Interplay of Anisotropic Exchange Interactions and Single-Ion Anisotropy in Single-Chain Magnets Built from Ru/Os Cyanidometallates(III) and Mn(III) Complex

Two novel 1D heterobimetallic compounds {[MnIII(SB2+)MIII(CN)6]·4H2O}n (SB2+ = N,N′-ethylenebis(5-trimethylammoniomethylsalicylideneiminate) based on orbitally degenerate cyanidometallates [OsIII(CN)6]3− (1) and [RuIII(CN)6]3− (2) and MnIII Schiff base complex were synthesized and characterized structurally and magnetically. Their crystal structures consist of electrically neutral, well-isolated chains composed of alternating [MIII(CN)6]3− anions and square planar [MnIII(SB2+)]3+ cations bridged by cyanide groups. These -ion magnetic anisotropy of MnIII centers. These results indicate that the presence of compounds exhibit single-chain magnet (SCM) behavior with the energy barriers of Δτ1/kB = 73 K, Δτ2/kB = 41.5 K (1) and Δτ1/kB = 51 K, Δτ2 = 27 K (2). Blocking temperatures of TB = 2.8, 2.1 K and magnetic hysteresis with coercive fields (at 1.8 K) of 8000, 1600 Oe were found for 1 and 2, respectively. Theoretical analysis of the magnetic data reveals that their single-chain magnet behavior is a product of a complicated interplay of extremely anisotropic triaxial exchange interactions in MIII(4d/5d)–CN–MnIII fragments: −JxSMxSMnx−JySMySMny−JzSMzSMnz, with opposite sign of exchange parameters Jx = −22, Jy = +28, Jz = −26 cm−1 and Jx = −18, Jy = +20, Jz = −18 cm−1 in 1 and 2, respectively) and single orbitally degenerate [OsIII(CN)6]3− and [RuIII(CN)6]3− spin units with unquenched orbital angular momentum in the chain compounds 1 and 2 leads to a peculiar regime of slow magnetic relaxation, which is beyond the scope of the conventional Glaubers’s 1D Ising model and anisotropic Heisenberg model.

In this context, a synthesis of the first heterobimetallic magnetic chain compound involving orbitally degenerate [Os III (CN) 6 ] 3− complex and high-spin Mn III Schiff-base complex [128], which exhibits distinct SCM behavior with enhanced U eff and T b parameters, is worth mentioning. This system is of particular interest for understanding the underlying physical mechanism of slow magnetic relaxation in a magnetic chain with highly anisotropic non-Ising spin coupling. Theoretical analysis for the discrete [Os III (CN) 6 ] 3−based trinuclear clusters with similar local cyanide-bridging topology [119,121] showed that the spin coupling in the Os III -CN-Mn III fragments is described by an extremely anisotropic triaxial spin Mn z with opposite signs of the exchange parameters, such as J x = −18, J y = +35, J z = −33 cm −1 in the Mn III 2 Os III cluster [119]. This points at a special regime of magnetic relaxation in the {NC-Os III -CN-Mn III -NC-} n chains resulting from a complicated interplay of highly anisotropic non-Ising exchange interactions and single-ion ZFS anisotropy of Mn III ions, which is even more sophisticated due to the noncollinear orientation of the local magnetic axes. Obviously, such a scenario is the subject of new magnetic physics, which can be considered neither within the existing Ising theory nor within the anisotropic Heisenberg model.
Based on these considerations, we prepared new heterometallic chain compounds involving [ 3+ cationic complexes. We present the results of static and dynamic magnetic measurements and a detailed theoretical interpretation based on the anisotropic spin coupling model. The presence of orbitally degenerate magnetic units with unquenched L is shown to lead to a peculiar regime of magnetic relaxation in the chain compounds 1 and 2 that goes far beyond the usual Ising and anisotropic Heisenberg models.

Synthetic Approach
Describing an approach chosen for the preparation of the new SCMs based on hexacyanidometallates as metalloligands, we would like point out that no SCM containing [Ru III (CN)6] 3− synthon has been obtained so far. The reason for this lies in the instability of hexacyanidoruthenate(III) anion in solution during slow-diffusion crystallization of heterobimetallic assemblies, unlike its iron and osmium congeners. With the latter, the anionic 1D polymers [Mn III acacen(Fe III /Os III )(CN)6] 2− exhibiting SCMs properties were successfully obtained and studied [78,128], whereas for [Fe III (CN)5NO] 2− and [Re IV (CN)7] 3− 0D÷3D, assemblies with Mn(III) complexes were obtained depending on synthetic conditions [49,92,101,127,129,130]. In such cases, the only way to obtain lowdimensional heterometallic complexes is to create conditions in which electroneutrality is a driving force of self-assembly. In order to guarantee a 1:1 stoichiometry in a chain or binuclear compound, identically charged counterions must be used as precursors. For cyanidometallates, this approach has been successfully developed and applied to the synthesis of neutral  [131,133].
