Effect of spin–orbit coupling on the magnetic susceptibility of polynuclear complexes of 3d metals containing a Co²⁺ ion

Using a program we developed allowing us to take into account zero-field splitting, spin-orbit, and exchange interactions in polynuclear complexes, we have interpreted the temperature dependences of the magnetic susceptibility for the complexes Fe₂CoO(CF₃CO₂)₆(H₂O)₃·H₂O (1) and [Co(H₂O)₆][Cu₃L₂(H₂O)]·3H₂O (2, H₄L = 1,9-dicyano- 1,9-bis(hydroximino)-3,7-diazanonane-2,8-dione). We have analyzed the differences between the values of the simulation parameters determined within different theoretical models.

Inseparably connected with the name of Academician V. D. Pokhodenko is the study of fine electronic effects in the physical chemistry of free radicals that are due to the presence of an unpaired electron in the molecule [1,2]. It is interesting to note that back in the 1980s [3], the Pokhodenko scientific school focused special attention on the study of exchange interactions between spin moments in metal radical systems. In the 1990s, owing to the initiative of V. D. Pokhodenko, at the L. V. Pisarzhevskii Institute of Physical Chemistry (National Academy of Sciences of Ukraine) research began in the field of spin-correlated systems based on polynuclear complexes of transition metals and polynuclear complexes with free radicals, with the aim of obtaining molecular magnets.

Study of polynuclear and multiple-spin systems containing magnetically anisotropic transition metal ions is a timely problem in modern physical inorganic chemistry, since magnetic anisotropy is one of the factors facilitating the appearance of “molecular magnet” properties [4]. The appearance of magnetic anisotropy is connected with a difference between magnetic characteristics (for example, magnetic susceptibility χ) along different crystallographic axes. Such a difference may arise due to anisotropy of the electronic structure of an individual paramagnetic ion in the compound, i.e., splitting of the electronic levels in which unpaired electrons may be found that is different along different coordinate axes. In turn, anisotropy of the electronic structure of the ion may arise as a consequence of various factors, where the major factors are spin–orbit coupling (SOC) (and its special case: zero-field splitting (ZFS)), and also the action of a crystal field. While interpretation of the magnetic properties of mononuclear compounds of transition metals taking into account spin–orbit coupling or zero-field splitting is rather trivial [5], taking into account the simultaneous effect of spin–orbit coupling (or ZFS) and exchange interactions in a polynuclear compound is much more complicated, since in the overwhelming majority of cases the corresponding spin Hamiltonians do not have analytical solutions.

The aim of this work included determination of the contribution of spin–orbit coupling and zero-field splitting of the Co²⁺ ion to the magnetic properties of the polynuclear complexes Fe₂CoO(CF₃CO₂)₆(H₂O)₃·H₂O (compound 1) and [Co(H₂O)₆][Cu₃L₂(H₂O)]·3H₂O (compound 2, H₄L = 1,9-dicyano-1,9-bis(hydroximino)-3,7-diazanonane-2,8-dione) and comparison of results obtained using models taking into account ZFS and spin–orbit coupling and also models neglecting the anisotropy of the electronic structure in these compounds.

Spin–orbit coupling is numerically characterized by a spin–orbit coupling constant λ, which is introduced by means of a Hamiltonian of the following form [5]:

$$ {\hat{H}_{\rm{SOC}}} = A{\rm{\kappa \lambda }}\hat{L}\hat{S} + {{\rm{\mu }}_{\rm{B}}}\left( {A{\rm{\kappa }}\hat{L} + {g_e}\hat{S}} \right)H $$

(1)

where A is the ratio of the angular momentum of the terms in the ground states of the ion in the considered compound and in the free ion (for the high-spin Co²⁺ ion, A = −3/2); κ is the orbital contraction parameter, and the rest of the symbols have their conventional meanings.

ZFS is splitting of the levels in the absence of an external magnetic field under the influence of weak direct coupling between spins, the electrostatic field of the ligands [5], and also can arise due to the appearance of spin–orbit coupling [6]. Numerically, ZFS is characterized by the parameters D and E, which are introduced through the corresponding spin Hamiltonian [5–7]

$$ {\hat{H}_{\rm{ZFS}}} = D\left[ {\hat{S}_z^2 - \frac{1}{3}S\left( {S + 1} \right)} \right] + E\left( {\hat{S}_x^2 - \hat{S}_y^2} \right) $$

(2)

where D and E are the axial and rhombic ZFS parameters respectively.

