# Zero-Field Splitting in Hexacoordinate Co(II) Complexes

^{*}

## Abstract

**:**

_{1}> 1.9, the composition of KDs from the spin states |±1/2> and |±3/2> with the dominating percentage p > 70%, and the first transition energy at the NEVPT2 level

^{4}Δ

_{1}. Just the latter quantity causes a possible divergence of the second-order perturbation theory and a failure of the spin Hamiltonian. The data set was enriched by the structural axiality D

_{str}and rhombicity E

_{str}, respectively, evaluated from the metal–ligand distances Co-O, Co-N and Co-Cl corrected to the mean values. The magnetic data (temperature dependence of the molar magnetic susceptibility, and the field dependence of the magnetization per formula unit) were fitted simultaneously, either to the Griffith–Figgis model working with 12 spin–orbit kets, or the SH-zero field splitting model that utilizes only four (fictitious) spin functions. The calculated data were analyzed using statistical methods such as Cluster Analysis and the Principal Component Analysis.

## 1. Introduction

^{n}that equals $\left(\begin{array}{c}10\\ n\end{array}\right)=\frac{10\cdot 9\cdot \dots}{1\cdot 2\cdot \dots n}$: 10 for d

^{1}and d

^{9}, 45 for d

^{2}and d

^{8}, 120 for d

^{3}and d

^{7}, 216 for d

^{4}and d

^{6}, and 252 for d

^{5}configurations. Working in such a space requires computer-aided efforts that, on the other hand, reduce transparency. However, the contemporary software based upon CASSCF + NEVPT2 + SOC calculations represents a useful tool in determining the spin–orbit multiplets using the fine-structure energy levels and the energy differences among them [14,15,16]. Alternate routes represent the ab initio Ligand Field Theory and the Generalized Crystal Field Theory [17,18,19].

_{S}>: 2 to 6 kets for d

^{1}to d

^{9}electron configurations. For instance, for the d

^{7}configuration, the complete (active) space consists of 120 spin–orbit kets labelled as |(νLS), J, M

_{J}> or |Γ′,γ′,a′>, but the SH formalism works only with four spin kets |S, M

_{S}> = |3/2, ±1/2> and |3/2, ±3/2>.

**Λ**-tensor, which serves for the calculation of the spin–spin interaction

**D**-tensor, the magnetogyric-ratio

**g**-tensor, the temperature-independent susceptibility

**X**-tensor, and eventually the hyperfine interaction

**A**-tensor [20,21]. This assumption is fulfilled when the expression ${H}^{\prime}=\u23290\left|{\widehat{H}}^{\mathrm{so}}\right|K\u232a/({E}_{0}-{E}_{K})$ is not too large, provided by a large enough denominator as the energy gap between the ground term |0> and excited terms |K>. As an effect of small perturbation, the content of the original spin functions in the multiplets is high; in other words, SOC does not mix the spin states significantly. We will see later that this requirement alone often fails. Within the SH formalism, the energy gaps between spin–orbit multiplets are expressed in terms of the axial and rhombic zero-field splitting parameters. Notably, these D and E parameters are not observables introduced as eigenvalues of a quantum-mechanical operator. They serve as descriptive parameters and thus they should be handled with care.

## 2. Theoretical Analysis

^{n}: ν, L, M

_{L}, S, M

_{S}>, where ν is the seniority number for repeated terms. On passage to the molecular systems belonging to a point group of symmetry G, the orbital part spans an irreducible representation Γ, its eventual component γ, and the branching index a, i.e., |Γ, γ, a; S, M

_{S}>; the spin part stays untouched. It is assumed that the effects of the configuration interaction are covered by the operator of the interelectron repulsion. For the irreducible representations (IRs) of the electron terms, the Mulliken notation is used, as appearing in the standard character tables of the point groups; this contains A, B, E, and T labels, with some subscripts identifying symmetry details. These definitions are compiled in Table 1.

_{J}>. In molecules, the spin–orbit multiplets (crystal-field multiplets) are labelled according to IRs Γ′, their components γ′ and branching index a′ within the double point group G′: |Γ′, γ′, a′> [22]. Here, the Bethe notation (Γ

_{1}to Γ

_{8}) is applied as found in character tables of the double point groups [23,24]. For Kramers systems (possessing the half-integral spin S = 1/2, 3/2, 5/2, 7/2), belonging to double groups with an order less than cubic, all IRs are doubly degenerate Γ

_{i}(2); for the cubic groups, a four-fold degenerate IR also exists: Γ

_{8}(4).

^{7}systems exemplified by the hexacoordinate Co(II) complexes. On symmetry descent from the octahedral geometry, the orbitally triply degenerate ground term is split

^{4}T

_{1g}→

^{4}E

_{g}$\oplus $

^{4}A

_{2g}(D

_{4h}), and the excited term as

^{4}T

_{2g}→

^{4}B

_{2g}$\oplus $

^{4}E

_{g}; the orbitally non-degenerate term transforms as

^{4}A

_{2g}→

^{4}B

_{1g}. On further symmetry descent to the D

_{2h}(isomorphous with C

_{2v}), the additional splitting yields

^{4}E

_{g}(D

_{4h}) →

^{4}B

_{3g}$\oplus $

^{4}B

_{2g}, whereas the non-degenerate term transforms as

^{4}A

_{2g}→

^{4}B

_{1g}. The corresponding irreducible representations for spin–orbit multiplets, depending upon the respective double point group, are shown in Figure 1, Figure 2 and Figure 3.

_{1g}or E

_{g}), the SH formalism cannot be applied. This is a frequent mistake: sometimes the D and E values are reported; however, they are undefined when the ground state is E

_{g}. Note that the ground electron term for the Co(II) complexes in the geometry of an elongated tetragonal bipyramid is

^{4}E

_{g}(the above case) and the set of the spin–orbit multiplets is labelled as Γ

_{6}, Γ

_{6}, Γ

_{7}, and Γ

_{7}. The differences among these Kramers doubles, abbr. as δ

_{1,2}, δ

_{3,4}, δ

_{5,6}, and δ

_{7,8}, cannot be expressed with the help of D- and E-parameters. The splitting between the ground term

^{4}E

_{g}and the first excited

^{4}A

_{2g}is denoted as Δ

_{ax}, and for axial elongation it is negative. Then, the asymmetry parameter ν = Δ

_{ax}/λ is positive, since λ = −ξ/2S < 0 for d

^{7}systems.

_{4h}→ D

_{2h}→ C

_{2v}, the daughter terms B

_{3g}and B

_{2g}stay quasi-degenerate. Formally, the spin Hamiltonian can be applied in such a case. However, when the energy denominator in 1/(E

_{0}–E

_{K}) is small, the second-order perturbation theory can suffer divergence, which manifests itself in overestimated D values and also in high asymmetry of the g-tensor components, sometimes unacceptable g

_{i}< 2.

^{4}A

_{2g}(D

_{4h}) produces two Kramers doublets (KDs) separated by δ

_{3,4}. In this case, the crystal-field splitting parameter is positive, Δ

_{ax}> 0, and then the asymmetry parameter ν < 0. Now the spin Hamiltonian can be applied, assuming that Δ

_{ax}is not too small (when the quasi degeneracy again occurs).

_{L}, S = 3/2, M

_{S}> allows a comparison of three cases [25]. (i) The case of a perfect octahedron (rather hypothetical due to the Jahn–Teller effect), with the ground term

^{4}T

_{1g}for which ν = 0, displays a round maximum at the μ

_{eff}vs. kT/|λ| curve. (ii) With ν > 10 (the case of an elongated bipyramid), the maximum is much reduced and the high-temperature tail almost disappears for very negative Δ

_{ax}; then, the effect of the low-lying excited state

^{4}A

_{2g}is filtered off and the magnetic properties are dominated only by the eight members (4 KDs) originating in the

^{4}E

_{g}term. (iii) For ν < 10, the ground term is orbitally non-degenerate

^{4}A

_{2g}and the μ

_{eff}curve falls down at low temperature due to a depopulation of δ

_{3,4}vs. the ground multiplet δ

_{1,2}. The high-temperature tail is represented by a straight line, reflecting some temperature-independent paramagnetism. The situation is well described by the SH formalism when Δ

_{ax}is not too small.

## 3. Methods and Modelling

#### 3.1. Spin Hamiltonian

_{z}, g

_{x}). Additionally, only the Cartesian components are often considered: x{π/2, 0}, y(π/2, π/2}, z{0, 0}. The diagonalization of the Hamiltonian matrix $\u2329I\left|{\widehat{H}}^{\mathrm{zfs}}+{\widehat{H}}_{kl}^{Z}({B}_{m})\right|J\u232a$ yields energy levels (two KDs for d

^{7}systems) ε

_{kl}(B

_{m}) that depend upon discrete (at least three) values of the magnetic field. They enter the partition function Z

_{kl}(B

_{m}, T) from which the magnetization M

_{kl}(B, T) and magnetic susceptibility χ

_{kl}(B, T) are evaluated via the first and second (numerical) derivatives with respect to the magnetic field. In addition to this universal method, there are also some simpler procedures; for instance, based upon the van Vleck equation for magnetic susceptibility. The powder average is a simple arithmetic average of the grid-dependent M

_{kl}and χ

_{kl}.

_{z}(B) and M

_{xy}(B). More informative are 3D graphs, as shown in Figure 5, where the value of D > 0 leads to the easy plane and D < 0 to easy-axis magnetism.

_{0}< 0.5 T, and field dependence of the magnetization per formula unit M

_{1}at T

_{0}< 5 K) has been fitted simultaneously by minimizing the error functional F(χ, M) → min. Several forms of the error functional have been applied; for instance, $F={w}_{1}\cdot E(\chi )+(1-{w}_{1})\cdot E(M)$, $F=E(\chi )\times E(M)$, and $F={w}_{1}\cdot C(\chi )+(1-{w}_{1})\cdot C(M)$, where the relative error E(P) and the “city-block” factor C(P) for individual observable P = χ or M are

#### 3.2. Griffith–Figgis Model

_{L}, S, M

_{S}> [25,26].

_{ax}(Δ

_{rh})—axial (rhombic) crystal-field splitting energy, g

_{L}= —Aκ effective orbital magnetogyric factor (negative owing to the T-p isomorphism), κ—orbital reduction factor accounting to some degree of covalency. This formula has been extended by considering the asymmetry of the Zeeman term

_{ax}< 0 (easy axis) or Δ

_{ax}> 0 (easy plane) is essentially the same as obtained by the SH-zfs model. However, the key parameter has a completely different physical origin: in the SH-zfs model, it is the anisotropy of the fictitious spin angular momentum $[D({\widehat{S}}_{z}^{2}-{\overrightarrow{S}}^{2}/3)+E({\widehat{S}}_{x}^{2}-{\widehat{S}}_{y}^{2})]$; in the GF model, it is the anisotropy of the orbital angular momentum ${[\Delta}_{\mathrm{ax}}({\widehat{L}}_{\mathrm{p},z}^{2}-{\overrightarrow{L}}_{\mathrm{p}}^{2}/3){+\Delta}_{\mathrm{rh}}({\widehat{L}}_{\mathrm{p},x}^{2}-{\widehat{L}}_{\mathrm{p},y}^{2})]$. It can be concluded that the negative axial crystal-field splitting parameter causes the easy-axis magnetization (Figure 6).

