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Quantum vibrational states are quantized: E_v = ℏω(v + 1/2) where v = 0, 1, 2,... IR-active transitions require Δv = ±1 and a change in dipole moment along the vibration. Overtones (Δv = ±2, ±3,...) are typically forbidden or very weak. Hot bands from thermally populated excited states appear at lower frequency than fundamental transitions.
Measure IR spectrum of a diatomic or small polyatomic molecule; identify fundamental, overtone, and hot band transitions. Relate intensities to Franck-Condon factors and dipole moment derivatives.
From the harmonic oscillator model, you know that a vibrating diatomic molecule behaves approximately like a mass on a spring, with the potential energy rising parabolically as the bond stretches or compresses. Quantum mechanics tells us that such a system cannot vibrate with arbitrary energy — its energy is quantized into discrete levels given by E_v = ℏω(v + ½), where v is the vibrational quantum number (0, 1, 2, ...) and ω is the angular frequency determined by the bond's force constant and the reduced mass. The ½ in the formula means that even at v = 0, the molecule has zero-point energy — it never stops vibrating entirely, a direct consequence of the Heisenberg uncertainty principle.
The spacing between adjacent vibrational levels is uniform in the harmonic approximation: ΔE = ℏω regardless of which level you start from. This sets the stage for the selection rule Δv = ±1, which says that in a harmonic oscillator, only transitions between neighboring levels are allowed. The physical basis is that the transition dipole moment integral vanishes for Δv ≠ ±1 when the potential is exactly parabolic. The transition from v = 0 to v = 1 is the fundamental, and it dominates the IR spectrum.
But there is a second requirement: the vibration must cause a change in dipole moment. This is why homonuclear diatomics like N₂ and O₂ are IR-invisible — stretching the bond does not change the dipole moment (which is zero by symmetry at all bond lengths). Heteronuclear diatomics like HCl are IR-active because stretching the bond changes the charge separation. For polyatomic molecules, each normal mode is independently IR-active or inactive depending on whether that particular vibration modulates the molecular dipole.
Real molecules are not perfect harmonic oscillators. The true potential is anharmonic — it flattens out as the bond stretches toward dissociation and steepens at very short distances. Anharmonicity has two consequences: it makes the energy levels progressively closer together at higher v, and it relaxes the Δv = ±1 selection rule, allowing weak overtone transitions (Δv = ±2, ±3). Overtones appear in the spectrum at roughly twice, three times, etc., the fundamental frequency, but with rapidly decreasing intensity. Hot bands arise from transitions originating in thermally populated excited states (e.g., v = 1 → v = 2). Because anharmonicity compresses the spacing, hot bands appear at slightly lower frequency than the fundamental. Their intensity increases with temperature as more molecules occupy higher vibrational states, providing a direct spectroscopic thermometer for molecular vibration.