Why one arrow was never enough
In the previous guides a d-d transition was a tidy single arrow: one electron hops from t2g up to eg across the gap delta-o, the absorbed wavelength sets the color, and we move on. That cartoon is honest for the simplest case, but the moment you put a real spectrum of, say, [Cr(H2O)6]3+ or [Co(H2O)6]2+ in front of a student, it betrays you: there are often two or three distinct absorption bands, not one. A single-electron jump can only ever predict one band. Something the cartoon hid is doing the work.
The thing hidden is electron-electron repulsion. When two or more d electrons share an ion, they do not occupy orbitals independently; they jostle, and the energy of the ion depends on exactly how their spins and orbital motions are arranged. A configuration like d2 is not a single energy at all — it is a small family of distinct energy levels called terms, born purely from the electrons repelling one another. The orbital-splitting story of the last rung ignored this entirely; to read a real spectrum we have to put it back. That is the honest reason this guide exists, and fair warning up front: this is the deep end of the rung, sketched in concept rather than derived line by line.
Free-ion terms: the configuration's hidden family
Start with the bare ion, no ligands yet. A free-ion term bundles the electrons' total orbital angular momentum L and total spin S into a compact label, the term symbol, written as a superscript-then-letter like 3F or 4F. The letter (S, P, D, F, G...) encodes L = 0, 1, 2, 3, 4..., exactly mirroring the s, p, d, f naming of single orbitals but now for the whole group of electrons. The superscript is the spin multiplicity, 2S+1, which counts how many ways the spins can line up: a superscript of 3 means two unpaired electrons (a triplet), 4 means three unpaired (a quartet). The term with the most unpaired spins and, among those, the largest L sits lowest in energy — that is just Hund's rules speaking in this richer language.
How many terms does a configuration spawn? More than you might guess, because you must count every distinct way of distributing the electrons among the orbitals, respecting the Pauli exclusion principle. A d2 ion, for instance, generates five terms — 3F, 3P, 1G, 1D, 1S — with 3F lying lowest because it has the most parallel spins. We almost never need all of them: only the ground term and the small handful of excited terms that share its spin multiplicity matter for the visible spectrum, because of the spin rule we meet shortly. A genuinely useful piece of bookkeeping is that the ground term of dn and d(10-n) is the same letter — d2 and d8 both have an F ground term, d3 and d7 both share another — which is why one diagram can quietly do double duty.
Switching on the field: terms split too
Now lower the ion into an octahedral ligand field. Just as a single d orbital split into t2g and eg, a free-ion term splits into a set of new ligand-field levels — and the rules are the same symmetry rules in disguise. A D term, for example, splits into two pieces, much like the five d orbitals split into two sets; an F term splits into three. Each resulting level keeps the spin multiplicity of its parent term, so a 3F parent yields only triplet daughters. The energy spacings between these new levels grow as the field gets stronger, while the spacing between different parent terms stays anchored to electron repulsion (to B). That competition — field strength pulling one way, repulsion the other — is the whole plot.
Free-ion term --(octahedral field, strength grows ->)--> ligand-field levels
d2 ground term = 3F -> 3A2g , 3T2g , 3T1g(F) (F splits into 3)
3P -> 3T1g(P) (P does not split)
d3 ground term = 4F -> 4A2g (ground), 4T2g, 4T1g(F)
4P -> 4T1g(P)
band energy ~ depends on BOTH delta_o (field) and B (repulsion)This finally explains the multiple bands. The visible absorptions of a complex are transitions from the lowest ligand-field level up to the other levels that share its spin multiplicity. For d3 [Cr(H2O)6]3+ there are three such upper levels, so we expect — and find — three bands. The spin selection rule is why we only count same-multiplicity levels: a transition that would have to flip a spin is hundreds of times weaker, often invisible, so jumping from a quartet ground level to a doublet excited level barely registers. Those forbidden transitions are exactly why d-d colors are pale to begin with, a thread from the previous guide; here the same rule tells us which bands to even look for.