_{ax}= ±500 cm

^{−1}, the susceptibility and magnetization curves are almost the same; some differences are seen at the effective magnetic moment. For Δ

_{ax}= ±1000 cm

^{−1}, the differences in the magnetization become visible; the μ

_{eff}for Δ

_{ax}= +1000 cm

^{−1}rises according to the straight line, which reflects the presence of the excited state manifesting itself in the temperature-independent paramagnetism. For Δ

_{ax}= ±3000 cm

^{−1}, the differences are substantial in magnetization, susceptibility, and effective magnetic moment. For Δ

_{ax}= +3000 cm

^{−1}, the effect of the excited state is filtered off and the system behaves like a typical zfs system.

#### 3.3. Ab Initio Calculations

_{k}and the wavefunctions of the states projected to the model space ${\tilde{\Psi}}_{k}$. Using a singular value decomposition procedure, the elements of D- and g-tensor are extracted [35]. In case of the low norm of the projections (N << 1), the trial model Hamiltonian is inapplicable to the given system.

#### 3.4. Generalized Crystal-Field Theory

^{7}system like in the CAS method. By applying the irreducible tensor algebra, the matrix elements of the interaction operators are evaluated: the interelectron repulsion ${\widehat{H}}^{\mathrm{ee}}$, crystal field ${\widehat{H}}^{\mathrm{cf}}$, spin–orbit coupling ${\widehat{H}}^{\mathrm{so}}$, Zeeman orbital, and Zeeman spin interactions [18,19,37]. The parameters dependent upon the radial functions are expressed by the Racah B and C parameters for the interelectron repulsion, and the crystal-field poles F

_{2}(L) and F

_{4}(L) for individual ligands L, respectively. The angular parts of the matrix elements are integrated using coefficients of fractional parentage and vector coupling coefficients (3j- and 6j-symbols). The positions of ligands involve the spherical harmonic functions owing to which the matrix elements are complex. The spin–orbit interaction is characterized by the spin–orbit coupling constant ξ

_{Co}. The diagonalization of the matrix $\u2329J\left|{\widehat{H}}^{\mathrm{ee}}(B,C)+{\widehat{H}}^{\mathrm{cf}}({F}_{2},{F}_{4})+{\widehat{H}}^{\mathrm{so}}(\xi )\right|I\u232a$ yields the crystal-field multiplets $\left|K\right.\u232a=\left|{(\mathrm{d}}^{n}\nu LS);{{\mathsf{\Gamma}}^{\prime}}_{a},{{\gamma}^{\prime}}_{a},{a}^{\prime}\right.\u232a$, where we used labelling of the irreducible representations (IRs) and their components {${{\mathsf{\Gamma}}^{\prime}}_{a},{{\gamma}^{\prime}}_{a},{a}^{\prime}$} within the double point group of symmetry.

**Λ**-tensor by means of the second-order perturbation theory

**Λ**-tensor is used in the definition of the

**D**-tensor ${{D}^{\prime}}_{ab}=-{\lambda}^{2}{\Lambda}_{ab}$. In the traceless form

_{4}(L) and eventually F

_{2}(L) for individual ligands. This method is suitable for the modelling of SH parameters over a wide range of geometries and crystal-field strengths.

## 4. Results and Discussion

#### 4.1. Geometry of Complexes

_{3})

_{6}]

^{2+}, [Co(H

_{2}O)

_{6}]

^{2+}and [CoCl

_{6}]

^{4−}complex units, giving rise to $\overline{d}(\mathrm{Co}-\mathrm{N})=$ 2.185 Å, $\overline{d}(\mathrm{Co}-\mathrm{O})=$ 2.085 Å, and $\overline{d}(\mathrm{Co}-\mathrm{Cl})=$ 2.475 Å [38]. This procedure is effective for complexes with a heterogeneous donor set with different averaged metal–ligand distances. Sometimes it is not clear which distances should be selected as axial, and which as equatorial. Hereafter, a constraint is utilized: E

_{str}/|D

_{str}| < 1/3. (Analogous constraints are used for axial and rhombic zero-field splitting parameters in SH theory.) Some of the studied complexes possess the form of a pincer-type: there are severe deviations of four donor atoms from the equatorial plane due to the rigidity of the organic ligand. Therefore, the values of D

_{str}* need to be handled with care.

^{O,O})

_{3}][Co(NCS)

_{4}] (

**J**), [Co(dppm

^{O,O})

_{3}][CoBr

_{4}] (

**K**) and [Co(dppm

^{O,O})

_{3}][CoI

_{4}] (

**L**) with the chromophores {CoO

_{2}O’

_{2}O”

_{2}}, the values of D

_{str}vary as −1.65, −1.45, and +2.15 pm; [Co(dppm

^{O,O})

_{3}][CoCl

_{4}] (

**U**) has a different symmetry of the chromophore {CoO

_{3}O’

_{3}}. The complex [Co(pydm)

_{2}](dnbz)

_{2}(

**O**) is analogous to [Co(pydm)

_{2}](mdnbz)

_{2}(

**P**), but differing in D

_{str}= −20.1 and −17.9 pm. Finally, [Co(bzpy)

_{4}Cl

_{2}] (

**F**) contains two crystallographic different units with D

_{str}= +7.05 and −3.5 pm.

#### 4.2. Elongated Tetragonal Bipyramid

_{str}> 0. In such a case, it is expected that the temperature dependence of the effective magnetic moment passes through a round maximum (which not necessarily is visible until room temperature). The effective magnetic moment exceeds μ

_{eff}> 5 μ

_{B}and the magnetization per formula unit saturates close to M

_{1}= M

_{mol}/(N

_{A}μ

_{B}) ~ 3. In some cases, the magnetic data have been refitted by a more appropriate model than published previously. Note that the magnetization data can suffer of some orientation effect in higher fields especially when the rso- or vsm-mode of detection is used in the modern SQUID apparatus. Then, the detected magnetization data could be a bit higher than calculated by the fitting procedure which equally weights the susceptibility data taken in small DC field. The free parameters also cover some temperature-independent (para)magnetism χ

_{TIM}that influences the high-temperature tail of the magnetic susceptibility, and the molecular field correction zj effective at the lowest temperatures (not listed here).

_{1}, introduced as follows:

_{1}= N(KD1)·g

_{1}·[E(KD3) − E(KD2)]·Δ

_{1}/10,000

_{1}, the separation of the two subsets of KDs [E(KD3) − E(KD2)], and the first transition energy at the NEVPT2 level

^{4}Δ

_{1}(scaled to smaller numbers). Values of S

_{1}> 50 refer to class 5—fulfilled; S

_{1}< 10 span class 1—invalid. Limiting value for the fulfillment is S

_{1}= 0.7 × 1.9 × 600 × 600/10,000 = 48. The above parameter can be extended by considering the mixing of the spin states: S

_{2}= S

_{1}× P/100 where P—percentage of the greater portion of spins | ± 1/2> or | ± 3/2> in the first KD (ideally p > 70).

_{2}O)

_{6}]

^{2+}(OHnic)

^{−}

_{2}(

**A**). The complex cation adopts the geometry of an elongated tetragonal bipyramid with a considerable axiality and zero rhombicity, D

_{str}= +7.1 pm and E

_{str}= 0. The orbitally degenerate ground term

^{4}T

_{1g}(O

_{h}) causes the Jahn–Teller effect, leading to the symmetry descent, and consequently to the splitting of the ground mother term. The energies of the daughter terms lie at {0, 199, 2468} cm

^{−1}, which is consistent with the ground term

^{4}E

_{g}and the excited

^{4}A

_{2g}(Δ

_{2}= 2468 cm

^{−1}). Small splitting of the ground term {0, 199} cm

^{−1}is caused by “innocent” hydrogen atoms that disturb the ideal D

_{4h}symmetry (the ground state is orbitally quasi-degenerate). As valuable results, comparable with experiments, serve the energies of Kramers doublets δ{0, 209, 526, 814} cm

^{−1}, the transitions among them can be identified, for instance, by the FAR-infrared spectra. Rather unexpected is the fact that the compositions of these KDs contain almost equal contributions from | ± 1/2> and | ± 3/2> spins. This means that the spin–orbit coupling seriously mixes the states of the different spin projections, or in other words, ${\widehat{H}}^{\mathrm{so}}$ cannot be considered as a small perturbation. Therefore, all “products” of the spin Hamiltonian can be false. Indeed, g

_{1}= 1.76 << 2, D = −101 cm

^{−1}are artefacts and this conclusion is also supported by the small norm of the projected state N(KD1) = 0.61 << 1. The attempts to fit the magnetic data with the GF model were successful: the round maximum at the effective magnetic moment was perfectly reproduced with λ

_{eff}= Aκλ = −188 cm

^{−1}, g

_{L}= −1.10, and Δ

_{ax}= −112 cm

^{−1}.

^{II}Co

^{III}(L

^{1}H

_{2})

_{2}(H

_{2}O)(ac)]·(H

_{2}O)

_{3}(

**B**) contains the homogeneous donor set {CoO

_{4}O’

_{2}} with averaged distances Co-O

_{eq}= 2.061 and Co-O

_{ax}= 2.150 Å owing to which D

_{str}= +8.9 pm. The energies of KDs are δ{(0, 220), (736, 1006)} cm

^{−1}, and they are better separated owing to higher Δ

_{2}= 1516 cm

^{−1}. The secondary splitting of

^{4}E

_{g}term is Δ

_{1}= 444 cm

^{−1}so that the orbital degeneracy is partly removed. However, there are critical indicators warning that the spin-Hamiltonian theory is questionable in the present case: N(KD1) = 0.71, g

_{1}= 1.84, and severe mixing of the spin states with the principal contribution < 0.7; D = −101 cm

^{−1}. GF Hamiltonian was used in fitting the magnetic data with Δ

_{ax}< 0 resulting in λ

_{eff}= −198 cm

^{−1}, g

_{Lz}= −1.64, g

_{Lx}= −1.11, and Δ

_{ax}= −774 cm

^{−1}.

_{2}(H

_{2}O)

_{2}(nca)

_{2}] (

**C**), after the corrections to the heterogeneous donor set {CoO

_{2}O

_{w2}N

_{2}}, possesses D

_{str}= +7.75; its room-temperature effective magnetic moment reaches μ

_{eff}~ 5 μ

_{B}and the magnetization saturates close to M

_{1}~ 3. These features are typical for systems with the ground electronic term

^{4}E

_{g}for which the spin-Hamiltonian formalism could fail. The energies of KDs are not well separated into two subsets of KDs δ{0, 256, 525, 850} cm

^{−1}. The critical indicators are: N(KD1) = 0.58, g

_{1}= 1.51,

^{4}Δ

_{1}= 117 cm

^{−1}(almost degenerate ground state), and a boundary mixing of the spin states; D = −113 cm

^{−1}could be an artefact. The fitting of magnetic data based upon the GF Hamiltonian gave λ

_{eff}= −172 cm

^{−1}, g

_{Lz}= −2.06, g

_{Lx}= −1.50, and Δ

_{ax}= −739 cm

^{−1}.