Owing to the low stability of the [Ru(CN)6] 3− anion in solution [106,107,134] compound 2 was obtained through rapid precipitation of the coordination polymer by mixing solutions, a procedure earlier used for the preparation of 3 [131]. Chain 1 was synthesized using a process similar to one described in Reference [77]. A layering of H2O:MeCN solutions containing [Mn(SB 2+ )(H2O)2](ClO4)3•H2O and (Ph4P)3[Os(CN)6] in a 1:1 ratio after a few days gave fern-like dark crystals slightly powdered by a white precipitation of Ph4PClO4, which was removed via washing in a few milliliters of acetonitrile. The data of IR, CHN analysis, and powder XRD confirmed the good quality and purity of the samples.

Synthetic Approach
Describing an approach chosen for the preparation of the new SCMs based on hexacyanidometallates as metalloligands, we would like point out that no SCM containing [Ru III (CN) 6 ] 3− synthon has been obtained so far. The reason for this lies in the instability of hexacyanidoruthenate(III) anion in solution during slow-diffusion crystallization of heterobimetallic assemblies, unlike its iron and osmium congeners. With the latter, the anionic 1D polymers [Mn III acacen(Fe III /Os III )(CN) 6 ] 2− exhibiting SCMs properties were successfully obtained and studied [78,128], whereas for [Fe III (CN) 5 NO] 2− and [Re IV (CN) 7 ] 3− 0D÷3D, assemblies with Mn(III) complexes were obtained depending on synthetic conditions [49,92,101,127,129,130]. In such cases, the only way to obtain low-dimensional heterometallic complexes is to create conditions in which electroneutrality is a driving force of self-assembly. In order to guarantee a 1:1 stoichiometry in a chain or binuclear compound, identically charged counterions must be used as precursors. For cyanidometallates, this approach has been successfully developed and applied to the synthesis of neutral  [131,133].
Owing to the low stability of the [Ru(CN) 6 ] 3− anion in solution [106,107,134] compound 2 was obtained through rapid precipitation of the coordination polymer by mixing solutions, a procedure earlier used for the preparation of 3 [131]. Chain 1 was synthesized using a process similar to one described in Reference [77] 6 ] in a 1:1 ratio after a few days gave fern-like dark crystals slightly powdered by a white precipitation of Ph 4 PClO 4 , which was removed via washing in a few milliliters of acetonitrile. The data of IR, CHN analysis, and powder XRD confirmed the good quality and purity of the samples.

Crystal Structure
For 1, we were able to grow crystals suitable for single-crystal X-ray structure analysis. This study has shown that Os-Mn polymer is isostructural to its earlier studied congener [Mn(SB 2+ )Fe III (CN) 6 ]·4H 2 O [77], and the crystal unit cell parameters of 1 determined at 110 K are a = 11.2510(1), b = 16.7747 (2), and c = 18.6078(2) Å β = 96.53(6) • , space group P2/c (#15) (Table S1, see Supplementary Materials)). The asymmetric unit is shown in Figure 1. The view of the 1D chain motif is presented in Figure 2. Selected geometric parameters for 1 compared to its Fe congener are listed in Table 1. More bond lengths and bond angles are presented in Table S2. The coordination environment of the Mn ion is an elongated tetragonal bipyramid because of the Jahn-Teller distortion. The 2O and 2N donor atoms of the SB 2+ ligand in the basal plane of the pyramid form shorter bonds of 1.88-1.98 Å, while two N atoms of trans-disposed CN ligands form much longer Mn-N CN bonds of 2.28 Å with an NC N-Mn-N CN angle of 173.5 • , which is larger than in Mn-Fe analog ( Table 1). The Mn-N-C bond angles are much more acute than 180 • and equal to 142.5 • , being slightly less than 144.4 • for the Fe-containing chain. In 1, Os ion coordinates four terminal CN groups, forming two hydrogen bonds N2···O1W of 2.94 Å and two N3···O2W of 2.88 Å. The additional contacts O1···O2W of 2.94 Å and O1W···O2W of 2.85 Å connect the neighboring chains into an H-bonded 3D structure ( Figure S1).

