# Spin Symmetry in Polynuclear Exchange-Coupled Clusters

^{1}

^{2}

^{*}

## Abstract

**:**

_{N}} and cyclo-[A

_{N}] systems were modeled. Magnetic data for 20 actually existing endohedral clusters were analyzed and interpreted.

## 1. Introduction

_{3}B (

**C**), the exchange Hamiltonian with two coupling constants

_{3v}**D**), and when the coupling constants are J

_{3h}_{a}= J

_{b}= J, the formula collapses into a tetrahedron (

**T**) with

_{d}- Spherical-tensor matrices:

- Shift-operator matrices:

- Cartesian matrices:

_{10}, S

_{A}= 1/2] centers can be handled (K = 1024); [A

_{7}, S

_{A}= 1] yields K = 729; [A

_{6}, S

_{A}= 3/2] yields K = 1024; [A

_{4}, S

_{A}= 2] yields K = 625; and [A

_{4}, S

_{A}= 5/2] yields K = 1296. When we increase our limit to K ~ 5000, then [A

_{12}, S

_{A}= 1/2] yields K = 4096; [A

_{6}, S

_{A}= 1] yields K =2187; [A

_{6}, S

_{A}= 3/2] yields K = 4096; [A

_{5}, S

_{A}= 2] yields K = 3125; and [A

_{4}, S

_{A}= 5/2] yields K = 1296.

^{III}

_{4}}, has K = 65,536 magnetic states and can be handled by diagonalizing the largest spin block for J = 10 of the size n(J) = 171.

_{A}, of the centers in any order and any size and the topological matrix, T(A,B), which specifies the pairwise interactions. Everything else can be viewed as a black box prepared by the programmer. The only limitations are the memory and speed of the user’s (personal) computer.

_{N}] systems compared with closed finite cyclo-[A

_{N}] rings (N = 4–9, or 13) for spins S

_{A}= 1/2, 1, 3/2, 2, and 5/2. Section 5 deals with the modeling of selected convex polyhedrons [A

_{N}] (N = 4, 5, 6), for spins S

_{A}= 1/2, 1, 3/2, 2, and 5/2. Section 6 deals with real complexes of Mn(III), Mn(II), Fe(III), Co(II), Er(III), and Dy(III), which have already been published elsewhere [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52]. They are ordered in a way that allows for comparison and certain generalizations.

- Isotropic exchange constants are uniformly defined through the form $-{J}_{AB}({\overrightarrow{S}}_{A}\cdot {\overrightarrow{S}}_{B}){\hslash}^{-2}$.
- The angular momentum operators yield eigenvalues in units of the reduced Planck constant, $\hslash $, when operating on the corresponding wave function (ket).
- The Condon–Shortley phase convention is used together with the pseudo-standard phase system for irreducible tensor operators.
- It is assumed that the energy quantities, E (like ε, J, D), are in the form of the corresponding wavenumber; i.e., E/hc are provided in units of cm
^{−1}. - SI units are used consistently through the paper; χ
_{mol}[SI] = 4π × 10^{−6}χ_{mol}[cgs&emu]. - Fundamental physical constants (μ
_{0}, N_{A}, k_{B}, μ_{B}, $\hslash $) adopt their usual meaning. The reduced Curie constant ${C}_{0}={N}_{\mathrm{A}}{\mu}_{0}{\mu}_{\mathrm{B}}^{2}/{k}_{\mathrm{B}}$ = 4.7141997 × 10^{−6}K m^{3}mol^{−1}is met in the contribution. - The temperature evolution of the magnetic susceptibility is often displayed through the product function, χT, given in units of cgs&emu [cm
^{3}K mol^{−1}]. This old-fashioned representation can be equivalently expressed as χT/C_{0}. This dimensionless product function has some advantages as its values for Curie paramagnets (χ = C_{0}g^{2}S(S + 1)/3T with g = 2) are 1, 8/3, 5, 8, 35/3, 16, and 21 for S = 1/2 to 7/2. This quantity is additive unlike the effective magnetic moment, so it is more suitable for polynuclear systems. Conversion to non-SI units: χT[cgs&emu] = C_{0}/(4π × 10^{−6}) × (χT/C_{0}) = 0.3751 × χT/C_{0}. The conversion of the effective magnetic moment to a dimensionless product function is χT/C_{0}= (μ_{eff}^{2})/3 when μ_{eff}is given in the unit of the Bohr magneton, μ_{B}.

## 2. Methodology

#### 2.1. Spin Symmetry

_{A}+ 1, ν = S

_{max}− M, and $\mathrm{INT}(\nu /m)$ is the largest integer that is less than or equal to ν/m. For example, for the tetrad of S

_{A}= 1/2, the individual dimensions are K

_{−2}= K

_{+2}= 1, K

_{−1}= K

_{+1}= 4, and K

_{0}= 6, so K = 16. For non-equivalent centers, the formula is more complex:

_{max}. For example, for the tetrad of S

_{A}= 1/2, $n(0,0)=6-4=2$ and $n(1,1)=4-1=3$, whereas $n({S}_{\mathrm{max}},{S}_{\mathrm{max}})=(2,2)=1$ is trivial.

#### 2.2. Matrix Elements

_{3}, for the bra- and ket-vectors, together with the tensor rank of the added spins, k

_{3}= 1; and in the third row, there are intermediate spins of the bra-vector, ${\tilde{S}}_{3}^{\prime}={S}_{123}^{\prime}$; ket-vector, ${\tilde{S}}_{3}={S}_{123}$; and the intermediate rank of the operator, ${\tilde{k}}_{3}={k}_{123}$.

- (a)
- Bilinear isotropic exchange:

- (b)
- Zeeman operator:

#### 2.3. Density of State Function

#### 2.4. Implementation

_{max}is even or odd. This is a trivial task.

_{A}{1, 1, ½, ½}. To avoid handling half-integral values (1/2), they are all doubled: D

_{A}{2, 2, 1, 1}. (Calculations with integers are much faster than with real numbers.) Now the spins are summed: the minimum value is |D

_{i}− D

_{i+1}|, and the maximum is |D

_{i}+ D

_{i+1}|, with all values in between in step 2. For example, the range of D

_{12}is |D

_{1}− D

_{2}| = 0 to (D

_{1}+ D

_{2}) = 4, so D

_{12}= 0, 2, 4. The range of D

_{123}is |D

_{12}− D

_{3}| to (D

_{12}+ D

_{3}), which involves 1, 3, and 5; the range of D

_{1234}is |D

_{123}− D

_{4}| to (D

_{123}+ D

_{4}), which yields 0 (twice), 2 (four times), 4 (three times), and 6 (once). The scheme is shown in Table 2 and defines the “coupling history matrix”, hereafter, the CHM. All necessary information for the decoupling process is encoded in the CHM; i.e., it contains all intermediate spins.

_{A}= 1. With 4 centers, there are 6 pairwise interactions for which the tensor ranks of the involved centers, A

_{1}through A

_{4}(independent of spins), ${\widehat{T}}_{k=0}({\overrightarrow{S}}_{A}\otimes {\overrightarrow{S}}_{B})$, are provided in Table 3. The last assignment of the tensor ranks refers to the ranks of the intermediate operator, ${\tilde{k}}_{i}={k}_{12\dots i}$, for each pair (see also the explanation for Equation (27)).

_{i}= 1/2.