_{2}(H

_{2}O)

_{2}] (

**D**) with the chromophore {CoO

_{4}O

_{w2}}, the longest distance is Co-O

_{w}= 2.157 Å gave D

_{str}= +12.0 pm. The energies of KDs are δ{(0, 155), (915, 1153)} cm

^{−1}and they form two well separated sub-sets. Additionally, the energy of the first electronic transition

^{4}Δ

_{1}= 763 cm

^{−1}suggests that the ground state is well separated from the excited counterpart. Therefore, the spin-Hamiltonian formalism could work, which is also supported by the critical indicators N(KD1) = 0.81 and g

_{1}= 1.943, yielding the score S1 = 91. The calculated D = +73 cm

^{−1}and E/D = 0.23 yield an estimate of the energy gap G

_{3,4}= 2(D

^{2}+ 3E

^{2})

^{1/2}= 2|D|[1 + 3(E/D)

^{2}]

^{1/2}= 155 cm

^{−1}that equals the energy of the KD2 δ

_{3,4}= 155 cm

^{−1}. The calculated magnetic susceptibility passes through the experimental points and the magnetization per formula unit amounts to M

_{mol}/(N

_{A}μ

_{B}) = 2.35 at B = 7.0 T.

^{2}

_{2}Cl

_{2}] with the {CoO

_{2}N

_{2}Cl

_{2}} chromophore (

**E**), after correction to the heterogeneity of the donor set, displays D

_{str}= +9.45 pm with large rhombicity E

_{str}= 2.65 pm (E

_{str}/D

_{str}= 0.28) close to the critical value of 0.33 when the sign of the D

_{str}is uncertain. The alternate structural parameters are D

_{str}= −8.7, E

_{str}= 3.4 pm, and E

_{str}/D

_{str}= 0.39. The ab initio calculations confirm that the SH theory for this system is appropriate since N(KD1) = 0.91, g

_{1}= 2.03,

^{4}Δ

_{1}= 1217 cm

^{−1}, and a weak mixing of spin states exists: D = +43 cm

^{−1}. The estimated energy gap G

_{3,4}=2(D

^{2}+ 3E

^{2})

^{1/2}= 94 cm

^{−1}matches perfectly the energy of the KD2 δ

_{3,4}= 94 cm

^{−1}. The magnetic data were fitted with the SH-zfs model using D = 75, °E = 4.8 cm

^{−1}, and

**g**{2.51, 2.36, 2.0}.

_{4}Cl

_{2}] contains two crystallographic independent molecular complexes with different axiality D

_{str}= +7.05 and −3.5 pm, respectively. The unit

**Fa**possesses two well-separated subsets of KDs δ{(0, 179), (633, 911)} cm

^{−1}. The first transition energy

^{4}Δ

_{1}= 448 cm

^{−1}indicates that the orbital degeneracy is partly removed. Critical indicators classify the SH as acceptable and calculated D = 88 cm

^{−1}as reasonable. The unit

**Fb**displays different properties: not separated groups of KDs δ{0, 252, 455, 787} cm

^{−1}, N(KD1) = 0.49, orbital degeneracy

^{4}Δ

_{1}= 130 cm

^{−1}, and subnormal g

_{1}= 1.60 that approves classification of SH as invalid. Though the ab initio calculations were performed for individual units separately, the magnetic data reflect some average of their response. The GF model gave λ

_{eff}= −175 cm

^{−1}, g

_{Lz}= −1.02, g

_{Lx}= −1.28, and Δ

_{ax}= −424 cm

^{−1}, whereas the SH-zfs model yields D = +106 cm

^{−1}and g

_{x}= 2.53.

#### 4.3. Nearly Octahedral Systems

_{str}were included in this group. The results of ab initio calculations and magnetic data fitted either with the GF or SH-zfs model are presented in Table 3.

_{2}(etpy)

_{2}] contains two independent crystallographic units, and thus ab initio calculations were performed for both of them. The unit

**Ga**shows four KDs at δ{0, 237, 461, 804} cm

^{−1}with a serious mixing of spin states. The critical indicators show that the SH theory is invalid: N(KD1) = 0.50; g

_{1}= 1.66 (subnormal),

^{4}Δ

_{1}= 109 cm

^{−1}(quasi degeneracy). The unit

**Gb**possesses a better separation of the two subgroups of KDs δ{(0, 196), (568, 873)} cm

^{−1}owing to increased transition energy

^{4}Δ

_{1}= 359 cm

^{−1}. The critical indicators are N(KD1) = 0.64; g

_{1}= 1.93 (SH is still problematic). Both complexes have small negative D

_{str}= −2.00 and −2.45 pm, which prefer the application of the GF model for the magnetic data fitting with λ

_{eff}= −159 cm

^{−1}, g

_{Lz}= −1.96, g

_{Lx}= −1.79, and Δ

_{ax}= −771 cm

^{−1}.

_{2}(bzpyCl)

_{2}] (

**H**) shows D

_{str}= −2.45 pm with well-separated subgroups of KDs δ{(0, 188), (582, 883)} cm

^{−1}. The set of indicators is still critical: N(KD1) = 0.58; g

_{1}= 1.95,

^{4}Δ

_{1}= 392 cm

^{−1}(degeneracy is partly lifted), and the mixing of spin states is rather weak. The SH formalism is problematic; D = +91 cm

^{−1}. The GF model for the magnetic data fitting gave λ

_{eff}= −170 cm

^{−1}, g

_{Lz}= −1.83, g

_{Lx}= −1.11, and Δ

_{ax}= −643 cm

^{−1}.

_{2}(tcm)

_{2}] (

**I**) displays small D

_{str}= −2.0 pm. All critical indicators confirm that the SH formalism is fulfilled: δ{(0, 131), (862, 1066)} cm

^{−1}, N(KD1) = 0.86; g

_{1}= 2.04,

^{4}Δ

_{1}= 900 cm

^{−1}(orbital degeneracy lifted), weak mixing of spin states. Then, the evaluated D = +50 cm

^{−1}can be considered as a valid parameter. The composition of the ground KD1 is {20·| ± 1/2> + 79·| ± 3/2>} with dominating contributions of | ± 3/2>; for D > 0, just | ± 1/2> is expected as a dominating component of the ground multiplet Γ

_{6}. Perhaps large rhombicity E/D = 0.29 causes this feature.

^{O,O})

_{3}][CoX

_{4}], X = NCS

^{−}, Br

^{−}and I

^{−}possess the same cationic complex (with 154 atoms) with small axiality D

_{str}= −1.65, −1.45, and +2.15 pm, respectively. (The fourth member with X = Cl

^{−}has a different geometry of the chromophore {CoO

_{3}O’

_{3}}.) The presence of the complex anions was not involved in calculations; however, the experimental data reflect their effect on the increased values of the effective magnetic moment and magnetization.

^{O,O})

_{3}][Co(NCS)

_{4}] (

**J**) possesses δ{(0, 211), (562, 966)} cm

^{−1}, g

_{1}= 1.97,

^{4}Δ

_{1}= 445 cm

^{−1}(degeneracy partly lifted), but a serious mixing of spin states. Therefore, it is classified as SH–problematic; D(O

_{h}) = +105 cm

^{−1}. Note that the solid-state magnetic data were fitted assuming the presence of both nearly octahedral and nearly tetrahedral units with D(O

_{h}) = 91, D(T

_{d}) = −5.0 cm

^{−1}.

^{O,O})

_{3}][CoBr

_{4}] (

**K**) behaves analogously to its NCS analogue: δ{(0, 211), (562, 966)} cm

^{−1}, g

_{1}= 1.97,

^{4}Δ

_{1}= 445 cm

^{−1}(degeneracy partly lifted), N(KD1) = 0.61 and a serious mixing of spin states; D(O

_{h}) = +105 cm

^{−1}. The SH is classified as problematic and the fitting of magnetic data gave D(O

_{h}) = +122 and D(T

_{d}) = +15 cm

^{−1}.

^{O,O})

_{3}][CoI

_{4}] (

**L**) with δ{(0, 223), (508, 874)} cm

^{−1}shows different critical parameters:

^{4}Δ

_{1}= 258 cm

^{−1}(near degeneracy), subnormal g

_{1}= 1.86 and again a strong mixing of spin states. The SH is classified as invalid; calculated D(O

_{h}) = +107 cm

^{−1}and fitted D(O

_{h}) = +99 and D(T

_{d}) = +19 cm

^{−1}.

#### 4.4. Compressed Tetragonal Bipyramid

_{str}<< 3 (Table 4). In general, the magnetic data for them can be fitted with the SH-zfs model which assumes g

_{z}= 2, g

_{x}>> 2, D >> 0. Alternatively, the GF model can also be used with Δ

_{ax}> 0.

_{6}](fm)

_{2}(

**M**) with the homogeneous ligand sphere contains the {CoN

_{6}} chromophore that can be classified as a compressed tetragonal bipyramid with considerable, but negative axiality and small rhombicity: D

_{str}= −6.10 and E

_{str}= 0.71 pm. The energies of the spin–orbit multiplets δ{0, 256, 450, 836} cm

^{−1}are quite similar to the complex [Co(H

_{2}O)

_{6}](OHnic)

_{2}(

**A**). There is a set of critical indicators warning that the spin-Hamiltonian theory fails: N(KD1) = 0.46,

^{4}Δ

_{1}= 35 cm

^{−1}(orbital degeneracy), g

_{1}= 1.30, g

_{2}= 1.83 (subnormal values), and severe mixing of the spin states; D = +124 cm

^{−1}. Nevertheless, the magnetic data were fitted with the SH-zfs model with parameters D = +69 cm

^{−1}and g

_{x}= 2.75.

_{4}(NCS)

_{2}] contains two crystallographic independent molecular complexes with D

_{str}= −11.75 and −11.05 pm, respectively. The electronic properties of them are similar: two subgroups of KDs δ{(0, 187), (646, 965)} cm

^{−1},

^{4}Δ

_{1}= 473 cm

^{−1}and g

_{1}= 1.93. Therefore, the SH is classified as questionable; D = +89 cm

^{−1}for

**Na**(and similar for

**Nb**). The magnetic data fitting using the SH model gave D = +95 cm

^{−1}and g

_{x}= 2.52.

_{2}](dnbz)

_{2}(

**O**) contains the pincer-type ligands pydm which, owing to a rigidity, do not coordinate on the axes of the equatorial plane, so that the values of D

_{str}* = −20.15 pm need be considered with care. Two sub-set of KDs are well separated δ{(0, 188), (864, 1099)} cm

^{−1}owing to increased

^{4}Δ

_{1}= 614 cm

^{−1}. The critical parameters indicate that the SH might be fulfilled: N(KD1) = 0.71, g

_{1}= 1.98, weak mixing of spin states. The only disturbance is the negative value of D = −92 cm

^{−1}, since positive value is expected for the compressed tetragonal bipyramid. This point will be explained later using the GCFT calculations.

_{2}](mdnbz)

_{2}(

**P**) with the same pincer ligand but slightly modified counter anion; D

_{str}* = −17.9 pm. Again, two groups of KDs are well separated δ{(0, 145), (870, 1099)} cm

^{−1}and the first transition energy is

^{4}Δ

_{1}= 708 cm

^{−1}. However, N(KD1) = 0.68 and increased mixing of the spin states cause the classification of the SH—close to fulfilled. Again, negative D = −69 cm

^{−1}was calculated for this system. With this data, the energy gap G

_{3,4}= 145 cm

^{−1}matches the energy of the first excited KD, δ

_{3,4}= 145 cm

^{−1}. The magnetic data were fitted almost perfectly using the SH-zfs model with D = −50 cm

^{−1}.

