Interpreting Spectra with Tanabe-Sugano Diagrams

Tanabe–Sugano analysis remains the everyday quantitative backbone of transition-metal electronic spectroscopy, used any time a chemist needs $\Delta_o$ and $B$ from a solution or solid-state UV-vis/near-IR spectrum rather than from crystallography or computation alone. In bioinorganic chemistry, it underlies the interpretation of $d$–$d$ bands in metalloprotein active sites — for example, distinguishing high-spin from low-spin heme iron states in cytochromes and peroxidases from their visible absorption fine structure, and tracking $\Delta_o$ changes upon substrate or ligand binding to Cu(II) and Ni(II) active sites. In materials chemistry, the same diagrams (adapted for tetrahedral and other symmetries via the appropriate hole-formalism or reversed-sign relationships) are used to assign the color and photophysics of ruby and emerald (Cr³⁺ in different host lattices — the classic textbook contrast in $\Delta_o$ and hence color between the two, both readable from exactly this kind of analysis), transition-metal-doped phosphors and laser gain media (Ti³⁺:sapphire, Cr³⁺:alexandrite), and pigment/glass colorants. In catalysis and organometallic chemistry, monitoring how $\Delta_o$ and $B$ shift as ligands are varied around a metal center provides an experimentally cheap, first-pass probe of how strongly a new ligand donates or backbonds, feeding directly into the qualitative reasoning of the spectrochemical series and, more quantitatively, into the Nephelauxetic Effect as a covalency metric. Modern instrumentation (fast, high-resolution diode-array and FTIR-UV-vis spectrometers) has mostly automated peak-picking, but the underlying diagram-matching logic — ratio first, absolute scale second — is unchanged from Tanabe and Sugano's original 1954 tables, and remains the standard first analysis performed on any new octahedral transition-metal chromophore before more elaborate ligand-field or DFT modeling is attempted.

:::quiz[High-spin d⁵ complexes (e.g. [Mn(H2O)6]2+) show very weak, sharp visible bands rather than a strong broad band because:]

• Δo is unusually small for Mn²⁺ aqua complexes
• The ground term ⁶A1g has no spin-allowed transitions available; all visible bands are spin-forbidden (ΔS ≠ 0) ✓
• B is anomalously large for Mn²⁺, shifting all bands into the UV
• The complex is diamagnetic, suppressing all transitions

Ask the tutor: How do I adapt this ratio-matching method when my complex is tetragonally distorted and shows split, overlapping $d$–$d$ bands instead of clean single peaks?

A free gaseous $d^n$ ion has its electrons distributed among five degenerate $d$ orbitals, and electron-electron repulsion alone splits the configuration into Russell–Saunders terms ($^3F$, $^3P$, $^1G$, etc.), with energy spacings parameterized by the Racah parameters $B$ and $C$. Turning on an octahedral ligand field lifts the fivefold orbital degeneracy into $t_{2g}$ and $e_g$ sets split by $\Delta_o$, and each free-ion term further splits into components labeled by their $O_h$ irreducible representation. A Tanabe–Sugano diagram is the completed marriage of these two effects: it plots the energy of every resulting state, in units of $B$, as a continuous function of the octahedral field strength, also in units of $B$, interpolating smoothly from the weak-field (free-ion, term-based) limit at $\Delta_o/B=0$ to the strong-field (orbital, configuration-based) limit at large $\Delta_o/B$. The diagram's defining convention is that the ground term is always plotted as the horizontal axis itself — its energy is defined as zero at every field strength. This means every vertical distance up to an excited-state curve, at the experimental value of $\Delta_o/B$, is a spin-allowed (or spin-forbidden) transition energy directly comparable to an absorption band. For $d^1$ and $d^9$ there is only one excited term, so the diagram is a single straight line and the diagram is almost trivial: one band, and $\Delta_o$ is read off directly (times $B$ isn't even needed, since there's no ratio to form; you simply read $E=\Delta_o$). For every other configuration except $d^{10}$, multiple excited terms of different multiplicities and symmetries appear, and terms of the same symmetry label (same $O_h$ irreducible representation, same spin multiplicity) are forbidden by the non-crossing rule from actually crossing as functions of $\Delta_o/B$ — instead they repel each other, bending away, sometimes swapping character entirely (an avoided crossing). This is precisely why $d^2$–$d^8$ diagrams look like a tangle of curved lines rather than a fan of straight ones, and why some diagrams (notably $d^4,d^5,d^6,d^7$) show a discontinuous kink along the x-axis: the ground term itself changes at a critical $\Delta_o/B$, marking the high-spin-to-low-spin crossover. Reading real spectra against this backdrop requires three deliberate steps: identify how many spin-allowed bands should be visible for the given $d^n$ and spin state, assign each observed band maximum to a specific excited term using the diagram's term ordering, and then solve — usually by ratio-matching or by iterative graphical/numerical fitting — for the single value of $\Delta_o/B$ consistent with all the assigned bands simultaneously. The self-consistency check (does one $\Delta_o/B$ value reproduce every band, not just two of three?) is what separates a rigorous extraction from a guess.

