Magneto-Structural Relationship of Tetragonally-Compressed Octahedral Iron(II) Complex Surrounded by a pseudo-S6 Symmetric Hexakis-Dimethylsulfoxide Environment

Since the octahedral high-spin iron(II) complex has the 5T2g ground term, the spin-orbit coupling should be considered in magnetic analysis; however, such treatment is rarely seen in recent papers, although the symmetry-sensitive property is of interest to investigate in detail. A method to consider the T-term magnetism was well constructed more than half a century ago. On the other hand, the method has been still improved in recent years. In this study, the octahedral high-spin iron(II) complex [Fe(dmso)6][BPh4]2 (dmso: dimethylsulfoxide) was newly prepared, and the single-crystal X-ray diffraction method revealed the tetragonal compression of the D4-symmetric coordination geometry around the iron(II) ion and the pseudo-S6 hexakis-dmso environment. From the magnetic data, the sign of the axial splitting parameter, Δ, was found to be negative, indicating the 5E ground state in the D4 symmetry. The DFT computation showed the electronic configuration of (dxz)2(dx2−y2)1(dyz)1(dxy)1(dz2)1 due to the tetragonal compression and the pseudo-S6 environment of dmso π orbitals. The electronic configuration corresponded to the 5E ground term, which was in agreement with the negative Δ value. Therefore, the structurally predicted ground state was consistent with the estimation from the magnetic measurements.


Introduction
Octahedral high-spin iron(II) complexes have the 5 T 2g ground terms, and their fundamental magnetic theory was constructed in the 1960s and 1970s [1][2][3]. On the other hand, the theory does not seem to be often used recently, probably because the contribution of the orbital angular momentum is difficult to be treated. However, consideration of the spin-orbit coupling is important for understanding the correct electronic state, which leads to an excellent physical property prediction and material design. In this article, for the purpose of showing how to analyze magnetic data of octahedral high-spin iron(II) complexes, we introduce some of the improved new techniques: (1) How to determine the sign of the ligand field splitting, (2) how to simulate field-effect in the low-temperature region, and (3) how to simulate saturation behavior of magnetization.
Metal complexes possessing the T-term ground states often show unusual magnetic behavior due to the spin-orbit coupling. After Kotani proved the contribution of the spinorbit coupling for such compounds [4], the effects of the low-symmetry ligand field and the orbital reduction factor were found to play an important role [5][6][7][8], and the temperature dependence of the effective magnetic moment came to be successfully simulated for mononuclear T-ground-term complexes [2,7,8]. Here, we want to emphasize that the spinorbit splitting caused by the spin-orbit interaction is different from the normal zero-field splitting. The normal zero-field splitting is caused by the spin-spin interaction when the number of unpaired electrons is equal to or larger than two. The spin-orbit splitting is actually a splitting in the zero-field, but is caused by the spin-orbit coupling when the orbital angular momentum remains unquenched. The magnetic behavior caused by the spin-orbit splitting is sometimes approximately analyzed by the theory of zero-field splitting, but it is just an approximation and the data should be analyzed by the theory considering the spin-orbit coupling.
For the octahedral high-spin iron(II) complexes, Griffiths obtained a 15 × 15 secular matrix for the 5 T 2g term [1], considering the spin-orbit coupling. He obtained an energy diagram of the 15 states from the 5 T 2g term with respect to the distortion around the iron(II) ion. Figgis also derived the secular matrix for the 5 T 2g term and successfully simulated the temperature dependence of the effective magnetic moment [2]. Later, Long introduced the orbital reduction factor for the octahedral high-spin iron(II) complexes [3]. Recently, single-ion magnet behavior was observed in high-spin iron(II) complexes [9,10], although they are not octahedral, and the interests to the high-spin iron(II) complexes are increasing. The study on the magneto-structural relationship is expected to give important insight in designing magnetic materials.