_{2}O contains two crystallographic independent units, both with negative axiality D

_{str}= −22.65 pm. The site

**Qa**possesses well-separated subgroups of KDs δ{(0, 162), (813, 1046)} and

^{4}Δ

_{1}= 614 cm

^{−1}. The critical indicators signalize that the SH is fulfilled: N(KD1) = 0.78, g

_{1}= 1.99; only the mixing of the spin states is stronger. The unit

**Qb**exhibits similar characteristics with D = −97 cm

^{−1}in comparison with D = −77 cm

^{−1}for

**Qa**. The fitting of the magnetic data with the SH-zfs model gave D = −89 cm

^{−1}, g

_{z}= 2.50, and g

_{x}= 2.42.

_{2}(H

_{2}O)

_{2}(MeIm)

_{2}] (

**R**) possesses D

_{str}= −11.9 pm and the ab initio data confirm a separation of the two subsets of KDs δ{(0, 156), (1030, 1230)} cm

^{−1}owing to large

^{4}Δ

_{1}= 878 cm

^{−1}. The critical indicators show that the SH is fulfilled: N(KD1) = 0.84 and g

_{1}= 1.91. With expectations, D = +75 cm

^{−1}is positive for compressed tetragonal bipyramid. There is a serious mixing of spin states. The same quality of the magnetic data fits was obtained using the GF model (λ

_{eff}= −217 cm

^{−1}, g

_{Lz}= −1.23, g

_{Lx}= −1.37, Δ

_{ax}= +568 cm

^{−1}) and/or the SH-zfs model (D = +82 cm

^{−1}, g

_{x}= 2.54).

_{2}Cl

_{2}] (

**S**) possesses D

_{str}= −7.03 and the critical indicators warn that the SH fails: N(KD1) = 0.50, subnormal g

_{1}= 1.43, and g

_{2}= 1.90, very small transition energy

^{4}Δ

_{1}= 76.6 cm

^{−1}which causes the two subgroups of KDs to not be separated δ{0, 262, 461, 800} cm

^{−1}. Then, the calculated D = +121 cm

^{−1}is false and the magnetic data fitting is not satisfactory when the SH is used. The GF model gave acceptable fit with λ

_{eff}= −181 cm

^{−1}, g

_{Lz}= −1.5, g

_{Lx}= −1.3, and Δ

_{ax}= +377 cm

^{−1}.

#### 4.5. Miscellaneous Geometry

_{str}is either undefined or oddly defined (Table 5). It shows a versatility of the magnetic behavior of hexacoordinate Co(II) complexes.

^{O,O})

_{3}][CoCl

_{4}] (

**T**) spans the series [Co(dppm

^{O,O})

_{3}][CoX

_{4}], but unlike the NCS

^{−}, Br

^{−}, and I

^{−}members, it displays different geometry of the {CoO

_{3}O

_{3′}} chromophore so that axiality D

_{str}is not defined in this case. The critical indicators warn that the SH is not fulfilled, since g

_{1}= 1.78,

^{4}Δ

_{1}= 150.6, and

^{4}Δ

_{2}= 150.9 cm

^{−1}, strong mixing of spin states are demonstrated, and not separated subsets of KDs δ{0, 314, 393, 926} cm

^{−1}. Therefore, the calculated D = +157 cm

^{−1}could be false. However, the magnetic data were satisfactorily fitted with D = +77 cm

^{−1}.

_{2}(dca)

_{2}] exists as two polymorphs (

**Ua**,

**Ub**) and again does not fulfil the definition of axiality D

_{str}. According to the critical indicators for

**Ua**, the SH is classified as invalid: N(KD1) = 0.54, g

_{1}= 1.49, very low transition energy

^{4}Δ

_{1}= 110 cm

^{−1}, and the two subsets of KDs not separated δ{0, 243, 495, 838} cm

^{−1}. The calculated value of D = +108 cm

^{−1}seems be overestimated. The magnetic data were fitted with D = 91 cm

^{−1}and g

_{x}= 2.66; however, the fit was not satisfactory for the magnetization data. For the polymorph

**Ub**, the situation was completely different with a high score of S

_{1}= 51 (mainly due to the high first transition energy

^{4}Δ

_{1}= 618 cm

^{−1}) that allows a classification of the SH as fulfilled. At the same time, both the susceptibility and magnetization data were fitted excellently using D = 85 cm

^{−1}and g

_{x}= 2.60.

_{2}](tcm)

_{2}(

**V**) possesses the considerable axiality D

_{str}= −8.2, E

_{str}= 0. However, the deviations of four N-donor atoms from the equatorial plane are not negligible owing to the rigid geometry of the pincer-type ligand. The energies of KDs are split into two well-separated subsets δ{(0, 159), (717, 1003)} cm

^{−1}, owing to the removal of the orbital degeneracy,

^{4}Δ

_{1}= 571 cm

^{−1}. The critical indicators confirm that the SH is fulfilled: N(KD1) = 0.74, g

_{1}= 1.99, weak mixing of spin states; the value of D = +72 cm

^{−1}is fully acceptable. However, the calculations were performed for a free complex cation abstracting from the environment. The environment alone plays a critical role, since the tcm

^{−}ligands link several cationic units into a complex network which shows features of the exchange interaction of a ferromagnetic nature. The magnetic susceptibility passes through a maximum that is typical for tetragonal systems with positive axiality, and at the same time the magnetization per formula unit exceeds a value of M

_{1}> 3. The magnetic data cannot be fitted by a reliable set of parameters using both GF and SH-zfs models for a single magnetic center.

_{2}O)] dca (

**W**) has structure of a 1D chain decorated by free dca

^{−}ions. The ab initio calculations were not performed; the magnetic data were fitted with the GF model.

## 5. Statistical Analysis

^{4}Δ

_{1}(Figure 9). For Δ

_{1}< 300 cm

^{−1}, the SH data are barely reliable (class 1) because of the quasi-degeneracy. For Δ

_{1}> 600 cm

^{−1}, the SH data are highly reliable (class 4 or 5). A numerical correlation including the correlation coefficient is presented in Figure 10.

^{2}− 1/4) for Kramers systems [1]. A deeper analysis of experimental data shows: (i) slow magnetic relaxation exists also in systems with D—positive, negligible, or in systems where D is undefined (S = 1/2); (ii) quantitatively, the above paradigm is not true, at least for the hexacoordinate Co(II) complexes. In the cases of hexacoordinate Co(II) complexes when the SH is fulfilled (D

_{4h}, D

_{2h}symmetry of the chromophore), D > 0 generally holds true. Ab initio calculations can indicate some D < 0; however, in the cases when HS fails (D

_{str}>> 0).

_{i,i+1}and SH parameters (D, g

_{z}, g

_{x}, χ

_{TIP}) depending upon the strength of the crystal field poles F

_{4}(ax) and F

_{4}(eq); the results are presented in Figure 12. For the elongated tetragonal bipyramid, the D-values are undefined, since the ground term is orbitally degenerate

^{4}E

_{g}and the two lowest multiplets span the irreducible representations Γ

_{6}and Γ

_{6}.

_{0}exp(U/k

_{B}T) is appropriate only for the Orbach relaxation process. Using a few high-temperature data, the evaluation of the extrapolated relaxation time (for infinite temperature) τ

_{0}, and the barrier to spin reversal U is often possible; however, it can yield incorrect values when the slow relaxation at the highest edge of the data taking is not attenuated. The collection of {D, U, τ

_{0}} data is of little value when the relaxation proceeds according to the alternate mechanisms such as Raman, phonon bottleneck, and direct relaxation mechanisms. (v) The plot lnτ vs. lnT brings information about the temperature coefficient in the above mechanisms proceeding via eqn. τ

^{−1}= CT

^{m}: m ~ 1 for the direct process, m ~ 2 for the phonon bottleneck process, or m = 5–9 for the Raman process. The Orbach process requires m > 9, which, as a rule, is not the case. Data in Figure 13 confirm that the HF relaxation mode at elevated temperatures proceeds via the Raman mechanism with the temperature coefficient m = 5.9; at low temperature, a reciprocating thermal behavior applies, m = −0.64 when on cooling the relaxation time decreases [61,62].

^{−1}(144 K), then the Boltzmann population of KD2 at T = 10 K is P

_{3,4}= 2 × (5.6 × 10

^{−7}), i.e., negligible. This discriminates the Orbach relaxation mechanism and related U and τ

_{0}as unrealistic parameters. In the light of these findings, the value of the collection of the published data on D and their relationship to U in hexacoordinate Co(II) complexes is questionable [63]. There are several original and review articles about the impact of the zero-field splitting on the DC and AC magnetic properties of hexacoordinate Co(II) complexes; some of them are accompanied by ab initio calculations; however, a deeper validity assessment of the spin-Hamiltonian formalism is missing [64,65,66,67,68,69,70].

## 6. Conclusions

_{str}: (i) complexes with large positive values referring to the elongated tetragonal bipyramid (with some o-rhombicity); (ii) nearly octahedral complexes with small |D

_{str}|; (iii) complexes with large negative D

_{str}referring to the compressed tetragonal bipyramid; and (iv) complexes with miscellaneous geometry. The first type possesses the ground electronic terms orbitally (nearly) degenerate

^{4}E

_{g}(with corresponding daughter terms on symmetry lowering). The spin Hamiltonian, as a rule, fails, and thus the magnetic data need to be fitted by employing the extended Griffith–Figgis model working in the space of 12 spin–orbit kets. The GF theory is an intermediate step between the spin Hamiltonian recognizing only 4 (fictitious) spin kets and the complete active space of 120 kets generated by the d

^{7}configuration.

^{4}Δ

_{1}< 300 cm

^{−1}, low projection norm N(KD1) < 0.7, subnormal value of the lowest g-factor g

_{1}< 1.9, and large mixing of the spin components into multiplets with the highest portion p < 70%. In such a case, the second-order perturbation theory tends to diverge and the calculated D parameters are overestimated. The statistical methods (Cluster Analysis, Principal Component Analysis) bring information which parameters mutually correlate.

_{3,4}can be reconstructed by assuming G = 2|D|, or G = 2(D

^{2}+ 3E

^{2})

^{1/2}. Thus, the successful fit itself is not a guarantee that the spin-Hamiltonian formalism is fulfilled for the given case. The main obstacle lies in the fact that for hexacoordinate Co(II) complexes, six Kramers doublets result from the ground electronic term

^{4}T

_{1g}; four of them can be close in energy while the spin-Hamiltonian formalism recognizes only two of them.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

abpt | 4-amino-3,5-bis(2-pyridyl)-1,2,4-triazol |

ac | acetato(1-) ligand |

ampyd | 2-aminopyrimidine |

bz | benzoato(1-) ligand |

bzpy | 4-benzylpyridine |

bzpyCl | 4-(4-Chlorobenzyl)pyridine |

dca | dicyanamide(1-) |

dmphen | 2,9-dimethyl-1,10-phenanthroline |

dnbz | 3,5-dinitrobenzoato(1-) |

dppm^{O,O} | bis-(diphenylphosphanoxido)methane |

etpy | 4-ethylpyridine |

fm | formiate(1-) ion |

hfac | hexafluoroacetylacetonato(1-) |

im, iz | 1H-imidazole |

L^{1}H_{2} | 2-{[(2-hydroxy-3-methoxyphenyl)-methylene]amino}-2-(hydroxymethyl)-1,3-propanediol |

L^{2} | 2-[(2,2-diphenylethylimino)methyl]pyridine-1-oxide |

mdnbz | 3,5-dinitrobenzoato(1-) |

MeIm | N-methylimidazole |

OHnic | 6-hydroxynicotinate |

pydca | pyridine-2,6-dicarboxylato(1-) |

pydm, dmpy | 2,6-pyridinedimethanol |

pypz | 2,6-bis(pyrazol-1-yl)pyridine |

tcm | tricyanomethanide(1-) |

w | aqua ligand |

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**Figure 1.**Development of the crystal field terms (T, A, E, B) and spin–orbit multiplets (Γ

_{i}) under symmetry lowering for hexacoordinate Co(II) high-spin systems.

