Rigid-rotor levels and rotational spectra

For nuclei of masses $m_A$ and $m_B$ separated by a fixed distance $R_e$, the moment of inertia about a perpendicular center-of-mass axis is $I=\mu R_e^2$, with reduced mass $\mu=m_Am_B/(m_A+m_B)$. Using the total molecular mass or placing one nucleus immovably at the origin gives an incorrect inertia except in an appropriate heavy-partner limit. The rigid approximation neglects vibration and centrifugal stretching, and initially assumes an electronic state with no angular-momentum complications.

The rotor Hamiltonian is squared angular momentum divided by twice I. Its eigenfunctions are spherical harmonics, with integer J and projection M from minus J to plus J. The energy depends on J but not M in zero field, giving degeneracy $2J+1$. Molecular spin, electronic orbital motion, hyperfine interactions, and identical-nucleus restrictions can modify the accessible levels or split them; these are additional structures, not failures of angular-momentum quantization itself.

A heteronuclear diatomic with a nonzero permanent electric dipole can undergo pure rotational electric-dipole transitions. In the elementary closed-shell rotor model the rule is $\Delta J=\pm1$, with polarization also constraining M. Absorption from J to J plus one has wavenumber $2\widetilde B(J+1)$. Thus ideal adjacent rotational lines are equally separated even though the levels themselves are not. Confusing level energies with line differences obscures this simple pattern.

A homonuclear molecule has no permanent electric dipole in its isolated symmetric state, so its pure rotational electric-dipole spectrum is absent in this model. Its rotation can still be observed by Raman scattering through anisotropic polarizability. Rotational Raman shifts follow different rules, including $\Delta J=\pm2$ for the anisotropic part, while zero shift is elastic. Absence of one dipole spectrum therefore does not imply absence of rotational levels or absence of every spectroscopic probe.

Keeping the electronic equilibrium geometry fixed while doubling reduced mass halves the rotational constant. This leading isotope rule is stronger than merely saying “heavier molecules rotate more slowly” because it identifies the correct mass and the square of bond length. A measured constant in vibrational state v is usually an average over that state’s bond-length distribution rather than exactly the equilibrium constant. Extracting an equilibrium distance therefore requires specifying which rotational constant is used.

At higher J, rotation stretches a flexible bond, raises its inertia, and lowers energies relative to the rigid prediction. The leading negative quartic correction captures that trend over a limited range; extrapolating it indefinitely would eventually produce unphysical decreasing energies. Spectroscopic constants can be fitted to measured lines or computed from a potential and nuclear model. The examples below use hypothetical constants to teach assignments and scaling, without presenting them as measured properties of a named molecule.