When using the theory for the 2 T 2g -ground term complexes [1][2][3], the 15 × 15 secular matric were to be solved each time for an octahedral high-spin iron(II) complex. However, recently, the secular matrix was successfully solved to obtain Zeeman energy equations expressed as the functions of three independent parameters, ∆, λ, and κ, where ∆ is the axial ligand-field splitting parameter, λ is the spin-orbit coupling parameter, and κ is the orbital reduction factor [11]. With these Zeeman energy equations, magnetic simulation can be performed by simply substituting numerical values for parameters without solving the secular matrix each time. In this study, magnetic analysis was conducted for a newly prepared octahedral high-spin iron(II) complex, [Fe(dmso) 6 ][BPh 4 ] 2 (dmso: dimethylsulfoxide), using the equations, for the purpose of gaining further insight into magneto-structural relationship.  (1) was determined by the single-crystal Xray method. Crystallographic data are summarized in Table 1, and the structures of the complex cation, [Fe(dmso) 6 ] 2+ , are shown in Figure 1. The compound consists of the complex cations and the tetraphenyl borate anions in a 1:2 molar ratio. In the complex cation, the six dmso molecules coordinate to the central iron(II) ion through oxygen atoms, forming an octahedral coordination geometry with the O 6 donor set. The cation is centrosymmetric, and tetragonally compressed along the Fe(1)-O(7) direction. The bond distances, Fe(1)-O(5), Fe(1)-O(6), and Fe(1)-O(7), were 2.1408(7), 2.1586(9), and 2.0899(8) Å, respectively. The central FeO 6 S 6 unit can be approximated as the S 6 symmetry (Figure 1a). Four inter-ligand CH···O hydrogen bonds were found in a complex cation (Figure 1b), affording a 16-membered chelating ring perpendicular to the tetragonal compression axis. Earlier, the crystal structure of [Fe(dmso) 6 ][SnCl 6 ] 2 (2) was reported [12], and phase transition behavior was investigated for [Fe(dmso) 6 ][ClO 4 ] 2 [13], but complex 1 has not been reported so far.   Among the six dmso moieties of the [Fe(dmso)6] 2+ complex cation in 1, four of t were found to be disordered, suggesting the sulfur-inversion motion as well as th balt(II) and zinc(II) derivatives [14,15] [16], and [Fe(dmso)6][BPh4]2 (1) complexes cobalt(II) and zinc(II) complexes form isomorphous crystals, while the iron(II) com (1) is isomorphous to the magnesium (II) complex. In the magnesium (II) complex ca a tetragonal compression was observed similar to 1.

Results and Discussion
In the related iron(II) complex 2, the [Fe(dmso)6] 2+ cation was more symmetrical that in 1, and the cation in 2 exactly belongs to the S6 point group. Each dmso moieti 2 showed the similar disorder at 213 K, suggesting the sulfur-inversion motion in the Among the six dmso moieties of the [Fe(dmso) 6 ] 2+ complex cation in 1, four of them were found to be disordered, suggesting the sulfur-inversion motion as well as the cobalt(II) and zinc(II) derivatives [14,15]. Among the crystals of [Co(dmso) 6 [16], and [Fe(dmso) 6 ][BPh 4 ] 2 (1) complexes, the cobalt(II) and zinc(II) complexes form isomorphous crystals, while the iron(II) complex (1) is isomorphous to the magnesium (II) complex. In the magnesium (II) complex cation, a tetragonal compression was observed similar to 1.
In the related iron(II) complex 2, the [Fe(dmso) 6 ] 2+ cation was more symmetrical than that in 1, and the cation in 2 exactly belongs to the S 6 point group. Each dmso moieties in 2 showed the similar disorder at 213 K, suggesting the sulfur-inversion motion in the crystal. The Fe-O distances in 2 [2.121(3) Å] was comparable to the average Fe-O distance in 1 [2.1298(8) Å]. In 2, the octahedral FeO 6 coordination geometry showed the slight trigonal compression along the S 6 axis. Using the conformation notation in [16], the conformer of the main [Fe(dmso) 6 ] 2+ structure in 2 was the "α 6 " conformation, which was considered to be the most stable one. On the other hand, the conformer of the main [Fe(dmso) 6 ] 2+ structure in 1 was "trans-β 2 γ 4 " conformation, which was not so stable on its own. The reason for this unstable conformation is considered to be due to the crystalpacking effect of the bulky tetraphenylborate anions [15][16][17] as discussed in our previous paper on [Mg(dmso) 6 ][BPh 4 ] 2 [16].