The entire quantitative machinery reduces to two linked statements about how a Tanabe–Sugano diagram encodes energy.

where: $E_i$ is the energy of excited term $i$ above the ground term, $B$ is the Racah repulsion parameter, and $f_i$ is the (generally curved) function plotted as the $i$-th line on the diagram. This is a reading, not an approximation: the diagram is nothing but a tabulation/plot of $f_i$ for each term, obtained by exact diagonalization of the electrostatic-plus-crystal-field Hamiltonian in the relevant $d^n$ basis.

where: the $B$ dependence cancels between numerator and denominator, leaving a ratio that depends on $\Delta_o/B$ alone. This is the entire trick of the method — reading a ratio off two experimental bands lets you locate $\Delta_o/B$ on the diagram without knowing $B$ in advance, because you are matching a shape (the ratio-vs-field curve), not an absolute scale.

where: once $(\Delta_o/B)_{\text{fit}}$ is known from ratio-matching, any single experimental band energy $E_i^{\text{obs}}$ (in cm⁻¹) divided by the corresponding dimensionless diagram value $f_i$ gives $B$ in cm⁻¹, and multiplying back by $(\Delta_o/B)_{\text{fit}}$ gives $\Delta_o$ in cm⁻¹. Reading: this equation is the "unlocking" step — it converts the dimensionless field strength into two physically meaningful numbers.

where: with only one excited term, there is no ratio to form and no $B$ to solve for from the spectrum alone; the single band energy is $\Delta_o$ directly (this is the trivial, degenerate limit of the general method, included explicitly as a sanity check).

Example 1 (foundational): $[\text{Cr(NH}_3)_6]^{3+}$, a clean $d^3$ case. Chromium(III) is $d^3$, always high-spin (there is no spin-state ambiguity for $d^3$ in any real octahedral field), with ground term $^4A_{2g}$. The $d^3$ Tanabe–Sugano diagram shows three spin-allowed excited quartet terms relevant in the visible/near-UV: $^4T_{2g}$, $^4T_{1g}(F)$, and $^4T_{1g}(P)$. The complex $[\text{Cr(NH}_3)_6]^{3+}$ shows two clear spin-allowed bands at

assigned respectively to $^4A_{2g}\to{}^4T_{2g}$ and $^4A_{2g}\to{}^4T_{1g}(F)$ (the third band, to $^4T_{1g}(P)$, often overlaps a charge-transfer edge and is not used here). A defining feature of the $d^3$ (and $d^8$) diagram is that the $^4T_{2g}$ line is essentially straight and passes through the origin with unit slope in these units — a consequence of $t_{2g}^3$ having no first-order repulsion-energy difference from the ground $t_{2g}^3$ configuration for a one-electron $e_g \leftarrow t_{2g}$ promotion. This gives directly