_{1}, D

_{12}, D

_{123},…., D

_{1…N}= 2S} or {S

_{1}, S

_{12}, S

_{123},…., S

_{1…N}= S}

- Define the topological matrix, T(A,B).
- Determine the total number of zero-field states, M; limit S
_{min}and S_{max}and the size of the matrices with the same spin dim(S). - For the final spin states, S, prepare the coupling history matrix: CHM = {D
_{1}, D_{12}, D_{123},…., D_{1…N}= 2S}. - For pairs of centers, prepare operator ranks, OR = {k
_{1},…, k_{N}}, and intermediate operator ranks, IOR = {${\tilde{k}}_{2}={k}_{12}$, …, ${\tilde{k}}_{N}={k}_{1\dots N}$}. - Open a loop over the molecular spins, S = S
_{min}to S_{max}, and fill matrix elements of the blocks for the same spin and all intermediate spins, ${M}_{{\alpha}^{\prime}{S}^{\prime},\alpha S}(A,B)$, for each relevant pair, {A,B}. The row and column indices of such a matrix use the set of intermediate spins contained in the CHM. - The final block, ${H}_{{\alpha}^{\prime}{S}^{\prime},\alpha S}$, is the sum of all relevant matrices, ${M}_{{\alpha}^{\prime}{S}^{\prime},\alpha S}(A,B)$, multiplied by a non-zero topological matrix, T(A,B), containing the current value of J(A,B).
- The final block is diagonalized (only eigenvalues are searched). The zero-field eigenvalues are enriched with a Zeeman term in the form of $\epsilon (S,B)={\epsilon}_{0}(S)+{\mu}_{\mathrm{B}}{g}_{\mathrm{eff}}B{M}_{S}$, where uniform g
_{eff}-factors occur. (This approximation is either a weakness or a strength of the whole procedure.) - Magnetic energy levels, $\epsilon (S,B)$, enter the statistical partition function, Z(B,T), from which the magnetization and susceptibility are calculated using Equations (33)–(37).
- The calculated susceptibility, χ
^{c}(B,T), and magnetization, M^{c}(B,T), together with the experimental points enter the error functional, F(B,T), which is processed by advanced minimization procedures such as simulated annealing or genetic algorithms to obtain an optimized set of magnetic parameters, J_{AB}and g_{eff}.

#### 2.5. Utilization of Symmetry

**G**, contains spatial operations, i.e., identity, E; rotation axes, C

_{n}reflection planes, σ

_{a}; inversion, i; and indirect rotations, S

_{n}. Elements of symmetry intersect in at least one point in space. In addition, the double group contains the “half-identity”, Q, which means a rotation by an angle of 2π while the identity, E, indicates a rotation by an angle of 4π.

**S**, is formed from all permutations between N-members of the group; their number is equal to N!. The symmetry group applies to many body systems and abstracts from the spatial views. Work with the symmetry group is described in detail elsewhere

_{N}- Spatial symmetry of atomic coordinates within the point group,
**G**; - Angular momentum symmetry within the fully rotational group in three dimensions,
**R**, and the special unitary group,_{3}**SU**in (2j + 1), dimensions, which contains 4j(j + 1) tensor operators ${T}_{q}^{(k)}$ for $1\le k\le 2j$ and $-k\le q\le k$;_{2j+1} - Permutation symmetry, which corresponds to permutations of individual particles (spins) within the symmetry group,
**S**._{N}

**S**, a key role is played by the partition, $\lambda =[{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{N}]$—the decomposition of the number, N, into natural numbers (0, 1, 2, …). Each partition defines classes and irreducible representations (IRs), Г

_{N}_{λ}, of the

**S**group. For a given partition, the dimension of the IRs in

_{N}**SU**(m = 2s +1 is the multiplicity) is given by the formula

_{m}**S**group, and then, the overall dimension is $K={\displaystyle \sum _{\lambda}n\{{\Gamma}_{\lambda}\in {S}_{{\rm N}}\}}\cdot d({\Gamma}_{\lambda}\in {S}_{{\rm N}})$. Now, the theory tells us exactly how the S-blocks in Table 1 can be further divided into blocks of lower dimensions; this is exemplified in Table 4 for four centers with the spin value of s = 2.

_{N}**S**, is isomorphous to the point group,

_{4}**T**(they have the same character table), which allows for the direct identification of relationships between their irreducible representations. This means that blocks of the given S can be further decomposed to blocks according to the IRs of the

_{d}**T**group. For example, spin-blocks are decomposed as follows:

_{d}_{1}, and T

_{2}, respectively.

**SU**, is that, in addition to the symmetry operations (permutations), there are rotations leading to new constraints on the objects (tensors). The subduction of

_{m}**R**into point groups,

_{3}**G**, is well known and has been reported in many sources.

**G**, can be generated as follows:

**G**. The superscript a in $\left|{\mathit{\Gamma}}_{j,\lambda}^{(a)}\right.\u232a$ distinguishes between repeated representations (it is an ordering number). Matrices of irreducible representations, ${[{\mathit{\Gamma}}_{j}^{}(R)]}_{\lambda \mu}^{}$, are tabulated elsewhere; only their diagonal elements refer to the projection operator. The transformation of matrix elements into a basis set of symmetry-adapted functions is performed using a formula:

- The basis set consists of uncoupled kets, i.e., $\left|I\right.\u232a=\left|\dots {S}_{A}{M}_{A}\dots \right.\u232a$; this approach is applicable to the general case, which includes other interactions besides isotropic exchange, such as asymmetric exchange, etc.
- The basis set is represented by coupled kets, $\left|I\right.\u232a=\left|{S}_{1},{S}_{2},{\tilde{S}}_{12},\dots ,{\tilde{S}}_{N-1},{S}_{N},SM\right.\u232a$; this is appropriate for isotropic exchange alone with a uniform Zeeman term (all g-factors are equivalent.

_{4}] system can be mapped within the point group,

**D**(h = 4), and the subgroup of the symmetry group,

_{2}**S**(whose dimension is $h=4!=24$), as seen in Table 5.

_{4}D_{2} (h = 4) | $\widehat{\mathit{E}}$ | ${\widehat{\mathit{C}}}_{2(\mathit{z})}^{}$ | ${\widehat{\mathit{C}}}_{2(\mathit{y})}^{}$ | ${\widehat{\mathit{C}}}_{2(\mathit{x})}^{}$ |
---|---|---|---|---|

${\mathit{\pi}}_{4}\subset {\mathbf{S}}_{4}$ | $\widehat{\mathit{P}}(1234)$ | $\widehat{\mathit{P}}(3412)$ | $\widehat{\mathit{P}}(4321)$ | $\widehat{\mathit{P}}(2143)$ |

A | +1 | +1 | +1 | +1 |

B_{1} | +1 | +1 | –1 | –1 |

B_{3} | +1 | –1 | –1 | +1 |

B_{2} | +1 | –1 | +1 | –1 |

^{a}Symmetry elements defined in Figure 1.

_{A}, in the trial kets

_{4}] system,

^{III}

_{8}(µ

_{4}-O)

_{4}(µ-pz)

_{12}Cl

_{4}] (No

**12**) will be discussed. In this system, one can identify three two-fold rotation axes, C

_{2}(z), C

_{2}(y), and C

_{2}(x), belonging to the symmetry point group,

**D**. This has four irreducible representations: E, B

_{2}_{1}, B

_{2}, and B

_{3}. Spin permutation symmetry uses a coupling scheme that is left invariant under the symmetry operations of the point group. This condition is fulfilled for the coupling scheme ${S}_{12}={S}_{1}+{S}_{2}$, ${S}_{34}={S}_{3}+{S}_{4}$, ${S}_{56}={S}_{5}+{S}_{6}$, ${S}_{78}={S}_{7}+{S}_{8}$, ${S}_{1234}={S}_{12}+{S}_{34}$, ${S}_{5678}={S}_{56}+{S}_{78}$, $S={S}_{1234}+{S}_{5678}$.The required coupling coefficients are shown in Table 6.

_{1}states is

## 3. Modeling of Finite Chains

_{AB}, appear just above the diagonal, as in Equation (46). Although the coupling constants are different (at least those at the ends of the chain), the approximation of the uniform coupling constants and uniform g-factors can be accepted. In general, however, it is not necessary to have uniform spin centers as, for example, in the catena-[Mn

^{II}

_{2}Mn

^{III}

_{2}(dipic)

_{6}(H

_{2}O)

_{4}] with the chain Mn

^{III}-Mn

^{II}-Mn

^{II}-Mn

^{III}[34].

## 4. Modeling of Finite Rings

## 5. Modeling of Convex Polyhedra

_{4}], [A

_{5}], and [A

_{6}] systems is presented in Table 17, Table 18 and Table 19. The following conditions were used: all, g = 2.0; B

_{0}= 10

^{−6}T; and J = −1 cm

^{−1}. The situation with different negative J values can be covered by a simple rescaling of the energy axis.

_{6}], where all six vertices are joined by 15 exchange-coupling constants, which also provide a highly degenerate energy spectrum, ε = S(S + 1). However, often, a diamagnetic atom, X, sits in the very center of the octahedron, [A

_{6}X], so the three trans-AB linkages through X can be neglected, and the remaining 12 cis-AB contacts cause the degeneracy to be lifted.

_{4}A], is a special case, which also provides a rotational band thanks to 10 J-constants.

## 6. Exchange Interaction in Real Clusters

_{max}= 60/2 for the cluster {Dy

_{4}}. The decomposition of the zero-field Hamiltonian matrix into blocks of smaller sizes is presented in Table 20.

