at this frequency. The magnetic field range where the U-peak should be observed is shown with a yellow shade, and a feature in the experimental spectrum is observed in this magnetic field region. R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 19 non-axially symmetric contributions are determined by the ratio jE=Djbetween the principal spin–spin coupling parameters in the matrix D[101], and by the difference jg x g y jbetween the principal components of the g-matrix of the Zeeman coupling in the perpendicular plane. The ratio jE=Djdoes not change the anisotropy with the microwave frequency of the EPR measurements, while the Zeeman contribution to the anisotropy is changed. Thus, one needs to choose the microwave frequency that minimizes the superposition of the contributions to the spectra of the entangled units, and also it is convenient to have measurements in a wide range of microwave frequencies to be able to distinguish the two non-axial contributions to the anisotropy, that we outline for a general case in subsection 5.2. Table 1 shows that in an important fraction of the listed works double quantum transitions (dqt) [101,113–115] have been invoked as the source of the observed U-peaks. This occurs because both contributions occur at the magic angles. One way to distinguish the origin of these contributions is to perform power saturation measurements. The U-peak is a fraction of the normal dimeric signal and the fraction Rshould not depend on the microwave power, while the dqt arises from a second-order process [113] and its intensity grows with the square of the microwave power. We verify in our results of Fig. 16 that within the experimental uncertainty the dqt does not contribute to the U-peak observed for CAH. We cannot prove that this is the case for the other cases mentioned in Table 1 where this information does not exist but in our opinion, that considers that most studied compounds are very similar, this result applies also to the other cases and suggest that simple power saturation measurements for the powder spectra are strongly advised in future investigations. Rigamonti et al.[35] studied the structural properties of several compounds having structures similar to that of copper acetate where the apical water ligand is replaced by other molecules, producing a change in the bridging scheme between the dimeric units. A continuation of our study of interdimeric couplings in two copper benzoates [90] with different chemical bridges between dimeric molecules may be benefited from following these more formal chemical steps. The simple approach for the description of the spin dynamics of arrays of dimeric units used in this review was chosen to attract the attention of a wide group of people interested in magnetic excitations and phase transitions displayed by EPR, the technique that opened the field 70 years ago. We wanted to show here the interesting features of these arrays that arise from EPR data using a formalism that follows naturally the historical treatment. Higherlevel descriptions of the elementary excitations and phase transitions may be needed as new experimental results and new theoretical ideas appear, as may be expected. A consistent result for the exchange frequency obtained from the four independent measurements of merging and narrowing of the single crystal peaks, B ex ¼—h x dd ex =g hi l B [mT] 10 mT (in magnetic field units) supports well the validity of the model. Data availability Data will be made available on request. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments ORN is grateful to CNPq, Brasil, for its support of this investigation. PN would like to thank the Czech Science Foundation for the support through the GAC ˇR EXPRO grant number 21-20716X. AS acknowledges the support of the Czech Ministry of Education, Youth and Sports (MŠMT C ˇR, grant number LTAUSA19060). MŠ and VTS acknowledge the ERC Starting Grant (No. 714850) under the European Union’s Horizon 2020 Research and Innovation Program. MŠ also acknowledges the BUT IGA project FEKT S 206215. We are grateful to Prof. Oswaldo Baffa for allowing us to use the K-band spectrometer. RC is a member of CONICET, and would like to acknowledge Mireille Perec for discussions in the initial stages of our work with dimeric systems. Appendix Matrices of the operators S dd  2 ,H dd ex;0 , and H dd ex;1 defined in eq (16) of the main text of the paper for the double dimeric dd-unit, in the base of 16 monomeric-product states u dd i are: S dd  2 ¼ 6000000000000000 0310100010000000 0130100010000000 0002011001100000 0110300010000000 0001021001001000 0001012000101000 0000000300010110 0110100030000000 0001010002101000 0001001001201000 0000000100030110 0000011001102000 0000000100010310 0000000100010130 0000000000000006 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A R. Calvo, R.P. Sartoris, O.R. Nascimento et al. Coordination Chemistry Reviews 480 (2023) 215007 20 H dd ex;0 ¼J 0 1=2000000000000000 001=20000000000000 01=200000000000000 001=20000000000000 000000001=20000000 0 0 000 1=21=20 01=2000000 0 0 0001=21=20001=200000 000000000001=20 0 0 0 00001=200000000000 0 0 0001=20001=21=200000 0 0 000 01=20 01=21=200000 00000001=200000000 0000000000001=20 0 0 000000000000001=20 00000000000001=20 0 0000000000000001=2 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A H dd ex;1 ¼J 1 1=4000000000000000 01=400000000000000 001=401=200000000000 0001=401=20000000000 001=20 1=400000000000 0001=20 1=40000000000 000001=21=4000000000 00000001=400000000 000000001=40000000 0000000001=4000000 00000000001=401=20 0 0 000000000001=401=20 0 00000000001=20 1=40 0 0 000000000001=20 1=40 0 000000000000001=40 0000000000000001=4 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A The transformation matrix Tfrom the monomeric-product states of eq (16) to the eigenvectors of H dd ex;0 given in Table 1, and the 16 16 matrix of H dd ex;1 rotated to the base of eigenvectors of H dd ex;0 using the transformation ðH dd ex;1 Þ T ¼T 0 H dd ex;1 Tare: T¼ 10000000 0 000 0000 01=20 001=20 0 0ffiffiffiffiffiffiffiffi 1=2 p00 0000 01=20 001=20 0 0 ffiffiffiffiffiffiffiffi 1=2 p00 0000 00p600 0 ffiffiffiffiffiffiffiffi 1=2 p0ffiffiffiffiffiffiffiffi 1=3 p000 0000 01=20 001=200 0 000ffiffiffiffiffiffiffiffi 1=2 p000 00p600000ffiffiffiffiffiffiffiffiffiffiffi 1=12 p01=20 01=20 1=2 00p600000ffiffiffiffiffiffiffiffiffiffiffi 1=12 p01=20 01=201=2 00 01=20 0 0 1=20 000 00ffiffiffiffiffiffiffiffi 1=2 p0 01=20 001=200 0 000 ffiffiffiffiffiffiffiffi 1=2 p000 00p600000ffiffiffiffiffiffiffiffiffiffiffi 1=12 p01=20 0 1=201=2 00p600000ffiffiffiffiffiffiffiffiffiffiffi 1=12 p01=20 01=201=2 00 01=20 0 0 1=20 000 00ffiffiffiffiffiffiffiffi 1=2 p0 00p600 0 ffiffiffiffiffiffiffiffi 1=2 p0ffiffiffiffiffiffiffiffi 1=3 p000 0000 00 01=20 0 0 1=20 0 0ffiffiffiffiffiffiffiffi 1=2 p0000 00 01=20 0 0 1=20 0 0 ffiffiffiffiffiffiffiffi 1=2 p0000 00001000 0 000 0000 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A ðH dd ex;1 Þ T ¼T 0 H dd ex;1 T¼ R. 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