Where a d–d band falls

A ligand field splitting is an energy difference between two sets of orbitals. Put an electron in the lower set, give it that energy, and it can go to the upper one. Since the energy of light is fixed by its wavelength, the splitting says where a complex absorbs — and the arithmetic connecting them is a reciprocal and nothing more.

Splittings are quoted in wavenumbers, which are already inverse wavelengths, so the conversion is a division: 10⁷ divided by a splitting in cm⁻¹ gives a wavelength in nanometres. That is the whole calculation, and it is worth doing because of what it says about a claim everybody has heard.

The claim is that a transition-metal complex is coloured because its d electrons absorb, and that the colour seen is the complement of what is absorbed. The first half is often true. The second half is somebody else’s subject. And there is a large and familiar class of compounds for which the first half is false in a way that is easy to check.

The spread, and what falls where #

The measured splittings for six-coordinate complexes run from about 7,000 cm⁻¹ for iodide to about 34,000 for carbon monoxide — a factor of nearly five in energy, and the same factor in wavelength the other way, from 1,429 nm to 294 nm.

The visible range is roughly 380 to 750 nm, which is 13,300 to 26,300 cm⁻¹. Of the nine ligands, water at 730 nm, ammonia at 463 and ethylenediamine at 457 fall inside it. The four halides fall below it in energy — their bands are in the near infrared, invisible to an eye and perfectly visible to a spectrometer. Cyanide at 376 nm and carbon monoxide at 294 nm fall above it, in the ultraviolet.

So the series straddles the visible rather than sitting inside it, and this was worth checking rather than assuming: the expectation written down before the computation was that most of them would land in the visible, and the arithmetic refused it. Three of nine.

That refusal is more useful than the expectation was. It means the statement “transition-metal complexes are coloured because of d–d transitions” cannot be a general explanation, because for two thirds of the ligands in the standard list the d–d transition is not in the visible at all. Something else is colouring a great many coloured compounds.

What the band energy actually is #

Here care is needed, because the identification of the absorption energy with Δ is exact in exactly two cases and approximate in the rest.

For a d¹ ion — titanium(III) in water is the standard example — there is one electron, it sits in the lower set, and exciting it to the upper set costs the splitting. Initial and final states differ by one electron moving between two one-electron levels, so the band energy is Δ, and measuring the band measures the splitting directly.

For a d⁹ ion the same holds by the hole argument: one hole in the upper set moves to the lower, and the energy is again Δ.

For everything in between it is not. A d³ ion excited from the lower set to the upper has three electrons before and three after, and the repulsion between them differs in the two arrangements. The band energy therefore contains interelectronic repulsion terms, which is why the analysis of a real d–d spectrum uses Tanabe–Sugano diagrams with two parameters — the splitting and a Racah parameter measuring repulsion — rather than one.

The second parameter is not computed here, and the reason is simple: repulsion between electrons is not in a one-electron model. What repulsion does when it is put in, on a system small enough to solve exactly, is the subject of the smallest many-electron calculation, and the honest position here is that the conversion from a splitting to a wavelength is exact arithmetic performed on a quantity that only equals the band energy in two special cases.

The bands are broad, and that is structural #

Every band drawn in the hero figure has a width, and the widths are not instrumental. A d–d band is typically two to four thousand wavenumbers across — a fifth of its own energy — which is enormous by the standards of the sharp lines of what a photoelectron spectrum measures or a rotational spectrum.

The reason is structural and is the Franck–Condon argument. Moving an electron from an orbital pointing between the ligands to one pointing at them changes the equilibrium metal–ligand distance: the excited state wants to be bigger. The electronic transition is fast compared with nuclear motion, so the molecule arrives in the excited electronic state still at the ground state’s geometry, which is not the excited state’s equilibrium — and the transition therefore reaches a spread of vibrational levels rather than one.

So the width of a d–d band is a measure of how much the geometry changes, which makes a broad band a structural datum rather than a nuisance. Bands from transitions between orbitals with the same spatial character are narrow; bands that move an electron between the two sets are broad. The same reasoning appears from the other direction in a bond length out of a spectrum, where the geometry is what is being extracted.