**Figure 2.**Scheme of 12 energy levels for elongated square bipyramid (D

_{4h}), o-rhombic bipyramid (D

_{2h}), and symmetry lower polyhedron (C

_{2v}) in hexacoordinate Co(II) complexes.

**Figure 4.**Temperature dependence of the effective magnetic moment for the ground

^{4}T

_{1g}term within the Figgis theory. Note: $v={\Delta}_{\mathrm{ax}}/\lambda $ and λ = −ξ/2S < 0 for d

^{7}systems.

**Figure 5.**The 3D model of magnetization M(x,y,z) within the zfs model at T = 2.0 K and B = 1.0 T. Left—for D = +20 cm

^{−1}(easy-plane magnetism); right—for D = −20 cm

^{−1}(easy-axis magnetism).

**Figure 6.**The 3D model of magnetization M(x,y,z) within the Griffith model at T = 2.0 K, B = 1.0 T, λ = −170 cm

^{−1}and g

_{L}= −1.5. Left—for Δ

_{ax}= +500 cm

^{−1}(easy plane); right—for Δ

_{ax}= −500 cm

^{−1}(easy axis).

**Figure 7.**Modelling of the magnetic functions using various Δ

_{ax}in the GF model; λ = −170 cm

^{−1}, g

_{L}= −1.5. Left—temperature dependence of the effective magnetic moment (inset—molar magnetic susceptibility); right—magnetization per formula unit.

**Figure 8.**Results of the statistical analysis. Top and center—cluster analysis, Wards method, squared Euclidean distance. Bottom—biplot of principal component analysis. Codes: K2, K3, K4—energies of Kramers doublets; D1—transition energy

^{4}Δ

_{1}; g1—the lowest g-factor; N—projection norm N(KD1); Pg (Pl)—greater (lower) portion of spins in multiplets of KD1; P12—portion of ±1/2 spins in multiplets of KD1; S1 and S2—score of SH; C—classification factor of SH (1 = invalid, 5 = fulfilled); D—axial zero-field splitting parameter; ED—ratio E/D; Ds—axiality D

_{str}; Es—rhombicity E

_{str}.

**Figure 9.**Classification of the spin Hamiltonian by qualitative score: 5—fulfilled, 4—acceptable, 3—questionable, 2—problematic, 1—invalid. S

_{1}= N(KD1) × g

_{1}× [E(KD3) − E(KD2)] × Δ

_{1}/10,000 for individual complexes. Limiting value S

_{1}= 0.7 × 1.9 × 600 × 600/10,000 ~ 50. Values S1 > 50 refer to the class 5—fulfilled; S

_{1}< 10 span the class 1—invalid.

**Figure 10.**Correlations among ab initio calculated parameters: KD2(KD3) = b

_{0}+ b

_{1}·Δ

_{1}. The greater the first transition energy Δ

_{1}: (i) the lower the energy of the second Kramers doublet (KD2 ~ 2D); (ii) the greater the energy of KD3. For Δ

_{1}< 300 cm

^{−1}, the SH data are barely reliable because of the quasi-degeneracy (C = 1). For Δ

_{1}> 600 cm

^{−1}, the SH data are highly reliable (C = 5).

**Figure 11.**Calculated energies of the crystal field terms (A, E) and multiplets (G6, G7) on angular distortion of square bipyramid D

_{4h}to D

_{2d}via angle α bisecting O-Co-O. Expt.: 2α = O2-Co1-O1 = 151.88 and O4-Co1-O3 = 154.16 deg for the complex Co(pydca)(dmpy)] (

**O**) with the pincer-type ligands.

**Figure 12.**Three-dimensional diagram of D vs. F

_{4}(z)-F

_{4}(xy) calculated by GCFT for hexacoordinate Co(II) complexes. δ

_{3}= E

_{3}(Γ

_{7}) − E

_{1}(Γ

_{6}) for compressed form (~2D); δ

_{3}= E

_{3}(Γ

_{6}) − E

_{1}(Γ

_{6}) for elongated form not matching the spin Hamiltonian. Manifold co-ordinate points for g

_{z}, g

_{x}and χ

_{TIP}refer to different 10Dq.

**Figure 13.**AC susceptibility data for [Co(pydca)(dmpy)]·0.5H

_{2}O. Left: frequency dependence of AC susceptibility at various temperatures and applied field B

_{DC}= 0.4 T showing three relaxation channels; solid lines—fitted with the three-set Debye model. Right: dependences of the relaxation time and their fit to the exponential Arrhenius-like equation lnτ = b

_{0}+ b

_{1}T

^{−1}and power equation lnτ = b

_{0}+ b

_{1}lnT [53].

Free Atom/Ion | Molecule/Complex | |||
---|---|---|---|---|

Operators | ${\widehat{H}}^{\mathrm{ee}}$ | ${\widehat{H}}^{\mathrm{ee}}+{\widehat{H}}^{\mathrm{so}}$ | ${\widehat{H}}^{\mathrm{ee}}+{\widehat{H}}^{\mathrm{cf}}+{\widehat{H}}^{\mathrm{so}}$ | ${\widehat{H}}^{\mathrm{ee}}+{\widehat{H}}^{\mathrm{cf}}+{\widehat{H}}^{\mathrm{so}}$ |

Wave function | Atomic term | Atomic multiplet | Multielectron term | Spin–orbit multiplet |

Notation | |d^{n}: ν, L, M_{L}, S, M_{S}> | |(νLS), J, M_{J}> | |Γ, γ, a; S, M_{S}> | |Γ′, γ′, a′> |

Irreducible representations ^{b} | D^{(L)}(2L + 1):S, P, D, F, G, H, I | ^{2S + 1}D_{J}(2J + 1) | ^{m}A(1), ^{m}B(1), ^{m}E(2), ^{m}T(3) ^{b} | Γ_{i}(1, 2, 3, 4) |

-for Kramers systems | S = 1/2, 3/2, 5/2, 7/2 | J = |L − S|,…L + S | m = 2S + 1 = 2, 4, 6, 8 | Γ_{i}(2), Γ_{8}(4) |

^{a}${\widehat{H}}^{\mathrm{ee}}$—interelectron repulsion, ${\widehat{H}}^{\mathrm{cf}}$—crystal-field operator, ${\widehat{H}}^{\mathrm{so}}$—spin–orbit coupling.

^{b}Orbital degeneracy in parentheses.

A, [Co(H_{2}O)_{6}]^{2+} (OHnic^{−})_{2}, [CoH_{12}O_{6}]^{2+} 2(C_{6}H_{4}NO_{3})^{−} | CAS Theory: Spin–Orbit Multiplets | ||||
---|---|---|---|---|---|

CCDC FONQUV, 295 K, R _{gt} = 0.054 [39,40] | {CoO_{4}O’_{2}}Co-O’ 2.113 Å Co-O 2.042 Å D _{str} = +7.1 pmE _{str} = 0 | KD1, 0.61 ^{a} | KD2, 0.78 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 209 | δ_{5,6} = 526 | δ_{7,8} = 814 | ||

41·| ± 1/2> + 57·| ± 3/2> | 58·| ± 1/2> + 40·| ± 3/2> | 55·| ± 1/2> + 42·| ± 3/2> | 42·| ± 1/2> +57·| ± 3/2> | ||

Magnetic data, SMR–n.a. | SH theory: score S_{1} = 7, S_{2} = 4, classification 1–invalid | ||||

GF model λ _{eff} = −188 cm^{−1}g _{L} = −1.10Δ _{ax} = −112 cm^{−1} | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 199^{4}Δ_{2} = 2468 | D = −100.9 D _{1} = −119.9D _{2} = +10.5 | E/D = 0.16 E _{1} = −0.01E _{2} = −10.8 | g_{1} = 1.762g_{2} = 1.906g_{3} = 3.104g_{iso} = 2.258 | |

B, [Co^{II}Co^{III}(L^{1}H_{2})_{2}(H_{2}O)(ac)]·(H_{2}O)_{3}, [C_{26}H_{35}Co_{2}N_{2}O_{13}] 3(H_{2}O) | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 1440294, 100 K, R_{gt} = 0.039 [41] | {CoO_{4}O’_{2}}Co-O’ 2.150 Å Co-O 2.061 Å D _{str} = +8.9 pmE _{str} = 0 | KD1, 0.71 | KD2, 0.88 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 220 | δ_{5,6} = 736 | δ_{7,8} = 1006 | ||

56·| ± 1/2> + 43·| ± 3/2> | 38·| ± 1/2> + 58·| ± 3/2> | 36·| ± 1/2> + 62·| ± 3/2> | 64·| ± 1/2> +34·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 30, S_{2} = 17, classification 3–questionable | ||||

GF model λ _{eff} = −198 cm^{−1}g _{Lz} = −1.64g _{Lx} = −1.11Δ _{ax} = −774 cm^{−1} | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 444^{4}Δ_{2} = 1516 | D = −100.9 D _{1} = −114.8D _{2} = +20.0 | E/D = 0.25 E _{1} = −0.01E _{2} = −20.0 | g_{1} = 1.842g_{2} = 2.293g_{3} = 3.102g_{iso} = 2.412 | |

C, trans-[Co(bz)_{2}(H_{2}O)_{2}(nca)_{2}], [C_{26}H_{26}CoN_{4}O_{8}] | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 804191, 293 K, R_{gt} = 0.038 [42] | {CoO_{2}O’_{2}N_{2}}Co-N 2.147 Å Co-O 2.084 Å Co-O’ _{w} 2.143 ÅD _{str} = +7.75 pmE _{str} = 1.85 pm | KD1, 0.58 | KD2, 0.73 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 256 | δ_{5,6} = 525 | δ_{7,8} = 850 | ||

65·| ± 1/2> + 34·| ± 3/2> | 31·| ± 1/2> + 67·| ± 3/2> | 31·| ± 1/2> + 67·| ± 3/2> | 72·| ± 1/2> +27·| ± 3/2> | ||

Magnetic data, SMR–n.a. | SH theory: score S_{1} = 3, S_{2} = 2, classification 1–invalid | ||||

GF model λ _{eff} = −172 cm^{−1}g _{Lz} = −2.06g _{Lx} = −1.50Δ _{ax} = −739 cm^{−1} | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 117^{4}Δ_{2} = 1138 | D = −113.3 D _{1} = −131.0D _{2} = +26.9 | E/D = 0.31 E _{1} = −0.08E _{2} = −27.0 | g_{1} = 1.507g_{2} = 2.042g_{3} = 3.160g_{iso} = 2.237 | |