Crystal Structure
For 1, we were able to grow crystals suitable for single-crystal X analysis. This study has shown that Os-Mn polymer is isostructural to its congener [Mn(SB 2+ )Fe III (CN)6]•4H2O [77], and the crystal unit cell pa determined at 110 K are a = 11.2510(1), b = 16.7747 (2), and c = 18.6078(2) Å space group P2/c (#15) (Table S1, see Supplementary Materials)). The asym shown in Figure 1. The view of the 1D chain motif is presented in Fig  geometric parameters for 1 compared to its Fe congener are listed in Tabl lengths and bond angles are presented in Table S2. The coordination envi Mn ion is an elongated tetragonal bipyramid because of the Jahn-Teller dis and 2N donor atoms of the SB 2+ ligand in the basal plane of the pyramid form of 1.88-1.98 Å, while two N atoms of trans-disposed CN ligands form much l bonds of 2.28 Å with an NCN-Mn-NCN angle of 173.5°, which is larger than in ( Table 1). The Mn-N-C bond angles are much more acute than 180° and being slightly less than 144.4° for the Fe-containing chain. In 1, Os Figure S1).

Powder X-ray Diffraction Investigations
The powder samples for the neutral heterobimetallic (Mn-Ru/Os) 1D polymers obtained via precipitation are crystalline, and their XRD patterns correspond well to the simulated diffractogram of the Mn-Fe chain, with the exception of one Bragg reflection (−1, 1, 2) (see Figures S2 and S3). This peak was calculated for the Mn-Fe compound significantly contributed by iron centers at 2θ = 12.867 • (d = 6.8745 Å), but it is not observed due to a very low intensity. However, as the atomic weight of the central cyanidometallate atom increases, the corresponding peak emerges in the powder diffraction pattern for Ru and Os chains (Figures S2 and S3 and Table S3). The PXRD patterns confirmed that all three 1D coordination polymers containing hexacyanidometallates(III) of the iron group are isomorphic.

. Static Magnetic Properties and Their Theoretical Analysis
The temperature dependences of the magnetic susceptibility measured at 1 kOe are shown in Figure 3 as the χT product. At 300 K, χT = 3.32 and 3.23 cm 3 K/mol for 1 and 2, respectively, which agrees well with the Curie constant of 3.30 cm 3 K/mol expected for magnetically uncoupled Mn III spin S = 2 with g = 2.0 and (Ru/Os) III spin S = 1/2 with g = 1.8. As the temperature decreases starting from room temperature, the χT product of 2 slowly decreases, passing through a flat minimum around 100 K, and then rapidly rises at low temperatures to reach a sharp maximum of~8 cm 3 K/mol at~6 K. In contrast, χT of 1 increases monotonically, without a minimum, forming a much higher maximum of 19 cm 3 K/mol at~8 K. It is noteworthy that the peak value of χT decreases with increasing field ( Figure S4   The magnitude of the spin coupling constant J in heterometallic chains of compounds 1 and 2 can approximately be estimated in terms of an isotropic Heisenberg model for alternating spins S Os/Ru = 1/2 and S Mn = 2, which is described by the Hamiltonian (1): A solution of this Hamiltonian, in the approximation that large spins S Mn are treated classically, was obtained by Seiden as an analytical formula for susceptibility [135]. The only J value was adopted equally for all Ru-Mn or Os-Mn pairs and g Ru/Os = 2 was fixed for both chains. The best fit to the magnetic data in the range of 30-300 K has resulted in and g Mn = 1.76 and J/k B = +25.4 K for 1 and g Mn = 2.23, J/k B = −62.8 K for 2 (inset in Figure 3). To improve the fit, an additional parameter zJ' was added in order to take into account interchain interactions, yielding and 0.51 K for 1-1.39 K for 2. The obtained values imply antiferromagnetic interactions within the chain for 2, with weak ferromagnetic interchain coupling, while for 1, the interactions within the chain are strong ferromagnetic, with weaker antiferromagnetic coupling between the chains.  