This chapter develops the rigid rotor model as the quantum mechanical description of molecular rotation and applies it to the interpretation of rotational and rotation–vibration spectra. By solving the Schrödinger equation in spherical polar coordinates, the chapter connects angular momentum, molecular structure, and spectroscopic observables such as bond lengths and rotational constants.
Rotational motion is most naturally described using spherical polar coordinates. The chapter introduces the coordinates r, θ, and φ, their physical interpretation, and their relationship to Cartesian coordinates. This coordinate system greatly simplifies the description of rotational motion and the form of the Hamiltonian for a rotating molecule.
For a rigid rotor, all rotational energy is kinetic energy and the potential energy is zero. Fixing the bond length reduces the Hamiltonian to an angular operator involving the moment of inertia, I = μr². This highlights how molecular masses and bond lengths control rotational energy levels.
Solving the Schrödinger equation by separation of variables leads to angular equations in θ and φ. The requirement that wavefunctions be single valued introduces quantization of the magnetic quantum number, while boundary conditions on the polar equation lead to quantization of total angular momentum.
The rigid rotor energy levels are given by E = BJ(J + 1), where J is the rotational quantum number. Each level has a degeneracy of (2J + 1) due to the allowed values of the magnetic quantum number. The spacing between energy levels increases with increasing J.
The angular wavefunctions are expressed in terms of Legendre and associated Legendre polynomials, which combine with azimuthal functions to form the spherical harmonics. These functions are eigenfunctions of the angular momentum operators and appear in all quantum mechanical problems with spherical symmetry, including the hydrogen atom.
The chapter develops the operators corresponding to angular momentum and shows their relationship to the rigid rotor Hamiltonian. While the squared angular momentum operator commutes with each component, the components do not commute with one another. This explains why only the magnitude of angular momentum and one of its components can be known simultaneously.
Expressing rotational energies in spectroscopic units introduces the rotational constant B, which is directly related to bond length. High-resolution rotational spectra therefore provide extremely precise molecular structural information, including equilibrium bond distances.
Real molecules are not perfectly rigid. At high rotational energies, bonds stretch due to centrifugal forces, requiring correction terms to the rigid rotor energy expression. Introducing distortion constants improves agreement with experimental spectra and provides insight into bond flexibility.
The rigid rotor model is applied to microwave spectra, where pure rotational transitions obey the selection rule ΔJ = ±1. In infrared spectra, vibrational transitions are accompanied by rotational fine structure, producing characteristic P- and R-branch patterns that can be analyzed to extract spectroscopic constants.
Combination differences provide a powerful method for isolating rotational energy spacings and eliminating dependence on one vibrational state. This technique enables highly accurate determination of rotational and distortion constants from experimental spectra.
The intensity of rotational lines depends on both quantum mechanical line strengths and the thermal population of rotational levels. The Maxwell–Boltzmann distribution explains the characteristic intensity envelope of rotational spectra, while Hönl–London factors describe branching ratios between P- and R-branch transitions.
Overall, Chapter 5 shows how rotational motion, angular momentum, and spectroscopy are tightly linked. The rigid rotor model provides one of the most direct bridges between quantum mechanics and experimentally measurable molecular structure, making rotational spectroscopy a cornerstone of molecular spectroscopy.