This chapter develops the quantum mechanical harmonic oscillator as a model for molecular vibrations and shows how its predictions connect directly to infrared and Raman spectroscopy. Although approximate, the harmonic oscillator provides a powerful framework for understanding vibrational energy levels, wavefunctions, selection rules, and experimentally measurable molecular properties.
The chapter begins by examining the potential energy surface of a diatomic molecule, highlighting the steep repulsive wall at short bond lengths and the softer restoring force at longer distances. Expanding the potential about the equilibrium bond length using a Taylor series and truncating at the quadratic term leads naturally to the harmonic oscillator potential, equivalent to a Hooke’s law spring.
By transforming to center-of-mass coordinates, the vibrational motion of a diatomic molecule is reduced to the motion of a single particle with a reduced mass. This transformation clarifies why vibrational frequencies depend on both atomic masses and bond stiffness, and it explains limiting cases such as equal masses or one atom being much heavier than the other.
The Schrödinger equation for the harmonic oscillator is constructed using the reduced mass and quadratic potential. Applying physically meaningful boundary conditions leads to quantized vibrational energy levels labeled by the quantum number v = 0, 1, 2, …. Unlike the particle-in-a-box, these levels are evenly spaced, with the spacing determined by the force constant and reduced mass.
Vibrational energies are expressed as term values in units of cm-1, allowing direct comparison with spectroscopic data. Experimental vibrational frequencies provide a route to determining molecular force constants, revealing systematic trends such as stronger bonds corresponding to larger force constants and higher vibrational frequencies.
The harmonic oscillator wavefunctions are shown to consist of a Gaussian exponential multiplied by Hermite polynomials. These polynomials form an orthogonal set and possess well-defined symmetry (even or odd) under inversion. Their mathematical properties simplify normalization and integration and play a central role in evaluating expectation values.
Using symmetry alone, it is shown that the expectation values of position and momentum are zero for all vibrational eigenstates, while the expectation value of energy is simply the corresponding eigenvalue. These results reinforce the probabilistic interpretation of quantum mechanics and emphasize the role of symmetry in simplifying calculations.
The finite extent of the harmonic oscillator wavefunctions beyond classical turning points introduces the concept of tunneling. This behavior, impossible in classical mechanics, allows a system to be found in regions forbidden by classical energy considerations and illustrates one of the most striking consequences of quantum theory.
The limitations of the harmonic oscillator are addressed by introducing the Morse potential, which allows for molecular dissociation and unequal spacing of vibrational levels. Anharmonicity corrections explain overtone transitions and enable more accurate modeling of real molecular spectra, including the determination of dissociation energies.
Infrared and Raman spectroscopy are presented as experimental probes of vibrational motion. Selection rules derived from the harmonic oscillator and group theory determine which vibrational modes are active or inactive in each technique. These tools allow vibrational spectra to be interpreted in terms of molecular structure, symmetry, and bonding.
For polyatomic molecules, vibrational modes are classified using group theory. By constructing reducible representations and decomposing them into irreducible components, the symmetries of normal modes can be identified and their infrared and Raman activity predicted. Examples such as H2O, NH3, and SF4 illustrate the power of symmetry-based analysis.
Overall, Chapter 4 connects quantum mechanics, molecular vibrations, spectroscopy, and group theory into a unified framework that explains how vibrational structure arises and how it is observed experimentally. These ideas form the foundation for interpreting infrared and Raman spectra throughout molecular spectroscopy.