Tanabe-Sugano Diagrams: Concept

Tanabe-Sugano diagrams remain the default tool for assigning d-d absorption bands in transition-metal complexes across research and teaching laboratories, precisely because they let a single measured spectrum yield both $\Delta_o$ and $B$ simultaneously by matching two or more observed transition energies against the diagram's predicted ratios — the procedure carried out explicitly in the worked examples above and developed further under Interpreting Spectra with Tanabe-Sugano Diagrams. The extracted $B$ value feeds directly into the Nephelauxetic Effect series, which ranks ligands by how much they delocalize d electron density onto themselves (covalency), using $\beta = B_{\text{complex}}/B_{\text{free ion}}$. The ruby ${}^4A_2 \to {}^2E$ spin-forbidden line, whose near-independence of $\Delta_o$ was explained mechanically in Worked Example 3, is used today as the standard internal pressure calibrant in diamond-anvil-cell high-pressure physics and materials science, because its sharp fluorescence shifts almost linearly and reproducibly with pressure while remaining largely insensitive to the surrounding chemical environment. The $d^4$–$d^7$ spin-crossover region of these diagrams is the direct theoretical foundation of molecular spin-crossover materials — Fe(II) and Fe(III) complexes engineered so that $\Delta_o/B$ sits extremely close to the crossover ratio, allowing modest changes in temperature, pressure, or light irradiation to flip the compound between high-spin and low-spin states. These materials are actively studied for molecular memory, display, and sensor applications precisely because the transition is so sharp — a direct macroscopic consequence of the genuine (non-avoided) level crossing analyzed in the second derivation. The same crossover logic underlies the spin-state behavior of heme iron centers in biological oxygen transport and enzymatic catalysis, where the protein's ligand field tunes iron between high- and low-spin configurations to modulate reactivity. Finally, Tanabe-Sugano-style diagonalization is the direct ancestor of modern ligand-field density-functional and ab initio multireference (CASSCF/NEVPT2) treatments of transition-metal electronic structure: contemporary computational chemistry reproduces and refines exactly the same secular-matrix diagonalization, now including tetragonal and lower-symmetry distortions, spin-orbit coupling, and covalency corrections, but the conceptual backbone — competing repulsion and field terms, non-crossing rule, spin-state competition — is unchanged from the 1954 construction.

Ask the tutor: How do I use a Tanabe-Sugano diagram, together with two observed absorption band energies, to actually solve for both Δ₀ and B for an unknown octahedral complex?

Start from what an Orgel diagram already shows you: a free ion in a $d^n$ configuration has a set of Russell-Saunders terms (${}^3F$, ${}^3P$, ${}^1G$, etc.) fixed by electron-electron repulsion alone, spaced by combinations of the Racah parameters $B$ and $C$. Turn on an octahedral field and each term splits according to how its orbital degeneracy decomposes under $O_h$ symmetry — an F term splits into $A_2 + T_1 + T_2$, for instance — with splittings governed by $\Delta_o$. Orgel diagrams draw this splitting as if it were the whole story, with straight or gently diverging lines radiating from the free-ion term energies. The full story requires more care for one reason: symmetry labels, not parent terms, are conserved once the field is on. Two different free-ion terms that happen to produce a component of the same $O_h$ irreducible representation (say, two different ${}^3T_1$ states, one descended from ${}^3F$ and one from ${}^3P$) are not independent in the presence of the field — they are coupled by off-diagonal matrix elements of the electrostatic repulsion operator. Quantum mechanically, the true eigenstates are mixtures of the two parent terms, and their energies must be found by diagonalizing a small secular matrix, not by drawing two lines that blithely cross. This is the non-crossing rule: two energy levels of the same symmetry, as a function of a continuously tunable parameter, never actually cross; they repel and exchange character instead. This repulsion is exactly why the two ${}^3T_1$ curves in the $d^2$ or $d^3$ diagrams bend sharply as $\Delta_o/B$ increases, and it is exactly why, for $d^4$ through $d^7$, a low-spin term can descend rapidly enough with increasing field to overtake the high-spin ground term. Where two terms of different spin and different symmetry cross, there is no mixing (the non-crossing rule applies only within a given symmetry label), so a genuine crossing — a spin crossover — is allowed and appears as a kink where the diagram's zero-line (the ground state) switches identity. The normalization is the second half of the concept. Racah showed that all electrostatic (Coulomb + exchange) matrix elements within a $d^n$ configuration can be written as integer or half-integer combinations of just two parameters, $B$ and $C$ (with a third, $A$, contributing only a configuration-independent shift). Crystal-field splittings scale with $\Delta_o$. Because every energy in the problem is some combination of $B$, $C$, and $\Delta_o$, dividing all energies by $B$ and fixing a typical ratio $C/B \approx 4$–$4.5$ (appropriate for first-row transition-metal ions) collapses the entire family of possible diagrams — one per metal, per oxidation state — onto a single universal curve for each $d^n$. Plotting $E/B$ vertically against $\Delta_o/B$ horizontally makes the diagram literally material-independent within a given $d^n$ and spin multiplicity family.