In the case of many 3d metals, the effect of spin–orbit coupling on the magnetic characteristics of the compound can be neglected, due to the phenomenon of “freezing” of the orbital moment [8]. However, for compounds containing the Co²⁺ ion, generally the model neglecting spin–orbit coupling does not allow us to describe the experimental data, i.e., the orbital moment of Co²⁺ cannot be considered to be equal to zero. The accidental coincidence of the results of simulation neglecting spin–orbit coupling and the experimental curve (which is sometimes called “the λ = 0 andD= 0 case”) does not have physical meaning.

⁴ T _1g is the ground-state term for the high-spin Co²⁺ ion in an octahedral environment. In most Co²⁺ compounds, the symmetry of this ion can be represented as axial (D _4h ), usually the ground state is the orbital singlet (M _L = 0) ⁴ A _2g while the excited state is the orbital doublet (M _L= ±1) ⁴ E _g . These levels are split by the effect of spin–orbit coupling, in particular the ⁴ A _2g level is split into two Kramers doublets [5].

The spin levels of the Co²⁺ ion in an axially distorted octahedral environment is described by a Hamiltonian taking into account axial splitting of the levels by the crystal field, spin–orbit coupling, and Zeeman splitting:

$$ {\hat{H}_{{\rm{Co - full}}}} = \Delta \left[ {\hat{L}_z^2 - \frac{1}{3}L\left( {L + 1} \right)} \right] - \frac{3}{2}{\rm{\kappa \lambda }}\hat{L}\hat{S} + {{\rm{\mu }}_{\rm{B}}}\left( { - \frac{3}{2}{\rm{\kappa }}\hat{L} + {g_e}\hat{S}} \right)H. $$

(3)

The basis functions of Hamiltonian (3) for the Co²⁺ ion have the form $\left| {{M_L},\left. {{M_S}} \right\rangle } \right.$ , , whereM _L = 0, ±1,M _S = ±1/2, ±3/2. In the case when the value of ∆ is sufficiently large, the thermal population is significant for only two lower Kramers doublets arising from the state ⁴ A _2g . The energy difference between these doublets (2D _eff) can be considered as the ZFS of the quartet state ⁴ A _2g of the ion with “effective” L = 0. In this case, the Hamiltonian for the cobalt ion can be represented in the form of Eq. (2), in whichDcorresponds toD _eff, E = 0 (the full Hamiltonian also includes the term ${\hat{H}_{\rm{Zee}}} = g{{\rm{\mu }}_{\rm{B}}}H\hat{S}$). In this case, the effect of spin–orbit coupling is apparent in the presence of a nonzero D _eff, and also in the difference between g _∥ and g _⊥, where the values of g_∥ are about 2 and the g _⊥ values range from 2 to 4.33 (in the case of axial symmetry, g _x = g _y ). The values of D _eff for the Co²⁺ ion usually are several tens of cm^–1.

In the case of a polynuclear compound, exchange interactions also can lead to the appearance of levels with different energies. The splitting due to exchange interactions between the i-th and j-th paramagnetic centers is numerically characterized by the parameter J _i,j , which is introduced as follows [5]:

$$ {\hat{H}_{\rm{ex}}} = - 2\sum\limits_{i,j} {{J_{i,j}}{{\hat{S}}_i}{{\hat{S}}_j}} $$

(4)

where i ≠ j, and each interaction ${\hat{S}_i}$ and${\hat{S}_j}$ is taken into account once.

The Hamiltonian presented above can be rewritten in the form of matrices in a basis of orthonormalized Ψ functions, differing only in the projections of the spin and/or orbital moment of the individual ions. These functions can be represented in the form $\left| {{M_{{S_1}}},{M_{{S_2}}},\,\,\,,{M_{{S_N}}},{M_{{L_1}}},\,\,\,,\left. {{M_{{L_M}}}} \right\rangle } \right.$ (M _L for the atoms, where L ≠ 0), and in such a case, the eigenfunctions of the Hamiltonian are some linear combinations of all possible Ψ functions of this paramagnetic system. Determination of the energies of the spin levels of the system involves calculating the eigenvalues of the Hamiltonian. In order to solve this problem, we developed the program Mjöllnir. The program allows us to calculate the χT values for polynuclear systems, taking into account isotropic exchange interactions and ZFS (Eq. (5)) or isotropic exchange interactions and spin–orbit coupling (Eq. (6)) while neglecting mixing of high-lying states (except for temperature-independence paramagnetism, which is introduced as conventionally in [8]):

$$ {\hat{H}_{{\rm{ex + ZFS}}}} = - 2\sum\limits_{i,j} {{J_{i,j}}{{\hat{S}}_i}{{\hat{S}}_j} + \sum\limits_i {{D_i}\hat{S}_{zi}^2 + \sum\limits_i {{{\rm{\mu }}_{\rm{B}}}{g_i}{{\hat{S}}_i}H,} } } $$

