Anharmonicity & Overtones — Anharmonicity and overtones (conceptual)

Historical Context & Motivation

The study of molecular vibrations began in earnest with the development of infrared spectroscopy in the late nineteenth century, when physicists first noticed that absorption spectra of diatomic molecules were far more complex than a single fundamental frequency would predict. Early attempts to model vibrational motion relied on Hooke's law — the same restoring-force model used for macroscopic springs — which predicts equally spaced energy levels and a single absorption frequency. Yet experimentalists consistently observed weaker absorption bands at frequencies near twice, three times, and higher integer multiples of the fundamental, along with systematic deviations in the spacing of rotational–vibrational lines. These overtone bands and convergent energy-level spacings could not be explained within the harmonic framework, motivating the search for a more realistic potential energy function.

The central question that anharmonicity addresses is deceptively simple: why does a real molecular bond not behave like a perfect spring? Answering this question requires moving beyond the harmonic oscillator to a potential that accounts for bond weakening at large internuclear separations and the repulsive wall at short distances. The consequences are profound — non-uniform energy-level spacing, selection-rule relaxation permitting overtone transitions, and a natural route to calculating bond dissociation energies directly from spectroscopic data.

Core Principles & Definitions

Before diving into the mathematical details, it is essential to establish the conceptual foundations that distinguish the anharmonic oscillator from its harmonic counterpart. The harmonic oscillator assumes a perfectly parabolic potential energy curve, V(x) = ½kx², which yields uniformly spaced energy levels separated by ℏω. In reality, no chemical bond is perfectly parabolic: the potential well is asymmetric, steep on the repulsive side and gradually flattening as the bond stretches toward dissociation. This departure from parabolicity is what we call anharmonicity.

Visual Explanation — Harmonic vs. Anharmonic Potential

The diagram above encapsulates the essential physics. In the harmonic approximation, the potential energy curve is symmetric about the equilibrium bond length r_e, and it extends to infinity in both directions — implying that a bond can be stretched indefinitely without breaking, which is clearly unphysical. The Morse potential corrects this by incorporating a dissociation plateau at energy D_e above the minimum. As the vibrational quantum number v increases, the spacing between adjacent levels shrinks — the levels converge toward the dissociation limit. This convergence is the spectroscopic fingerprint of anharmonicity and has the practical consequence that the first overtone (v = 0 → 2) does not appear at exactly twice the fundamental frequency, but rather at a frequency slightly less than 2ν̃₀.

Mathematical Framework

The quantitative treatment of anharmonicity begins with the energy eigenvalues of the Morse oscillator. While the harmonic oscillator produces a single-term expression for E_v, the anharmonic treatment introduces a correction term that is quadratic in the vibrational quantum number. In spectroscopic convention, energies are expressed in wavenumber (cm⁻¹) units, and the vibrational term value G(v) replaces the absolute energy.

The transition frequency for a jump from vibrational level v to v + Δv is obtained by taking the difference ΔG = G(v + Δv) − G(v). For the fundamental transition (Δv = 1, starting from v = 0), this yields a frequency slightly below ω_e. For overtones, the pattern is systematic.

Overtone Transitions & Intensity Patterns

A natural question arises: if the harmonic oscillator selection rule restricts transitions to Δv = ±1, how do overtones (Δv = ±2, ±3, …) become observable at all? The answer lies in the fact that the anharmonic potential distorts the harmonic wave functions, leading to non-zero overlap integrals for transitions that would vanish in the harmonic limit. Moreover, the dipole moment function μ(r) of a real molecule is not a perfectly linear function of displacement; its higher-order Taylor expansion terms contribute transition dipole matrix elements for Δv > 1. However, these matrix elements decrease rapidly with |Δv|, so overtone intensities fall off dramatically — typically by a factor of 10–100 for each additional quantum of Δv.

The spectrum above illustrates the key experimental observables. The fundamental transition near 2886 cm⁻¹ dominates the spectrum by far. The first overtone at ≈5668 cm⁻¹ has only about 5% of the fundamental's intensity, and higher overtones decay even more steeply. In purely harmonic theory, the first overtone would appear at exactly 2 × 2886 = 5772 cm⁻¹; the fact that it is observed at 5668 cm⁻¹ — some 104 cm⁻¹ lower — is a quantitative measure of the anharmonicity of the HCl bond. This discrepancy can be used, in conjunction with the fundamental frequency, to extract both ω_e and ω_ex_e for the molecule.

Worked Example — Determining ωₑ and ωₑxₑ from Spectral Data

Suppose you observe the fundamental absorption of a diatomic molecule at ν̃_0→1 = 2143.3 cm⁻¹ and the first overtone at ν̃_0→2 = 4260.0 cm⁻¹ (these are approximate values for ¹²C¹⁶O). Determine ω_e and ω_ex_e.

Harmonic vs. Anharmonic Models — Strengths & Limitations

Connection to Advanced Theory — Beyond the Morse Model

While the Morse potential is an enormous improvement over the harmonic approximation, it is not the end of the story. For high-precision spectroscopy — where line positions are known to 10⁻³ cm⁻¹ or better — additional anharmonic corrections are required. The Dunham expansion generalizes the vibrational–rotational term value as a double power series in (v + ½) and J(J + 1), with empirically determined coefficients Y_ij. For polyatomic molecules, the situation becomes considerably richer: combination bands and Fermi resonances arise when two modes of similar frequency interact through anharmonic coupling terms in the potential, producing intensity borrowing and level repulsion that can dramatically alter the spectrum.

A particularly important application of anharmonicity is the Birge–Sponer extrapolation, which uses the systematic decrease in adjacent energy-level spacings ΔG_v+½ = G(v+1) − G(v) to estimate the dissociation energy. By plotting ΔG_v+½ versus (v + ½), one obtains a linear graph (for a Morse-like potential) whose area under the curve equals D₀, the dissociation energy measured from the zero-point level. This elegant connection between spectroscopic observables and thermodynamic quantities underscores why anharmonicity is far more than a theoretical curiosity — it is a practical tool for extracting bond energies from spectral data.

Practice Problems

Summary — Anharmonicity & Overtones