Ligand Field Theory and the Fascination of Colours: Oxidic Iron(III) Solids as the Omnipresent Examples

Abstract

The treatment of the high-spin d⁵-configurated iron(III) cation in 6- and 4-coordinate ligand fields is a highly complex matter. The ⁶A₁-ground state allows only spin-forbidden transitions, here of relevance to ten spin-reduced ⁴A₁-, ⁴A₂-, ⁴E(2x)-, ⁴T₁(3x)-, ⁴T₂(3x)-states, which are not easy to handle. Though the literature offers a series of carefully prepared solids with beautifully resolved ligand field spectra, the philosophy of utilising these in terms of their binding character, particularly in respect to the d-electron cloud density between cation and the anions, diverges. Accordingly, the magnitudes of reported ligand field parameters Δ and of the Racah parameters of interelectronic repulsion B and C, which parameterise the mentioned effects, differ, and comparisons become difficult. In this review we propose a well-founded and comprehensible calculational procedure, in order to, as the main matter of concern, convince the readers that the ligand field spectra also sensibly reflect finer perturbational details of local or even cooperative binding quality. The origin of the latter effects is from the chemical environment beyond the first coordination sphere of a central, say Fe^IIIL _n -complex (n = 6,4) in an extended solid. Already subtle disturbances of this kind will modify the shade of colour. An essential point of the discussion is the symmetry analysis, which provides rigorous constraints and sets strict conditions on what can be experimentally observed. We will first discuss manganese(II) in oxidic solids. Because a divalent cation is associated with rather ionic binding properties towards oxygen, crystal field theory is the appropriate analytical instrument. Iron(III) provides a situation, which requires more sophistication and a refinement of the theory by taking also bond covalency into account. A symmetry-based reinterpretation of the additional absorptions in the d–d-spectra of Fe^III and Cr^III in corundum-type solids is presented. This treatment sheds new light on the finer roots of the impressive red-to-green colour change of Cr³⁺ in mixed crystals Al_2−x Cr _x O₃ with increasing x. Particular examples are discussed, where absorptions due to octahedral and tetrahedral iron(III) overlap in the ligand field spectra of spinel- and garnet-type solids, which model the hue in a predictable way. Finally, though not directly related to the primary topic, the charge-transfer properties of oxidic iron(III) are briefly examined. These absorptions often stray far into the visible region, with a very frequently significant influence on the apparent hue.

Dirk Reinen is the only surviving member of the founding editorial board of Structure and Bonding in 1966. C.K. Jorgensen, J.B. Neilands, R.S. Nyholm and R.J.P. Williams sadly are no longer with us.

Appendices

Appendix 1: The Crystal Field Treatment of a d⁵-Cation in a Cubic Environment

The state energies of a free d⁵-cation, in relation to its high-spin ⁶S-ground state, are, in terms of the Racah parameters B ₀ and C ₀ which parameterize the interelectronic repulsion [2]:

$$ \begin{array}{l}\to {}^4\mathrm{G}\kern0.5em 10{B}_0 + 5\ {C}_0;\kern1.75em \to {}^4\mathrm{P}\kern0.5em 7\ {B}_0 + 7{C}_0;\\ {}\to {}^4\mathrm{D}\kern0.5em 17{B}_0 + 5{C}_0;\kern2.23em \to {}^4\mathrm{F}\kern0.5em 22\ {B}_0 + 7{C}_0\end{array} $$

(6)

These states transform and split, by symmetry, in an octahedral crystal field into the components: ⁴A_1g, ⁴E_g, ⁴T_1g, ⁴T_2g; ⁴T_1g; ⁴E_g, ⁴T_2g; ⁴A_2g, ⁴T_1g, ⁴T_2g – where ⁶A_1g is the (high-spin) ground state. The energies are (see also Eq. 6) [2]:

$$ \begin{array}{l}{}^4{\mathrm{A}}_{1\mathrm{g}}\kern0.5em 10B+5C;{\kern2.6em }^4{\mathrm{E}}_{\mathrm{g}}\ 10B+5C\\ {}{}^4{\mathrm{A}}_{2\mathrm{g}}\kern0.5em 22B+7C;\kern2.5em 17B+5C\end{array} $$

