2.1. Crystal Structure of 1
Crystal structure of [Fe(dmso)
6][BPh
4]
2 (
1) was determined by the single-crystal X-ray 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
6S
6 unit can be approximated as the
S6 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 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][BPh
4]
2, [Zn(dmso)
6][BPh
4]
2, [Mg(dmso)
6][BPh
4]
2 [
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
S6 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
S6 axis. Using the conformation notation in [
16], the conformer of the main [Fe(dmso)
6]
2+ structure in
2 was the “
α6” 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-
β2γ4” conformation, which was not so stable on its own. The reason for this unstable conformation is considered to be due to the crystal-packing 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.
2.2. Magnetic Properties
Magnetic susceptibility (
χA) was measured in the temperature range of 2–300 K, and the
χAT versus T plot is shown in
Figure 2. The observed
χAT 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
χAT 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
χAT maximum, is typical of
5T2g-term magnetism for octahedral high-spin iron(II) complexes [
2].
Three typical theoretical curves are depicted in
Figure 3, based on the Hamiltonian,
H =
Δ(
Lz2 − 2/3) +
κλL·
S +
β(
κLu +
geSu)·
Hu (
u =
x,
y,
z) [
11], where
Δ 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 =
Δ/(
κλ), was introduced, and a relationship can be seen between the
v value and the maximum
χAT temperature (
Tmax). That is, the larger the
v value, the higher the maximum temperature,
Tmax. When the
Tmax value is in the range of 138–150 K, the |
v| value is considered to be close to zero; when the
Tmax value is lower than 138 K, the
v value is considered to be positive; when the
Tmax value is higher than 150 K, the
v value is considered to be negative. Therefore, since the
Tmax value of the observed data is ~120 K, the
v value is considered to have a positive sign and the
Δ value is considered to be negative, indicating the
5E ground state (
Figure 4). That is, the
5T2 ground term in the
O symmetry splits into
5E and
5B2 terms in the
D4 symmetry, and the
5E term is lower in energy than the
5B2 term.
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 =
Δ/
λ, but in this study, we used the axiality parameter
v =
Δ/(
κλ) [
11], because it has some advantages in expressing coefficients. It is noted that Kahn used a parameter
v =
Δ/|
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
5T2g 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
5T2g term. The zero-field magnetic susceptibility was obtained as the ordinary Van Vleck equation, and in addition, the field-dependent 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.
In the analysis, at first, the
χAT 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
Rχ = 3.9 × 10
−3 and
RχT = 1.6 × 10
−3 in the range of 10–300 K, and
Rχ = 2.4 × 10
−1 and
RχT = 5.1 × 10
−3 in the range of 2–300 K. The
Δ 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
χAT versus T curve. The data in the range of 10–300 K were well reproduced with reasonable parameters; however, the decrease in
χAT below 10 K and the magnetization were not reproduced.
For the decrease in χAT, in this case, there are two possible reasons, the field-saturation 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 (Rχ = 3.8 × 10−3 and RχT = 1.5 × 10−3 in the range of 10–300 K; Rχ = 2.4 × 10−1 and RχT = 4.3 × 10−3 in the range of 2–300 K). By the field effect of 3000 Oe, the calculated χAT 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 χAT 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
χAT 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
Rχ = 2.6 ×10
−3 and
RχT = 4.3 ×10
−4 (in the full temperature range). The
Δ 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
χAT 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
χAT below 10 K.
Using the Figgis basis set [
2], the lowest three states from the
5T2g term are described by the wave functions expressed in Equations (8)–(10), where
(
= 0, ±1, ±2 and
= 0, ±1, ±2) are wave functions and
ck (
k = 0 − 35) are coefficients. It is noted that the coefficients
ck depend only on two parameters
κ and
v. When the
Δ value is negative, the ground state corresponds to the wave functions
and
, and the coefficients
c4 and
c7 will be dominant. Therefore, the ground state can be approximated as the
= ±2 states. This enables us to calculate the
g values of the ground state from the first-order Zeeman coefficients as
gz = 2.21 and
gx = 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.
2.3. 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
dσ orbitals are
dz2 and
dxy, and the
dπ orbitals are
dx2−y2,
dxz, and
dyz. From the calculation, the
dxz 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
dxz 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
dσ orbitals were
dxy <
dz2, 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
dπ 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
dπ orbitals were
dxz <
dx2−y2 <
dyz, and this seems to be consistent with the orientation of the dmso π orbitals arranged in the
pseudo-
S6 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 (
dxz)
2(
dx2−y2)
1(
dyz)
1(
dxy)
1(
dz2)
1, and this electronic configuration is consistent with the combination of the tetragonal compression and the orientation of the
pseudo-
S6 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 (
dxz,
dyz) orbitals in the tetragonally compressed
D4 coordination geometry were filled with three electrons, the ground state became
5E in the
D4 approximation. This is consistent with the negative
Δ value found from the magnetic analysis.
2.4. Magneto-Structural Relationship
Now we discuss the magneto-structural relationship especially between the structure and the
Δ value. In the crystal structure, the tetragonal compression was observed; however, the order of the
dπ 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-
S6 symmetric hexakis-dmso environment, and the combination of the tetragonal compression and the
pseudo-
S6 environment was found to generate the
d-orbital splitting in
Figure 5, generating the electronic configuration of (
dxz)
2(
dx2−y2)
1(
dyz)
1(
dxy)
1(
dz2)
1. In the ideal
D4 symmetric coordination geometry due to the tetragonal-compression, the (
dxz,
dyz) orbitals are degenerate. However, in the (
dxz,
dyz) orbitals, the
dxz orbital becomes lower in energy due to the less overlap with the dmso π orbitals, and the (
dxz,
dyz) orbitals became filled with three electrons. This electronic configuration corresponded to the
5E ground state in the
D4 symmetric coordination geometry. The
5E ground state directly indicated the negative
Δ value, which was consistent with the magnetic measurements. Judging from the negative sign of the
Δ 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 (
dxz,
dyz) orbitals in the cobalt(II) complex were considered to be higher than the
dxy orbital, affording the
4E ground state. This leads to the negative
Δ value and the easy-axis anisotropy along the
z axis.