D, [Co(acac)_{2}(H_{2}O)_{2}], [C_{10}H_{18}CoO_{6}] | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 1842364, 100 K, R_{gt} = 0.024 [43] | {CoO_{2}O’_{2}O_{w}}Co-O 2.040Å Co-O’ 2.034 Å Co-O _{w} 2.157 ÅD _{str} = +12.0 pmE _{str} = 0.30 pm | KD1, 0.81 | KD2, 0.93 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 155 | δ_{5,6} = 915 | δ_{7,8} = 1153 | ||

53·| ± 1/2> + 45·| ± 3/2> | 46·| ± 1/2> + 52·| ± 3/2> | 56·| ± 1/2> + 40·| ± 3/2> | 40·| ± 1/2> +57·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 91, S_{2} = 48, classification 5–fulfilled | ||||

SH-zfs model from ab initio calculations | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 763^{4}Δ_{2} = 1398 | D = +72.0 D _{1} = +39.7D _{2} = +22.7 | E/D = 0.23 E _{1} = −39.6E _{2} = −22.8 | g_{1} = 1.943g_{2} = 2.462g_{3} = 2.804g_{iso} = 2.403 | |

E, [CoL^{2}_{2}Cl_{2}]·3.5H_{2}O, [C_{40}H_{36}Cl_{2}CoN_{4}O_{2}]·3.5H_{2}O | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 796703, 150 K, R_{gt} = 0.045 [44] | {CoN_{2}O_{2}Cl_{2}}Co-N 2.081 Å Co-O 2.034 Å Co-Cl 2.492 Å D _{str} = +9.45 pmE _{str} = 2.65 pmE _{str}/D_{str} = 0.28 | KD1, 0.91 | KD2, 0.96 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 94 | δ_{5,6} = 1238 | δ_{7,8} = 1441 | ||

88·| ± 1/2> + 9·| ± 3/2> | 8·| ± 1/2> + 89·| ± 3/2> | 11·| ± 1/2> + 86·| ± 3/2> | 87·| ± 1/2> +9·| ± 3/2> | ||

Magnetic data, SMR–n.a. | SH theory: S_{1} = 257, S_{2} = 226, classification 5–fulfilled | ||||

SH-zfs model D = 75.1 cm ^{−1}E = 4.8 cm ^{−1}g _{z} = 2g _{x} = 2.51g _{y} = 2.36 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 1217^{4}Δ_{2} = 2039 | D = +43.3 D _{1} = +21.5D _{2} = +13.9 | E/D = 0.24 E _{1} = +13.6E _{2} = −3.8 | g_{1} = 2.032g_{2} = 2.341g_{3} = 2.566g_{iso} = 2.313 | |

Fa, [Co(bzpy)_{4}Cl_{2}], [C_{48}H_{44}Cl_{2}CoN_{4}] | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 1497488, 120 K, R_{gt} = 0.027 [45] | {CoN_{4}Cl_{2}}Unit A Co-Cl 2.443 Å Co-N 2.235 Å Co-N 2.176 Å D _{str} = +7.05 pmE _{str} = 1.15 pm | KD1, 0.69 | KD2, 0.89 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 179 | δ_{5,6} = 633 | δ_{7,8} = 911 | ||

69·| ± 1/2> + 29·| ± 3/2> | 24·| ± 1/2> + 73·| ± 3/2> | 36·| ± 1/2> + 62·| ± 3/2> | 73·| ± 1/2> +24·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 27, S_{2} = 19, classification 3–questionable | ||||

GF model/11 λ _{eff} = −175 cm^{−1}g _{Lz} = −1.02g _{Lx} = −1.28Δ _{ax} = −424 cm^{−1} | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 448^{4}Δ_{2} = 993 | D = +87.6 D _{1} = +43.2D _{2} = +31.6 | E/D = 0.13 E _{1} = + 43.1E _{2} = −31.4 | g_{1} = 1.948g_{2} = 2.498g_{3} = 2.779g_{iso} = 2.408 | |

SH-zfs model D = +106 cm ^{−1}g _{x} = 2.53g _{z} = 2 | |||||

Fb, [Co(bzpy)_{4}Cl_{2}], [C_{48}H_{44}Cl_{2}CoN_{4}] | CAS Theory: Spin–Orbit Multiplets | ||||

Structure as above for Fa | {CoN_{4}Cl_{2}}Unit B Co-Cl 2.433 Å Co-N 2.187 Å Co-N 2.169 Å D _{str} = −3.5 pmE _{str} = 0.9 pmE/|D| = 0.26 | KD1, 0.49 | KD2, 0.68 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 252 | δ_{5,6} = 455 | δ_{7,8} = 787 | ||

54·| ± 1/2> + 44·| ± 3/2> | 47·| ± 1/2> + 51·| ± 3/2> | 34·| ± 1/2> + 65·| ± 3/2> | 66·| ± 1/2> +33·| ± 3/2> | ||

Magnetic data as above for Fa | SH theory: score S_{1} = 2, S_{2} = 1, classification 1–invalid | ||||

^{4}Δ_{0} = 0 ^{4}Δ_{1} = 130^{4}Δ_{2} = 804 | D = +120.9 D _{1} = + 56.2D _{2} = +34.8 | E/D = 0.17 E _{1} = + 56.2E _{2} = −34.7 | g_{1} = 1.604g_{2} = 2.163g_{3} = 2.942g_{iso} = 2.237 |

^{a}Explanation: | ± 1/2> means a cumulative percentage of the spin contributions in the given spin–orbit multiplet arising from the lowest roots referring to the block of the spin multiplicity m = 4 (sum of contributions > 1%);

^{m}Δ

_{i}—transition energies between terms at NEVPT2 level; δ—spin–orbit multiplets; D

_{i}(E

_{i})—contributions to the D (E) parameter from the lowest excitations; all energy data in cm

^{−1}. For cations and the solvent containing species, the calculations run for atoms in square brackets in the chemical formula moiety. Critical data—Italic.

Ga, [Co(hfac)_{2}(etpy)_{2}], [C_{24}H_{20}CoF_{12}N_{2}O_{4}] | CAS Theory: Spin–Orbit Multiplets | ||||
---|---|---|---|---|---|

CCDC 2223471, 100 K, R_{gt} = 0.050 | A: {CoO_{4}N_{2}}Co-N 2.132 Å Co-O 2.056 Å Co-O 2.048 Å D _{str} = −2.0 pmE _{str} = 0.4 pm | KD1, 0.50 | KD2, 0.73 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 237 | δ_{5,6} = 461 | δ_{7,8} = 804 | ||

49·| ± 1/2> + 50·| ± 3/2> | 50·| ± 1/2> + 49·| ± 3/2> | 52·| ± 1/2> + 46·| ± 3/2> | 46·| ± 1/2> + 52·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 2, S_{2} = 1, classification 1–invalid | ||||

GF model λ _{eff} = −159 cm^{−1}g _{Lz} = −1.96g _{Lx} = −1.79Δ _{ax} = −771 cm^{−1} | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 109^{4}Δ_{2} = 785 | D = +112 D _{1} = +59.8D _{2} = +33.4 | E/D = 0.20 E _{1} = 58.8E _{2} = −33.4 | g_{1} = 1.661g _{2} = 2.043g _{3} = 2.932g _{iso} = 2.212 | |

Gb, [Co(hfac)_{2}(etpy)_{2}], [C_{24}H_{20}CoF_{12}N_{2}O_{4}] | CAS Theory: Spin–Orbit Multiplets | ||||

B: {CoO_{4}N_{2}}Co-N = 2.151Å Co-O 2.040 Å Co-O 2.058 Å D _{str} = −1.45 pmE _{str} = 0.35 pm | KD1, 0.64 | KD2, 0.87 | KD3 | KD4 | |

δ_{1,2} = 0 | δ_{3,4} = 196 | δ_{5,6} = 568 | δ_{7,8} = 873 | ||

37·| ± 1/2> + 63·| ± 3/2> | 45·| ± 1/2> + 55·| ± 3/2> | 55·| ± 1/2> + 45·| ± 3/2> | 40·| ± 1/2> + 60·| ± 3/2> | ||

Magnetic data as above | SH theory: S_{1} = 16, S_{2} = 10, classification 2–problematic | ||||

^{4}Δ_{0} = 0 ^{4}Δ_{1} = 359^{4}Δ_{2} = 901 | D = +94 D _{1} = +47.9D _{2} = +29.7 | E/D = 0.18 E _{1} = −47.9E _{2} = 29.7 | g_{1} = 1.931g _{2} = 2.351g _{3} = 2.808g _{iso} = 2.364 | ||

H, [Co(hfac)_{2}(bzpyCl)_{2}], [C_{34}H_{22}Cl_{2}CoF_{12}N_{2}O_{4}] | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 2223472, 100 K, R_{gt} = 0.036 | {CoO_{4}N_{2}}*Co-N 2.137 Å Co-O 2.061 Å Co-O 2.062 Å D _{str} = −2.45 pmE _{str} = 0.05 pm | KD1, 0.58 | KD2, 0.83 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 188 | δ_{5,6} = 582 | δ_{7,8} = 883 | ||

24·| ± 1/2> + 74·| ± 3/2> | 78·| ± 1/2> + 20·| ± 3/2> | 79·| ± 1/2> + 20·| ± 3/2> | 6·| ± 1/2> + 90·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 17, S_{2} = 13, classification 2–problematic | ||||

GF model λ _{eff} = −170 cm^{−1}g _{Lz} = −1.83g _{Lx} = −1.11Δ _{ax} = −643 cm^{−1} | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 392^{4}Δ_{2} = 905 | D = +91 D _{1} = +45.9D _{2} = +29.6 | E/D = 0.16 E _{1} = 45.7E _{2} = −29.0 | g_{1} = 1.954g _{2} = 2.372g _{3} = 2.781g _{iso} = 2.369 | |

I, [Co(abpt)_{2}(tcm)_{2}], [C_{32}H_{20}CoN_{18}] | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 997721, 173 K, R_{gt} = 0.036 [47] | {CoN_{4}N’_{2}}*Co-N’ 2.133 Å Co-N 2.109 Å Co-N 2.125 Å D _{str} = −2.0 pmE _{str} = 0.4 pm | KD1, 0.86 | KD2, 0.96 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 131 | δ_{5,6} = 862 | δ_{7,8} = 1066 | ||

20·| ± 1/2> + 79·| ± 3/2> | 76·| ± 1/2> + 17·| ± 3/2> | 85·| ± 1/2> + 13·| ± 3/2> | 5·| ± 1/2> + 92·| ± 3/2> | ||

Magnetic data, SMR–yes [47] | SH theory: S_{1} = 115, S_{2} = 91, classification 5–fulfilled | ||||

SH-zfs model D = +55 cm ^{−1}E = 14.6 cm ^{−1}g _{x} = 2.53g _{z} = 2 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 900^{4}Δ_{2} = 1878 | D = +50.3 D _{1} = +28.5D _{2} = +17.6 | E/D = 0.29 E _{1} = +28.5E _{2} = −17.6 | g_{1} = 2.037g _{2} = 2.333g _{3} = 2.636g _{iso} = 2.335 | |

