Rigid-Rotor Spectrum

Formula

I=\mu R_e^2, \qquad H_{\mathrm{rot}}=\frac{L^2}{2I},

E_J=B_EJ(J+1), \qquad B_E=\frac{\hbar^2}{2I}, \qquad g_J=2J+1.

The common rotational-constant conventions are

B_f=\frac{B_E}{h}=\frac{h}{8\pi^2I}, \qquad \widetilde B=\frac{B_E}{hc}=\frac{h}{8\pi^2Ic}.

For ideal electric-dipole absorption by a polar linear rotor,

\Delta J=+1, \qquad \nu_{J\to J+1}=2B_f(J+1), \qquad \widetilde\nu_{J\to J+1}=2\widetilde B(J+1).

Assumptions and Conventions

$J=0,1,2,\ldots$ for scalar wavefunctions on $S^2$ .
$B_E$ , $B_f$ , and $\widetilde B$ have units of energy, frequency, and inverse length, respectively.
$g_J=2J+1$ is orbital degeneracy only; spin and nuclear-spin weights are not included.

Symbols

Symbol	Meaning
$I$	moment of inertia
$\mu$	reduced mass
$R_e$	fixed bond length in the ideal model
$J,M$	total rotor angular momentum and projection labels
$\theta_{\mathrm{rot}}=B_E/k_B$	rotational temperature

Validity and Warnings

A tabulated value in $\mathrm{cm}^{-1}$ is $\widetilde B$ , not $B_E$ .
Energy levels are not equally spaced; the ideal adjacent-transition lines are equally spaced.
Homonuclear diatomic molecules have no permanent electric dipole, so the ideal dipole line rule does not guarantee an observed spectrum.
Centrifugal distortion, vibration, fields, and spin couplings modify the rigid-rotor baseline.

Canonical Treatment

Configuration space, eigenfunctions, spectroscopy conventions, isotope and thermal scaling, nonrigid corrections, exercises, and references are at Rigid Rotor.