A gap left over from the last rung
You ended the coordination rung holding a small but powerful picture: when six ligands close in along the axes of an octahedron, the metal's five d orbitals stop being equal. The two that aim their lobes straight down the axes at the incoming ligands — dz2 and dx2-y2, the pair called eg — are shoved up in energy, because an electron sitting there is nose-to-nose with a ligand lone pair. The three that point into the gaps between the axes — dxy, dxz, dyz, the set called t2g — sink down, because an electron there dodges the ligands. The energy gap between the low t2g and the high eg is the [[crystal-field-theory|crystal field splitting]], written delta-o for an octahedral field. This guide cashes that gap in for something you can literally see: color.
Here is the thing worth pausing on. The gap delta-o is tiny on the scale of chemical energies — far smaller than a bond, far smaller than the gaps inside an isolated atom. It happens to land in an energy range that corresponds to visible light. That is the whole coincidence behind the famous fact that [[colored-transition-metal-compounds|transition-metal compounds are colored]] while a sodium or magnesium salt is a boring white. NaCl is colorless because its ions have no partly filled d shell and no small gap for visible light to bridge; [Cu(H2O)6]2+ is sky-blue because it has exactly such a gap, and a photon of just the right energy can hoist an electron across it.
The electron's leap: a d-d transition
Picture the simplest possible case, the d1 ion [Ti(H2O)6]3+. Titanium(III) has a single d electron, and in the octahedral field it sits in the lower t2g set, with the eg orbitals empty above it. Shine white light through the solution and most of it sails through untouched — but a photon whose energy exactly matches delta-o can be absorbed, and when it is, it lifts that lone electron from t2g up into eg. That promotion of an electron from a lower d orbital to a higher one is a [[d-d-transition|d-d transition]] (also called a ligand-field transition), and it is the engine of color for the great majority of transition-metal complexes.
The link from gap to wavelength is the bridge that physics built. A photon's energy and its wavelength are inversely tied: E equals hc divided by lambda. A large gap demands a high-energy, short-wavelength photon (toward the blue and violet end); a small gap is bridged by a low-energy, long-wavelength photon (toward the red end). So the size of delta-o directly chooses which slice of the rainbow gets swallowed. [Ti(H2O)6]3+ absorbs in the green-yellow around 500 nm, which is why its solution is the soft purple you may have seen in a lab. Make the gap bigger and the absorption marches toward the violet; make it smaller and it slides toward the red.
The twist: we see the opposite color
Now the single most misremembered point in the whole subject. The complex does not look like the color it absorbs — it looks like the color that is left over. White light is the full rainbow; if the complex drinks the red photons out of it, what reaches your eye is everything except red, and your brain reads that leftover blend as green. The color you see is the [[complementary-color|complementary color]] of the light absorbed. Absorb red, look green. Absorb orange, look blue. Absorb green-yellow, look purple — which is exactly what [Ti(H2O)6]3+ does.
ABSORBED light -> COLOR YOU SEE (its complement)
red (~700 nm) -> green
orange (~600 nm) -> blue
yellow (~580 nm) -> violet
green (~530 nm) -> red / purple
blue (~470 nm) -> orange
violet (~420 nm) -> yellow
A BIGGER delta -> shorter absorbed wavelength (toward violet)
-> seen color shifts the OTHER way (toward yellow)This single inversion explains an apparent paradox students trip over. A bigger gap means the complex absorbs higher-energy, bluer light — yet it can end up looking warmer (yellow or orange), not bluer, because what you see is the complement of what vanished. So 'stronger field, bigger delta' does not translate to 'looks more blue'; it means 'absorbs more toward the blue, therefore looks more toward the yellow.' Keep the absorbed-versus-seen distinction straight and the rest of this rung clicks into place.
Three dials that change the color
Since color tracks delta, anything that changes delta changes the color. There are three dials you can turn. The first is the ligand. Some ligands clamp a strong field on the metal and force a big gap; others give a weak field and a small gap. Ranking them from weakest to strongest gives the [[spectrochemical-series|spectrochemical series]], an experimental order that runs roughly I- < Br- < Cl- < F- < OH- < H2O < NH3 < en < CN- < CO. Swap water for ammonia around the same metal and the gap widens and the color shifts: [Ni(H2O)6]2+ is green while [Ni(NH3)6]2+ is blue-violet, because ammonia sits higher in the series and opens a wider delta.
The second dial is the metal itself, and the third is its [[oxidation-state|oxidation state]]. Going down a group the d orbitals reach out farther and feel the ligands more strongly, so a third-row metal gives a much bigger delta than its first-row cousin. And for one metal, raising the oxidation state pulls the ligands in tighter against a more positive center, again widening the gap: Fe2+ versus Fe3+ complexes, or the deep purple of manganese(VII) in permanganate versus the pale pink of manganese(II), are everyday demonstrations. A useful rule of thumb is that the dial sizes line up as ligand and oxidation state and going-down-a-group all pulling in the same direction — toward a larger gap and bluer absorption.
Honest limits: faint d-d, vivid charge transfer
Two honest caveats keep this picture from misleading you. First, d-d transitions are intrinsically weak. A selection rule (the Laporte rule) says a pure d-to-d jump within the same metal is formally forbidden, so it only sneaks through because vibrations briefly bend the perfect symmetry. That is why most d-d colors are pastel — the gentle blue of copper sulfate, the pale pink of Mn2+ — rather than blazing. If you ever meet a transition-metal compound with a savage, intense color, suspect a different mechanism is at work.
That other mechanism is the [[charge-transfer-band|charge-transfer transition]], where a photon does not just shuffle an electron between the metal's own d orbitals — it hurls an electron from the ligands onto the metal, or from the metal out to the ligands. These transitions are allowed, not forbidden, so they are hundreds of times more intense. The blinding purple of permanganate [MnO4]- and the orange of chromate [CrO4]2- are charge-transfer colors, not d-d colors at all — note that permanganate's manganese is d0, with no d electrons to shuffle, so its color cannot be d-d. We will give charge transfer its own guide; for now just hold the contrast: d-d is faint and tells you about the gap, charge transfer is vivid and tells you about the metal-ligand redox relationship.
One last note for the road, tying back to the coordination rung. The whole story above assumed an octahedron, where t2g lies below eg. A [[tetrahedral-field-splitting|tetrahedral field]] inverts the pattern — the two-orbital set drops below the three-orbital set — and the splitting is much smaller, only about four-ninths of the octahedral gap for the same metal and ligands. A smaller gap means longer-wavelength absorption, which is part of why tetrahedral complexes such as the brilliant blue [CoCl4]2- look so different from their octahedral, pinkish [Co(H2O)6]2+ relatives. Same metal, same idea, different stage — and a different color to show for it.