Zero-Field Splitting in Hexacoordinate Co(II) Complexes

Abstract

A collection of 24 hexacoordinate Co(II) complexes was investigated by ab initio CASSCF + NEVPT2 + SOC calculations. In addition to the energies of spin–orbit multiplets (Kramers doublets, KD) their composition of the spins is also analyzed, along with the projection norm to the effective Hamiltonian. The latter served as the evaluation of the axial and rhombic zero-field splitting parameters and the g-tensor components. The fulfilment of spin-Hamiltonian (SH) formalism was assessed by critical indicators: the projection norm for the first Kramers doublet N(KD1) > 0.7, the lowest g-tensor component g₁ > 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 ⁴Δ₁. 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

Zero-field splitting (zfs) is a phenomenon of existence of fine-structure energy levels that are further split only owing to the external magnetic field. In order to avoid confusions, one has to distinguish between the experimentally verified zfs as existing energy gap(s) and the tools used for the description of zfs based upon theoretical assumptions and approximate methods. ZFS is also a source of the magnetic anisotropy reflected in the different evolution of the magnetization components, such as the easy axis or easy plane. It is generally accepted that the magnetic anisotropy of the easy axis is the crucial factor that prevents the fast magnetic relaxation via the Orbach mechanism, and secures a slow magnetic relaxation as a prerequisite of the single molecule/ion magnetism [1,2].

ZFS can be experimentally determined by various techniques, such as (i) magnetometry, (ii) susceptometry, (iii) electron paramagnetic resonance and its variants (high-field/high-frequency electron magnetic resonance), (iv) far-infrared spectroscopy and its variants in the magnetic field (FIRMS, FDMRS), (v) magnetic circular dichroism, (vi) low-temperature calorimetry, and (vii) inelastic neutron scattering [3,4,5,6,7,8,9,10,11,12,13].

ZFS can be treated as an effect of the spin–orbit interaction that splits the multielectron terms—the energy states referring to an antisymmetric wave function that accounts for the interelectron repulsion (configuration interaction) and the effects of the ligands on the central atom. The active space of kets relevant to the central atom is given by all members of the electron configuration dⁿ 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 d¹ and d⁹, 45 for d² and d⁸, 120 for d³ and d⁷, 216 for d⁴ and d⁶, and 252 for d⁵ 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].

The most common approximate concept utilized for the description of the zfs is the spin-Hamiltonian (SH) theory. This is a drastic simplification based upon reduction of the active space of kets to only a few spin functions |S, M_S>: 2 to 6 kets for d¹ to d⁹ electron configurations. For instance, for the d⁷ 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>.

The basic assumption of the SH formalism is that the effect of the spin–orbit coupling can be treated as a small perturbation. Then, the second-order perturbation theory for non-degenerate ground multielectron term offers an explicit formula for the 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 $H^{\prime }=〈0\left|{\widehat{H}}^{\mathrm{so}}\right|K〉/(E_0-E_K)$ 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.

The core of this review article is to show the limitations of the spin-Hamiltonian approach in treating the zero-field splitting for the difficult cases of hexacoordinate Co(II) and analogous Fe(I) and Ni(III) complexes.

2. Theoretical Analysis

Multielectron terms in atoms are labelled by the angular momentum quantum numbers as |dⁿ: 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. Involved operators and wave functions ^a.

	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	|dn: ν, L, ML, S, MS>	|(νLS), J, MJ>	|Γ, γ, a; S, MS>	|Γ′, γ′, a′>
Irreducible representations b	D(L)(2L + 1): S, P, D, F, G, H, I	2S + 1DJ(2J + 1)	mA(1), mB(1), mE(2), mT(3) b	Γi(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	Γi(2), Γ8(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.

.

The spin–orbit coupling in atoms causes a splitting of the atomic terms into a set of atomic multiplets; these are characterized by the total angular momentum quantum numbers |(νLS), J, M_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 (Γ₁ to Γ₈) 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: Γ₈(4).

The spin–orbit multiplets can be considered as observables since they are eigenvalues/eigenvectors of the operator $\widehat{H}={\widehat{H}}^{\mathrm{ee}}+{\widehat{H}}^{\mathrm{cf}}+{\widehat{H}}^{\mathrm{so}}$. In the variation method, they result from the diagonalization of the interaction matrix $H_{IJ}=〈I\left|\widehat{H}\right|J〉$ where the matrix elements need to be evaluated in an appropriate basis set. The simplest approach utilizes the atomic terms as a basis set for the calculations of multiplets; other bases can be considered as a result of the unitary transformation and thus the final eigenvalues stay invariant.

Let us focus on the d⁷ systems exemplified by the hexacoordinate Co(II) complexes. On symmetry descent from the octahedral geometry, the orbitally triply degenerate ground term is split ⁴T_1g → ⁴E_g $\oplus$ ⁴A_2g (D_4h), and the excited term as ⁴T_2g → ⁴B_2g $\oplus$ ⁴E_g; the orbitally non-degenerate term transforms as ⁴A_2g → ⁴B_1g. On further symmetry descent to the D_2h (isomorphous with C_2v), the additional splitting yields ⁴E_g (D_4h) → ⁴B_3g $\oplus$ ⁴B_2g, whereas the non-degenerate term transforms as ⁴A_2g → ⁴B_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.

There are several important consequences of the ground and the first excited electronic terms on the SH theory. When the ground multielectron term is orbitally degenerate (T_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 ⁴E_g (the above case) and the set of the spin–orbit multiplets is labelled as Γ₆, Γ₆, Γ₇, and Γ₇. 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 ⁴E_g and the first excited ⁴A_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 d⁷ systems.

On further symmetry descent, such as D_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/(E₀–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.

In the opposite distortion to the compressed tetragonal bipyramid, the ground electronic term ⁴A_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).

The impact of the lowest terms on the magnetic properties can be visualized by plotting the effective magnetic moment against the (reduced) temperature as displayed in Figure 4. The Griffith–Figgis theory working in the space of 12 spin–orbit kets |L = 1, M_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 ⁴T_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 ⁴A_2g is filtered off and the magnetic properties are dominated only by the eight members (4 KDs) originating in the ⁴E_g term. (iii) For ν < 10, the ground term is orbitally non-degenerate ⁴A_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

Let us recapitulate the key formulae of the spin Hamiltonian appropriate to the zero-field splitting. The spin Hamiltonian contains the axial D (rhombic E) zfs parameters

$${\widehat{H}}^{\mathrm{zfs}}=D({\widehat{S}}_z^2-{\overrightarrow{S}}^2/3){\hslash }^{-2}+E({\widehat{S}}_x^2-{\widehat{S}}_y^2){\hslash }^{-2}$$

(1)

and it is enriched by the Zeeman term

$${\widehat{H}}_{kl}^Z={\mu }_{\mathrm{B}}B{\hslash }^{-1}(g_x{\widehat{S}}_x\sin {\vartheta }_k\cos {\phi }_l+g_y{\widehat{S}}_y\sin {\vartheta }_k\sin {\phi }_l+g_z{\widehat{S}}_z\cos {\vartheta }_k)$$

(2)

that depends on the polar angles $\{{\vartheta }_k,{\phi }_l\}$ distributed uniformly over a sphere in order to mimic a powder average correctly. There are also higher-order zfs parameters expressed with the help of the Stevens operators [21]. Depending upon the situation, a reduced set of parameters is often utilized (D, g_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 $〈I\left|{\widehat{H}}^{\mathrm{zfs}}+{\widehat{H}}_{kl}^Z(B_m)\right|J〉$ yields energy levels (two KDs for d⁷ 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.