The right-hand region of a Tanabe-Sugano diagram (large $\Delta_o/B$) is the strong-field limit, where it is more natural to build states from $t_{2g}^x e_g^y$ electron configurations and turn on electron repulsion as the perturbation; the left-hand region (small $\Delta_o/B$) is the weak-field limit, where free-ion terms are the natural starting basis and the octahedral field is the perturbation. A full diagonalization interpolates smoothly (and, at spin crossovers, not-so-smoothly) between the two.

Notice the key graphical trick: because the diagram always plots energy relative to whichever term is lowest, the ground state is a straight horizontal line at $E/B = 0$ across the whole diagram, but its identity (its Russell-Saunders/strong-field label) can switch abruptly at the crossover. Everything above it is redrawn, term by term, relative to that new zero — which is precisely why published $d^4$–$d^7$ diagrams show a visible vertical break: the slopes of every excited-state line change discontinuously in derivative at the crossover, because the reference state itself changed.

The starting Hamiltonian for the $d^n$ configuration in an octahedral field is

where:

• $H_0$ is the one-electron (kinetic + spherical potential) part, common to all terms of a configuration and dropped as an overall constant,
• $H_{ee}$ is interelectronic (Coulomb + exchange) repulsion, whose matrix elements within $d^n$ are expressed via Racah parameters $A$, $B$, $C$,
• $H_{CF}$ is the octahedral crystal-field operator, diagonal in the strong-field ($t_{2g}$, $e_g$) basis with splitting $\Delta_o$ between the two orbital sets.

Racah's parametrization gives every free-ion term energy (relative to the lowest term) as an integer combination of $B$ and $C$; for example, within $d^2$,

where: $B$ measures the difference between parallel- and antiparallel-spin repulsion (roughly, how strongly the $d$ electrons avoid each other via exchange), and $C$ enters mainly in spin-flip, high-multiplicity-difference terms. Both are empirical parameters obtained by fitting free-ion spectral data, and both decrease from their free-ion values once the ion is placed in a real ligand field — the fractional decrease is quantified later by the Nephelauxetic Effect. The crystal-field part, restricted to $O_h$ symmetry, is captured by one parameter:

where: $\Delta_o$ (also called $10Dq$) is the one-electron orbital energy gap created by the octahedral ligand array, with the barycenter rule fixing $E(t_{2g}) = -0.4\,\Delta_o$ and $E(e_g) = +0.6\,\Delta_o$ relative to the spherical (free-ion) average. A full state of the $d^n$ configuration with a given total spin $S$ and $O_h$ symmetry label $\Gamma$ is, in general, a linear combination of every free-ion term that (a) has that spin multiplicity and (b) contains a component transforming as $\Gamma$. If there are $k$ such contributing terms, the energies of the resulting $k$ true eigenstates are the $k$ roots of a $k\times k$ secular determinant:

where: the matrix is built in the basis of the $k$ free-ion (or strong-field) basis functions sharing spin $S$ and symmetry $\Gamma$; diagonal entries are Racah-parameter combinations plus crystal-field energies, off-diagonal entries are the configuration-interaction matrix elements (also combinations of $B$ and $C$) that couple terms of different parentage but identical $\Gamma$. Once every root $E_i(\Delta_o, B, C)$ is found for every $(S,\Gamma)$ combination that occurs in $d^n$, Tanabe and Sugano do two more things: fix $C/B$ at a representative ratio (commonly $C/B = 4.42$ or $4.8$ depending on the ion family, though many textbook diagrams use $C/B=4.0$ near the ionic limit for simplicity), and re-express every energy relative to the numerically lowest root at each value of $\Delta_o/B$:

where: this difference, plotted against $\Delta_o/B$, is exactly the y-value drawn for term $i$ on the published Tanabe-Sugano diagram; the ground term itself plots identically as the zero line by this construction, and the diagram is genuinely universal for a fixed $d^n$ and fixed $C/B$ ratio because both axes carry only the parameter combination $\Delta_o/B$ (or the derived energies in units of $B$), independent of which actual metal/oxidation-state pair produced that value of $B$.