#### 6.1. Mn Complexes

^{II}

_{2}Mn

^{III}

_{2}(dipic)

_{6}(H

_{2}O)

_{4}] (

**1**) has a core, {Mn

^{II}Mn

^{III}Mn

^{III}Mn

^{II}}, and given its chain structure, two coupling constants occur: J

_{t}(terminal) and J

_{i}(inner), with separation, Mn

^{III}-Mn

^{III}= 3.827 Å, and two angles, Mn

^{III}-O-Mn

^{III}= 109.6°. The spin Hamiltonian, along with the corresponding topological matrix selecting the exchange coupling constants, is contained in Figure 2, together with the topological function that defines the pairwise interactions. The experimental DC magnetic functions (the temperature dependence of the product function χT and the field dependence of the magnetization per formula unit) are also included and superimposed by the fitted data.

^{II}

_{2}Mn

^{III}

_{2}(HBuDea)

_{2}(BuDea)

_{2}(DMBA)

_{4}] (

**2**) with a {Mn

^{II}(Mn

^{III}Mn

^{III})Mn

^{II}} core is somehow analogous to

**1**, but with a different coupling path: the peripheral Mn

^{II}centers are coupled to both inner Mn

^{III}ones. The crystallographic Mn1 centers related to Mn

^{III}are labeled as 1 and 1′, while the Mn2 centers corresponding to Mn

^{II}are labeled as 2 and 2′ centers. Then, J

_{i}refers to the inner diad J(Mn

^{III}-Mn

^{III}) with separation, Mn1-Mn1 = 3.17 Å; J

_{a}and J

_{b}correspond to two different pairs, Mn

^{II}-Mn

^{III}, for separations Mn2-Mn1 = 3.24 and 3.41 Å, respectively. The magnetic data are shown in Figure 3.

^{III}–O–Mn

^{III}superexchange pathways with a bond angle of 98° transmit an exchange coupling of a ferromagnetic nature with positive J

_{i}. The slightly positive J

_{a}reflects two superexchange pathways with Mn

^{III}–O–Mn

^{II}bond angles of 89° and 107°. The slightly negative J

_{b}relates to two superexchange pathways with Mn

^{III}–O–Mn

^{II}bond angles of 103° and 109°. A simplified model that merges J

_{a}= J

_{b}provides g

_{eff}= 1.95, J

_{i}= 25.0 cm

^{−1}, J

_{a}= 0.97 cm

^{−1}, χ

_{TIM}= 9 × 10

^{−9}m

^{3}mol

^{−1}; R(χ) = 0.041, R(M) = 0.054. The spectrum of energy levels for

**2**is completely different from

**1**, as the ferromagnetic interaction between the inner pair of Mn

^{III}centers, J

_{i}>> 0, now dominates (Figure 3).

^{II}

_{4}(abpt)

_{4}(μ

_{1,1}-N

_{3})

_{8}(H

_{2}O)

_{2}] (

**3**) is a typical chain of uniform spins whose Hamiltonian contains two different coupling constants J

_{t}(terminal) and J

_{i}(internal). The magnetic functions presented in Figure 4 were fitted assuming the ferromagnetic exchange; the ground state, S

_{0}= S

_{max}= 10, is also confirmed by the saturation of the magnetization.

^{II}

_{3}Cr

^{III}

_{4}(NCS)

_{6}(Htea)

_{6}] (

**4**) with a {Mn

^{II}

_{3}Cr

^{III}

_{4}} core has the shape of a plaquet, where the trinuclear chain Mn

^{III}…Mn

^{III}…Mn

^{III}is decorated with two pairs of Cr

^{III}centers. An appropriate Hamiltonian, together with the topological matrix for the chosen numbering, is shown in Figure 5. The energy spectrum of

**4**has an “irregular structure” with a ground state of neither the lowest nor the highest spin.

^{II}(poxap)Mn

^{II}(ac)

_{4}Mn

^{II}(poxap)] (

**5**) show that the magnetic susceptibility gradually decreases upon cooling but then rises abruptly (Figure 6). A simultaneous fitting of susceptibility and magnetization for

**5**yielded J = −4.56 cm

^{−1}, g = 1.96, D = −0.02 cm

^{−1}, and zj/hc = +0.054 cm

^{−1}. This is also the case for the irregular energy spectrum.

^{III}

_{8}(μ

_{3}-O)

_{4}(μ-pz)

_{8}(μ-OMe)

_{4}(OMe)

_{4}] (

**6**) with a {Mn

^{III}

_{8}} core has a complex architecture. The coupling path involves three exchange constants (Figure 7).

#### 6.2. Fe(III) Complexes

^{III}

_{4}(μ

_{4}-O)

_{4}Mn

^{III}

_{4}(L)

_{8}(DMF)

_{4}] ·2DMF (

**7**) contains an {Fe

^{III}

_{4}Mn

^{III}

_{4}} core. Its architecture is represented by the central Fe

^{III}

_{4}unit arranged in a tetrahedron, which is further decorated by four peripheral Mn

^{III}centers (a tetrahedron within a tetrahedron); this is somewhat similar to

**5**. There might be two distinct coupling pathways J

_{a}(Fe

^{III}-Fe

^{III}) and J

_{b}(Fe

^{III}-Mn

^{III}) of an antiferromagnetic nature (Figure 8). The averaged bond angles are Fe-O-Fe = 103° and Mn-O-Fe = 113°.

^{III}

_{6}Fe

^{III}

_{6}(HL)

_{2}(L)

_{10}(μ-Cl)

_{2}]·8DMF (

**8**) contains a magnetoactive {Fe

^{III}

_{6}} chain decorated with six diamagnetic Co

^{III}centers. This means there are two coupling constants: J

_{t}(terminal) and J

_{i}(internal). The profile of the product function suggests antiferromagnetic exchange (Figure 9).

^{II}(CN)

_{6}{Fe

^{III}(salpet)}

_{6}]Cl

_{2}complex (

**9**) contains a central unit {Fe

^{II}(CN)

_{6}} with six {Fe

^{III}(salpet)}

^{+}moieties attached. The Fe

^{II}center in a strong crystal field is non-magnetic. Six S

_{A}= 5/2 centers provide the resulting spins, S = 0 through 15. However,

**9**shows a thermally induced spin crossover, as shown in Figure 10, probably from three low-spin plus three high-spin states. The susceptibility upon cooling only increases and does not show the maximum typical of the S

_{0}= 0 ground state. However, this could be masked by paramagnetic impurity because of S = 5/2 mononuclear fragments.

_{a}, L

_{b}, and L

_{c}, of which [Fe

^{II}(CN)

_{6}{Fe

^{III}(L

_{b})}

_{6}]Cl

_{2}·H

_{2}O (

**10b**) is analyzed below. The core {Fe

^{III}

_{6}Fe

^{II}} of the complex is identical to

**9**, but the complex is high-spin over the entire temperature range (Figure 11). The first model considers fifteen J constants; the second considers twelve J

_{c}(cis)-constants and omits three J(trans).

**D**point group of symmetry. Consequently, the entire interaction matrix is divided according to irreducible representations into blocks A

_{6}_{1}(K = 4291), A

_{2}(K = 3535), B1 (K = 4145), B

_{2}(K = 3605), E

_{1}(K = 15,470), and E

_{2}(K = 15,610).

^{III}

_{7}O

_{3}(O

_{2}CPh)

_{9}(mda)

_{3}(H

_{2}O)] (

**11**) was prepared and investigated in depth elsewhere [44]. Its architecture suggests a combination of a ring and a star with magnetic data indicating antiferromagnetic coupling. Magnetostructural correlations (MSCs) predict that there are four coupling constants in play, J

_{a}= −45.0, J

_{b}= −12.6, J

_{c}= −6.2, and J

_{d}= −30.0 cm

^{−1}(for notation −2J between spins), yielding the ground state S

_{0}= 5/2. To confirm this prediction, a spin-Hamiltonian with empirical (MSC) coupling constants (rescaled to −J notation) was worked out, and the predicted magnetic functions are plotted in Figure 12. The course of the product function matches the experimental findings [44]. The resulting energy levels confirm the ground state, S

_{0}= 5/2 (not 1/2), separated from the lowest excited state, S = 7/2, by ΔE = 157 cm

^{−1}

^{III}

_{8}(µ

_{4}-O)

_{4}(µ-pz)

_{12}Cl

_{4}] (

**12**) has a tetrahedro@tetrahedro-{Fe

^{III}

_{8}} core. Susceptibility data are shown in Figure 13 and indicate massive antiferromagnetic coupling. The topological function suitable for the data fitting contains two coupling constants, J

_{i}(inner tetrahedron, 6×) and J

_{o}(outer 12×). There is an obstacle—the large size of the largest block n(S = 5) = 16576. The problem of diagonalization of such matrices was avoided by using the symmetry point group,

**D**.

_{2}_{i}= −2.1, J

_{o}= −50.6 cm

^{−1}, and g = 2.0 (fixed); the energy spectrum is drawn in Figure 13. Only a very limited part of the energy spectrum is thermally populated; this explains the temperature evolution of the product function. The small value of J

_{i}refers to six coupling pathways inside the inner tetrahedron with an average angle of 12 bonds: Fe-O-Fe = 96.7°. The bond angles of Fe

_{i}-O-Fe

_{o}average 119.7°, rationalizing the much more negative J

_{o}.