Technically, it is too ambitious to obtain reliable values of the E-parameter from the magnetic data taken above 2 K (usual range). Therefore, E as a rule is neglected. Then, the grids of the magnetic field can be limited to only a few points (e.g., 11) distributed uniformly over the half of the meridian with $\phi =0$.

The magnetic anisotropy can be visualized in the 2D graphs as the separate curves M_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.

The experimental set of DC magnetic data (temperature dependence of the molar magnetic susceptibility χ at B₀ < 0.5 T, and field dependence of the magnetization per formula unit M₁ at T₀ < 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

$$E(P)=\frac{1}{N}{\displaystyle \sum _i^N\left|\frac{P_i^{\mathrm{obs}}-P_i^{\mathrm{calc}}}{P_i^{\mathrm{obs}}}\right|}$$

(3)

$$C(P)=\frac{1}{N}\frac{{\displaystyle \sum _i^N\left|P_i^{\mathrm{obs}}-P_i^{\mathrm{calc}}\right|}}{{\displaystyle \sum _i^N{P}_i^{\mathrm{obs}}}}$$

(4)

In order to balance the dominating low-temperature susceptibility data against the high-temperature tail, the product $P_i={\chi }_i\cdot T_i$ or the effective magnetic moment $P_i=\mu {(\mathrm{eff})}_i$ were also used.

3.2. Griffith–Figgis Model

The GF model is based upon the Hamiltonian working in the space of twelve spin–orbit kets |L = 1, M_L, S, M_S> [25,26].

$$\begin{array}{c}{\widehat{H}}^{\mathrm{GF}}=\underset{\mathrm{spin}–\mathrm{orbit} \mathrm{coupling}}{\underbracket{(-{\lambda }_{}^{\mathrm{sf}}A\kappa )({\overrightarrow{L}}_{\mathrm{p}}\cdot \overrightarrow{S}){\hslash }^{-2}}}+\underset{\mathrm{spin} \mathrm{and} \mathrm{orbital} \mathrm{Zeeman} \mathrm{terms}}{\underbracket{{\mu }_{\mathrm{B}}\overrightarrow{B}\cdot (g_{\mathrm{e}}\overrightarrow{S}+g_{\mathrm{L}}{\overrightarrow{L}}_{\mathrm{p}}){\hslash }^{-1}}}\\ +\underset{\mathrm{axial} \mathrm{distortion}}{\underbracket{{\Delta }_{\mathrm{ax}}({\widehat{L}}_z^2-{\overrightarrow{L}}^2/3){\hslash }^{-2}}}+\underset{\mathrm{rhombic} \mathrm{distortion}}{\underbracket{{\Delta }_{\mathrm{rh}}({\widehat{L}}_x^2-{\widehat{L}}_x^2){\hslash }^{-2}}}\end{array}$$

(5)

where Aλ—spin–orbit splitting parameter modified by the Figgis CI factor A (3/2 for the weak crystal field), Δ_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

$$\begin{array}{l}{\widehat{H}}_{kl}^{\mathrm{GF}}=-A\kappa \lambda ({\overrightarrow{L}}_{\mathrm{p}}\cdot \overrightarrow{S}){\hslash }^{-2}+[{\Delta }_{\mathrm{ax}}({\widehat{L}}_{\mathrm{p},z}^2-{\overrightarrow{L}}_{\mathrm{p}}^2/3){+\Delta }_{\mathrm{rh}}({\widehat{L}}_{\mathrm{p},x}^2-{\widehat{L}}_{\mathrm{p},y}^2)]{\hslash }^{-2}\\ \hspace{1em}\hspace{1em}\hspace{0.17em}+{\mu }_{\mathrm{B}}B{g}_{\mathrm{e}}(\cos {\vartheta }_k{\widehat{S}}_z+\sin {\vartheta }_k\cos {\phi }_l{\widehat{S}}_x+\sin {\vartheta }_k\sin {\phi }_l{\widehat{S}}_y){\hslash }^{-1}\\ \hspace{1em}\hspace{1em}\hspace{0.17em}-{\mu }_{\mathrm{B}}B(A{\kappa }_z\cos {\vartheta }_k{\widehat{L}}_{\mathrm{p},z}+A{\kappa }_x\sin {\vartheta }_k\cos {\phi }_l{\widehat{L}}_{\mathrm{p},x}+A{\kappa }_y\sin {\vartheta }_k\sin {\phi }_l{\widehat{L}}_{\mathrm{p},y}){\hslash }^{-1}\end{array}$$

(6)

where k, l define positions of the grids distributed uniformly over the polar angles $\{{\vartheta }_k,{\phi }_l\}$. In practice, k = 11 distributed along half of the meridian secures the correct powder average for the sample with an axial character. Formally, the above Hamiltonian is isomorphous to the exchange-coupled dimer, possessing the axial zero-field splitting adapted for the powder average. The energy levels obtained by the diagonalization are treated as above. This form allows a reproduction not only of the magnetic susceptibility but also the field dependence of magnetization.

The magnetic anisotropy for either Δ_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).

The general procedure for evaluating the magnetic susceptibility and magnetization has been employed in order to model the temperature dependence of the effective magnetic moment and the field dependence of the magnetization for realistic parameters of the FG model relevant to Co(II) complexes (Figure 7). With Δ_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

The only input for this kind of first-principle calculations is the molecular geometry (atomic coordinates) and the associated basis set. As the basis set, the Gaussian type functions are exclusively used with large enough angular momentum (for example, f- and g-functions for d-orbitals).

In the CASSCF (Complete Active Space Self Consistent Field) method, which is beyond the Hartree–Fock approximation, the coefficients of the linear combination of atomic orbitals (LCAO MO) and the coefficients of the configuration interaction (CI) are evaluated by the variation method. In this version of the multi-configuration method, the electrons in the active space (N, M) have variable occupations unlike those in the inactive space [27]. For transition metal complexes, the complete active space contains N d-electrons in M d-orbitals (10 spinorbitals). For a given CAS, the average energy of all states is minimized, whereas each state is weighted equally (SA-CASSCF). This version of the multi-configuration method accounts partly for the correlation energy (static correlation).

Dynamic electron correlation energy is calculated from excited configuration state functions (CSF) in which electrons are excited into empty orbitals. These CSF $\left|{\tilde{\Phi }}_K\right.〉$ are used in a linear combination [28,29,30].

$$\left|{\tilde{\Psi }}_I\right.〉={\displaystyle \sum _{K\in \mathrm{CAS}}{C}_{KI}\left|{\Phi }_K\right.〉}+{\displaystyle \sum _{K\cancel{\in }\mathrm{CAS}}{T}_{KI}\left|{\tilde{\Phi }}_K\right.〉}$$

(7)

where the expansion coefficients (C and T) can be determined by application of the perturbation theory, such as second-order N-electron valence perturbation theory (NEVPT2). This method provides a second-order correction $\Delta E_I^{\mathrm{PT}2}$ of the total CAS energy for each state. It is a good reference for the interpretation of excitation energies in the electronic d-d spectra.