The magnitude of the spin coupling constant J in heterometallic chains of co 1 and 2 can approximately be estimated in terms of an isotropic Heisenberg m alternating spins SOs/Ru = 1/2 and SMn = 2, which is described by the Hamiltonian  These data indicate that the Seiden's model results in unreliable and inconsistent magnetic parameters that are difficult to expect for the isostructural and isoelectronic chain compounds 1 and 2. This is particularly apparent from the opposite sign of the exchange parameters (F in 1 and AF in 2) and the large scatter in the effective g-factor of Mn III ions. The basic reason behind these issues is that the isotropic Heisenberg model does not account for the anisotropic magnetic interactions associated with single-ion ZFS anisotropy of Mn III ions and anisotropic exchange interactions of Os III and Ru III ions due to unquenched orbital angular momentum in the ground state. The origin of highly anisotropic exchange interactions in the Os III -CN-Mn III and Ru III -CN-Mn III exchangecoupled pairs was examined in detail in Ref. [119] for cyanide-bridged trinuclear clusters [Mn III 2 (5-Brsalen) 2 (MeOH) 2 M III (CN) 6 ] (M = Os, Ru) whose structures are very close to the local structure of the chains in compounds 1 and 2. The ground state of the Os III ion in the octahedral ligand field of the [Os III (CN) 6 ] 3− complex is an isotropic Kramers doublet Г 7 resulting from the spin orbit splitting of the ground orbital triplet 2 T 2g (5d 5 ). The energy separation ∆E = 3/2ζ Os = 4500 cm −1 to the first excited state Г 8 is determined by the spin orbit coupling constant (ζ Os ≈ 3000 cm −1 ). The Ru III ion in complex [Ru (CN) 6 ] 3− behaves similarly but with a smaller spin orbit coupling constant (ζ Ru ≈ 880 cm −1 ). It has been shown that the spin coupling between the ground Г 7 state of Os III (corresponding to the effective spin S = 1/2) and Mn III ions (S = 2) is described by an anisotropic triaxial spin Hamiltonian, S Mn JS Os = −J x S Mn x S Os x −J y S Mn y S Os y −J z S Mn z S Os z , with opposite sign of exchange parameters, J x = −18, J y = +35, J z = −33 cm −1 [119].
Hence, given close similarity in the molecular structure of trinuclear clusters [Mn III 2 (5-Brsalen) 2 (MeOH) 2 M III (CN) 6 ] and chain compounds 1 and 2, the same anisotropic spin Hamiltonian can be applied to the alternating heterometallic Os-Mn and Ru-Mn chains (2): where the sum <ij> runs over the neighboring Os(i) and Mn(j) cyanide-bridged exchangecoupled ions in the chain; the tensor of anisotropic spin coupling (J) has a three-axis structure, S Mn JS Os

Magnetic Relaxation Parameters of 1 and 2 Derived from Static Magnetic Measurements
Low-field dc measurements were used to determine the ∆ ξ parameter for 1 and 2. Anisotropic 1D magnetic systems have a gap in the spin excitation energy spectrum, which leads to the susceptibility dependence χT ≈ C eff Exp(∆ ξ /T) [26]. In this equation C eff is an effective Curie constant, which takes into account the averaging of anisotropic magnetic susceptibility in the powder sample. ∆ ξ designates an energy of the domain wall, which is the lowest excitation of the ground state in the chain of correlated spins. To estimate ∆ ξ , the susceptibility data measured in 1 kOe were plotted as ln(χT) vs. T −1 ( Figure 5). The linear part of the plot in the region from 10 to 30 K was used to obtain ∆ ξ /k B = 20.09 K and 9.45 K, and C eff = 2.77 and 2.71 cm 3 K/mol for 1 and 2, respectively. If the linear sections of the curves at the temperatures below 8 K will be drawn, the intersection of these straight lines with the linear parts of the plots between 10 and 30 K will give the crossover temperatures. The latter determine a crossover between infinite chain regime and finite chain regime in the relaxation dynamics. In our case, they are 10 and 8 K for 1 and 2 respectively.