^{III}

_{10}(bdtbpza)

_{10}(MeO)

_{20}] (

**13**) is a typical ring system (wheel) that has only one J-constant between neighboring members. Its magnetic functions are presented in Figure 14, indicating an exchange coupling of an antiferromagnetic nature. Fitting 10-membered Fe

^{III}systems is an unrealistic task because there are M = 4,395,456 zero-field states, and the biggest S-block has a dimension of n(S = 5) = 484,155. Therefore, we restricted ourselves to the cyclo-{Fe

^{III}

_{8}} model with M = 135,954 and n(S = 5) = 16,576, hoping it would work satisfactorily. The construction of the topological matrix and the spin-Hamiltonian are straightforward; the fitting procedure yielded J = −8.58 cm

^{−1}, g = 2.0; x

_{PI}= 0.0049. For a more simplified cyclo-{Fe

^{III}

_{6}} model with M = 4332 and n(S = 4) = 609, the calculated parameters were J = −8.64 cm

^{−1}, x

_{PI}= 0.0064. The energy levels for a such model are displayed in Figure 14.

#### 6.3. Co(II) Complexes

^{II}

_{6}Co

^{III}(thmp)

_{2}(acac)

_{6}(ada)

_{3}] (

**14**) with a {Co

^{II}

_{6}Co

^{III}} core is a ring system indicating only a single exchange coupling constant. However, there is some asymmetry in the Co…Co separations (3.034, 3.145, 3.032, 3.169, 3.174, 3.038 Å), and therefore, two different J-constants were considered (Figure 15).

**14**(see Equation (11)). The magnetic data are shown in Figure 15, and the fitting procedure yielded J

_{a}= 0.95, J

_{b}= 5.11 cm

^{−1}, g = 2.65, and D = 79 cm

^{−1}. The modeled energy levels for D = 0 are plotted in Figure 15. (The D-value makes the spin no longer a good quantum number, and the classification of zero-field states using spin becomes meaningless).

^{II}

_{11}Co

^{III}

_{2}(thmp)

_{4}(Me

_{3}CCOO)

_{4}(acac)

_{6}(OH)

_{4}(H

_{2}O)

_{4}] (Me

_{3}CCOO)

_{2}·H

_{2}O (

**15**) with a {Co

^{II}

_{11}Co

^{III}

_{2}} core is the result of the fusion of two ring systems. The magnetic functions are shown in Figure 16. Treating the eleven S = 3/2 centers is a serious problem because there are now K = 4,194,304 energy levels in play. The task was simplified by considering a ring of only seven cyclo-[A

_{7}] centers with K = 16,384 levels in the basis of uncoupled functions. The use of permutation symmetry made it possible to divide the entire matrix into subblocks. A new set of symmetry-adapted spin basis sets was created using the

**D**point group. The total interaction matrix is split into submatrices A

_{7}_{1}(N = 1300), A

_{2}(N = 1044), E

_{1}(N = 4680), E

_{2}(N = 4680), and E

_{3}(N = 4680). Then, a fitting procedure based on both temperature and field-dependent magnetic data resulted in J = 3.34 cm

^{−1}, D = 63.8 cm

^{−1}, and g = 2.64. The increased value of the g-factor relative to the free-electron, g

_{e}= 2.0, reflects the presence of the orbital angular momentum; it is manifested by a magnetization per formula unit that exceeds the value of M

_{1}~ 21 μ

_{B}. The energy levels in the approximation of the [Co

_{7}] ring are drawn in Figure 16.

^{II}

_{9}Co

^{III}

_{3}(μ

_{3}-O)

_{3}-(μ

_{1,1,1}-N

_{3})(μ

_{1,1}-N

_{3})

_{3}(μ

_{3}-L)

_{9}(μ-L)

_{6}](ClO

_{4})

_{2}·H

_{3}tea·9.5H

_{2}O (

**16**) contains a {Co

^{II}

_{9}Co

^{III}

_{3}} core and it has the architecture of a plaquet. The magnetic data are plotted in Figure 17, which shows the hook at the lowest temperature of the product function. Three inner Co

^{II}centers forming a triangle are coupled by double bridges with angles i{90.6(N), 103.5(O)}, and the inner-outer paths include double bridges of o{100.2, 101.2} and o{94.1, 104.5}. S-blocking offered the highest matrix to be diagonalized at a size of n(S = 7/2) = 5300, and the fitting procedure yielded J

_{i}= −10.3 cm

^{−1}, J

_{o}= +0.98 cm

^{−1}, and g = 2.58. The zero-field splitting was omitted, which prevents a reliable reconstruction of the low-temperature data.

#### 6.4. Ln(III) Complexes

_{z}—g

_{x}

_{y}). Because of the addition of spin and angular momenta, the total angular momentum, J = L + S, is in play, giving rise to spin–orbit multiplets. The crystal field is a minor effect because the f-orbitals are effectively screened against point charges generated by the ligands. Stevens operators, ${\widehat{O}}_{k}^{q}$, however, imitate the effect of a crystal field, where the contributions cover several ${B}_{k}^{q}{\widehat{O}}_{k}^{q}$ members (k—tensor rank, q—component). These operators cause J-multiplets to be split and mixed into zero-field-slitting levels. The omission of these factors has an important effect on the reconstruction of the magnetization at low temperatures: the calculated magnetization is much higher than the experimental data.

^{III}

_{3}Cl(L)

_{3}(OH)

_{2}(H

_{2}O)

_{5}]Cl

_{3}(

**17**) has an {Er

^{III}

_{3}} core referring to an isosceles triangle (there is one additional Er-Cl bond). The Er-Er distances are 3.505, 3.509, and 3.478 Å, indicating two coupling constants. The bond angles along the path a{97, 99} and along b{95, 98} deg predict that J

_{a}and J

_{b}will be small. The product function gradually decreases upon cooling, reflecting the prevailing antiferromagnetic exchange (Figure 18). The susceptibility data were fitted using a Hamiltonian that includes the total angular momentum, J = S + L, instead of the net spin, where each Er(III) center offers J

_{A}= 15/2 (the free-atom multiplet is

^{4}I

_{15/2}with g

_{J}= 6/5). The molecular J-value varies between 1/2 and 45/2, yielding an irregular energy spectrum; the ground state is J

_{0}= 21/2 (Figure 18). When only the ground state is populated at a sufficiently low temperature, the estimated magnetization will be M

_{0}~ gJ

_{0}~ 12.3 μ

_{B}.

^{III}

_{4}(L)

_{4}(μ

_{2}-OH)

_{2}Cl

_{4}]Cl

_{2}·EtOH (

**18**) with a {Dy

^{III}

_{4}} core is topologically analogous to

**2**(rhombus). It has three coupling paths with J

_{a}{107, 112} and J

_{i}{109} for an inner diad and J

_{c}{100, 99, 80-Cl}, where data in parentheses refer to bond angles. Upon cooling, the product function remains almost constant and then decreases (Figure 19). The energy spectrum ranges from J

_{min}= 0 to J

_{max}= 4·(15/2) = 60/2; the ground state is J

_{0}= 0 (Figure 19). The leading coupling is an inner diad, J

_{i}< 0, and is responsible for the overall antiferromagnetic exchange. This is the main difference compared with the

**2a**with an analogous topology.

^{III}

_{4}(HL)

_{4}(H

_{2}L)

_{2}(NO

_{3})

_{4}](NO

_{3})

_{4}·4MeOH (

**19**) can be considered a four-membered ring with a single J-constant. The Dy-O-Dy bond angles are 110—115°, so a negative J is expected. The magnetic data are shown in Figure 20; they were reconstructed with J = −0.03 cm

^{−1}[51]. The fitting procedure yielded J = −0.041 cm

^{−1}, g = 1.27.