Magnetic parameters are calculated through the quasi-degenerate perturbation theory, within which the spin–orbit coupling (SOC) operator is diagonalized over the manifold of all CAS states corrected by the NEVPT2 [31]

$$H_{{}_{IJ}}^{\mathrm{QDPT}}={\delta }_{IJ}\left(E_{I,\mathrm{CAS}}+\Delta E_I^{\mathrm{PT}2}\right)+\left.〈{\Psi }_{I,\mathrm{CAS}}^{SM}\right|{\widehat{H}}_{\mathrm{SOC}}+\dots \left|{\Psi }_{J,\mathrm{CAS}}^{S^{\prime }{M}^{\prime }}\right.〉$$

(8)

The SOC operator has the Breit–Pauli form in which the spin–orbit mean-field (SOMF) approximation is used [32]

$${\widehat{H}}_{\mathrm{SOC}}^{\mathrm{SOMF}}={\displaystyle \sum _i{\widehat{z}}_i^{\mathrm{SOMF}}{\widehat{s}}_i}$$

(9)

where ${\widehat{z}}_i^{\mathrm{SOMF}}$ is an appropriately defined effective one-electron operator.

Application of the partitioning technique and/or quasi-degenerate perturbation theory yields the matrix elements of the effective Hamiltonian spanned by the spin manifold of the orbitally non-degenerate ground electronic state as follows

$$H_{M{M}^{\prime }}^{\mathrm{eff}}=E_0{\delta }_{M{M}^{\prime }}+〈{\Psi }_0^{SM}\left|{\widehat{H}}_1\right|{\Psi }_0^{S{M}^{\prime }}〉-{\displaystyle \sum _{I{S}^{″}{M}^{″}}{\Delta }_I^{-1}}〈{\Psi }_0^{SM}\left|{\widehat{H}}_1\right|{\Psi }_I^{S^{″}{M}^{″}}〉\cdot 〈{\Psi }_I^{S^{″}{M}^{″}}\left|{\widehat{H}}_1\right|{\Psi }_0^{S{M}^{\prime }}〉$$

(10)

with the denominator defined by excitation energies ${\Delta }_I^{-1}={(E_I-E_0)}^{-1}$. Then, the D-tensor and g-tensor components arising from the spin–orbit coupling are obtained using closed formulae.

An alternate way of calculating magnetic parameters (used here) is provided by the effective Hamiltonian theory [33,34]. This method is based on the construction of a model Hamiltonian which is projected into the complete space of the CAS Hamiltonian ${\widehat{H}}^{\mathrm{rel}}$ in the sense of des Cloizeaux definition of the effective Hamiltonian

$${\widehat{H}}_{\mathrm{C}}^{\mathrm{eff}}={\displaystyle \sum _k\left|{\tilde{\Psi }}_k^{\mathrm{C}}\right.〉}{E}_k\left.〈{\tilde{\Psi }}_k^{\mathrm{C}}\right|$$

(11)

Here, the effective Hamiltonian reproduces the energy levels of the CAS Hamiltonian E_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.

All calculations were conducted using the ORCA package [14,15,36] in the experimental geometry of Co(II) complexes (neutral or charged). As a basis set, ZORA-def2-SV(P) was used for non-metal atoms, and ZORA-def2-TZVPP for the Co(II) center.

3.4. Generalized Crystal-Field Theory

This method (GCFT) works in the basis set of atomic terms characterized by the angular momentum quantum numbers $\left|I\right.〉=\left|{\mathrm{d}}^n:\nu ,L,M_L,S,M_S\right.〉$. The space covers 120 kets for a d⁷ 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 F₂(L) and F₄(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 $〈J\left|{\widehat{H}}^{\mathrm{ee}}(B,C)+{\widehat{H}}^{\mathrm{cf}}(F_2,F_4)+{\widehat{H}}^{\mathrm{so}}(\xi )\right|I〉$ yields the crystal-field multiplets $\left|K\right.〉=\left|{(\mathrm{d}}^n\nu LS);{ {\mathsf{\Gamma }}^{\prime }}_a,{{\gamma }^{\prime }}_a,a^{\prime }\right.〉$, 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.

The evaluation of the spin-Hamiltonian parameters represents an approximation based upon the construction of the Λ-tensor by means of the second-order perturbation theory

$${\Lambda }_{ab}={\hslash }^{-2}{\displaystyle \sum _{J\ne 0}^{}〈0\left|{\widehat{L}}_a\right|J〉〈J\left|{\widehat{L}}_b\right|0〉/(E_J-E_0)}$$

(12)

where the summation runs over all excited terms $\left|J\right.〉$. The Λ-tensor is used in the definition of the D-tensor ${D^{\prime }}_{ab}=-{\lambda }^2{\Lambda }_{ab}$. In the traceless form

$$D_{ab}={D^{\prime }}_{ab}-{\delta }_{ab}({D^{\prime }}_{xx}+{D^{\prime }}_{yy}+{D^{\prime }}_{zz})/3$$

(13)

the axial and rhombic zero-field splitting parameters are expressed as

$$D=(-{D^{\prime }}_{xx}-{D^{\prime }}_{yy}+2{D^{\prime }}_{zz})/2$$

(14)

$$E=({D^{\prime }}_{xx}-{D^{\prime }}_{yy})/2$$

(15)

In the GCFT calculations, the key role plays an appropriate choice of the crystal field poles F₄(L) and eventually F₂(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

The complexes under study have the shape of an elongated or compressed tetragonal bipyramid with some o-rhombic component in the equatorial plane (

Table 2. Elongated tetragonal bipyramid, D_str > +3 pm.