In the crystal structure of 1, the complex cation was surrounded by eight tetraphenylborate anions, and the distinct CH···π interactions were observed between the dmso methyl groups and the phenyl rings of the tetraphenylborate anions.

Magnetic Properties
Magnetic susceptibility (χ A ) was measured in the temperature range of 2-300 K, and the χ A T versus T plot is shown in Figure 2. The observed χ A T value at 300 K (3.46 cm 3 K mol −1 ) was larger than the spin-only value for the S = 2 state (3.00 cm 3 K mol −1 ), and this suggests a contribution of the orbital angular momentum. When lowering the temperature, the observed χ A T value slightly increased until at 120 K (3.47 cm 3 K mol −1 at 120 K), and decreased until at 2 K (1.42 cm 3 K mol −1 at 2 K). This behavior, possessing a χ A T maximum, is typical of 5 T 2g -term magnetism for octahedral high-spin iron(II) complexes [2]. Three typical theoretical curves are depicted in Figure 3, based on the Hamiltonian, H = ∆(L z 2 − 2/3) + κλL·S + β(κL u + g e S u )·H u (u = x, y, z) [11], where ∆ is the axial splitting parameter, κ is the orbital reduction factor, and λ is the spin-orbit coupling parameter. In addition, the axiality parameter v, defined as v = ∆/(κλ), was introduced, and a relationship can be seen between the v value and the maximum χ A T temperature (T max ). That is, the larger the v value, the higher the maximum temperature, T max . When the T max value is in the range of 138-150 K, the |v| value is considered to be close to zero; when the T max value is lower than 138 K, the v value is considered to be positive; when the T max value is higher than 150 K, the v value is considered to be negative. Therefore, since the T max value of the observed data is~120 K, the v value is considered to have a positive sign and the ∆ value is considered to be negative, indicating the 5 E ground state ( Figure 4). That is, the 5 T 2 ground term in the O symmetry splits into 5 E and 5 B 2 terms in the D 4 symmetry, and the 5 E term is lower in energy than the 5 B 2 term.  In the earlier works [1][2][3], the Hamiltonian had been s respect to handling the orbital reduction factor. Figgis and c bital reduction factor in the third term (Zeeman term) of th and coworker further introduced the orbital reduction fact orbit coupling term) of the Hamiltonian [3]. Long and cowor but in this study, we used the axiality parameter v = Δ/(κλ) vantages in expressing coefficients. It is noted that Kahn used  In the earlier works [1][2][3], the Hamiltonian had been respect to handling the orbital reduction factor. Figgis and c bital reduction factor in the third term (Zeeman term) of th and coworker further introduced the orbital reduction fact orbit coupling term) of the Hamiltonian [3]. Long and cowor but in this study, we used the axiality parameter v = Δ/(κλ) vantages in expressing coefficients. It is noted that Kahn used octahedra high-spin cobalt(II) complexes [18]. Since our tr In the earlier works [1][2][3], the Hamiltonian had been slightly being modified with respect to handling the orbital reduction factor. Figgis and coworkers introduced the orbital reduction factor in the third term (Zeeman term) of the Hamiltonian [2], and Long and coworker further introduced the orbital reduction factor in the second term (spin-orbit coupling term) of the Hamiltonian [3]. Long and coworker used a parameter v = ∆/λ, but in this study, we used the axiality parameter v = ∆/(κλ) [11], because it has some advantages in expressing coefficients. It is noted that Kahn used a parameter v =∆/|λ| for an octahedra high-spin cobalt(II) complexes [18]. Since our treatment is slightly different from the others, the simulation results are also slightly different from the earlier works.