For the real SCMs, below a certain temperature, the χT(1/T) dependence deviates from the exponential function and saturates ( Figure 5)-even for a low applied field-due to the finite chain length caused by crystal imperfection [26]. This occurs when the rising correlation length surpasses an average chain length n. For 1, it is visible below 8 K, where (χT) max = 22.1 cm 3 K/mol, and it is visible below 6 K for 2, for which (χT) max = 8.56 cm 3 K/mol (data measured at 3 and 15 Oe, respectively). The average chain length estimated using the relation (χT) max = nC eff , gives n ≥ 7.94 nm for 1 and n ≥ 3.17 nm for 2, which correspond to 16 Mn-Os and 6 Mn-Ru units, respectively (7.94 or 3.17 nm /5.166 Å). However, such an estimation of n should be treated with precaution because two other effects can also decrease the measured susceptibility in this temperature range: (a) possible antiferromagnetic inter-chain interaction and (b) demagnetization leading to a decrease in the measured χ. For this reason, the estimation given above is a lower limit of n. magnetic susceptibility in the powder sample. ∆ξ designates an energy of the which is the lowest excitation of the ground state in the chain of correla estimate ∆ξ, the susceptibility data measured in 1 kOe were plotted as ln(χT) 5). The linear part of the plot in the region from 10 to 30 K was used to obta K and 9.45 K, and Ceff = 2.77 and 2.71 cm 3 K/mol for 1 and 2, respectively sections of the curves at the temperatures below 8 K will be drawn, the these straight lines with the linear parts of the plots between 10 and 30 K crossover temperatures. The latter determine a crossover between infinite and finite chain regime in the relaxation dynamics. In our case, they are 10 and 2 respectively. For the real SCMs, below a certain temperature, the χT(1/T) depend from the exponential function and saturates ( Figure 5)-even for a low appl to the finite chain length caused by crystal imperfection [26]. This occurs w correlation length surpasses an average chain length n. For 1, it is visible bel (χT)max = 22.1 cm 3 K/mol, and it is visible below 6 K for 2, for which (χT)max = (data measured at 3 and 15 Oe, respectively). The average chain length es the relation (χT)max = nCeff, gives n ≥ 7.94 nm for 1 and n ≥ 3.17 nm for 2, whi to 16 Mn-Os and 6 Mn-Ru units, respectively (7.94 or 3.17 nm /5.166 Å). H an estimation of n should be treated with precaution because two other e decrease the measured susceptibility in this temperature range: Figure 5. Magnetic dc susceptibility measured at one kOe for 1 (green) and 2 (orange), respectively. Straight dotted lines were fitted (see text). T* represents a crossover temperature.

Dynamic Magnetic Properties
To examine the magnetic dynamics of 1 and 2, the temperature-dependent and frequency-dependent ac susceptibilities were measured. Below 6 and 5 K for 1 and 2, respectively, the ac susceptibility shows a distinction between different frequencies, indicating slow magnetization relaxation. In Figure 6, the temperature dependence of the ac susceptibility measured at zero dc field for different ac field frequencies is presented. The imaginary part of the ac susceptibility χ (T) shows maxima below 5 and 6 K, shifting with the change of the ac drive field frequency υ, and retains the shape that is usual for a temperature induced relaxation process. The Mydosh parameter α, defined as the temperature shift of χ (T) peak position on a decade of frequency ∆T m /[T m ∆log(υ)], remains around 0.10 in both cases. Such a value is above the range typical for spin glasses and is closer to the values for superparamagnets [136].
For a deeper understanding of the relaxation processes, the ac susceptibility was studied over the frequency range 0.1-1000 Hz at low temperatures. These data are presented in Figure 7. The frequency dependent susceptibility measured at constant temperatures was used to determine the relaxation time at each temperature. The generalized Debye relaxation model [137] was used to fit χ (υ) and χ (υ) simultaneously (Equation (3)) (solid lines in Figure 7).
temperature dependence of the ac susceptibility measured at zero dc field for different ac field frequencies is presented. The imaginary part of the ac susceptibility χ''(T) shows maxima below 5 and 6 K, shifting with the change of the ac drive field frequency υ, and retains the shape that is usual for a temperature induced relaxation process. The Mydosh parameter α, defined as the temperature shift of χ'(T) peak position on a decade of frequency ΔTm/[TmΔlog(υ)], remains around 0.10 in both cases. Such a value is above the range typical for spin glasses and is closer to the values for superparamagnets [136].
(a) (b) For a deeper understanding of the relaxation processes, the ac susceptibility was studied over the frequency range 0.1-1000 Hz at low temperatures. These data are presented in Figure 7. The frequency dependent susceptibility measured at constant temperatures was used to determine the relaxation time at each temperature. The generalized Debye relaxation model [137] was used to fit χ'(υ) and χ''(υ) simultaneously (Equation (3)) (solid lines in Figure 7). . ac susceptibility measured for 1 (a) and 2 (b) at selected temperatures versus ac frequency. Solid lines were fitted simultaneously to χ'(ν) and χ''(ν) curves using a generalized Debye relaxation model.