Orgel and Tanabe-Sugano: the same story, two charts
Chemists do not re-derive all this by hand each time; they read it off a graph. An **Orgel diagram** plots the energies of the ligand-field levels (vertical axis) against field strength (horizontal axis), with the free-ion terms anchored at the left edge where the field is zero. As you slide rightward into a stronger field, the lines fan out, and the vertical gaps from the ground line up to each excited line are exactly the band energies you would measure. Orgel diagrams are the friendly, intuitive version — but they only handle the weak-field, high-spin cases and they say nothing about spin-state changes.
The **Tanabe-Sugano diagram** is the professional's tool and fixes both limitations with two clever choices. First, it makes the ground state the flat horizontal baseline at zero, so every line on the chart is plotted as an energy gap straight up from the ground state — exactly what a spectrometer measures. Second, it divides everything by B, plotting E/B against delta-o/B, so the chart becomes dimensionless and one diagram serves every metal and ligand of that dn count. Where it really earns its name: for d4 through d7 there is a sharp vertical line on the diagram marking the high-spin/low-spin crossover, the field strength at which the ground term abruptly changes. To the left it is high-spin, to the right low-spin — the spin-state fork from the previous guide, drawn as a place on a graph.
Reading a real spectrum backwards: extracting delta-o and B
Here is the payoff that makes all this worth the climb. The diagram is usually run forward — pick a field strength, read off the bands. But a measured spectrum lets you run it backward: from the band positions you can recover the two numbers that define the complex's electronic structure, the splitting delta-o and the Racah parameter B. You are turning a strip of colored glass into hard quantum-mechanical numbers, with nothing more than a spectrometer and a chart.
- Record and assign the bands: measure the absorption maxima (in cm-1, the natural unit here), figure out the dn count from the metal's oxidation state, and match the number of spin-allowed bands you see to the upper levels on the right diagram.
- Use the simplest band to get delta-o: for several configurations the lowest-energy band corresponds directly to a transition whose energy equals delta-o, handing you the splitting almost for free.
- Take a ratio to find the operating point: form the ratio of two band energies, slide along the dimensionless diagram until the predicted lines hit that same ratio, and read off the value of delta-o/B sitting underneath — that fixes where on the chart this complex lives.
- Solve for B, then delta-o: at that operating point the diagram gives each band as a known multiple of B, so dividing a measured band energy by its E/B value yields B in cm-1; multiply delta-o/B by that B and you have delta-o too.
The value of B you extract carries a quiet, satisfying lesson. The free ion has a textbook B, but the B you measure inside a complex is almost always smaller — the electron cloud has expanded a little, so the electrons repel each other less fiercely. That shrinkage is the nephelauxetic effect ('cloud-expanding'), and it is hard physical evidence that the metal-ligand bond is partly covalent: the d electrons are spending real time out on the ligands, not sitting on point charges. So the very same data that gives you a crystal-field number, delta-o, also quietly refutes crystal field theory's central fiction. That is the kind of honesty good models invite.
Honest limits of the deep end
Keep this guide in proportion. Everything above is sketched in concept, not derived in full — counting terms rigorously, working out exactly how each splits, and computing the slopes of those lines is a serious quantum-mechanics exercise that fills chapters, and we have deliberately skipped the algebra. The diagrams themselves rest on assumptions: they treat delta-o and B as the only knobs, ignore the smaller spin-orbit coupling that actually matters for the heavier metals and the f-block, and assume clean octahedral symmetry. Real bands are also broad and often overlapping, thanks to vibronic coupling and Jahn-Teller distortions, so reading a precise number off a fuzzy hump is as much craft as calculation.
None of that diminishes the achievement. With nothing but a list of band positions and a chart, you can now name the electronic states of a transition-metal complex, predict how many d-d bands to expect, and extract two real, tabulated quantum numbers — delta-o, the strength of the ligand field, and B, the residual electron repulsion. That is the quantitative heart of d-block spectroscopy, and it connects directly back to color, to spin state, and to the spectrochemical series you already know. If the algebra one day calls to you, this is the conceptual map you will be filling in.