How many bands there are, which is a counting question #

A splitting diagram with two levels invites the expectation of one band, and real d–d spectra usually show two or three. The reason is the same one that has run through this site since how many frequencies, not how many modes: the number of transitions is a count over states, not over orbital levels, and the two counts differ as soon as there is more than one electron.

A d³ ion in an octahedral field has three electrons distributed over five orbitals, and the arrangements available to it are more numerous than the two levels suggest. Each distinct arrangement is a state with its own energy, the energies differ by repulsion terms, and every allowed transition between them is a band. Chromium(III) complexes show three, at positions that are not multiples of anything simple.

So a spectrum counts states and a splitting diagram draws orbitals, and getting from the first to the second is the work Tanabe–Sugano diagrams exist to do. The count itself is symmetry — it comes from reducing the states in the point group, which is the same argument what an absence proves makes about a vibrational spectrum — and the positions need the repulsion parameter a one-electron model does not have.

Colourless is not the same as transparent #

Four of the nine ligands put the band in the near infrared, and complexes of those ligands are often described as pale or colourless. That description is about an eye and not about the compound.

A hexaiodido complex absorbs at 1,429 nm as strongly as an aqua complex absorbs at 730 — the transition is the same kind of transition, moved. An instrument sees both; the word “colourless” records that one of them lands where human receptors do not respond. The site’s practice of stating what was measured rather than what was noticed applies here as much as it does to say what it encloses: the datum is a wavenumber, and the perceptual consequence is a separate claim with its own conditions.

The same care applies at the other end. Carbon monoxide complexes absorb at 294 nm, in the ultraviolet, and metal carbonyls are often colourless or faintly yellow — but they are strongly absorbing objects, and their photochemistry is the reason a great deal of organometallic chemistry is done in the dark.

Permanganate has no d electrons #

The most intensely coloured transition-metal compound in ordinary use is potassium permanganate, and its colour is not a d–d transition, because manganese(VII) has no d electrons at all.

Its absorption is a charge transfer: an electron moves from an orbital that is mostly on the oxygen ligands to one that is mostly on the metal. That is a transition between the levels of the molecular orbital diagram rather than within the metal’s d set, and it is subject to none of the restrictions that make d–d bands weak.

A charge-transfer transition lives in a different diagram entirely: the lower block mostly ligand, the block above it mostly metal, and a transition between the two blocks rather than within one. Those bands are orders of magnitude stronger than the d–d ones and they are not what this essay is locating.

The intensity difference is the decisive evidence and it is enormous. A d–d band has a molar absorption coefficient of order 1 to 100; a charge-transfer band, 10³ to 10⁴. So a charge-transfer absorption a hundredth as concentrated still dominates the colour, and a compound with both will be coloured by its charge transfer.

That is why so many deeply coloured inorganic compounds are coloured for reasons unconnected to ligand field theory: permanganate, dichromate, mercury(II) iodide, the iron(III) thiocyanate used as a blood substitute in films. Ligand field theory is the right tool for the pale ones.

Why d–d bands are weak in the first place #

The intensity comparison above is not an accident of magnitudes; it follows from symmetry, and the symmetry argument is short.

An electric dipole transition is allowed only if the product of the two states’ representations with the dipole’s contains the totally symmetric one. In a centrosymmetric complex both d orbitals are even — g — and the dipole is odd, so the product is odd and the integral vanishes. Exactly. Not approximately.

That is Laporte’s rule, and why a d–d band is weak computes it with the same selection-rule argument selection rules are one theorem sets out. What makes the bands weak rather than absent is that a vibration can destroy the centre of symmetry momentarily, and the essay works out how much intensity that buys.

The rule also predicts a difference between geometries, and the prediction is met. A tetrahedron has no centre of symmetry at all, so nothing forbids the transition and tetrahedral complexes are markedly more intensely coloured than octahedral ones — the deep blue of tetrahedral cobalt(II) against the pale pink of the octahedral aqua ion is the standard demonstration, and the two differ by about a factor of a hundred in intensity.

Where this site stops #

The arithmetic here converts an energy to a wavelength. It does not say what colour anything looks, and the boundary is worth stating plainly rather than being crossed quietly.