At each temperature, the fitted parameters were χ0 and χ, the relaxation time τ and the parameter α, which describes the distribution of relaxation times. The values of α were in the range 0.11-0.58 for 1 and 0.09-0.47 for 2 (Tables S4 and S5), confirming the good quality of the sample and indicating the SCM nature of both compounds.
The relaxation times in the temperature range from 2.0 to 5.0 K for 1 and 1.8 to 4.2 K for 2, obtained from the ac (Figure 7) and dc ( Figure S7) susceptibility analysis, are presented in Figures 8 and S8. The dependence ln(τ)-(1/T) deviates from the straight line of the Arrhenius law. This is a feature of experimentally studied SCMs with finite chains, for which, below the crossover temperature T*, the probability of relaxation arising from the ends of chains becomes important, changing the relaxation barrier [138]. Above T*, where the correlation length ξ is lower than the average chain length l = na, the relaxation barrier is equal to Δτ1 = ΔA + 2Δξ, where ΔA is the anisotropy energy of a single spin unit. Below T*, the relaxation barrier is reduced and equal Δτ2 = Δτ1 + Δξ in the low temperature limit. The values of Δτ1 and Δτ2 are usually obtained from two linear regions of the ln(τ)-(1/T) dependence much above-and much below-T*, respectively. To obtain both  (Tables S4 and S5), confirming the good quality of the sample and indicating the SCM nature of both compounds.
The relaxation times in the temperature range from 2.0 to 5.0 K for 1 and 1.8 to 4.2 K for 2, obtained from the ac (Figure 7) and dc ( Figure S7) susceptibility analysis, are presented in Figure 8 and Figure S8. The dependence ln(τ)-(1/T) deviates from the straight line of the Arrhenius law. This is a feature of experimentally studied SCMs with finite chains, for which, below the crossover temperature T*, the probability of relaxation arising from the ends of chains becomes important, changing the relaxation barrier [138]. Above T*, where the correlation length ξ is lower than the average chain length l = na, the relaxation barrier is equal to ∆τ 1 = ∆ A + 2∆ξ, where ∆ A is the anisotropy energy of a single spin unit. Below T*, the relaxation barrier is reduced and equal ∆τ 2 = ∆τ 1 + ∆ξ in the low temperature limit. The values of ∆τ 1 and ∆τ 2 are usually obtained from two linear regions of the ln(τ)-(1/T) dependence much above-and much below-T*, respectively. To obtain both relaxation barriers using all data points also close to T*, we used the relation derived by Luscombe et al. for the finite Ising chain [138]. The finite length l of the chain shortens the relaxation time by the factor f (l/ξ): where f(x) = (1 + w 2 /x 2 ) −1 , and w is the solution of the equation, Molecules 2023, 28, x FOR PEER REVIEW 13 of Together with temperature dependence of the correlation length exp Equation (4) allows the calculation of τ(T) as the function of parameters Δξ, Δτ1, τ01, an l/a. Equation (5) was solved numerically using the bisection method for x > 10 −3 , while t approximation w = (2x) 1/2 was used for small values of x < 10 −3 . It is worth noting that the crossover temperatures T* of 2.7 and 2.3 K for 1 and respectively-which were obtained from the intersection of the linear lines in Figures and S8-are very close to those of 3.7 and 2.65 K, respectively, calculated from ln(χT) v T −1 dependencies for the susceptibility data collected at the 0 dc and 3 Oe ac fields at 1 H below 9 and 4.2 K for 1 and 2, respectively ( Figure S9). Moreover, they are in go agreement with the magnetization blocking temperatures TB ≈ 2.8 and 2.1 K found wh registering hysteresis at different temperatures see Figure S10.
The appearance of slow relaxations in 1 and 2 was also confirmed by presence of t hysteresis loops opens below 2.8 and 2.1 K respectively. Their coercive field grows wi decreasing temperature. There are no additional steps on the M(H) curves ( Figure S1 Such a behavior is expected for SCMs, contrary to SMMs, where the quantum tunneli leads to a faster relaxation at the specific fields. Again, this effect is typically observed f SCMs [77]. The parameters related to SCM behavior of 1 and 2 are summarized in Table 2 an compared with those of 3 [77]. Together with temperature dependence of the correlation length ξ = a 2 exp ∆ξ kB T Equation (4) allows the calculation of τ(T) as the function of parameters ∆ξ, ∆ τ1 , τ 01 , and l/a. Equation (5) was solved numerically using the bisection method for x > 10 −3 , while the approximation w = (2x) 1/2 was used for small values of x < 10 −3 . It is worth noting that the crossover temperatures T* of 2.7 and 2.3 K for 1 and 2, respectively-which were obtained from the intersection of the linear lines in Figure 8 and Figure S8-are very close to those of 3.7 and 2.65 K, respectively, calculated from ln(χT) vs. T −1 dependencies for the susceptibility data collected at the 0 dc and 3 Oe ac fields at 1 Hz below 9 and 4.2 K for 1 and 2, respectively ( Figure S9). Moreover, they are in good agreement with the magnetization blocking temperatures T B ≈ 2.8 and 2.1 K found when registering hysteresis at different temperatures see Figure S10.
