rigidrotor

Rigid rotor is a model in physics and physical chemistry used to describe a rotating system whose internal distances are fixed, i.e., the body maintains a rigid shape while it spins. It provides a simplified framework for understanding rotational motion for molecules and other small rigid bodies.

In classical mechanics, a rigid rotor has a fixed moment of inertia I about its axis of rotation. The kinetic energy is T = L^2/(2I), where L is the angular momentum. Rotation can occur about any axis through the center of mass, and the dynamics depend on the geometry of the body through its principal moments of inertia.

In quantum mechanics, rotational energy is quantized. For a simple linear rigid rotor, such as a diatomic molecule with bond length r and reduced mass μ, the moment of inertia is I = μ r^2. The energy levels are

E_J = ħ^2 J(J+1) / (2I),

with J = 0, 1, 2, ... and degeneracy 2J+1. Transitions follow the selection rule ΔJ = ±1, producing a microwave or far-infrared rotational spectrum. In spectroscopic units, the levels are often written as E_J/hc = B J(J+1), where B is the rotational constant.

For non-linear rigid rotors, the body has three principal moments of inertia I1, I2, I3. If two moments are equal, the system is a symmetric top; if all three are equal, it is a spherical top. Energy levels then depend on quantum numbers J and K (for symmetric tops) and are more complex, typically requiring approximate formulas or numerical solutions. A common form for a symmetric top is E_JK ≈ h c [A J(J+1) + (C−A) K^2], where A and C are rotational constants related to the moments of inertia.

Rigid rotor models are foundational in rotational spectroscopy and molecular structure analysis, providing a first-order description of molecular rotation and its spectral fingerprints.