A, [Co(H2O)6]2+ (OHnic−)2, [CoH12O6]2+ 2(C6H4NO3)−		CAS Theory: Spin–Orbit Multiplets
CCDC FONQUV, 295 K, Rgt = 0.054 [39,40]	{CoO4O’2} Co-O’ 2.113 Å Co-O 2.042 Å Dstr = +7.1 pm Estr = 0	KD1, 0.61 a	KD2, 0.78	KD3	KD4
		δ1,2 = 0	δ3,4 = 209	δ5,6 = 526	δ7,8 = 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 S1 = 7, S2 = 4, classification 1–invalid
	GF model λeff = −188 cm−1 gL = −1.10 Δax = −112 cm−1	4Δ0 = 0 4Δ1 = 199 4Δ2 = 2468	D = −100.9 D1 = −119.9 D2 = +10.5	E/D = 0.16 E1 = −0.01 E2 = −10.8	g1 = 1.762g2 = 1.906g3 = 3.104giso = 2.258
B, [CoIICoIII(L1H2)2(H2O)(ac)]·(H2O)3, [C26H35Co2N2O13] 3(H2O)		CAS Theory: Spin–Orbit Multiplets
CCDC 1440294, 100 K, Rgt = 0.039 [41]	{CoO4O’2} Co-O’ 2.150 Å Co-O 2.061 Å Dstr = +8.9 pm Estr = 0	KD1, 0.71	KD2, 0.88	KD3	KD4
		δ1,2 = 0	δ3,4 = 220	δ5,6 = 736	δ7,8 = 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: S1 = 30, S2 = 17, classification 3–questionable
	GF model λeff = −198 cm−1 gLz = −1.64 gLx = −1.11 Δax = −774 cm−1	4Δ0 = 0 4Δ1 = 444 4Δ2 = 1516	D = −100.9 D1 = −114.8 D2 = +20.0	E/D = 0.25 E1 = −0.01 E2 = −20.0	g1 = 1.842g2 = 2.293g3 = 3.102giso = 2.412
C, trans-[Co(bz)2(H2O)2(nca)2], [C26H26CoN4O8]		CAS Theory: Spin–Orbit Multiplets
CCDC 804191, 293 K, Rgt = 0.038 [42]	{CoO2O’2N2} Co-N 2.147 Å Co-O 2.084 Å Co-O’w 2.143 Å Dstr = +7.75 pm Estr = 1.85 pm	KD1, 0.58	KD2, 0.73	KD3	KD4
		δ1,2 = 0	δ3,4 = 256	δ5,6 = 525	δ7,8 = 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 S1 = 3, S2 = 2, classification 1–invalid
	GF model λeff = −172 cm−1 gLz = −2.06 gLx = −1.50 Δax = −739 cm−1	4Δ0 = 0 4Δ1 = 117 4Δ2 = 1138	D = −113.3 D1 = −131.0 D2 = +26.9	E/D = 0.31 E1 = −0.08 E2 = −27.0	g1 = 1.507g2 = 2.042g3 = 3.160giso = 2.237
D, [Co(acac)2(H2O)2], [C10H18CoO6]		CAS Theory: Spin–Orbit Multiplets
CCDC 1842364, 100 K, Rgt = 0.024 [43]	{CoO2O’2Ow} Co-O 2.040Å Co-O’ 2.034 Å Co-Ow 2.157 Å Dstr = +12.0 pm Estr = 0.30 pm	KD1, 0.81	KD2, 0.93	KD3	KD4
		δ1,2 = 0	δ3,4 = 155	δ5,6 = 915	δ7,8 = 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: S1 = 91, S2 = 48, classification 5–fulfilled
	SH-zfs model from ab initio calculations	4Δ0 = 0 4Δ1 = 763 4Δ2 = 1398	D = +72.0 D1 = +39.7 D2 = +22.7	E/D = 0.23 E1 = −39.6 E2 = −22.8	g1 = 1.943g2 = 2.462g3 = 2.804giso = 2.403
E, [CoL22Cl2]·3.5H2O, [C40H36Cl2CoN4O2]·3.5H2O		CAS Theory: Spin–Orbit Multiplets
CCDC 796703, 150 K, Rgt = 0.045 [44]	{CoN2O2Cl2} Co-N 2.081 Å Co-O 2.034 Å Co-Cl 2.492 Å Dstr = +9.45 pm Estr = 2.65 pm Estr/Dstr = 0.28	KD1, 0.91	KD2, 0.96	KD3	KD4
		δ1,2 = 0	δ3,4 = 94	δ5,6 = 1238	δ7,8 = 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: S1 = 257, S2 = 226, classification 5–fulfilled
	SH-zfs model D = 75.1 cm−1 E = 4.8 cm−1 gz = 2 gx = 2.51 gy = 2.36	4Δ0 = 0 4Δ1 = 1217 4Δ2 = 2039	D = +43.3 D1 = +21.5 D2 = +13.9	E/D = 0.24 E1 = +13.6 E2 = −3.8	g1 = 2.032g2 = 2.341g3 = 2.566giso = 2.313
Fa, [Co(bzpy)4Cl2], [C48H44Cl2CoN4]		CAS Theory: Spin–Orbit Multiplets
CCDC 1497488, 120 K, Rgt = 0.027 [45]	{CoN4Cl2} Unit A Co-Cl 2.443 Å Co-N 2.235 Å Co-N 2.176 Å Dstr = +7.05 pm Estr = 1.15 pm	KD1, 0.69	KD2, 0.89	KD3	KD4
		δ1,2 = 0	δ3,4 = 179	δ5,6 = 633	δ7,8 = 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: S1 = 27, S2 = 19, classification 3–questionable
	GF model/11 λeff = −175 cm−1 gLz = −1.02 gLx = −1.28 Δax = −424 cm−1	4Δ0 = 0 4Δ1 = 448 4Δ2 = 993	D = +87.6 D1 = +43.2 D2 = +31.6	E/D = 0.13 E1 = + 43.1 E2 = −31.4	g1 = 1.948g2 = 2.498g3 = 2.779giso = 2.408
	SH-zfs model D = +106 cm−1 gx = 2.53 gz = 2
Fb, [Co(bzpy)4Cl2], [C48H44Cl2CoN4]		CAS Theory: Spin–Orbit Multiplets
Structure as above for Fa	{CoN4Cl2} Unit B Co-Cl 2.433 Å Co-N 2.187 Å Co-N 2.169 Å Dstr = −3.5 pm Estr = 0.9 pm E/|D| = 0.26	KD1, 0.49	KD2, 0.68	KD3	KD4
		δ1,2 = 0	δ3,4 = 252	δ5,6 = 455	δ7,8 = 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 S1 = 2, S2 = 1, classification 1–invalid
		4Δ0 = 0 4Δ1 = 130 4Δ2 = 804	D = +120.9 D1 = + 56.2 D2 = +34.8	E/D = 0.17 E1 = + 56.2 E2 = −34.7	g1 = 1.604g2 = 2.163g3 = 2.942giso = 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%); ^mΔ_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.

). Distances on the trans-ordinate were averaged and further processed as follows. Assuming that the highest eccentricity lies along the z-axis, the structural distortion parameters are defined as

$$D_{\mathrm{str}}={(d_i-{\overline{d}}_i)}_z-[{(d_i-{\overline{d}}_i)}_x+{(d_i-{\overline{d}}_i)}_y]/2$$

(16)

$$E_{\mathrm{str}}=[{(d_i-{\overline{d}}_i)}_x-{(d_i-{\overline{d}}_i)}_y]/2$$

(17)

where ${\overline{d}}_i$ is the mean distance for a given bond (i = N, O, Cl). These have been taken from compounds containing the [Co(NH₃)₆]²⁺, [Co(H₂O)₆]²⁺ and [CoCl₆]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.

In some cases, the compound contains two crystallographic independent complex units. The ab initio calculations were performed for each of them. Some compounds contain identical or similar structural units. In the series [Co(dppm^O,O)₃][Co(NCS)₄] (J), [Co(dppm^O,O)₃][CoBr₄] (K) and [Co(dppm^O,O)₃][CoI₄] (L) with the chromophores {CoO₂O’₂O”₂}, the values of D_str vary as −1.65, −1.45, and +2.15 pm; [Co(dppm^O,O)₃][CoCl₄] (U) has a different symmetry of the chromophore {CoO₃O’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)₄Cl₂] (F) contains two crystallographic different units with D_str = +7.05 and −3.5 pm.

4.2. Elongated Tetragonal Bipyramid

Some hexacoordinate Co(II) complexes possess the geometry of the chromophore close to the elongated tetragonal bipyramid with eventual small o-rhombic component. Their structural parameter (axiality) is D_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 M₁ = M_mol/(N_Aμ_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).

The key results of the ab initio calculations and fitted magnetic data (susceptibility and magnetization) are presented in Table 2. Certain calculations were redone with respect to the published data in order to keep the same basis set. The success of the spin-Hamiltonian theory is classified as either 5—fulfilled, 4—acceptable, 3—questionable, 2—problematic, or 1—invalid. However, such a classification is rather subjective; a more objective classification is the quantitative evaluation of the spin Hamiltonian according to the score S₁, introduced as follows:

S₁ = N(KD1)·g₁·[E(KD3) − E(KD2)]·Δ₁/10,000

(18)

The set of critical parameters involves the norm of the projected state for the first Kramers doublet N(KD1), the smallest g-tensor component g₁, the separation of the two subsets of KDs [E(KD3) − E(KD2)], and the first transition energy at the NEVPT2 level ⁴Δ₁ (scaled to smaller numbers). Values of S₁ > 50 refer to class 5—fulfilled; S₁ < 10 span class 1—invalid. Limiting value for the fulfillment is S₁ = 0.7 × 1.9 × 600 × 600/10,000 = 48. The above parameter can be extended by considering the mixing of the spin states: S₂ = S₁ × P/100 where P—percentage of the greater portion of spins | ± 1/2> or | ± 3/2> in the first KD (ideally p > 70).

The simplest complex containing the hexa-aqua ligands is [Co(H₂O)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 ⁴T_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 ⁴E_g and the excited ⁴A_2g (Δ₂ = 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, g₁ = 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.

