Orgel Diagrams

Orgel-diagram reasoning remains the fastest first-pass tool for reading a transition-metal UV-visible spectrum, and it underlies practical work across several fields. In gemology and materials color science, the different colors of ruby (Cr³⁺ in Al₂O₃) versus emerald (Cr³⁺ in beryl) are explained by exactly the same $d^3$ Orgel branches shifting because the surrounding oxide lattice changes $\Delta_o$ and $B$, even though the chromophore ion is identical — a direct, visually vivid confirmation of the correlation-diagram logic. In bioinorganic chemistry, the characteristic blue color of oxidized blue copper proteins and the spectra of iron centers in heme and non-heme enzymes are first rationalized with weak-field-style term diagrams before moving to more detailed ligand-field or DFT treatments; the qualitative assignment of which $d$–$d$ transition is responsible for a given absorption feature in a metalloenzyme active site often starts with an Orgel-style sketch. In industrial pigment and phosphor chemistry, Cr³⁺- and Mn²⁺/Mn⁴⁺-doped oxide phosphors used in LED lighting and display technology are tuned by adjusting the host lattice to shift $\Delta_o$, moving Orgel-diagram branches to place emission or absorption bands at technologically useful wavelengths. In analytical and instrumental chemistry, the nephelauxetic ratio extracted via Orgel-diagram-guided band assignment remains a standard, simple diagnostic for the degree of covalency in a metal–ligand bond, complementing more expensive spectroscopic and computational methods, and forms the conceptual stepping-stone toward the fully quantitative Tanabe-Sugano Diagrams: Concept used whenever strong-field or low-spin complexes must be treated rigorously.

Ask the tutor: How do I decide the correct sign and magnitude of the $T_{1g}(F)$–$T_{1g}(P)$ mixing matrix element $H_{12}$ for a specific $d^n$ configuration when I want numeric, not just qualitative, band-curve positions?

Consider a free gaseous Cr³⁺ ion ($d^3$). Its ground state, found from Hund's rules and Russell–Saunders term symbols, is ⁴F, with excited quartet terms ⁴P lying higher, plus various doublet terms even higher. These terms are (2L+1)(2S+1)-fold degenerate in the free ion because there is no preferred direction in space. The moment six ligands approach along $\pm x, \pm y, \pm z$ to form an octahedral complex, that spherical symmetry is destroyed, and the terms must transform according to the irreducible representations of $O_h$. An F term (L = 3, sevenfold orbitally degenerate) splits into $A_{2g} + T_{1g} + T_{2g}$; a P term (L = 1, threefold degenerate) transforms as $T_{1g}$ and does not split further. This is exactly analogous to how a single d-orbital set splits into $t_{2g}$ and $e_g$, because the microstates making up a term are built from d-orbital occupations, so the same group-theoretic machinery from Splitting of Terms in an Octahedral Field applies directly to L = 2, 3 terms. The Orgel diagram is the energy-level correlation diagram that results when you plot the energies of the split components of the ground term and next higher term of the same spin multiplicity as a function of field strength (expressed via $\Delta_o$), and join them smoothly from the free-ion limit ($\Delta_o = 0$) on the left to the octahedral-field limit on the right. Two features make the diagram compact and generalizable. First, terms of a given orbital character but different total spin barely interact — spin-orbit coupling is a small perturbation next to the crystal field and interelectronic repulsion — so within Orgel's weak-field, spin-conserving scope, each spin multiplicity gets its own independent branch structure. Second, a beautiful hole–electron duality lets one diagram serve four configurations: filling n electrons into d-orbitals produces crystal-field splitting patterns that are the mirror image of putting (10 − n) holes into the same orbitals, and inverting the sign of $\Delta$ converts octahedral behavior into tetrahedral behavior (because $T_d$ has no inversion center and the $t_2/e$ ordering is reversed relative to $t_{2g}/e_g$). Concretely: the diagram used for $d^1$ (octahedral) is also the diagram for $d^9$ (tetrahedral), $d^4$ (tetrahedral) and $d^6$ (octahedral) high-spin cases, all built from a single S/D-term correlation picture; and the diagram for $d^2$ octahedral doubles for $d^3$ tetrahedral, $d^7$ octahedral, and $d^8$ tetrahedral, all sharing the F/P correlation picture.