J, [Co(dppm^{O,O})_{3}][Co(NCS)_{4}],[C_{75}H_{66}CoO_{6}P_{6}]^{2+} Co(NCS)_{4}^{2−} | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 1526142, 100 K, R_{gt} = 0.041 [48] | {CoO_{2}O’_{2}O”_{2}}Co-O 2.094 Å Co-O’ 2.089 Å Co-O” 2.074 Å D _{str} = −1.65 pmE _{str} = 0.35 pm | KD1, 0.61 | KD2, 0.86 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 211 | δ_{5,6} = 562 | δ_{7,8} = 966 | ||

51·| ± 1/2> + 48·| ± 3/2> | 47·| ± 1/2> + 51·| ± 3/2> | 47·| ± 1/2> + 51·| ± 3/2> | 57·| ± 1/2> + 41·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 21, S_{2} = 11, classification 2–problematic | ||||

SH-zfs model D = +93 cm ^{−1}g _{x} = 2.76g _{z} = 2 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 445^{4}Δ_{2} = 539 | D = +105.5 D _{1} = +46.3D _{2} = +42.0 | E/D = 0.03 E _{1} = +45.7E _{2} = −41.9 | g_{1} = 1.972g _{2} = 2.592g _{3} = 2.688g _{iso} = 2.417 | |

K, [Co(dppm^{O,O})_{3}][CoBr_{4}], [C_{75}H_{66}CoO_{6}P_{6}]^{2+} CoBr_{4}^{2−} | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 1526141, 100 K, R_{gt} = 0.044 [49] | {CoO_{2}O’_{2}O”_{2}}Co-O 2.109 Å Co-O’ 2.102 Å Co-O” 2.091 Å D _{str} = −1.45 pmE _{str} = 0.35 pm | KD1, 0.61 | KD2, 0.86 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 211 | δ_{5,6} = 562 | δ_{7,8} = 966 | ||

51·| ± 1/2> + 47·| ± 3/2> | 46·| ± 1/2> + 53·| ± 3/2> | 47·| ± 1/2> + 52·| ± 3/2> | 59·| ± 1/2> + 39·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 19, S_{2} = 10, classification 2–problematic | ||||

SH-zfs model D = +122 cm ^{−1}g _{x} = 2.68g _{z} = 2 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 445^{4}Δ_{2} = 539 | D = +105.5 D _{1} = +46.3D _{2} = +42.53 | E/D = 0.03 E _{1} = +45.7E _{2} = −41.9 | g_{1} = 1.972g _{2} = 2.592g _{3} = 2.688g _{iso} = 2.417 | |

L, [Co(dppm^{O,O})_{3}][CoI_{4}], [C_{75}H_{66}CoO_{6}P_{6}]^{2+} CoI_{4}^{2−} | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 1526143, 100 K, R_{gt} = 0.028 [48] | {CoO_{2}O’_{2}O”_{2}}Co-O 2.092 Å Co-O’ 2.076 Å Co-O” 2.065 Å D _{str} = +2.15 pmE _{str} = 0.55 pm | KD1, 0.57 | KD2, 0.80 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 223 | δ_{5,6} = 508 | δ_{7,8} = 874 | ||

45·| ± 1/2> + 54·| ± 3/2> | 56·| ± 1/2> + 41·| ± 3/2> | 49·| ± 1/2> + 49·| ± 3/2> | 49·| ± 1/2> + 48·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 8, S_{2} = 4, classification 1–invalid | ||||

SH-zfs model D = +99 cm ^{−1}g _{x} = 2.70g _{z} = 2 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 258^{4}Δ_{2} = 732 | D = +107.9 D _{1} = +54.6D _{2} = +34.1 | E/D = 0.15 E _{1} = +54.6E _{2} = −34.1 | g_{1} = 1.860g _{2} = 2.319g _{3} = 2.868g _{iso} = 2.349 |

M [Co(iz)_{6}]^{2+}(fm^{−})_{2}, [C_{18}H_{24}CoN_{12}]^{2+} 2(CHO_{2})^{−} | CAS Theory: Spin–Orbit Multiplets | ||||
---|---|---|---|---|---|

CCDC 624939, 296 K, R_{gt} = 0.034 [39,50] | {CoN_{4}N’_{2}}Co-N’ 2.211 Å Co-N 2.197 Å Co-N’ 2.143 Å D _{str} = −6.10 pmE _{str} = 0.71 pm | KD1, 0.46 | KD2, 0.54 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 256 | δ_{5,6} = 450 | δ_{7,8} = 836 | ||

60·| ± 1/2> + 38·| ± 3/2> | 34·| ± 1/2> + 64·| ± 3/2> | 35·| ± 1/2> + 64·| ± 3/2> | 72·| ± 1/2> + 26·| ± 3/2> | ||

Magnetic data, SMR–n.a. | SH theory: S_{1} = 0.4, S_{2} = 0.2, classification 1–invalid | ||||

SH-zfs model D = +69.2 cm ^{−1}g _{x} = 2.75g _{z} = 2 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 35^{4}Δ_{2} = 591 | D = +124.0 D _{1} = +62.4D _{2} = +39.0 | E/D = 0.15 E _{1} = +61.9E _{2} = −37.0 | g_{1} = 1.302g _{2} = 1.829g _{3} = 2.974g _{iso} = 2.035 | |

Na, [Co(bzpy)_{4}(NCS)_{2}], [C_{50}H_{44}CoN_{6}S_{2}] | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 1497489, 120 K, R_{gt} = 0.036 [45] | {CoN_{4}N’_{2}}Unit A Co-N’ 2.086 Å Co-N 2.217 Å Co-N 2.180 Å D _{str} = −11.7 pmE _{str} = 1.35 pm | KD1, 0.68 | KD2, 0.88 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 187 | δ_{5,6} = 646 | δ_{7,8} = 965 | ||

79·| ± 1/2> + 21·| ± 3/2> | 13·| ± 1/2> + 85·| ± 3/2> | 33·| ± 1/2> + 67·| ± 3/2> | 78·| ± 1/2> + 20·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 28, S_{2} = 22, classification 3–questionable | ||||

SH-zfs model D = +90.5 cm ^{−1}g _{x} = 2.52g _{z} = 2 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 473^{4}Δ_{2} = 838 | D = +88.9 D _{1} = +47.2D _{2} = +30.7 | E/D = 0.17 E _{1} = +47.0E _{2} = −30.4 | g_{1} = 1.932g _{2} = 2.446g _{3} = 2.823g _{iso} = 2.400 | |

Nb, [Co(bzpy)_{4}(NCS)_{2}], [C_{50}H_{44}CoN_{6}S_{2}] | CAS Theory: Spin–Orbit Multiplets | ||||

Unit B Co-N’ 2.094 Å Co-N 2.213 Å Co-N 2.196 Å D _{str} = −11.0 pmE _{str} = 0.85 pm | KD1, 0.67 | KD2, 0.88 | KD3 | KD4 | |

δ_{1,2} = 0 | δ_{3,4} = 189 | δ_{5,6} = 638 | δ_{δ7,8} = 975 | ||

79·| ± 1/2> + 21·| ± 3/2> | 12·| ± 1/2> + 85·| ± 3/2> | 33·| ± 1/2> + 66·| ± 3/2> | 80·| ± 1/2> + 18·| ± 3/2> | ||

Magnetic data as above | SH theory: S_{1} = 28, S_{2} = 22, classification 3–questionable | ||||

^{4}Δ_{0} = 0 ^{4}Δ_{1} = 481^{4}Δ_{2} = 776 | D = +91.7 D _{1} =+47.0D _{2} = +32.1 | E/D = 0.15 E _{1} = +46.6E _{2} = −31.6 | g_{1} = 1.938g _{2} = 2.466g _{3} = 2.806g _{iso} = 2.403 | ||

O, [Co(pydm)_{2}]^{2+}(dnbz)^{−}_{2}, [C_{14}H_{18}CoN_{2}O_{4}]^{2+}·2(C_{7}H_{3}N_{2}O_{6})^{−} pincer type | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 1533249, 100 K, R_{gt} = 0.037 [51] | {CoO_{4}N_{2}}Co-N 2.039 Å Co-O 2.110 Å Co-O 2.171 Å D _{str}* = −20.15E _{str}* = 3.05 | KD1, 0.71 | KD2, 0.89 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 188 | δ_{5,6} = 864 | δ_{7,8} = 1099 | ||

22·| ± 1/2> + 75·| ± 3/2> | 75·| ± 1/2> + 21| ± 3/2> | 59·| ± 1/2> + 36·| ± 3/2> | 37·| ± 1/2> + 61·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 58, S_{2} = 44, classification 5–fulfilled | ||||

SH-zfs model D = −62 cm ^{−1}g _{z} = 2.13g _{x} = 2 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 615^{4}Δ_{2} = 2199 | D = −91.8 D _{1} = −103.0D _{2} = +8.9 | E/D = 0.13 E _{1} = −0.3E _{2} = −11.4 | g_{1} = 1.983g _{2} = 2.169g _{3} = 3.058g _{iso} = 2.403 | |

P, [Co(pydm)_{2}]^{2+}(dmnbz)^{−}_{2}, [C_{14}H_{18}CoN_{2}O_{4}]^{2+}·2(C_{8}H_{5}N_{2}O_{6})^{−}; pincer type | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 1945478, 100 K, R_{gt} = 0.042 [52] | {CoO_{4}N_{2}}Co-N 2.038 Å Co-O 2.120 Å Co-O 2.114 Å D _{str}* = −17.9 pmE _{str}* = 0.30 pm | KD1, 0.68 | KD2, 0.88 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 145 | δ_{5,6} = 870 | δ_{7,8} = 1099 | ||

40·| ± 1/2> + 57·| ± 3/2> | 57·| ± 1/2> + 38·| ± 3/2> | 63·| ± 1/2> + 35·| ± 3/2> | 32·| ± 1/2> + 65·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 72, S_{2} = 41, classification 5–fulfilled | ||||

SH-zfs model D = −50.0 cm ^{−1}g _{z} = 2.30g _{x} = 2 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 708^{4}Δ_{2} = 1831 | D = −69.0 D _{1} = −86.9D _{2} = +11.7 | E/D = 0.19 E _{1} = −0.02E _{2} = −11.8 | g_{1} = 2.047g _{2} = 2.213g _{3} = 2.878g _{iso} = 2.379 | |

Qa, [Co(pydca)(dmpy)], [C_{14}H_{12}CoN_{2}O_{6}]; pincer type | CAS Theory: Spin–Orbit Multiplets | ||||

[Co(pydca)(dmpy)]·0.5 H_{2} OCCDC 1585697, 100 K, R _{gt} = 0.041 [53] | A: {CoO_{4}N_{2}}Co-N 2.031 Å Co-O 2.152 Å Co-O 2.163 Å D _{str}* = −22.6 pmE _{str}* = 0.55 pm | KD1, 0.78 | KD2, 0.93 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 162 | δ_{5,6} = 813 | δ_{7,8} = 1046 | ||

64·| ± 1/2> + 34·| ± 3/2> | 37·| ± 1/2> + 61·| ± 3/2> | 20·| ± 1/2> + 77·| ± 3/2 > _{2} | 77·| ± 1/2> + 21·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 62, S_{2} = 40, classification 5–fulfilled | ||||