The complex [Co^IICo^III(L¹H₂)₂(H₂O)(ac)]·(H₂O)₃ (B) contains the homogeneous donor set {CoO₄O’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 Δ₂ = 1516 cm⁻¹. The secondary splitting of ⁴E_g term is Δ₁ = 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, g₁ = 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.

The complex trans-[Co(bz)₂(H₂O)₂(nca)₂] (C), after the corrections to the heterogeneous donor set {CoO₂O_w2N₂}, possesses D_str = +7.75; its room-temperature effective magnetic moment reaches μ_eff ~ 5 μ_B and the magnetization saturates close to M₁ ~ 3. These features are typical for systems with the ground electronic term ⁴E_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, g₁ = 1.51, ⁴Δ₁ = 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.

In the complex [Co(acac)₂(H₂O)₂] (D) with the chromophore {CoO₄O_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 ⁴Δ₁ = 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 g₁ = 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(D² + 3E²)^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_Aμ_B) = 2.35 at B = 7.0 T.

The complex [CoL²₂Cl₂] with the {CoO₂N₂Cl₂} 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, g₁ = 2.03, ⁴Δ₁ = 1217 cm⁻¹, and a weak mixing of spin states exists: D = +43 cm⁻¹. The estimated energy gap G_3,4 =2(D² + 3E²)^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}.

The compound [Co(bzpy)₄Cl₂] 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 ⁴Δ₁ = 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 ⁴Δ₁ = 130 cm⁻¹, and subnormal g₁ = 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

Since the distance-corrected values $\overline{d}$(Co-O) and $\overline{d}$(Co-N) can vary [46], complexes with small negative or small positive D_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. Nearly octahedral systems, |D_str| < 2.5 pm.

Ga, [Co(hfac)2(etpy)2], [C24H20CoF12N2O4]		CAS Theory: Spin–Orbit Multiplets
CCDC 2223471, 100 K, Rgt = 0.050	A: {CoO4N2} Co-N 2.132 Å Co-O 2.056 Å Co-O 2.048 Å Dstr = −2.0 pm Estr = 0.4 pm	KD1, 0.50	KD2, 0.73	KD3	KD4
		δ1,2 = 0	δ3,4 = 237	δ5,6 = 461	δ7,8 = 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: S1 = 2, S2 = 1, classification 1–invalid
	GF model λeff = −159 cm−1 gLz = −1.96 gLx = −1.79 Δax = −771 cm−1	4Δ0 = 0 4Δ1 = 109 4Δ2 = 785	D = +112 D1 = +59.8 D2 = +33.4	E/D = 0.20 E1 = 58.8 E2 = −33.4	g1 = 1.661 g2 = 2.043 g3 = 2.932 giso = 2.212
Gb, [Co(hfac)2(etpy)2], [C24H20CoF12N2O4]		CAS Theory: Spin–Orbit Multiplets
	B: {CoO4N2} Co-N = 2.151Å Co-O 2.040 Å Co-O 2.058 Å Dstr = −1.45 pm Estr = 0.35 pm	KD1, 0.64	KD2, 0.87	KD3	KD4
		δ1,2 = 0	δ3,4 = 196	δ5,6 = 568	δ7,8 = 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: S1 = 16, S2 = 10, classification 2–problematic
		4Δ0 = 0 4Δ1 = 359 4Δ2 = 901	D = +94 D1 = +47.9 D2 = +29.7	E/D = 0.18 E1 = −47.9 E2 = 29.7	g1 = 1.931 g2 = 2.351 g3 = 2.808 giso = 2.364
H, [Co(hfac)2(bzpyCl)2], [C34H22Cl2CoF12N2O4]		CAS Theory: Spin–Orbit Multiplets
CCDC 2223472, 100 K, Rgt = 0.036	{CoO4N2}* Co-N 2.137 Å Co-O 2.061 Å Co-O 2.062 Å Dstr = −2.45 pm Estr = 0.05 pm	KD1, 0.58	KD2, 0.83	KD3	KD4
		δ1,2 = 0	δ3,4 = 188	δ5,6 = 582	δ7,8 = 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: S1 = 17, S2 = 13, classification 2–problematic
	GF model λeff = −170 cm−1 gLz = −1.83 gLx = −1.11 Δax = −643 cm−1	4Δ0 = 0 4Δ1 = 392 4Δ2 = 905	D = +91 D1 = +45.9 D2 = +29.6	E/D = 0.16 E1 = 45.7 E2 = −29.0	g1 = 1.954 g2 = 2.372 g3 = 2.781 giso = 2.369
I, [Co(abpt)2(tcm)2], [C32H20CoN18]		CAS Theory: Spin–Orbit Multiplets
CCDC 997721, 173 K, Rgt = 0.036 [47]	{CoN4N’2}* Co-N’ 2.133 Å Co-N 2.109 Å Co-N 2.125 Å Dstr = −2.0 pm Estr = 0.4 pm	KD1, 0.86	KD2, 0.96	KD3	KD4
		δ1,2 = 0	δ3,4 = 131	δ5,6 = 862	δ7,8 = 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: S1 = 115, S2 = 91, classification 5–fulfilled
	SH-zfs model D = +55 cm−1 E = 14.6 cm−1 gx = 2.53 gz = 2	4Δ0 = 0 4Δ1 = 900 4Δ2 = 1878	D = +50.3 D1 = +28.5 D2 = +17.6	E/D = 0.29 E1 = +28.5 E2 = −17.6	g1 = 2.037 g2 = 2.333 g3 = 2.636 giso = 2.335
J, [Co(dppmO,O)3][Co(NCS)4], [C75H66CoO6P6]2+ Co(NCS)42−		CAS Theory: Spin–Orbit Multiplets
CCDC 1526142, 100 K, Rgt = 0.041 [48]	{CoO2O’2O”2} Co-O 2.094 Å Co-O’ 2.089 Å Co-O” 2.074 Å Dstr = −1.65 pm Estr = 0.35 pm	KD1, 0.61	KD2, 0.86	KD3	KD4
		δ1,2 = 0	δ3,4 = 211	δ5,6 = 562	δ7,8 = 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: S1 = 21, S2 = 11, classification 2–problematic
	SH-zfs model D = +93 cm−1 gx = 2.76 gz = 2	4Δ0 = 0 4Δ1 = 445 4Δ2 = 539	D = +105.5 D1 = +46.3 D2 = +42.0	E/D = 0.03 E1 = +45.7 E2 = −41.9	g1 = 1.972 g2 = 2.592 g3 = 2.688 giso = 2.417
K, [Co(dppmO,O)3][CoBr4], [C75H66CoO6P6]2+ CoBr42−		CAS Theory: Spin–Orbit Multiplets
CCDC 1526141, 100 K, Rgt = 0.044 [49]	{CoO2O’2O”2} Co-O 2.109 Å Co-O’ 2.102 Å Co-O” 2.091 Å Dstr = −1.45 pm Estr = 0.35 pm	KD1, 0.61	KD2, 0.86	KD3	KD4
		δ1,2 = 0	δ3,4 = 211	δ5,6 = 562	δ7,8 = 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: S1 = 19, S2 = 10, classification 2–problematic
	SH-zfs model D = +122 cm−1 gx = 2.68 gz = 2	4Δ0 = 0 4Δ1 = 445 4Δ2 = 539	D = +105.5 D1 = +46.3 D2 = +42.53	E/D = 0.03 E1 = +45.7 E2 = −41.9	g1 = 1.972 g2 = 2.592 g3 = 2.688 giso = 2.417
L, [Co(dppmO,O)3][CoI4], [C75H66CoO6P6]2+ CoI42−		CAS Theory: Spin–Orbit Multiplets
CCDC 1526143, 100 K, Rgt = 0.028 [48]	{CoO2O’2O”2} Co-O 2.092 Å Co-O’ 2.076 Å Co-O” 2.065 Å Dstr = +2.15 pm Estr = 0.55 pm	KD1, 0.57	KD2, 0.80	KD3	KD4
		δ1,2 = 0	δ3,4 = 223	δ5,6 = 508	δ7,8 = 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: S1 = 8, S2 = 4, classification 1–invalid
	SH-zfs model D = +99 cm−1 gx = 2.70 gz = 2	4Δ0 = 0 4Δ1 = 258 4Δ2 = 732	D = +107.9 D1 = +54.6 D2 = +34.1	E/D = 0.15 E1 = +54.6 E2 = −34.1	g1 = 1.860 g2 = 2.319 g3 = 2.868 giso = 2.349

.