Using the Figgis basis set [2] for the 5 T 2g term, the secular matrices were constructed [11]. The shapes of the resulting matrices are essentially the same as the Griffiths matrices [1] except for the orbital reduction factor. The exact solution was successfully obtained for the matrices [11], and the zero-field energies and the first-and the second-order Zeeman coefficients were obtained for 15 sub-states of the 5 T 2g term. The zero-field magnetic susceptibility was obtained as the ordinary Van Vleck equation, and in addition, the fielddependent magnetic susceptibility equation was obtained as expressed in Equations (1)- (3).
The powder average of the magnetization is generally expressed by Equation (4). In this study, the expanded equation [Equation (5) with Equation (6)] [19] by calculating the integrals for the axial symmetry was used. We calculated the powder average of the magnetization using Equation (7) with m = 90.
n,θ H θ exp(−E n,θ /kT) ∑ n exp(−E n,θ /kT) , In the analysis, at first, the χ A T versus T data in the range of 10-300 K were analyzed by the zero-field equation (Figure S1), and the obtained parameter set was obtained as (λ, κ, v) = (−100 cm −1 , 0.66, 5.6) with the discrepancy factors of R χ = 3.9 × 10 −3 and R χT = 1.6 × 10 −3 in the range of 10-300 K, and R χ = 2.4 × 10 −1 and R χT = 5.1 × 10 −3 in the range of 2-300 K. The ∆ value was calculated to be −370 cm −1 . The obtained λ value is consistent with the −ζ/4 value with ζ = 400 cm −1 for iron(II) ion [20], where ζ is the single-electron spin-orbit coupling parameter. The small κ value is thought to be due to the π orbital interaction with the dmso ligands, which will be discussed in the following density functional theory (DFT) computation section. The obtained positive v value is consistent with the initial estimation from the χ A T versus T curve. The data in the range of 10-300 K were well reproduced with reasonable parameters; however, the decrease in χ A T below 10 K and the magnetization were not reproduced.
For the decrease in χ A T, in this case, there are two possible reasons, the fieldsaturation effect and the intermolecular antiferromagnetic interactions. If the magnetic field effect of 3000 Oe was considered, using the field-dependent susceptibility equation [Equations (1)-(3)], the obtained parameter set was the same, but the discrepancy factors became slightly better (R χ = 3.8 × 10 −3 and R χT = 1.5 × 10 −3 in the range of 10-300 K; R χ = 2.4 × 10 −1 and R χT = 4.3 × 10 −3 in the range of 2-300 K). By the field effect of 3000 Oe, the calculated χ A T value became 6% smaller at 2 K, and 1.5% smaller at 4 K Therefore, the small contribution of field-saturation effect was confirmed, indicating that the field-saturation effect was not dominant in the χ A T decrease.
In the next approach, the intermolecular interaction was also considered in addition to the field effect, introducing the Weiss constant, θ, for the intermolecular interaction. The full χ A T versus T data (2-300 K) and the field-dependent data of the magnetization were simultaneously analyzed, and both data were successfully fitted as shown in Figure 2. The best-fitting parameter set was obtained as (λ, κ, v, θ) = (−100 cm −1 , 0.66, 5.5, −1.5 K) at H = 3000 Oe, with discrepancy factors of R χ = 2.6 ×10 −3 and R χT = 4.3 ×10 −4 (in the full temperature range). The ∆ value was calculated to be −363 cm −1 , and the obtained θ value corresponded to zJ =~−0.5 cm −1 . The zJ value is consistent with the intermolecular antiferromagnetic interaction through CH···π interactions observed in the crystal structure. Both the full χ A T versus T data and the field dependent data of the magnetization were successfully analyzed with reasonable parameters, and the intermolecular interaction was found to be the dominant factor for the decrease in χ A T below 10 K.