Now use $\nu_2$ to extract $B$. From tabulated $d^3$ Tanabe–Sugano data, at $\Delta_o/B \approx 33$ (a value we must find self-consistently, since $B$ is unknown at the outset — but published $d^3$ tables give $E(^4T_{1g}(F))/B \approx 43$ at this field, a number read from the diagram/table rather than re-derived here), the ratio

is matched on the diagram at $\Delta_o/B \approx 32.5$. Using $\Delta_o = 21{,}550\ \text{cm}^{-1}$,

This compares to the free-ion value for Cr³⁺, $B_0 = 918\ \text{cm}^{-1}$, giving a nephelauxetic ratio $\beta = B/B_0 = 663/918 = 0.72$ — a physically reasonable amount of covalent delocalization for an ammine complex, consistent with NH₃ sitting in the middle of the spectrochemical/nephelauxetic series. Limiting-case check: if the ligand field were so weak that $\Delta_o/B \to 0$, the $^4T_{2g}$ and $^4T_{1g}(F)$ lines would converge toward the free-ion $^4F$ term energy (both curves emanate from the same point on the y-axis at $x=0$), and the ratio $\nu_2/\nu_1$ would diverge rather than approach a finite constant — signaling that ratio-matching breaks down exactly where it should, in the unphysical zero-field limit, since real octahedral complexes never sit at $\Delta_o/B\approx 0$.

Example 2 (applied, with a genuine assignment ambiguity): $[\text{Co(H}_2\text{O)}_6]^{2+}$, high-spin $d^7$. Cobalt(II) aqua complex is high-spin $d^7$ (weak-field $\text{H}_2\text{O}$, confirmed by a magnetic moment near the spin-only $d^7$ value of $3.87\ \mu_B$, observed close to $4.7$–$5.2\ \mu_B$ once orbital contribution is included — the point here is simply that it is high-spin), with ground term $^4T_{1g}(F)$. This is one of the trickier diagrams to use directly because the ground term itself is orbitally degenerate and triply degenerate spin-allowed transitions are predicted: $^4T_{1g}(F)\to{}^4T_{2g}$, $^4T_{1g}(F)\to{}^4A_{2g}$, and $^4T_{1g}(F)\to{}^4T_{1g}(P)$. The observed spectrum shows a weak band near $8{,}000$–$9{,}000\ \text{cm}^{-1}$ (often only seen as a shoulder or in the near-IR), a moderate band around $19{,}400\ \text{cm}^{-1}$, and — the diagnostic one — a band near $21{,}000\ \text{cm}^{-1}$ that in high-spin $d^7$ octahedral complexes is famously split or shows a shoulder due to spin-orbit coupling and to the close approach of $^4T_{1g}(P)$ to spin-forbidden doublet terms ($^2G$) nearby. Take as data

(the middle transition to $^4A_{2g}$, being a formally two-electron / spin-allowed-but-orbitally-doubly-excited process, is often very weak and is skipped here — this is itself a diagnostic point: if a "band" were mistakenly assigned to $^4A_{2g}$ where in reality it is a doublet spin-forbidden feature, the ratio-matching below would simply fail to converge). Forming the ratio,

Matching this ratio on the high-spin $d^7$ Tanabe–Sugano diagram locates $\Delta_o/B \approx 9.5$ (this configuration's diagram has $^4T_{1g}(F)$ as a curved, non-straight ground-term axis, so the fit is read graphically rather than from a closed formula — this is precisely the curvature effect the diagram exists to capture). Reading the corresponding dimensionless value for the $^4T_{2g}$ transition at that field strength, $E(^4T_{2g})/B \approx 8.2$, gives

which is larger than the free-ion value ($B_0(\text{Co}^{2+}) \approx 1{,}120\ \text{cm}^{-1}$ gives $\beta\approx 0.88$, chemically sensible for a weak-field O-donor ligand) and then

a value close to typical tabulated $\Delta_o$ for $[\text{Co(H}_2\text{O)}_6]^{2+}$ ($\sim 9{,}300\ \text{cm}^{-1}$), confirming the assignment was self-consistent. Limiting-case check: in the strong-field limit relevant to low-spin $d^7$ (not this complex, but the other side of the diagram's kink), the ground term would instead be $^2E_g$, and none of the quartet-based ratio logic above would apply at all — which is exactly why identifying the spin state from the magnetic moment before opening the diagram is not optional bookkeeping but a logical prerequisite.

