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A nonlinear molecule with N atoms has 3N−6 vibrational degrees of freedom, each described as a normal mode — a collective, synchronized motion of all atoms. Normal modes are found by diagonalizing the mass-weighted Hessian (second derivative of potential energy). A mode is IR-active if it changes the molecular dipole moment (selection rule: ∂μ/∂Q ≠ 0) and Raman-active if it changes the polarizability (∂α/∂Q ≠ 0). The mutual exclusion rule states that for centrosymmetric molecules, no mode can be both IR and Raman active. Anharmonicity introduces overtones (Δv = ±2) and combination bands in observed spectra.
Work through the normal mode analysis of CO₂ and H₂O to see how symmetry governs activity. Apply the mutual exclusion rule to CO₂, then use group theory to categorize modes of larger molecules.
When a molecule vibrates, all of its atoms move simultaneously in coordinated patterns. Rather than thinking of each bond as an independent spring, quantum mechanics shows that the natural modes of vibration — normal modes — are collective motions of the entire molecule. Each normal mode has all atoms moving with the same frequency and in phase, but with different amplitudes at different atomic positions. These are the eigenvectors of the mass-weighted Hessian (the matrix of second derivatives of the molecular potential energy surface), and they are the fundamental units of molecular vibration.
The count of normal modes follows directly from the degrees of freedom argument. A molecule of N atoms has 3N total degrees of freedom (three Cartesian coordinates per atom). Three of these describe the center-of-mass translation, and three describe rotation (two for a linear molecule, which cannot rotate about its own axis). The remaining degrees of freedom must be vibrational: 3N − 6 for nonlinear molecules, 3N − 5 for linear ones. This formula is worth internalizing — students frequently forget the linear-molecule exception, which leads to wrong mode counts for molecules like CO₂ and HCN.
Whether a given normal mode appears in an IR spectrum depends on the selection rule: the vibration must change the molecular dipole moment (∂μ/∂Q ≠ 0). Physically, an IR photon is absorbed when its oscillating electric field can couple to an oscillating dipole in the molecule. Symmetric stretches of centrosymmetric molecules — like the symmetric C=O stretch of CO₂ — do not alter the dipole moment and are therefore IR-inactive. Asymmetric stretches and bends that distort the charge distribution are IR-active. Raman spectroscopy uses a complementary selection rule: a mode is Raman-active if the vibration changes the molecular polarizability (∂α/∂Q ≠ 0). The two techniques are thus complementary detectors of different aspects of molecular geometry.
For centrosymmetric molecules, the mutual exclusion rule applies: no mode can be both IR-active and Raman-active simultaneously. This is a consequence of inversion symmetry — modes that are symmetric (gerade, g) with respect to inversion are Raman-active but IR-inactive, while antisymmetric (ungerade, u) modes are IR-active but Raman-inactive. CO₂ is the canonical example: its symmetric stretch is Raman-active and IR-inactive; its asymmetric stretch and bends are IR-active and Raman-inactive. This complementarity makes IR and Raman spectroscopy together more informative than either alone.
Real molecular vibrations deviate from the ideal harmonic oscillator in ways that show up in spectra. Anharmonicity — the departure of the true potential well from a perfect parabola — means vibrational energy levels are not perfectly equally spaced, and transitions with Δv = ±2 (overtones) and Δv = 0 for one mode combined with Δv = ±1 for another (combination bands) become weakly allowed. These extra features in an IR spectrum are often diagnostic but require the harmonic selection rules as a starting point to interpret.