Molecular spectroscopy exploits the quantized nature of molecular energy levels to extract structural and dynamic information. The Born-Oppenheimer approximation allows separation of electronic, vibrational, and rotational motion based on their vastly different timescales and energy spacings. Electronic transitions occur in the UV-visible region (10⁴-10⁵ cm⁻¹), vibrational transitions in the infrared (10²-10⁴ cm⁻¹), and rotational transitions in the microwave region (1-10² cm⁻¹). The total molecular energy is approximately E_total = E_electronic + E_vibrational + E_rotational, and a single electronic transition encompasses a rich structure of vibrational and rotational sub-bands.

Rotational Spectroscopy

The rigid rotor model treats a diatomic molecule as two masses rotating about their center of mass. The rotational energy levels are E_J = J(J+1)h²/(8π²I) = J(J+1)hcB, where I = μr² is the moment of inertia, μ is the reduced mass, r is the bond length, and B is the rotational constant in cm⁻¹. The selection rule ΔJ = ±1 produces a series of equally spaced lines separated by 2B in the rotational spectrum. The intensity distribution follows the Boltzmann population of levels, peaking at J_max = √(kT/2hcB) - 1/2. Centrifugal distortion, which becomes significant at high J, causes deviations from the ideal rigid rotor by adding a small correction term -D_J J²(J+1)².

Vibrational Spectroscopy

The harmonic oscillator approximation describes vibrational motion with energy levels E_v = (v + ½)hν, where v = 0, 1, 2… is the vibrational quantum number and ν is the fundamental frequency. The zero-point energy ½hν means molecules never come to rest even at absolute zero. The selection rule Δv = ±1 yields a single fundamental absorption band for a harmonic oscillator. Real bonds are anharmonic, described by the Morse potential V(r) = D_e[1 - e^{-a(r-r_e)}]², where D_e is the dissociation energy. Anharmonicity allows Δv = ±2, ±3 overtones and combination bands, and causes the spacing between adjacent levels to decrease as v increases. The Boltzmann distribution ensures most molecules occupy v = 0 at room temperature, making the fundamental v = 0 → 1 transition the strongest.

IR and Raman Spectroscopy

Infrared absorption requires a change in the molecular dipole moment during vibration. For a diatomic molecule, this depends on the asymmetry of charge distribution — homonuclear diatomics (O₂, N₂) are IR-inactive, while heteronuclear diatomics (HCl, CO) are IR-active. Group frequencies in the mid-IR (4000-400 cm⁻¹) provide characteristic absorption patterns: O-H stretch (~3600-3200 cm⁻¹), C=O stretch (~1750-1680 cm⁻¹), N-H bend (~1650-1580 cm⁻¹), and fingerprint region (1500-400 cm⁻¹) unique to each molecule. Raman spectroscopy complements IR by measuring inelastically scattered light. The Raman selection rule requires a change in polarizability during vibration — symmetric vibrations like the C=C stretch in ethene are Raman-active but IR-inactive. Together, IR and Raman provide complete vibrational characterization.

Electronic Spectroscopy and the Franck-Condon Principle

Electronic transitions involve promotion of an electron from a ground-state orbital to an excited-state orbital, typically a π → π* or n → π* transition. According to the Franck-Condon principle, because electronic transitions occur much faster (~10⁻¹⁵ s) than nuclear motion (~10⁻¹³ s), the molecular geometry remains unchanged during the transition. The transition probability between vibrational levels is governed by the Franck-Condon factor, the square of the overlap integral between the initial and final vibrational wavefunctions. The resulting band structure reveals whether the excited-state equilibrium geometry differs from the ground state — a long vibrational progression indicates a significant geometry change.

Fluorescence, Phosphorescence, and the Jablonski Diagram

The Jablonski diagram illustrates the photophysical processes following light absorption. Absorption promotes an electron from S₀ (singlet ground state) to S₁ or S₂ (singlet excited states). Internal conversion (IC) rapidly relaxes excess vibrational energy. Fluorescence is the spin-allowed radiative transition S₁ → S₀, with lifetimes of 1-100 ns. Intersystem crossing (ISC) to the triplet state T₁ is formally forbidden but occurs through spin-orbit coupling. Phosphorescence is the spin-forbidden T₁ → S₀ transition, with much longer lifetimes (µs to seconds). Kasha’s rule states that emission occurs from the lowest vibrational level of the lowest excited state of each multiplicity. The fluorescence quantum yield Φ_F = k_r/(k_r + k_nr) and lifetime are key parameters characterizing fluorophores, important for applications in sensing and imaging.