Why a d–d band is weak

A selection rule is simple to state: an integral vanishes unless the product of the representations of everything inside it contains the totally symmetric representation. That is a theorem about the group, it is exact, and selection rules are one theorem applied it to vibrational spectra.

Applied to a coordination complex it gives a result that is visible on any laboratory bench. The transition between the two halves of a split d shell — the transition whose energy the whole of ligand field theory is about — is forbidden.

The argument is three lines. In an octahedral complex the d orbitals span $E_g \oplus T_{2g}$: both even, both unchanged by inversion, because a d function is even and the complex has a centre. The electric dipole operator transforms as x, y and z, which span $T_{1u}$: odd. The product of even, odd and even is odd, an odd representation contains no totally symmetric part, and the integral is zero.

Not small. Zero, in the same sense as the forbidden overlaps of exactly zero, where the integrand is odd about a plane and the contributions cancel in pairs. There is no quantity to estimate.

The integral the rule is about #

It is worth writing down what is being claimed to vanish, because a selection rule stated as a rule about labels can look like bookkeeping.

The strength of an electric dipole transition between two states is fixed by the integral of (final state) × (dipole operator) × (initial state) over all space. The dipole operator is a position — x, y or z — so the integrand is a product of three functions, and the question is whether that product integrates to something other than zero.

A function integrates to zero over all space whenever the space can be divided into regions that cancel in pairs. That is the argument exactly zero makes for a forbidden overlap: the integrand is odd about a plane, every contribution from one side has a partner of equal size and opposite sign on the other, and the total is not small but exactly nothing.

Parity is that argument in its simplest form. Inversion sends every point to its opposite; a d function is unchanged by it; a coordinate changes sign. So the integrand of a d–d transition moment changes sign under inversion, every point pairs with its opposite, and the integral vanishes for the same reason a symmetric function integrates to zero against an antisymmetric one.

The group-theoretical version — reduce the product and look for the totally symmetric representation — is that argument generalised past inversion to every operation the molecule has. It is the same statement made mechanical, and being mechanical is what lets a program do it for a group of forty-eight operations without anybody having to hold the geometry in mind.

The computation, and what it refuses #

It can be done as a character product and a reduction rather than as a piece of reasoning, so that it can be checked and so that it can fail.

A proper check tests the forbidden product and two other things, and the third is the one that gives the first two content: it asks the same question about a tetrahedral complex, where there is no centre of symmetry, and requires the answer to come back allowed. A rule that returned “forbidden” everywhere would pass a check that only tested the octahedron.

In a tetrahedron the d orbitals span $E \oplus T_2$ and the dipole spans $T_2$. The product $E \times T_2 \times T_2$ contains the totally symmetric representation, and the transition is allowed along all three polarisations.

What the rule predicts, and what is measured #

That difference between two geometries is a prediction about intensities, and it is met by a factor that is hard to miss.

A d–d band in an octahedral complex has a molar absorption coefficient of order 1 to 100 litres per mole per centimetre. The same transition in a tetrahedral complex of the same metal runs to several hundred. Charge-transfer bands, which are forbidden by nothing, run to 10⁴.

The visible consequence is a classroom demonstration: cobalt(II) in water is octahedral and a pale pink, and adding concentrated hydrochloric acid converts it to a tetrachloridocobaltate ion, which is tetrahedral and an intense blue. Both are d⁷ cobalt. The tetrahedral form absorbs at lower energy — four ninths of the splitting, as two models, one ratio computes — and far more strongly, because the parity restriction has gone.

The absence of an inversion centre is a property of the arrangement rather than a label attached to it, and this site’s point-group search finds it without being told: handed a tetrahedral tetrachlorido complex’s coordinates it returns Td, and every group a molecule can fall to is what the search is built for.

How a forbidden band appears at all #

An octahedral complex is not colourless, so the forbidden transition happens. The rule is exact and the molecule is escaping it, and the escape is worth computing rather than waving at.

Laporte’s rule applies to a molecule at its equilibrium geometry. A real complex is vibrating, and a vibration that is itself odd destroys the centre of symmetry for as long as it is displaced. The transition moment then has to be evaluated for the distorted molecule, and the relevant product includes the vibration.

Computing the product with each vibrational species in turn gives a clean answer. The odd species — $T_{1u}$ and $T_{2u}$ — make the transition allowed. Every even species — $A_{1g}$, $E_g$, $T_{2g}$ — leaves it forbidden.