(7)

${}^4{\mathrm{T}}_{1\mathrm{g}}\left|\begin{array}{ccc}\hfill -\Delta -E\hfill & \hfill 3B\sqrt{2}\hfill & \hfill -C\hfill \\ {}\hfill 3B\sqrt{2}\hfill & \hfill 9B+C-E\hfill & \hfill -3B\sqrt{2}\hfill \\ {}\hfill -C\hfill & \hfill -3B\sqrt{2}\hfill & \hfill \Delta -E\hfill \end{array}\right|=0,\kern0.84em \mathrm{with}\ 10B + 6\ \mathrm{C}\ \mathrm{t}\mathrm{o}\ \mathrm{be}\ \mathrm{added}\ \mathrm{t}\mathrm{o}\ \mathrm{t}\mathrm{he}\ \mathrm{t}\mathrm{hree}\ \mathrm{solutions}$

${}^4{\mathrm{T}}_{2\mathrm{g}}\left|\begin{array}{ccc}\hfill -\Delta -E\hfill & \hfill -\sqrt{6}B\hfill & \hfill -\left(4B+C\right)\hfill \\ {}\hfill -\sqrt{6}B\hfill & \hfill -5B-C-E\hfill & \hfill -\sqrt{6}B\hfill \\ {}\hfill -\left(4B+C\right)\hfill & \hfill -\sqrt{6}B\hfill & \hfill \varDelta -E\hfill \end{array}\right|=0,\kern0.85em \mathrm{with}\ \mathrm{diagonal}\ \mathrm{additions}\ \mathrm{of}\ 18B + 6C$

Here, the ligand field parameter Δ and the modified Racah parameters B and C account for the energy modifications due to placing the cation into an external crystal field. Accordingly, ten S = 5/2 – to – S = 3/2-transitions may be observed at the most, from which one (⁴A_2g) and those with a positive Δ dependence (_c ⁴T_1g, _c ⁴T_2g) are expected only at very high energies; furthermore, two (⁴A_1g, _a ⁴E_g) are degenerate in energy. The ⁴T_g-matrices indicate that – according to a (near-to) (−Δ)- or (+Δ)-dependence – the spin pairing may occur in the t_2g-(t_2g ⁴e_g ¹) or in the e_g-subshell(t_2g ²e_g ³), respectively, while the _b ⁴T_1g- and _b ⁴T_2g-transitions are expected to depend only weakly on Δ. The specific symmetry of the two ⁴T_g matrices in respect to Δ is the reason why the Eq. (7) can be used in the case of tetrahedral crystal fields as well. Though usually a change of sign due to the reverse splitting into a lower-energy e- and a higher-energy t₂-orbital set in T_d as compared to O_h has to be applied, this is needless here.

Appendix 2: A Critical Review Towards the Numerical Instruments of Ligand Field Theory in Respect to Mn²⁺ and Fe³⁺ in O_h- and T_d-Coordination

As has been discussed in detail elsewhere [7], spectral data for the free manganese(II)-cation [32], specifically the electronic transitions from ⁶S to ⁴G, ⁴P, ⁴D and ⁴F, can be rather precisely fitted to the Racah parameters B ₀ and C ₀, with numerical values of B ₀ = 750 cm⁻¹ and C ₀/B ₀ = 5.2 – though with the exception of the ⁶S → ⁴P-transition, which is calculated to occur at a, by about 10% too low energy. Thus – refraining from the doubtful-debatable procedure to introduce one additional parameter in order to achieve an improved fit to the experiment (Trees correction, see [7]) – one may confidently use the crystal field energies of Eq. (7) in Appendix 1 to parameterize experimental data, thereby leaving aside the ⁶A_1g → ⁴T_1g-transitions in the numerical calculation, however, for obvious reasons. The following master energy equations (utilising the B/C = B ₀/C ₀-ratio of 5.2 and referring to Δ/B-ratios in the range of those, deduced for oxidic manganese(II)-solids in T_d(left) and O_h(right) crystal fields) are easily derived – restricting to the six lowest energy transitions [7]:

$$ {}^6{\mathrm{A}}_1{\to}^4{\mathrm{A}}_1,\;{\;_{\mathrm{a}}}^4\mathrm{E}\kern0.75em 36B\kern1.75em {{}_{\mathrm{b}}}^4\mathrm{E}\kern1em 43B $$

(8)

and, for Δ/B ≅ 7 (left) and ≅14 (right), respectively:

$$ \begin{array}{llll}{}^6{\mathrm{A}}_1\to \hfill & {{}_a}^4{\mathrm{T}}_1\hfill & -0.84\Delta +38.0B\hfill & -0.94\Delta +39.1B\hfill \\ {}\hfill & {{}_a}^4{\mathrm{T}}_2\hfill & -0.34\Delta +37.3B\hfill & -0.70\Delta +41.0B\hfill \\ {}\hfill & {{}_b}^4{\mathrm{T}}_2\hfill & -0.28\Delta +43.5B\hfill & -0.14\Delta +42.0B\hfill \\ {}\hfill & {{}_b}^4{\mathrm{T}}_1\hfill & +0.68\Delta +42.0B\hfill & +0.48\Delta +44.5B\hfill \end{array} $$

After all, one is in the comfortable situation, that there are frequently four experimental data (⁶A₁ → ⁴A₁, _a ⁴E; _b ⁴E; _a ⁴T₂; _b ⁴T₂), from which only two unknown parameters have to be deduced. The interelectronic repulsion parameter B can be derived from the Δ-independent transitions, where the minimum positions of the ground- and excited-state potential curves coincide. These minima are displaced in respect to each other, if Δ is involved. Though the Stoke shift, which is the energy difference between a transition in absorption and emission, is rather small in the case of a divalent cation such as Mn²⁺, the derived Δ parameter is nevertheless – though modestly – semi-empirical in nature. It is hence expected that – within the discussed limits – the nephelauxetic ratio β = B/B ₀ images the covalency properties near to reality and that also Δ is a parameter, which reflects changes in the binding quality in a sensitive manner, if one sticks to the outlined critical calculation procedure. Having this in mind, both parameters, B and Δ, constitute as reliable members of the nephelauxetic and spectrochemical series of ligands [6].

Iron(III) differs from Mn(II) in its binding properties, by the distinctly enhanced covalent overlap between the metal and the ligands, and demands to classify the t_2g- and e_g-electrons in O_h according to their π- and σ-anti-bonding character, respectively; similarly, the e- and t₂-electrons in T_d are π-anti-bonding in the former and intermixed between σ- and π-anti-bonding in the latter MOs. Accordingly, the Racah parameters differ, depending on whether they refer to t₂- or e-electrons. For example, in the case of Cr³⁺ in the octahedral coordination of fluoride elpasolites, B varies considerably between 790, ≅685 and ≅620 cm⁻¹, for a t_2g ³-, t_2g ²e_g ¹- and t_2g ¹e_g ²-configuration, respectively [11]. Having covalence phenomena of this type in mind, a coarser approach than the one introduced for Mn²⁺ seems appropriate – and this with the clear intention to not overload the treatment with too many not sufficiently meaningful parameters. We will accordingly follow an approach, in which only a singular B-value is used and in which also the ⁶A₁ → ⁴T₁-transitions are included into the fitting procedure, thereby utilising a C/B-ratio of 5.0. Though well defined, this approach certainly implies a larger uncertainty in the numerical fixation of the ligand field parameters. Nevertheless, “quod est demonstrandum”, rather satisfactory fits to the iron(III)-ligand field spectra are possible, with consistent interpretations of the derived binding parameters Δ and B. The basic demand is, as always, the assistance of an appropriate symmetry classification. As for Mn²⁺, we use in the following master equations for the transition energies of iron(III) in O_h- and T_d-ligand fields at (for the Δ-dependent absorption bands) Δ/B-ratios, which are particularly applicable to the oxidic iron(III) solids, considered here, with C/B = 5.0:

$$ {}^6{\mathrm{A}}_1\to {}^4{\mathrm{A}}_1,\;{\;_{\mathrm{a}}}^4\mathrm{E}\ :\kern0.5em 35B\kern2.5em \to {\;_{\mathrm{b}}}^{\mathrm{a}}\mathrm{E}\ :\ 42B $$

(9)

and, for Δ/B ≅ 12 (left) and 20 (right), respectively:

$$ \begin{array}{llll}{}^6{\mathrm{A}}_1\to \hfill & {{}_{\mathrm{a}}}^4{\mathrm{T}}_1\hfill & -0.93\Delta +37.7B\hfill & -0.97\Delta +38.3B\hfill \\ {}\hfill & {{}_{\mathrm{a}}}^4{\mathrm{T}}_2\hfill & -0.77\Delta +41.7B\hfill & -0.90\Delta +43.7B\hfill \\ {}\hfill & {{}_{\mathrm{b}}}^4{\mathrm{T}}_2\hfill & -0.02\Delta +38.3B\hfill & -0.01\Delta +38.1B\hfill \\ {}\hfill & {{}_{\mathrm{b}}}^4{\mathrm{T}}_1\hfill & +0.57\Delta +41.8B\hfill & +0.23\Delta +47.2B\hfill \end{array} $$

Appendix 3: The Symmetry Relations and the d-Electron Density Distribution of Chromium(III) and Iron(III) in the Corundum Structure

If one marks the t_2g- and e_g-wave functions in O_h with ζ, η, ξ and Θ, ε, respectively, the following linear combinations result, if the axis of quantisation is now C₃ instead of C₄ (see column C_3v (1)):

$$ \begin{array}{lll}{\mathrm{O}}_{\mathrm{h}}\hfill & {\mathrm{C}}_{3\mathrm{v}}(1)\hfill & {\mathrm{C}}_{3\mathrm{v}}(2)\hfill \\ {}{\mathrm{t}}_{2\mathrm{g}}\kern1.5em \zeta \hfill & 1/\sqrt{3}\left(\zeta +\eta +\xi \right)\kern1.5em {\mathrm{a}}_1\hfill & {{\mathrm{d}}_z}^2\hfill \\ {}\kern2.62em \eta \hfill & 1/\sqrt{6}\left(2\zeta -\eta -\xi \right)\kern1.12em \mathrm{e}\hfill & {\mathrm{d}}_{x^2-{y}^2}\hfill \\ {}\kern2.62em \xi \hfill & 1/\sqrt{2}\left(\eta -\xi \right)\hfill & {\mathrm{d}}_{xy}\hfill \\ {}{e}_g\kern1.5em \varTheta \hfill & \kern2em \varTheta \kern5.8em \mathrm{e}\hfill & {\mathrm{d}}_{xz}\hfill \\ {}\kern2.62em \varepsilon \hfill & \kern2em \varepsilon \kern6.1em \hfill & {\mathrm{d}}_{yz}\hfill \end{array} $$

(10)