The appearance of slow relaxations in 1 and 2 was also confirmed by presence of the hysteresis loops opens below 2.8 and 2.1 K respectively. Their coercive field grows with decreasing temperature. There are no additional steps on the M(H) curves ( Figure S10). Such a behavior is expected for SCMs, contrary to SMMs, where the quantum tunneling leads to a faster relaxation at the specific fields. Again, this effect is typically observed for SCMs [77]. The parameters related to SCM behavior of 1 and 2 are summarized in Table 2 and compared with those of 3 [77]. For the chain 1, the total energy barrier for an infinite chain is ∆τ 1 = ∆ A + 2∆ ξ = 73 K at high temperature ( Figure 8). However, at low temperature for a finite size chain regime, ∆τ 2 = ∆ A + ∆ ξ = 41.5 K, giving ∆ ξ = 31.5 K, which is considerably larger than ∆ ξ = 20.09 K obtained from the ln(χT) vs. T −1 plot ( Figure 5). On the other hand, the intrinsic anisotropic barrier ∆ A = ∆τ 1 − 2∆ ξ = 10 K obtained from the plot in Figure 8 is much smaller than ∆ A = ∆τ 1 −2∆ ξ = 32.82 K, calculated with ∆ ξ = 20.09, resulting from the ln(χT) vs. T −1 dependence. This indicates that the SCM relaxation mechanism in 1 cannot be described in the frame of the traditional anisotropic Heisenberg SCM model or Glauber model. The reason lies in the interplay of two independent sources of magnetic anisotropy, i.e., single-ion ZFS anisotropy of Mn III ions and strong three-axis exchange anisotropy in the Os-CN-Mn linkages. The same also applies to the ruthenium chain of compound 2, and quantitative estimates of its SCM parameters ∆ A and ∆ ξ made from the data of Figure S8 are presented in Table 2. Similar unconventional SCM behavior was previously reported for Mn II (H 2 Dapsc)-Fe III (CN) 6 chain complex based on [Fe III (CN) 6 ] 3− units with unquenched orbital angular momentum featuring highly anisotropic spin coupling [80].  6 ] 3− complexes with unquenched orbital angular momentum. Anisotropic exchange parameters J x , J y , J z obtained from our theoretical calculations (J x = −22, J y = +28, J z = −26 cm −1 for osmium compound 1) are remarkably consistent with those previously reported for discrete trinuclear Mn III 2 Os III clusters with similar molecular structure, such as J x = −18, J y = +35, J z = −33 cm −1 [119] and J x = −23.5, J y = +32.0, J z = −25.9 cm −1 [121]. Our theoretical calculations and analysis of magnetic relaxation parameters ∆ A and ∆ ξ have distinctly showed that these new 1D coordination polymers 1 and 2 are SCMs beyond the Glauber model and the anisotropic Heisenberg SCM model.

Materials and Methods
All chemicals were of reagent grade and were used as purchased.  6 ] were prepared according to procedures in the literature [55,128]. Elemental analyses were performed by means of the Euro-Vector 3000 analyzer (Eurovector, Redavalle, Italy). IR spectra were recorded using a Scimitar FTS 2000 spectrophotometer (Digilab LLC, Canton, MA, USA) (KBr pellets) and Nicolet 300 FT-IR spectrometer in reflectance mode (Thermo Electron Scientific Instruments LLC, Madison, WI, USA). Powder X-ray measurements were performed with CuKα radiation (λ = 1.5418 Å) with an Expert-Pro powder diffractometer (PANalytical Inc., Almelo, The Netherlands). Magnetic measurements were performed using the QD MPMS 5XL magnetometer (Quantum Design, Inc., San Diego, CA, USA). The magnetic signal of the sample holder and the diamagnetic correction of the sample were taken into account. A check for small ferromagnetic impurities was performed at room temperature. The powder sample was restrained in cyanoacrylate glue for low-temperature ac measurements.