That is the result stated as a rule, and it is worth noticing that it was not assumed. The calculation asked the same question of all five species and three of them said no. A check that only asked about the odd ones would have proved nothing, because a broken product routine that always returned “allowed” would have passed it.

The intensity that results is proportional to how much odd vibration is present, which is why d–d bands in octahedral complexes get stronger as the temperature rises: more of the odd modes are excited, and the molecule spends more of its time away from the symmetry that forbids the transition. That temperature dependence is the experimental signature of a vibronically allowed band and distinguishes it from a band that is allowed outright.

Where the borrowed intensity comes from #

Saying that a vibration makes the transition allowed is correct and slightly opaque. The mechanism has a better description, and it explains a pattern in the numbers.

An odd vibration mixes states of opposite parity: while the complex is displaced along it, what was a pure d state acquires a small admixture of an odd state — typically a charge-transfer state, which is allowed and intense. The forbidden transition then borrows intensity from the allowed one in proportion to how much of it is mixed in.

That predicts something checkable. The closer in energy the allowed state is, the more of it is mixed in for a given displacement, and the stronger the forbidden band becomes. Complexes with low-lying charge-transfer bands do have relatively strong d–d bands, and complexes whose charge transfer is far into the ultraviolet have very weak ones — which is a correlation between two features of the same spectrum, and it is the kind of evidence that distinguishes a mechanism from a restatement.

Two complexes of one metal, a hundredfold apart #

The cleanest demonstration of the rule uses one element in one oxidation state and changes only the arrangement.

Cobalt(II) in water is octahedral, six water molecules around a d⁷ ion, and it is a pale pink whose absorption coefficient is a few units. Add concentrated chloride and the coordination changes to four chlorides in a tetrahedron, and the solution turns an intense blue whose absorption coefficient is several hundred.

Everything about the metal is the same. The splitting has fallen, to four ninths of what it was, which moves the band to lower energy; the number of ligands has fallen; and the intensity has risen by about a hundredfold. Nothing in a picture of orbital energies predicts that rise. What predicts it is that the tetrahedron has no centre of symmetry, so the parity restriction that costs the octahedral complex two orders of magnitude is simply absent.

The tetrahedral character table, generated from the molecule’s own operations in why a character table stops where it stops, has no inversion among them and no g or u subscript anywhere in its species labels — which is the whole reason a tetrahedral complex is intensely coloured and its octahedral counterpart is not.

The other selection rule, which is about spin #

Parity is not the only thing forbidding transitions here. Spin is, and its consequence is a compound most people have seen without noticing what it demonstrates.

An electric dipole operator does nothing to spin, so a transition between states of different total spin has a vanishing moment for a second and independent reason. Nearly every d–d transition preserves spin, so the rule is usually silent — but there is one filling where it is not.

A high-spin d⁵ ion — the arrangement the pairing energy decides the moment computes the condition for — has five electrons in five orbitals, all with parallel spins. Every rearrangement of those electrons must pair two of them, because there is nowhere else for an electron to go, and pairing changes the total spin. So every d–d transition of a high-spin d⁵ ion is spin-forbidden as well as parity-forbidden.

The prediction is that such compounds are almost colourless, and manganese(II) salts are: the faint pink of a manganese(II) solution has a molar absorption coefficient of a few hundredths, some three orders of magnitude below an ordinary Laporte-forbidden band and five below a charge-transfer band. Two independent restrictions, each costing about a hundredfold, and a compound that is the palest thing in the transition series.

The contrast with permanganate is the sharpest available. The same element, in a different oxidation state with no d electrons at all, gives one of the most intensely coloured compounds in ordinary use, because its absorption is a charge transfer that neither rule touches. Two manganese compounds, five orders of magnitude apart in absorption strength, and the whole difference is which transition is available.

What “forbidden” is worth as a word #

This essay sits at the end of a line of argument that began with a character table, and it is a good place to say what the exactness of these results is worth, because there is a temptation to discount them.

A selection rule can look like a formality when the forbidden band is plainly there in the spectrum. It is not a formality, and the reason is quantitative: the rule does not say the intensity is zero in practice, it says the intensity is zero for the stated symmetry, and everything that appears is therefore a measure of the departure from that symmetry. A forbidden transition is a meter for the thing forbidding it.