The compound [Co(hfac)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; g₁ = 1.66 (subnormal), ⁴Δ₁ = 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 ⁴Δ₁ = 359 cm⁻¹. The critical indicators are N(KD1) = 0.64; g₁ = 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.

An analogous complex [Co(hfac)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; g₁ = 1.95, ⁴Δ₁ = 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.

The molecular complex [Co(abpt)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; g₁ = 2.04, ⁴Δ₁ = 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 Γ₆. Perhaps large rhombicity E/D = 0.29 causes this feature.

Three complexes of the type [Co(dppm^O,O)₃][CoX₄], 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 {CoO₃O’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.

The complex cation in [Co(dppm^O,O)₃][Co(NCS)₄] (J) possesses δ{(0, 211), (562, 966)} cm⁻¹, g₁ = 1.97, ⁴Δ₁ = 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.

The complex cation in [Co(dppm^O,O)₃][CoBr₄] (K) behaves analogously to its NCS analogue: δ{(0, 211), (562, 966)} cm⁻¹, g₁ = 1.97, ⁴Δ₁ = 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.

The complex [Co(dppm^O,O)₃][CoI₄] (L) with δ{(0, 223), (508, 874)} cm⁻¹ shows different critical parameters: ⁴Δ₁ = 258 cm⁻¹ (near degeneracy), subnormal g₁ = 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.

In summary, the ab initio calculations for nearly octahedral Co(II) complexes predicted D > 0 when the spin Hamiltonian was appropriate and matching the magnetic data fitting.

4.4. Compressed Tetragonal Bipyramid

This numerous group involves complexes with a considerable negative axiality of D_str << 3 (

Table 4. Compressed tetragonal bipyramid, D_str < −3 pm.

M [Co(iz)6]2+(fm−)2, [C18H24CoN12]2+ 2(CHO2)−		CAS Theory: Spin–Orbit Multiplets
CCDC 624939, 296 K, Rgt = 0.034 [39,50]	{CoN4N’2} Co-N’ 2.211 Å Co-N 2.197 Å Co-N’ 2.143 Å Dstr = −6.10 pm Estr = 0.71 pm	KD1, 0.46	KD2, 0.54	KD3	KD4
		δ1,2 = 0	δ3,4 = 256	δ5,6 = 450	δ7,8 = 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: S1 = 0.4, S2 = 0.2, classification 1–invalid
	SH-zfs model D = +69.2 cm−1 gx = 2.75 gz = 2	4Δ0 = 0 4Δ1 = 35 4Δ2 = 591	D = +124.0 D1 = +62.4 D2 = +39.0	E/D = 0.15 E1 = +61.9 E2 = −37.0	g1 = 1.302 g2 = 1.829 g3 = 2.974 giso = 2.035
Na, [Co(bzpy)4(NCS)2], [C50H44CoN6S2]		CAS Theory: Spin–Orbit Multiplets
CCDC 1497489, 120 K, Rgt = 0.036 [45]	{CoN4N’2} Unit A Co-N’ 2.086 Å Co-N 2.217 Å Co-N 2.180 Å Dstr = −11.7 pm Estr = 1.35 pm	KD1, 0.68	KD2, 0.88	KD3	KD4
		δ1,2 = 0	δ3,4 = 187	δ5,6 = 646	δ7,8 = 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: S1 = 28, S2 = 22, classification 3–questionable
	SH-zfs model D = +90.5 cm−1 gx = 2.52 gz = 2	4Δ0 = 0 4Δ1 = 473 4Δ2 = 838	D = +88.9 D1 = +47.2 D2 = +30.7	E/D = 0.17 E1 = +47.0 E2 = −30.4	g1 = 1.932 g2 = 2.446 g3 = 2.823 giso = 2.400
Nb, [Co(bzpy)4(NCS)2], [C50H44CoN6S2]		CAS Theory: Spin–Orbit Multiplets
	Unit B Co-N’ 2.094 Å Co-N 2.213 Å Co-N 2.196 Å Dstr = −11.0 pm Estr = 0.85 pm	KD1, 0.67	KD2, 0.88	KD3	KD4
		δ1,2 = 0	δ3,4 = 189	δ5,6 = 638	δδ7,8 = 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: S1 = 28, S2 = 22, classification 3–questionable
		4Δ0 = 0 4Δ1 = 481 4Δ2 = 776	D = +91.7 D1 =+47.0 D2 = +32.1	E/D = 0.15 E1 = +46.6 E2 = −31.6	g1 = 1.938 g2 = 2.466 g3 = 2.806 giso = 2.403
O, [Co(pydm)2]2+(dnbz)−2, [C14H18CoN2O4]2+·2(C7H3N2O6)− pincer type		CAS Theory: Spin–Orbit Multiplets
CCDC 1533249, 100 K, Rgt = 0.037 [51]	{CoO4N2} Co-N 2.039 Å Co-O 2.110 Å Co-O 2.171 Å Dstr* = −20.15 Estr* = 3.05	KD1, 0.71	KD2, 0.89	KD3	KD4
		δ1,2 = 0	δ3,4 = 188	δ5,6 = 864	δ7,8 = 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: S1 = 58, S2 = 44, classification 5–fulfilled
	SH-zfs model D = −62 cm−1 gz = 2.13 gx = 2	4Δ0 = 0 4Δ1 = 615 4Δ2 = 2199	D = −91.8 D1 = −103.0 D2 = +8.9	E/D = 0.13 E1 = −0.3 E2 = −11.4	g1 = 1.983 g2 = 2.169 g3 = 3.058 giso = 2.403
P, [Co(pydm)2]2+(dmnbz)−2, [C14H18CoN2O4]2+·2(C8H5N2O6)−; pincer type		CAS Theory: Spin–Orbit Multiplets
CCDC 1945478, 100 K, Rgt = 0.042 [52]	{CoO4N2} Co-N 2.038 Å Co-O 2.120 Å Co-O 2.114 Å Dstr* = −17.9 pm Estr* = 0.30 pm	KD1, 0.68	KD2, 0.88	KD3	KD4
		δ1,2 = 0	δ3,4 = 145	δ5,6 = 870	δ7,8 = 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: S1 = 72, S2 = 41, classification 5–fulfilled
	SH-zfs model D = −50.0 cm−1 gz = 2.30 gx = 2	4Δ0 = 0 4Δ1 = 708 4Δ2 = 1831	D = −69.0 D1 = −86.9 D2 = +11.7	E/D = 0.19 E1 = −0.02 E2 = −11.8	g1 = 2.047 g2 = 2.213 g3 = 2.878 giso = 2.379
Qa, [Co(pydca)(dmpy)], [C14H12CoN2O6]; pincer type		CAS Theory: Spin–Orbit Multiplets
[Co(pydca)(dmpy)]·0.5 H2 O CCDC 1585697, 100 K, Rgt = 0.041 [53]	A: {CoO4N2} Co-N 2.031 Å Co-O 2.152 Å Co-O 2.163 Å Dstr* = −22.6 pm Estr* = 0.55 pm	KD1, 0.78	KD2, 0.93	KD3	KD4
		δ1,2 = 0	δ3,4 = 162	δ5,6 = 813	δ7,8 = 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: S1 = 62, S2 = 40, classification 5–fulfilled
	SH-zfs model D = −89.5 cm−1 gx = 2.42 gz = 2.50	4Δ0 = 0 4Δ1 = 614 4Δ2 = 2228	D = −77.2 D1 = −93.6 D2 = +11.0	E/D = 0.18 E1 = −0.01 E2 = −11.4	g1 = 1.992 g2 = 2.226 g3 = 2.945 giso = 2.388
	GF model λεff = −141 cm−1 gL = −1.13 Δax = −811 cm−1
Qb, [Co(pydca)(dmpy)], [C14H12CoN2O6]; pincer		CAS Theory: Spin–Orbit Multiplets
	B: {CoO4N2} Co-N 2.028 Å Co-O 2.133 Å Co-O 2.176 Å Dstr* = −22.6 pm Estr* = 2.15 pm	KD1, 0.82	KD2, 0.95	KD3	KD4
		δ1,2 = 0	δ3,4 = 147	δ5,6 = 968	δ7,8 = 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: S1 = 107, S2 = 96, classification 5–fulfilled
		4Δ0 = 0 4Δ1 = 786 4Δ2 = 2692	D = −97.1 D1 = −112.0 D2 = +9.6	E/D = 0.10 E1 = −0.08 E2 = −6.9	g1 = 2.022 g2 = 2.112 g3 = 2.898 giso = 2.377
R, [Co(ac)2(H2O)2(MeIm)2], [C12H22CoN4O6]		CAS Theory: Spin–Orbit Multiplets
CCDC 618142, 100 K, Rgt = 0.033 [54,55]	{CoO2O’2N2} Co-N 2.127 Å Co-O’ 2.122 Å Co-Ow 2.170 Å Dstr = −11.9 pm Estr = 2.4 pm	KD1, 0.84	KD2, 0.93	KD3	KD4
		δ1,2 = 0	δ3,4 = 156	δ5,6 = 1030	δ7,8 = 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: S1 = 123, S2 = 70, classification–5 fulfilled
	GF model λeff = −217 cm−1 gLz = −1.23 gLz = −1.37 Δax = +568 cm−1	4Δ0 = 0 4Δ1 = 878 4Δ2 = 1593	D = +75.1 D1 = +35.0 D2 = +23.2	E/D = 0.16 E1 = +35.0 E2 = −23.0	g1 = 1.910 g2 = 2.508 g3 = 2.764 giso = 2.394
	SH-zfs model D = +82 cm−1 gx = 2.54 gz = 2
S, [Co(ampyd)2Cl2], [C16H20Cl2CoN12]		CAS Theory: Spin–Orbit Multiplets
CCDC SEQFUQ, 293 K, Rgt = 0.023 [56,57]	{CoN4Cl2} Co-Cl 2.450 Å Co-N 2.233 Å Co-N 2.233 Å Dstr = −7.03 pm Estr = 0	KD1, 0.50	KD2, 0.63	KD3	KD4
		δ1,2 = 0	δ3,4 = 262	δ5,6 = 461	δ7,8 = 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: S1 = 1, S2 = 0.6, classification 1–invalid
	GF model λeff = −181 cm−1 gLz = −1.5 gLx = −1.3 Δax = +377 cm−1	4Δ0 = 0 4Δ1 = 77 4Δ2 = 869	D = +121 D1 = +60.9 D2 = +30.9	E/D = 0.23 E1 = +60.9 E2 = −30.9	g1 = 1.434 g2 = 1.900 g3 = 3.066 giso = 2.133
	SH-zfs model D = +146 cm−1 gx = 2.91 gz = 2

