Theoretical and computational chemistry

Selection rules, group theory and symmetry

Use molecular symmetry and representations to determine which spectroscopic transitions can occur and to interpret the symmetry of molecular vibrations.

1 Lower secondary 2 Upper secondary 3 Undergraduate 4 Advanced 5 Research

IntuitionSymmetry predicts what can be seen

A transition needs a nonzero interaction between initial and final states. Symmetry can force that transition moment to vanish, making the transition forbidden in the idealized model, even when its energy matches the light.

Rotate H₂O and identify its C₂ axis and mirror planes; these operations define the C₂v point group.

SchoolDipole transition rule

Definition: Transition dipole moment

The electric-dipole transition amplitude is the matrix element $\langle\psi_f|\boldsymbol\mu|\psi_i\rangle$ . It must be nonzero for an electric-dipole-allowed transition.

M_{fi}=\langle\psi_f|\boldsymbol\mu|\psi_i\rangle;\quad M_{fi}\ne0\;\Longleftrightarrow\;\Gamma_f\otimes\Gamma_{\mu}\otimes\Gamma_i\supset\Gamma_{\mathrm{tot.sym.}}

The integral over all molecular coordinates can be nonzero only if the direct product of the final-state, dipole-component, and initial-state representations contains the totally symmetric representation.

UndergraduatePoint groups and character tables

A point group collects symmetry operations that leave a molecule unchanged. Irreducible representations label how orbitals, normal coordinates, and vector components transform. Character tables list those transformations compactly.

Example: Classify the water bend

For H₂O in C₂v, the in-plane bend ν₂ leaves every atom motion unchanged under E, C₂ and both mirror planes: it is totally symmetric. Which IR dipole component can couple to it?

Solution

The bend transforms as A₁ and z also transforms as A₁; the dipole component along the C₂ axis is allowed. (The asymmetric stretch ν₃ is the B₂ mode, coupled to the y-polarized component.)

Probe	Operator	Typical rule
IR absorption	Electric dipole	Vibration changes dipole

AdvancedFrom symmetry to observable intensity

A forbidden transition has zero intensity only under the approximations used to assign the symmetry. Vibronic coupling, static distortion, magnetic-dipole or electric-quadrupole terms can lend weak intensity. Selection rules are therefore powerful filters, not universal statements that a line can never appear.

ResearchFrontier: symmetry breaking and intensity borrowing

References

Molecular Symmetry and Group Theory · Alan Vincent, 1988
Chemical Applications of Group Theory · F. Albert Cotton, 1990