Example 1 — Foundational: reading $\Delta_o$ and $B$ from a $d^3$ (weak-field, non-crossover) spectrum. $[\text{Cr(NH}_3)_6]^{3+}$ is $d^3$, always high-spin (no crossover complication — $d^3$ has only one reasonable ground term, ${}^4A_2$, across the entire diagram, since it comes from the single configuration $t_{2g}^3$). Two spin-allowed d-d bands are typically observed near $21{,}550\ \text{cm}^{-1}$ and $28{,}500\ \text{cm}^{-1}$, assigned respectively to

For $d^3$, the ${}^4A_2 \to {}^4T_2$ transition energy equals $\Delta_o$ directly, because ${}^4T_2$ arises from promoting one electron $t_{2g}\to e_g$ with no change in repulsion energy relative to the ground configuration in the simplest (single-configuration) approximation:

The second band, ${}^4A_2 \to {}^4T_1(F)$, is not simply $\Delta_o$ plus a Racah term — it bends with configuration interaction from the higher ${}^4T_1(P)$ state — but the Tanabe-Sugano $d^3$ diagram gives the ratio $E({}^4T_1(F))/B$ as a function of $\Delta_o/B$ directly. Reading the diagram at trial values of $\Delta_o/B$ and matching both observed transition energies simultaneously (an iterative graphical fit) gives the self-consistent solution

which checks against $\Delta_o \approx 33 \times 650 = 21{,}450\ \text{cm}^{-1}$, matching the first band within rounding, and against the free-ion value $B_{\text{free}}(\text{Cr}^{3+}) = 918\ \text{cm}^{-1}$ giving a nephelauxetic ratio

a sensible value (covalency reduces $B$ to roughly 70% of the free-ion value for an N-donor ligand), foreshadowing the quantitative treatment given under the Nephelauxetic Effect. Limiting check: if $\Delta_o \to 0$ (isolated free Cr³⁺ ion), the ${}^4T_2$ band would collapse to zero excitation energy and the ${}^4T_1(F)$ band would relax to its free-ion ${}^4P$-derived splitting of $15B$ above ${}^4F$ — consistent with the diagram's requirement that all curves converge to the correct free-ion term spacings as $\Delta_o/B\to 0$.

Example 2 — Applied: locating the $d^6$ spin crossover for an iron(II) complex. $[\text{Fe(H}_2\text{O)}_6]^{2+}$ is high-spin $d^6$ ($S=2$, term ${}^5T_2$), while $[\text{Fe(CN)}_6]^{4-}$ is low-spin $d^6$ ($S=0$, term ${}^1A_1$). Both are the same metal and oxidation state, so $B$ is comparable (though reduced somewhat more by the strong-field cyanide ligands via the Nephelauxetic Effect); the difference in ground-state spin is a Tanabe-Sugano crossover question, not a difference in $B$ alone. Typical experimental values: $\Delta_o(\text{H}_2\text{O}) \approx 10{,}400\ \text{cm}^{-1}$; $\Delta_o(\text{CN}^-) \approx 33{,}000\ \text{cm}^{-1}$, with $B\approx 900$–$1000\ \text{cm}^{-1}$ for the aqua complex (somewhat reduced from the free Fe²⁺ value of $\approx 1058\ \text{cm}^{-1}$). Using $B\approx 900\ \text{cm}^{-1}$ for the aqua ion:

Using $B\approx 600\ \text{cm}^{-1}$ (heavily nephelauxetic-reduced, typical for a strong π-acceptor like CN⁻) for the cyanide complex:

Both numbers straddle the $d^6$ crossover value of $\Delta_o/B\approx 20$–$22$ derived above, exactly as required: the aqua complex sits well to the left (high spin, ${}^5T_2$ ground term, weak field) and the cyanide complex sits far to the right (low spin, ${}^1A_1$ ground term, strong field). This is the mechanistic reason [Fe(H₂O)₆]²⁺ is paramagnetic with four unpaired electrons while [Fe(CN)₆]⁴⁻ is diamagnetic, despite identical metal oxidation state and identical d-electron count — the only variable that moved is $\Delta_o/B$, and the Tanabe-Sugano diagram is precisely the map of which side of the crossover a given ratio falls on. Limiting check: if $\Delta_o/B$ were tuned continuously from 11.6 to 55 (e.g., by a family of ligands of intermediate field strength — as happens in real Fe(II) spin-crossover materials), the diagram predicts an abrupt ground-state change near $\Delta_o/B\approx 20$–$22$, which is exactly the physical basis of thermally- and pressure-induced spin-crossover phenomena in Fe(II) coordination polymers used in molecular switches.