_{2}O)

_{6}Dy

^{III}

_{2}(μ

_{2}-L)

_{2}(μ

_{3}-O)

_{4}Cu

^{II}

_{5}(μ

_{2}-Cl)

_{2}] (

**20**) with a plaquet architecture contains two Dy

^{III}centers that are non-coordinated: two tetracoordinate Cu

^{II}, two pentacoordinate Cu

^{II}, and one hexacoordinate Cu

^{II}center. The magnetic functions are shown in Figure 21 with an unusual course of product function: it gradually decreases during cooling but then rises sharply. The width of the energy spectrum is J = 1/2 to 35/2, and the ground state is J

_{0}= 25/2. The irregular shape of the energy spectrum causes the product function to increase upon cooling since the magnetically productive ground state is then increasingly more populated; this is regardless of antiferromagnetic exchange (J

_{a}and J

_{b}< 0) when one would (erroneously) expect J

_{0}= 1/2.

## 7. Conclusions

^{III}

_{8}} system with spins with S

_{A}= 5/2, the largest block for S = 5 has a dimension of 16,576.

- A number of magnetic centers, N, and spins, S
_{A}, on individual centers in any order and size; - The topological matrix T(A,B), which defines the coupling path; this contains a trial set of exchange-coupling constants, J(A,B), that will be optimized; their number is less or equal to N(N—1)/2;
- The value of the g-factor, which must be uniform (g
_{eff}), to correctly exploit the blocking of the Hamiltonian matrix according to molecular spin.

^{o}, M

^{c}; χ

^{o}, χ

^{c}), composed of observed (o) and calculated (c) magnetic functions. Susceptibility and magnetization data are optimized simultaneously. This is a very strict requirement: sometimes excellent fits are obtained only for the susceptibility and magnetization data.

_{A}), zero-field splitting (D

_{A}, E

_{A}), asymmetric exchange (D

_{AB}, E

_{AB}), and antisymmetric exchange (

**a**

_{AB}) cause the key advantage of the blocking to collapse.

_{0}= 0 or ½. This is not true in general, as there are systems that have an irregular energy spectrum where the ground state falls between S

_{min}and S

_{max}. This is always the case for star-like architectures and odd catena-[A

_{N}] chains for N = 3, 5, 7, 9, … S = 3/2, and 5/2.

_{N}] system, the ground state is four-fold degenerate (two Kramers doublets with S = 1/2) if Ns is a half-integer. If Ns is an integer, the ground state is non-degenerate (S = 0).

_{9}, s = 1/2], the ground state is doubly degenerate S

_{0}= 1/2 (twice). The ground state of catena-[A

_{9}, s = 1/2] is S

_{0}= 1/2 (×1), and the first excited state, S

_{1}= 1/2 (×1), lies at an energy of −0.75J.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Structure and magnetic functions for

**1**. Magnetic susceptibility is expressed as a dimensionless product function, χT/C

_{0}; magnetization per formula unit in Bohr magnetons. According to [34].

A_{N} System | Magnetic States, K | Zero-Field States, M | Dimension n(S) from the Lowest Spin, S_{min} = 0 or 1/2, to the Highest Spin, S_{max} = N·S_{A} |
---|---|---|---|

S_{A} = ½ | |||

A_{3} | 8 | 3 | 2, 1 |

A_{4} | 16 | 6 | 2, 3, 1 |

A_{5} | 32 | 10 | 5, 4, 1 |

A_{6} | 64 | 20 | 5, 9, 5, 1 |

A_{7} | 128 | 35 | 14, 14, 6, 1 |

A_{8} | 256 | 70 | 14, 28, 20, 7, 1 |

A_{9} | 512 | 126 | 42, 48, 27, 8, 1 |

A_{10} | 1024 | 252 | 42, 90, 75, 35, 9, 1 |

A_{11} | 2048 | 462 | 132, 165, 110, 44, 10, 1 |

A_{12} | 4096 | 924 | 132, 297, 275, 154, 54, 11,1 |

A_{13} | 8192 | 1716 | 429, 572, 429, 208, 65, 12, 1 |

A_{14} | 16,384 | 3432 | 429, 1001, 1001, 637, 273, 77, 13, 1 |

A_{15} | 32,768 | 6435 | 1430, 2002, 1638, 910, 350, 90, 14 1 |

S_{A} = 1 | |||

A_{3} | 27 | 7 | 1, 3, 2, 1 |

A_{4} | 81 | 19 | 3, 6, 6, 3, 1 |

A_{5} | 243 | 51 | 6, 15, 15, 10, 4, 1 |

A_{6} | 729 | 141 | 15, 36, 40, 29, 15, 5, 1 |

A_{7} | 2187 | 393 | 36, 91, 105, 84, 49, 21, 6, 1 |

A_{8} | 6561 | 1107 | 91, 232, 280, 238, 154, 76, 28, 7, 1 |

A_{9} | 19,683 | 3139 | 232, 603, 750, 672, 468, 258, 111, 36, 8, 1 |

A_{10} | 59,049 | 8954 | 603, 1585, 2025, 1890, 1398, 837, 405, 155, 45, 9, 1 |

S_{A} = 3/2 | |||

A_{3} | 64 | 12 | 2, 4, 3, 2, 1 |

A_{4} | 256 | 44 | 4, 9, 11, 10, 6, 3, 1 |

A_{5} | 1024 | 155 | 20, 34, 36, 30, 20, 10, 4, 1 |

A_{6} | 4096 | 580 | 34, 90, 120, 120, 96, 64, 35, 15, 5, 1 |

A_{7} | 16,384 | 2128 | 210, 364, 426, 400, 315, 210, 119, 56, 21, 6, 1 |

A_{8} | 65,536 | 8092 | 364, 1000, 1400, 1505, 1351, 1044, 700, 406, 202, 84, 28, 7, 1 |

A_{9} | 262,144 | 30,276 | 2400, 4269, 5256, 5300, 4600, 3501, 2352, 1392, 720, 321, 120, 36, 8, 1 |

A_{10} | 1,048,576 | 116,304 | 4269, 11,925, 17,225, 19,425, 18,657, 15,753, 11,845, 7965, 4785, 2553, 1197, 485, 165, 45, 9, 1 |

S_{A} = 2 | |||

A_{3} | 125 | 19 | 1, 3, 5, 4, 3, 2, 1 |

A_{4} | 625 | 85 | 5, 12, 16, 17, 15, 10, 6, 3, 1 |

A_{5} | 3125 | 381 | 16, 45, 65, 70, 64, 51, 35, 20, 10, 4, 1 |

A_{6} | 15,625 | 1751 | 65, 180, 260, 295, 285, 240, 180, 120, 79, 35, 15, 5, 1 |

A_{7} | 78,125 | 8135 | 260, 735, 1085, 1260, 1260, 1120, 895, 645, 420, 245, 126, 56, 21, 6, 1 |

A_{8} | 390,625 | 38,165 | 1085, 3080, 4600, 5460, 5620, 5180, 4340, 3325, 2331, 1492, 868, 454, 210, 84, 28, 7, 1 |

A_{9} | 1,953,125 | 180,325 | 4600, 13,140, 19,845, 23,940, 25,200, 23,925, 20,796, 16,668, 12,356. 8470, 5355, 3108, 1644, 783, 330, 120, 36, 8, 1 |

A_{10} | 9,765,625 | 856,945 | 19,845, 56,925, 86,725, 106,050, 113,706, 110,529, 98,945, 82,215, 63,645, 45,957, 30,933, 19,360, 11,220, 5985, 2913, 1277, 495, 165, 45, 9, 1 |

S_{A} = 5/2 | |||

A_{3} | 216 | 27 | 2, 4, 6, 5, 4, 3, 2, 1 |

A_{4} | 1296 | 146 | 6, 15, 21, 24, 24, 21, 15, 10, 6, 3, 1 |

A_{5} | 7776 | 780 | 45, 84, 111, 120, 115, 100, 79, 56, 35, 20, 10, 4, 1 |

A_{6} | 46,656 | 4332 | 111, 315, 475, 575, 609, 581, 505, 405, 300, 204, 126, 70, 35, 15, 5, 1 |

A_{7} | 279,936 | 24,017 | 1050, 1974, 2666, 3060, 3150, 2975, 2604, 2121, 1610, 1140, 750, 455, 252, 126, 56, 21, 6, 1 |

