Selection Rules for Electronic Transitions

Introduction

An energy-level diagram shows candidate transitions, but not whether light can drive them strongly. The probability of absorption depends on the transition electric-dipole moment, which symmetry and spin can force to zero in an idealised model. Coordination spectra therefore contain intense, moderate and extremely weak features even when the photon energies match electronic gaps. Selection rules turn term labels into predictions about band intensity, and their controlled breakdown explains why “forbidden” does not mean invisible.

Core explanation

For an ordinary electric-dipole transition, the spin selection rule is ΔS=0: the total spin of initial and final states should match. A quartet-to-quartet transition is spin-allowed; quartet-to-doublet is spin-forbidden. The photon electric-dipole operator acts on spatial coordinates and does not directly flip electron spin. In an approximate LS-coupled picture, states of different S are orthogonal in their spin parts, making the transition moment vanish. Spin–orbit coupling can mix states of different nominal spin multiplicities and lend weak intensity to a spin-forbidden band, especially for heavier metals where this interaction is stronger.

The Laporte rule concerns inversion parity. In a centrosymmetric complex such as ideal O h, an electric-dipole operator is ungerade, so it connects a gerade state with an ungerade state. The matrix element between two purely gerade states vanishes. Metal-centred dⁿ states carry g labels, so an ideal d–d transition is Laporte-forbidden even when ΔS=0. It may nevertheless appear weakly because vibrations distort the complex away from exact inversion symmetry at an instant, mixing odd-parity character into the electronic states. This intensity borrowing is vibronic coupling. Static distortion that removes the inversion centre and mixing of metal d with suitable ligand or metal p character can similarly relax the parity prohibition.

Spin and Laporte restrictions are separate. A spin-allowed octahedral d–d band is still Laporte-forbidden; a spin-forbidden band in a tetrahedral complex is not rescued simply because tetrahedral symmetry lacks an inversion centre. The two effects compound. High-spin d⁵ complexes, for example, often have very pale colours because their d–d excitations from a sextet ground state reach quartet states, violating spin conservation, while an octahedral geometry also imposes parity restriction.

Tetrahedral complexes have no centre of inversion, so g/u parity labels and the strict Laporte rule do not apply. Their d-derived orbitals can mix with p-like components of matching tetrahedral symmetry, giving spin-allowed d–d bands substantially greater electric-dipole intensity than analogous ideal octahedral bands. This is a tendency, not an absolute guarantee: intensity also depends on the particular transition moment, covalency, geometry and band overlap. Charge-transfer transitions move electron density between ligand and metal, typically involve orbitals of different character, and can be much more intense than metal-centred d–d transitions.

The mathematical test is whether the direct product of initial-state symmetry, dipole-operator symmetry and final-state symmetry contains the totally symmetric representation. In O h, x,y,z transform as T₁u. A g initial state multiplied by a g final state and T₁u remains ungerade, so it cannot contain A₁g. This formal rule is more reliable than saying a transition is allowed merely because one orbital box changes. Spin symmetry is tested separately.

Experimental labels “allowed” and “forbidden” describe leading-order probabilities, not binary observations. A weak forbidden band may sharpen or strengthen at low temperature, and a nominally allowed band can be obscured by a stronger charge-transfer feature. The integrated intensity and vibronic pattern, alongside energy and term assignments, help distinguish mechanisms. One should also avoid equating the observed colour solely with the lowest calculated d–d gap; intense absorptions and spectral tails influence appearance.

Step-by-step reasoning

Assign the initial and final many-electron terms. Compare their spin multiplicities to test ΔS=0. If the complex has an inversion centre, compare parity: electric-dipole g↔u is permitted, whereas g↔g or u↔u is Laporte-forbidden. Then consider how vibrations, noncentrosymmetric distortion or spin–orbit coupling might lend intensity. Finally compare predicted relative intensity with measured absorbance before deciding whether a feature is d–d or charge transfer.