(5)

$$ {\hat{H}_{{\rm{ex + SOC}}}} = - 2\sum\limits_{i,j} {{J_{i,j}}{{\hat{S}}_i}{{\hat{S}}_j} + \Delta \hat{L}_z^2 - \frac{3}{2}{\rm{\kappa \lambda }}\hat{L}\hat{S} + {{\rm{\mu }}_{\rm{B}}}\left( { - \frac{3}{2}{\rm{\kappa }}\hat{L} + {g_e}\hat{S}} \right)H.} $$

(6)

The Mjöllnir program allows us to take into account the contribution of a paramagnetic impurity with a certain spin and the effect of the molecular field (the parameter zJ'). The number of spin levels in the complex compound should not be greater than 5·10³. This program can be obtained from the authors of this paper on request.

As noted above, the expression taking into account the contribution of ZFS allows us to a certain extent to describe the effect of spin–orbit coupling on the electronic structure of the metal ion. In many cases, this makes interpretation of the experimental data considerably easier, since in calculating the χ values for polynuclear compounds, taking into account ZFS is considerably simpler than fully taking into account spin–orbit coupling. In the literature, examples are described for use of a model including only ZFS (for example, [9]), and also models taking into account spin–orbit coupling (for example, [10]).

In the program, the magnetic susceptibility is calculated by numerical diagonalization of the matrices of the spin Hamiltonians of the corresponding system described above. In the models, the two nonequivalent axes are considered individually: the z axis (parallel to the field) and the x axis (perpendicular to the field; the x and y axes are equivalent), which is due to the anisotropic nature of ZFS and axial splitting by the crystal field. In the interpretation of the measurement results for a polycrystalline sample, the values of χ in the direction of the different axes are averaged using the expression χ = (χ_∥ + 2χ_⊥ )/3.

The initial step in running the Mjöllnir program is analysis of the user-specified model, containing a list of ions in the complex compound, the values of the total spin, the total angular momentum, the symmetry of each ion (isotropic symmetry corresponds to the absence of ZFS for L = 0 or the absence of a contribution from axial splitting by the crystal field (delta) for L ≠0, while axial symmetry corresponds to taking them into account). Based on the model, a set of basis psi functions is created ${\Psi_i} = \left| {{M_{{S_1}}},{M_{{S_2}}},\,\,\,,{M_{{S_N}}},{M_{{L_1}}},\,\,\,,\left. {{M_{{L_M}}}} \right\rangle } \right.$ (M _L for the atoms where L ≠ 0). For each term in the spin Hamiltonian, its matrix $\left\langle {{{\Psi_j}}} \mathrel{\left | {\vphantom {{{\Psi_j}} {{\rm{\hat{A}}}\left| {{\Psi_i}} \right.}}} \right. } {{{\rm{\hat{A}}}\left| {{\Psi_i}} \right.}} \right\rangle$ is calculated in explicit form (each element of which is either zero or contains an expression with the parameters of this term). The matrices are summed (symbolic summation) and are retained for calculation of the energies of the spin levels in calculating the dependence of χT on T for concrete values of the simulation parameters.

In simulation of the curve, the user inputs into the program the desired simulation parameters (J, D, g, etc.) and the magnetic field induction. These values are substituted into the matrix of expressions calculated in the preceding step, which gives a numerical matrix; such a matrix is numerically diagonalized using the procedure implemented in the LAPACK library [11]. The eigenvalues obtained are equal to the energies E _i of the levels in the system. Then the eigenvalues of the analogous matrix are calculated but with a field value equal to H' = H + dH, where dH is specified by the user. Based on the results obtained, the values of the derivatives $\partial {E_i}/\partial {H}$ are found by numerical differentiation and then are used to determine the dependence of χT on T:

$$ \chi T = \frac{{{N_{\rm{A}}}{\rm{\mu }}_{\rm{B}}^2}}{{3kH}}\frac{{\sum\limits_i {\left( { - \partial {E_i}/\partial H} \right)\exp } \left( { - {E_i}/kT} \right)}}{{\sum\limits_i {\exp \left( { - {E_i}/kT} \right)} }}. $$