Using the Figgis basis set [2], the lowest three states from the 5 T 2g term are described by the wave functions expressed in Equations (8) It is noted that the coefficients c k depend only on two parameters κ and v. When the ∆ value is negative, the ground state corresponds to the wave functions Ψ 2 and Ψ 3 , and the coefficients c 4 and c 7 will be dominant. Therefore, the ground state can be approximated as the M S = ±2 states. This enables us to calculate the g values of the ground state from the first-order Zeeman coefficients as g z = 2.21 and g x = 0.00, although these values are not ascertained by the electron spin resonance (ESR), because the compound is unfortunately ESR-silent. Further investigation has not been conducted on the single molecule magnet properties.

Density Functional Theory (DFT) Computation
The restricted open-shell Hartree-Fock (ROHF) DFT calculation was conducted for the [Fe(dmso) 6 ] 2+ complex cation, using the crystal structure. The ROHF calculation is suitable for considering d-orbitals with unpaired electrons. The energy levels of five molecular orbitals related to the d-orbitals are shown in Figure 5 together with the depiction of the molecular orbitals. The tetragonal compression axis was taken as the principal axis along the z direction. The x and y axes were taken along the two bisecting directions of the adjacent equatorial donor atoms, because the dmso π orbitals were considered to be oriented parallel to the bisecting directions. Therefore, the resulting d σ orbitals are d z 2 and d xy , and the d π orbitals are d x 2 −y 2 , d xz , and d yz . From the calculation, the d xz orbital was found to be the lowest among the five d-orbitals and to be filled with two electrons, whereas other four orbitals were found to be singly occupied. The energy level of the lowest d xz orbital may look unusually low compared with the d-d separation in the octahedral coordination geometry, but this is normal because only this orbital is doubly occupied. The order of the d σ orbitals were d xy < d z 2 , and this is consistent with the tetragonal compression along the z axis. From the point of view of the crystal field theory, the splitting of the d π orbitals is not so much because the tetragonal compression is very small in this case and the effect of the π orbitals is generally less than 10% of the σ orbital. The order of the calculated d π orbitals were d xz < d x 2 −y 2 < d yz , and this seems to be consistent with the orientation of the dmso π orbitals arranged in the pseudo-S 6 symmetry. That is, the larger the overlap between the d-orbital and the dmso π orbitals, the higher in energy. Therefore, the electronic configuration becomes (d xz ) 2 (d x 2 −y 2 ) 1 (d yz ) 1 (d xy ) 1 (d z 2 ) 1 , and this electronic configuration is consistent with the combination of the tetragonal compression and the orientation of the pseudo-S 6 symmetric dmso π orbital environment. The small orbital reduction factor (κ = 0.66), estimated from the magnetic data, was shown to be consistent with the significant interaction with the dmso π orbitals. Since the (d xz , d yz ) orbitals in the tetragonally compressed D 4 coordination geometry were filled with three electrons, the ground state became 5 E in the D 4 approximation. This is consistent with the negative ∆ value found from the magnetic analysis.

Magneto-Structural Relationship
Now we discuss the magneto-structural relationship especially between the structure and the ∆ value. In the crystal structure, the tetragonal compression was observed; however, the order of the d π orbitals was found to be determined by the orientation of the dmso π orbitals, which was significantly influenced by the orientation of the dmso moieties. In this complex cation, the central iron(II) ion was surrounded by the pseudo-S 6 symmetric hexakis-dmso environment, and the combination of the tetragonal compression and the pseudo-S 6 environment was found to generate the d-orbital splitting in Figure 5, generating the electronic configuration of (d xz ) 2 (d x 2 −y 2 ) 1 (d yz ) 1 (d xy ) 1 (d z 2 ) 1 . In the ideal D 4 symmetric coordination geometry due to the tetragonal-compression, the (d xz , d yz ) orbitals are degenerate. However, in the (d xz , d yz ) orbitals, the d xz orbital becomes lower in energy due to the less overlap with the dmso π orbitals, and the (d xz , d yz ) orbitals became filled with three electrons. This electronic configuration corresponded to the 5 E ground state in the D 4 symmetric coordination geometry. The 5 E ground state directly indicated the negative ∆ value, which was consistent with the magnetic measurements. Judging from the negative sign of the ∆ value, the magnetic anisotropy is considered to be uniaxial, and the tetragonal compression axis (z axis) is considered to be the easy axis.