Visual explanation

Draw a four-cell grid. Across the top put “same S” and “different S”; down the side put “parity changes” and “parity unchanged.” Shade same-S/parity-changing as strongest in the ideal electric-dipole model, and different-S/parity-unchanged as doubly forbidden. Add arrows from the forbidden cells labelled “spin–orbit mixing” and “odd vibration” toward small but nonzero intensity.

Real-world analogy

A building may require both the right access badge and the right door direction. Passing one condition does not override the other. Spin conservation and parity are similarly independent checks on a transition. A vibrating doorway can momentarily create a narrow opening, but the entry remains much less probable than through a fully permitted route.

Real-world example

The octahedral [Ti(H₂O)₆]³⁺ ion has d¹ and a ^2T₂g→^2E g ligand-field excitation. Both terms are doublets, so it is spin-allowed; both are g, so it is Laporte-forbidden in ideal O h. Its observed band can arise through vibronic intensity borrowing. By comparison, tetrahedral [CoCl₄]²⁻ has no inversion centre and commonly shows more intense spin-allowed d–d absorption.

Why?

Why can a formally Laporte-forbidden octahedral band be recorded? The rule assumes perfect inversion symmetry and an electric-dipole transition between unmixed electronic states. Odd vibrations or static asymmetry introduce u character, making the transition moment small rather than exactly zero.

Common misconception

“Forbidden means the transition cannot occur under any circumstance.” It means the idealised leading electric-dipole matrix element vanishes under a specified symmetry or spin approximation. Real complexes vibrate and may have spin–orbit or orbital mixing, so weak bands can be observed.

Worked example

Compare ^4A₂g→^4T₂g and ^4A₂g→^2E g in an ideal octahedral d³ complex. The first preserves quartet multiplicity but is g→g, making it spin-allowed and Laporte-forbidden. The second changes quartet to doublet and is also g→g, so it is both spin- and Laporte-forbidden. Expect the second to be much weaker if both are observed. This relative prediction does not locate either band energy; term-energy information is needed for that.

Quick check

1. Is an octahedral ^3T₁g→^3T₂g transition spin-allowed? Answer: Yes, both states are triplets and have S=1. It remains Laporte-forbidden as a g→g electric-dipole transition in ideal O h.

Exam focus

Check spin and inversion parity independently. State the assumed ideal geometry, then identify the relevant relaxation mechanism rather than claiming any measured band disproves a rule. For tetrahedral complexes, omit g/u labels; for octahedral d–d bands, expect parity restriction even if spin is conserved.

Advanced insight

The transition intensity is proportional to the squared magnitude of a transition moment, ⟨ψ f μ̂ ψ i⟩ ². Symmetry sets the integral to zero only when the states and operator retain exact symmetry. Perturbative admixture of allowed-state character makes a forbidden moment proportional to the mixing coefficient; its intensity is then roughly proportional to that coefficient squared.

Summary

Spin-allowed electric-dipole transitions conserve S; Laporte-allowed transitions change inversion parity in centrosymmetric complexes. Octahedral d–d bands are typically parity-forbidden, while spin-forbidden bands can be even weaker. Vibrations, symmetry lowering and spin–orbit coupling explain observed exceptions.

Practice questions

1. Why is a tetrahedral spin-allowed d–d band often stronger than an analogous octahedral one? Answer: A tetrahedral complex has no inversion centre, so a strict g→g Laporte prohibition does not apply and d/p character can mix. Spin must still be conserved for a fully spin-allowed transition. 2. Classify an octahedral ^6A₁g→^4T₁g transition under the two elementary rules. Answer: Sextet to quartet changes S, making it spin-forbidden; g to g makes it Laporte-forbidden. It can still have weak borrowed intensity through spin–orbit and vibronic effects. 3. What does a very intense visible band suggest if a proposed assignment is a pure octahedral d–d transition? Answer: Reconsider the assignment or model. Charge transfer or strong mixing may better explain high intensity because a pure centrosymmetric d–d electric-dipole transition is parity-forbidden. 4. What physical interaction most directly lends intensity to a nominally spin-forbidden band? Answer: Spin–orbit coupling mixes states of different nominal S, allowing a small transition moment that the pure-spin model lacks.