(7)

The experimental data and the data calculated from the parameters input by the user are compared using the value of R ², calculated from the formula R ² =∑ (χ_exp T - χ_calc T)2/ ∑(χ_exp T)²

In order to test the validity of the program operation, we analyzed previously published results of measurement of the temperature dependence of the magnetic susceptibility for the polynuclear complexes [${\rm{Fe}}_2^{\rm{III}}$Ni^IIO(CH₃CO₂)₆(H₂O)₃]·3H₂O,[${\rm{Fe}}_2^{\rm{III}}$M^IIO(CH₃CO₂)₆(py)₃]·py (M = Mg²⁺, Mn²⁺) [12], [${\rm{Fe}}_3^{\rm{III}}$O(CH₃CO₂)₆(H₂O)₃]Cl·2H₂O [12,13],[${\rm{Fe}}_2^{\rm{III}}$Mn^IIO(CF₃CO₂)₆(H₂O)₃]·H₂O, [${\rm{Fe}}_2^{\rm{III}}$Ni^IIO(CF₃CO₂)₆(H₂O)₃]·H₂O [14], not containing anisotropic ions, and also data for the Co²⁺ complexes, interpreted taking into account ZFS [15], and in all cases we obtained agreement between the simulation parameters and the values presented by the authors of the corresponding papers.

The approach presented above was used for interpretation of previously published results from measurement of χ for complexes 1 [14] and 2 [16] taking into account spin–orbit coupling of the Co²⁺ ion. The structures of these compounds are described in detail in [14,16].

In complex 1, the two Fe²⁺ ions and the Co²⁺ ion are located at the vertices of an isosceles triangle and are connected by a μ₃-O bridge and bridging trifluoracetate ions (Fig. 1a). Table 1 gives the results of simulation of the χT vs. T dependence using the different models: only exchange interactions (the Hamiltonian ${\hat{H}_{\rm{ex}}}{ = } - {2}{J_{\rm{FeCo}}}\left( {{{\hat{S}}_{\rm{Fe1}}}{{\hat{S}}_{\rm{Co}}}{ + }{{\hat{S}}_{\rm{Co}}}{{\hat{S}}_{\rm{Fe2}}}} \right) - {2}{J_{\rm{FeFe}}}{\hat{S}_{\rm{Fe1}}}{\hat{S}_{\rm{Fe2}}}{, }$, which is a special case of Eq. (4)), exchange interactions plus ZFS (Eq. (5)), and exchange interactions plus spin–orbit coupling (Eq. (6)). In all three cases, the values of J _FeFe proved to be within the range from −39 to −43 cm^–1. At the same time, taking into account ZFS or spin–orbit coupling led to a decrease in the absolute value of J _FeCo compared with the model taking into account only exchange interactions, from −17.8(1.4) to −12.0(5) cm^–1. The calculated dependence of χT on T for 2 is shown in Fig. 1b.

Fig. 1

a) Structure of the complex Fe₂CoO(CF₃CO₂)₆(H₂O)₃;b) χT vs. T for Fe₂CoO(CF₃CO₂)₆(H₂O)₃(points) and the calculated curve (curves for both models used visually coincide).

Table 1 Results of Simulation of Magnetic Properties of Complexes Fe₂CoO(CF₃CO₂)₆(H₂O)₃·H₂O (1) and [Co(H₂O6][Cu₃L₂(H₂O)]·3H₂O (2)

In compound 2, the Co²⁺ ion is found in an environment of six coordinated water molecules, and the cation obtained [Co(H₂O)₆]²⁺ is connected with the trinuclear unit [Cu₃L₂(H₂O]²– by a system of hydrogen bonds, which can transmit exchange interactions (Fig. 2a). The magnetic properties of this compound were interpreted previously in the range 200–300 K, neglecting spin–orbit coupling and ZFS for the Co²⁺ ion and assuming that there are no exchange interactions between the Co²⁺ and Cu2+ ions [16]. Although from the shape of the dependence of χT on T we can hypothesize that there is ferromagnetic exchange between Co²⁺ and Cu²⁺, it was impossible to test this hypothesis due to the complexity of the calculation of the χT vs. T dependence taking into account spin–orbit coupling of the Co²⁺ ion. In the first approximation, taking into account the different Co—Cu distances in compound 2, in this compound we can identify tetranuclear fragments containing three Cu²⁺ ions and a Co²⁺ ion at the vertices of a rhombus. Exchange interactions in such a system can be described by the Hamiltonian