A_{8} | 1,679,616 | 135,954 | 2666, 7700, 11,900, 14,875, 16,429, 16,576, 15,520, 13,600, 11,200, 8680, 6328, 4333, 2779, 1660, 916, 462, 210, 84, 28, 7, 1 |

A_{9} | 10,077,696 | 767,394 | 26,775, 50,904, 70,146, 83,000, 88,900, 88,200, 82,005, 71,904, 59,661, 46,920, 34,980, 24,696, 16,478, 10,360, 6111, 3360, 1707, 792, 330, 120, 36, 8, 1 |

A_{10} | 60,466,176 | 4,395,456 | 70,146, 204,050, 319,725, 407,925, 463,155, 484,155, 473,670, 437,590, 383,670, 320,166, 254,639, 193,095, 139,545, 95,985, 62,712, 38,808, 22,660, 12,420, 6345, 2993, 1287, 495, 165, 45, 9, 1 |

^{a}For the N-spins, s = 1/2: $n(S)=(2S+1)\cdot N!/[(N/2+S+1)!(N/2-S)!]$. Size of the maximum block is in bold type.

**Table 2.**Scheme for the addition of doubled spins D

_{A}{2, 2, 1, 1} and D

_{A}{1, 2, 2, 1}, yielding the coupling history matrix: CHM = {D

_{1}, D

_{12}, D

_{123}, …, D

_{1…N}= 2S}

^{a}.

Coupling Scheme 1 | Coupling Scheme 2 | |||||||
---|---|---|---|---|---|---|---|---|

D_{A} | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 1 |

State | D_{1} | D_{12} | D_{123} | D_{1234} = 2S | D_{1} | D_{12} | D_{123} | D_{1234} = 2S |

1 | 2 | 0 | 1 | 0 | 1 | 1 | 1 | 0 |

2 | 2 | 0 | 1 | 2 | 1 | 1 | 1 | 2 |

3 | 2 | 2 | 1 | 0 | 1 | 1 | 3 | 2 |

4 | 2 | 2 | 1 | 2 | 1 | 1 | 3 | 4 |

5 | 2 | 2 | 3 | 2 | 1 | 3 | 1 | 0 |

6 | 2 | 2 | 3 | 4 | 1 | 3 | 1 | 2 |

7 | 2 | 4 | 3 | 2 | 1 | 3 | 3 | 2 |

8 | 2 | 4 | 3 | 4 | 1 | 3 | 3 | 4 |

9 | 2 | 4 | 5 | 4 | 1 | 3 | 5 | 4 |

10 | 2 | 4 | 5 | 6 | 1 | 3 | 5 | 6 |

^{a}${\tilde{S}}_{i}={S}_{12\dots i}$, ${\tilde{D}}_{i}={D}_{12\dots i}=2{S}_{12\dots i}$; D

_{1234}/2 is the final spin state S

_{1234}= S. In total, there are M = 10 zero-field states.

**Table 3.**Tensor ranks for spins (vectors of 1st rank) occurring in the scalar products ${\widehat{T}}_{k=0}({\overrightarrow{S}}_{A}\otimes {\overrightarrow{S}}_{B})$

^{a}.

Operator Ranks, OR | Intermediate Operator Ranks, IOR | ||||||
---|---|---|---|---|---|---|---|

Pair A, B | k_{1} | k_{2} | k_{3} | k_{4} | ${\tilde{\mathit{k}}}_{2}={\mathit{k}}_{12}$ | ${\tilde{\mathit{k}}}_{3}={\mathit{k}}_{123}$ | ${\tilde{\mathit{k}}}_{4}={\mathit{k}}_{1234}=\mathit{k}$ |

1, 2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

1, 3 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |

2, 3 | 0 | 1 | 1 | 0 | 1 | 0 | 0 |

1, 4 | 1 | 0 | 0 | 1 | 1 | 1 | 0 |

2, 4 | 0 | 1 | 0 | 1 | 1 | 1 | 0 |

3, 4 | 0 | 0 | 1 | 1 | 0 | 1 | 0 |

^{a}The data printed in bold represent the example discussed below.

Partition | Young Diagram | IR ^{a}Г _{λ}(d) | Dimension n × d | Spin, S, in R_{3} ^{b}0–8 | Dimension of Blocks | Reduced Blocks Free of Projections ^{c} | IR T _{d} |
---|---|---|---|---|---|---|---|

[4000] = [4] | Г_{1}(1) | 70 × 1 = 70 | 0, 2^{2}, 4^{2}, 5, 6, 8 | 1, 10, 18, 11, 13, 17 = 70 | 1, 2, 2, 1, 1, 1 | A_{1} | |

[1111] = [14] | Г_{2}(1) | 5 × 1 | 2 | 5 | 1 | A_{2} | |

[2200] = [22] | Г_{3}(2) | 50 × 2 = 100 | 0^{2}, 2^{2}, 3, 4^{2}, 6 | (2, 10, 7, 18, 13) = 50 × 2 | (2, 2, 1, 2, 1) × 2 | E | |

[3100] | Г_{4}(3) | 105 × 3 = 315 | 1^{2}, 2^{2}, 3^{3}, 4^{2}, 5^{2}, 6, 7 | (6, 10, 21, 18, 22, 13, 15) = 105 × 3 | (2, 2, 3, 2, 2, 1, 1) × 3 | T_{2} | |

[2110] = [212] | Г_{5}(3) | 45 × 3 = 135 | 1^{2}, 2, 3^{2}, 4, 5 | (6, 5, 14, 9, 11) = 45 × 3 | (2, 1, 2, 1, 1) × 3 | T_{1} | |

sum | K = 625 magnetic states | 119 | K = 625 magnetic states | 85 zero-field states |

^{a}Г

_{1}—one-dimensional fully symmetric representations; Г

_{2}—one-dimensional fully antisymmetric.

^{b}In this special notation, the exponent denotes multiple occurrences of a given spin; e.g., 4

^{2}means S = 4 twice.

^{c}The reduced block with the maximum dimension represents a 9 × 9 matrix for Г

_{4}and S = 3.

Symmetry Operation | E | C_{2}(z) | C_{2}(x) | C_{2}(y) |
---|---|---|---|---|

Permutation | O(12345678) | O(21436587) | O(34128765) | O(43217856) |

Coupling of centers | <1,2,12> | <2,1,12> | <3,4,34> | <4,3,34> |

<3,4,34> | <4,3,34> | <1,2,12> | <2,1,12> | |

<5,6,56> | <6,5,56> | <8,7,78> | <7,8,78> | |

<7,8,78> | <8,7,78> | <6,5,56> | <5,6,56> | |

Coupling of diads | <12,34,1234> | <12,34,1234> | <34,12,1234> | <34,12,1234> |

<56,78,5678> | <56,78,5678> | <78,56,5678> | <78,56,5678> | |

Coupling of tetrads | <1234,5678,S> | <1234,5678,S> | <1234,5678,S> | <1234,5678,S> |

S | A_{1} | B_{1} | B_{2} | B_{3} | Total Number |
---|---|---|---|---|---|

0 | 776 | 630 | 630 | 630 | 2666 |

1 | 1820 | 1960 | 1960 | 1960 | 7700 |

2 | 3080 | 2940 | 2940 | 2940 | 11,900 |

3 | 3625 | 3750 | 3750 | 3750 | 14,875 |

4 | 4201 | 4076 | 4076 | 4076 | 16,429 |

5 | 4066 | 4170 | 4170 | 4170 | 16,576 |

6 | 3958 | 3854 | 3854 | 3854 | 15,520 |

7 | 3340 | 3420 | 3420 | 3420 | 13,600 |

8 | 2860 | 2780 | 2780 | 2780 | 11,200 |

9 | 2128 | 2184 | 2184 | 2184 | 8680 |

10 | 1624 | 1568 | 1568 | 1568 | 6328 |

11 | 1057 | 1092 | 1092 | 1092 | 4333 |

12 | 721 | 686 | 686 | 686 | 2779 |

13 | 400 | 420 | 420 | 420 | 1660 |

14 | 244 | 224 | 224 | 224 | 916 |

15 | 108 | 118 | 118 | 118 | 462 |

16 | 60 | 50 | 50 | 50 | 210 |

17 | 18 | 22 | 22 | 22 | 84 |

18 | 10 | 6 | 6 | 6 | 28 |

19 | 1 | 2 | 2 | 2 | 7 |

20 | 1 | 0 | 0 | 0 | 1 |

catena-[A_{4}], J_{n}(3×) | catena-[A_{5}], J_{n}(4×) | catena-[A_{6}], J_{n}(5×) |

catena-[A_{7}], J_{n}(6×) | catena-[A_{8}], J_{n}(7×) | catena-[A_{9}], J_{n}(8×) |

catena-[A_{10}], J_{n}(9×) | catena-[A_{11}], J_{n}(10×) | catena-[A_{12}], J_{n}(11×) |