SH-zfs model D = −89.5 cm ^{−1}g _{x} = 2.42g _{z} = 2.50 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 614^{4}Δ_{2} = 2228 | D = −77.2 D _{1} = −93.6D _{2} = +11.0 | E/D = 0.18 E _{1} = −0.01E _{2} = −11.4 | g_{1} = 1.992g _{2} = 2.226g _{3} = 2.945g _{iso} = 2.388 | |

GF model λ _{εff} = −141 cm^{−1}g _{L} = −1.13Δ _{ax} = −811 cm^{−1} | |||||

Qb, [Co(pydca)(dmpy)], [C_{14}H_{12}CoN_{2}O_{6}]; pincer | CAS Theory: Spin–Orbit Multiplets | ||||

B: {CoO_{4}N_{2}}Co-N 2.028 Å Co-O 2.133 Å Co-O 2.176 Å D _{str}* = −22.6 pmE _{str}* = 2.15 pm | KD1, 0.82 | KD2, 0.95 | KD3 | KD4 | |

δ_{1,2} = 0 | δ_{3,4} = 147 | δ_{5,6} = 968 | δ_{7,8} = 1179 | ||

8·| ± 1/2> + 90·| ± 3/2> | 90·| ± 1/2> + 7·| ± 3/2> | 83·| ± 1/2> + 12·| ± 3/2> | 14·| ± 1/2> + 85·| ± 3/2> | ||

Magnetic data as above | SH theory: S_{1} = 107, S_{2} = 96, classification 5–fulfilled | ||||

^{4}Δ_{0} = 0 ^{4}Δ_{1} = 786^{4}Δ_{2} = 2692 | D = −97.1 D _{1} = −112.0D _{2} = +9.6 | E/D = 0.10 E _{1} = −0.08E _{2} = −6.9 | g_{1} = 2.022g _{2} = 2.112g _{3} = 2.898g _{iso} = 2.377 | ||

R, [Co(ac)_{2}(H_{2}O)_{2}(MeIm)_{2}], [C_{12}H_{22}CoN_{4}O_{6}] | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 618142, 100 K, R_{gt} = 0.033 [54,55] | {CoO_{2}O’_{2}N_{2}}Co-N 2.127 Å Co-O’ 2.122 Å Co-O _{w} 2.170 ÅD _{str} = −11.9 pmE _{str} = 2.4 pm | KD1, 0.84 | KD2, 0.93 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 156 | δ_{5,6} = 1030 | δ_{7,8} = 1230 | ||

41·| ± 1/2> + 57·| ± 3/2> | 58·| ± 1/2> + 40·| ± 3/2> | 54·| ± 1/2> + 44·| ± 3/2> | 41·| ± 1/2> + 57·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 123, S_{2} = 70, classification–5 fulfilled | ||||

GF model λ _{eff} = −217 cm^{−1}g _{Lz} = −1.23g _{Lz} = −1.37Δ _{ax} = +568 cm^{−1} | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 878^{4}Δ_{2} = 1593 | D = +75.1 D _{1} = +35.0D _{2} = +23.2 | E/D = 0.16 E _{1} = +35.0E _{2} = −23.0 | g_{1} = 1.910g _{2} = 2.508g _{3} = 2.764g _{iso} = 2.394 | |

SH-zfs model D = +82 cm ^{−1}g _{x} = 2.54g _{z} = 2 | |||||

S, [Co(ampyd)_{2}Cl_{2}], [C_{16}H_{20}Cl_{2}CoN_{12}] | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC SEQFUQ, 293 K, R_{gt} = 0.023 [56,57] | {CoN_{4}Cl_{2}}Co-Cl 2.450 Å Co-N 2.233 Å Co-N 2.233 Å D _{str} = −7.03 pmE _{str} = 0 | KD1, 0.50 | KD2, 0.63 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 262 | δ_{5,6} = 461 | δ_{7,8} = 800 | ||

54·| ± 1/2> + 45·| ± 3/2> | 41·| ± 1/2> + 57·| ± 3/2> | 50·| ± 1/2> + 49·| ± 3/2> | 53·| ± 1/2> + 46·| ± 3/2> | ||

Magnetic data, SMR–n.a. | SH theory: S_{1} = 1, S_{2} = 0.6, classification 1–invalid | ||||

GF model λ _{eff} = −181 cm^{−1}g _{Lz} = −1.5g _{Lx} = −1.3Δ _{ax} = +377 cm^{−1} | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 77^{4}Δ_{2} = 869 | D = +121 D _{1} = +60.9D _{2} = +30.9 | E/D = 0.23 E _{1} = +60.9E _{2} = −30.9 | g_{1} = 1.434g _{2} = 1.900g _{3} = 3.066g _{iso} = 2.133 | |

SH-zfs model D = +146 cm ^{−1}g _{x} = 2.91g _{z} = 2 |

*****denotes the “pincer”-type complexes possessing deviations from the equatorial plane.

T, [Co(dppm^{O,O})_{3}]^{2+}·CoCl_{4}^{2−}, [C_{75}H_{66}CoO_{6}P_{6}]^{2+} CoCl_{4}^{2−} | CAS Theory: Spin–Orbit Multiplets | ||||
---|---|---|---|---|---|

CCDC 296003, 100 K, R_{gt} = 0.066 [48] | {CoO_{3}O’_{3}}Co-O 2.112 Å Co-O’ 2.074 Å | KD1, 0.40 | KD2, 0.72 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 317 | δ_{5,6} = 389 | δ_{7,8} = 925 | ||

50·| ± 1/2> + 49·| ± 3/2> | 49·| ± 1/2> + 49·| ± 3/2> | 49·| ± 1/2> + 50·| ± 3/2> | 48·| ± 1/2> + 50·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 0.7, S_{2} = 0.4, classification 1–invalid | ||||

SH-zfs model D = +77 cm ^{−1}g _{x} = 2.55g _{z} = 2 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 142^{4}Δ_{2} = 142 | D = +158.3 D _{1} = +59.1D _{2} = +59.1 | E/D = 0.00 E _{1} = +59.1E _{2} = −59.1 | g_{1} = 1.781g _{2} = 2.505g _{3} = 2.505g _{iso} = 2.263 | |

Ua, cis-[Co(phen)_{2}(dca)_{2}], [C_{28}H_{16}CoN_{10}] α-polymorph | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 997503, 193 K, R_{gt} = 0.029 [58] | {CoN_{4}N’_{2}}Co-N 2.153 Å Co-N’ _{dca} 2.076 Å | KD1, 0.54 | KD2, 0.65 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 243 | δ_{5,6} = 495 | δ_{7,8} = 838 | ||

19·| ± 1/2> + 79·| ± 3/2> | 82·| ± 1/2> + 17·| ± 3/2> | 82·| ± 1/2> + 15·| ± 3/2> | 8·| ± 1/2> + 90·| ± 3/2> | ||

Magnetic data, SMR–n.a. | SH theory: S_{1} = 2, S_{2} = 2, classification 1–invalid | ||||

SH-zfs model D = +91 cm ^{−1}g _{x} = 2.66g _{z} = 2 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 110^{4}Δ_{2} = 961 | D = 108.2 D _{1} = 63.7D _{2} = 27.3 | E/D = 0.30 E _{1} = 63.7E _{2} = −27.2 | g_{1} = 1.487g _{2} = 1.956g _{3} = 3.085g _{iso} = 2.176 | |

Ub, cis-[Co(phen)_{2}(dca)_{2}], [C_{28}H_{16}CoN_{10}] β-polymorph | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 997504, 293 K, R_{gt} = 0.040 [58] | {CoN_{4}N’_{2}}Co-N 2.153 Å Co-N’ _{dca} 2.071 Å | KD1, 0.76 | KD2, 0.91 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 168 | δ_{5,6} = 737 | δ_{7,8} = 1029 | ||

51·| ± 1/2> + 48·| ± 3/2> | 46·| ± 1/2> + 51·| ± 3/2> | 52·| ± 1/2> + 46·| ± 3/2> | 47·| ± 1/2> + 50·| ± 3/2> | ||

Magnetic data, SMR–n.a. | SH theory: S_{1} = 51, S_{2} = 26, classification 5–fulfilled | ||||

SH-zfs model D = +85 cm ^{−1}g _{x} = 2.60g _{z} = 2 | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 618^{4}Δ_{2} = 1041 | D = 81.2 D _{1} = 40.3D _{2} = 25.7 | E/D = 0.16 E _{1} = 40.1E _{2} = −25.7 | g_{1} = 1.923g _{2} = 2.460g _{3} = 2.768g _{iso} = 2.383 | |

V, [μ-(dca)Co(pypz)(H_{2}O)] dca ^{a} | |||||

CCDC 1973544, 295 K, R_{gt} = 0.033 [59] | {CoN_{3}N’_{2}O}Co-N 2.155 Å Co-N’ 2.076 Å Co-O 2.134 Å | ||||

Magnetic data, SMR–yes | |||||

GF model λ _{eff} = −131 cm^{−1}g _{L} = −2.00Δ _{ax} = −2000 cm ^{−1} | |||||

W, [Co(pypz)_{2}]^{2+}(tcm)^{−}_{2}, [C_{22}H_{18}CoN_{10}]^{2+}·2(C_{3}N_{4})^{− a} | CAS Theory: Spin–Orbit Multiplets | ||||

CCDC 1973546, 295 K, R_{gt} = 0.037 [59] | {CoN_{4}N’_{2}}Co-N’ 2.082 Å Co-N 2.164 Å Co-N 2.164 Å D _{str}* = −8.2 pmE _{str} = 0 | KD1, 0.74 | KD2, 0.91 | KD3 | KD4 |

δ_{1,2} = 0 | δ_{3,4} = 159 | δ_{5,6} = 717 | δ_{7,8} = 1003 | ||

14·| ± 1/2> + 85·| ± 3/2> | 85·| ± 1/2> + 11·| ± 3/2> | 88·| ± 1/2> + 11·| ± 3/2> | 3·| ± 1/2> + 95·| ± 3/2> | ||

Magnetic data, SMR–yes | SH theory: S_{1} = 47, S_{2} = 40, classification 4–acceptable | ||||

GF model λ _{eff} = −87 cm^{−1}g _{L} = −2.77Δ _{ax} = −4000 cm ^{−1} | ^{4}Δ_{0} = 0 ^{4}Δ_{1} = 571^{4}Δ_{2} = 1179 | D = +72.2 D _{1} = +43.4D _{2} = +21.0 | E/D = 0.26 E _{1} = +43.6E _{2} = −21.0 | g_{1} = 1.990g _{2} = 2.349g _{3} = 2.818g _{iso} = 2.384 |

^{a}No ab initio calculations for the chain complex.

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**MDPI and ACS Style**

Boča, R.; Rajnák, C.; Titiš, J.
Zero-Field Splitting in Hexacoordinate Co(II) Complexes. *Magnetochemistry* **2023**, *9*, 100.
https://doi.org/10.3390/magnetochemistry9040100

**AMA Style**

Boča R, Rajnák C, Titiš J.
Zero-Field Splitting in Hexacoordinate Co(II) Complexes. *Magnetochemistry*. 2023; 9(4):100.
https://doi.org/10.3390/magnetochemistry9040100

**Chicago/Turabian Style**

Boča, Roman, Cyril Rajnák, and Ján Titiš.
2023. "Zero-Field Splitting in Hexacoordinate Co(II) Complexes" *Magnetochemistry* 9, no. 4: 100.
https://doi.org/10.3390/magnetochemistry9040100