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

). 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.

The complex [Co(iz)₆](fm)₂ (M) with the homogeneous ligand sphere contains the {CoN₆} 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(H₂O)₆](OHnic)₂ (A). There is a set of critical indicators warning that the spin-Hamiltonian theory fails: N(KD1) = 0.46, ⁴Δ₁ = 35 cm⁻¹ (orbital degeneracy), g₁ = 1.30, g₂ = 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.

The compound [Co(bzpy)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, ⁴Δ₁ = 473 cm⁻¹ and g₁ = 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.

The cationic complex of [Co(pydm)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 ⁴Δ₁ = 614 cm⁻¹. The critical parameters indicate that the SH might be fulfilled: N(KD1) = 0.71, g₁ = 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.

An analogous compound contains the same cationic complex [Co(pydm)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 ⁴Δ₁ = 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.

The compound [Co(pydca)(dmpy)]·0.5H₂O 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 ⁴Δ₁ = 614 cm⁻¹. The critical indicators signalize that the SH is fulfilled: N(KD1) = 0.78, g₁ = 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.

The complex [Co(ac)₂(H₂O)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 ⁴Δ₁ = 878 cm⁻¹. The critical indicators show that the SH is fulfilled: N(KD1) = 0.84 and g₁ = 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).

The complex trans-[Co(ampyd)₂Cl₂] (S) possesses D_str = −7.03 and the critical indicators warn that the SH fails: N(KD1) = 0.50, subnormal g₁ = 1.43, and g₂ = 1.90, very small transition energy ⁴Δ₁ = 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

This section involves data for complexes that do not span the above three classes: the structural axiality D_str is either undefined or oddly defined (

Table 5. Miscellaneous geometry.