[Mn(SB 2+ )Os(CN) 6   The single crystals covered by a drop of the oil were directly placed into a stream of cold nitrogen with the precentered goniometer head with CryoMount ® (Chelan County, WA, USA) and attached to the goniometer of a diffractometer. The data for 1 were collected on an Agilent Technologies Gemini diffractometer equipped with an Atlas S2 CCD detector and a CuKα microfocus source using 0.5 • ω scans. The data processing was performed with the CrysAlis software package (CrysAlisPro 1.171.40.47a, Rigaku Oxford Diffraction, 2019). Empirical absorption correction was applied based on the equivalent reflections. The structure was solved by direct methods with SHELXS [139] and refined by full-matrix least squares method against F 2 in anisotropic approximation using the SHELXL-2014/6 package (Shelx, Göttingen, Germany). All non-hydrogen atoms were refined with anisotropic displacement parameters. Hydrogen atoms on the organic part were placed in idealized positions and refined isotropically according to the riding model. The hydrogen atoms of the water molecule were located from the electron density map and refined in a riding model (U iso (H) = 1.5 U iso (O)) with only one distance of 4 (O2w-H21) restrained to 0.82(2) Å. The residual electron density has to chemical meaning. Crystallographic data and further details of the diffraction experiments are given in Table S1.
Theoretical calculations details. Magnetic properties of chain compounds 1 and 2 were analyzed in terms of the anisotropic spin Hamiltonian in Equation (2) Table S1: Experimental details for 1; Table S2: Selected geometric parameters; Figure S1: Hydrogen bonding system in 1; Table S3: Some crystallographic parameters for [Mn(SB 2− )M(CN) 6 ]·4H 2 O; Figure S2: Simulated (red) and experimental X-ray powder pattern for neutral Mn-Os chain polymer; Figure S3: Simulated and experimental X-ray powder patterns (red) for neutral Mn III -M III (CN) 6 chain polymers (M III = Fe, Ru, Os) with additional reflections (green rectangle) from (-1, 1, 2) plane consisted of metal atoms; Figure S4: Magnetic susceptibility times temperature vs. T at 1 kOe and lower fields for 1 in 3 Oe (left) and 2 in 15 Oe (right); Figure S5: Calculated components χ x , χ y , χ z of magnetic susceptibility of the Mn-Os chain (1). Below 50 K, magnetic susceptibility is strongly anisotropic; Figure S6: Experimental (yellow circles) and simulated (solid blue line) magnetic susceptibility χT of 2. The χT curve was simulated with the spin Hamiltonian (Equation (2)) involving anisotropic 3-axes spin coupling S Mn JS Ru = −J x S Mn x S Ru x −J y S Mn y S Ru y −J z S Mn z S Ru z . The best fit is obtained at the set of parameters J x = −18, J y = +20, J z = −18 cm −1 , g Mn = 2.00, g Ru =1.80, D Mn = -4.0 cm −1 . Calculations are performed for a six-membered fragment {Mn-Ru} 3 of the heterometallic chain of 2 with cyclic boundary conditions for the Mn-Ru spin coupling, as shown in the inset; Table S4: Cole-Cole fits parameters for 1; Table S5: Cole-Cole fits parameters for 2; Figure S7: Time dependence of magnetization relaxation for 1 (left) and 2 (right) following the field change from 10 to 0 kOe at constant temperatures of 1.8 ÷ 2.5 K; Figure S8: Relaxation time of 2 derived from the ac data (left) and time dependent dc magnetization (right). The dotted lines correspond to the linear fit according to the Arrhenius law: τ = τ 01 Exp(∆ τ1 /k B T); Figure S9: Crossover temperatures T* obtained from the ln(χT) vs. T −1 dependencies for the susceptibility data collected at 0 dc and 3 Oe ac field at 1 Hz below 9 and 4.2 K for 1 (left) and 2 (right), respectively; Figure S10: Magnetization versus field for 1 (top) and 2 (bottom)-hysteresis loops. Solid lines are to guide the eye; Figure S11: Zero-field cooling/field cooling magnetic susceptibility vs. temperature for 1 and 2 in 15 Oe with a temperature sweep rate of 2 K/min; Figure S12: FTIR (ATR) spectra of 1 (top) and 2 (bottom).