The two levels between which nothing is allowed to happen are the ordinary ones, drawn in two models, one ratio: three below two, separated by the most-quoted gap in coordination chemistry. The transition that measures that gap is forbidden by parity, weakened further by spin where the two states differ in it, and observed anyway — which is three facts about one absorption band and the reason the number is quoted more confidently than it is obtained.

That is the same lesson what an absence proves drew from a vibrational spectrum, where a missing band is evidence of a centre of symmetry rather than an experimental failure. An exactly vanishing quantity is more informative than a small one, because anything measured in its place has a single identifiable cause.

What this says about the counting results #

Four earlier results were about counting: how many vibrations there are, how many bands a spectrum can show, what an absence proves, and why a degeneracy is a dimension. All four were answers to how many.

This one is an answer to how strongly, and it is worth noticing that the same method gave it. Nothing new was added to the character tables; the transition moment is an integral like any other, the product rule applies to it as it applies to a vibrational overlap, and the difference between an allowed band and a forbidden one is a reduction returning a totally symmetric part or not returning one.

That reach is the argument for doing the reduction in the first place. A table of which transitions are allowed for which point group would have served just as well for the octahedron and would have had nothing to say the first time an unusual geometry turned up. Reducing a product computed from the molecule’s own operations works for whatever the molecule is, including coordination complexes whose groups are recovered from their coordinates rather than declared.

Who found it, and when #

Otto Laporte stated the parity rule in 1925, for atomic spectra and before quantum mechanics had the form it now has: the observation was that atomic lines connect terms of opposite parity, and the explanation followed. Applying it to complexes came with the ligand field work of the 1950s, and the vibronic escape — the argument that an odd vibration lends intensity to a parity-forbidden transition — is due to Albert Liehr and Carl Ballhausen, in 1957.

The order matters. The rule was an empirical regularity in atomic spectra, then a theorem about parity, then an explanation of why a whole class of compounds is pale, and then a tool: because the intensity measures the departure from centrosymmetry, a band that is too strong for a Laporte-forbidden transition is evidence that the complex is not centrosymmetric — which is a structural conclusion drawn from an intensity — the same move what an absence proves makes with a missing band, and chirality is a symmetry statement makes with an optical rotation.

How much the mechanism is worth, measured #

The argument says a forbidden transition becomes weakly allowed because the molecule is never quite centrosymmetric, and the size of weakly is measurable — by comparing two geometries of the same metal, one with a centre of inversion and one without.

An octahedral complex has a centre. Its d–d bands are forbidden by parity, appear only through the vibrational mechanism, and have extinction coefficients of about one to twenty.

A tetrahedral complex of the same metal has no centre at all. The parity rule does not apply, its d–d bands are allowed by the same argument that forbids the octahedron’s, and its extinction coefficients are in the hundreds.

A factor of roughly a hundred between the two, on the same metal, in the same oxidation state, with the same kind of ligand — and the only difference is whether the arrangement has a centre of inversion.

That is the vibrational mechanism priced. It supplies about one per cent of the intensity that the transition would have if it were allowed outright, which is what it should: the transition borrows its strength from the departure of the molecule from its own symmetry, and that departure is a small fraction of the molecule.

The comparison is also visible without an instrument. Cobalt in an octahedral aqua complex is a pale pink; the same cobalt as a tetrahedral chloride is an intense blue, and the two solutions interconvert in a demonstration that is done in every teaching laboratory. The colour change is a hundredfold change in extinction, and its cause is a symmetry element rather than anything about the electrons.

What the argument cannot reach #

Three limits, stated as they always are here.

The rule says zero and nothing else. It does not say how weak a vibronically allowed band will be, only that its strength is proportional to a vibrational amplitude. Computing the coefficient requires the coupling between the electronic and vibrational parts, which is beyond a symmetry argument.

The spin rule is exact only without spin–orbit coupling. Spin and space are separable in a light atom and progressively less so in a heavy one, which is why spin-forbidden bands are far more visible for a second- or third-row metal than for a first-row one.

Symmetry says nothing about the energy. The pattern of levels comes from the reduction of the splitting is a symmetry statement and their separation from a model. Everything here is about whether a transition can happen, not about where it is, which is the standard division of labour: the group decides what is allowed, and a model with energies in it decides where. Where a d–d band falls is the other half, and it is the half that needs a parameter.