In the case of the related cobalt(II) complex, [Co(dmso) 6 ][BPh 4 ] 2 [14], tetragonal elongation (along the z axis) was observed, and judging from the large orbital reduction factor, close to the free-ion value, the effect of the dmso π orbitals was thought to be smaller than that in the present iron(II) complex. The pseudo-degenerate (d xz , d yz ) orbitals in the cobalt(II) complex were considered to be higher than the d xy orbital, affording the

Materials and Methods
All the chemicals were commercial products and were used as supplied. Elemental analyses (C, H, and N) were obtained on Elemental Analyzers, Yanaco (Tokyo, Japan) CHN Corder MT-5 and MT-6, at the Elemental Analysis Service Centre of Kyushu University (Fukuoka, Japan). IR spectra were recorded on a JASCO (Tokyo, Japan) FT/IR-4100 FT-IR spectrometer. Differential scanning calorimetry (DSC) measurements were conducted on PerkinElmer DSC 8500 in the temperature range of 303-203 K at scan rate of 30, 10, 5, and 1 K/min. Magnetic susceptibility measurements were made on Quantum Design (San Diego, CA, USA) MPMS XL5min SQUID susceptometer with scan rates of 10.0 K /min for 40-300 K and 0.5 K/min 2-40 K at 0.3 T. The isothermal magnetization was measured on Quantum Design MPMS XL5min SQUID susceptometer at 2 K in an applied field of 0-5 T. The magnetic correction for the sample holder was performed by measurement for the empty capsule. The susceptibilities were corrected for the diamagnetism of the constituent atoms using Pascal's constant.

Crystallography
Colorless single-crystals suitable for X-ray analysis were obtained by slow diffusion of 2-propanol to a dmso solution of 1 under nitrogen. Single-crystal diffraction data were measured on a Rigaku (Tokyo, Japan) XtaLAB PRO MM007 diffractometer. The structure was solved by direct methods and expanded using Fourier techniques. The non-hydrogen atoms were refined anisotropically except for the disordered dmso ligand, and hydrogen atoms were refined using the riding model. The final cycle of full-matrix least squares refinement on F 2 was let to satisfactory converge with R 1 = 0.0327 [I > 2σ(I)]. Final R(F), wR(F 2 ), and goodness of fit agreement factors, as well as details on data collection and analysis are in Table 1.

Conclusions
The octahedral high-spin iron(II) complex [Fe(dmso) 6 ][BPh 4 ] 2 (1) was newly prepared, and the single-crystal X-ray diffraction method revealed the tetragonal compression of the D 4 -symmetric coordination geometry around the iron(II) ion and the pseudo-S 6 hexakisdmso environment. From the magnetic data, the sign of the axial splitting parameter, ∆, was found to be negative, indicating the 5 E ground state in the D 4 symmetry. That is, the 5 T 2 ground term in the O symmetry splits into 5 E and 5 B 2 terms in the D 4 symmetry, and the 5 E term is lower in energy than the 5 B 2 term. The DFT computation showed the electronic configuration of (d xz ) 2 (d x pression and the pseudo-S 6 environment of dmso π orbitals. The electronic configuration corresponded to the 5 E ground term, which was in agreement with the negative ∆ value. Therefore, the structurally predicted ground state was consistent with the estimation from the magnetic measurements. Supplementary Materials: The following are available online at https://www.mdpi.com/2312-748 1/7/2/30/s1, Figure S1: The χT versus T plot and the M versus H plot (insertion).