$$ {\hat{H}_{\rm{ex}}} = - 2{J_{{\rm{Cu - Cu}}}}\left( {{{\hat{S}}_{\rm{Cu1}}}{{\hat{S}}_{\rm{Cu2}}} + {{\hat{S}}_{\rm{Cu2}}}{{\hat{S}}_{\rm{Cu3}}}} \right) - 2{J_{\rm{CuCu}}}\left( {{{\hat{S}}_{\rm{Cu1}}}{{\hat{S}}_{\rm{Co}}} + {{\hat{S}}_{\rm{Cu3}}}{{\hat{S}}_{\rm{Co}}}} \right) $$

(see Fig. 2a, insert). The value of J, describing the interaction between the Cu1 and Cu3 ions, has practically no effect on the shape of the χT vs. T dependence (which matches the conclusions of [16]).

Fig. 2

a) Fragment of the crystal structure of [Co(H₂O)₆][Cu₃L₂(H₂O)]·3H₂O (the O—Hdistances are indicated; the insert gives the symbols for the exchange parameters J). b) χT vs. T for [Co(H₂O)₆][Cu₃L₂(H₂O)]·3H₂O (points) and the curve calculated with the parameters for Eq. (5) (see Table 1) (solid line) and the curve calculated with the same parameters but J _CuCo = 0 (dotted line). c) Curve calculated with the parameters for Eq. (6) (see Table 1) (solid line) and the curve calculated with the same parameters but J _CuCo = 0 (dashed line).

Taking into account exchange interactions between the Cu²⁺ and Co²⁺ ions (J _CuCo) allows us to achieve agreement between the calculated curves for the χT vs. T dependence and the experimental data over the entire studied temperature range, with R ² on the order of 10^–3 to 10^–4 (Fig. 2b,c, Table 1, the formula for calculating R ² is given above). Note that on the experimental χT vs. T curve for 2, we observe a discontinuity at a temperature of about 80 K, which can be a sign of a phase transition (a similar discontinuity was also observed on the χT vs. T curve for the manganese-containing analog [16]). In the case of both models (taking into account ZFS or spin–orbit coupling), the parameter J _CuCo is positive and has a substantial effect on the shape of the calculated χT vs. T dependence (Fig. 2b,c). As proposed in [16], the presence of ferromagnetic exchange interactions in 2 makes it possible to explain the high values of χT for 2 at low temperatures. Use of our program allowed us to confirm the hypothesis that Cu—Co ferromagnetic exchange interactions occur in 2 and let us estimate the numerical value of J _CuCo.

In the case of 1 and 2, the values of λ and ∆ lie within a range typical for octahedral complexes of cobalt(II) [17]. The greater value of ∆ in complex 1 compared with 2 is consistent with the differences in the coordination spheres of the Co²⁺ ions (negatively charged ligands in 1 create a stronger crystal field than the water molecules in 2). In the case of both complexes, the use of models with spin-orbit coupling made it possible to achieve better agreement between the calculated and experimental χT vs. T dependences (according to the R ² criterion). At the same time, when using this model, good agreement is achieved for values of g(Co) greater than 2.0023, which is not consistent with theory (in Eq. (3), we should use g values equal to the g factor for the free electron) and can be the consequence of effects not taken into account by the given model.

Thus in our program Mjöllnir, spin–orbit coupling and/or ZFS of 3d metals in a polynuclear complex are taken into account by introducing the corresponding terms into the Hamiltonian, with its subsequent solution by numerical full-matrix diagonalization. The parameters obtained taking into account ZFS or spin–orbit coupling are different from the parameters obtained within a model describing only exchange interactions. In the case of complex 1 with Δ = 900 cm^–1, the values of J obtained using the different models are close, which shows that the model taking into account the “effective” ZFS is applicable in the case of large Δ. In the case of complex 2 with Δ = 150 cm^–1, the parameters J determined are different, and the magnetic properties of this compound are most correctly described taking into account spin–orbit coupling of the Co²⁺ ion.

We would like to thank V. A. Bogaenko (Candidate of Technical Sciences) and S. A. Litvinenko for helping develop the program. This work was partially supported by a joint project of the National Academy of Sciences of Ukraine and the Russian Foundation for Basic Research (project 7/2R).