^{a}These are true (open) chains; no cyclic boundary has been applied.

catena-[A_{4}], J_{n}(3×) | catena-[A_{5}], J_{n}(4×) | catena-[A_{6}], J_{n}(5×) |

catena-[A_{7}], J_{n}(6×) | catena-[A_{8}], J_{n}(7×) | catena-[A_{9}], J_{n}(8×) |

^{a}Odd-member chains (e.g., catena-[A

_{3}], catena-[A

_{5}], catena-[A

_{7}], catena-[A

_{9}]) have an irregular energy spectrum; i.e., S

_{0}= 1 is the ground state irrespective of the antiferromagnetic exchange.

catena-[A_{4}], s = 3/2, J_{n}(3×) | catena-[A_{4}], s = 2, J_{n}(3×) | catena-[A_{4}], s = 5/2, J_{n}(3×) |

catena-[A_{5}], s = 3/2, J_{n}(4×) | catena-[A_{5}], s = 2, J_{n}(4×) | catena-[A_{5}], s = 5/2, J_{n}(4×) |

catena-[A_{6}], s = 3/2, J_{n}(5×) | catena-[A_{6}], s = 2, J_{n}(5×) | catena-[A_{6}], s = 5/2, J_{n}(5×) |

^{a}Odd-member chains catena-[A

_{5}] have an irregular energy spectrum; i.e., S

_{0}= s (3/2, 2, 5/2) is the ground state.

cyclo-[A_{4}] | cyclo-[A_{5}] | cyclo-[A_{6}] |

cyclo-[A_{7}] | cyclo-[A_{8}] | cyclo-[A_{9}] |

cyclo-[A_{10}] | cyclo-[A_{11}] | cyclo-[A_{12}] |

^{a}Cyclic boundary has been applied. For the cyclo-[A

_{N}] system coupled in an antiferromagnetic manner, the ground state is four-fold degenerate (two Kramers doublets with S = 1/2) if Ns is a half-integer. If Ns is an integer, the ground state is non-degenerate (S = 0). For instance, cyclo-[A

_{9}, s = 1/2] possesses the doubly degenerate ground state S = 1/2 (twice). The ground state of catena-[A

_{9}, s = 1/2] is S = 1/2 (×1), and the first excited state, S = 1/2 (×1), lies at energy −0.75J.

cyclo-[A_{4}] | cyclo-[A_{5}] | cyclo-[A_{6}] |

cyclo-[A_{7}] | cyclo-[A_{8}] | cyclo-[A_{9}] |

cyclo-[A_{4}], s = 3/2 | cyclo-[A_{4}], s = 2 | cyclo-[A_{4}], s = 5/2 |

cyclo-[A_{5}], s = 3/2 | cyclo-[A_{5}], s = 2 | cyclo-[A_{5}], s = 5/2 |

cyclo-[A_{6}], s = 3/2 | cyclo-[A_{6}], s = 2 | cyclo-[A_{6}], s = 5/2 |

^{a}All cyclo-[A

_{N}]s have a regular energy spectrum; i.e., S

_{0}= 0 or 1/2 is the ground state.

**Table 14.**Normalized density of states for catena-[A

_{N}] s = 1/2 and cyclo-[A

_{N}] s = 1/2 systems, J = −1 cm

^{−1}.

Catena-[A_{13}], S_{0} = 1/2 (1×) | n(S) = 429, 572, 429, 208, 65, 12, 1 ^{a} | cyclo-[A_{13}], S_{0} = 1/2 (2×) |

catena-[A_{14}], S_{0} = 0 (1×) | n(S) = 429, 1001, 1001, 637, 273, 77, 13, 1 | cyclo-[A_{14}], S_{0} = 0 (1×) |

catena-[A_{15}], S_{0} = 1/2 (1×) | n(S) = 1430, 2002, 1638, 910, 350, 90, 14, 1 | cyclo-[A_{15}], S_{0} = 1/2 (2×) |

^{a}n(S)–numerosity of the spin states.

**Table 15.**Normalized density of states for catena-[A

_{N}] s = 1 and cyclo-[A

_{N}] s = 1 systems, J = −1 cm

^{−1}.

Catena-[A_{9}], S_{0} = 1 (1×) | n(S) = 603, 750, 672, 468, 258, 111, 36, 8, 1 | cyclo-[A_{9}], S_{0} = 0 (1×) |

catena-[A_{10}], S_{0} = 0 (1×) | n(S) = 603, 1585, 2025, 1890, 1398, 837, 405, 155, 45, 9, 1 | cyclo-[A_{10}], S_{0} = 0 (1×) |

**Table 16.**Comparison of zero-field energy levels for chain and ring systems, s = 1/2, J = −1 cm

^{−1}.

A_{9} | A_{10} | A_{11} | A_{12} | A_{13} |
---|---|---|---|---|

Energy spectrum for a chain—aligned left (black); for a ring—aligned right (blue) ^{a} | ||||

Normalized density of states: for a chain—black; for a ring—blue | ||||

^{a}For systems coupled in a ferromagnetic manner, the energy diagram is inverted. The A

_{10}-ring is still a crude approximation to the true A

_{10}-chain. The DOS for a ring shows waves; the DOS for a chain is smoother.

[A_{3}B], trigonal pyramid, s = 1/2J_{b}(3×), J_{a}(3×) = J_{b}/2, T_{1} | [A_{2}B_{2}], bisphenoid, s = 1/2J_{a}(2×), J_{c}(4×) = J_{a}/2, T_{2} | [A_{3}B], star, s = 1/2, J_{c}(5×), s = 1/2 T_{3} |

[A_{3}B], trigonal pyramid, s = 1 | [A_{2}B_{2}], bisphenoid, s = 1 | [A_{3}B], star, s = 1 |

[A_{3}B], trigonal pyramid, s = 3/2 | [A_{2}B_{2}], bisphenoid, s = 3/2 | [A_{3}B], star, s = 3/2 |

[A_{3}B], trigonal pyramid, s = 2 | [A_{2}B_{2}], bisphenoid, s = 2 | [A_{3}B], star, s = 2 |

[A_{3}B], trigonal pyramid, s = 5/2 | [A_{2}B_{2}], bisphenoid, s = 5/2 | [A_{3}B], star, s = 5/2 |

Topological matrices that define pair interactions of the centers: | ||

${T}_{1}(3-\mathrm{pyramid})=\left(\begin{array}{cccc}-& b& b& a\\ & -& b& a\\ & & -& a\\ & & & -\end{array}\right)$ | ${T}_{2}(\mathrm{bisphenoid})=\left(\begin{array}{cccc}-& c& a& c\\ & -& c& a\\ & & -& c\\ & & & -\end{array}\right)$ | ${T}_{3}(\mathrm{star})=\left(\begin{array}{cccc}-& 0& 0& c\\ & -& 0& c\\ & & -& c\\ & & & -\end{array}\right)$ |

[A_{4}B], tetragonal pyramid, s = 1/2J _{b}(4×), J_{a}(4×), J_{t}(2×) = 0, T_{1} | [A_{3}B_{2}], trigonal bipyramid, s = 1/2J _{b}(3×), J_{a}(6×), J_{t}(1×) = 0, T_{2} | [A_{4}B], star, s = 1/2 J _{c}(4×), T_{3} |

[A_{4}B], tetragonal pyramid, s = 1 | [A_{3}B_{2}], trigonal bipyramid, s = 1 | [A_{4}B], star, s = 1 |

[A_{4}B], tetragonal pyramid, s = 3/2 | [A_{3}B_{2}], trigonal bipyramid, s = 3/2 | [A_{4}B], star, s = 3/2 |

[A_{4}B], tetragonal pyramid, s = 2 | [A_{3}B_{2}], trigonal bipyramid, s = 2 | [A_{4}B], star, s = 2 |