T, [Co(dppmO,O)3]2+·CoCl42−, [C75H66CoO6P6]2+ CoCl42−		CAS Theory: Spin–Orbit Multiplets
CCDC 296003, 100 K, Rgt = 0.066 [48]	{CoO3O’3} Co-O 2.112 Å Co-O’ 2.074 Å	KD1, 0.40	KD2, 0.72	KD3	KD4
		δ1,2 = 0	δ3,4 = 317	δ5,6 = 389	δ7,8 = 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: S1 = 0.7, S2 = 0.4, classification 1–invalid
	SH-zfs model D = +77 cm−1 gx = 2.55 gz = 2	4Δ0 = 0 4Δ1 = 142 4Δ2 = 142	D = +158.3 D1 = +59.1 D2 = +59.1	E/D = 0.00 E1 = +59.1 E2 = −59.1	g1 = 1.781 g2 = 2.505 g3 = 2.505 giso = 2.263
Ua, cis-[Co(phen)2(dca)2], [C28H16CoN10] α-polymorph		CAS Theory: Spin–Orbit Multiplets
CCDC 997503, 193 K, Rgt = 0.029 [58]	{CoN4N’2} Co-N 2.153 Å Co-N’dca 2.076 Å	KD1, 0.54	KD2, 0.65	KD3	KD4
		δ1,2 = 0	δ3,4 = 243	δ5,6 = 495	δ7,8 = 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: S1 = 2, S2 = 2, classification 1–invalid
	SH-zfs model D = +91 cm−1 gx = 2.66 gz = 2	4Δ0 = 0 4Δ1 = 110 4Δ2 = 961	D = 108.2 D1 = 63.7 D2 = 27.3	E/D = 0.30 E1 = 63.7 E2 = −27.2	g1 = 1.487 g2 = 1.956 g3 = 3.085 giso = 2.176
Ub, cis-[Co(phen)2(dca)2], [C28H16CoN10] β-polymorph		CAS Theory: Spin–Orbit Multiplets
CCDC 997504, 293 K, Rgt = 0.040 [58]	{CoN4N’2} Co-N 2.153 Å Co-N’dca 2.071 Å	KD1, 0.76	KD2, 0.91	KD3	KD4
		δ1,2 = 0	δ3,4 = 168	δ5,6 = 737	δ7,8 = 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: S1 = 51, S2 = 26, classification 5–fulfilled
	SH-zfs model D = +85 cm−1 gx = 2.60 gz = 2	4Δ0 = 0 4Δ1 = 618 4Δ2 = 1041	D = 81.2 D1 = 40.3 D2 = 25.7	E/D = 0.16 E1 = 40.1 E2 = −25.7	g1 = 1.923 g2 = 2.460 g3 = 2.768 giso = 2.383
V, [μ-(dca)Co(pypz)(H2O)] dca a
CCDC 1973544, 295 K, Rgt = 0.033 [59]	{CoN3N’2O} Co-N 2.155 Å Co-N’ 2.076 Å Co-O 2.134 Å
Magnetic data, SMR–yes
	GF model λeff = −131 cm−1 gL = −2.00 Δax = −2000 cm−1
W, [Co(pypz)2]2+(tcm)−2, [C22H18CoN10]2+·2(C3N4)− a		CAS Theory: Spin–Orbit Multiplets
CCDC 1973546, 295 K, Rgt = 0.037 [59]	{CoN4N’2} Co-N’ 2.082 Å Co-N 2.164 Å Co-N 2.164 Å Dstr* = −8.2 pm Estr = 0	KD1, 0.74	KD2, 0.91	KD3	KD4
		δ1,2 = 0	δ3,4 = 159	δ5,6 = 717	δ7,8 = 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: S1 = 47, S2 = 40, classification 4–acceptable
	GF model λeff = −87 cm−1 gL = −2.77 Δax = −4000 cm−1	4Δ0 = 0 4Δ1 = 571 4Δ2 = 1179	D = +72.2 D1 = +43.4 D2 = +21.0	E/D = 0.26 E1 = +43.6 E2 = −21.0	g1 = 1.990 g2 = 2.349 g3 = 2.818 giso = 2.384

^a No ab initio calculations for the chain complex.

). It shows a versatility of the magnetic behavior of hexacoordinate Co(II) complexes.

The complex [Co(dppm^O,O)₃][CoCl₄] (T) spans the series [Co(dppm^O,O)₃][CoX₄], but unlike the NCS⁻, Br⁻, and I⁻ members, it displays different geometry of the {CoO₃O_3′} chromophore so that axiality D_str is not defined in this case. The critical indicators warn that the SH is not fulfilled, since g₁ = 1.78, ⁴Δ₁ = 150.6, and ⁴Δ₂ = 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.

The complex cis-[Co(phen)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, g₁ = 1.49, very low transition energy ⁴Δ₁ = 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 S₁ = 51 (mainly due to the high first transition energy ⁴Δ₁ = 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.

The complex cation in [Co(pypz)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, ⁴Δ₁ = 571 cm⁻¹. The critical indicators confirm that the SH is fulfilled: N(KD1) = 0.74, g₁ = 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 M₁ > 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.

The complex [μ-(dca)Co(pypz)(H₂O)] 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

The calculated ab initio data were used to form a worksheet for modern statistical analysis [60]. The Cluster Analysis divides the observables according to the “distance” into four or five groups; for codes, see Figure 8.

Complementary information brings the biplot of the Principal Component Analysis: (i) with decreasing D1, the energy of K2 increases and simultaneously the energies of K3 and K4 decrease; (ii) at the same time, the critical indicators g1 and N decrease, thus showing a failure of the spin-Hamiltonian formalism; (iii) D correlates with Ds; and (iv) C anticorrelates with D (with increasing calculated D, the classification factor C decreases from 5 to 1).

The classification score of the spin Hamiltonian (between 1 to 5) correlates with the first transition energy ⁴Δ₁ (Figure 9). For Δ₁ < 300 cm⁻¹, the SH data are barely reliable (class 1) because of the quasi-degeneracy. For Δ₁ > 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.

The sign of the axial zero-field splitting parameter D attracts great attention, mainly in the light of the D-U paradigm, according to which the barrier to spin reversal U for the Orbach process of slow magnetic relaxation fulfills the relationship U = |D|(S² − 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).

There is an exception for complexes containing the pincer-type ligands when the donor set occupies sites outside the axes of the equatorial plane. This point has already been modelled by using GCFT calculations, as depicted in Figure 11 [51].

The GCFT allows a wide-range modelling of the energies of spin–orbit multiplets (KDs) δ_i,i+1 and SH parameters (D, g_z, g_x, χ_TIP) depending upon the strength of the crystal field poles F₄(ax) and F₄(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 ⁴E_g and the two lowest multiplets span the irreducible representations Γ₆ and Γ₆.

A majority of hexacoordinate Co(II) complexes investigated in this work were tested for a slow magnetic relaxation (SMR), and all of the tested cases confirm the presence of SMR. The existence of SMR is independent of the geometry—whether the complex belongs to the elongated or compressed tetragonal bipyramid, the nearly octahedral, or miscellaneous geometry. The contemporary state of the art is as follows (Figure 13). (i) A more careful data selection at low frequencies of the oscillating AC magnetic field reveals the second (LF) and eventually third (IF) relaxation channel in addition to the high-frequency (HF) one. These channels are strongly dependent upon the applied DC field. (ii) The low-frequency relaxation channel attenuates on heating more progressively than the HF one. (iii) The temperature ranges, in which the maximum (maxima) on the out-of-phase susceptibility is visible, are often limited to T < 6 K. (iv) The analysis of the relaxation data according to the Arrhenius-like equation τ = τ₀exp(U/k_BT) is appropriate only for the Orbach relaxation process. Using a few high-temperature data, the evaluation of the extrapolated relaxation time (for infinite temperature) τ₀, 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, τ₀} 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].

The final critical remark is addressed to the spin-Hamiltonian formalism that considers the existence of only two KDs separated by 2D. If this gap, for instance, is only G = 100 cm⁻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 τ₀ 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

The hexacoordinate Co(II) complexes can be classified into four groups according to their structural axiality (tetragonality) D_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 ⁴E_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 d⁷ configuration.

The activation of the spin-Hamiltonian formalism in the first-principle calculations yields the D-parameters that could be false. Typically, D > 0 holds true for hexacoordinate Co(II) complexes with the exception of those with pincer-type ligands. The perfect fulfilment of the spin-Hamiltonian formalism is rather rare; some critical indicators allow a classification as 5—fulfilled, 4—acceptable, 3—questionable, 2—problematic, and 1—invalid. Of 24 compounds containing 28 hexacoordinate complexes studied by ab initio method, only 10 span the categories 5 and 4, and 9 the category 1. The failure of the SH manifests itself in low first transition energy° ⁴Δ₁ < 300 cm⁻¹, low projection norm N(KD1) < 0.7, subnormal value of the lowest g-factor g₁ < 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.

The D values obtained by fitting the magnetic data are risky to accept without deep theoretical analysis. The first energy gap given by the energy of the second Kramers doublet G = δ_3,4 can be reconstructed by assuming G = 2|D|, or G = 2(D² + 3E²)^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 ⁴T_1g; four of them can be close in energy while the spin-Hamiltonian formalism recognizes only two of them.