That single crossing pair — one branch rising with slope $+0.6\,\Delta_o$, the other falling with slope $-0.4\,\Delta_o$ — is the entire content of the $d^1/d^4/d^6/d^9$ Orgel diagram, and it explains why Ti(III) (d¹, an aqua ion) shows one broad absorption band (with a shoulder from Jahn–Teller distortion) rather than a complex multiplet. The richer diagrams for $d^2, d^3, d^7, d^8$ involve an F term and a P term of the same spin multiplicity, and because both give rise to a $T_{1g}$ representation, the two $T_{1g}$ branches are not allowed to cross — they repel each other via configuration interaction, curving away from a naive straight-line extrapolation. This non-crossing rule (an application of the general quantum-mechanical avoided-crossing behavior for states of like symmetry) produces the diagnostic curvature that assigns the "double-humped" $\nu_1$ or $\nu_3$ bands seen in Ni(II) and Cr(III) spectra.

The quantitative backbone of the Orgel diagram rests on how many-electron term energies in an octahedral field are expressed as linear combinations of the crystal-field splitting parameter $\Delta_o$ and the interelectronic-repulsion (Racah) parameters $B$ and $C$. For the simplest, non-interacting terms:

where:

• $E(^{2S+1}E_g)$, $E(^{2S+1}T_{2g})$ are the energies of the crystal-field components of the D term,
• $\Delta_o$ is the octahedral crystal-field splitting energy (often written $10Dq$).

Reading: the D term (from a single d-electron or single d-hole) splits into exactly two levels whose separation is $\Delta_o$ by definition; the barycenter (weighted average) of the two components stays at the free-ion energy, giving the characteristic $+0.6\Delta_o / -0.4\Delta_o$ split relative to that barycenter. For terms that undergo configuration interaction (two terms of the same symmetry label mixing), the energies are no longer simple linear functions of $\Delta_o$; instead they satisfy a secular (mixing) equation:

where:

• $E_1, E_2$ are the unperturbed (uncoupled) energies of the two same-symmetry terms (e.g. the two $^4T_{1g}$ levels arising from F and P),
• $H_{12}$ is the off-diagonal matrix element coupling them, which for octahedral d-term mixing is commonly taken as $H_{12} = 4B\sqrt{15}$ times a numerical coefficient depending on convention (frequently quoted compactly for the $T_{1g}(F)$–$T_{1g}(P)$ pair),
• $E_\pm$ are the true, mixed eigenvalues that appear as curved branches on the diagram.

Reading: whenever $E_1$ and $E_2$ approach each other, the $\pm\sqrt{\cdots}$ term keeps $E_+$ and $E_-$ apart by at least $2H_{12}$ — this is precisely the non-crossing rule and is why the $T_{1g}(F)$ branch bends over rather than crossing the rising $T_{1g}(P)$ branch as $\Delta_o$ increases. Finally, the observed band energies in a spin-allowed d–d spectrum are read directly off the vertical separations between the ground-term branch and each excited-term branch at the experimental value of $\Delta_o$:

where:

• $h\nu_i$ is the photon energy of the $i$-th spin-allowed absorption band,
• the branch energies are read at the $\Delta_o$ appropriate to the real complex, which is precisely the calibration step that lets a measured spectrum be inverted to yield $\Delta_o$ and $B$.

Example 1 — Foundational: reading $\Delta_o$ directly from a $d^1$ spectrum. $[\text{Ti}(\text{H}_2\text{O})_6]^{3+}$ is the textbook $d^1$ ion. Its single (Jahn–Teller-split) absorption band peaks near 20,300 cm⁻¹ (frequently quoted as a broad band centered around 20,000–20,300 cm⁻¹ with a shoulder near 17,000–17,500 cm⁻¹ from the tetragonal distortion of the excited $^2E_g$ state). Because the $d^1$ Orgel diagram has exactly one branch pair, $^2T_{2g}$ (ground) and $^2E_g$ (excited), separated by definition by $\Delta_o$, the transition energy is the crystal-field splitting directly:

No repulsion parameter is needed because there is only one excited state of the correct spin to consider and no configuration interaction (there is no second quartet/doublet term of matching symmetry nearby). This is the cleanest possible Orgel-diagram reading: one band, one branch gap, one number. Limiting check: if the ligand field were replaced by nothing at all ($\Delta_o \to 0$, e.g. imagining the bare gas-phase Ti³⁺ ion), the $^2E_g$ and $^2T_2g$ branches converge and the transition energy correctly goes to zero — consistent with a free ion having no such absorption band at all, only fine-structure (spin-orbit) splitting on the order of a few hundred cm⁻¹.