[A_{4}B], tetragonal pyramid, s = 5/2 | [A_{3}B_{2}], trigonal bipyramid, s = 5/2 | [A_{4}B], star, s = 5/2 |

Topological matrices that define pair interactions of the centers: | ||

${T}_{1}(4-\mathrm{pyramid})=\left(\begin{array}{ccccc}-& b& 0& b& a\\ & -& b& 0& a\\ & & -& b& a\\ & & & -& a\\ & & & & -\end{array}\right)$ | ${T}_{2}(3-\mathrm{bipyramid})=\left(\begin{array}{ccccc}-& b& b& a& a\\ & -& b& a& a\\ & & -& a& a\\ & & & -& 0\\ & & & & -\end{array}\right)$ | ${T}_{3}(\mathrm{star})=\left(\begin{array}{ccccc}-& 0& 0& 0& c\\ & -& 0& 0& c\\ & & -& 0& c\\ & & & -& c\\ & & & & -\end{array}\right)$ |

[A_{6}] octahedron, s = 1/2 J _{c}(12×), J_{t}(3×) = 0, T_{1} | A_{6}, trigonal prism, s = 1/2J _{b}(6×) = J_{a}(3×), J_{a}_{2}(6×) = 0, T_{2} | A_{5}B, star, s = 1/2, J_{c}(5×), T_{3} |

[A_{6}] octahedron, s = 1 | A_{6}, trigonal prism, s = 1 | A_{5}B, star, s = 1 |

[A_{6}] octahedron, s = 3/2 | A_{6}, trigonal prism, s = 3/2 | A_{5}B, star, s = 3/2 |

[A_{6}] octahedron, s = 2 | A_{6}, trigonal prism, s = 2 | A_{5}B, star, s = 2 |

[A_{6}] octahedron, s = 5/2 | A_{6}, trigonal prism, s = 5/2 | A_{5}B, star, s = 5/2 |

Topological matrices that define pair interactions of the centers: | ||

${T}_{1}=\left(\begin{array}{cccccc}-& c& 0& c& c& c\\ & -& c& 0& c& c\\ & & -& c& c& c\\ & & & -& c& c\\ & & & & -& 0\\ & & & & & -\end{array}\right)$ | ${T}_{2}=\left(\begin{array}{cccccc}-& b& b& a& 0& 0\\ & -& b& 0& a& 0\\ & & -& 0& 0& a\\ & & & -& b& b\\ & & & & -& b\\ & & & & & -\end{array}\right)$ | ${T}_{3}(\mathrm{star})=\left(\begin{array}{cccccc}-& 0& 0& 0& 0& c\\ & -& 0& 0& 0& c\\ & & -& 0& 0& c\\ & & & -& 0& c\\ & & & & -& c\\ & & & & & -\end{array}\right)$ |

${T}_{1\mathrm{a}}=\left(\begin{array}{cccccc}-& b& b& 0& e& e\\ & -& b& e& 0& e\\ & & -& e& e& 0\\ & & & -& b& b\\ & & & & -& b\\ & & & & & -\end{array}\right)$ Trigonal antiprism T is equivalent to _{1a}T when J_{1}_{b}(6×) = J_{e}(6×), J_{t}(3×) = 0 |

No | Core | K | M | Size of Blocks {S_{min} through S_{max}} ^{a} |
---|---|---|---|---|

1, 2 | {Mn^{II}_{2}Mn^{III}_{2}} | 6^{2}·5^{2} = 900 | 110 | {S = 0–9}: 5, 13, 18, 29, 19, 15, 10, 6, 3, 1 |

3 | {Mn^{II}_{4}} | 6^{4} = 1296 | 146 | {S = 0–10}: 6, 15, 21, 24, 24, 21, 15, 10, 6, 3, 1 |

4 | {Mn^{II}_{3}Cr^{III}_{4}} | 6^{3}4^{4} = 55,296 | 5737 | {S =1/2–27/2}: 326, 661, 852, 915, 862, 726, 550, 375, 228, 122, 56, 21, 6, 1 |

5 | {Mn^{II}_{3}} | 6^{3} = 216 | 216 | {S =1/2–15/2}: 2, 4, 6, 5, 4, 3, 2, 1 |

6 | {Mn^{III}_{8}} | 5^{8} = 390,625 | 38,165 | {S = 0–16}: 1085, 3080, 4600, 5460, 5620, 5180, 4340, 3325, 1492, 868, 454, 210, 84, 28, 7, 1 |

7 | {Fe^{III}_{4}Mn^{III}_{4}} | 6^{4}·5^{4} = 810,000 | 71,346 | {S = 0–18}: 1650, 4735, 7221, 8844, 9500, 9250, 8290, 6890, 5326, 3829, 2555, 1576, 892, 458, 210, 84, 28, 7, 1 |

8 | {Fe^{III}_{6}Co^{III}_{6}} | 6^{6} = 46,656 | 4332 | {S = 0–15}: 111, 315, 475, 575, 609, 581, 505, 405, 300, 204, 126, 70, 35, 15, 5, 1 |

9, 10 | {Fe^{III}_{6}Fe^{II}} | 6^{6} = 46,656 | 4332 | {S = 0–15}: 111, 315, 475, 575, 609, 581, 505, 405, 300, 204, 126, 70, 35, 15, 5, 1 |

11 | {Fe^{III}_{7}} | 6^{7} = 279,936 | 24,017 | {S = 1/2–35/2} 1050, 1974, 2666, 3060, 3150, 2975, 2604, 2121, 1610, 1140, 750, 455, 252, 126, 56, 21, 6, 1 |

12, 13 | {Fe^{III}_{8}},{Fe ^{III}_{10}}→ {Fe ^{III}_{8}} | 6^{8} = 1,679,616 | 135,954 | {S = 0–20}: 2666, 7700, 11,900, 14,875, 16,429, 16,576, 15,520, 13,600, 11,200, 8680, 6328, 4333, 2779, 1660, 916, 462, 210, 84, 28, 7, 1 |

14 | {Co^{II}_{6}Co^{III}} | 4^{6} = 4096 | 580 | {S = 0–9}: 34, 90, 120, 120, 96, 64, 35, 15, 5, 1 |

15 | {Co^{II}_{11}Co^{III}_{2}} → {Co^{II}_{7}} | 4^{7} = 16,384 | 2128 | {S = 1/2–21/2}: 210, 364, 426, 400, 315, 210, 119, 56, 21, 6, 1 |

16 | {Co^{II}_{9}Co^{III}_{3}} | 4^{9} = 262,144 | 30,276 | {S = 1/2–27/2}: 2400, 4269, 5256, 5300, 4600, 3501, 2352, 1392, 720, 321, 120, 36, 8, 1 |

17 | {Er^{III}_{3}} | 16^{3} = 4096 | 192 | {J = 1/2–45/2}: 2, 4, 6, 8, 10, 12, 14, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 |

18, 19 | {Dy^{III}_{4}} | 16^{4} = 65,536 | 2736 | {J = 0–30}: 16, 45, 71, 94, 114, 131, 145, 156, 164, 169, 171, 170, 166, 159, 149, 136, 120, 105, 91, 78, 66, 55, 45, 36, 28, 21, 15, 10, 6, 3, 1 |

20 | {Dy^{III}_{2}Cu^{II}_{5}} | 16^{2}·2^{5} = 8192 | 482 | {J = 1/2–35/2}: 0, 30, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 31, 26, 16, 6, 1 |

^{a}The maximum-sized block is in bold type; J = S + L—total angular momentum. The symbol → means a simplification of the ring system into a smaller ring.

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**MDPI and ACS Style**

Boča, R.; Rajnák, C.; Titiš, J.
Spin Symmetry in Polynuclear Exchange-Coupled Clusters. *Magnetochemistry* **2023**, *9*, 226.
https://doi.org/10.3390/magnetochemistry9110226

**AMA Style**

Boča R, Rajnák C, Titiš J.
Spin Symmetry in Polynuclear Exchange-Coupled Clusters. *Magnetochemistry*. 2023; 9(11):226.
https://doi.org/10.3390/magnetochemistry9110226

**Chicago/Turabian Style**

Boča, Roman, Cyril Rajnák, and Ján Titiš.
2023. "Spin Symmetry in Polynuclear Exchange-Coupled Clusters" *Magnetochemistry* 9, no. 11: 226.
https://doi.org/10.3390/magnetochemistry9110226