Example 3 (exam/competition level): locating the high-spin/low-spin boundary for a hypothetical $d^6$ complex. A $d^6$ complex $[\text{M(L)}_6]^{n+}$ is measured to have $B = 700\ \text{cm}^{-1}$ (from an independent low-energy spin-forbidden band assignment) and shows a single dominant spin-allowed band at $\nu = 20{,}300\ \text{cm}^{-1}$, with the complex confirmed diamagnetic (i.e., low-spin, $^1A_{1g}$ ground term, all six electrons paired in $t_{2g}^6$). The question: is this single-band assignment ($^1A_{1g}\to{}^1T_{1g}$) consistent with the complex actually being low-spin, or could the data equally fit a high-spin $^5T_{2g}$ ground term picture?

Step 1. For low-spin $d^6$, the lowest spin-allowed transition is $^1A_{1g}\to{}^1T_{1g}$, and near $\Delta_o/B \sim 25$–$35$ (typical for strong-field low-spin $d^6$ complexes such as Co(III) amines), the diagram gives $E(^1T_{1g})/B$ in the range $30$–$36$. Testing the hypothesis $\Delta_o/B = 29$: this would require, from the definition of the ratio,

which is within a few hundred cm⁻¹ of the observed $20{,}300\ \text{cm}^{-1}$ — consistent within the reading precision of a hand-matched diagram (real published tables carry roughly this much graphical uncertainty).

Step 2. Cross-check against the definition of $\Delta_o$ implied: $\Delta_o = 29 \times 700 = 20{,}300\ \text{cm}^{-1}$ (taking the ratio value that reproduces the observed band exactly). For low-spin $d^6$ octahedral complexes, this $\Delta_o$ is entirely typical (compare $[\text{Co(NH}_3)_6]^{3+}$, $\Delta_o \approx 22{,}900\ \text{cm}^{-1}$, or $[\text{Fe(CN)}_6]^{4-}$, $\Delta_o \approx 33{,}800\ \text{cm}^{-1}$ — both comfortably low-spin), so the assignment is self-consistent.

Step 3. Now test whether the high-spin branch could instead fit the same single band. For high-spin $d^6$ ($^5T_{2g}$ ground term), there is only one spin-allowed transition possible, $^5T_{2g}\to{}^5E_g$ (a $t_{2g}\to e_g$ single-electron promotion), whose energy is by definition equal to $\Delta_o$ itself (exactly analogous to the $d^1$ trivial case, since a high-spin $d^6$ ion is, in terms of the single "extra" electron picture relative to a half-filled $t_{2g}^3e_g^2$ core, isoelectronic in symmetry structure with $d^1$). If we tried to force $\Delta_o = 20{,}300\ \text{cm}^{-1}$ directly for a high-spin assignment, we would need $B$ separately from a different (spin-forbidden) band to check consistency — but the critical field strength for the $d^6$ high-spin/low-spin crossover is known to be $(\Delta_o/B)_{\text{crit}} \approx 20$–$22$ depending on $C/B$. Since the fitted $\Delta_o/B = 29$ from Step 1 lies clearly above this crossover value, the low-spin assignment is the internally consistent one, and the high-spin hypothesis is excluded by the diagram itself, not merely by the (already-known) diamagnetism.

Step 4 — the exam-level punchline. This demonstrates the deeper point: the position of a single band relative to the diagram's kink is itself a spin-state diagnostic, independent of any external magnetic measurement — a beautiful internal consistency check that Tanabe–Sugano analysis provides for free, precisely because the ground-term line is discontinuous at $(\Delta_o/B)_{\text{crit}}$.