Example 2 — Applied: extracting $\Delta_o$ and $B$ for $[\text{Ni}(\text{H}_2\text{O})_6]^{2+}$ ($d^8$, using the $d^3$-type diagram by hole–electron correspondence). $[\text{Ni}(\text{H}_2\text{O})_6]^{2+}$ shows three spin-allowed bands, typically reported near

assigned respectively to $^3A_{2g} \to {}^3T_{2g}$, $^3A_{2g} \to {}^3T_{1g}(F)$, and $^3A_{2g} \to {}^3T_{1g}(P)$. On the $d^8$ Orgel diagram the lowest-energy transition, to the uncoupled $^3T_{2g}$ level, gives $\Delta_o$ immediately because that branch (unlike the two $T_{1g}$ branches) does not mix with anything of the same symmetry:

To extract the Racah parameter $B$, use the standard combination derived from the secular-equation treatment above (the well-known Ni²⁺ working formula relating the three band energies):

A commonly used simplified estimate (treating $\nu_2$ as approximately midway once mixing is accounted for) gives, on substitution of the numbers above,

Since a negative result signals that the simplified additive formula (which neglects second-order mixing corrections) is being pushed outside its safe regime here, the standard practical route instead uses tabulated Tanabe–Sugano-consistent ratios; nevertheless, published fits for $[\text{Ni}(\text{H}_2\text{O})_6]^{2+}$ converge on

Comparing to the free Ni²⁺ ion value $B_{\text{free}} \approx 1{,}041\ \text{cm}^{-1}$, the nephelauxetic ratio is

indicating modest covalent delocalization of d-electron density onto water ligands (a nephelauxetic effect), consistent with water sitting in the middle of the spectrochemical/nephelauxetic series. Limiting check: if the ligand were a purely ionic, electron-density-localizing field (an idealized point-charge limit), $\beta$ would approach 1 exactly, since the metal-centered electron repulsion would be unperturbed by covalency — real ligands like water and especially sulfur/phosphorus donors always pull $\beta$ below 1, more so for more covalent ligands.

Example 3 — Exam/competition level: distinguishing octahedral $\text{Co(II)}$ from tetrahedral $\text{Co(II)}$ using Orgel-diagram slopes. A student is given two cobalt(II) complexes, one a pale pink octahedral aqua complex $[\text{Co}(\text{H}_2\text{O})_6]^{2+}$ and one an intense blue tetrahedral complex $[\text{CoCl}_4]^{2-}$, and asked to explain (a) why the tetrahedral complex is far more intensely colored, and (b) why its single main visible-region band sits at lower energy despite chloride being a weaker-field ligand in the sense of $\Delta$ magnitude comparisons within the same geometry. Both are $d^7$ systems (Co²⁺), so both fit on the shared $d^2/d^3/d^7/d^8$ Orgel diagram; going from octahedral to tetrahedral flips the sign of $\Delta$ in the correlation (the hole-formalism inversion mentioned earlier), which swaps which branch is the ground term. In $O_h$, high-spin $d^7$ has ground term $^4T_{1g}(F)$, and the relevant near-UV/visible spin-allowed transitions are to $^4T_{2g}$, $^4A_{2g}$, and $^4T_{1g}(P)$ — with the ground term itself being one of the mixing $T_1$ states, this is the case where Laporte-forbidden $d$–$d$ transitions in a centrosymmetric ($O_h$) environment are weak. In $T_d$, there is no inversion center, so the Laporte selection rule is relaxed by direct (not merely vibronic) mixing of d and p character on cobalt; this alone increases the molar absorptivity by roughly one to two orders of magnitude, explaining the far deeper color of $[\text{CoCl}_4]^{2-}$ (part a). For part (b): tetrahedral splitting is intrinsically smaller than octahedral splitting for comparable ligands and metal–ligand distances, following the well-known relationship

so even though $\Delta_o(\text{H}_2\text{O})$ and $\Delta_t(\text{Cl}^-)$ involve different ligands and cannot be compared by this formula directly across different systems, the factor-of-(4/9) reduction inherent to tetrahedral geometry compounds with chloride's weaker field (chloride sits lower than water in the spectrochemical series) to push all the tetrahedral $\text{Co}^{2+}$ bands to substantially lower energy than the corresponding octahedral $\text{Co}^{2+}$ bands — consistent with the experimentally observed visible band of $[\text{CoCl}_4]^{2-}$ near 14,000–15,000 cm⁻¹ region (with fine structure) versus the higher-energy, weaker bands of $[\text{Co}(\text{H}_2\text{O})_6]^{2+}$ split around 8,000 and 19,000–20,000 cm⁻¹. Limiting check: if $\Delta_t$ were allowed to grow arbitrarily large (an unphysical limit for a genuinely tetrahedral field, since $\Delta_t$ is capped near $\tfrac{4}{9}\Delta_o$ for matched ligands/distances), a tetrahedral complex could never show $d$–$d$ bands at lower energy than its octahedral analogue — the qualitative color/energy trend observed here is therefore a direct, testable consequence of the $4/9$ geometric factor, not a coincidence of ligand choice alone.