Adopting the C₃- as the new molecular z-axis, the d-wave functions transform as listed under C_3v (2); the latter are the ones used in the main text and in Fig. 11. According to the presence of the threefold axis, any cyclic permutation of the wave functions under C_3v (1) is also a valid choice. The d _z ²-wave function models the metal pair interaction in the corundum lattice via a₁. The two e-doublets come out to be non-bonding (from t_2g) and σ-anti-bonding (from e_g), respectively, if otherwise the octahedral symmetry is retained – as this is nearly so in Cr₂O₃. In difference, the structural C_3v distortion is very significant in α-Fe₂O₃, and the two e-orbital sets are expected to intermix. The distance between the two, displaced, octahedral cations within the O₃MO₃MO₃-binuclear pairs (Fig. 10) along the C₃-axis is about 2.12, 2.17 and 2.27 Å for M being Al(III), Cr(III) and Fe(III), respectively. These values coarsely follow the sequence of the octahedral ionic radii, which are reported to be [33, 34] 0.53₅, 0.61₅ and 0.64₅ Å, only in the case of the two 3d-cations. For Al(III) this spacing is about 0.1 Å larger, which becomes apparent, if one corrects the internuclear distances in respect to the ionic radii differences. This observation indeed matches with the lack of intercationic binding in that case. Focussing on Cr(III) and Fe(III) accordingly, one further notes that the bond lengths towards the oxygen ligands in the boundary faces of the binuclear pairs (1.97 and 1.94₅ Å, respectively) indicate a by nearly 3% (again radii-corrected) reduced effective value in the latter case. Recalling the spectral results in addition, one has to conclude as follows: The distinct C_3v- structural component, considering α-Fe₂O₃, has introduced, via the e(t_2g) – e(e_g)-interaction (Fig. 11, left), particularly a more pronounced t_2g-splitting; this energy effect is nicely imaged via the by a factor of two enhanced parameter δ (see Eq. (3) and Table 3 in Sect. 3.3), and attended by the conversion of the two symmetry-equivalent orbital doublets into linear combinations (see Eqs (10)). Also the derived ligand field parameters (Δ, in 10³ cm⁻¹) for the here and in [21] discussed spinel-, garnet- and corundum-type phases indicate the exceptional position of α-Fe₂O₃:

$$ \begin{array}{lll}\mathrm{Compound}\hfill & \Delta \hfill & B\hfill \\ {}{\mathrm{ZnCr}}_2{\mathrm{O}}_4\hfill & 17.4\hfill & 0.64\hfill \\ {}{\mathrm{ZnFe}}_2{\mathrm{O}}_4\hfill & 15.7\hfill & 0.65\hfill \\ {}{\mathrm{Y}}_3{\mathrm{Cr}\mathrm{Ga}}_4{\mathrm{O}}_{12}\hfill & {16.1}_5\hfill & 0.63\hfill \\ {}{\mathrm{Y}}_3{\mathrm{Fe}\mathrm{Ga}}_4{\mathrm{O}}_{12}\hfill & 15.3\hfill & 0.65\hfill \\ {}\upalpha \hbox{-} {\mathrm{Cr}}_2{\mathrm{O}}_3\hfill & 15.8\hfill & 0.57\hfill \\ {}\upalpha \hbox{-} {\mathrm{Fe}}_2{\mathrm{O}}_3\hfill & =11.7\hfill & =0.60\hfill \end{array} $$

(11)

It is, induced by the dramatic rise of the intercationic overlap parameter δ, the pronounced decrease of Δ, which mirrors the binding landscape specifically in the α-Fe _x Al_2−x O₃-mixed crystals (see Table 3). Δ becomes smaller by at least 20%, when replacing chromium(III) by iron(III) in the corundum structure, in comparison to 10% or even less in spinel- or garnet-type phases. Considering the above outlined model arguments, this is an expected experimental result. Already discussed is the reduction of B (in 10³ cm⁻¹, see Eq. 11) by about 10%, which is observed, if the within-pair overlap comes up in addition (Table 3). We think that – though the iron(III) bonding properties in the corundum lattice are highly complex – the proposed model is an acceptable symmetry-adjusted approach to what one may call the truth.