What a wavelength does after it reaches an eye — which receptors respond, how three responses become a colour, why the answer depends on the illuminant and on what is next to the object — is a subject of its own. The pictures here show a spectrum and never a colour, and that is deliberate: the standard observer, colour matching and appearance modelling belong to colour science, and drawing a swatch would be borrowing an argument not made here.

There is a practical reason for the discipline as well. The complement rule — absorb green, look red — is a rough guide that fails on exactly the cases a chemist cares about: a compound with two absorption bands, a compound whose band is broad enough to cover half the visible, a compound absorbing at the edge of the range. Cobalt(II) in water absorbs around 510 nm and looks pink; nickel(II) in water absorbs at three places and looks green; and neither colour can be got from one wavelength by a rule of thumb. The chemistry done here stops at the wavenumber, and stopping there is what keeps the wavenumber trustworthy.

The line is easy to state in chemical terms. Chemistry decides where the absorption is. It does not decide what is seen. A compound absorbing at 500 nm is not thereby “purple”: that word describes a perceptual response to whatever light is left, which depends on the width and shape of the absorption, on what else the compound absorbs, on how concentrated it is, and on the light it is being looked at in.

Who found it, and when #

The connection between ligand field splittings and the colours of complexes was made in the early 1950s, when Leslie Orgel and others took Bethe’s and Van Vleck’s crystal field arithmetic — written twenty years earlier for magnetic susceptibilities — and applied it to visible spectra. Yukito Tanabe and Satoru Sugano’s diagrams, which handle the many-electron cases a one-electron model cannot, are from 1954.

What is striking in retrospect is how long the correlation had been available without being drawn. The spectra were measured, the colours were known, the crystal field theory was published in 1929, and the connection waited two decades — largely because the theory arrived as an account of magnetism, and the community that measured magnetic susceptibilities was not the community that measured absorption spectra.

What the intense bands measure instead #

The compound with no d electrons is not an exception to be noted and set aside. Its band is the more informative of the two kinds, and what it reports is worth stating, because it is not a ligand field at all.

An intense band in a transition-metal compound is almost always a charge transfer: an electron moving from a ligand-based orbital to a metal-based one, or the reverse. Being allowed rather than forbidden, such bands are a hundred to a thousand times stronger than a d–d band — the extinction coefficient of a d–d transition is single or double digits, and a charge-transfer band’s runs to tens of thousands.

The two directions measure two different things and both are electrochemistry read off a spectrum.

Ligand to metal. An electron moves from the ligands onto a metal in a high oxidation state, so the energy required measures how badly the metal wants an electron — its oxidising power. Permanganate absorbs at about 525 nanometres and chromate at about 370, and permanganate is the stronger oxidant. The more oxidising the metal, the lower the energy of its charge-transfer band, and the purple of permanganate is that statement in the visible.

Metal to ligand. An electron moves the other way, from a metal in a low oxidation state into an empty π* orbital on the ligand, so the energy measures how readily the metal gives an electron up. Tris(bipyridyl)ruthenium(II) is the standard example: an intense band near 450 nanometres, an extinction coefficient near fifteen thousand, and an excited state that is a genuine oxidised metal beside a reduced ligand — which is why compounds of that family are the workhorses of photochemistry that has to do something with the excited electron.

So the field’s two kinds of absorption divide cleanly. A d–d band is weak, and its position measures the ligand field. A charge-transfer band is strong, and its position measures a redox potential. Nothing about the first predicts the second, the compounds people find striking are almost all the second, and a rule about splittings has no business being applied to them.

What the conversion is worth #

Stated carefully, the result of this essay is narrow and useful.

The ligand field splitting fixes the position of the d–d absorption, exactly for d¹ and d⁹ and approximately otherwise; the position runs from the near infrared to the ultraviolet across ordinary ligands; three of nine common ligands put it in the visible; the band is broad because the excited state has a different equilibrium geometry; and it is weak because it is Laporte-forbidden in a centrosymmetric complex.

None of that adds up to “complexes are coloured because of d–d transitions”. Some are. The intensely coloured ones usually are not, and the compound most people would name first has no d electrons to excite. A rule that explains the pale compounds and gets overruled in the vivid ones is worth having with its scope attached — which is the whole difference between a rule and a slogan.