Raman Selection Rules #
The quantum mechanical condition for Raman activity: a vibration is Raman active if it produces a change in the molecular polarizability tensor (∂α/∂Q ≠ 0, where Q is the normal coordinate). Contrast this directly with the IR selection rule, which requires a change in dipole moment. These are fundamentally different physical requirements, and a given vibration need not satisfy both.
Polarizability described physically: the ease with which the electron cloud is distorted by an electric field. Symmetric stretches that expand and compress the electron cloud periodically satisfy this condition well. Antisymmetric stretches in symmetric molecules often do not.
Classical picture: the induced dipole moment p = αE, where E is the incident electric field. If the nuclear motion modulates α, it modulates p at the beat frequency ω_laser ± ω_vib, producing the Stokes and anti-Stokes Raman lines.
α is a rank-2 tensor with components α_ij (i, j = x, y, z), giving nine elements (reduced to six by symmetry for non-absorbing media). The Raman scattering intensity depends on the matrix element:
⟨ν+1 | (∂α_ij/∂Q) | ν⟩
This must be non-zero for the transition to be Raman active. The tensor character has practical consequences: different tensor components are probed by different polarisation combinations of incident and scattered light, forming the basis of polarised Raman measurements.
Formal treatment using molecular symmetry (point groups). A vibrational mode is Raman active if it belongs to a symmetry species (irreducible representation) that transforms as one or more of the quadratic functions x², y², z², xy, xz, yz in the character table of the molecule’s point group.
Worked through for key symmetry cases:
- CO₂ (D∞h): The symmetric stretch (Σg⁺) transforms as x²+y², z² — Raman active, IR inactive. The antisymmetric stretch (Σu⁺) and bending modes (Πu) are IR active, Raman inactive. A clean illustration of mutual exclusion.
- H₂O (C₂v): Both stretching modes (A₁ and B₁) and the bending mode (A₁) are both Raman and IR active — no centre of inversion, so mutual exclusion does not apply.
- Benzene (D₆h): Several modes are Raman active (A₁g, E₂g), including the prominent 992 cm⁻¹ ring-breathing mode and the 1582 cm⁻¹ C=C stretch. Others are IR active only.
Character table column headers explained: the columns labelled with quadratic functions identify Raman-active species directly.
For molecules possessing a centre of inversion (centrosymmetric molecules), no vibrational mode can be simultaneously Raman and IR active. Modes that are Raman active (gerade, g symmetry) are IR inactive, and vice versa (ungerade, u symmetry). This is the mutual exclusion rule.
Practical significance: if a band appears in both Raman and IR spectra of a material, the molecule or structural unit cannot have a centre of inversion. This is a direct symmetry diagnostic — used, for example, to distinguish centro- from non-centrosymmetric polymorphs, or to confirm local symmetry in crystalline materials.
Examples of centrosymmetric species where the rule applies: CO₂, ethylene, homonuclear diatomics (N₂, O₂). Examples where it does not: H₂O, SO₂, CH₄.
Formal selection rules apply strictly to fundamental transitions (Δν = ±1). Overtones (Δν = ±2, ±3) and combination bands are in principle forbidden under the harmonic approximation but appear weakly due to mechanical and electrical anharmonicity.
Resonance Raman: when the excitation wavelength approaches an electronic absorption, Franck-Condon overlap dominates intensity, and modes that are totally symmetric with respect to the excited state geometry are strongly enhanced — sometimes modes that are weak under normal conditions become dominant features.
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