Molecular Symmetry, Group Theory, and Spectroscopy

Manual: General · Subject: Molecular Science

Use symmetry to classify molecular states, predict selection rules, and interpret rotational, vibrational, and electronic spectra.

Symmetry as a predictive tool

Why symmetry matters

Symmetry reduces complexity by organizing orbitals, vibrational modes, and electronic states into irreducible representations. In molecular science, group theory is not decorative; it determines degeneracies, selection rules, and allowed couplings.

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Practical payoff

Symmetry can tell you which transitions are forbidden before you run a calculation, and which degeneracies may split under perturbation.

Point groups and irreducible representations

Representative molecular symmetries

Point group	Example	Key consequence
$C_{2v}$	Water	All irreps are one-dimensional
$D_{3h}$	Trigonal planar molecules	Degenerate $E'$ and $E''$ modes may appear
$O_h$	Octahedral complexes	Strong orbital degeneracies and selection-rule structure

Which statement is generally true for a molecule in the $C_{2v}$ point group?

It has only degenerate irreducible representations All irreducible representations are one-dimensional It must be linear It must have inversion symmetry

What does a symmetry selection rule tell you in spectroscopy?

Rotational and vibrational spectroscopy

Rigid rotor and harmonic oscillator limits

For a diatomic rigid rotor, rotational energies scale as $E_J = B J(J+1)$ , where $J$ is the rotational quantum number. In the harmonic approximation, vibrational levels are $E_v = \hbar\omega(v+1/2)$ . Real molecules deviate through centrifugal distortion, anharmonicity, and rovibrational coupling.

For a harmonic oscillator, why do vibrational overtones have weak intensity in the ideal limit?

Because the oscillator has no energy levels Because the dipole moment transition moment favors

\Delta v = \pm 1

Because all vibrations are forbidden Because the selection rule is

\Delta J = 0

only

Why are anharmonic corrections important in infrared spectroscopy?

Electronic spectra and vibronic structure

Key terms

Franck-Condon principle

Electronic transitions are fast relative to nuclear motion, so geometry changes shape vibrational progressions

Vibronic coupling

Coupling between vibrational and electronic degrees of freedom

Selection rules

Constraints imposed by symmetry and angular momentum

Zero-phonon line

A transition without accompanying vibrational excitation

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