Chemistry · Lesson

Selection rules, transition intensities and symmetry

Ask which interaction connects two states before naming a transition allowed or forbidden. Use symmetry and transition integrals within their stated approximations, and keep strength separate from a measured peak height.

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By the end, you can

Distinguish an energy match from a nonzero transition matrix element.
Compare box, harmonic, IR and Raman selection conditions.
Apply model-limited symmetry and spin statements without treating weakly enabled transitions as impossible.

A transition requires the relevant coupling

In a weak-field one-photon electric-dipole model, the transition matrix element is M_fi = ⟨f|μ̂|i⟩, where μ̂ is the electric-dipole operator. An energy-matched transition can still have zero M_fi. Within that interaction model, the strength contains |M_fi|². Population, frequency factors, orientation, line profile and detection also affect the measured signal.

The electric-dipole approximation assumes the spatial light-field variation is small across the molecule. Magnetic dipole, higher multipole and other interactions have different operators and rules. A transition forbidden in one approximation can gain strength through other couplings, symmetry breaking or state mixing.

Do not copy a rule from one Hamiltonian to another

For the one-dimensional infinite-well position operator, off-diagonal ⟨m|x|n⟩ is nonzero when m+n is odd, equivalently when Δn is odd. Thus 1→4 can be dipole allowed even though it changes n by 3; 1→3 has a zero integral in this model. Signs of off-diagonal matrix elements depend on chosen state phases, while their squared magnitudes do not.

For the exact harmonic oscillator with a dipole expansion linear in its displacement, the relevant coordinate matrix elements instead give Δv = ±1. Anharmonic state mixing or nonlinear dipole dependence can enable other bands. These two rules belong to different model wavefunctions and coupling operators.

The box position operator permits odd changes beyond one step

Box transition	Δn	⟨final\|x\|initial⟩ / L	Relative squared matrix element
1 → 2	1	−0.18012655	1
1 → 3	2	0	0
1 → 4	3	−0.01441012	0.00640

Original exact 1D infinite-well position integrals using normalized sine states. Matrix elements are reported divided by L. The final column compares only squared matrix-element magnitudes to the 1→2 value; it is not a complete absorption-intensity ratio. The sign is phase-convention dependent.

A dipole derivative and a polarizability derivative test different activity

Ordinary vibrational IR activity requires a nonzero change of molecular dipole along the normal coordinate. First-order vibrational Raman activity requires the appropriate polarizability derivative. A mode can therefore appear through one interaction and be absent through another. For centrosymmetric molecules under these standard approximations, the mutual-exclusion symmetry result separates IR-active and Raman-active modes.

In Stokes Raman scattering, the scattered photon has lower frequency and the molecule gains excitation. Anti-Stokes scattering produces a higher-frequency photon from an initially excited molecule. Populations, scattering frequency factors and instrument response matter for observed intensity ratios; a Boltzmann population ratio alone is not the complete raw signal ratio.

An approximation-limited zero is useful information

Symmetry can force a transition integral to vanish when the combined state and operator symmetries cannot produce the required invariant. The answer depends on the actual molecular symmetry, polarization and interaction, not only a label attached to a state. Degenerate modes can share a frequency, and overlapping bands can hide distinct transitions.

A spin-independent electric-dipole approximation preserves spin multiplicity. Spin–orbit coupling, including spin-vibronic mechanisms involving it, can enable nominally spin-forbidden transitions. Vibronic mixing can also relax spatial-symmetry rules; spin-free vibronic mixing alone does not remove the spin-multiplicity rule. Franck–Condon vibrational overlaps add another factor to an electronic transition, within the Born–Oppenheimer and Condon assumptions rather than as a universal exact intensity law.

Worked examples

Compare squared matrix elements under the same model

The supplied box integrals are x₁₂/L≈−0.18012655 and x₁₄/L≈−0.01441012. Find their squared-magnitude ratio.

Divide the two unrounded analytic magnitudes; their ratio is 0.0800.
Square that ratio: 0.0800² = 0.00640.
A phase change can reverse a matrix-element sign without changing its square.
A complete observed intensity comparison would also require frequency, population and measurement information.

The 1→4 squared matrix element is 0.00640 times the 1→2 value. It is nonzero despite Δn=3.

Convert a Raman shift with its correct sign

Use a 532 nm vacuum laser wavelength and a stipulated positive vibrational shift of 1.00 × 10³ cm⁻¹. Find the Stokes and anti-Stokes vacuum wavelengths.

The incident wavenumber is 10⁷/532 ≈ 18796.99248 cm⁻¹.
Stokes: subtract1000 cm⁻¹, then take the reciprocal length.
Anti-Stokes: add1000 cm⁻¹, then take the reciprocal length.
The corresponding wavelengths are approximately 561.893 and 505.127 nm.

Stokes λ≈ 562 nm; anti-Stokes λ≈ 505 nm. Subtracting a wavenumber does not mean subtracting the same numerical value from wavelength.

Try it yourself

In the exact box position-operator model, is 1→4 forbidden merely because Δn is 3?
Show the answer and reasoning
No.
Its odd Δn gives a nonzero off-diagonal position integral.
What Δv does the harmonic linear-displacement electric-dipole model permit?
Show the answer and reasoning
±1.
The rule is model-specific and differs from the box rule.
Which derivative controls ordinary first-order vibrational Raman activity?
Show the answer and reasoning
A polarizability derivative along the normal coordinate.
The corresponding IR condition instead uses the dipole derivative.
Does ordinary IR/Raman mutual exclusion apply to every molecule?
Show the answer and reasoning
No.
It is a centrosymmetric-molecule result under the stated interaction approximations.

What to carry forward

Choose the coupling operator before using a selection rule.
An energy match alone does not supply a transition strength.
IR and Raman test different molecular responses.
Read forbidden labels within their symmetry and interaction assumptions.

References

Teaching resources and reference material supporting this lesson’s explanations and examples.