Example 3 — Exam/competition level: predicting the number of spin-allowed bands and identifying a spin-forbidden line in Cr(III), ruby-style, without a diagram in hand. Consider $[\text{Cr(H}_2\text{O)}_6]^{3+}$, $d^3$, with observed bands at $17{,}400\ \text{cm}^{-1}$, $24{,}000\ \text{cm}^{-1}$, and a weak, sharp line at $14{,}400\ \text{cm}^{-1}$ that does not fit either of the two spin-allowed progressions. Step 1 — Predict how many spin-allowed transitions to expect. For $d^3$ high-spin, the ground term is ${}^4A_2$ (from $t_{2g}^3$, the only quartet configuration). The only other quartet ($S=3/2$) terms accessible by promoting electrons among $t_{2g}$ and $e_g$ while keeping the free-ion parentage ${}^4F$ or ${}^4P$ are ${}^4T_2$, ${}^4T_1(F)$, and ${}^4T_1(P)$ — three excited quartets, giving three spin-allowed bands in principle, though the third (${}^4A_2\to{}^4T_1(P)$) often lies in the UV and is weak/obscured. The two visible bands at $17{,}400$ and $24{,}000\ \text{cm}^{-1}$ correspond to ${}^4T_2$ and ${}^4T_1(F)$. Step 2 — Assign $\Delta_o$ from the first band. As in Example 1,

Step 3 — Recognize the spin-forbidden line. The line at $14{,}400\ \text{cm}^{-1}$, lower in energy than either spin-allowed band and far too sharp (narrow linewidth, weak intensity) to be a spin-allowed d-d band, is the fingerprint of a transition to a doublet ($S=1/2$) term — spin-forbidden by the $\Delta S = 0$ selection rule, hence weak (borrowed intensity via spin-orbit coupling) and sharp (the doublet terms, derived purely from ${}^2G$, ${}^2H$, etc., are nearly independent of $\Delta_o$ because they arise from a $t_{2g}^3$ configuration with different spin-pairing, so their energy is nearly flat versus field strength — they run almost horizontal on the Tanabe-Sugano diagram). This is assigned as

This is precisely the ruby $R$-line transition (${}^4A_2 \to {}^2E$) used in the ruby laser, and its near-independence of $\Delta_o$ (hence of host lattice and pressure) is exactly why ruby fluorescence is used as an internal pressure standard in diamond-anvil-cell experiments: the sharp ${}^2E$ line barely moves even though the surrounding spin-allowed bands shift substantially with pressure-induced changes in $\Delta_o$. Limiting check: the near-flatness of the ${}^2E$ curve versus $\Delta_o/B$ is not an approximation of convenience — it follows directly from the fact that ${}^2E$ arises from the same $t_{2g}^3$ configuration as the ${}^4A_2$ ground term (both are pure $t_{2g}^3$, differing only in the spin-coupling of the three $t_{2g}$ electrons), so neither state's energy depends on $e_g$ occupation, and thus neither depends strongly on $\Delta_o$ at all — the two curves run essentially parallel and close together across the entire diagram, exactly as drawn on every published $d^3$ Tanabe-Sugano plot.

Yukito Tanabe and Satoru Sugano developed these diagrams in Japan in 1954, building directly on the ligand-field matrix elements tabulated a few years earlier by Giulio Racah for free ions, and on the emerging crystal-field theory of transition-metal complexes being developed contemporaneously by Leslie Orgel, Carl Ballhausen, and others in the West. Tanabe and Sugano's contribution was distinctly computational for its era: rather than sketching qualitative splitting diagrams by symmetry arguments alone, they explicitly set up and numerically diagonalized the full electrostatic-plus-crystal-field secular matrices for every $d^n$ configuration from $d^2$ through $d^8$, by hand and with early mechanical calculation aids, and tabulated/plotted the resulting eigenvalues as functions of $\Delta_o/B$. Their two-part 1954 paper in the Journal of the Physical Society of Japan became one of the most cited references in inorganic spectroscopy, and "Tanabe-Sugano diagram" entered the standard vocabulary of every subsequent coordination chemistry textbook.